This book grew out of a set of notes that I supplied to the audience of a series of
lectures on "The Thomas-Fermi Method in Atomic Physics and Its Refinements" ...
Lecture Notes in Physics Edited by H. Araki, Kyoto, J. Ehlers, MSnchen, K. Hepp, Z~irich R. Kippenhahn, MLinchen, H.A. WeidenmSIler, Heidelberg J. Wess, Karlsruhe and J. Zittartz, K61n
300 Berthold-Georg Englert
Semiclassical Theory of Atoms
Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo
Author Berthold-Georg Englert Universit~.t MLinchen, Sektion Physik Am Coulombwall 1, D-8046 Garching, F.R.G.
ISBN 3-540-19204-2 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-19204-2 Springer-Verlag NewYork Berlin Heidelberg Library of Congress Cataloging-in-Publication Data. Englert, B.-G. (Berthold-Georg), 1953Semiclassical theory of atoms / B.-G. Englert. p. cm.-(Lecture notes in physics; 300) Includes bibliographies and indexes. ISBN 0-38?-19204-2 (U.S.) 1. Atoms-Models. 2. Thomas-Fermi theory. I. Title. I1. Series. O.C 173.E56 1988 539'. 14-dc 19 88-4949 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1988 Printed in Germany Printing: Druckhaus Beltz, Hemsbach/Bergstr. Binding: .1. Sch~ffer GmbH & Co. KG., GrL~nstadt 2158/3140-543210
PREFACE
This of
book
grew
a series
and
Its
of
and
sequence,
this
presentation
to
supply
cite
only
manded
been the
The
thorough
is
consists Both
skipped
during
reasonable
It
try
Garching,
of
that
are
paper
on
Thus
papers.
to
On
of
reading.
1985.
lectures[
ago.
As
Julian
a con-
step-by-step
a few
ignored. many
is
of
over
intention
subject,
as
was
as
not
this
selective
and
honesty
de-
occasions work[
In p a r -
pieces
I hope
that
these
any
mind,
previous
knowledge
the
prerequisite
quantum of
atoms
general
mechanics, is
some
that
and
certainly
concepts
although I trust
only
of
about is
some
helpful.
and
technical
the
latter
readers
a
can
can
perform
be a
themselves.
for
the
of
people.
to
list
many Being
insights afraid
of
gained
in d i s c u s s i o n s
forgetting
somebody,
with
a
I shall
them. to
thank
Mrs.
E.
Figge,
who
typed
the
manuscript
skill.
February
I
put-downs.
have
elementary
the
else's
as
open
absorbed,
these
scattered my
referencing
someone
an
not the
every
a mixture
a first
a pleasure
enviable
model
phenomenology
to be
selection
number
is
of
need
I am g r a t e f u l
with
the
is
organize
hand,
expected
Physics
in
Professor
detailed
other
about
by
audience
Atomic
Munich
during
years
the
value.
to
eight
in
of
the
implications.
and
misunderstood
understanding
text
even
be not
with
of
relevant
In a d d i t i o n
detail.
large
list
role
researchers
Thomas-Fermi
remarks
not
reader
familiarity
other
to
Method
developed
complete,
collect
little
most
will
subject.
not
of
a minor
their
On
I supplied
University
about
first
publications.
critical
comments
The
the
the
approach
and
of to
that
Thomas-Fermi at
played
the
work
a complete
have
"The
novel
ideas
tried
about
original
would
the
the
notes
beginning
is
our
I have
knowledge
the
book
Naturally, ticular,
on
myself,
of
on
of
delivered
material
was
Schwinger
a set
lectures
textbook
emphasis
many
of
Refinements"
Standard the
out
1988
B.-G.
Englert
TABLE OF
Chapter
One.
Atomic Bohr
2 4
with
effective of
Chapter
potential
........................................
13
24
Thomas-Fermi
Model ..................................
27
.............................................
27
model ..................................................
Neutral
TF
Maximum
property
electrostatic
TF
density
Minimum
of
property bounds
on
Lower
bounds
on
energy
function
Scaling
the
TF
analogy
functional
Upper
Binding
33
a t o m s .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
An
TF
10 11
......................................................
formalism
TF
..............................
.....................................
21
Two.
General
integrals
shielding
atoms .................................................
Problems
The
phase-space
atoms
Size
I
units ..................................................
and
Bohr
........................................
atoms ....................................................
Traces
The
Introduction
CONTENTS
of
potential
functional
37
...............
39
......................................
41
.........................................
43
the
TF
density
functional
.................
45
B .............................................
46
B .............................................
50
of
neutral
TF
atoms ............................
54
F(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
properties
of
the
TF
56
model ............................
66
Highly
ionized
TF
atoms .......................................
73
Weakly
ionized
TF
atoms ........................................
80
Arbitrarily
ionized
Validity
the
Density
of and
Relation Problems
Chapter
atoms ..................................
96
the
functionals TF
99
.............................
approximation
and
Hartree's
method
104 ....
......................................................
Three.
Scott's
TF
model ......................................
potential
between
Qualitative First
TF
Strongly
Bound
argument
quantitative original
Electrons
..........................
.......................................... derivation
argument
of
Scott's
correction
...........
.....................................
TFS
energy
functional
.........................................
TFS
density
...................................................
119 124
130 130 131 138 139 146
VI
Consistency ................................................... Scaling Second Some
properties
of the TFS model ...........................
quantitative
implications
Electron
density
derivation
concerning at t h e
of Scott's
c o r r e c t i o n ...........
energy ...........................
site of the nucleus ...................
148 155 158 162 164
Numerical
procedure ............................................
168
Numerical
results
170
for neutral
mercury .........................
Problems ......................................................
Chapter
Four.
Quantum
Qualitative
Corrections
Exchange ...................
175
arguments ..........................................
176
Quantum
corrections
I
Quantum
corrections
II
The yon Weizs~cker Quantum Airy
and
(time t r a n s f o r m a t i o n (leading
energy
f u n c t i o n ) ...........
177
correction) ............
190
term .......................................
corrections
III
(energy) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
averages .................................................
Validity
173
of the TF approximation ................................
194 196 199 207
Quantum
corrected
EI(V+~) .....................................
210
Quantum
corrected
density .....................................
213
Exchange
I
(general) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
221
S e l f e n e r g y .......................................................
225
Exchange
225
II
(leading
correction) ..............................
History ........................................................
230
Energy
232
corrections
Ionization Minimal
for ions ....................................
energies ..............................................
binding
Shielding
energies
of the
Simplified
nuclear
(chemical magnetic
new differential
An application
(?) ES m o d e l .
Exchange
III.
(Exchange
moment .......................
equation
o f t h e ES m o d e l .
Improved
potentials) ................
Electric
(ES m o d e l ) . . . . . . . . . . . . . . .
Diamagnetic
susceptibilities..
polarizabilities ..............
236 239 241 244 255 260
potential) .............................
275
equation ......................................
278
Small distances .................................................
282
New differential
Large
distances ................................................
Numerical
results ...............................................
Problems .......................................................
Chapter
Five.
Shell
Experimental Qualitative Bohr
Structure ....................................
evidence ........................................... arguments .........................................
atoms ....................................................
283 286 293
295 297 299 300
VII
TF
quantization
Fourier Isolating Lines
the
of
TF
...........................................
contribution
degeneracy
Classical
314
orbits ..............................................
320
in
degeneracy
the
TF
and
potential
the
features
of
N
General
features
of
E
Linear
degeneracy.
Perturbative
terms.
.................................
systematics qu
of
the
Periodic
Table .......
.......................................
qu Scott
approach
....................................... correction
Leading
integrals
I oscillations
to
...........................
................................. osc TF model ...........................................
Z-quantized
Fresnel
312
315
General
j~0
.................................
305
...........................................
Degeneracy TF
...............................................
formulation
E
energy
oscillation
.........................
323 327 329 333 335 336 338 342
.............................................
347
................................................
351
v oscillations
................................................
358
Semiclassical
prediction
360
Other
manifestations
of
for shell
E
.............................
osc structure
.......................
Problems ......................................................
Chapter
Six.
Miscellanea
Relativistic Kohn-Sham Wigner's Problems
Footnotes
.........................................
corrections
equation
......................................
............................................
phase-space
functions
................................
......................................................
363 367
370 370 377 379 381
.........................................................
383
Chapter
One ...................................................
383
Chapter
Two ...................................................
384
Chapter
Three .................................................
387
Chapter
Four ..................................................
389
Chapter
Five ..................................................
394
Chapter
Six ...................................................
395
Index ............................................................
397
Names .........................................................
397
Subjects
398
......................................................
Chapter One
INTRODUCTION
Atoms that contain m a n y electrons possess a degree of complexity
so high that it is impossible to give an exact answer even w h e n
we are asking simple questions. a p p r o x i m a t e descriptions. o r e t i c a l atomic physics. finements;
We are t h e r e f o r e c o m p e l l e d to resort to
Two m a i n approaches have been pursued in theOne is the Hartree-Fock(HF)
m e t h o d and its re-
it can be v i e w e d as a g e n e r a l i z a t i o n of S c h r ~ d i n g e r ' s des-
c r i p t i o n of the h y d r o g e n atom to m a n y - e l e c t r o n systems; struction,
it is, by con-
the more r e l i a b l e the smaller the number of electrons.
o t h e r one is the T h o m a s - F e r m i
The
(TF) t r e a t m e n t and its improvements;
this
one uses the picture of an electronic a t m o s p h e r e s u r r o u n d i n g the nucleus; it is the b e t t e r the larger the number of electrons.
For this reason,
the TF method is f r e q u e n t l y called the "statistical t h e o r y of the atoms."
Throughout these lectures we shall be c o n c e r n e d with the TF approach,
t h e r e b y c o n c e n t r a t i n g on more recent developments.
The repe-
tition of m a t e r i a l that has been p r e s e n t e d in textbooks I already will be limited to the m i n i m a l amount n e c e s s a r y to make the lectures self-contained.
The d e r i v a t i o n of k n o w n results will, w h e r e v e r feasible,
differently,
and - I believe - more elegantly,
be done
than in standard texts on
the subject. It should be r e a l i z e d that the methods of the TF a p p r o a c h are in no way limited to atomic physics.
Besides the i m m e d i a t e m o d i f i c a t i o n s
for a p p l y i n g the f o r m a l i s m to m o l e c u l e s or solids,
there exists the pos-
s i b i l i t y of e m p l o y i n g the technics
and in nuclear phy-
in astrophysics
sics. The latter a p p l i c a t i o n n a t u r a l l y requires a p p r o p r i a t e changes ref l e c t i n g the t r a n s i t i o n
from the C o U l o m b i n t e r a c t i o n of the electrons
to the m u c h more c o m p l i c a t e d n u c l e o n - n u c l e o n forces. In these lectures we shall confine the d i s c u s s i o n to atoms, however.
This has the a d v a n t a g e of k e e p i n g the c o m p l e x i t y of most cal-
culations at a rather low level,
so that we can fully focus on the pro-
perties of the TF m e t h o d w i t h o u t being d i s t r a c t e d by the t e c h n i c a l comp l i c a t i o n s that arise from the c o n s i d e r a t i o n s of m o l e c u l a r s t r u c t u r e or from our i n c o m p l e t e k n o w l e d g e of the nuclear forces,
for instance.
R e s t r i c t i n g o u r s e l y e s to atoms is further a d v a n t a g e o u s because it enables us to compare p r e d i c t i o n s of TF theory w i t h those of other methods, like HF calculations.
The u l t i m a t e test of a t h e o r e t i c a l d e s c r i p t i o n is,
of course,
the c o m p a r i s o n of its implications w i t h e x p e r i m e n t a l data.
W h e n e v e r possible,
we shall t h e r e f o r e m e a s u r e the a c c u r a c y o f the TF
predictions by c o n f r o n t i n g it w i t h e x p e r i m e n t a l results. Lack of e x p e r i m e n t a l data sometimes upon HF results
for comparison.
forces us into relying
The same s i t u a t i o n occurs w h e n quanti-
ties of a more t h e o r e t i c a l nature are d i s c u s s e d
(as, e.g., the nonrela-
tivistic b i n d i n g energy, w h i c h is not a v a i l a b l e from experiments).
Such
a p r o c e d u r e must not be m i s u n d e r s t o o d as an attempt of r e p r o d u c i n g HF p r e d i c t i o n s by TF theory. The TF m e t h o d is not an a p p r o x i m a t i o n to the HF description, sics.
but an i n d e p e n d e n t approach to t h e o r e t i c a l atomic phy-
[Incidentally,
it is the h i s t o r i c a l l y older one: TF theory origi-
nated in the years 1926
(Thomas)
did not exist prior to 1928
and 1927
(Hartree)
(Fermi), whereas the HF model
and 1930
(Fock).] 2 The two appro-
aches should not be r e g a r d e d as c o m p e t i n g w i t h each other, p l e m e n t i n g one another.
but as sup-
Each of the two methods is well suited for stu-
dying certain properties of atoms. the i o n i z a t i o n energy of oxygen,
For example,
if one is interested in
a HF c a l c u l a t i o n will p r o d u c e a reli-
able result; but if you want to know how the total b i n d i n g energy varies over the entire Periodic Table,
the TF model will tell you. Tersely:
the
HF m e t h o d for specific i n f o r m a t i o n about a p a r t i c u l a r atom, the TF method for the systematics of all atoms. There is, of course, o v e r l a p of the two approaches,
a certain
and they are not c o m p l e t e l y unrelated.
We shall discuss their c o n n e c t i o n to some extent in Chapter Two.
Atomic units.
All future algebraic m a n i p u l a t i o n s
are eased significant-
ly w h e n atomic units are used for m e a s u r i n g distances,
energies,
etc.
Let us b r i e f l y c o n s i d e r the m a n y - p a r t i c l e H a m i l t o n o p e r a t o r
N
1
'=
+2
N 9=I
Ze 2
i ~
j e2 (I-I)
j,k=1
rjk
of an atom w i t h nuclear charge Ze and N electrons,
each of mass m and
c a r r y i n g charge -e. The third sum is primed to denote the o m i s s i o n of the term w i t h j = k. Obviously,
r. stands for the d i s t a n c e between the 3 nucleus and the j-th electron, whereas rjk is the d i s t a n c e from the j-th to the k-th electron,
and pj the m o m e n t u m of the j-th electron.
This
Hmp is a c c o m p a n i e d by the c o m m u t a t i o n relations
[~j,Pk ] = i ~ ? 6jk
(1-2)
and the i n j u n c t i o n s obey. ~.
Equations
c a u s e d by the F e r m i
(I) and
(2) c o n t a i n
But n o n e of t h e m can p o s s i b l y
perturbation m i c scale.
series
because
To see this
statistics
that the e l e c t r o n s
three dimensional
parameters:
be u s e d as e x p a n s i o n
together
in detail,
variable
m, e,
of a
t h e y do no m o r e t h a n set the ato-
let us r e w r i t e
(I) and
(2) w i t h the
aid of the B o h r r a d i u s ~2
ao = m--~-fe = 0.5292
and t w i c e the R y d b e r g
Eo
Equations
(I-3)
- 27.21eV.
(I-4)
energy
2 _ e ao _ me~2
(I) and
~
(2) n o w a p p e a r
as
Z-/ 1
7-
/~
)
_
Hmp/E° = 3 ~(PJ a°
7,
+
1>'
~ j,k
j ~
(I-5)
(rjk/ao)
and
[(rj/a o)
If w e t h e n Hmp/E o
, (pk/~{]
introduce
as r e l e v a n t
U s i n g the same
.
the d i m e n s i o n l e s s objects,
letters
Hmp =
= i 1 6jk
(I-6)
quantities
~./ao, ÷ 3 Pj/ao' and to m, e, and ~ d i s a p p e a r s .
all r e f e r e n c e
as for the d i m e n s i o n a l
y pj
~
j
j
IZ
+
y
J
quantities,
we now have
.... rjk
j ,k
(I-7)
and ÷ _~ [rj,Pk]
Equations
(7) and
=
e9 i 1 6jk
(I-8)
(8) are i d e n t i c a l
i n s t e a d of the m a c r o s c o p i c s Formally,
.
the t r a n s i t i o n
units
from
w i t h Eqs.
(cm, erg,
(I) and
(2) to
" s e t t i n g e = { = m = I," but t h e m e a n i n g is m a d e p r e c i s e
by the a r g u m e n t
presented
(I) and
etc.)
atomic
(7) and
of this above.
(2) e x c e p t untis
that
are used.
(8) can be d o n e by
colloquial
procedure
Besides s i m p l i f y i n g the algebra,
the use of atomic units also
prevents us from trying such foolish things like "expanding the energy in powers of ~ , " a literature.
phrase that one meets s u r p r i s i n g l y frequently in the
The energy is n o t h i n g but Eo times a d i m e n s i o n l e s s
func-
tion of Z and N, it depends on ~ only through E o ~ I / ~ 2. We shall see later, what is really meant w h e n the foregoing phrase is used. The m a n y p a r t i c l e problem defined by Eqs. (7) and be solved exactly.
It is m u c h too complicated.
(8) cannot
This is true even w h e n
the number of electrons is only two, the s i t u a t i o n of h e l i u m - l i k e atoms. There is a branch of r e s e a r c h 3 in w h i c h rigorous theorems about the system
(7) and
(8) are proved,
the total binding energy.
such as
(disappointingly rough)
One can show for example,
limits on
that for N = Z ÷ ~ the
m a n y p a r t i c l e problem reduces to the o r i g i n a l TF model, w h i c h we shall d e s c r i b e in the next Chapter. those highly m a t h e m a t i z e d
In these lectures, we shall not follow
lines.
I prefer rather simple physical argu-
ments instead of e m p l o y i n g the m a c h i n e r y of functional analysis. it is my i m p r e s s i o n that those "rigorous" methods
Also,
are of little help
w h e n it comes to improving the d e s c r i p t i o n by going beyond the o r i g i n a l TF model. (7) and
Finally,
let us not forget that m a t h e m a t i c a l theorems about
(8) are not absolute k n o w l e d g e about real atoms,
ting down the H a m i l t o n o p e r a t o r proximations:
because in put-
(7) we have already made p h y s i c a l ap-
the finite size and mass of the nucleus is disregarded;
so are all r e l a t i v i s t i c effects i n c l u d i n g m a g n e t i c interactions quantum
e l e c t r o d y n a m i c a l corrections;
other than electric interactions
are n e g l e c t e d - no r e f e r e n c e is made to g r a v i t a t i o n a l Of course,
both attitudes,
physical one,
are valuable,
and
and weak forces.
the highly m a t h e m a t i c a l one and the m o r e but there is danger in judging one by the
standards of the other.
Bohr atoms. We continue the i n t r o d u c t o r y remarks by studying a v e r y simple model in order to i l l u s t r a t e a few basic concepts.
This primi-
tive t h e o r e t i c a l model neglects the i n t e r - e l e c t r o n i c interaction, t r e a t i n g the electrons as i n d e p e n d e n t l y bound by the nucleus.
thus
But even
if fermions do not interact they are aware of each other t h r o u g h the Pauli principle.
Therefore,
such n o n i n t e r a c t i n ~ electrons
(NIE) will
fill the s u c c e s s i v e Bohr shells of the Coulomb p o t e n t i a l w i t h two electrons in each o c c u p i e d orbital state. For the present purpose it w o u l d be sufficient to c o n s i d e r the s i t u a t i o n of m full Bohr shells.
But w i t h an eye on a later dis-
cussion of shell effects, in Chapter Five, let us a d d i t i o n a l l y suppose
t h a t t he tiplicity
(m+1)th s h e l l
is f i l l e d by a f r a c t i o n
of the s h e l l w i t h
the t o t a l number,
principal
N, of e l e c t r o n s
quantum
t h e n is
~, o ~ < I . n u m b e r m'
(see P r o b l e m
S i n c e the mulis 2 m ' 2 - f o l d , I)
m
N =~
2m '2 + B2(m+1) 2
(I-9)
m'=1 = ~2 (re+l) 3 _ 61(m+1 ) + 2 B ( m + 1 ) 2
The total
binding
energy
for a n u c l e u s
of c h a r g e
Z is e v e n simpler,
m
-E = > 2m '2 Z2 m'=l ~
=
which
zg(m+~)
uses the s i n g l e
+ B2(m+1)2
particle
s t a n d Eq. (9) as d e f i n i n g -E(Z,N).
Towards
dence
explicit
we p r o c e e d
binding
if y s o l v e s
Z2/(2m'2).
of N, t h e n Eq.(10)
of m a k i n g
this
the e q u a t i o n
,
(I-12)
We use the s t a n d a r d
there
Gaussian
[y+I/2]
that is
y - = integer,
is just one solu-
notation,
(I-13)
the i n t r o d u c t i o n
= y -
dis-
depen-
(I-11)
m = [y - I/2]
For t h e s e q u e l
functional
(m+~)3 _ ~(m+~) I 3
I p a r t of y - ~ . (For N>0,
I/2.)
If we u n d e r -
from n o t i n g that
I 3 y3 _ ~ y = ~ N
tion larger than
energy
m and ~ as f u n c t i o n s
the o b j e c t i v e
< ~3N < (m+~) 3 - ~1(m+½ ) =
t h e n m is the i n t e g e r
(1-10)
,
plays
Consequently,
Z2 2(m+l)Z
,
of ,
defined
by
(I-14)
I
< =
-5
will
prove
) 3
into
Eq. (9),
_ I (y_l_) (I-18)
+ 2~(y-) 2 and
solve
for
b. The
result
I = 2 + AS a c o n s e q u e n c e easily cally
checks from
The
-E
with
y(N)
solved
jumps
from m + ½ to m+~,
Also,
~ grows
one
monotoni-
of Eqs. (I0), (16),and
I _ ~ +
2 Y-3 } (y_) 2
1 (2_)
Let
us
first
observe
(19)
now p r o d u c e s
,
that
(i-20)
this
binding
f u n c t i o n of y - and t h e r e f o r e of N - a l t h o u g h I I +~ to -~. Next, w e n o t e t h a t for l a r g e N,
from
energy o c c a Eq. (12)
is
by y(N)
so t h a t
the
quently,
=
the
(3N)I/3
oscillatory
the b i n d i n g
-E = Z 2
where
is n o n z e r o .
as it should.
combination
from Eq. (12).
here
3
as y i n c r e a s e s
= z2{y
(I-19)
(y_)2
the d e n o m i n a t o r
to one,
is a c o n t i n u o u s sionally
2 Y-~
(2_~)
of y>~,
that,
zero
is I
+
,
ellipsis
+ I~ (3N)-I/3
contribution
energy
of N I E
{(~N) I/3 - ~I +
indicates
+ ....
in
(20)
(1-21)
is of o r d e r
N -I/3.
Conse-
is
....
}
oscillatory
,
terms
(I-22)
of o r d e r
N -I/3
and
smal-
ler. The physical origin of these terms is the process of shells.
We shall disregard
of the filling
them here with the promise of returning
later when we shall engage in a more detailed d i s c u s s i o n of shell effects. Expansion
(21) is expected
just the two terms displayed
to be good for large N. However,
explicitly
form a p r a c t i c a l l y
mula even for small N. An impressive way of d e m o n s t r a t i n g
perfect
for-
the high qua-
lity of this two-term a p p r o x i m a t i o n
is to look at the values predicted 357 for N, at which closed shells occur. For y=~,~,~ .... , the exact ans-
wer of Eq. (12) is N=2,10,28, 9.999974,
27.999991,
.... whereas
Eq. (21) produces N=1.99987,
... ; even for the first shell the agreement
is bet-
ter than one hundredth of a percent. We have just learned an important asymptotic
expansion
like Eq.(21)
accurate a p p r o x i m a t i o n derations
may be,
a few terms of an
and frequently
are,
a highly
even for very m o d e r a t e values of N. Such consi-
based upon large numbers
are the origin of the label
cal" that is attached to TF theory, tion is, however,
lesson:
The fundamental
rather a semiclassical
This will become
physical
"statisti-
approxima-
one.
clearer when we now answer the question
how
one can find the leading term in
(22) somewhat more directly,
utilizing our detailed knowledge
of the energy and d e g e n e r a c y of bound
states
without
in the Coulomb potential. The count of electrons
the m u l t i p l i c i t i e s
summed over all occupied
> N =
is evaluated
of all occupied
all state
in Eq. (9) as the sum of
shells.Equivalently,
we could have
states,
sI0 if the state is °ccupied if the state is not occupied
Since a given state is occupied is larger than a certain amount,
(or not)
}
if its binding energy,
(i-23)
-E
state' ~ , (or less), we can employ Heaviside's
unit step function,
~(x)
=
{~ for x > o for x < o
'
(I-24)
in w r i t i n g
N(~)
= > all states
Such a sum over all cle Hamilton operator
~(-Estat e - ~)
eigenstates for NIE,
of an operator,
(I-25)
here the single-parti-
1 2 Z HNI E = ~ p - ~ is m o r e
concisely
N(~)= [Do not w o r r y equals
the
assigning the
expressed
tr
about
the
the
any v a l u e
The
of the
between
we
0 and
can
of p a r t l y
respective
of a f r a c t i o n a l l y
single-particle
E(~)
We t h e n
have
(I-27)
possibility
energy
t e r Five,] A n a l o g o u s l y , over
as a trace.
u ( - H N I E - ~)
binding
situation
(I-26)
,
I to D(x=o)
filled express
energies
shell. the
filled
shell,
shells.
and the
enables More
energy
of all o c c u p i e d
Then
freedom
of
us to d e s c r i b e
about
this
of Eq. (I0)
in C h a p -
as the
sum
states,
= tr HNI E ~ ( - H N I E - ~)
1-28)
~(x))
1-29)
identity
d ~(x
used
in the
= ~(x)
form
co
-
/dx'
~(-x')
1-30)
= x ~(-x)
X
can be u s e d
to r e l a t e
E(~)
= tr
E(~)
to N(~):
(HNI E + ~) ~
(-HNI E - ~) - ~ tr ~ ( - H N I E - ~) (I-31)
o0 =
-
tr
fd~'
~(-HNI E - ~')
- ~ N(~)
,
or, oo
E(~)
We
see t h a t
values
energy of
-
E(~)
is no s u r p r i s e . binding
=
~ N(~)
- fd~'
is i m m e d i a t e l y Recall
larger
~ equal
that than
to the
N(~)
N(~')
(I-32)
available signifies
as soon
the n u m b e r
~. C o n s e q u e n t l y ,
binding
energy
as we k n o w N(~).
of a
N(~)
of s t a t e s
with
is d i s c o n t i n u o u s
(Bohr)
shell,
This
and t h e
at all size
of the jump of N(~) respective
shell.
at such a d i s c o n t i n u i t y
So N(~),
the energy and the m u l t i p l i c i t y The problem in an appropriate, of Eq. (22) only, gument.
space volume
approximate way.
[Remember,
the trace in Eq. (27)
it is the leading term
that we want to derive simply.] states.
of the
of all shells.
is now reduced to evaluating
The count of states
count of orbital
is the m u l t i p l i c i t y
regarded as a function of ~, tells us both
First an intuitive
is the spin m u l t i p l i c i t y
of two,
There is roughly one orbital
(2v~) 3 [=(2E) 3 in atomic units],
state per phase-
so that
n(-1½p ,2- ~-r)z
2
ar-
times the
~)
,
(I-33)
(2~) 3 where primes
have been used to d i s t i n g u i s h
step function phase-space
equals
treme semiclassical
classical
unity in the classically
and vanishes
the p o s s i b i l i t y
numbers
from operators.
allowed domain of the
outside. Therefore, Eq.(33)
(or should we say:
The
semiquantal?)
of finding the q u a n t u m - m e c h a n i c a l
represents limit,
the ex-
in which
system outside the
allowed region is ignored. Some support
implications.
for the a p p r o x i m a t i o n
After p e r f o r m i n g
N(~)
~ f(d~') ~ I
the m o m e n t u m
[2( ~, _ ~)]3/2
where the square root is understood Then the r' integration
N(~)
=
(the integral
produces,
~ (2Z2/~)3/2
to vanish
we have
,
(I-34)
for negative
with x = ~r'/Z
arguments.
,
3--~4fdx x I/2 (l-x) 3/2 o
2 tZ 2 , 3/2 ~,~
has the value ~/16.)
-E(~)
(33) is supplied by its integration,
=- Z 2 ( ~ ) I/2
which combined with
(I-35)
Insertion
into Eq. (32) results
in
(I-36)
(35) is
-E ~ Z2(3N) I/3
(I-37)
10
Indeed,
here is the leading term of Eq. (22), now very simply r e p r o d u c e d
by the s e m i c l a s s i c a l counting of states.
Please note that the steps from
Eq. (32) to Eq. (37) did not require any k n o w l e d g e about the energy and m u l t i p l i c i t y of bound states in the Coulomb potential.
Traces and p h a s e - s p a c e integrals. tion alone w h e n
One does not have to rely upon intui-
w r i t i n q down Eq. (33). A general way of e v a l u a t i n g the
trace of a f u n c t i o n of p o s i t i o n o p e r a t o r ~ and the conjugate m o m e n t u m o p e r a t o r p is -9,-
tr F(r,p)
->-
= S(d~')
-)-
(1-38)
We have left out the factor of two for the spin m u l t i p l i c i t y here, because i t
dered
is
irrelevant
=
F (~' ,p'
[ 5'>
)
and inserted into
tr F ( ~ , p )
=
-i r', p'
1
(2tO 3 / 2
e
(38) produces
= f (d~-')-(d~'')
F(~',p')
(i-41)
(2~) 3
Equation
(41) is an exact statement
the operator,
for an ordered o p e r a t o r F(~,p).
of w h i c h the trace is desired,
sometimes do the o r d e r i n g explicitly.
-i[~p - F.r)t e
=
is not ordered,
An example is
one can
(see Problem 3)
If
11
i + F-r÷ t -
where of
-i ~I( p÷ + I ~ t)zt _ i F 2 t 3 / 2 4
e
e
F is a c o n s t a n t
e
vector.
W i t h the aid of
½3 2 _ ÷F-r÷ can be o r d e r e d if e x p r e s s e d
integral.
W e shall have
a use
W i t h the e x c e p t i o n ordering
of an o p e r a t o r
then Eq. (41)
when
can be used and p
manifests
--- ; (d~') (d~') (2~) 3
as the basis
a highly
position
and momentum. therefore
withstanding
the
The s t a r t i n g
point
of
This
fact that
is clearly
involves of p,
terms.
Under
(I-43)
Since the n o n c o m m u t a t i v i t y
in the s e m i c l a s s i c a l
limit,
Eq. (43)
approximation.
improvements is a l r e a d y
"quantum
corrections,"
a quantum mechanical
the s e m i c l a s s i c a l
theory
even
process
is at the heart of q u a n t u m mechanics,
(41)
the
functions
(43) are due to the n o n c o m m u t a t i v i t y
call these
not the c l a s s i c a l
However,
are small,
for approximations.
semiclassical
Four.
instances,
impossible.
of r w i t h
functions Fourier
F(~' ,pl ÷ }
insignificant
we shall
in C h a p t e r simple
as the o r d e r i n g
of functions
commutators
All r e f i n e m e n t s
ture,
Inasmuch
all other
from F(r',p') by c o m m u t a t o r
these
becomes
later,
is p r a c t i c a l l y
of c o m m u t a t o r s
÷ ÷ tr F(r,p)
of r
for Eq. (42)
F(r,p)
differs circumstances
(42),
(I-42)
as the a p p r o p r i a t e
of a few r e l a t i v e l y
is not useless.
the e v a l u a t i o n
,
of the atom,
(or,
and not-
result.
semiquantal)
w h i c h does
of
not exist
picin the
first place. For the trace cal e v a l u a t i o n
of Eq. (33) will
Broglie
wavelength
typical
distance
This
condition
clear c h a r g e
sible
In this
electron
electron
the s e m i c l a s s i -
approximation, is small
potential
in this
situation
sense the s e m i c l a s s i c a l
if the de-
compared
varies
to the
significantly. and the nu-
the e l e c t r o n i c
approximation
cloud
is equi-
one.
The p r i m i t i v e
for h i g h l y
ionized
potential
is negligible.
not be t a k e n
that
if both the number of electrons
a statistical
by the C o u l o m b
interaction
should
be a r e l i a b l e
the C o u l o m b
because
shielding.
approximation
is g o v e r n e d
above
over w h i c h
is s a t i s f i e d
to a large-N,
Bohr atoms w i t h
all this means
of an i n d i v i d u a l
are large,
is very dense. valent
of Eq. (27)
seriously,
mo d e l
atoms
of NIE
only,
of the nucleus, Consequently, unless
constitutes
in w h i c h
a sen-
the d y n a m i c s
and the e l e c t r o n -
the results
obtained
N 2 m"2
+ (Z-Zm,+I)] (I-73)
+ 2m ,2)
m" =I
= 2m '2 Rm,
which
uses
Eqs. (45) and
N2 _
(Z-Zm+I)2
m'-1 (2> Um,,(Rm,) m" =I (52),
as well
+ Urn, (Rm,))
as
[Eqs. (45), (52)
m = B2(m+1)212y---2m"2+
and
9)]
~2(m+I) 2]
m"=1 (I-74)
m
= ~2(m+1) 2Rm+ I ( 2 >
turns
Urn. (Rm+ I) + ~Um+ I (Rm+ I)) m" = I
(72) into m
E 2
=
-
m' -I 2m'2~m Um,,(Rm,) m '=I "=I
+ ~2(m+I)
2 m > Um,,(Rm+ I m" = 1
(i-75)
19
-
{m
~I y 2 m ' 2 U m ,
(Rm,) + ~I[ ~ 2 (m+1) 2] [bUm+ I (Rm+ I ]
}
m' =I The contents of the two curly brackets
are immediately
recognized as
the interaction energy of the pairs of shells and the self energy E of the individual
shells,
sse Indeed, E 2 is the negative of
respectively.
the electron-electron interation energy, as it should be. Before adding EIp of (65) and E 2 of (75) to get the total energy itself,
it is useful to rewrite EiP,pot.
EIP,pot
m Z = ~---2m'2[-R--~ m'=1
From
m + ~---Um,,(Rm,) m"=1
(64) we get
+ ~Um+1(Rm,)]
m + ~2(m+i)2[_ " Z + > Um' (Rm+ I) + ~Um+ I (Rm+ I)] Rm+ I m' =I = ,\m
2m' 2
/m,..=lRm,
m'-1
m
(-Z+7 2m ''2) m"=1
*~
2m'ZU m, (Rm,)
m' =I
m m + 7--- 7-~2m'2Um,, (Rm,) m'=1 m"=m' +I
(I-76)
m +
2m,ZUm+1(Rm, ) m' =I
+ ~
2(m+i)2 Rm+ I
m (-Z+ ~----2m ,2) m'=l
+ [~2(m+I) 2] x [~Um+ I (Rm+ I) ]
After using Eq. (59) and the m'-m" reaction symmetry that we observed
symmetry of 2m '2 Um,,(Rm,)
[the action-
in Eq. (53)], this reads
m = -2> EIP'p°t
Z 2 - 2;~ Z 2 m+1 m'=1 m'
+"I m P 1 ~m"-1 2m"2Um, "= m'=1
(Rm,,) + #(2m+I)2> m U m, (Rm+ I m'=1
+
20
+ 2
~-----2m'2Um, (Rm,)+ ~[~,2(m+I) mt=1
] [~Um+ I (Rm+l)
m
(I-77)
= - 2 >
Z 2 - 2~ Z 2 m' m+1
- E2 + E
sse
m'=1 F i n a l l y,
we o b t a i n t h e t o t a l
-E = -
=~
binding
(Eki n + E i P , p o t m --z2m'
Z2 m+1
+ ~
energy
+ E 2)
(I-78)
- ESS e
m'=1 We c o m p a r e
this w i t h Eq. (46) and n o t i c e t h at the m o r e
m e n t of the e l e c t r o n s This
leads to an a d d i t i o n a l
is v e r y s a t i s f a c t o r y
because
electrons
in the s a m e s h e l l
w h i c h was
left out w h e n Equation
E
term,
symmetrical
Ess e , in the energy.
is the i n t e r a c t i o n
sse (plus t h e i n n o c u o u s
treat-
electron
e n e r g y of self-energy),
(46) was d e r i v e d .
(78) can be s i m p l i f i e d .
First,
we use
2m '2 = Zm, - Zm,+1
(I-79)
and
~2(m+I)2
which
=
Zm+l
are c o n s e q u e n c e s
-
(Z-N)
,
(I-80)
of Eqs. (45) and
(9), to r e w r i t e Ess e of
(60)
as
m Ess e = >
(Zm,- Zm,+l)Zm,
+ ~ z m+1 2
- 5(Z-N)Zm+I
(I-81)
m'=1 T h e n we i n s e r t this
into
m -E = > Zm, m' =] W e can now e v a l u a t e given
Zm,+1
(19),
is
1-82)
+ ~(Z-N) Zm+ I
the sum o v e r m',
in Eqs. (16) and
to -E. T h e y are
(78). T h e o u t c o m e
e x p r e s s m and ~ in terms of y, as
and p i c k out the two
leading
contributions
21 4 7 ) _ ~Z 1 2 + 0(Z5/3~Zy2~yS) - E = (Z2y - ~1Z y , + ~-~y To t h i s
order,
y is
simply
3 -E = Z2( N) I/3(I
aiven
by
(3N) 1/3
[Eq. ( 2 1 ) ] ,
(1-83)
so t h a t
I N + I N 2) _ 1 2 2 Z ~( ) ~Z
+ 0(ZN2/3~
(1-84)
N5 / 3 )
The n e u t r a l - a t o m b i n d i n g energy p r e d i c t e d by our i m p r o v e d model of Bohr atoms w i t h s h i e l d i n g is, consequently,
-E = 0.736 Z 7/3 - ~1Z 2 + 0 (Z5/3)
W i t h o u t shielding, interaction,
(I-85)
that is: w i t h o u t a c c o u n t i n g for the e l e c t r o n - e l e c t r o n
the result was
[Eq. (22) for N = Z]
-E = 1 • 145 Z 7/3 - ~1Z 2 + 0 (Z5/3 )
(I-86)
W h e r e a s the s c r e e n i n g of the nuclear p o t e n t i a l by the inner electrons reduces the leading term by 5/14 ~ I/3, it does not affect the Z 2 term at all. We shall see later,
in C h a p t e r Three,
that this n e x t - t o - l e a d i n g
term is a c o n s e q u e n c e of the C o u l o m b shape of the e f f e c t i v e p o t e n t i a l for small r. It is the same for all p o t e n t i a l s w i t h V ~ - Z/r for r ÷ o, for w h i c h reason it is i n d e p e n d e n t of N
[Eqs. (22) and
The two examples that we looked at so far, and the V(r)
of
Size of atoms. d i n g consists
(84) confirm this].
the C o u l o m b p o t e n t i a l of
(26)
(56), both have this property.
A last a p p l i c a t i o n of our model of Bohr atoms w i t h shielin s t u d y i n g the Z d e p e n d e n c e of the size of neutral atoms.
The i n d i v i d u a l Bohr shells shrink p r o p o r t i o n a l to I/Zm, as Z increases [see Eq.(59)].
This w o u l d m e a n that the size of an atom is roughly gi-
ven by I/Z, if there w e r e not the n e c e s s i t y of filling additional shells to c o m p e n s a t e
for the
growth of n u c l e a r charge. Clearly, a q u a l i f i e d state-
ment about atomic size requires the e v a l u a t i o n of some average of r over the atom. A c c o r d i n g to Eq. (59), it is the inverse of Rm, that is easy to handle. We shall t h e r e f o r e m e a s u r e the size R of an atom as
22
5=5 R
where
N
r
denotes
(i-87)
the e x p e c t a t i o n
value
of I/r.
In our m o d e l
it is
given by
m 2m I 2 < > = >m'=1 R m , + It has
a simple
physical
of the e l e c t r o n s of the nucleus, unit
charge
2 (m+ I ) 2
(I-88)
Rm+1
significance:
is the e l e c t r o s t a t i c
in the field of a unit r=o;
or,
equivalently,
charge
I
r
=
potential
Z ~) (r=o)
(v+
at the l o c a t i o n
the e l e c t r o s t a t i c
in the field of the electrons.
in terms of the e f f e c t i v e
situated
As such
energy
energy
of this
it can be e v a l u a t e d
V:
m =>
(I-89)
Um, (o) + ~Um+1(o) m'=1
Indeed,
Eq. (52) After
assures
us of the e q u i v a l e n c e
employing
Eq. (59) to r e w r i t e
of
(88)
and
(89).
(88),
m
= 2>
Zm,
+ 2~ Zm+ I
m'=1
:2>
(I-90)
m (Z- 2m' (m'-1) (m'
_
I
5))+
2~(Z-N)
+ 4~2(m+I) 2
m'=1 [the latter identify
the
equality leading
(21). The r e s u l t
also uses Eq. (80)], contributions
for n e u t r a l
we can sum over m'
and then
the aid of Eqs. (16),
(19),
and
is
= 2Z( N ) I / 3 ( I r which
with
atoms
N ~I ~)
N - Z(I- 3)
+ 0(N 2/ 3% ZN- I/3)
,
(I-91)
reads
(I-92)
Consequently,
the atomic
size
is
[Eq. (87)]
23
R ~ Z -I/3
(I-93)
Heavier atoms are g e o m e t r i c a l l y simple model will remain valid A remarkable Z derivative
smaller,
This prediction of our rather
in more realistic
observation
treatments.
is the agreement of Eq. (92) with the
of Eq. (84) to leading order,
~ ~z(_E)
Its physical
(I-94)
significance
becomes
transparent when we exhibit the change
in the binding energy that is caused by increasing by the infinitesimal
6(-E)
the nuclear charge Z
amount 6Z:
= ~z(-E)6Z
~ r
(I-95)
This says that the change in the binding energy is mainly given by the electrostatic
energy of the extra nuclear charge;
of the shell radii Rm, do not contribute
to 8(-E)
the induced alterations to leading order.
result of the model must be contrasted with the corresponding of the exact treatment based upon the m a n y - p a r t i c l e (7). In general, operator
causes
an infinitesimal
tion value of the respective present discussion,
&(-E)
reads
= = j rj r
which says that the change in the bindin~ the electrostatic
energy is entirely given by
in our model of Bohr atoms with shielding,
exactly,
but approximately.
This minor deficiency
possibly be removed by a slightly different (59)].
(I-96)
'
energy of the extra nuclear charge 8Z.
As we see, is not obeyed
in a Hamilton
which is equal to the expecta-
change of the Hamilton operator 5. In the
this statement
=
Hamilton operator
change of a parameter
a change in the energy,
This
implication
It is not worth the trouble, The models
definition
Eq. (96) could
of the Rm,
[Eq.
though.
studied in this Introduction
not only provided
a first insight into the genera] characteristics of complex atoms, but also made us somewhat familiar with a few important ideas: the concept of the effective potential, through p h a s e - s p a c e
the semiclassical
integrals,
and relations
by Eq. (32) are the central ones.
evaluation
of traces
of the kind illustrated
The next Chapter,
devoted to the Tho-
24
mas-Fermi model,
will use them for a first self-consistent
description.
Problems
I-I. Sums of powers of m' , as, e.g., evaluated
in Eq. (9), can be conveniently
following the pattern of this example:
>
m
m'
m I I 2_(m,_ I 2] = ~ - - ~ 5 [ (m'+ ~) ~)
m'=i
m'=1 I TM = ~ >
I 1 ~ -1 (m'+ 5) 2 - _ /
m' =I
= l(m+~)
Show that the other sums,
1 2 (m'+ 5)
m' =o
-
(0+
)2 = 21 (m+5)I 2 __8
that occur in this Chapter,
are given by
I 3 _ 1 ~ ( m + y) I m' 2 = l(m + 5) m' =I
m
m'3 =
1
1 4
(m+5)
-
1(m+1
5)
z
1
+ 6--4
m' =I m
~ ---m' 4
: { ( m + 5 )1 5 _ 1z ( m + Y )1 3 + ~ (7m + 5 )
1
m'=1 m 's =
(m+ I 6
-
m'=1
"•4 ( m + ~ )1
~. "~6 +
1 2
(m+~)
1
128
'
and m
m,6
1 5 + T8 (m +5) 1 3: 7I( ~ + 1 ) 7 _ z1( m +5)
31 (m+ I 1344 5)
m' =I I-2. Use the periodicity
of
[see Eq. (17)] to write
series, (-I)m ~{n" m=1
sin(2~my)
= -
it as a Fourier
25
Integrate
this
repeatedly
to e v a l u a t e
(-1)m (~m) 2 cos (2umy)
~-' ~_
m=1
(-1)m (~m) 3 sin (2~my)
,
m=1
(-1)m (~m) ~ cos (2~my) m=1
1-3.
In o r d e r
to e s t a b l i s h
Eq. (43),
first
use
the one-dimensional
sta-
tements
_i _, p2 2
e
which
are
F X = e
-iFxt
p e
iFxt
illustrations
e
-i f(p)
xe
6F
(-Fx)
e
ie! 6F
=p+Fxt
,
of i f (p)
d = x - ~
f(p)
i f(x)
d = p + ~
f (x)
and e
to s h o w
-i f(x)
p e
that I p2_ -i(~ Fx)t
P8 -i ~
e
= e
e
I-4.
The
e
to t h r e e
average
value _
where
m'
-i F2t3/24
and
arrive
at Eq. (42).
o f r ~ for an o r b i t a l m '2 ~T[5m'2-
is the p r i n c i p a l quantum
6F
e
dimensions
(r--f)m,,Z ,
momentum
i e
-i ~I( p + ~ FIt ) 2 t
iFxt = e
i e
I (p+F t) 3 -i ~-~
iFxt = e
Generalize
iFxt e
number.
quantum Average
state
3Z ' (Z' + 1 ) ]
number this
in the B o h r
is
,
and £ ' = o , . . . , m ' - I
over
atom
the
£' v a l u e s
the
angular
to find
26 m,2 = ~r
--
(r2) m,
A measure
(7m'2 + 5)
for the size R of the atom is the average of r 2 , m
N R 2 = (r2)atom = ~---2m'2(r2)m, + ~2(m+I) 2 (r2)m+ I m' =I
Show that
R--- ~ Z
[ 3 N) 2/3 I (Y + Y]
for large N. Compare with Eqs. (87) and
I-5. The contribution
(92).
of a full Bohr shell, with principal quantum num-
ber m', to no, the electron density at the site of the nucleus,
is gi-
ven by (2Z)3 4~
(
)'
Show that no -
(2Z) 4~ 3 [ ~
(m1__[ ) 3 _ 2 (1N )3 - 2 / 3
+ 0(N -4/ 3 ) ]
,
m,=1 for a Bohr atom
(without shielding)
that contains N electrons.
I-6. Derive the identity m
m f(m') - [ dyf(y) m'=1 I (which, Eq.(47).
incidentally,
was
=
m f dy < y - g > I
dy
first proven by Euler)
and use it to confirm
Chapter
Two
THOMAS The m a y be u s e f u l were
moving
this
idea very
crude
effective
neral
s t r u c t u r e 1,2 Z
V = - -- + r challenge
trostatic
way.
_
V
V2
Ves,
due
part]
n
tool
the
us t h a t
We s h a l l explicit
that
it
as if t h e y n o w take
assumptions
V possesses
the
ge-
,
(2-I)
electron-electron
for achieving
this
part
aim
in a
is the
elec-
,
(2-2)
es
relates
b o t h Ves
=
making
It is c l e a r
in f i n d i n g
taught
(or ion)
potential.
however,
V.
fundamental
Chapter
in an a t o m
[electron-electron
equation
4~ which
without,
Poisson
I
preceding
electrons
potential,
consists
The
of the
MOD£L
in an e f f e c t i v e
seriously,
the
consistent
the
independently
about
and the
models
to t r e a t
- FERMI
the
to the
electron
electrons.
and n in t e r m s
density, As
of V,
soon
n,
to the
as we
shall
Eq. (2) w i l l
electrostatic
potential,
have managed
to e x p r e s s
determine
the
effective
poten-
tial.
General
formalism.
The
independent-particle
H = lp2
The
electrons
all
states
pied, thus
equals
the
the
energy
with
by the
number
electrons
is c o n t r o l l e d
by the
(2-3)
eigenstates
binding
those
determined
of the
operator
+ V(~)
fill
with
whereas
dynamics
Hamilton
less
larger binding
requirement
of e l e c t r o n s
of H s u c c e s s i v e l y
N.
than
Just
a certain
energy
that
the
in such value,
are not. count
The
a way
{, are o c c u -
parameter
of o c c u p i e d
as in Eq. (I-27)
that
this
~ is
states
is e x p r e s s e d
as
N = tr
where trace.
we r e m e m b e r
~ (-H-{)
that
the
,
spin mulitplicity
(2-4)
of two
is i n c l u d e d
in the
28
The
The
sum of i n d e p e n d e n t - p a r t i c l e
Eip
= tr H~(-H-~)
combination
function
~,
invites
Eip
which,
with
H+~,
that
appears
rewriting
aid of
is,
analogously,
(2-5)
in the
Eip
= tr(H+~)u(-H-~)
the
energies
argument
of H e a v i s i d e ' s
step
as
- ~tr ~ (-H-~)
,
(2-6)
(4) and the d e f i n i t i o n
E I { tr(H+~)~(-H-~)
(2-7)
,
reads
Eip
In this E I are ding
= E I - ~N
equation,
N is the
function(al)s
energy
(2-8)
of the
given
number
effective
of e l e c t r o n s ,
potential
and b o t h
Eip
V and the m i n i m u m
and bin-
~.
Let us m a k e
contact
with
Eqs. (I-27)
and
(I-32),
in t h a t w e
write co
E I (~)
= - Sd~ ' N(~')
N(~')
= tr ~(-H-~')
,
(2-9)
where
is the
count
appears
now
of states
with
(2-10)
binding
energy
exceeding
~'.
Equation
as
N = N(~)
Equation
(2-11)
(9) can be e q u i v a l e n t l y
If ~ d e v i a t e s
from
by the
8~,
amount
its then
~E I & ~ E I = ~--~ 8~
This
has
the
(4)
important
correct
presented
value
[which
as a d i f f e r e n t i a l is d e t e r m i n e d
statement.
by Eq. (11)]
E I is off by
= N(~)~
implication
= N&~
that
(2-12)
E i p of Eq. (8) is s t a t i o n a r y
un-
29
der variations
of
8( Eip
~
= 8
In a d d i t i o n of b o t h
energies
density
n:
first
this
E I - N6(
= 8V E I =
follows
b e r of e l e c t r o n s
and
second
expresses
as t h e t r a c e derivative trace
the
of the
of t h a t
operation,
8 H f(H)
Under
the trace
also
of the
we
change
let us
from
need
of the
8H
that
possible
the
response
electron
a formal
because
we r e g a r d
proof.
N is the as
independent
numof
identity:
(2-15)
of a f u n c t i o n
change
in the (15)
with
if
The
given
,
commutes
only
local
(2-14)
following
[Note t h a t
f' (H)
The
exhibits
supply
(8),
the
f' (H)
on V.
n(~')
in the t r a c e
product
= 6H
depend
potential
( is a p a r a m e t e r
function.
the
(2-13)
obvious,
= tr 6H
of course):
= o
immediately
unless
value,
f(d~')SV(~')
equality,
8 H tr f(E)
which
correct
to (, E I and E i p
is i n t u i t i v e l y
equality
V. F o r t h e
its
to v a r i a t i o n s
6 V Eip Although
(around
of an o p e r a t o r
operator,
8H,
H
and the
is not t r u e w i t h o u t
the
~:
[6H,H]
noncommutativity
= o.
does
(2-16)
not m a t t e r . ]
In o u r
application,
f(H)
=
(H+()D(-H-() (2-17)
f' (H)
[compare
with
= q(-H-()
Eq. (I-29)],
and 8H
= 6V.
Accordingly,
8 v E 1 = tr 8V q(-H-~)
(2-18) = 2
W e use, of two
again, is,
f(d~')
primes
o n c e more,
= ; (d~')Vee (~')n (~')
;
(2-29)
32
the last step uses Eq. (20). C o n s e q u e n t l y ,
the c o r r e c t
energy expression
is E = Eip - / ( d ~ ) V e e n The s e c o n d t e r m r e m o v e s interaction
contained
+ Eee
the i n c o r r e c t
in Eip,
stationary
6
account
for the e l e c t r o n - e l e c t r o n
and the last t e r m adds the c o r r e c t
The e n e r g y of Eq. (30) of b e i n g
(2-30)
is e n d o w e d w i t h the i m p o r t a n t
under variations
of b o t h V and
~,
E = 8 v E = o.
In o r d e r to see this,
(2-31)
first
6( - / ( d ~ ) V e e n
amount.
property
appreciate
. Eee ) (2-32)
= - ;(d~)(6Veen
=
which and
-
+ VeeSn)
+ f ( d ~ ) S n Vee
f(d~)nSVee
is an i m p l i c a t i o n
of Eq. (25). Further,
a consequence
of Eqs. (13)
(14) is
8Eip
= 8~Eip
+ 8 V Eip
= f (d~)nSV
Then,
the c h a n g e
(2-33)
in E is
5E -- S ( d ~ ) n ( ~ V - S V e e ) = f ( d ~ ) n S ( V - V e e ) In v i e w of
[Eq. (I)]
V = " Z + V r ee the v a r i a t i o n
6(V-Vee)
It is u s e f u l part, change
Ees'
(2-34)
,
(2-35)
vanishes,
and Eq. (34) i m p l i e s
to s e p a r a t e
E
ee of Eq. (26), and the r e m a i n d e r
interaction
and p o s s i b l y
Eee = Ees + E'ee
'
Eq. (31), indeed.
into the c l a s s i c a l
electrostatic
E' w h i c h c o n s i s t s of the exee' o t h e r effects. A c c o r d i n g l y , we w r i t e
(2-36)
33
and l i k e w i s e V
ee
= V
+ V' ee
es
The electrostatic
(2-37)
contribution
w i t h the aid of the P o i s s o n field -~V
es
to the e n e r g y
equation
(30) can be r e w r i t t e n ,
(2), in terms
of the e l e c t r o s t a t i c
:
f ( d ~ ) n Ves
+ E
as
=
I - ~ f ( d ~ ) n Ves (2-38)
I f(d~) (V2Ves)Ves = 8--~ [The s u r f a c e
t e r m of the p a r t i a l
for l arge r.
] Fur t h e r ,
V
thereby
es
=
V
+
expressing
V'
r
es
integration
w e c o m b i n e Eqs. (35)
_Z _
V
I f(dr) ÷ (~Ves) 2 = - 8-~ is zero, and
because
V
(37) into
,
ee
N/r
es
(2-39)
in t e r m s of V, as n e e d e d
in
(2). The e n e r g y now
reads
E = Eip - ~
I
Z f (d~) [~(V+ ~ - V e e ) ] 2 (2-40)
- f (d~)n V'ee + E'ee This
expression
models
for the e n e r g y
emerge depending
[Eqs. (7) and
(8)]
is e v a l u a t e d ,
t a k e n into account. lanced treatment
The TF model.
is our b a s i s
u p o n the a c c u r a c y
Of course,
for a p p r o x i m a t i o n s .
to w h i c h the t r a c e
and u p o n the e x t e n t a consistent
Various
in Eip
to w h i c h E' is ee r e q u i r e s a ba-
description
of both.
The s i m p l e s t
model based
u p o n Eq. (40)
is the TF model.
It
neglects
E' e n t i r e l y [then V' also d i s a p p e a r s from (40)] and e v a l u ee ee ' ates the t r a c e of Eq. (7) in the h i g h l y s e m i c l a s s i c a l a p p r o x i m a t i o n of
Eq. (I-43).
The TF e n e r g y
expression
is t h e r e f o r e
I 2- v - e ) ETF = 2 f ( d ~ ) (d~) i~_p2+V÷~)~l_ yp
- ~N
(2~) 3
I
- 8-~ f (d~)[~(V+ We recognize
the last t e r m as the q u a n t i t y
Z ]2 ~) E 2 of E q . ( I - 6 7 ) ,
(2-41)
w h i c h was
34
there introduced to remove the doubly counted
(electrostatic)
inter-
action energy; the term plays the same role here. The p h a s e - s p a c e integral is the TF v e r s i o n of El, p r o p e r l y denoted by however,
(EI)TF. We shall,
suppress the subscript TF until it will become a n e c e s s a r y
d i s t i n c t i o n from other models. The step function cuts off the m o m e n t u m integral at the dependent)
maximal momentum
P = -/~ (V+ ~ )
(the so-called
(r-
"Fermi momentum")
,
(2-42)
so that P (2~) 3
0
(2-43)
rg2 or,
square roots of negative arguments being zero, E I = f(d~) (-
I ) [-2(V+~)] 5/2 15~ 2
(2-44)
This is the T h o m a s - F e r m i result for ~I" The entire e n e r g y functional in the TF model is then
ETF = E I + E 2 - ~N (2-45)
= f(d~)(-
I )[_2(v+~)15/2
a~f(d~)[~(V+
Z ~)12_~
.
15~ 2 Is there any reality to it? Yes. Look back to Chapter One, w h e r e has been used u n c o n s c i o u s l y situation,
E 2 equals zero,
[see Eqs.(1-26)
through
for the Coulomb potential V=-Z/r.
(45)
In this
and ETF gives the leading term of Eq.(I-22)
(I-37)]. Since V is e s s e n t i a l l y equal to the
C o u l o m b potential in a'highly ionized atom, we conclude
ET F
m
-
ZZ(~N) I/3
for
N -~, this d e n s i t y
There is a sharp b o u n d a r y assigned to atoms in the TF model.
In contrast,
in an exact q u a n t u m m e c h a n i c a l d e s c r i p t i o n the t r a n s i t i o n
from the c l a s s i c a l l y allowed to the c l a s s i c a l l y forbidden region is smooth. We have just learned about one of the d e f i c i e n c i e s of the TF model.
It is going to be removed later w h e n we shall i n c o r p o r a t e q u a n t u m
corrections of the sort d i s c u s s e d b r i e f l y after Eq. (I-43). The d i f f e r e n t i a l
equation
(48) for V, known as the TF e q u a t i o n
for V, is s u p p l e m e n t e d by the c o n s t r a i n t
(49) and the short d i s t a n c e be-
36
havior
of V,
r V ÷ - Z
It s i g n i f i e s tial
the
is m a i n l y
tron with ness
the
for
r + o
physical
given
requirement
nucleus;
formally,
is large
ses the p o t e n t i a l
square
root
in
ty is zero, law,
with
V = and the
radius
--
negative, less
region, negative;
Z-N r
for
r o of the
V ( r = r o)
and
= - ~
it e q u a l s beyond
(52),
to
is
and the
Poisson
the
finite-
for s m a l l
large;
distance,
r,
as r i n c r e a at the e d g e
argument ro,
of the
the d e n s i -
equation.
Gauss's
implies
,
=
poten-
of an e l e c -
ensure
finally,
-~,
this
then
r > rO
atom
the e f f e c t i v e energy
situation:
less n e g a t i v e ;
is the h o m o g e n e o u s
Eqs. (49)
~
following
and the d e n s i t y
and
turns (48)
for r÷o, potential
is n e c e s s a r y
the
allowed
(51)
so t h a t
combined
(52)
we have
becomes
of the c l a s s i c a l l y
that
by the e l e c t r o s t a t i c
of E 2. C o n s e q u e n t l y ,
the p o t e n t i a l
(2-52)
(2-53)
is d e t e r m i n e d
by
,
(2-54)
or, Z-N
-
r
The
electric
an atom);
field
-~V
is
in p a r t i c u l a r ,
d
Neutral
at the
r 2 o
o
ly large,
do not h a v e
It is u s e f u l
argument x
N=Z,
just
of w h i c h = Z1/3r/a
have
~=o,
have
learned
to m e a s u r e
charged
surfaces
in
V+~
b o t h V and d V / d r
Eq.(48),
a solution that
requires
satisfying
neutral
TF atoms
only
"inside".
an
as as m u l t i p l e
a function
these are
vanish
ro=~, bounda-
infinite-
of t h e p o t e n t i a l
f(x),
,
is r e l a t e d ,
so that
for V,
an "outside!,,
by introducing
V + ~ = - Z f(x) r the
no
(2-56)
r e it c a n n o t
We h a v e
of the n u c l e u s
are
edge w e h a v e
the TF e q u a t i o n
ry c o n d i t i o n s . they
(there
r o
systems,
at r=r o. C o n s e q u e n t l y , for a f i n i t e
continuous
Z-N
v (r) I r
since
(2-55)
o
(2-57)
to the p h y s i c a l
I~3~%2/3 a = ~,-~,
distance
= 0.8853...
r by (2-58)
37
The
constant
a is c h o s e n
d2
called dary
f (x)
=
f(o)
= I ,
q,
(58).
f(x)
for
f(x),
(52),(54),
introduces
to r ° t h r o u g h
notice
and
f(x o)
that
Consequently,
tion,
q,
so that
for
f(x)
is s o l e l y
all ions
with
Z;
served
the
for
f(x),
through
the The
the
same
in C h a p t e r
shrinking One,
translate
N = I - ~
ionization.
Of course,
now
as
appears
appear
The boun-
- q
,
(2-60)
x ° is r e l a t e d
(2-61)
individually
determined same
Z/r,
x of Eq.
in Eqs. (59)
b y the d e g r e e
q possess
potential
of h e a v i e r
when
factors.
into
d -Xo d--~ f(xo) o
factor
of the TF v a r i a b l e
of n u m e r i c a l
x > x°
Z and N do not
depend
implies
of (53)
and of t h e d e n s i t y . first
(56)
the d e g r e e Equation
potential on
equation
(2-59)
is free
= o ,
= q ( 1 - x / x o)
(60).
Z dependence
differential
,
I/2
t h e TF e q u a t i o n
Please
the
x
2
conditions
which
that
[f (x) ] 3/2
-
dx
such
a common
V itself but
(58).
atoms
then The
also
Bohr
atoms
Fig.1
a sketch
with
of the
of course,
because
factor
that we have
considering
shape
does,
and
of i o n i z a -
of the
Z I/3 t h e r e already
shielding,
obsee
Eq. (I-93). For illustration, for w h i c h (60)
x ° ~ 3. T h e
geometrical
shows
significance
of
f(x)
of the
for q = I/2,
third
equation
in
is i n d i c a t e d .
1
Z ....
±
)X
0
1
I,
2
Xo
4
Fig. 2-I. Sketch of f ( x ) for q = 1 / 2 .
Neutral
T F atoms.
For the
solution
of Eqs. (59)
and
(60)
that
belongs
to
38
q=o,
we write
F(x)
and call
F(x)
=
d2
= I
slope
F(x)
additive
tely write
with
the main
additional
The
the known
=
simultaneous
has no e f f e c t
significance.
We
-
for
body
the change
energy
6ETF
x o
;
= o
only for
An(~)=o
for all
~
;
(2-106)
the TF d e n s i t y functional of Eq. (95) has an absolute m i n i m u m at the correct density,
p r o v i d e d Eq. (I05) holds.
In general, trial densities
satisfying
(105) will mean to c o n s i d e r only such
that obey the constraint
(98),
since then
S(d~)An = o
(2-107)
The m a i n e x c e p t i o n are neutral atoms, sequently, Eq. (I05)
trial values
about w h i c h we k n o w that ~=o. Con-
for ~ need not be chosen,
so that A~=o. Then
is satisfied without r e s t r i c t i n g the density according to
(107).
This o b s e r v a t i o n will prove useful, when seeking lower bounds on the constant B.
Upper bounds on B. We pick up the story at Eq. (69). The c a l c u l a t i o n is c o n s i d e r a b l y simplified by employing the TF variables x, x o, and f(x), which have been introduced in Eqs. (57) through potential functional appears as
E = -
2 (Z7/3/a) {~
[f' (x) E~xf(X)]
co
;dx If(x)]512 I/2
0
(60). In these,
X
co
+
I Sdx[f'(xl÷ 0
J2+
the TF
47
(2-108)
+ q !!-q)_ } Xo
where,
replacing
Whereas
V and
arbitrary
~, it is n o w
variations
f(x)
of x o m a y
and x o t h a t
have
be c o n s i d e r e d ,
to be
f(x)
found. 7
is s u b j e c t
to f(o)
= 1
(2-I 09)
and f' (x)
The
first
of t h e s e
~, in Eq. (57), ensure
the
the trial
= - -~xo
[see(60)].
into
functions
the
of t h e
do not
of
second
have
,
(2-110)
second
the definition
= o
This,
x ÷~
is Eq. (52),
finiteness
f(x o)
for
f(x).
In
integral.
from the (108)
Note,
inclusion
it is n e e d e d in p a r t i c u l a r ,
-Xof' (Xo)
= q
to that
(2-111)
equation
[f(x)]3/2 I/2
=
of
to o b e y
and t h e d i f f e r e n t i a l
f" (x)
comes
(2-112)
X
[see (59)] is h o w
are i m p l i c a t i o n s
it works:
of the
infinitesimal
stationary
variations
of
property f(x)
of
cause
(108).
a change
Here in E,
o = 6[-(a/Z7/3)E]
8fix){EfIX)f312 I/2 -
co
0
co
f" Ix) } +
X
;dx
£ {6f(x)
[f' ( x ) +
1}
0
(2-113) oo
=
;dx 5f(x) { o
where
the
first
uses
(109)
and
differential. clude
that
equality (110)
Thus
beyond
[f (x) ] 3/2 i/2
is the
in f i n d i n g (112)
-
f" (x) }
x
stationary the
is implied.
a certain
property
null value We combine
(yet u n s p e c i f i e d )
and the
of the
~,
it w i t h f(x)
last
integrated (110)
one total
to con-
is n e g a t i v e
and
linear:
f(x)
X--X = q Xo
for
x >_- ~
(2-114)
48
Next,
we
consider
variations
of x o.
They
produce
o = 8[-(a/Z7/3)E]
co
,
(2-115)
o
implying of
the vanishing
(114),
the
of the
integration
contents
stops
o = fdx[f' (x)+ ~q] o x°
+
of the
curly
brackets.
In v i e w
at x:
(l-q) (2-116)
= f(~)
the
last
that
- f(o)
step makes
(114)
+ q~-~ + I -
use of
becomes
(61)
Let us n o w t u r n of the
functional
binding
energy
(109)
and
(108),
and
where
the
together of X o = ~ is a m o n g
equal with
for
to n e u t r a l
trial
f(o)
and o b t a i n the
right
functions
= I
q=o.
The m a x i m u m
property
form of the
neutral
to
that
f(x)=F(x). need
(109)
subject
we can
(~>o)
function.
to use
and
f' (x ÷ ~)
,
trial
for
are
f(x),
÷ f(~x)
,
(2-117)
(110),
Note
that
explicitly
0
[f~x)]5/2 1/2
x
x o disappeared our knowledge
the c o m p e t i t i o n
in
(117)
to
= o.
always
(2-118)
change
the
scale,
,
(2-119)
The
optimal
choice
of
= g
atom
o
w e do not
,
3 B < 2 fdx
7
only
According
side
so
the k n o w n
I fdx[f' (X)] 2
I/2 +3
holds
so that
another hand
x = x o,
reads
x
For any trial
f(x)
see t h a t
co
sign
q,
atoms,
with
[f(x)]5/2
o
q=o.
N o w we
(111).
co
2 fdx
7B--o.
We are now ready to invent trial functions and produce upper bounds on B. Before doing so, let us make a little observation. equals F, the optimal ~ in holds only for f(x)=F(x). in
(121)
is
(3/7)B. We c o n c l u d e
(120)
Consequently,
are equal for f=F.
2 Sd x o
is unity,
(117)
the n u m e r a t o r and d e n o m i n a t o r
In this s i t u a t i o n the related sum in
[F(x)]5/2 _ 2 B x I/2 -
If f
since the equal sign in
,
(117)
(2-123)
and co
I fdx[F' (x)]2 = "~ BI
(2-124)
o
An i n d e p e n d e n t
(and rather clumsy)
the d i f f e r e n t i a l
partial integrations.
Equations
(123)
i m m e d i a t e check of the e q u a l i t y in
7 B = ~
d e r i v a t i o n of these equations uses
e q u a t i o n obeyed by F(x)
(~-
I/2 2 2/3 7 B)
More about relations
like
on the scaling p r o p e r t i e s
(123)
[Eq.(62)],
and
can be employed
for an
(122) for f(x)=F(~x) :
2 1/3 (~ 7 B) and
(124)
combined w i t h some
(2-125)
(124) will be said in the section
of the TF model.
A v e r y simple trial f u n c t i o n is
f(x)
I
= i;x
"
(2-126)
50
for w h i c h oo
2
co
of
dx
[the i n t e g r a l ,
[f(x)]5/2 xi/2
in t e r m s
= 2 fdx x - I / 2 ( 1 + x ) - 5 / 2 5 o
of E u l e r ' s
co
Beta
function,
15 :-8
(2-127)
I is B(~,2}
4 = ~], j-
and
co
;dx[f' (x)] 2 = f o o
dx (1+x) 4
=
I 3
--
(2-128)
Accordingly,
B < ~
A better
value
5 -2/3
= 1.596
is o b t a i n e d
(2-129)
for
r
f (x) =
(~]__)4/3
,
(2-130)
I ~ X
when B < 2-19/9
This
number
is no p o i n t
Lower
bounds
in t e r m s
is,
(--737--)I/3
as w e
shall
in c o n s i d e r i n g
on B.
=
see, more
In o r d e r
of TF v a r i a b l e s ,
n(~)
-
[~ (21 :]2/3 [(1) :]-4/3 close
to e x p r e s s
to the trial
the d e n s i t y
1.5909
actual
(2-131)
one;
so t h e r e
functions.
functional
of Eq. (95)
we w r i t e
I
V2 (V+ ~) Z
I
I d2
~-~
very
complicated
=
Z [r(V+ ~)]
4~ r dr 2 I
I
d2 - -
(2-132)
(zf(x))
4~ r dr 2
=
or,
more
I Z 2 f" (x) 4n a 3 x
conveniently
here,
'
51
I
n
The
function
constant.
g(x),
constant
= f' (x) + -qxo
in v i e w of
1110)
g(x ÷~)
With
-/(d~)
and
to r e q u i r i n g
(135) we h a v e
for the i n t e r a c t i o n
Z7/3 ~ Z7/3 a Idx g' (x) = a g(o) o
~r n
equ a l s
(108), gives
(2-134)
e n e r g y bet-
and the e l e c t r o n s
energy
n(~)n(~') ÷ ÷,
Ir-r I
ty f u n c t i o n a l
by a
(2-135)
I f(d~) (d~')
of
from f' (x), at most,
,
is e q u i v a l e n t
the e l e c t r o n - e l e c t r o n
[This q u a n t i t y
differs
to be q / x o,
= o
(133)
w e e n the n u c l e u s
whereas
(2-133)
thus i n t r o d u c e d ,
We c h o o s e t h i s
g(x)
which
Z 2 g' (x) a
-E2,
this result.]
(2-136)
is
_ z 7/3 I fdx[gix)] 2 a 2
(2-137)
o
so t h a t Eq. (134),
can be e x p r e s s e d
,
The remaining in t e r m s
used
in the s e c o n d
contributions
of g(x)
integral
to the d e n s i -
immediately.
We a r r i v e
at
E -
Z7/3{ 3 ;dx x I/3 5/3 a 5 [g' (x)] + g(o) o
+
I ;dx[g(x)] 2 o (2-138)
co
- --q-It-q- fdx x g' (x)]} xo o
Again
arbitrary
variations
ted by the r e q u i r e m e n t
g' (x) > o
of x o m a y be c o n s i d e r e d ,
of n o n - n e g a t i v e
,
and b y Eq. (135). T o g e t h e r ,
whereas
g is r e s t r i c -
densities,
(2-139) they imply
52
g(x)< o
(2-140)
A c c o r d i n g to the discussion upper bounds on the energy,
of Eq.(103),
Eq.(138)
provided that Eq. (I05)
supplies
is obeyed.
Expressed
in terms of x o and g, it reads A(~) "'O
fdx x ~g' (x) = o o
We did notice already
[see the remark following Eq.(107)],
situation of neutral atoms, so that
(141)
(2-141)
our knowledge
is satisfied without
of Xo=~ results
further ado.
that in the in ~ ( ~ ) = o ,
In particular,
g~x)
need not be subject to
; dx x ~g' (x) = o o
or
,
(2-142)
[this is Eq. (98)], more precisely,
oo
fdx x g' (x) = 1-q ; = I
for
q=o
(2-143)
o
The m i n i m u m property of the functional the known form of the neutral
atom
(138), together with
(q=o) binding energy,
Eq.(67),
im-
plies 3 B ~ - {~ 3
;dx x I/3 [g' (X)] 5/3 + g(o) + ~I fdx[g(x)] o
2}
,
(2-144)
o
where the equal sign holds only for g(x) = F' (x). For this g(x), the 3 I value of the two integrals is ~B and ~B, respectively, as follows from Eqs. (123) and
(124), and the differential
equation
(62) obeyed by F(x).
Consequently, F' (o) = -B
,
(2-145)
which is nothing more than the original definition As in the p r e c e d i n g scale.
Here the possible
g(X) ÷ ~ig(~2 x)
section,
scalings
of B in
(64).
we can consider changes of the
are even more general,
, (~i,~2 > o)
,
(2-146)
53 because there is no analog to the restriction watch before.
f(o)=1,
that we had to
The optimal choices for ~I and ~2 maximize the right hand
side of oo
3 {3 ;dx x I/3 d 5/3 + #Ig(o) 7B > - 5 [~Id--xg (#2 x) ] o
+ I ;dx[#ig(#2x) ] 2 } 0
oo
5/3 I/3 3 [dx x ~/3 5/3 t~1 ~2 ~ [g' (x)] + ~ig(o) o
2 ~I I ;dx[g(x)] 2} co
+
- -
~2~o (2-147) They are co
(1) 413 ~1
=
[-g [o) ] 4/3
I ;dx[g(x) ] 2)-I/3
oo (y ~-fdx I / 3 [g, ix) ]5/3 o
2-148)
o
and co
~2
(1)4/3 =
[-g(o) ] 4/3
I ;dx[g(x) ] 2)2/3 I~
~
1;d x xl/3[g, (x)15/3
2-149)
o
0
Inserted into
(147) they produce the scale invariant version of
B > (1) 4/3 . [-g(o)]7/3 = co eo (l/dx xl/3[g ' (x)]5/3)(l[dx[g(x )]2)I/3 o o
144):
(2-150_)
,
where the equal sign holds only for g(x)=~iF' (~2x) with arbitrary ~i,~2>o.
Indeed,
for such a g(x), we get
B = (1) 4/3
[~I B]7/3 ..... .513 113 1B)(~I2 (IJ,~I
I~ 2
,u,2
1B)113
(2-151)
The main contribution to the energy of an atom comes from the vicinity of the nucleus. Now, Eqs. (62) and (63) imply,
54
F" (x)
Consequently,
-'- ~ x
for
a good
trial
1
An example
It t u r n s
(2-152)
x ÷ o
g(x)
for
has
to be
such
that
x ÷ o
(2-153)
is
g(x)
=-
out,
that
that w e m a y
(1+/x) -~
the
right
immediately
helps
in this
limit,
stead
of g(x)
for ~+~.
,
~>o
hand
side
consider
since
the
it a l l o w s
The l i m i t i n g
(2-154)
of
(150)
limit
increases
~÷~.
The
to e v a l u a t e trial
with
scaling
~,
so
invariance
g ( x / ~ 2) for ~÷~,
function
is a s i m p l e
in-
exponen-
tial:
g(x)
For this
= lim[-(1+ [Z÷oo
g(x),
B
>
we h a v e
I -a ] -K/~)
in
-
e
-/x
(2-155)
(I 50)
[-(-i)] 7/3 25 (.~) 4/3 --y (~5 2"2/3) (1) I/3 =
(1) 4/3
(2-156)
= 1.5682
Binding in
(131)
energy and
of n e u t r a l
a lower
1.5682
one
TF atoms. in
(156).
< B < 1.5909
We have N o w we
found
combine
an u p p e r the
two
bound
on B
and s t a t e
,
(2-157)
or B = 1.580
The margin know cal
B with effort
in
(158)
is about
a precision was
needed
physical
picture
entirely
sufficient.
(2-158)
-+ 0.012
that
of
1.5%
of the
average
0.75%.
Please
appreciate
in o b t a i n i n g
this
we
using,
are s t i l l
A higher
precision
result.
value,
little
In v i e w
of the
the v a l u e
is not
so that
how
for B in
called
we numericrude
(158)
for at t h i s
is stage
55
of the development. I n s e r t e d into
(67), this B v a l u e produces
-ETF = 0.765 Z 7/3 w h i c h is the TF p r e d i c t i o n
8
I
,
(2-159)
for the total b i n d i n g energy of neutral
I
I
I
1
TF 6 I-IF
/
.,.+
÷ ÷
OL
0
i
J
j
J
25
50
75
100
_I
125
Z Fig. 2-2. Compar~on of the TF prediction (160) with HF binding £n~gi~ (crosses).
56
atoms.
In Fig.2
-ETF 12
the q u a n t i t y
= 1.53
Z I/3
(2-160)
yz
is com p a r e d of Z. This atomic
to the c o r r e s p o n d i n g plot
binding
shows
is n e v e r t h e l e s s crudeness
Z values
viation and
than
fractional is 29,
120,
(160)
remarkable
TF curve
21,
duction. Z~20
energies?
they were
available
be m e a s u r e d particle misled
against
Hamilton
This
that
17,
is,
15,
and
13 percent with
energies
operator
by r e l a t i v i s t i c
for Z=I0,
of
based
upon,
are the more
de-
60,
90,
experimen-
in the Introonly up to
from the HF crosses).
of Z, the TF result
which
30,
and not w i t h
(I-7) ; this w a y we are
effects,
since it is relative
20,
are k n o w n e x p e r i m e n t a l l y
predictions
at small
Z.
are the ones m e n t i o n e d
for large values different
of this
it
the
In Fig.2
HF crosses
increasing
indistinguishable
trend of the
despite
a deception
The amount
values
is clear,
it represents.
w i t h HF predictions,
The reasons
are
that
however,
counts.
It d e c r e a s e s
Total b i n d i n g
(in Fig. 2 they
the TF model works
approximation
W h y do we compare tal b i n d i n g
the general
for r e f i n e m e n t s
is closer to the integer-Z
difference 24,
reproduce
the need
how well
at large ones.
respectively.
does
Altough
of the p h y s i c a l
the continuous
the
that
energies.
HF p r e d i c t i o n s 8 for integer
Even
should
e.g.,
if
still
the many-
sure to not be
significant
the
lar-
ger Z.
TF function
F(x).
the TF function. shall
We have
do so in this
(63). F(x)
d2 dx 2 F(x)
and the b o u n d a r y
F(o)
Upon u s i n g appears
as
= I
there
a lot about is m u c h more
the initial
slope B of
to say about
F(x).
We
section.
Let us p r o c e e d (62) and
learned
Naturally,
from r e c a l l i n g
obeys
the d e f i n i n g
the d i f f e r e n t i a l
= F" (x)
properties
of Eqs.
equation
[F(x) ] 3/2 I/2 x
(2-161)
conditions
,
F(~)
= o
/x as the m a i n variable,
(2-162)
the d i f f e r e n t i a l
equation
(161)
57
d
F(x)
= 2/x F' ix)
,
(2-163)
d F' (x) : 2[F(x)] 3"2 / d6x Whereas
(161)
is w e l l
behaved
panded
is s i n g u l a r at /x=o
in p o w e r s
F (x) =
F(x)
, F(x)
equations
can be ex-
,
(2-164)
as B a k e r ' s
s e r i e s 9 The c o m p a r i s o n
with
,
We gain
sI = o
,
s 2 = -B
calculation
(2-166)
of the Sk'S
it by i n s e r t i n g
left h a n d
F"(x)
(2-165)
shows
For the s u c c e s s i v e
The
s y s t e m of d i f f e r e n t i a l
t h a t a r o u n d x=o
= I - Bx + 0(x 3/2)
sO = I
(161).
s k x k/2
known
[this is Eq. (64)]
relation.
, this
of /x :
~ k=o
w h i c h has b e c o m e
at x=o
. We c o n c l u d e
(164)
for k>2, we need a r e c u r r e n c e
into the d i f f e r e n t i a l
equation
side is simple:
-
k k = ~ sk ~ ( ~ k=o
I
)x k-2-2/
(2-167) = ~3 s3 x -I/2
where
s1=o
has b e e n used,
right
hand
side of Eq. (161)
ries b e c o m e s
+
x (/-I)/2
a n d the s u m m a t i o n is n o n l i n e a r
more complicated.
[F (x) ] 3 / 2 / x 1 / 2
~ Z=I
(£+I) (£+3)s/+3 4
index s h i f t e d
in F(x),
(k=/+3).
The
so that the p o w e r
se-
We h a v e
= x-I/211
+ ~ k=2
s k xk/2] 3/2
(2-168) = x-I/211 + ~(
3/2
~
j ) (7k=2 sk
xk/2) j
]
r
58
where
the b i n o m i a l
theorem
is employed.
of the sum over k e x p l i c i t k I , k 2 , ....
81,k
kj
=
is used to c o l l e c t
by w r i t i n g
Next,
1
for
k = 1
for
k ~ I
all terms
[F(x)]3/2/x1/2
the
it as the p r o d u c t
; then the K r o n e c k e r
o
we make
Delta
symbol
j-th p o w e r
of j sums over
,
(2-169)
' x -I/2 x 1/2
of o r d e r
x (/-I)/2
=
= x-I1211+~(3o)
(2-207)
The TF model itself is not invariant under such a scaling, b o u n d a r y c o n d i t i o n f(o)=1
fixes the scale.
because the
Therefore, we have to be
somewhat more careful w h e n i n v e s t i g a t i n g the scaling p r o p e r t i e s of the TF model. W h e n we were looking for bounds on B, we found it a d v a n t a g e o u s to exploit certain scaling properties of the r e s p e c t i v e functionals.
The
scaling t r a n s f o r m a t i o n s that we c o n s i d e r e d then, were, Eq. (119):
f(x) ÷
f(~x)
(2-208)
and, Eq. (146) w i t h g(x)=f' (x)+q/Xo=f' (x) for q=o:
f(x) ÷ ~q f(~2 x)
,
w h e r e ~, ~I' and ~2 w e r e a r b i t r a r y
(2-209)
(positive)
numbers.
mine the implications of t r a n s f o r m a t i o n s as general as to the TF p o t e n t i a l functional. We return to Eq. (45),
Let us now exa(209) a p p l i e d
67
ETF = ] (d~) ( -
I
) [_2(V+~)15/2
15~ 2 (2-210)
=
E I
+
E 2
-
{N
and consider V(r) ÷ v
Since,
V(~r)
for the existence
rV(r)
,
(~>o)
(2-211)
of E2, we need
÷ - Z
for
such a scaling t r a n s f o r m a t i o n
[Eq. (52)]
r ÷ o
(2-212)
,
of V has to be accompanied by a scaling
of Z, Z ÷ ~-I
For convenience, ÷ v
Z
(2-213)
we also scale ~ by E
,
(2-214)
so that the structure V+E is conserved. In terms of f(x), f(x) + v - 1
which identifies whereas
f(~x)
(207) and
(211) and
(214), without
(213), read
,
(2-215)
(208)
as the special
situations
(209) is realized by ~2=~ and ~i=~ ~. However,
just have
(208), as we should have,
ring f(o)=1;
and only
Under
(211),
since
with
v=4 and v=1, (213), we
(212) is equivalent
to requi-
(208) is consistent with this constraint. (213), and
(214)
the various
contributions
to ETF
scale according to E 1 ÷ E I (~) = f(dr) (-
I ) [_2 V(V(~r)+~)]5/2 15~ 2 (2-216)
= 55v/2-3
El
•
and
E2 ÷ E2(I~)
= _ 8~f(d~ ) [~(llVV(gr)
IIVZ.
+ _~)]2
=
68
(2-217) =
29-I E2
,
as w e l l as ~N +
V
~N
(2-218)
Consequently
ET F ÷ ETF(~)
For ~=I, this the e n e r g y first o r d e r
= ~5~/2-3
is just ETF;
is s t a t i o n a r y
El + 2 v - I
for ~=I+5~,
under
~ETF ~
=
O n the o t h e r hand,
6
from
( v-1
(2-219)
w e have ETF+6 ETF . Now,
infinitesimal
changes must originate
5 ETF
E2 _ ~V~N
variations
in the s c a l i n g of Z. T h a t
is
~ETF = (~-I)Z--~--5~
Z)
since
of V and ~, all
(2-220)
(219) w e get
5 ETF = [(5v-3)E I +
(2~-I)E 2 - ~ N ] 6 ~
,
(2-221)
so t h a t w e c o n c l u d e
(-~-3)E I ÷ (2~-I)~ 2 - ~
= (v-1)z~z~T~
T h i s is a l i n e a r e q u a t i o n
in v. It has to h o l d
two i n d e p e n d e n t
among the different
relations
"virial theorems."
Besides
(215)],
choice
for any v. So we o b t a i n energy quantities
- two
v=1, w h e n
I - ~ E I + E 2 - ~N = o the o t h e r n a t u r a l
(2-222)
,
(2-223)
is t h e TF s c a l i n g
v=4
[see the c o m m e n t
to Eq.
for w h i c h
7E I + 7E 2 - 4~N = 3 Z ~ z E T F
The latter combines with ETF=EI+E2-~N
8-~ ETF = - ~
(2-224)
and
(2-225)
69
to give (2-226)
~ ETF (z ,N) 7ETF (Z ,N) = 3 (Z-~+Ny~)
We have made explicit here,
that the energy of an atom is a function
of Z and N. For N=Z,
Eq. (226) has the simple implication
7ETF(Z,Z)
= 3Z~zETF(Z,Z)
,
(2-227)
or
ETF(Z,Z)
= - C Z 7/3
,
(2-228)
where the constant C is yet undetermined. knowledge of
C=- ~ETF/~N=o
for
(dr) ( V + )
N=Z
It is found by combining our
with
V 2Z r
~
(2-229) :
-
z(v+
)1
;:
-
B Z7/3
for
N = Z
r=o
The third step here is a partial zes -ZS(r) valid that
the fourth one recogni-
as the source of the Coulomb potential
for N=Z only, uses Eq. (65). (229)
integration;
identifies
Z/r; the last one,
[The comment to that equation says
the interaction
energy between the nuclear charge
and the electrons:
ENe = Z~ZETF which,
according to
(2-230)
,
(I-96),
TF model in its validity.]
ETF(Z,Z) -- 71Z
÷N
is a general
statement,
)~T~(Z,m[
: ~Z~ET;(Z,N) [ N=Z
3 B Z7/3
=-TK which
identifies
not limited to the
Now,
the constant C. This is, of course,
N=Z
(2-231)
the result that
70
we had
found
earlier
The
first
in Eq. (67).
of our
"virial"
theorems,
Eq. (223),
has
the
conse-
quence E I = 2(E2-%N)
= 2(ETF , E I
,
(2-232)
orr 2
E I = ~ ETF
,
(2-233)
and I
E 2 - ~N = ~ E T F
For
a Coulomb
usual
system,
theorems
they
about
the
essential + E
to
emerge
(2-234)
like
Eki n = - ETF Indeed,
.
the
one
kinetic
,
from
we
and
are the
considering, potential
one
expects
energy:
Epo t = 2ETF the
relations
the
(2-235) that
r e m e m b e r how E 1 and E 2 a r e
we
have
found
so
far.
It
is
c o m p o s e d o f E k i n and E p o t = ENe
:
ee
E I = Eki n + ENe
+ 2Eee
+ ~N
, (2-236)
E 2 = _ Eee Note
in p a r t i c u l a r
in E I. W i t h
(230)
the
double
counting
of t h e
electron-electron
energy
we have
Epo t = ENe
+ Eee
= Z~TETF~_ - E 2 (2-237)
which
makes
produce
the
use
of
second
(225).
Now
statement
Eqs. (226) of
(235),
and
(234)
which
then
can
be
implies
employed the
to
first
one
immediately. The
relative
E
: Eki n
ee
sizes
: (-ENe)
are
:
(- ~--ETF-~N): (-ETF): (- 7 E T F + ~ N )
= I : 3 : 7
for
N = Z, w h e n
; ~ = o
.
(2-238)
71
In words:
the electron-electron
of the kinetic
energy of a neutral TF atom is one third
energy and one seventh of the
(negative of the)
nucleus-
electron energy. For ions, there is less specific merely
implies that ETF(Z,N)
introducing
tion of q=I-N/Z,
e(o)
It
in the form
(2-239)
a reduced binding energy,
e(q),
that is a func-
the degree of ionization: Z7/3 = - ~ e(q)
ETF(Z,N) We know e(q)
can be written
in Eq. (226).
N = Z 7/3 x [function of ~]
ETF(Z,N) This invites
information
for q=o, 3 = 7B
(2-240)
i.e., N=Z
:
,
(2-241)
which is simply Eq. (231). The factor m u l t i p l y i n g same as the one in Eqs. (108) and of the TF potential
(density)
e(q)
(138). The m a x i m u m
functional
in
(240)
(minimum)
is, therefore,
is the property
here expressed
as 2 fax {5 o
[f(x)]5/2 x I/2
I fdx[f' (x)+ q ] 2 + ~ o Xo
+ q(1-q)} x° (2-242)
e(q) oo _
{! fdx x I/3
5
'
[g ( x ) ]
5/3
+ g(o)
o
The competing
g(x)'s are hereby restricted
by
+ yI fdx[g(x) ] 2 } . o
[Eq. (98)]
co
fdx xg' (x) = 1-q
,
(2-243)
o
whereas f(x=o)
= 1
The equal signs in Eqs. (59) and
(2-244) (242) hold only for g(x)=f' (x)+q/x o, when f(x) obeys
(60), w h i c h also determine x o.
We can relate e(q)
to Xo(q)
by recognizing
that Eq. (225) says
72
Z 7/3 ~ e(q))
~--~(-
-
Z 4/3 d a dq e(q)
_--
_
Z4/3 ~
--
a
(2-245) q x o (q)
thus, d e (a) = dq -
q x O (q)
(2-246)
Consequently, q q, ,
fdq' Xo~q')
e (q) = 73 B
- o
(2-247)
and e(q)
of w h i c h
q' xo(q')
whereas
q~1).
e(q=1)
has been used;
to w e a k l y
one is d e s i g n e d
the o b v i o u s
ionized
for h i g h l y
systems ionized
(2-249)
it says:
7Z
ETF
=
no e l e c t r o n s
Eq. (229)
-
- no b i n d i n g
energy.
gives
Z Z(V+~+ ~)I + ~Z r=o (2-250)
Z7/3 -
so
that Eq. (226)
a
[ f " (o)
q
translates
+
-~-q]
xo
into
q2 7e(q) By w r i t i n g
atoms
statement
= o
For ions,
Z
be a p p l i e d
the second
In Eq. (248)
(2-248)
,
the first one should
(N~Z, q~o), (N dk I "''dk j 61,k1+. • .+k j j=2 k I =I kj =I (2-354)
This we recognize
to be the recursion for the c k of Eq. (196), after y
and o are interchanged.
Consequently,
appear in the expansion of Eq. (310).
{(t) =
144/x°t ~ [ I+
the d~s are the coefficients
that
That is
dk(at°)k]
(2-355)
k=1
This connection between the fz(x)'s and ¢(t), which is, of course, not accidental, is the clue to computing el, e2, ... of Eq. (338). We reveal its significance by inserting Eqs. (349) and (352) into the ansatz (316),
fq(X)
= 14411+ ~ ( - [ f q ( 0 ) + B ] - X 3
k=1
3 (12) 4
[x° (q) ] O k x ~ ) ( _ _ ~ ,)kOdk ] +
Xolq,
90 + ....
(2-356)
where the ellipsis indicates the terms containing "powers of y(x)." After introducing h by q I = B + 7(12) 4~8y (y+G) [hq/X O(q)]°
-fq(O)
(2-357)
Eq. (355) is employed: co
fq(X)
= 1443 [ I+ Z dk(a[h q ~ ]X x k=1
O)k]
+...
(2-358)
A 2h3 #(hq ~ ) = q [i-~] q where
+ ...
[l(q)]2=q[Xo(q)] 3 is used. What is exhibited in (358) is the part
of f (x) that goes with the zeroth power of y(x). Likewise, q ry power of y(x), say [y(x)] m contributes [y(x)]m 144 [ cm + (powers of ~(hqX/Xo(q))°)
x3
an arbitra-
] (2-359)
~ a[~@q)]2 hq
Cm[Y(X)]m ~m(hq x--~) x
to fq(X). The functions ~m(t) thus defined are such that 144/A 2 ~m(t) -
[ I + (powers of ~t°)]
(2-360)
t3
We can now make explicit what supplements Eq. (358):
x
fq(X) = q[l--~q)] 2 h~[~(hq x--~-~)+
j
Cm[Y(x)]m~m(hq x ~ q ) ) ]
m=1 (2-361). Whereas
(316) is an expansion that is expected to converge rapidly for
x not too close to Xo(q), the series of (361) is the faster convergent the smaller y(x) is. This identifies
large values of x (i.e., x~x o) as
91 the domain of application. It is instructive
to make contact with the original defini-
tion of ~(t), _ qI f q (tXo(q)) ] #(t) : ~l(q) (t) I q+o q÷o
(2-362)
[see the comment to Eq. (305)]. Since = o y(tx o(q) ) J : ~(tx o(q))-Y[ q÷o q+o the combination of Eqs. (361) and
,
(2-363)
(362) reads
(t) = h 3 ¢ (hqt) ] q q÷o
(2-364)
from which we learn that hq
]
= I
(2-365)
q÷o
This tells us what e I is. Equations el + 2e 2 qy/3 +...
=
(342) and
-[fq(O)+B]
ly
(357) together say A2/3
q-a~3 (2-366)
7(12)4
B(y+a) A2/3
:
hq [ql/3Xo (q)
or, after making use of ql/3xo(q)=[l(q)]2/3 ,
el + 2e 2 qy/3
7 (~) 4
+...
= ~
~B(¥+o)
A-2Y/3(hq
A
[77~71
2/3
) (2-367)
Now the limit q÷o identifies 7 eI = ~
With ~, B, and
(_~)4 .e,13(y+a) A-2y/3
A from
(313),
rical value of e I is roughly
(204), and
(2-368)
(312), respectively,
10% larger than the estimate of
the nume(343):
92 e I = 0.825908
(2-369)
Note that Eq. (368) reveals the physical of B and
significance
A has been clear since Eqs. (67) and The requirement
yo(q)
fq(X=Xo(q))=o
of ~ and 6; that
(301).
relates hq to yo(q),
given by
-- Y(Xo(q) ) = 6[Xo(q)]-Y
(2-370)
= BA-2y/3 q y / 3 [ ~ ] 2 y / 3
inasmuch as x = Xo(q)
{(hq)
in Eq. (361) yields
+ f Cm[Yo(q)] m ~m(hq) m=1
With the aid of ~' (I)= -I in yo(q),
or, qy/3,
(2-371)
= o.
[which is Eq. (307)], we find to first order
respectively:
h q = I + c I Yo (q) ~I (I) + 0(Yo (q))2) (2-372) = I - ~ A -2Y/3 ~I (1)qy/3 + 0(q2y/3) where ci= -I has been used.
In conjunction
with Eq. (340) this has the
consequence (hq[
a I _BA-2¥/3 ]2/3)o = (i+(7e
~I (I) )qy/3 +. . .)o (2-373)
o
= 1 + o(.Te 1
_6A-2y/3
¢1(1))_
ay/3
+
.
.
.
.
so that the order qy/3 in Eq. (367) is o _BA-2y/3 2e 2 = oe1( 7 e I
To proceed
~i(I))
(2-374)
further, we need to know ~i(I). The insertion of Eq. (361) into the differential
eauation
obeyed by fq(X) produces d2 [-dt 2
3 t-y 2 A /~-~/~] ~1 (t) = o
,
(2-375)
93 when terms linear in y(x) are identified. This is quite analogous to Eq. (319) , so that 9o(t ) - tY[9(t)+ ~It
(2-376)
%'(t)]
is one solution of (375), the one corresponding to fo(X) of (320). The Wronskian of 91 (t) with 9o(t) is ,
I ,12,
9o(t)~1'(t) - ~o(t)91 (t) = ~-~-;4~o(o+X)
t2
,
Y
(2-377)
which makes use of Eq. (360)~ and the small t form of 9o(t), 9o (t) = t ¥ I d 3q 2 d--{(t3%(t))
= 13t-(2-Y) (Y) 2~t[1-c~t°+d2(°~t°)2+''']
(2-378)
I ~_2) 2~ o to+Y-3 [I-2d 2 ~t°+..] = - 7(
Equation
(377) now implies I ,12, I t '¥ ~1(t) = 9°(t) [91(I)/~°(I) - ~l-A-;4 cx°(°+Y)fdt' (t ~ )
2] ' (2-379)
where 91(I) is determined by the t÷o form of 91 (t), statet in Eq. (360). In connection with (378) this requires that the square brackets in (379) possess the form - 3t-(O+Y)
~o
[I+
(powers
o f 0~t°)]
1
1 (y)
91 (1)/90(I) - 3(°+Y)fdt't [- ~
2
c~o
t,Y ~ ]
(2-380)
I
~I (I)/9o(I) - ~ ( o + y ) f d t ' t
2
t -2(O-3) (1+4d2~t°)
~O (G+¥) fdt'{[I ~I ( ~ ) 2 ~ O ~ t1'Y2 _ t
t-2 (O-3) (I+4d20~ t°) }.
94
In the latter t=o,
version,
since the
the
integrand
second
integrates I Idt' t
is no longer
singular
at
has the s t r u c t u r e
{ ... } = a2t ' s [const. which
integral
+ (powers
(2-381)
of o t ° )]
to I = fdt' { ...} o
{ ...}
+ ~ztT[const.+(powers
of at°)]
I
The first
= ;dt' o
{...}
integral
in
+ t -(m+x) (~tm) 2 [ c o n s t . + ( p o w e r s
of at°)] (2-382)
(380)
_ ~(o+¥)[
is
1-t7-2° 7-20
-t 7-° 17-o ]
+ 4d20
(2-383) _ ~(i+~d2
The c o n s e q u e n c e
91
of
o+x a) -
-V(380)
+
(I)/~o(I)
3
a-6
t-(o+x) (i+4d2 o+x at 0)
y
is t h e r e f o r e 3
o+¥
--~--o,)
~-~(I+4d 2
(2-384) I
1,12,
tY
1+4d2at°
~o ~
t °+Y+l
- --~-3(O+¥) ] d t { [- ~-A-; 20° ~lq--T~T] 2 ~0
With
o
the aid of 9o(I)
= -I/3
9 d 2 = 12+4o2
this
and,
9/2 = o(2o+y)
from
(310)
3 X = 4 2o+y
or
} =
0
(354),
,
(2-385)
says
1 (1+3 ~ a ) 91(1) = ~6 (2-386) 1 l+3~t + o+x [dt { ~o o t ~+¥-I
This
expresslon
does
not
lend itself
°
"1"12"2~0 ~ - ~t -Y~ ] z } -[3~x-~;
to further
algebraic
simplifica-
tions. The n u m e r i c a l
value
of 91(I),
obtained
by a m e t h o d
analo-
95
gous to the one that produced
#1(I)
and or in Eqs. (312) and
= 0.3216868353717
w h i c h illustrates Wronskian
A
,
(2-387)
once more the high precision of the algorithm.
(377), at t=1,
~1'(I)
is employed
= (4+¥)~i (I) -
the d i f f e r e n t i a l
(~)4 ~o(o+~)
~1"(1)
equation
= 2Y41'11 ) -
The
in finding
= 0.2869164052321
whereas
(313), is
(2-388)
,
(375)
supplies
yly+1)41(1 ) (2-389)
= 0.002936027410 The a l g e b r a i c
statements
41(1)
where,
because
0.15% o f t h e
= 21
of the
sum.
of
(388)
)4c~(o+y)
smallness
In conjunction
and
(389)
can be c o m b i n e d i n t o
+ g ~1"(1)
,
o f ~1"(1),
the
with
(2-390)
latter
Eq. ( 3 6 8 ) ,
this
[SA- 2 ~ / 3 41 (1) = 76 el + g1 6 A-2,~/3 ~1"(1) w h i c h we insert into
(374)
_ 0+6
e2
here,
14
resulting
0+6 - ~ - e12
about
implies ,
;
(2-391)
(2-392)
a 0.5% correction
= 0.671015
,
to
(2-393)
in e 2 = 0.667554
Naturally,
only
~1 ''(I)
.12,~ ] ~-6 2 [-Ej ~(o+y)
the ~i"(I) term represents
e2
is
to find I
e1211
part
the subsequent
(2-394) coefficients
in
(338)
can be computed the
96
same way. The results of this section are summarized -ETF(Z,N)
= 0.768745
in
Z 7/3
-0. 047310 (Z-N) 7/3 [I+0. 825908 (I-N/Z) 0. 257334 +0. 667554 (I-N/Z) 0.514668 + ... ]
which is the w e a k - i o n i z a t i o n Eq. (289) [and its supplement
result of
7].
One last remark is in order. (361)
(2-395)
analog to the h i g h - i o n i z a t i o n of Problem
making explicit use of the requirement to
,
How could we get around without -Xo(q)f~(Xo(q))
= q ? As applied
it reads
OO
hq~' (hq) + >
Cm[Yo(q)]m[hq~m(hq)
- Y m ~m(hq)]
m=l
(2-396) = - [ I(_~A) ]2 h-3 q
Indeed,
this together with
which can be converted
(371)
into %(q)
gives
%(q)
as a function of yo(q),
as a function of q, w h e r e a f t e r
Eq. (340)
identifies
el, e2, etc. Fortunately,
ween f~(o)
and hq in Eq. (356), so that we could avoid the more tedious
(though,
of course,
Arbitrarily analytic
equivalent)
we came to know the relation bet-
procedure based upon Eq.(396).
ionized TF atoms. We have spent some time on studying the
form of such quantities
like e(q), Xo(q),
and -f~(o)
as func-
tions of q - both for q~1 and for q~o, which are the situations ghly and weakly
ionized atoms,
respectively.
ever, did not tell us how good are few-terms approximations (289) and
(395). Let us, therefore,
sults of numerical
integrations
fq(X)
values of q.
for various
We present the nineteen q values
of hi-
These considerations,
make the comparison with the re-
of the differential
equation obeyed by
in Table 2 the outcome of such calculations 0.95,0.90, .... 0.05
for q=1 and q=o. The fractional
how-
as in Eqs.
, supplemented
binding energy
for
by what we know
97
e(q)/e(o)
= ETF(Z,N)/ETF(Z,Z)
is a d d i t i o n a l l y plotted,
(2-397)
as a f u n c t i o n of q, in Fig.4. We o b s e r v e that
1.0
O.8
0.6 0
S
0.&
0.2
00
I
I
0.2
I
0./-,
0.6
I
O.B
1.0
q Fig.2-4.
The f r a c t i o n a l
binding
r e m o v i n g 30% of the e l e c t r o n s
e n e r g y e(q)/e(o) as a f u n c t i o n
from the neutral atom,
of q.
reduces the b i n d i n g
energy o n l y by I%; a r e d u c t i o n by 10% requires the removal of 65% of the electrons.
Even when only 5% of the electrons are left, the binding ener-
gy is still m o r e than 50% of the n e u t r a l - a t o m one. Here is the quantitative v e r s i o n of the q u a l i t a t i v e remark that the innermost electrons c o n t r i b u t e most to the b i n d i n g energy, From Eq. (286),
e(q)/e(o) 1.489(N/Z)1/3
(241)
the o u t e r m o s t least.
and P r o b l e m 7 we find that,
N N 2 = I - 0.4236 ~ + 0.0909(~) + ...
The s u c c e s s i v e a p p r o x i m a t i o n s that this r e p r e s e n t s
for N
(A more careful d i s c u s s i o n hereof will be presented The effective stage,
potential V(~)
(2-424) in Chapter Four.)
that appears here is unspecified
except for remarking that it is a functional
at this
of the density,
÷
n(r),
because the density matrix on the left-hand
tional.
The factor of two is the spin m u l t i p l i c i t y
to make explicit
side is such a funcwhich we now choose
instead of further assuming that a trace on spin indi-
ces is left implicit. adding a constant,
Note that V is determined without the option of
since Eq. (423)
has to hold for the given density.
The diagonal version of Eq. (424) showed up earlier,
in Eq.
(20). We are clearly back to the picture of particles m o v i n g independently in an effective potential V. The notation established useful here,
too.
In particular,
we introduce
the independent
then is particle
Hamilton operator ÷ ÷ H(r,p)
= ~I p2 + V(~)
(2-425)
just as in Eq. (3). The kinetic
energy of
(421)
is then rewritten
as
Eki n = f(d~') (d~") [I~,.~,, 5(~'-~")]n (I)(~' ;~")
= f(d 'l = tr
I
~p
1½p 2 2
D(-H-~)
(2-426)
,
where we remember that the trace operation spin factor.
2
includes m u l t i p l y i n g
E I = tr(H+~)q(-H-~) is a functional energy
by the
The quantity E I of Eq. (7),
of V+~,
(2-427)
thus a functional of n, as V=V(n).
(426) is contained
in
The kinetic
(427),
Eki n = E I - t r ( V + ~ ) ~ ( - H - ~ ) (2-428) = E I - f (d~')(V(~') +~)n (~')
109
This we insert
into
E(n,~)
(419)
and arrive
= E I (V+~)
at
- ](d~') (V(~')
- Vext(~'))n(~') (2-429)
+ Eee(n) In the present fore,
(429)
context,
V is still regarded
is the same functional
than reorganize
the right-hand
is stationary
under variations
just as
is stationary.
(419)
a change
- ~N
in E(n,~)
(419), we have done no more
Consequently,
of n and ~ around
An infinitesimal
= ( ~ E I (V+~)
- N)8~
= o
zero for the same reasons
sider a variation
of n. There-
the functional their correct
variation
(429)
values,
of ~ induces
given by
8~E(n,~) it is, indeed,
as in
side.
as a functional
;
(2-430)
that implied
Eq. (13). Now con-
of the density:
6nE(n,~)
= ~(d~')SnV(~')n(~')
- ~ (d~')6nV(~')n(~')
- / (d~') (V(~') - V e x t(~'))6n(~') (2-431) + f (d~')6n(~')
Vee(~')
= ~(d~')[-V(~')+
where Eqs. (14) and E(n,~)
thus
Vext(r
) +Vee
(25) have been employed.
(~')]Sn(~')
The stationary
property
implies
V = Vex t + Vee In words:
the effective
tial and the effective
6nEee(n) Note in particular, dent)
of
(2-432)
potential interaction
equals
the sum of the external
potential,
Vee , defined
+ = f (dr' ÷ )Sn(r,)Vee(~, ) that V is always
a local
poten-
by Eq.(25),
(2-433) (i.e., momentum
indepen-
potential. So far, V has been regarded
Because
of the circumstance
as a functional
that the contributions
in
of the density. (431), that ori-
110
g i n a t e in v a r i a t i o n s of V, take c a r e of t h e m s e l v e s , we can e q u a l l y t r e a t V as an i n d e p e n d e n t
E(V,n,() is o b v i o u s l y V,n,
= E I (V+~)
stationary
The e n e r g y
- f(d~')(V-Vext)n
under
independent
functional
+ Eee(n)
infinitesimal
- (N
(2-434)
variations
and (. If we do not w a n t to h a v e b o t h V and n as i n d e p e n d e n t
tities,
we have the o p t i o n of e l i m i n a t i n g
(434) b a c k to expressing V(n)
variable.
well
in
(419)
is d o n e by f i r s t
the p o t e n t i a l
(434).
Likewise,
one has to use Eq. (432), to e x p r e s s sity from
in t e r m s of t h e d e n s i t y , to o b t a i n
l e a v i n g us w i t h a p o t e n t i a l
Starting
these
The
step from
for V, t h e r e b y
and t h e n u s i n g this
of the p o t e n t i a l
is a f u n c t i o n a l
ee of V. T h i s n(V)
Let us i l l u s t r a t e tionals.
a functional
in w h i c h V
n as a f u n c t i o n a l (434)
one of the two.
s o l v i n g Eq. (20)
of quan-
alone,
of the d e n s i t y ,
then eliminates functional
the den-
E(V,C).
ideas w i t h the r e s p e c t i v e
TF func-
from
ETF(V,n,()
= f(d~) (-
I ) [_2 (V+() ] 5/2 15~ 2
Z f (d~) (V+ ~) n (2-435)
÷
I ÷ ÷ n(r)n(~') + ~ f (dr) (dr') I~-~' I w h e r e V e x t is n o w t h e p o t e n t i a l -Z/r,
we get the d e n s i t y
(N
,
e n e r g y of an e l e c t r o n w i t h the nucleus,
funcitonal
of Eq. (95), a f t e r
first
inverting
[Eq.(51)] I n =--[-2(V+()] 3~ 2
3/2
(2-436)
to V =
_
which then allows dingly.
~I ( 3 ~ 2 n ) 2/3 - (
to r e w r i t e the f i r s t
On the o t h e r hand,
v is s o l v e d
,
(2-437)
and s e c o n d t e r m in
(435)
accor-
if
= - -z , [ (dr' ÷) r
n(r') ÷ ÷ Ir-r' I
(2-438)
for n, n = - ~1
V 2 (V+
Z ~)
(2-439)
111
(this is, of course,
Poisson's
equation),
we can eliminate n from the
second and third term in (435) and are led to the TF potential functional of Eq. (45). Of course, within the framework of the TF model, three functionals
E(V,n,~),
E(n,~),
and E(V,~)
lent, but I repeat:
as a basis for improvements
tion, the potential
functional
over the TF approxima-
is the preferable one.
We have investigated TF model.
the
are perfectly equiva-
earlier the scaling properties
of the
Let us now see, what one can state about the behavior of
the exact density functionals mations of the density, n(~')+n
Ekin(n)
and Eee(n)
under scale transfor-
(~') : b3 n ( ~ ' )
T h e TF approximations intuitively expect: (&in(n))TF
(2-440)
to Eki n and Eee scale in the manner that one would
= f (d~')
1 (3~2n(~,))5/3 I 0~ 2
÷ ~ -3 f(d~' )
I ( 3 ~ob 3 n (~r') ÷ ) 5/3 I 0~ 2
= 2z (Eki n (n))TF
(2-441)
'
and likewise I (d~" n (~') n (~") (Eee(n.))TF = 2 ] (d~') ) IF'-~''I (2-442) + a[Eee(n)]TF For the exact functionals,
the equations
Eki n(n ) = ~2 Ekin(n)
(2-443)
Eee(n ) = # Eee(n)
(2-444)
and
do not hold, however;
even their combination
Ekin(n ) + E e e ( n ) = ~2 Ekin(n ) + ~ Eee(n)
(2-445)
112
is only true if ~=1+e with an infinitesimal vation has been made only recently, that the statement der to derive
(445)
the virial
2 Ekin(n) from the minimum
is, indeed,
uniquely
= - Eee(n)
property
(445), we recall a certain
ground-state
only needed
potential
Vex t and
where E ~ is the ground-state o I~o > according to
and a certain
I~o~>
I~>
about Eqs. (443)
÷ ,. . . , ~ 1 ~ o " and "o, x I is sufficiently
and
large to jus-
128
tify the use of the asymptotic implies 4 X 2 = ~X I
form
(179) for F(X~Xl).
Show that this
432 x14 = q-7~O
and
Then derive co
2 ~dx If(x)]5/2 5 I/2 O x
2 B 7
=
2 (12) 5 + 2(X_~o)5/2 (4)3.~13,5~-9/3,~; 35 7 xI
and 1 ~dx[f' (x) + 20
]2 =
B
56
Xo +7 ( )2 xl
x17
Putting everything together you should have
(12) s 30 x17- [7
3 > 7B-
8~ + 3 Xo 4x~I ]
3/3
3 A-2/3 q7/3 7
,
for
q÷o .
It is then useful to switch from x ° to a new independent parameter, I, by setting XoE12/3q -I/3. Check that then xi=(432)I/4 1 I/6 q-I/3, so that, for all I>o, A-2/3
Optimize
7 f2/3 > ~
8 ~
(56~_ i-7/6 3/3 30)
I and find the lower bound on A-2/3 of
this optimal
(303). Show that, for
I, the ratio x2/x ° does not equal unity. Consequently,
the
trial f(x) does not change its sign at X=Xo, as the actual f(x) does. Impose x2=x ° and demonstrate that a lower bound on A -2/3 emerges, which is worse than the previous one. For an upper bound on A -2/3 use the trial function
g(x)
~F' (x)
for o~x~x I
+ q/x O
= ~ - q / X o ~ 1 - x / x 2)
for Xl~X~X 2 for x2~x
Make sure that g is continuous functional of
and obeys Eq. (243). Then evaluate the g-
(242). You should get
129
A-2/3
6t ]7/3 I )I/3[ I ( ~ (5t'1)(3-t) (I - ~ t ~ - - - 16 + 191t - 74t 2 )
< I[
_74tl/3(1 _ 1 t ) 1 / 3 (1 - t 4/3 )] ,
+ I
where the range of t=xl/x 2 is ~ < t ~ 1. Find
(numerically)
value
for t and thus the upper bound on A -2/3 of
2-11.
Insert Eq. (316)
the optimal
(303).
into
(t) = ~ f (t x o(q)) I q q q÷o and derive
(352).
2-12.
Derive Eq. (462)
2-13.
Because
dimensional appear
directly
from Eqs. (433), (432),
of the homogeneity
space,
the density
and isotropy
functionals
finitesimally
translated
this with the stationary
(428).
of the physical
Ekin(n)
in Eq. (417), have the same numerical
and
value
and Eee(n), for n(~')
which
and the in-
and rotated
n(r') = n(~' + 6g + 6~×r').
property
(417)
of
three-
Combine
to show that there is no net
force,
[ (dr' )n (r')
(-V Vex t IF')) --o
and no net torque, f (d~')n(~')~'
× (-~'V ext (~')) = o
exerted
on the system by the external
Problem
2?
2-14.
Show that the density
functional
potential.
Are you reminded
of the kinetic
of
energy is given
by Eki n(n) if there
= f (d~)
1 (~n)Z/n
is only one electron.
no contradiction
to the general
,
This does scale statement
like
(443). Why is there
that Eki n does not obey
(443)?
Chapter Three
STRONGLY BOUND
ELECTRONS
In the p r e c e d i n g Chapter there was a section entitled ty of the TF model," TF model:
(i) the inner region of strong
ceed ~I/Z; and of the atom
"Validi-
in w h i c h we found two regions of failure of the binding,
where r does not ex-
(ii) the outer region of weak binding around the edge
(r larger than ~I, for a neutral
atom).
Of these the first
one is more important because of the enormous binding energies trons that are close to the nucleus.
Consequently,
of elec-
the leading correc-
tion to the TF model consists of an improved handling of the strongly bound electrons.
Qualitative
This is the topic of the present Chapter.
argument.
If we simply exclude the critical v i c i n i t y of
the nucleus when evaluating
the r-integral
of Eq. (2-44), then the TF
version of E I is replaced by I 5/2 IEIlTF S ~ f(d~)(- I--~)[-2(V+~)] • r~1/Z I Z ] 5/2 1--~-~)[-2 (- ~)
~(EI)TF - f(d~)(r~l/Z
=
with C a constant
CEIl T F
+ c
z2
,
of order unity.
small r, which states, the n u c l e u s - e l e c t r o n
once more,
interaction
Here we have made use of V ~ - Z / r that the dynamics
a discussion
for Scott, who
small.
argument
to the TF
(which is a variant of the one
given by Schwinger 2) says that the TF energy is supplemented ditive term p r o p o r t i o n a l
The third
(in 1952) was the
of this leading correction
Our simple qualitative
for
are dominated by
if r is sufficiently
initial of the subscript TFS stands first to present 1 energy,
(3-I)
by an ad-
to Z 2, i.e., of relative order Z2/Z 7/3 = Z -I/3
as compared to the TF contribution.
This is consistent with the obser-
vations
of the Introduction,
22) and
(1-84), the numerical value of C being 1/2 in both equations.
More evidence
where we have seen such terms in Eqs. (I-
in favor of a Z 2 term is supplied by Fig.2-2, w h e r e the
smooth TF curve would be shifted down by the amount of 2 C , in which
131
event the agreement with the HF crosses would be significantly
impro-
ved. Scott's result,
E T F s ( Z , N ) = ETF(Z,N)
+ 1 Z2
,
(3-2)
is identical with the one of the Introduction. models of Bohr atoms,
with or without
In view of the primitive
shielding,
that are used there,
it may be p u z z l i n g that the numerical values of the coefficients This m y s t e r y
is easily resolved:
shape of the effective p o t e n t i a l tion are, certainly,
realistic
all that matters
agree.
is the Coulombic
for r+o. The models of the Introduc-
at these small distances.
But there is
even more to it: since one can easily imagine that the slight deviation of the e f f e c t i v e potential
from its limiting
irrelevant,
is anticipated
Scott's result
reasoning is abandoned the p r e s e n t a t i o n
in favor of a more convincing
of Scott's original
One remarkable
correction
N. This is, of course,
one. We postpone
is its independence
a consequence
that the small-r shape of the effective
not depend on N, or, again,
potential
the most strongly bound electrons
ly aware of the more w e a k l y bound ones because the Coulomb the nucleus
is
when his
argument until later.
feature of Scott's
of the number of electrons, circumstance
form -Z/r + constant
to remain valid,
of the does
are hard-
forces of
are so strong.
First q u a n t i t a t i v e
derivation
of Scott's
correction.
For a q u a n t i t a t i v e
treatment of the strongly bound electrons we split El(~)
[cf. Eq.(2-7)]
into two parts,
E I (~) = tr(H+~)~(-H-~) (3-3) =
tr (H+~) ~ (-H-~s) +tr(H+~) [17 (-H-~)-O (-H-~s) ]
-
E
+ s
E ~s
The separating b i n d i n g - e n e r g y
~s d i s t i n g u i s h e s
trons from the rest of the atom,
the respective
the strongly bound eleccontributions
to E I be-
ing E s and E ~ s. This
~s is not a u n i q u e l y defined quantity,
not a r b i t r a r y
It has to be small compared to the typical
either.
electron Coulomb energy
(%Z 2) but large on the TF scale:
but it is single-
132 Z4/3
~
+
,
denotes the larger
(3-103)
'
one of r and r'; the latter identity is
based upon the spherical
symmetry of the density.
square brackets of
can be written as
V + ~ -z
which,
~r
(101)
(rV) = - r ~ -~~ ( V + ~ )z
in connection with
I
0 ,
,
of the
(3-!04)
(103), draws our attention to
for
r
The contents
I
,
for
r>r'
Therefore,
Since the range of integration integration
over r' is limited by r and a further
over r w e i g h t e d by niME(r)
of r' contribute
for which niME(r')
is required,
only those values
is of significant
size. Consequent-
ly,
(106) is well approximated when n(r') = niME(r )+n(r') is replaced % by niME(r'), because n is effectively zero where niM E is large. After this replacement,
the integrand
can be symmetrized
between r and r',
so that
Z ~Z ~S E = 1 f(d~)(d~')niME(r)niME(r')
!r>
=
153
1 = ~ f (d~)(d~')
This
is the electrostatic
innermost of Zr
niM E (r) niM E (r')
energy of the charge distribution
Since n i M E ( r )
electrons•
(just as 0 s and
equals
"number"
We have,
being composed
thus,
Z 7/3
(TF) and Z 2
Z3 times
a function
of the particular
Z - perfectly
(Scott).
values
Z dependence negligible
In other words:
of the TFS model;
(3-108)
number } ,
found that the implicit
with an energy of order accuracy
a factor
due to the
i~n. 12 av do individually) , we have ]
Z ~Z AsE = Z x { a positive this
(3-107)
of ns, wj, and ~j.
of AsE is associated
on the scale set by
Eq. (92) is obeyed within
there is no internal
the
inconsistency.
Fine, correction,
but didn't we just blow it? Certainly, AsE is Scott's 1 2 it equals ~Z ; and, being independent of N, there is no dif-
ference between Shouldn't
its partial
and its total derivative
we, consequently,
Z ~ZAs E = Z 2
? the correct
- (92),(108),
This is so because rent meanings. formalism:
Equation
infinitesimally,
gral in
changes
do not contribute
general
answer
• 6VextE
[see Eq. (2-434)]
in the framework
N, and a given external
potential,
in energy when this external
here
Z to Z + 6z; of course, e.g.,
the second inte-
but the stationary
property
that these induced
to the change
by
potential
of V and ~ implies
of the energy
in energy to first order;
the
(3-110)
as Eq. (88) in the present
Eq. (92), which we now read correctly
is:
of the general is specified
= f(d~)6Vex t n
appears
answer
sides have diffe-
of V and ~, which
here done by varying
under variations
changes
left-hand
of V and ~ [so that,
(67) still exists],
functional
The puzzling
(109) - are true.
(92) is derived
then we ask for the change
is varied
and
we start with a functional
there are induced
statement?
the respective
a given number of electrons, -Z/r;
to Z.
(3-109)
Or is, after all, Eq.(92) all three equations
with respect
obtain
as the change
context;
it leads to
in AsE caused by var-
154
y i n g nothing but the strength of the C o u l o m b p o t e n t i a l of the nucleus. In the TFS e n e r g y functional this e x t e r n a l p o t e n t i a l -Z/r occurs o n l y in the second term of Eq. (67). Therefore,
the Z, w h i c h is c o n t a i n e d in
(68) both e x p l i c i t l y and i m p l i c i t l y in the C. and in the B o h r - a t o m den3 sity Ps' and w i t h respect to w h i c h we d i f f e r e n t i a t e in Eq. (I08), must posses a d i f f e r e n t significance.
It makes r e f e r e n c e not to the external
p o t e n t i a l -Z/r, but to the e f f e c t i v e p o t e n t i a l V, in the sense of
z =
(-rv) I = -S(d~)r(v+()8(~) r=o
w h e r e we have added the
(3-1 11)
,
(otherwise innocuous)
constant ~ in order to
e m p h a s i z e the d e p e n d e n c e of AsE on the sum V+~. ferentiation in Eqs. (I08), or (Coulomb part of the) Of course,
Consequently,
the dif-
(101), really means a v a r i a t i o n of the
effective potential,
for the actual V, the Z of
C o u l o m b p o t e n t i a l of the nucleus,
and not of the n u c l e a r charge.
(111) must equal the Z of the
but not so for the trial p o t e n t i a l s
that we are free to use in AsE of Eq. (68). Now that we have r e c o g n i z e d that the Z in
(68) changes w h e n the e f f e c t i v e p o t e n t i a l is varied, we
must also take into account the c o r r e s p o n d i n g a d d i t i o n a l c o n t r i b u t i o n to the density,
labelled n z. It emerges from Eqs. (I01) and
(111), when
c o m b i n e d into
f(d~)6Vn z ~
AsE ~vZ (3-112)
= S ( d ~ ) S V (-r6(~))~z AsE
as n Z(r)
Z = - r6(7) ~I S (d~')niM E (r') [V(r') + r'
and has almost n o significance, ly zero.
~ (r'V(r'))] ' ~r' (3-113)
because the product rS(r)
is effective-
Its only use consists in the p o s s i b i l i t y of e v a l u a t i n g the re-
sponse of E I to an a r b i t r a r y v a r i a t i o n of V in the standard w a y of Eq. (69), w h e r e the i n c l u s i o n of n z into the density enables one to consider v a r i a t i o n s of the kind 8V = - 6Z/r. We then obtain from tegrated version
(101) that we know already.
Note,
(69) the in-
in particular,
that
n Z integrates to zero, and that its o n l y c o n t r i b u t i o n to the p o t e n t i a l is an a d d i t i v e c o n s t a n t for r=o anyhow.
In short:
[see Eq. (103)], w h e r e V is infinite,
as long as we r e m e m b e r that Eq. (I01) m u s t be taken in-
to account w h e n v a r i a t i o n s of the l i m i t i n g C o u l o m b part of the effective
155
potential
are considered,
we can forget about n Z.
As to Eq. (109), we need only remark that it does not refer to the energy functional
(68), but to its numerical value for V=-Z/r.
deed, we have calculated
Scott's correction
In-
by simply inserting the Cou-
lomb potential
into A E. This raises the question, whether we can do s better than that. How does one account for the deviation of V from its limiting Coulomb
shape when evaluating
the Scott term? The clue is Eq.
(89) which relates the energy to the small-r the TF model we exploited properties
S calin@ properties
of the TFS model.
Z
÷
v
÷
~
÷
~-I
~
AS in Chapter Two, we consider
,
, Z
(~>o)
(3-114)
,
which repeat Eqs. (2-211),
(2-213),
and
property of the energy functional,
(2-214). Again,
here:
6
ET~FS
(89), to be precise].
= 6~(v-I)
as in Eq. (2-220) , or with
8
T•
E FS
where ~=I+6~ side of
Z ~Z ETFS
the stationary
the TFS functional
plies that all first order changes must originate [the Z of Eqs. (88) and
section we
that replace the actual V,~, and Z according to
V(~r) v
In the following
thing for the TFS model.
staling transformations V(r)
In
this equation in connection with the scaling
of the TF energy functional.
shall do the analogous
form of the potential.
(66), im-
in the scaling of Z
Thus
'
(3-115)
(89) ,8
= - 5~(v-1)
Z Z(V+~) [
,
(3-116)
r=o
with an infinitesimal
6~ is understood.
On the left-hand
(116),
6
E~F s
we already know 6
6
= 6
E~F s + 6
AsE~
,
(3-117)
ETF from the earlier investigations, 5
ET~F = {(~9-3)f(d~)
1-
i
1 5~
2) [-2(V+C)]
5/2
156
~!{/
-(2v-1)
(d~) [ ~ ( V + ~ )z]
2
- v(N}61~
,
(3-118)
which is the left-hand side of Eq. (2"222). For the evaluation of 6 AsE~, it is useful to prepare some tools first. The scaling of Z [either the explicit statement in (114) or the (identical) result of inserting the scaled V into (111)] has an effect on the Bohr-shell densities l~n Jf v~ and Ps' given by 3 2
(r)
+
3(v-1)
I~njla
( u-1 r )
(3-119)
J~n'l~v/3
A consequence thereof is {](d~)V(r)
Ps(r)}~ = 4 v - 3
=
#
4v-3
f(d~)V(~r)
-3(v-1)
ps(~ u-1 r)
[(d;V-1~)V(~r)Ps(V-lr)
(3-120)
= ~ v ~ (d~) V (#2-Vr) ps (r)
so that 6 {f(d~)VPs}~ : 6~{vf(d~)VPs + (2-v)~(d~)(r~rV)Ps } = 6#{2(v-1)J'(d~)VPs
+ (2-v)J'(d~)~r(rV)Ps
(3-121) }
An analogous statement holds for the integral appearing in Eq. (98), implying 6
Cj# = 6 # { 2 ( v - 1 ) ( j
or, more conveniently 5 (-u~.)
- ( 2 - v ) f (dr) ÷ ~-~ ~ (rV) @nj J2v~ '
(3-122)
for the sequel,
= 6~(V_2){~j + f(d~)~r(rV) j~n lav~2~ 3
(3-123)
It is used in exhibiting the scaling behavior of E ~ j , preferably studied in the concise form (19):
157
8 E~
~j
: 6{
- S
Id~l ~! d~' (~'-~v~)-~[-2(~VV(~r)+~v~)] I/2} 5v-3
÷
-V
.1~ 3d~ ' (~'-~)[-2(V(r)+~)] 1/2}
S (dr) S
(3-124)
5
= 8 ~ ( ~ - 3)E
j +
~E ~j. 8~j 6
With Eq. (72) and (123) this supplies 6 E ~~ J
= 6~{ (5v-3) E ~ j - (v-2)Qj [~j + f(d~)~r(rV)l~n. 12av] } 3
(3-125)
This combines with both Eq. (121) and 6 {Z2ns+~Ns }~ = 8~{2 (~-1) Z2ns + V~Ns}
(3-126)
to &S E~ = 6~{(5v-3)AS E - (~-2~.[7~,Q, ~.~ + ½S (d})Vps j 3 3 J
8
I
3
+5 (d~)niME~r(rV) +2 Z2ns +2 ~Ns]}
(3-127)
Together with Eq. (118), we then conclude from Eqs. (116) and (117) that 5
(~v-3)ETF S +
½
+ (v-2)
=
3
z
(~-2)~N+ (v-1)Z(V+r) 1 r=o
7, 2
(d~) [~ (V + ~) ]
J ÷ ~ I 2 3 (v-2) [~-j=lWjQj~j +1 S (d~)Vp s + f (dr)niME~-~(rV) +~Z n + ~ N
(3-128)
s]
which, finally, states the scaling behavior of the TFS model. Please observe that the terms on the right-hand side of Eq. (128) all refer to the specially treated strongly bound electrons. Therefore, replacing ETF S by ETF and setting the right-hand side equal to zero should reproduce the corresponding statement about the TF model. Indeed,
158
what is obtained noticed
is
(a rewritten
that the most useful
version
choice
of) Eq. (2-222).
There
it was
for v is v=4. This is still true.
After employing
(Z,N)
= - ~
[this has appeard
(3-129)
ETF S(Z,N)
earlier
as Eq. (2-225)]
and Eq. (89) , Eq. (128)
reads,
for v=4, (3-130) J = 2[
where
÷
wjQj~j + ~ ] ( d r ) V O s + f (dr) n i M S ~
it is made explicit
derivation
3
in the dependence
Eq. (2-226).
of Scott's
correction.
always been content with the approximation
Until now we have
V~-Z/r when evaluating
Scott correction.
It is time to pay attention
V and its small-r
Coulomb part.
We rewrite
1 2
(rV) + ~Z n s- ~NsT~ETF S (Z ,N)],
that we are now interested
of ETF S on Z and N. This generalizes Second quantitative
8
In general,
to the difference
V(r)
the
between
is given by Eq. (I03).
it by using the identity
1
1
1
~> = ~,~(r'-r)
1
1
1
+ ~q(r-r') =--r' - (~' -~)q(r-r')
,
(3-131)
obtaining V(r) = - r Z + 5(d~') The first integral
appeared
I _I~)~ (r-r') f (d~')n(r') (~,
n(r') r'
in Eq. (99), it equals
- ~zETFs(Z,N).
(3-132)
For
the second one we write v(r), I
1
v(r)
= f(d~')n(r') (?, -~)~(r-r')
V(r)
=
,
(3-133)
so that Z r
~ETFS ~Z
v(r)
(3-134)
In this 'form, we shall insert it into the right-hand It is thereby in V+Z/r,
not necessary
to keep track of more than the first order
since the TFS model
this is a small quantity
side of Eq. (130).
is based on the physical
for the strongly
argument
bound electrons
that
[see Eq.(37)]°
159
From Eq. (50) we get -
~.
3
Z 2 + ~ETF_____~ S + 2n2 ~Z 3
(3-135)
nj
with (3-136)
~n. - f(d~)v(r) [~n.[2v (r) 3 3
being the average of v (r) over the n.-th Bohr shell. When evaluating 3 the integral, that gives Q5 in (72), to first order in v, another average of v(r) is also met, 2n 2. --
f(d Iv(r) 4
nj
5~ 2 2n. 3
'
(3-137)
where the range of integration is r
6(n'-v)
n '=-~
,
identity
= ~ - - - e i2rumv = I + 2 > m=-~
p e r f o r m the v integration,
cos(2runp)
m= I
and arrive at Eq. (25). Repeat this p r o c e d u r e
to evaluate [Vs]
Tn2 n'=1
and d e r i v e Eq. (30).
3-2. AS an i l l u s t r a t i o n of Scott's N n o n - i n t e r a c t i n g particles, along the x-axis,
"boundary effect"
argument,
consider
r e s t r i c t e d to the o n e - d i m e n s i o n a l m o t i o n
and c o n f i n e d to the range 0Sx~a. These p a r t i c l e s occu-
py the N states w i t h least energy, one per state. Compare the TF approx i m a t i o n to the d e n s i t y and the energy w i t h the exact results. Note, particular,
in
that the r e q u i r e m e n t of v a n i s h i n g w a v e functions at x=0 and
x=a cause the exact energy to be larger than the TF result. Repeat for N p a r t i c l e s c o n f i n e d to the i n t e r i o r of a t h r e e - d i m e n s i o n a l sphere,
and
observe that t h e r e is m u c h less s t r u c t u r e in the d e n s i t y than in the previous one-dimensional
situation. Why?
3-3. S i m u l a t e the r e p l a c e m e n t
W(Vs)
(31) w i t h the aid of a w e i g h t f u n c t i o n
, 2 3 fdVs W ( g s ) ( ~ V s - Ns)
= 0.
174
Since N s is constant for ns7
dvf(v)
cos(2mnv)
m=1 Vs
Then integrate by parts repeatedly to find
>
f(n') =
n'=[Vs]+1
fdvf(v) VS
~ sin (2may s) - f(Vs)/ ~m m=1 I f, - ~
I f.
~
cos (2nmv s)
(Vs) m = I
(~m) 2
~
sin (2n/n~s)
(VS)m= I
(~m) 3
+
+ 4
=
fdvf(v)
+ oscillation.
Vs
Specify f(9) = ~
, and arrive at Eqs. (164) and
(163).
...
Chapter
Four
QUANTUM CORRECTIONS AND EXCHANGE In Chapter Two we learned tional to Z7/3; correction
that the TF energy of an atom is propor-
in Chapter Three it was established
to this TF energy is p r o p o r t i o n a l
that the leading
to Z 2 = Z7/3/Z I/3""
. In this
Chapter we shall be concerned with the second correction which, surprisingly,
supplies
energy of atoms. ger -Z
It will account
HF crosses
not
a term of order Z7/3/Z 2/3 = Z 5/3 to the binding for the difference
and the continuous
between the inte-
TFS curve in Fig.3-3.
where the phase
motion along,
di-
matters, say, the
(we choose to distribute
the
and z)
function
(4-29)
is obeyed,
=
, of which the one dimensional version
(33). With the Hamilton operator
÷ +
i ~
equation obeyed
is written
as Eq.
(10) it reads in three dimensions:
= (- ~1 q I'2 + V(r÷ I,))
= (- ~1 V 2' 2 + V(r+,)) 2
4-52)
Upon setting ÷
~,
= !+12} , 1
accompanied
by
÷,
1 ~,
Vl
r2')
'
+ ~
= 2
+, S
'
the sum and difference i ~
,
s = ~1' - ~2'
1 ~,
V2
_ ~
= 2
s
of the two versions
+ ~I V,2 + 21 Vs2_ ~
(4-53)
,
of
(4-54)
(52) appear as
21
e
= 0, (4-55)
and
q ' .V s
-
÷ + ~is ÷) [V(r'
- V(~'-~ I
where the time t r a n s f o r m a t i o n These differential
equations
(2_
13/2
e
-i~
function is inserted
= 0 ,
in the form
(4-56)
(51).
are to be solved subject to the initial
condition I
%3/2
2~--IT'
e-i~
+ 6(s)
,
(4-57)
for t ÷ 0
[cf. Eq. (34)], w h i c h is satisfied provided that T ÷ t
,
("tyme ÷ time")
+
s2 2t
For the sequel, and
for
t ÷ 0
,
(4-58)
t ÷ 0
(4-59)
and
'
for
it is helpful to carry out the differentiations
(56) formally,
in
(55)
and to separate the real and imaginary parts of the
resulting equations.
This
leads us to a system of four partial differ-
185 ential
equations
determining
_3
~ and T,
+9+
log •
- ~1 {3 V,2 l o g T
1-*
+ v ( ~ ' - 3 sI )+ ]
+ Z9 ~ ' l o g T
34 ~,.gslog T
• ~slOgT
= ~1[ V ( r÷' + 31~)
{3 ~ t log T
+ V s2
s
{
-3
(4-60a)
,
_ I ~,®.~ s
(4-60b)
_ v ( ~ ' - [1 5 ) ]
- 3 ~s~.~sZogT
} (4-60c)
+ ~ I {v' 2 ® -
~,.~
}
- ~9 (~,logT) 2 + (~'~) 2 }
[v(~' + yIs÷)
=
2
3
~'{.
~,logT} -- 0
+ 3 ~, ~ .Vslog T
_
3+ 3
,
.~
V'IogT
s
~ = 0
(4-60d)
,
(4-61)
N o w since
__~ ~ Z4/3
~
~t
'
we notice
that
in
~ Z2/3
~, m Z 1/3
s (60a)
' and
(60c)
terms
are of o r d e r
Z 4/3 and Z 2/3,
sides
of
(60d)
(60a)
is equal
1
~(
(60b)
1+.÷, e~-S V
and
and second
respectively,
are of o r d e r
whereas
Z 3/3 each.
curly-bracket the l e f t - h a n d
The r i g h t - h a n d
side of
to
÷e
= v(~')
which
t h e first
are terms
_1~.~, 2
) v ( r÷' )
1 1 ~ .~,)2
+ Y(3
of order
= c o s h ( y1s+ . ¢ , )
v(~')
Z 4"3, /
v(~,)
1 1 +.~,)4
+ ~7(~s
Z 2"3, /
Z 0"3, /
....
÷
v(r')
+ ...
Similarly,
(4-62)
we have
in
(60b)
_1~.~,
l~ .~, I (e 2
3
- e
= (~1 ~ . ~ , )
2
V(~')
] V(~')
11~.~,
+ 6(2
= s i n h ( 21 ~ .~,) V(~')
)
3
+
V(r')
+ ...
,
(4-63)
186
these being terms of order Z 3/3, of Z -1/3 is facilitated
Z I/3,
by introducing
plays the role of Z -I/3.
... . The counting of the powers a parameter
~ that essentially
It enters Eqs. (60a-d) , (61) , and
(62) via the
replacements 1
.-f£
~
÷-7-~
+
'
Vs÷-~
1 ~
,
s
~,÷
1~,
~-
(4-64) V ÷
V
,
s.V' ÷ ~ s-?'
Then the power of ~ m u l t i p l y i n g
any term indicates
its order in powers
of Z - I/3." All reference to b can finally be removed by setting
b equal
to unity. After common
factors of ~ are cancelled, the effect of
m u l t i p l y the second curly-bracket write,
in conjunction with
(I/b) s i n h ( ~
.~')V(~')
= {- ~s
and
(~s .~')V(~')
sides of
(and left to the reader)
are solved to order 2
÷ ~(~',s,t)
(60a)
(63), cosh
for the right-hand
is then straightforward equations
terms in
(62) and
(64) is to 2 (60c) by ~ and to
(60a)
and
and (60b). It
to verify that these
by
2
+ v(~')t} + ~2 t
{ (~.~')2v(~'l + [~'V (~')]2t2 } (4-65)
and l o g T (~' ,~,t)
= logt
- 2
the latter one being equivalent + T(~',s,t)
Inasmuch as
= t[1-~
(65) and
t 2 V,2
to
2 t 2 V,2 ]-~ V(~')]
(4-67)
(67) are identical with
soon as ~ is put equal to one, we are, ximations
(4-66)
(45),
indeed,
(47), and
assured that those appro-
are correct up to the relative order of Z -2/3.
A few comments
are in order.
The last reasoning,
powers of ~, shows that only even powers of ~ emerge; to corrections
of relative orders
there is no Z -I/3 term. tive.
Z -4/3,
and so on. Remarkably,
however,
is of relative
are exposed to the Coulomb potential
so that their energies
tiples of Z 4/3 but of Z 2
the counting or these correspond
fit in? The answer is both simple and instruc-
The strongly bound electrons any shielding,
Z -2/3,
The Scott correction,
size Z -I/3. How does this without
(50), as
and their distances
are m e a s u r e d not in mulare not ~Z -1/3 but %Z -1
187
which is also the m a g n i t u d e of their deBroglie wavelength. in the Coulomb part of the potential that,
instead of the TF relations V ~ Z2
,
the scale is changed to the effect
(37) and
r' ~ s ~ Z -I
,
In other words:
(61), we now have
~' ~ ÷Vs ~ Z (4-68)
t ~ Z -2
~3- ~
,
w h i c h has the c o n s e q u e n c e
Z2
that all terms
size, namely ~Z 4. This implies that, there is no expansion parameter
in Eqs. (60a-d)
available
for a systematic
calculation of ~ and T if V is the Coulomb potential. upon d i s r e g a r d i n g in a w r o n g answer.
if one simply extrapolates
(47), and
special treatment
approximate
Any scheme based
certain terms in Eqs.(60a-d) will inevitably result 2 Therefore, the v i c i n i t y of the nucleus will not be
dealt w i t h correctly of Eqs. (45),
are of the same
in contrast to the TF situation,
(50) into this region.
the quantum corrections
There is no way around the
of the strongly bound electrons
that we studied in the
p r e c e d i n g Chapter. A second comment is the following. p r o x i m a t i o n via the local o s c i l l a t o r
In arriving at the new ap-
potential
of Eqs. (26) and
(27),
terms of order t 3 w e r e kept in ~ and T, those of a higher order in t discarded.
Does this imply that one could regard the expansions
(46) as counting the powers of the time t? Of course, results
[Eqs. (45),
displaying
(47),
(50)], however,
all contribu£ions
(41) and
they do; the final
are not correct in the sense of
to order t 3. As a matter of fact,
they do
not even contain all terms of order t, since }(~, + ,s,t)
s2 =-
Isinh({ ~'~')] + - ----
2-£
is the small-t
I + ÷, ~s.V
I
[
V(~')t + 0(t 3)
(4-69)
J
form of }, as can be v e r i f i e d with the aid of Eqs. (60a-d).
The extra terms d i s a p p e a r
from
ferent for the c o n t r i b u t i o n
=
(69) for s = o ,
but the situation
is dif-
~t 3 # because
2
+
212v( ,it3 (4-70)
+ 0 (t5) replaces
(45) if powers of t instead of Z -I/3 are counted.
Thus,
it is
really Z -I/3, not t, what is the expansion parameter. 3 Here is a third comment.
If the t r a n s i t i o n
from the dimensio-
188
nal m a n y - p a r t i c l e is not made,
Hamilton operator
the gradients
(I-I)
to the dimensionless
in Eqs.(60a-d)
as well as
one
(45),(47),
(1-7)
and
(50)
come with a factor of ~ each. Then the Z -2/3 terms are all m u l t i p l i e d by ~2, so that they can be m i s u n d e r s t o o d
as the beginning of a series
in powers of M, or ~2. It has already been remarked, of the Introduction,
on the first pages
that M is not a parameter of the theory,
so it cer-
tainly cannot serve as a measure of the quality of an approximation. What is really meant by the phrase
"expanding in powers of ~" is the
process of counting the powers of the ~' operator.
Indeed,
terms are all d i s p l a y i n g two ~'s. This is not accidental;
the Z -2/3 there is a
simple reason why all terms of order Z -m/3 also contain the ~' differential operator exactly m times.
An immediate
implication of Eqs. (60a-d)
is that both # and T are even in ~ and odd in t, so that an arbitrary term in the expansion of either quantity is
(?,2)~(~.~,)2B V Y ti+26
where ~, B, ¥, and 6 are integers. (71) is of the order ~+¥,I-26 Z °, as discussed
(symbolically)
given by
,
(4-71)
Now,
if V is the Coulomb potential,
in Z 2, and since each such term is of order
above, we find
+ y = I + 26
With this restriction,
the most general
(t V ' 2 ) ~ ( s . ~ ' ) 2 ~ ( V
which, deed,
(4-72)
for the TF potential,
t) Y
(71) is
,
(4-73)
is of order 2(~+~)
the number of the ~' differential
lished that,
form of
in Z -1/3. This is, in-
operators.
for our application to large atoms,
We have thus estab-
expanding
in powers of
Z -I/3 is equivalent to counting the powers of ~', or, more c o l l o q u i a l l y of M.4 It must be emphasized when the approximation ready m e n t i o n e d
that the situation
is applied to other physical
the bare Coulomb potential,
meter for an expansion
is likely to be different
in the first place,
systems.
We have al-
for w h i c h there is no paraso that the m e c h a n i c a l
coun-
ting of the powers of ~' or ~ can only be qualified as nonsense. The final comment answers the question why it is advantageous to w r i t e
o (2~) 3
which uses the effective quantum-corrected
,
f(H) of the independent-particle
(19) is now approximated
tr f° (2~) 3
is the desired modification This new injunction
is now applied
E~ s
where
of Eq.(90).
for evaluating
traces of functions
to the trace in Eq. (97). The outcome
=
2 (d l(d l
we arrive
of -H-~
is
5(zlV2V
to ~' just like z is to ~ [Eq. (119)].
ming the ~' integrition,
E~S
l °
,
of course, 1 2 zs =-~p -V-
I ~s + ~ x
I~vl
2/3
= z - (~s-~)
(4-131)
199
Equations
(129) and (130) are the quantum corrected versions of the TF
expressions
(3-19) and (3-44), respectively, to which they reduce upon
neglecting the ~V and V2V dependences and performing the Airy averaging and the momentum integration.
It is a technical detail of enormous sig-
nificance that one can both Airy average and integrate over momentum in (129) and
(130) explicitly,
so that, as before, only the spatial inte+
gration is left. Because of the Delta functions the p-integrals are im12 mediate; the Airy averages, however, are nontrivial. The next section shows how to deal with them.
Airy Averages.
It is expedient to first consider
(4-132) = -~I o Tt where the second equality is based upon the momentum integral
(99). Let
us reexpress this in terms of the variable
2 (v+~)
y =
12 vl 2/3
,
(4-133)
which has the property of being negative allowed
(positive) in the classically
(forbidden) region, to get
2r (d_~__ O J (2T~) 3
2 12~Vi-I/3 Fo(Y ) = ~
(4-134) '
where Fo(Y) = 2~ °
(4-135)
Since the fundamental Airy average, Eq. (87), concerns an exponential function
of
x,
we e m p l o y
the
identity
(2-2/3 x - Y )-I/2 = 5 dT 6(r2 + y - 2 - 2 / 3 x )
,
in conjunction with the Fourier integral of the Delta function arrive at
(4-136)
(12), to
200 Fo(Y)
= (2~) 2 f d'[ ~ do < e - i ( m 2 + y - 2 - 2 / 3 x ) ° > --co
°
--co
(4-137)
fa~ao
= (~)2
the last step using
a = x+x' the integrand
e-i(T
2
+y)o-io
3
/12
,
(87). After the substitutions
I , T = ~(x-x
) , dTdo = dxdx'
,
(4-138)
factorizes,
co
Fo(Y)
= 2~ S dx e - i y x - ix3/3 (4-139) co e-iyx'-ix' 3/3 x 2~] d x . -co
so that the Fourier transformed
statement of Eq. (81), namely
co
Ai(y)
: 2~ S dx e -iyx e -ix3/3
,
(4-140)
can be used twice, with the final outcome
F o(y)
= [Ai(y) ]
2
(4-I 41)
Before proceeding tum-integrals
of 6(z),
to the evaluation of the A i r y - a v e r a g e d
~(z),...
, let us supply another,
oriented d e r i v a t i o n of this result. proximations potential. momentum
(47) and
Therefore,
integral
For this purpose
momen-
more physically
recall that the ap-
(50) are exact in the situation of a constant
force
by simply undoing the steps that introduced the
[Eq.(79)]
and the Airy average
[Eqs. (81) and
(87)],
we have
2 r (d~)
;(2g) 3
(4-142}
where z I and the Hamilton operator H I refer to the constant-force tial V I of Eq. (20). The eigenstates - E
and their transverse m o m e n t u m ~i
mensional vector k i is perpendicular
of HI, characterized
poten-
by their energy
(which is to say that the two-dito the constant
force -~V)
are,
201
properly normalized
ikl'~ 2(Vl+~)+ki 2 _ e (2E) 2 21/2 12~11÷, 1/6 Ai ('1~V112/3 I ,
w h i c h reminds us that the d i f f e r e n t i a l (123), is e s s e n t i a l l y linear potential.
the S c h r ~ d i n g e r
(4-143)
equation obeyed by Ai(x), equation
Eq.
for the one-dimensional
The m a t r i x element on the right-hand
side of
(142)
is
now evaluated by
2. t ' ~
co
°
(4-149)
~-m ~12~vi-I/3 Fo(Y)
where the last equality uses Eq. (134). The definition of y in Eq. (133) has the immediate consequence
d _ 212~vi-2/3 ~y
(4-150)
dz
which implies
2-(d_i[L )(2~) 3
o
The boundary at z=-~ in (148) clearly corresponds (149) and
(152), respectively,
so that
(~y)-1 f(y) = _ S dy' f(y') Y
to ~=~ and y=~ in
(d/dy) -I signifies
(4-153)
An immediate recurrence relation for the functions F m (y) is
203
dyd Fm(Y ) = Fm_ I (y)
(4-154)
To produce another relation we first remark that, for m~o, the second version of
(152) has the significance
(-1)! I < (2-2/3x _ y)m-I/2>o F m (Y) - (m_l) ! 2~
which can be checked against
(4-155)
(154). Now observe that
< ( 2 - 2 / 3 x - y)m+I/2>o = < ( 2 - 2 / 3 x - y) (2-2/3x_y)m-1/2>o (4-156) I d2 (4 2 dy
(2-2/3 x Y)
>1" and "y° ~V + 2 ~
,
(4-215)
where said approximations produce
- a -a~ f ~d t f ~ I
t 2 2 V ( ~ , ) ) e - i (V (~') +~.')t ]312(i + ~v' (4-216)
i + ÷ ]2t3 ] x exp[-i [V (~') +~' ) t _ ~--~[V'V (r') to first order in the Laplacian of the effective potential, as always. The factor of t 2 multiplying this Laplacian can equivalently be replaced by ( _ ~ / ~ , ) 2 which operation is advantageously performed after the integration. Then the remaining integral is simplified by means of the identities
(79) and (87), followed by explicitly integrating over
t and differentiating with respect to ~'. At this stage, we have
(4-217)
11_ i
,2
22 . v jf
= f
(dp') (4-232)
then n (I) (~,;~,,) = n (1) (r ÷, ;~,,)
(4-233)
223 if r'-r" is infinitesimal.
Clearly,
the equal sign in (233) cannot be
true for arbitrary +r' and ÷r", since 4 1 )
originates
in a Slater determi-
nant: ,~,
N
4 1) ( ~ ' ; ~ " )
= ~____~j(r')~j(~') j=l -~-
= N
f
!
--k
(4-234) .
÷
~
(dr2)...(drN')¢v(r",r2',
"~"
÷
"+
!
.... r~)~VIF' ,r2', .... r~)
with + ,, ÷ , I detI~j (~k,)] , ~V(rl "'"rN) = N~. j,k
(4-235) -~
÷
whereas the true ground state wave function ~o(r1',...,rN') is certainly not of this simple structure. the N lowest-energy
The ~j's in (234) and
eigenstates
(235) are, of course,
of the effective Hamilton operator H,
counting different spin states separately.
In Eq. (2-509), these were de-
noted by ~j. The fact that n (I) and n~ 1) agree for ~':=~" suggests the use •
of 4 I) as an approximatlon Likewise,
to n
(1)"
+,,
÷
for arbitrary values of r' and r .
4 2) should not differ much from n (2) if ÷r I'
= ~1" ~r÷ 2'= ~2"'
that
is for the range of arguments which contributes most to Eee. Upon inserting the contruction
(235) into
(229), we obtain N
4 2) (rl,r + ' ~ 2'-~i",~' . . ,. + ) ~j (r1')~k(r2') [~j*(÷r I. )~k(r2 , )= ~ ÷ ÷ j ,k=1 - ¢j*(~i,,)
* +
,,
÷
÷
(4-236)
,
~k(r2 ) ~k(r1') ~j(r 2 )]
Before proceeding it is necessary to recall that in both Eq. (234) and (236) tracing over spin indices is implicit.
For instance,
in
(234) the
spin matrix is 6o,o,, with o' and o" taking on the values + and - ("up" and "down")
each. Thus the trace in question is
Z 60, O, = 2 O'
,
(4-237)
which is, Of course, the factor of two that reflects the spin multiplicity in Eq. (232). Now the spin matrix for the first summand in
(236) is (4-238)
6~i~I"
602~ ~'
the trace of which is
224
>
,,,
8oW
= 2 × 2
14-2391
a1', a 2' two factors
of two.
In contrast,
the spin matrix of the second summand
is
6a~a~' 6a~a~
(4-240)
with the trace
> oi',02 '
6a½a1' 5a~a½
I = 2 = ~(2x2)
just one factor of two. This explains
(4-241)
why in combining
(234)
and
(236)
into 42)
+ I, ,+r2';+r,I, +,r2') , (r
= ~1)
+ +r1") 4 1 ) (r1';
+ ;r2" + ) (r2' (4-242)
1 V(1) (r ÷ ,I ;~2,, ) 4 1 ) - ~n
the factor of I/2 appears.
Its physical
factor expresses
that only half of the electron
their spins
the fact,
is obvious:
this
pairs have
parallel.
The approximation ation of E
significance
+ + (r2';r1")
for n (2) to be used in
(230)
for the evalu-
is then
ee
n (2) (r +' , ~"., ÷' +' , +r"; ÷r', ~" ) r , 7" ) ~ 4 2) (r (4-243) = 4 I) (r; +' 7' )n v(I) (r" ÷ ;~") - 1 4 1 )
or with Eqs.(233)
and ÷
I) (~';r')
(2-423) ÷
= n(r')
combined
(~' ;r") + 4 I) (~" +;r')
r
to
,
(4-244)
simply n (2) (r',r";r' + ÷ ÷ ,7" ) ~ n ( r÷, ) n (÷r " ) - ~ nI V(I)(7' ;r +'' ) 4 I) (r";r'). + ÷ (4-245) This inserted E
where E
ee
into
(230)
gives
~ E
+ E
,
es
ex
is the electrostatic es
(4-246)
energy
(227)
and E
the exchange ex
energy
225
E
~I) (r÷, ;7" )~I
I
+ + (r";r')
(4-247)
ex
Note that it is ~ I ) ,
not n (I), appearing
in this definition
of the ex-
change energy. The equality
in
(246)
is clearly approximate.
energy Eex takes into account the antisymmetry does this in a simple way by a p p r o x i m a t i n g Slater determinant.
the true wave function by a
incorporated
this way.
However,
hood of two electrons,
upon the interaction
in Eex. All three,
four,
tions to the energy are usually called seems to have as many definitions
Ecorr In particular,
of these two
five,..,
Self energy. because
(4-248) between the
because the HF po-
from the true effective potential V; the d e v i a t i o n betis, if course,
In Chapter One there appeared,
small.
between Eqs.(1-62)
and
(I-63),
that one should not worry about the electron self-energy,
"as soon as we shall have included the exchange
the picture,
For us,
E Eee - Ees - Eex
and this d i f f e r e n c e
the statement
a term that
between the two sides of Eq.(246),
true ground state energy and its HF approximation,
corr
contribu-
energy",
this Ecorr is not equal to the d i f f e r e n c e
tential differs
- are certain-
particle
"correlation
con-
in the neighbor-
as there are investigators.
it means no more than the d i f f e r e n c e
ween E
more subtle
- such as the influence of a third electron,
ly not contained
It
There is no doubt that the main effect of the anti-
symmetry is correctly sequences
The exchange
of the wave function.
the electronic
equally unphysical
interaction
self energy will be exactly canceled
self-exchange
into
by the
energy." We are now able to justify
this remark. The respective (236), the first summand these terms cancel
contributing
exactly in
the exchange self-energy ly, their sum vanishes. ding the self-energies fectly.
self-energies
(236)
originate
in the j=k terms of Eq.
to Ees , the second to Eex.
the electrostatic
are identical, In other words:
no room for explicit
and
Consequent-
the errors introduced
into Ees and Eex compensate
There is absolutely
self-energy
but differ in sign.
Since
by inclu-
for each other per-
self-energy
corrections,
once exchange is included into the description. 13
Exqhan~e II
(leadin@ correction).
lated to the time t r a n s f o r m a t i o n
The density matrix n$1)I is easily refunction of Eq. (51) by means of
226 n (1) (F';~") = 2
s ) 41)
1.+ +,+~sj 1+. , (~'-~s,r (4-254)
(252) draws our attention to the ~ integral
s2. I +t1__~)I = exp(i-~-~-T
227
= 2~
-
2 t_' +t_"~
f ds 2 exp(
s 2i
o t,t l' t' +t"
= 4~i
At this stage,
we have I
m
ex
(4-255)
t't"J
2(dt' [
I
]3/2 e-iV(~')t'
-~
x 2( d t ' ' r l ] 3 / 2 2~ 10, less than 5o for Z>25 less than I 7% for Z>62. C o n c e r n i n g the s h i e l d i n g factor 1-a m the two t h e o r e t i c a l values d i f f e r at for m e r c u r y
m o s t by 0.00003;
a small amount,
indeed. This happens
(Z=80), for w h i c h the HF c a l c u l a t i o n yields
I,am=0.99027,
w h e r e a s the s t a t i s t i c a l m o d e l gives 0.99030. A g a i n the s e m i c l a s s i c a l t r e a t m e n t has p r o v e n to not only supply an u n d e r s t a n d i n g of the analytical d e p e n d e n c e on the atomic number,
as e x p r e s s e d in Eq. (319), but
244
Table 4-I. C o m p a r i s o n of H F and S t a t i s t i c a l - M o d e l predictions
for
1 o5%. Z
HF
SM
1
1.8
2.2
30
252.2
251.3
2
5.99
5.74
35
312.1
310.9
3
10.15
10.11
40
374.2
373.9
4
14 93
15.13
440.0
439.9
5
20 20
20.68
50
508.6
508.6
6
26 07
26.70
56
591.9
594.4
7
32 55
33.13
62
681.7
683.8
8
39.51
39.94
68
776.4
776.3
9
47 07
47.09
74
873.6
871.9
10
55 23
54.55
80
973.0
970.4
15
96 12
96.02
86
1072.8
1071.6
Z
45
HF
SM
20
142 3
143.2
94
1210.
1210.5
25
194 2
195.2
102
1355.
1353.8
also to produce highly accurate n u m e r i c a l predictions,
not i n f e r i o r
to results o b t a i n e d by the m u c h m o r e involved HF method.
S i m p l i f i e d new d i f f e r e n t i a l e q u a t i o n
(ES model).
The e v a l u a t i o n of the
energy c o r r e c t i o n E as a small c o r r e c t i o n to the TFS energy has proven quite successful.
Let us now try to i n c o r p o r a t e the exchange energy and
the q u a n t u m c o r r e c t i o n to E I into the energy functional itself. At the present stage, we shall be s a t i s f i e d by keeping o n l y first o r d e r terms, so that we get a simplified p i c t u r e w i t h o u t the detail s u p p l i e d by the p l e t h o r a of A i r y functions that one meets, expression
for instance,
in the d e n s i t y
(210).
We are aiming at a d e s c r i p t i o n that enables us to study the o u t e r reaches of the atom. Consequently,
one s i m p l i f i c a t i o n w i l l con-
sist of paying little a t t e n t i o n to the details of the special t r e a t m e n t of the innermost electrons. sion of E I (V+~)
Iv÷ l
This is i l l u s t r a t e d by c o m b i n i n g the TF ver-
and the leading c o r r e c t i o n
f (dr) { + CSBE
(97) into
5/3 + 24-- E-2 ,
1/2v2v} (4-320)
245 where the initials CSBE are to remind us of the necessity of making corrections
for the strongly bound electrons
electron interaction
energy Eee(n)
form of the Dirac approximation
I f(d~) (d~')
The electron-
includes the exchange energy in the
(262)
who actually found this expression);
Eee (n)
eventually.
(do not forget that it was Jensen thus
n(~)n(~') (4-321)
- J'(d~) 4-~[3n.2n(~)] 4/3 The stationary property of the energy functional E(V,n,~)
= EI(V+~)-
(2-434),
S ( d ~ ) ( v + Z ) n + E e e ( n ) - ~N
under infinitesimal variations 4~I V 2 ( v + l ( 3 9 2 n ) I / 3
,
(4-322)
of both V and n implies +
.~)Z
I 3/2 n = 3--~[-2 (V+~) ]
1 -1/2 24~z [-2 (V+~) ] V2V (4-323)
What is subtracted
?2
I
+ ~
[-2(V+~)]
1/2
+ CSBE
.
from V on the left-hand side is, of course, the ex-
change potential Vex, defined by
6 n Eex = ;(d~)6n Vex , in the Dirac approximation,
(4-324)
that is
Vex ~ - ~I(3~2n) I/3 ,
(4-325)
as it results from the exchange energy term of
(321).
Consistent with the strategy of keeping the corrections to first order, we bring the Laplacian on the right-hand side of to the left-hand one, where it supplements
only (323)
the exchange potential by
one sixth. At this stage, we have - 4-~I V 2 ( V + ~ ( 3
2n) I/3 +~)Z
3/2
i
-3~z[-2(V+~)]
{I +
[_2(V+~)]3/2 3~ 2
+
CSBE
.
(4-326)
246
The inclusion
of Z/r into the Laplacian
change anything, an expression
because
that vanishes
mark that this would anyway.)
Now,
the resulting
at the origin.
(Incidentally,
the second term in the curly brackets
1 V2 Z 4~ (V +~)
it is added,
side does not
is multiplied
refer to the domain of the strongly
tion to the unity to which
is justified.
on the right-hand Delta function
one might
by re-
bound electrons
is already
a correc-
so that the TF evaluation
1 3/2 = 3-~Z!-2(V+~)]
(4-327)
Then we obtain
(4-328)
= 3~2[-2(V+~)] I 3/2
is to be used both to evaluate as a functional grations.
of the actual density
Going through
right-hand
side of
this procedure
(393) gives
electrons
tically
are irrelevant
constant,
(4-393)
the integrals
(for which
n(}l = ~
(392)
and to express
is well approximated
12~v(})IF2[y(}I]
at
for the strongly
of the potential
do not matter. by
for the
is unrealistic
There the corrections
and the gradient
E
ex after these inte-
~'=~)
with the TF approximation
so that higher derivatives
at the edge the density
of
(391). This TF density
the edge of the atom where V + ~ 0 . bound
is small,
of the exchange
is reliable;
(391)
of the atom.
the contribution
where the density
large to ensure that the Dirac is small compared
(262), (4-391)
in the dense interior
modification
func-
is prac-
Consequently,
[cf. Eq.(219)] 49 (4-394)
with
y(~) = 2>1
to the TF approxi-
of the atom, where V+~ is a form of F2(Y),
namely
[Eq. (161)]
,
(4-398)
is available. We are thus invited to insert king of V ° as a parameter
somewhat
of the atom - not, of course, this edge~
but merely
meaning
concerning
(397)
into
(392), thereby
related to the gradient that
(396)
becomes
the order of magnitude
thin-
at the e d g e
an identity
at
of V o. Just as
for the U of the ES model, there is no best value for V . To some exo o tent it can be regarded as an adjustable parameter. There is, again, a price to be paid for the simplicity (210)
[or (219)]
to
gained
in the transition
from Eq.
(397).
The differentiation that the new approximation
of F2(Y)
produces
for the exchange
F1(Y) , see Eq.(154),
energy is
so
(see also Problem
10) Eex(n)
where y(~)
(4-399)
~ - f(d~) 2~ V ~ y{~)dy' [F I (y,)]2
is given in terms of the density
n(r)
by means of
n(~) = ~ V3/2o F2(Y(~)~• This reduces which
to the Dirac expression
is the situation
= 0.371
(4-400)
for y ~ - 3 / 2
(391) where according
to
(398)
is applicable,
(171). Since F2(-3/2)
this requires n ~ (0.15 Vo)3/2
In particular,
for V ° ÷ 0, Eq.(391)
The exchange by considering
(4-401)
potential
infinitesimal
is regained
for all values
that corresponds
variations
to
(399)
of the density.
of n(~)
is obtained
They cause a
change of y by 6y, given by 6n =
_12z V3"2o ! FI (y) 8y
,
(4-402)
263
w i t h the c o n s e q u e n c e
6 n Eex(n)
= f(d~) 6y 2 ~ V ~
[F1(Y)] 2 (4-403)
VI/2 F = f (dr) 6n[- o I (Y)] This identifies the new e x c h a n g e p o t e n t i a l
Vex(n)
= -
[cf. Eq. (324)]
vl/2_ o rl (y(n)) (4-404)
= _ vl/2F o w h e r e we have,
for once,
I (2
v;3/2nl)
I inverted Eq. (400) formally. For d e n s i t i e s that
are large in the sense of
(401)
this reduces to the Dirac e x p r e s s i o n
(325), as it must do. So m u c h about an i m p r o v e d local t r e a t m e n t of the e x c h a n g e correction.
Before d i s c u s s i n g the c o r r e s p o n d i n g i m p r o v e m e n t of the indepen-
d e n t - p a r t i c l e energy EI-~N,
it is n e c e s s a r y to e s t a b l i s h a c r i t e r i o n for
our j u d g e m e n t w h e t h e r a modified, density and the e l e c t r o s t a t i c
local relation b e t w e e n the
(pseudo)
(pseudo)
p o t e n t i a l is more realistic at
the edge of the atom. A q u a n t i t y that is very sensitive to the dependence of the d e n s i t y upon the p o t e n t i a l is the electric p o l a r i z a b i l i t y of the atom. 50 It m e a s u r e s the e f f e c t i v e n e s s of a w e a k external elecP tric field E in i n d u c i n g a dipole m o m e n t
= [ (d~) ~ n(~)
(4-405)
of the charge d i s t r i b u t i o n inside the atom. We shall confine the present d i s c u s s i o n to the c i r c u m s t a n c e of no p e r m a n e n t electric dipole m o m e n t in the absence of external electric field. actual situation,
In isolated atoms,
this is the
since the d e n s i t y is s p h e r i c a l l y symmetric as long as
there are no external fields. The induced dipole m o m e n t is p r o p o r t i o n a l to the applied electric field,
if this field is s u f f i c i e n t l y weak,
and the factor expressing
this linear r e l a t i o n is the p o l a r i z a b i l i t y ~p,
= ~
~
(4-406)
P It is a v a i l a b l e e x p e r i m e n t a l l y from m e a s u r e m e n t s tric c o n s t a n t 6 of
(not too dense)
of atoms in the gas,
gases. W i t h n
the C l a u s i u s - M o s o t t i
of the static dielec-
b e i n g the d e n s i t y gas formula connects 6 to ~ accorP
264
ding to 6-I 6+2
-
4~ 3
The p o l a r i z a b i l i t y sphere cubed,
r P
n
gas
(4-407)
p
of a c o n d u c t i n g
sphere
is simply the radius
of the
so that we can i n t e r p r e t
H 1 1 3! P
as an "effective
(4-408)
polarization
radius"
of the atom.
Experimental
values
of r
(in atomic units) are listed in T a b l e 8 for t h o s e neutral atoms, P for w h i c h a is k n o w n somewhat accurately. 51 A l s o r e p o r t e d are HF pre45 p dictions, w h i c h o c c a s i o n a l l y agree v e r y well w i t h the e x p e r i m e n t a l data,
but d e v i a t e
substantially
for many
Z values
in the Table.
Let us now discuss which
the p h y s i c a l
a in the context of ES-type models, in P is c a l c u l a t e d from a pseudo d e n s i t y as in Eq.
density
(331), w h i c h m o r e g e n e r a l l y
n = p and
O is an
Table
reads
9/11 V2 (p) + CSBE 24----~ Uex
(algebraic)
4-8. E x p e r i m e n t a l
function
values
(4-409)
Of Ues + ~, as in Eq. (356)
for p o l a r i z a t i o n
radii
(EXP) , corn-
r P
pared w i t h HF predictions. HF
Z
EXP
HF
2
I .114
I .14
16
2.6
2.85
3
5.45
4.76
17
2.4
2.60
4
3.36
3.74
18
2.23
2.37
6.62
6.33
Z
EXP
5
2.71
2.85
19
6
2.28
2.27
20
5.54
6.11
7
I .96
I .89
21
5.37
5.65
8
I .75
I .70
36
2.56
2.76
9
I .56
I .53
37
6.84
6.77
10
I .39
I .38
38
5 7
6.72
11
5.45
5.01
54
3 00
3.31
4.2
4.57
55
7 4
7.61
13
3.9
4.20
56
6 5
7.70
14
3.3
3.58
8O
3 24
4.34
3.10
82
3 66
4.29
12
15
2.9
265
P(~) where
U
= P[Ues(~)+~],
and
es
~ are d e t e r m i n e d
I V2 4~ (Ues - Vext)
subject
(4-410)
,
to t h e u s u a l
(d~)p
by Poisson's
= p
boundary
(4-411)
'
conditions,
and
by the
normalization
= N
Now,
in p e r f o r m i n g
that
are
(4-412)
the CSBE
spherically
in
(409),
symmetric
(409),
when
the density
and do not
the Laplacian
of
result.
n c a n be e q u i v a l e n t l y
Thus,
equation
inserted
is m o d i f i e d
contribute
into
(405),
replaced
by
in
by t e r m s
(405).
integrates
Further,
to a n u l l
p in E q . ( 4 0 5 ) ,
= f ( d r ) r p (~) Considering field
a weak
of the
constant
nucleus,
as t h e
=
external
electric
field
E in a d d i t i o n
to t h e C o u l o m b
we have
Z
Vext
(4-413)
-~ -~
E.r
r
(4-414)
potential
in
(411).
Consequently,
U
consists
of t h e
es
= 0 term dratic,
U (0) and es in E:
U
= U (0) es
Quite
contributions
" (1) U(2) Ues' es'
"
.. w h i c h
are
linear,
qua-
...
+ U (I)
es
+ U (2)
es
analogously,
es
+
....
(4-415)
o n e has
P = P(O)
+ P(1)
+ P(2)
+ ...
( 4 - 4 1 6)
+
(4-417)
and = ~
In v i e w
(0)
+ ~
of Eq. (410),
p(°)
(1)
+ ~
(2)
...
we have
.. (0)
= P(Ues
+ ~
(0)
)
(4-418)
and
p(1)
-
9p ,. (0)
~u
L%s
es
*~
(0)
~. (1)
)X~%s
* ~
(1)
] ~
266
- p' (r) IU (I) ÷ (I) es (r) + E 1 Since
p(0)
already
integrates
(4-419)
to N, the s p a t i a l
i n t e g r a l of
p
(I) m u s t
(I) = 0, so that v a n i s h . This i m p l i e s ~ p(1) (~) = p , (r) U (I) (~) es
(4-420) +
If w e m e a s u r e
U (I) in m u l t i p l e s eS
U (I) (~) = - E . ~ v(r) es where,
÷
of E.r,
,
(4-421)
of course,
v(r) ÷ I
t h e n the i n d u c e d
for
r+~
,
(4-422)
dipole moment
is, to f i r s t o r d e r in the a p p l i e d
field,
g i v e n by
= /(d~)~ where
p(1) (~) = ~ • ~(d~) ~ [ - p '
~ ~ can be e q u i v a l e n t l y
replaced
(r)]v(r)
,
(4-423)
by ~I r 2 ++ I, so that we find
4K fdr r4[-p' (r)]v(r) ~p = 7 o B e c a u s e of the l a r g e - r b e h a v i o r must tend toward
and gives ensured
of v(r),
zero f a s t e r t h a n
p' (r) is n e g a t i v e ,
(4-424)
displayed
I/r 5 as r ÷ ~ .
provided
t h a t the f u n c t i o n
of
a smaller density
for l a r g e r v a l u e s
of U
to be p o s i t i v e , The r a d i a l
in the P o i s s o n
is d e t e r m i n e d
(411), w h i c h
I ?2 . (I) + + 4~ (Ues + E.r) and p roduce,
(422),
p' (r)
(410) es
that
is r e a s o n a b l e
(~) + ~. T h e n ~
p
is
as it m u s t be.
f u n c t i o n v(r)
equation
in
Further observe
=
after inserting
p(1)
by t h e first o r d e r terms
are
,
Eqs. (420)
(4-425) and
(421), the d i f f e r e n t i a l
equa-
tion
d2 [d-r-f + r The corresponding
+ 4Kp' (r)]v(r) integral
equation
= 0
(4-426)
267
IIdr r3r< r~ eo
v(r) incorporates
= I - ~ o
the boundary
[-
condition
(4-427)
p' (r')]v(r')
(422). Here r< and r> stand for the
smaller and the larger one of r and r', respectively.
Upon using the
identity r< r' r--f = ~-/ + ( ~ 2
r' ~)q
(r'-r)
(4-428)
>
in conjunction with Eq. (424), this integral
equation appears as
oo
v(r)
= 1 - ~P +r,
fdr'
r'((~)3-1)
[ - ~p'
(r')]v(r
')
(4-429)
.r Since
tends
p' (r) approaches
to
zero. faster v(r)
or, w i t h
than
1 / r 3, w i t h
~ I _ r 13 3
(421) and U(1) as
zero faster than I/r 5, the remaining
consequence
for large r
(4-430)
(406),
--- - ~ ' ~
w h i c h correctly
the
integral
+ ~'~--T
exhibits
(4-431)
•
the dipole potential.
At short distances,
it
is fitting to use the identity r < r --f r = r'2
(~
r' 1 ~ q(r-r')
2
(4-432)
>
instead of
(428), in
v (r)
=
v°
(427), which gives r + f dr o
,
r' 31 [---p ~ , (r')]v(r') r' (I-(-~-)
(4-433)
where co
V O : V(0)
: I - Sdr r [ - ~ p '
(r)]v(r)
(4-434)
o In particular, density
if the connection
U
es
and the
p has the TF form for r÷0, then 4 .2z.I/2
- --p' (r) ~ ~-~(~--) which,
(410) between the potential
inserted
into
for
r÷ 0
,
(433), implies the small-r form
(4-435)
268
16 ,~. 3 I/2 = Vo[1 + 2--~IzT.r ) + ... ]
v(r)
After picking
an a r b i t r a r y
for Vo,
one can t h e n use this
merical
integration
(4-436)
(positive)
initial
behavior
of t h e d i f f e r e n t i a l
t i o n is l i n e a r in v(r),
the s o l u t i o n v(r)
w h i c h has the d e f i n i t e
unity
Vo = ~o(I
Q(r) d ~(r) r~-~
w e e m p l o y the s c a l e =
to e x t r a c t means
~
- r--~3 )
3~ o p v-- "r ~ o
=
lim r÷~
ap
"}
for r + ~. I n a s m u c h
to start the nu-
(426).
S i n c e this equa-
thus o b t a i n e d w i l l be a m u l t i asymptotic
form
(430),
as
for r ÷ ~
invariant
of v(r)
equation
p l e of the a c t u a l v(r), approaching
v a l u e 9 ° as a guess
(4-437)
,
expression
{r 3 [I + 3 ~ ( r ) / r ] - I ~ d~ (r)/dr S
the p o l a r i z a b i l i t y .
(4-438)
In p r a c t i c e ,
that we h a v e to p i c k a d i s t a n c e
this l i m i t i n g p r o c e s s
r so large that
simply
p' (r) is e s s e n t i -
ally zero. Now, what
requirements
t h a t Ues+~
tends
Ues
while
a f t e r this p r e p a r a t o r y
p tends
e m e r g e on the
+ ~ ÷ ~ ~(Z,Z)
to zero.
which condition
(Ues+~)-dependence
of
let us see
p. F i r s t o b s e r v e
to ~ as r ÷ ~ ,
PIUes(~)+~l=
where
general discussion,
This
0
E ~o
'
(4-439)
implies
for
is s a t i s f i e d
U e s ( ~ ) + ~ ~ ~o
'
(4-440)
b o t h by the TF and by the ES relation,
~o=0 and ~o=Uo , r e s p e c t i v e l y .
As a c o n s e q u e n c e
of
(440),
atomic
ions h a v e an edge at r=r O w i t h
Ues(ro) ÷ ~(Z,N)
beyond which i n f i n i ty,
the d e n s i t y
Z-N + ~(Z,N) ro
equals
as is the s i t u a t i o n
v o l u m e as well, lization
=
as r e a l i z e d
of the TF r e l a t i o n ,
zero,
= ~o
whereas
neutral
in t h e TF model,
in t h e ES model. power
laws
'
(4-441)
atom m a y e x t e n d to
or be l i m i t e d to a f i n i t e
Then consider,
as a g e n e r a -
p(Ues + ()
~ ~ [ ~ o - ( U e s + C)]
for with
constants
equation
Ues
+
269 v+l (4-442)
( ~ ~o
~ and v. T h i s p r o d u c e s ,
in c o n j u n c t i o n
w i t h the P o i s s o n
(411), d 2
dr 2 (- r U e s )
as t h e d i f f e r e n t i a l the n e u t r a l - a t o m
= 4%# r-V( - r U e s )v+1
equation
governing
pseudo-potential
(4-443)
the l a r g e - r
Ues(r).
asymptotic
form of
For 0 < v < 2, this a s y m p t o t i c
f o m a is a l g e b r a i c ,
~1-v/2>1/Vr-2/v
Ues (r) --+ - k--~--
,
(4-444)
and for v = 0 it is e x p o n e n t i a l , Uco
Ues(r)
--+ ---~- e x p ( - / ~
w i t h an u n d e t e r m i n e d Pt (r) e m e r g e
constant
r)
U~.The
(4-445)
corresponding
asymptotic
forms
from
-p' (r)
for t h e s e n e u t r a l
~ #(v+1) [-Ues(r)]V
atoms,
(4-446)
with the outcome
I(v+l (1-v/2) 1 v2 rZ
~ ~-f ,
for 0 < v < 2 ,
-p' (r) --+
(4-447) = const.
Such cause
p' (r)'s r e s u l t p' (r) m u s t
n i t e ~p. (426)
(438), We thus
correct
in i n f i n i t e
,
for
v = 0
polarizabilities
t e n d to zero f a s t e r t h a n
This we o b s e r v e d
and
of
at Eq. (424);
is t h e s u b j e c t conclude,
potential-dependence
r a n g e v < 0 is i m m e d i a t e l y
for n e u t r a l
I/r 5 in o r d e r
a different
of P r o b l e m
argument,
of t h e d e n s i ty,
at least
of, b e c a u s e
a growth
of -p': (r) at l a r g e d i s t a n c e s .
potential
Ues i t s e l f d e c r e a s e s
behavior.
The i n f e r e n c e
is t h e r e f o r e ,
that
based upon
c a n n o t be the
for 0 ~ v < 2. The
t h e r e Eq. (446)
On the o t h e r hand,
slower than
bea fi-
11.
t h a t t h e p o w e r - l a w form (442)
disposed
atoms,
to p r o d u c e
I/r,
certainly
p must
approach
implies
for v ~ 2 the an u n r e a l i s t i c zero f a s t e r
270
t h a n any p o w e r of ~o-(Ues+~) t i v e values, [or
(363)]
of course).
ensures
is too rapid.
as this q u a n t i t y
In the ES model,
this.
But h e r e t h e t r a n s i t i o n
In p' (r) w e m e e t
in the d i f f e r e n t i a l
of d v ( r ) / d r w h e r e ly,
for n e u t r a l
atom,
(Ues +~))
equation
(426)
Ues + ~ = U o. For ions,
atoms,
(from posi-
through
in
(356)
the a t o m i c
edge
a term
-p' (r) = ... + Po 8 < U o which
tends to zero
the s t e p f u n c t i o n
however,
'
(4-448)
gives
rise to a d i s c o n t i n u i t y
this is at the edge at r=r ° on-
Ues + ~ = U ° in the e n t i r e
that is for all r ~ r o. No s e n s i b l e
interpretation
e x t e r i o r of the can be g i v e n to
s u c h a p' (r). The o n l y w a y out is to i n s i s t that so for n e u t r a l
ES atoms.
Then,
in v i e w of Ues ~
the edge of a n e u t r a l atom, the D e l t a f u n c t i o n 3 = 0, so that ap = ro, or w i t h (408) and (369): 2.2 x z 0"I ~ r p which
- surprisingly
order-of-magnitude
= r
~ 3.1
x z 0"08
o
enough
wise.
p': (r) = 0 for r > r al2 o (ro-r) just i n s i d e of
- roughly
Of course,
in
(448)
implies
v ( r = r o)
,
(4-449)
reproduces
the n u m b e r s
w e are not g o i n g to t a k e
of T a b l e (449)
8
seri-
ously. In s e a r c h creasing,
for a d e n s i t y - p o t e n t i a l
at the edge,
for instance, tion present
more rapidly
for t h e TF model) in th e ES model,
on the v i c i n i t y
exchange
insight
for an i m p r o v e d
about
exchange
- SCd l in the e n e r g y
t h a n the p o w e r law
one n a t u r a l l y
(characteristic, as the step func-
(219). W i t h the em-
an i m p r o v e d
a b o v e by r e p l a c i n g
4--~ (3~2p)
functional
edge,
t h a t is de-
the s m o o t h t r a n s i t i o n
treatment
of the
the TF r e l a t i o n by
v e r s i o n of the ES model,
and p e r f o r m
I
recalls
in Eqs. (210) or
of the a t o m i c
e n e r g y wa s a c h i e v e d
In s t r i v i n g
(410)
but not q u i t e as s u d d e n
a s s o c i a t e d w i t h the A i r y f u n c t i o n s phasis
relation
(397).
w e shall use this
the r e p l a c e m e n t
4/3 ÷ -
v2
o y (d~) ~-~
(342), w h e r e y(~)
cr1(ylj2
is d e t e r m i n e d
14-450)
by (4-451)
This
is, of course,
s i m p l y the e x c h a n g e
t h e - by now f a m i l i a r the p s e u d o d e n s i t y
(399),
supplied with
as a f u n c t i o n a l
of
p(~).
Concerning on t h e r i g h t - h a n d
e n e r g y of
- f a c t o r of 11/9 and t r e a t e d
the c o r r e s p o n d i n g
s i d e of
(342),
modification
the o b s e r v a t i o n
of
of the first i n t e g r a l
271
i
I v5/2 F3(2(u+~)/Vo) + o
15Z2 [-2 (U+~) ]
4~
for invites
V
+ 0
o
5/2
,
(4-452)
I V5/2 _ o )~3(2(U+{)/Vo)
(4-453)
,
the replacement (d~) ( - ~ )
[-2 (U+~) ]
5/2
÷ f(d~) ( - ~
with the consequence P = ~I V3/2 o
Although
this
looks quite natural,
does not approach F2(2~/Vo)>0.
F2(2 (U+~)/Vo)
(4-454) it is simply not good,
zero at large distances,
Instead
of
where U + 0 and F2(...)
(453), I would therefore
( -
15z2
because
p(~) +
like to propose
vs/2EF3(9 9/FI
+ f
(4-455) with the understanding -(U+~)
(455)
allowed
(4-456)
regime where
-(U+~)>0,
is set equal to zero for -(U+~) 0(.
[Re-
(50) are exact for a linear
potential.] 4-2. Find recurrence
relations
for the polynomials
in connection with Eqs. (158) and
(159).
4-3. Show that Fm(Y) obeys the differential
d3
[~and demonstrate (161),
and
4
+ 4m-2 Fm(Y)]
that are mentioned
= 0
equation
l
that it is consistent with the asymptotic
forms
166),
(170).
4-4. For which complex values of the integration variable x in Eq. (140) is the phase of the integrand path of integration
to contours
luation of the integral (163) and
stationary?
for y ~I
Use this insight to deform the
appropriate
for a stationary
and y
of the wave functions
: ~,Ri,E(r')
Y%m(@l~ ')
with the aid of the spherical
harmonics
then implies
a differential
equation
can be separated,
(5-39)
, Yim" The Schr~dinger
for R(r'),
I d2 £(Z+I)]~ - 2 dr--r 7-T~ V(r') + 2r-~-~TT--]~,Z,E (r') = E Ri, E (r') which determines
the energy eigenvalues
This differential the Schr~dinger operator
equation cannot,
The identification
,
(5-40)
E. as it stands,
equation associated with a one-dimensional
HiD(x,Px).
equation
be read as Hamilton
of r' with x and -id/dr'
with Px
does n o t work, because t h e range of r ' i s r e s t r i c t e d t o p o s i t i v e v a l u e s , a s i s i l l u s t r a t e d by t h e n o r m a l i z a t i o n i n t e g r a l
T h e r e f o r e , w e must, d u r i n g an i n t e r m e d i a t e s t a g e , i n t r o d u c e a new v a r i a b l e , x ' , t h a t ranges from -- t o m . The p h y s i c a l d i s t a n c e r ' i s now exp r e s s e d a s a f u n c t i o n of t h i s a u x i l i a r y v a r i a b l e : r ' = r 1 ( x ' ) . I t i s cert a i n l y r e a s o n a b l e t o r e s t r i c t o u r s e l v e s do such t r a n s f o r m a t i o n s f o r which d r r / d x ' > 0 and r g ( x ' + m ) = O r r 1 ( x ' + a ) = - . I n o t h e r words: a s x ' grows from -m t o m , r ' ( x r ) i n c r e a s e s monotonically from 0 t o malization i n t e g r a l (41) i s then
-.
The nor-
i s going t o be t h e wave f u n c t i o n i n t h e new v a r i a b l e x ' . The d i f f e r e n t i a l equation f o r X (x' ) i s
Now we can i d e n t i f y x ' + x and - i d / d x t + p x a s c o o r d i n a t e and momentum o f an e f f e c t i v e one-dimensional d e s c r i p t i o n . The a s s o c i a t e d Hamilton operat o r is '(''I)
where s ( x ) i s d e f i n e d by S
(x) =
XI
]'I2
d r ( x /dx
.
The i d e n t i t y
- -s 2 ( x )
2 s 2 ( x ) -[I I r ( x ) px +
+4s
d 2 s(x) ] r
(6-45
310
simplifies
(45) to
1[dr (x) ]-I HID(x'Px)
[dr (x) ]-I + V(r(x)]
= 2L d-~-~--]
px2 L dx
] (5-48)
(~+i/2) 2 + s 3 (x) d 2 s (x) 2r 2 (x) 2r 2 (x) dx 2
+
If w e i n s e r t this HID into the TF q u a n t i z a t i o n
rule
the r a d i a l m o t i o n
according
is
(approximately)
I dx'dpl nr + 2 = ~ 2u
quantized
(35), we
find that
to
HID (x' ,px' )]
~[E£,nr-
(5-49) = ~I fdx' dx'dr Ir / 2 r 2 ( E % , n r - V ( r ) ] •
- (Z+I/2) 2-s
3 d2s d--~2
,
• dr or s i nce dx' ~ - - ~ = d r , n r + I/2 : ~If ~
We c o m p a r e this w i t h w h a t we w o u l d as a o n e - d i m e n s i o n a l replaced
by
(5-5o)
/ 2 r 2 [E]~,nr_ v (r))-(£+I/z, .< , - --s ~2-{--
Schr6dinger
have obtained
equation
by w r o n g l y
and o b s e r v e
that
taking Z(%+I)
(I+I/2) 2 + s 3 d 2 s / d x '2. The l a t t e r t e r m is d i m e n s i o n l e s s
construction,
inasmuch
as
(46) shows t h a t s 2 has the d i m e n s i o n
Consequently,
as a f u n c t i o n of r this t e r m has to be a c o n s t a n t
e l s e w i s e we w o u l d be f o r c e d to i n t r o d u c e functions
(40) is
s(x'),
for w h i c h
an a r t i f i c i a l
s 3 d 2 s / d x '2 is i n d e p e n d e n t
scale of x',
by
of x. because
for r. The are g i v e n
by
slx,l = [So÷SI Ix,-Xol2]I12 where
,
(5-51)
s o > 0, s I > = 0, and x o are c o n s t a n t s .
x' + - ~ ; ~ ,
so that log r ' ÷ ± ~
for x' ÷ ±~.
S dx log r,x
Now recall
In terms
that r' ÷ 0; ~
of s(x')
for
this m e a n s
dX[s x ] 2
--co
--co
(5-52)
co
dx' =
-~f So+S1(~=Xo )z
The l a t t e r i n t e g r a l , equals
zero.
however,
T h e n s(x')
r(x)
results
= const.,
= rO ex/so
,
and
rO > 0 .
in a f i n i t e n u m b e r u n l e s s
sI
(46) i m p l i e s
(5-53)
311 The term s 3 d2s/dx 2 in HID of The particular both equal to unity.
= ~
1211--+V(r),
+(£+~)
and the resulting TF q u a n t i z a t i o n
nr
accordingly.
r = e
x
,
(5-54)
rule
+I 1 [ (E~'nr ~ = far 2 -V(r)
( £ + 1 / 2 ) 2 ]I/2 ~ j
(5-55)
has the familiar WKB form. This is how we shall relate Ei, n the q u a n t u m numbers
and the potential,
barrier"
Kramers 6 observed, functions however,
is quite different
from Kramers'
argument.
it is in is that
it is shown above that this
of ~(~+I) ÷ (Z+I/2) 2. But this is not
the only reason for being so explicit. the trace of qIE-HiD(X,Px)l (34) by,
one wants eventually
for instance,
For that purpose,
It is clear that in evaluating
introducing
to go beyond the TF the relevant Airy ave-
one must deal with the HID given in
thing of this has been worked out as yet,
(54). No-
and it would certainly be in-
t e r e s t i n g to find the resulting modifications (37) and
Instead,
the only reasonable m a p p i n g of 0 ~ r < ~ to -~ < x < ~.
So much about the justification
approximation
The reasoning given above,
in 1937. The main difference
for r=e x right away, whereas
is, in some sense,
since
in 1926, that the correct behavior of the WKB wave-
near r=0 is not obtained otherwise.
Langer settles
like to offer "centrifugal
(£+I/2)2/(2r 2) is, of course, w e l l - k n o w n
the spirit of Langer's 7 d e r i v a t i o n
rages.
I would
The necessity of replacing the naively expected I(£+I)/(2r 2) by
to both
for eventual use in ~q. (29).
Before p r o c e e d i n g with this development, a remark.
we can put
Then
11[ p2
HID(x,Px)
(48) vanishes
choice for r O and s o is irrelevant,
of the q u a n t i z a t i o n
rules
(55). For our present objective of studying Eos c it suffices
to employ Eq. (55). For large q u a n t u m numbers the TF
(or WKB)
~ and n q u a n t i z a t i o n according to r (55) is highly accurate by construction. What can
rule
we say about the small q u a n t u m numbers bound electrons? tential produces
(15) w i t h a
(small)
(see Problem
nr
+
1 ~ =
is basically a Coulombic
additive constant
E . Inserted into o
po(55) it
4) Z
~2 (Eo-Ei,nr) or
associated with the strongly
For these the potential
- (Z + I/2)
(5-56)
312
Z 2
Ei,nr This
is t h e
exact
pal q u a n t u m is also
cussed EZ, n
for s m a l l
strongly
in C h a p t e r of
about
Fourier
Three (55)
(29) r, t h e S c o t t More
quantum
dependence
in this
the q u a n t u m
asked
energy
the
fore
rewrite
(29)
First, gy E I to
N(E),
N(E)
because
princi(55)
the
correct
treatment
In p a r t i c u l a r ,
spectral
is t h e r e
dis-
when
the
s u m of E I in Eq.
without
further
ado.
Chapter.
expression
numbers
for E I ( V + ~ ) ,
Z and n r w i t h
Ei n
.
for a p r a c t i c a l
I
calculatlon
Eq. (29),
which
approximated r
by
is (55),
.
as it stands.
We t h e r e -
in a few steps. we r e c a l l
the
= tr
count
that
q(E-H)
[see Eqs. (209)
information
we can
of s t a t e s
and
about
shift
with
the
energy
emphasis less
than
from the
ener-
E,
,
(5-58
(2-20),
E I = tr(H+~)~(-H-~)
so that
gives
special
for.
,
suited
(55)
the
to e v a l u a t e
to the
The new
Since
electrons,
is no l o n g e r
are u s e d
later
formulation.
is n o t w e l l
o n the
numbers.
pay-off.
bound
correction
this
a sum o v e r
its w e l l - k n o w n
is an a d d i t i o n a l
for the
values
with
(5-57)
(nr+I/2) + (~+I/2) = n r +I+I . Thus , TF q u a n t i z a t i o n
good
Here
2(nr+Z+1)2
answer
number
very
energies
= E°
= -
N(E)
and n o t e
that
f ~ d E N(E)
,
is i m m e d i a t e l y
N(E)=N(~=-E)]
(5-59 turned
into
knowledge
of
E I -
Then, grate
over
I and
instead
v, g i v e n
i - i + I/2
and
of
,
summing
over
£ and n r, we
equivalently
inte-
by
v - n
r
+ I/2
,
(5-60
introduce El, v E E i , n r
which,
according
to
I = ~ ;~
After
the
,
(55),
(5-61) is r e l a t e d
[2r 2 (Ex,
introduction
to
-V(r))-12 ]
of D e l t a
I and
v by
I/2
functions
(5-62)
to s e l e c t
the discrete
quan-
313 rum numbers,
we arrive at
N(E)
= 2 ~
i,nr= 0
=
(21+1)q(
co S [dl t
4
o
E-E~,n r
)
1 6(Z+2-1)
(5-63)
Z=-~ oo
x
ofdv
n ~= 6 (nr
+ 1 -v)
~ (E-EI, v )
r Now, the twofold
application of the Poisson identity I
6 ( 1 + ~ ' - I) :
f
~=-~
(-I
6 ( n r + ~ _1
, (5-64)
v) =
(-I) j ei2~j v
n =-~ r
q =-~
produces
oo
N(E)
)ke i2~kl
k=-,~'
= 4>
oo
co
So far we have been reading Eq. (62) as implicitly defining more useful.
El, ~ for given
It understands
VE (I) = ~I5 ~
values
line of degeneracy, the c o r r e s p o n d i n g
(5-66)
in a l,v-diagram.
here because such lines connect
refer to
these states
(l,v)'s on the same
(orbital)
are degenerate
the general spin and angular m o m e n t u m multiplicity).
This is certainly
among the lines of d e g e n e r a c y that are straight,
so occur
for bent ones. The domain of integration
in
(65) selects,
rE(1).
therefore,
in
state with
(in addition to
possible
the line of d e g e n e r a c y
The
(l,v)
E. If it should happen that several
q u a n t u m numbers
then there is more than one
energy;
(55) before]
another view is
,
a "line of degeneracy"
belonging to the same energy (integer)
(5-65)
v as a function of I and E, V=VE(1),
is appropriate
(i,n r) pairs of
[and likewise
I and v. However,
[2r2 IE_V(r)I -I 2 ] I/2
which for each E defines term d e g e n e r a c y
.
(-I) k+j 5dlle i2~kl ~dve 12~jv q(E-EI, v) k ,j =-~ o o
but it can al-
(65) consists of all l,v below
For a fixed value of I, the step function
V~VE(;,),
q(E-El, v ) = qIVE(1)-v!
(5-67)
314 On the other hand, if I exceeds IE, defined by
1/2 X E - Max r
[2r2(E-V(r))]
(5-68)
then the argument of the square root in (66) is negative for all r, implying (5-69)
VE(~ ~ ~E) = 0 Consequently, we now have
N(E)
=
4~
,.!_l)k+ j ftE d~
k,j=-~ which, inserted into
~ e i2~k~ rE(X) fdv e i2Ejv o
o
•
(5-70)
(59), enables one to compute E I.
Isolatin~ the TF contribution. The j=k=0 term in (70) gives the result of integrating of I and v, without reference to the Delta functions that enforce the integral nature of I-I/2 and v-I/2. Therefore, we expect it to reproduce the TF version of N(E), (N(E))TF = 5(d~) ~ -I[ 2
(E-V) ]3/2
Indeed• this happens when rE(1) of (70)
(5-71)
(66) is put into the j=k=0 term of
;~E
:
(N(E))j=k=0
= 45 d l l rE(1) o 4 dr ~E I/2 = ~f-~- f dlI[2r2(E-V)-I 2] o 4 dr 3/2 = --3E~-r- [2r 2(E-V)]
which, in view of
(d~)= 4Er2dr, agrees with
(5-72)
(71).
This observation implies the decomposition of N(E) into its TF part and a supplement that represents quantum corrections,
N(E)
= (N(E))T F + Nqu(E)
,
(5-73)
where the term "quantum corrections" is used with the meaning given to it in the paragraph after Eq. (I-43). In the present context we approximate Nqu(E) by the right-hand side of Eq. (70) without the j=k=0 term. Quite analogously, E I is split into the TF expression plus a quantum cor-
315
rection 8 (not to be confused with AquF I of Chapter Four). We shall see that this c o r r e c t i o n its p e r t u r b a t i v e
is usually small compared to the TF part,
evaluation.
As a p r e p a r a t i o n we first collect
tion about the lines of d e g e n e r a c y in
allowing informa-
VE(I) w h i c h are the basic ingredients
(70).
Lines of degeneracy.
The m a x i m u m of Eq. (68) is located
at the distance
rE,
IE2
= 2r~[E,V(r E)]
This m a x i m u m property
(5-74)
implies that r E obeys
V(rE ) + d~Ed IrEV(rE)I
= 2E
,
(5-75)
which has the consequence 2 d rE[rEV
implication
is
= 2r2E
(5-77)
This shows that if we were given culate r E and then employ
for w h i c h
/
IE for some range of E, we could cal-
(74) to find V(r)
range of r E . As an illustration
1E = Z
(5-76)
~
,
hereof,
E < Eo
for r in the c o r r e s p o n d i n g
consider
,
(5-78)
(77) gives
rE
_ Z/2
Eo- E
rE > 0
'
(5-79)
so that E =
_
Z/2 r E + Eo '
2 IE =
Z rE
(5-80)
'
and, using Eq. (74), 2 IE V(r E) = E - 2r~
=
-
Z/rE+E
0
,
(5-81)
316
the Coulombic potential
rE(1) with l E from
(15) emerges.
= z//~T~-~-
I = IE-I
as required by
= 0
for
,
to
(5-82) by
I > IE
(5-83)
(69).
It is an important
lesson that IE, in its dependence
a lot of information
of l E is possibly
limited
about the potential
extend to infinity.
V(r).
(short-range potentials
with very large angular momenta), radius,
Eq. (56) is equivalent
(78), which is to be supplemented
v E(1)
contains
Indeed,
do not bind states
the c o r r e s p o n d i n g
Then I E determines
on E,
Since the range
range of r E need not
V(r) within a sphere,
of finite
around r=O. For values of I close to IE, the domain of integration
is a small neighborhood
of r E . There one can approximate
of the square root by a quadratic 2r2(E-V(r))-
12
polynomial
IE2 - I 2 - ~1E ( 2 r-rE)
in
(66)
the argument
in r-rE,
2
,
(5-84)
with 2
OJE
=
d 2
~
[2r 2[v(r)-E]]
r=r E
(5-85) dz d = 2 [r E ~ + dr E
1 ] [rEV(rE) ] rE
The relations 2
~°E indicate that,
4rE
drE/dE again,
d (I~)/dE = 8
(5-86)
d 2 (I~) ~dE 2
knowledge of 1 E is sufficient.
The integral that results
I f-~ dr ~E (I) ~ E
[I~ -I 2
from inserting
I 2 - y wE(r-r E
(84) into
2] I/2
has the structure of the one producing the VE(1)
(66), (5-87)
associated with the
Coulomb potential:
I89 -~ The evaluation
[2r2 (E+Z/r)-12] I/2 of
(87) is, therefore,
7.
(5-88)
¢_-y~ immediate.
We find,
for I S IE,
317
IE
(5-89)
rE(1) ~ / ~ ~ErE (IE-I) In the limit I ÷ I E this is exact,
v~ z
- ~
so that
IE
~E(1)I
Please note that,
~ErE
via Eqs. (77) and
mines the I d e p e n d e n c e
of rE(1)
Higher derivatives thod
(5-90)
= / ~ - 1=l E
(86), the E dependence
near I=I E-
of rE(1)
at I=I E can be calculated
(described in the Appendix to Ref.9)
ximation
(84). For example, V"
E-
-
of I E deter-
which improves
by a me-
upon the appro-
the second derivative
~ 2
~I z rE(1) I i=I E
(5-91)
equals 3, rE" = ~AE[I-(v~)2 + ~ E V 3
152 3 _ 8 _ V 3 +~v4l ,
(5-92)
)k[r V(rE) ]
(5-93)
where the coefficients k k! depend on
E,
(
and could be expressed
The recognition
that,
I d2 r dr 2 (rV) = V2V
in terms of E derivatives
for a spherically
of 1 E.
symmetric potential,
, (5-94)
I
(~-r)(rv)
= ~.~v
,
enables one to rewrite the right-hand ~
= [I + r 2 v 2 V ~.~V
I
]
side of
(90), the outcome being
-112
(5-95)
r=r E + +
The force -~V is towards the center at r=0,
so that r'VV is positive.
On the other hand, the Laplacian of an atomic potential the density negative.
(-V2V = 4~n, in the simplest
As a consequence,
are less than one,
v~ =
approximation)
the contents
is related to
and is therefore
of the square brackets
in
(95)
implying
~VE(~) ~I
I=IE > I "
(5-96)
318 The limit of unity is approached
for large binding energies
ging to strongly bound electrons
for which V ~ -Z/r. Note that
not true for any potential the oscillator potential Our interest,
however,
r + 0 and vanish
because ?2V can be positive.
V=(K/2)r 2 where v~=I/2
is in atomic potentials
for r + ~ .
The derivation
For these, of
-E
belon(96) is
An example is
for all energies which approach
E > 0.
-Z/r as
Eq. (96) holds.
(90) can also be done by using the general
expression
VE(l )
1 = "~5"~
- %'-'~"
t
together with the approximation (97) to find
~E/~l
(84)
(see Problem 5). Let us now employ
arises.
We isolate that part of the integral
ready is a good approximation.
7 dx
x2/(2z) /~Y9:7 ~ ,~+o
= I
(5-99
holds for all E,except contribution
This is the situation
E=0, where there is the possibili-
from the upper limit of the integral.
if V ~ r -m for r ÷ = ,
with m > 2. Potentials
m < 2 are long-range
potentials,
effective
of an ion where V = -(Z-N)/r
potential potential
of which the important
Vo(1)
for large r. In such a
is infinitely distant
On the other hand,
with
example is the
there is no limit to the quantum numbers,
entire line of d e g e n e r a c y in the l,v-diagram.
al-
2Zr = 12 (1+x 2) yields
2 '~E = ~ o ~
long-range
by intro-
of I and so small that V ~ - Z / r
1 Sr~-
i=0
ty of an additional
a fi-
At this stage we have
8rE(l)
Now the substitution
for in this
reaches down to r=0 from which neighborhood
ducing an upper limit r, independent
This statement
(5-97)
for l ÷ 0. No, the answer is not zero,
limit the integration nite contribution
,
[2r2(E_V(r))_12]I/2
and the
from the origin
for m > 2, we have a short-range
potential with a limit to the possible quantum numbers. Again we isolate this upper part of the integral, l-independent
lower limit ~,
(C> 0 , m > 2):
I f
(2c/12) I / (m-2) dr
now by a
large enough to justify V(r) ~ -c/r TM
I/r I/2 [2cr2-m-12 ]
1 [I÷0
=
319
•
oo
I 2 dx _ I m-2 ~ f 1+x 2 m-2
(5-100) '
o
where the s u b s t i t u t i o n
2cr 2 - m = 12 (1+x 2) has been made.
tials w i t h V ( r ÷ ~ ) ~ - I / r
'rE
~VE(l) 2t
=
~ I for E < 0 , = L 1 m-1 1 + m - ~ = m-----2>1 f o r
I t=0
For given I, rE(1) Therefore,
Thus,
for poten-
m, m > 2 we have
increases
(5-101)
E=O.
continuously with growing E.
the sudden increase of the initial slope at E=0 for short-
range potentials E approaches
must he accompanied
zero.
as
This is confirmed by the evaluation of I 5d r r = ~ [2r2(E_V)_1211/2
~--~ rE(1) for I=0, performed
~E(O) -
by a rapid change of ~E(I=0)
analogously
~
(5-102)
to Eq. (I00), w h i c h produces
v E (0) (5-103)
- C I/m 72--{
(I/m-I/2)! {_~i (1/m-1) !
(2+m) / (2m)
for E ~ 0. This has the consequence
VE(0)
c I/m ~ ~o (0) + /-~
for E ~ 0. Please note that, (103)
is positive,
nent of
(-E) in
whereas
(104)
(I/m-3/2)! (m-2)/(2m) (I/m-I) ! (-E)
(5-104)
because m > 2, the numerical it is negative
in
coefficient
in
(104), and that the expo-
is a positive number less than !/2.
Indeed,
rE(0)
grows rapidly as E + 0. Equations rE(0)
(103)
and
(104)
are illustrations
of the fact that
for E ~ 0 tests the outer reaches of the potential.
E-dependence
of rE(0)
is not converted
easily as the E-dependence
about V(r)
the as
of 1 E [recall the remark after Eq. (83)].
In Eq. (I03) we introduced w i t h respect to E. This notational the sequel.
into knowledge
However,
With this convention,
a dot symbolizing the derivative simplification
Eqs. (77) and
will prove useful in
(86) can be w r i t t e n
as
2 rE
=
2 =
~E
IEI E
,
"
4rE/rE
~
2"" "
= 8~E~E/(~E~E+IE
(5-105) )
'
320 and
(90) implies
-
4(v~)
-
I
(5-106)
"2
IE D i f f e r e n t i a t i o n of
(85) gives
d 2 = rE
d
2
4
a~EWE = - -
d
rE
2
~E
(5-107) ~[
=
2 d3 d2 rE ~ + 2rE dr~
d +r~][rEV(rE) ] dr E
of w h i c h a m o r e useful form is 2
1 d 2 ~( 4 dE ~E = 2 E\3+rE d~E>V2V(rE ) - (rE)2
(5-1 08)
This w i l l later be needed in ° I
tEVE
-
1 -2(v.)2E
4~vEjl' ,,2
-
d
~-E
~
2 g
(5-109)
'
the d e r i v a t i o n of w h i c h I leave to the reader.
C l a s s i c a l orbits.
Some of the equations of the p r e c e d i n g section pos-
sess an e l e m e n t a r y s i g n i f i c a n c e w h e n i n t e r p r e t e d as r e f e r r i n g to classical orbits of a p a r t i c l e in the s p h e r i c a l l y symmetric, tential V(r)°
For instance,
a t t r a c t i v e po-
the v e l o c i t y in a c i r c u l a r orbit of energy
E and radius r E is d e t e r m i n e d by the kinetic energy I
v
2
= E - V(rE)
,
(5 -110)
w h i c h c o m b i n e d w i t h the statement that the g r a d i e n t of V must supply the n e c e s s a r y centripal force, d - dr--~ V(rE)
reproduces
(110)
v2 rE
(5-111)
(75). In other words:
dius of the classical of
=
into
(74) identifies
this c i r c u l a r orbit.
rE, as o b t a i n e d from
circular orbit w i t h energy
(75), is the ra-
E. Further,
insertion
IE as the classical angular m o m e n t u m in
It is, indeed, well known that of all orbits to a
certain energy the c i r c u l a r one has the maximal angular momentum. 10 If the angular m o m e n t u m
I is less than
IE' the classical or-
bit is of the kind sketched in Fig.8. The radial m o t i o n is an o s c i l l a -
321
tion
between
lowed
domain.
(97),
and
changes
two d i s t a n c e s These
(102).
lar p e r i o d
are t h e
define
limits
equation
l
r as a f u n c t i o n
Fig.5-8: Sketch tric, attractive
of t h e
r(~+~)
corresponding
azimuth
= r(#),
with
responding
to
potential for the
second
in
i i
trajectory
in a spherically
symme-
r
(5-113) /2r2(E-V)-I 2
establishes
lines
;
(5-114)
of d e g e n e r a c y
orbit,
insight
and t h e
the d i f f e r e n c e
angu-
r2
~--7 ~ E (X)
classical
the
r = r I ÷ r = r2:
rI
= -
of the
This
in Eqs. (66), (the s i g n
In p a r t i c u l a r ,
is t w i c e
/
classical
2d~
(97)
¢.
/
rI
slope
al-
(5-112)
\\
of t y p i c a l potential.
f
the
is
;
~ of t h e orbit,
%/2~
of i n t e g r a t i o n
of t h e o r b i t
i
Comparison
classically
¢2r 2 0,
(5-132)
2 o' "'o- ' " o '
equation
obeyed
by the TF potential
V2V = (-4/3g) (-2V) 3 / 2 , Eq. (108) i s s i m p l i f i e d I d
2
4 dE
~E
r 2
8 -,-~[~3+r< d - i ] ( - 2 V ( r E ) ) 3 / 2 - ( v ~ ) 3~ ~E" ~ dr/, 8 r[2
I/2
I d
(75) are employed 2 = 16
4 dE ~E
which inserted •
into
to arrive
rE 2 ) ~~--7[wE
(109)
(5-133)
d
at
(v~) 2
,
(5-134)
produces
!
IEVE •
-~ E(
I
2
(~E+d--~E] [rEV(rE )] - ( r E )2
= ~ o3~ [-2V(rE)] Now Eqs. (74) and
to
-
[(v~)2-I] 2 + ~(-E)(rE/OJE)2{(IE/rE)z-2E
(5-135)
!
XEVE In particular,
for E=0 we get
~0 = -2(V0)2/~o
~2/r2 = -4 0 - o 3 Wo
= -~axo/F(x o) and
•
I [I - ~ X o / ~
(5-136) ~]
-3/2
Z-2/3
!
loVo • , loV o
_ [(Vo)2_I]2
-2 = ~I x 3 F(x o) [I - ~I Xo/XoF(Xo)]
(5-137)
326
The c o r r e s p o n d i n g
numerical
statements
ro/Z-I/3
=
1.86278
,
Io/ZI/3
=
0.927992
,
~o/Z 2/3
=
0.363593
,
V !
:
1.93768
-~o/Z -2/3
=
20.6527
=
7.58781
o
o o
are
(5-138)
(~2 - i~) ,
iv' o o In a d d i t i o n ,
from Eqs. (105)
io/Z -3/3
and
(106) we o b t a i n
= (ro/Z-1/3)2/(Io/Z1/3)
= 3.73920 (5-139)
Io~o
i2
-
4(v') o
2
-I
= 14.0184
,
o and
(92) leads to
(see P r o b l e m
I v" I O O = ~
7)
~, [(v.2)_i] [_5(~.)4 + 23(~_,)2 O u u c)
15]
(5-140) = 0.193647
These numbers
w i l l be u s e d below.
For the p u r p o s e ous q u a n t i t i e s large b i n d i n g
of i l l u s t r a t i o n ,
as a f u n c t i o n of energies,
cal for C o u l o m b i c
E/Z 4/3".
t h a t is -E ~ Z 4 / 3 , ~ E ( 0 )
potentials,
for w h i c h
~E(O)IZII3
=
IEIZ I13
rE/ZI/3
=
(IEIZI/3) 2
OOEIZ2/3
=
WY l
see Eqs. (78),
(79),
ly b o u n d e l e c t r o n s
and
we p r e s e n t ,
Observe
these
in Fig.10,
in p a r t i c u l a r equals
1 E. This
relations
vari-
that
for
is t y p i -
hold:
, ,
(5-141)
(IEIZI13)
(82). Of course,
is not u n e x p e c t e d ;
Coulombic
degeneracy
r e c a l l the d i s c u s s i o n
for s t r o n g -
a r o u n d Eqs.
327
(56) and
(57).
25 20
d 1.5
10 0.5 i
-2.0
i
-1.5
i
-1.0
-0.5
0
~lZ 413 Fig.5-10.
of E/Z 4/3 a r e shown: (a) rE~Z-I~3 , (b) (d) eE/Z2/3 ; f o r t h e n e u t r a l - a t o m TF p o t e n t i a l .
AS a f u n c t i o n
{c) vE(0)/zl/3,
TF d e g e n e r a c y and the systematics of the Periodic Table.
lE/Zl/3
Is there any
reality to the e n e r g e t i c d e g e n e r a c y as p r e d i c t e d via the n e u t r a l - a t o m TF p o t e n t i a l ? Our a f f i r m a t i v e answer begins w i t h p o i n t i n g out the s i m i l a r i t y b e t w e e n Figs.4 and 9. In q u a n t i t a t i v e terms we note that the curve conn e c t i n g 7s w i t h 5f in Fig.4 has t e r m i n a l slopes of about -I and -2, w h i c h agrees w i t h those of rE(1)
for E~0; in particular,
for E
m u l t i p l i c i t y × N(Z/Zmi n)
(5-148)
sequences It is technically
impossible
to perform this summation.
we can certainly use it to study the structure of N(E=0) of Z. Note that for large Z, N(Z/Zmi n) appears
as
Nevertheless, as a function
331
Table 5-I. city
Initial
state
(IS), c h a r a c t e r i z i n g
(MULT) of initial state,
for the first
15 sequences
The orbital states
and minimal
of states
ratio
v:l, multipli-
Z I/3 of initial state,
(ordered by increasing
3p, 5d, and 6p do not initialize
~I/3, amin J" a new se-
quence. IS
v:l
MULT
Is
1:1
2
0.822
2s
3:1
2
I .41
2p
1:3
6
1.90
3s
5:1
2
2.00
[3p
1-I
4s
7:1
2
Z I"3 / min
(Is sequence)
3 × 0..822 = 2.47] 2.60
3d
1:5
10
2.97
4p
5:3
6
3.05
5s
9:1
2
3.19
4d
3:5
10
3.54
5p
7:3
6
3.63
6s
11:1
2
3.79
4f
1:7
[5d
1.1
(Is sequence)
14
5 × 0 . 8 2 2 =4.11]
4.05
[6p
3.1
(2s sequence)
3 × 1.41
7s
13:1
2
4.39
5f
3:7
14
4.61
6d
7-5
10
4.69
= 4.23]
I N(Z/Zmi n) = ~ ( Z / Z m i n ) 2 / 3 (5-149) + (Z/Zmin)I/3 < l ( Z / Z m i n ) I/3 > + ... where the leading Z 2/3,
terms have been exhibited:
and an o s c i l l a t o r y
term of order
to turn the smooth term into the TF part terms.
has
IN(E=0)ITF = Z. Thus this smooth
term gains a factor of Z I/3 w h e n all sequences be equally true for the o s c i l l a t o r y
a smooth term of order
Z I/3. The sum over sequences
are summed.
This will not
An individual one has the pe-
riodicity
Z I/3 ÷ Z I/3 + 2 ~I/3 but as Table I shows the various sequences ami n , have what looks like randomly assigned values of zl~ 3. The amplitude of mln each o s c i l l a t o r y term is also d e t e r m i n e d by Z I/3 in conjunction w i t h the min m u l t i p l i c i t y of the sequence. There is nothing regular about these amplitudes as well.
Therefore,
we have to sum o s c i l l a t o r y
functions
that all
332
have the same shape but i r r e g u l a r amplitudes and periods. ence
of
these o s c i l l a t i o n s cannot be constructive.
The interfer-
We c o n c l u d e that
the r e s u l t i n g f l u c t u a t i n g function of Z I/3" has an a m p l i t u d e factor of Z I/3" as do the individual oscillations,
no e n h a n c e m e n t takes place.
What we have found is: Nqu(E=0 ) = Z I/3 x {fluctuating function of Z I/3}
(5-150)
As a matter of fact, the periods of the oscillations of the various sequences are not really assigned randomly. the shape of Vo(l). Accordingly, of the amplitude.
They are all d e t e r m i n e d by
there is a little bit of a m p l i f i c a t i o n
The d e t a i l e d analysis given below shows that the lea-
ding o s c i l l a t o r y term in N
is of the order Z I/2 = Z I/3" × Z I/6. However, qu for it to really d o m i n a t e the terms of order Z I/3, one needs the enor-
mous v a l u e of 5 × 1010 for Z. In the small-Z range of p h y s i c a l interest, this
"leading" o s c i l l a t i o n is u t t e r l y insignificant. Our algebraic results about N(E=0)
(E=0) as a f u n c t i o n qu of Z are confirmed by the plots p r e s e n t e d in Figs.12 and 13, of w h i c h the first one shows the s t a i r c a s e shape of straight-line
TF result,
and N
(145)
and compares it to the
and the second one illustrates
(150).
300
Z
150
0
150
0
300
t
Z Fig.5-12.
N(E=O) as a f u n c t i o n
The s t r a i g h t - l i n e
is
of Z for the
(N(E=O))T F = Z.
neutral-atom
TF p o t e n t i a l ,
333
5.0
2.5
N
0
:D O"
Z
-2.5
-5.0 0
1
2
3
/-.
5
6
7
Z1~3 Fig.5-13.
Nqu(E=O)/Z1/3 = [N(E=O)-Z]/Z 1/3 aS a f u n c t i o n of
Z1/3
f o r the
n e u t r a l - a t o m TF p o t e n t i a l .
General
features of E qu-. Armed with all this insight into the number of available states, we can now employ Eq. (59) and gain related information about the energy.
What we have said about N(E=0)
we keep the ratio
E/Z 4/3" fixed
of the p a r t i c u l a r
dependence
therefore
when changing
of
VE(1)/zl/3
on
E/Z 4/3
a consequence of the scaling properties This means that
IN(E)ITF equals
E/Z 4/3~ (for the TF potential,
of course,
p o t e n t i a l of the form "Z 4/3 times
holds also for E < 0 if
Z. This is an implication and l/Z I/3, and
of the TF model.
Z times a smooth function of or more generally
a function of zl/3r")."
for every
Indeed, we know
it does: 3/2 (N(E))T F = f (d~) 3---~[2 I (E-VTF (r))] = z of d x x l / 2 [ F l x l
The integration
thereof over
(El (~))TF = - ~ dE -oo
E,
(N{E))T F
+ ax
E ]3/2
(5-151)
334
5/2
= S(d~)(-15r~
5/2 52 Z 7/3a ofdX x -I /2 [F (x) -ax Z4--~3] ,,
results in
[-2(VTF(r)+~)]
( E I ) T F % Z 7/3 so that a factor Z 4/3 is acquired.
(5-152)
This is not
surprising since the differential dE is of this order:
dE = Z4/3 d(z4--~3 )
(5-153)
In addition to the smooth term tory contribution Nqu(E).
Generalizing
Z I/3 times a fluctuating function,
(151), N(E) has the oscilla-
(150) we state that it equals
the argument of which is the product
of Z I/3 and a smooth function of E/Z 4/3. An example for this structure is provided by
"Nqu(E)"
h E sin(l E)
=
(5-154)
Upon performing successive partial integrations,
this produces
"E qu (~)" = - -~S dE "N qu (H)"
(5-155)
=-f
-~
IE dE d-~[-__
cos(h E ) %
+ I_~ -~ d (A_~E) --- sin(IE) + ... ]
,
~E ~E differentiation with respect
where t h e d o t r e p r e s e n t s t o E, and t h e e l l i p s i s indicates further terms. The contents of the square brackets, evaluated at the upper limit, are oscillatory respective amplitudes
functions of Z I/3. Their
are of the orders
hE/~ E % ZI/3/Z -3/3 : Z 4/3
, (5-156)
I
d (hE] ~
I
I
Z4/3 = Z3/3
then Z 2/3, and so on. In short: when N qu ~ Z I/3, then the oscillatory part of Equ is ~Z 4/3. Oscillatory terms merely gain by a factor of Z 3/3. We infer that the binding-energy oscillation -Eos c is, for the TF potential, of amplitude Z 4/3 and (in some sense) periodic in Z I/3. Indeed, energy term
as promised,
this is small compared to the leading TF
(~Z 7/3) if only Z is sufficiently
large. A pertubative
335
treatment
of these oscillations
quantitative
statements
is fully justified.
about amplitude
We shall arrive at
and period below,
after a short
detour. We did not identify E nonoscillatory handles
with E , because E also contains qu osc qu Since the semiclassical spectral sum (29)
contributions.
the strongly bound electrons
represented
in
correctly,
dingly,
Nqu possesses
Fig.13,
somewhat
they are mis-
a related smooth term.
The slight
asymmetry
larger negative peaks than positive ones,
tent with the presence of such a term. shall exhibit the Scott correction tain
whereas
(EI)TF, the Scott correction must be part of Equ. Accor-
In the following section,
explicitly.
(at least part of) the quantum corrections
cussed in Chapter Four.
Further,
in
is consiswe
E qu must con-
to E I , that were dis-
So far it has not been d e m o n s t r a t e d
how one can
isolate them in E qu
Linear degeneracy.
Scott correction.
Let us briefly
energy of Bohr atoms with filled shells, leading o s c i l l a t o r y
look back at the
Eqs. (24) and
(25). There the
term has an amplitude of order Z 5/3. Why is the
chain of arguments that we applied to the TF potential not equally valid for a Coulombic
potential?
the lines of d e g e n e r a c y as for Coulombic states
~E(1)
potentials.
The reason is that for the TF potential are bent,
A straight
for w h i c h the respective
random character of the periods, tential
in Table
w h i c h we have observed
of
in phase.
The
for the TF po-
is a linear function of ~, a straight
In other words:
line in the
the existence of a principal Coulombic potentials
(or energy) from the TF
As a matter of fact, one can d e m o n s t r a t e 12 that linear de-
generacy near E = -~ always
leads to an energy o s c i l l a t i o n
The m a i n example of a physical neracy t h r o u g h o u t tial differs
is a highly ionized
bic degeneracy.
that selects
Z 5/3.
linear dege-
potential•
In neutral
by i n t r o d u c i n g
atoms,
As in Chap-
a separating bin-
the part of the spectrum w i t h Coulom-
Thus we write
E I (~) = [E 1 (~) - E I (~s)] + E I to
system d i s p l a y i n g
for the strongly bound electrons.
ter Three, we isolate these electrons ding energy ~s = - E s
of order
atom, w h e r e the effective poten-
but little from a Coulombic
one has linear d e g e n e r a c y
According
in sequences
N(Z/Zmi n) are perfectly
q u a n t u m number is what distinguishes potential.
line results
not straight
I, is absent in the situation of linear degeneracy,
that is when rE(1) l,~-diagram.
for 0 ~ -E/Z 4/3 ~ I ,
(24) and
(~s)
(25), E I (~s) is given by
(5-157)
336 EI(~s) where
:-Z2
[ 231Es 21 ;kE1 (2-1-~) + ...] s
,
(5-158)
[Eq. (78)]
IEs = Z/¢2 (E0-E s)
,
(5-159)
E° being the/ additive constantl of Eq. (15); for the TF potential it equals Z4/3B/a = 1.79374 Z 4/3. The leading term of E1(~s)
is, as always,
the TF contribution _ ~2 Z 21E s = _ 23 Z3/d2 (Eo-Es)
: f (d~) ( -
1 1--~-Z) [2 (E s + rz - Eo)]
= (E 1 I % - - -
when we insert
(15) into
Es))T F
(152)
5/2 (5-160)
,
[see also
(I-36)]. It combines with the
TF part of the integral
E I ( ~ ) - E1(~s)
= _
f
(5-I 61 )
dE N(E)
-~s to produce E1(~s).
IEI(~)ITF . More interesting is the next-to-leading
It equals ~Z 2 and does not depend on Es. Actually,
only part of E1(~s)
term of
being the
independent of ~s = -Es' this term is the only visi-
ble contribution of the strongly bound electrons to EI(~). All the other terms in
(158) cannot themselves
be present in El(~)
since El(~) does
not depend on ~s" We have thus identified the two leading contributions
E lit) = 6EI (~))T~+lz 2 + .--
to EI(~),
(5-162)
thereby rediscovering the Scott correction to which Chapter Three is dedicated.
Please note that at that earlier stage E ~
wise with the consequence
was evaluated TF s that the Bohr shell oscillations of (158) [or
(3-22)] had to be removed explicitly. aid of the semiclassical
sum
In (157) we compute E ~ s with the
(29), so that all Bohr shell artifacts
are
taken care of automatically.
Perturbative approach to Eosc~ After eliminating the density in favor
337
of the effective potential
from the energy functional
rating the electron-electron
interaction energy E
tic part and a remainder E' as in ee' the energy (2-40)
E(V,~)
(2-36)
(2-434) and sepa-
into its electrosta-
ee the potential
Z
functional of
2
!
= E 1 ( V * ~ ) - ~N- 8~ ~ (d~) [ ~ ( V + - ~ - V e e ) ] (5-163)
+ {E'ee(n) - / (dr) nVee(n)}n=n(V) emerges,
for Vex t = -Z/r. We recall that the density is expressed in
terms of the potential,
symbolically:
n=n(V),
by solving Eq. (2-432), in
which V
=V (n), for n. The potential V' is V minus the electrostaee ee ee ee tic potential (2-28) [see (2-37)]; its lion's share is the exchange po-
tential. We exhibit the TF part of
(163) by splitting E I into
(EI)TF
and E qu' EI
(V*~) = f I d a ) ( - l S - - ! ~ ) > 2
(v+~)] 5/2 . ~qu(V+~) (5-164)
= (E I (V+> I (5-213)
g(z)
The
leading
already also
~ ~-~
....
for z >> I
asymptotic
forms
represent
small
z. This
varying
our
statement
function
As d e f i n e d we take
+
for r e l a t i v e l y
illustrates
slowly
15 ~4z7
1
h(z)
in
(206),
to be an odd
h(z
(5-224)
(-1)kcos (2nklE){
k~0
v~.(~)
~
(-I) j
2
2
j=l
i= k~O
j ~0
where the invariance
of the summands
for rewriting the expressions.
>
-
j~0 I sinx twice and equate
under j ÷-j,
k÷-k
has been used
Now we can employ the identity
/
(- I ) j xlj j ....
x (5-225)
I x
(224) to
1E >
(-1)k cos(2zkl E)
k~0
~k
sin (~k/v~) (5-226)
= _ 21 E ~
(-1)k cos (2~klE) ~k sin (~k/~)
k=1 The leading
I oscillation
in
(222) is therefore
given by
354
(Nosc([))I = - 2 1 [ Z
(-1)k cos(2zkl[) zk sin (zklv ") E
k=1
(5-227)
+ 21[ ~
(-1)k+Jzk
{ (z [) cos (2zk 15)
k#0 j=1 +
where the ellipsis equals
likewise
I , - ~-~
and g(z).
of linear degeneracy term in
I oscillation
In passing,
For detail
consult
and f(z)
I ~2z3
(5-228)
,
we remark that in the situation
which identifies
A first partial
i oscillations,
term
z E ~ v ~ - I/2 ÷~,
for linear degeneracy.
directly.
I oscillation
asymptotic
~(z) - g(z)
[v~+0,
(227) vanishes,
strated
the nonleading
its leading
- f(z)
for g(z)
t
represents
f(z) without
f(z)
---
~(zE ) ~ - z~ 3] the second
the first one as the leading
This can, of course,
also be demon-
Ref.12.
[ integration
of
(227) supplies
the leading
of Eos c. It is given by
(-Eosc/Z4/3)l
:Z~k
sin(2~klo)
+ ....
(5-229)
k=1 with the Z dependent
coefficients
(~o IzI13 ~[
s k given by
(-I)k
~k = \iolZ-313)L - T ~ - ~ -
I sin(zkl~) (5-230)
1-11kI11J
I
j=1
Please
note that,
required
to obtain
bute significantly regarding
because
]]
~o'
=
" 93...,
purposes,
very large values
argument
to the Fourier
of the consequences
for all practical
I
a vanishing
of j and k are
of f. These terms do not contri-
sum of Eq. (229). Therefore,
of the discontinuity
harmless
of h(z)
(not to mention
from
(138)
and
(139)
dis-
the possibility
that v' is irrational); see also Problem 11. o For very large Z, the sum over j does not contribute (230), so that with the numbers
our
at z = 0 is,
to s k in
355
Sk
The
first
These
_
=
0.02518
,
s2
=
0.06232
,
s3
= -0.00283
,
s4
: -0.00783
,
S5
=
0.00104
,
s6
=
,
s7
= -0.00055
,
s8
= -0.00100
values
much with
for
reason
why
This is u n d e r s t o o d
requires out, to
k is even,
Table
Z I/3
5-2.
cannot
small-Z
2 show that
those
with
odd
from their
be used
difference
recalling
form,
is
that
large
in t h e
happens
f(z)
range
the
of physical
sk with
even
k chance
markedly
asymptotic
values
small-Z
range
only
according
S
I
Sl,S3,S5,...
the difference
for s m a l l
v O ~ 2 - 1/16,
j=k/2
Coefficients
{(z),
only
for e v e n
an approximation
we add the
between
!
jv~ ~ k. Now,
which
the relevant
in T a b l e
whereas
(5-232 )
k do and (232).
is t h e
sub-
12.
upon
find
for t h e
substantially
(232)
essential
asymptotic
0.00234
be u s e d
listed Z I/3,
ZI/3=I...5,
ject of Problem
insight
cannot
The numbers
differ,
picked
(5-231)
Z >> I
sI
Another
this
for
!
not c h a n g e
its
0.02515 sin(1 .621k)
f e w of t h e S k s are t h u s
large-Z
interest.
(-I)k+I k2
f o r S k.
t e r m of t h e
arguments
so t h a t
between
If k is odd,
f(z)
Let
2j = k is
us u s e t h i s
(231)
(230), w h e r e
will
do.
S
a n d s 4 for
2
ZI/3 = I,
S
3
1.5,
If
we evaluate
to
Sl,S2,S3,
and
z. In Eq. (230)
the term with
k, o f c o u r s e .
s u m in
s2,s4,s6,...
and
...,
S
5.
4
I
0.02467
0.00683
-0.00248
-0.00132
1.5
0.02490
0.00876
-0.00261
-0.00166
2
0.02500
0.01035
-0.00268
-0.00193
2.5
0.02506
0.01174
-0.00272
-0.00216
3
0.02509
0.01297
-0.00275
-0.00236
3.5
0.02511
0.01409
-0.00277
-0.00255
4
0.02513
0.01512
-0.00279
-0.00271
4.5
0.02514
0.01607
-0.00279
-0.00287
5
0.02515
0.01696
-0.00279
-0.00301
356
1
1
1
- ~ + ~ o/~Tk 2-v' o In addition
to the definition
f(z>I, the double
by the factor (5-244
Here one needs
~'( 0 ) = o
~
( t [ v [ :-1"22A[v[J I''" E=O
= ~o%
÷ ~o;o-
= ½t%~o+ which utilizes
(198)
~oio%
- ~Ao~ 12.,,°
(5-245
(~o+iI~o I = 32.9806 z -3/3
and the numbers
in
(138)
and
,
(139). The amplitude
360
of the leading
TF o s c i l l a t i o n is 0.015Z 4/3
see Fio. (14) and Eq. (200)
r
so that the leading v o s c i l l a t i o n has an amplitude of 0.001
i
Z 4/3" . This
is so small that we need not c o n s i d e r the n e x t - t o - l e a d i n g v oscillation.
S e m i c l a s s i c a l p r e d i c t i o n for Eos c. The time has come to put things together. We have i d e n t i f i e d various c o n t r i b u t i o n s to the b i n d i n g - e n e r g y oscillations,
-Eos c
=
+
(-Eosc)I, ~
(-Eos c ) I
+
(5-246)
(-Eosc)v
The separate calculations of the three types of o s c i l l a t i o n s Eqs. (195)
-
(199)
for the mixed l,v oscillations,
for the i o s c i l l a t i o n s (242)
(246)
are concerned,
range of p h y s i c a l interest. ~0.001
Figure
whereas the other two terms in
be dominant.
Z 4/3. As far as the mixed
this statement is true for the small-Z
In contrast, w h e n Z ~I,
Z 3/2 and the h - o s c i l l a t i o n is
needs at least Z I/6
(230)
and in Eq.
17(c) tells us that the a m p l i t u d e
is about 0.05 Z 4/3
both have an amplitude of about 0.001
l,v o s c i l l a t i o n s
in Eq. (237) w i t h
(which contain also the £TF terms),
for the ~ oscillations.
of the i o s c i l l a t i o n s
resulted in
this o s c i l l a t i o n is
~ s2 Z4/3 ~ 0.06 Z 4/3, so that one
60, or Z k 5×10 10 , for the mixed
l,v o s c i l l a t i o n s to
This is r i d i c u l o u s l y far beyond the domain of physics.
[In
p a s s i n g we remark that we have just d e l i v e r e d the j u s t i f i c a t i o n of the statements
f o l l o w i n g Eq. (150)]. Both the v and the l,v o s c i l l a t i o n s
shall neglect them completely. in
are very small, and we
I n a s m u c h as the subsequent
(237) are expected to be of larger amplitude,
justified. from
We must also not forget that the a p p r o x i m a t i o n
(191) introduces
an error,
~ oscillations
this is t h o u r o u g h l y (189) w i t h ~
in view of w h i c h it is quite u n n e c e s s a -
ry to pay attention to the small corrections that the ~ and the l,v oscillations
represent.
Consequently,
our s e m i c l a s s i c a l p r e d i c t i o n for
E0s c is given by the two terms on the r i g h t - h a n d side of Eq. (237), the sum of the leading and the n e x t - t o - l e a d i n g I oscillation:
k/-E°se~ z4/3]sc
=y
S k sin(2~kl o) + Z -I/3 ~
k=1
It is plotted in Fig.17(c).
c k cos (2~klo). (5-247)
k=1 We compare it w i t h the HF p r e d i c t i o n of Fig.2
in Fig.18. Both curves agree in a number of details.
First, they have the
same phase and Period, w h i c h is given by lo, the m a x i m u m value of the
361
0.06
OO&
002 N
0
o
LU !
-
0.02
- OO&
-OD6
1
Fig.5-18.
2
each sum of angular
really
number
Further,
maxima with
with the
HF
5
prediction
prediction.
(SC)
for the bin-
Ten t e r m s are added i n
(247).
quantum
what m o r e
4.
Comparison of our s e m i c l a s s i c a l
ding-energy oscillations
the same.
3 Z I/3
in the TF limit.
Then t h e i r amplitudes
they b o t h show r a t h e r
an e v o l v i n g
pronounced
double
sharp s t r u c t u r e l e s s
structure.
in the s e m i c l a s s i c a l
compare w i t h e x p e r i m e n t a l
data,
The
latter
curve.
there
are about minima,
phenomenon
and
is some-
But since we cannot
is no w a y of judging w h i c h
one i s right. The main d i f f e r e n c e semiclassical smooth
one
is that
term of o r d e r Our c a l c u l a t i o n
tions
and c o n s i s t e n t l y t e r m could
are i n d i c a t i o n s
the HF p r e d i c t i o n curve
Z 4/3 is o b v i o u s l y
mula.
missing
between
the s e c o n d
is shifted
missing
for Eos c and the
d o w n in Ref.18.
in the b i n d i n g - e n e r g y
A for-
of E
c o n c e n t r a t e d on the o s c i l l a t o r y c o n t r i b u osc d i s r e g a r d e d all s m o o t h c o n t r i b u t i o n s , so that this
not be found.
As we have r e m a r k e d
that the c o r r e l a t i o n
energy
(4-248)
around must
Eq. (4), there
be i n c l u d e d
into
362
the d e s c r i p t i o n in order to be able to find the correct smooth term of order Z 4/3 . A l t h o u g h the HF method produces the curve of Fig.2 speaking,
not even that,
since Estat of
vides no insight w h a t s o e v e r contrast,
[strictly
(I) is not a HF result],
for the origin of these oscillations.
it proIn
the s e m i c l a s s i c a l c a l c u l a t i o n supplies us w i t h an u n d e r s t a n -
ding of the o v e r - a l l amplitude factor Z 4/3
[recall that this is a conse-
quence of both the scaling properties of the TF p o t e n t i a l and the bent shape of the TF line of d e g e n e r a c y
Vo(%)]
and of the period
the largest angular m o m e n t u m in TF atoms);
it also gives an e x p l a n a t i o n
for the i n t r i g u i n g double structure of the maxima. of an e f f e c t i v e halving of the period: change e n o r m o u s l y as Z I/3 increases;
(given by
It is the b e g i n n i n g
the even-k coefficients
in
(247)
look again at Tables 2 and 3 as
well as at Eq. (236). To see how this effects the e v o l v i n g d o u b l e - p e a k structure of the maxima, we d e c o m p o s e
(247)
into
0.06
Q04
Q02
-4"
N o
1.1.1 I
-0.02
- 004
-0.06
1
2
3
4
5
Zl/3 Fig.5-19.
Thick
curve:
HF;
thin
curve;
SC,
odd;
dashed
curve;
sc,
even.
363
-Eosc h Z-~7 ~]
=k SC, odd
(Sk sin(2~klo) + Z-I/3akC°S(2~kio))
k=1,3,5... (5-248)
and (-Eosch 4-~7~) Z
=> (~k sin (2~kXo) + Z-I /3akCOS(2Zklo) ) " SC, even k=2,4,6,... (5-249)
These are plotted,
along with the HF oscillations,
serve that the relative phase of
(248) and
in Fig.19.
ma of the first coincide with the minima of the second. therefore,
We ob-
(249) is such that the extreThe sum has,
sharp structureless minima and broad doubly peaked maxima.
And the increase in amplitude of
(249) is responsible
for the growing
dip between the pairs of maxima.
Other manifestations
of shell structure.
ly discuss other quantities fects.
In this section we shall brief-
than the binding energy which show shell ef-
In Chapter Three we found that the density of electrons site of the nucleus is given by (2Z) ~
no/-4~
for a neutral atom; 0(I/Z)
= 1.2021
- 1.7937 Z -2/3 + 0(1/Z)
see Eq. (3-166).
~ 1.82/Z
that n o contains on oscillatory
(3-167),
(5-251)
of 0(I/Z)
in Fig.3-5,
where we notice
part. Since the amplitude of the oscil-
around the smooth curve in this plot decreases with Z whereby
the period gets longer, ment
(5-250)
- 0.82/Z 5/3
is compared to the HF prediction lations
The estimate of
at the
the natural surmise is that we have to supple-
(251) by a term periodic
insight by considering
no/
12z) 4r~
in Z I/3 with amplitude
Bohr atoms,
: 1.2021 - ~I I N)-2/3 + ~(~N) 5 3 -4/3 + 0 (N-5/3)
as obtained by calculating
Z -4/3. We gain some
for which (2 _I~)
,
(5-252)
the term of order N -4/3 in Problem
y is the solution of Eq. (6). For neutral
I-5. Here
Bohr atoms, the leading oscil-
364 latory contribution
to n
is, therefore,
o
/. /(2z)~h
V'o" 4~
1 •46 Z-4/3 Z
/osc =
T(-1)k ~E cos(2Ekx1.145
Z I/3)
k=1
w h i c h has the expected structure.
,
(5-253
For real atoms, the oscillations
in
n o are still w a i t i n g to be calculated. In Chapter Four we found the ionization by the statistical
model•
pared to experimental nounced shell effects,
Iosc(Z) One naturally
function of Z I/3 in Fig.20. as yet.
and
com-
there are very pro-
- Istat(Z)
that I
(5-254
(Z) equals
OSC
Z -I/3 times a fluctuating
This is confirmed by the experimental
A semiclassical
prediction
data presented
for Ios c has not been calculated
For the evaluation of Iosc(Z ) = Eosc(Z,N=Z-I)
-Eosc(Z,N=Z )
1./.,
(5-255
I
I
Kr
' t Ne Ar
1.1 He
TIL
Xe
Rn
HiI,. zo/ -;Ii At
Cd..i
N N .-.8 .o
in Eq. (4-294)
Obviously,
which we isolate by writing
= I(Z)
presumes
The result is reported
data in Fig.4-8.
energ [ as predicted
0.5 o.2 ~
-0J
Fig.5-20. See also
Li
i 2
No
Experimental values Fig.4-8.
K
, 3
Rb
Cs,
Fr 5
Z 1/3
of I O S C (Z)/Z -I/3 as a f u n c t i o n
of Z I/3
365
one
needs
TF lines those
E0s c for w e a k l y of d e g e n e r a c y
of n e u t r a l
effective
tistical
for
TF atoms
potential
of
ionized such
atoms.
ions,
because
This
which
of the
requires
differ
a study
of t h e
substantially
long-range
Coulomb
from
part
in t h e
an ion.
Also
in C h a p t e r
Four
we observed,
model
prediction
for t h e
in T a b l e
expectation
value
4-I, of
that
i/r,
the
sta-
for n e u t r a l
atoms, I r stat agrees 4-I of
well
with
indicate Z I/3.
that
For this
= 1.79374
the
Z 4/3
- Z + 0.44983
corresponding
the d i f f e r e n c e function
Z 2/3
HF p r e d i c t i o n . is
Z 2/3 t i m e s
an
(5-256)
The
numbers
of T a b l e
oscillatory
function
we w r i t e
I r osc In Fig.21
the
= r
_ r stat
HF p r e d i c t i o n s
for
(5-257)
are p l o t t e d .
Since,
see
Eq.
OSC
0.1:,).r
' He
' Ne
0.0~
Kr
Zn~
QO~ N
Cd
Rn
Ar ~
0
-0.0/.
-Q05
-0.12
Fig.5-21.
1 HF
2 prediction
3 Z~13
f o r r osc /~2/3 -~
as a f u n c t i o n
/-. of Z I/3
5
366
(4-31 8) ,
r
_
_ _ ~-~ E(Z,N) I' L
N=Z (5-258 _
d
dZ E(Z,Z)
+ --~ E(Z,N) i' N=Z
d dZ E(Z,Z)
- ~(Z,Z)
and r stat
_
d dZ E s t a t ( Z ' Z )
(5-259
we find
r osc
_ d dz(-Eosc(Z,Z))
(5-260
- C(Z,Z
0.15 0.10 Q05 0
-0.05 N
HF
.._~ -0.10 -Q15
SC
-020 -0.25
1
2
Fig.5-22: HF p r e d i c t i o n for O S C p r e d i c t i o n for < I / r > + ~. OSC
3 Z
4
compared to the s e m i c l a s s i c a l
5
(sc)
367
Inserting the semiclassical prediction for Eosc, Eq. (247), supplies the leading terms
( C)Iz2/3 r osc +
2~ Io < 3 Z I/3 k SkCOS (2~kl o) k=1
-~
+
I
~
4
(5-261)
2~
lo 3 Z 1 7 ] k % ) s i n ( 2 ~ k X o)
ZI/3 L _ _ ( ~ S k k=1
This is compared to (osc)HF_ in Fig.5-22. The agreement is as good as one could expect, but not better. Certainly, an improvement upon the semiclassical prediction for is asked for. Possibly the subseosc quent terms in (247) are significant, and presumably some knowledge of (Z,Z) is required. Nothing has been done along these lines so far.
Problems 5-I. Show that, more precisely than Eq. (9), Eq. (6) is solved by y = (~N)3I/3 + ]-21(~N)3 - I / 3 _~1'1t~)'3(~N)3-5/3 + ]I" 1t]_~j,4(~N)-7/3+...
observe,
in particular, that there is no term proportional to N -3/3.
5-2. Show that a potential V(r), which approaches -Z/r for r ÷ 0, must be of the form "Z 4/3 times a function of zl/3r", if Eq. (17) holds
(with
independent of Z, of course). 5-3. The Hamilton operator of a particle moving along the x-axis is
1 2 nip = ~Px + / ~
IxL
Show that the eigenvalues of HID are related to the zeros of F_1(y) , given in (4-158). Find approximations to these eigenvalues both by employing the TF quantization notice?
(37) and by utilizing
(Incidentally, this produced Fig.7.)
5-4. Show that, for r 2 > r I > 0,
r2 dr
f rI
:~1+r2
--r- /(r-r I) (r2-r) = ~k
rlV-C~q2) ,
(4-170). What do you
368 r2 f ~//(r-r rI
I) (r2-r)
= ~/~rlr 2
J"r2 dr r/~(r-rl) (r2-r) rI Use these integrals 5-5.
Use
Coulomb 5-6.
(84) in
to confirm
Eqs. (56),
(97), in conjunction
potential,
Show that
to derive
for the Tietz
r ° = R,
=
1°
5-7.
(115), and
(122).
with the known result
for the
(90). potential
~
'
Can you confirm the suspicion 2(ko-~)
r1+r 2 2
= ~
~o =
that
(4-222) ½¢Z-/-E
Vo(1)
'
one has,
for E=0,
Vo'
=
=
'v
o
is the straight
2,
v "o
=
0.
line Vo(1)
=
?
Evaluate
(92) for the neutral-atom
5-8. The Bernoulli
polynomials
Bm(X)
TF potential
are generated
and confirm
(140).
by
t e xt ~ t TM et_l - > B re(x) ~.t m=0 Show that Eq. (171)
is equivalent
to
~(_1)
[_Nosc(E))+£TF
= 4
m+1
(re+l)!
Bm+1 (½+ < I E > ) ( ~ )
m(IvE(l))
m=l (Footnote
4 to Chapter
Three may prove useful.)
5-9. Find a corresponding 5-10.
Suppose
amount? also,
side of
As a consequence
for this
that it exactly 5-11.
expression
for
(-Eosc)iT F.
E is such that for some j the number
Then the right-hand
(188)
If jv~ - k = E,
is discontinuous
of the discontinuity
E, a discontinuity cancels
I
t=t E
the one of
for this
of h(z)
of the right-hand
E. By which
at z=0, there is
side of
(218).
Show
(188).
161 for which we wrote
= 86, the
(less precise)
(Wiley, New York,
numbers of Ref.
11 of Chapter
Three w e r e used.
38
B.-G. Englert and J. Schwinger,
39
The p o t e n t i a l V in Eq. (13
Phys. Rev. A 26, 2322
of Ref.
38 equals U - Uex + ~; it is thus
an e l e c t r o s t a t i c p s e u d o - p o t e n t i a l .
W i t h this identification,
of Ref.38 is e q u i v a l e n t to Eq.(342) 40
(1982).
(13)
in the t e x t .
The external p o t e n t i a l -Z/r has to be r e p l a c e d by a sum over the C o u l o m b potentials of all the nuclei in the molecule; also the Scott 1 2 t e r m ~ Z becomes a sum over the c o n t r i b u t i o n s from individual nuclei. The level of s o p h i s t i c a t i o n in p e r f o r m i n g the CSBE in Eq. (331b) depends on the p a r t i c u l a r application.
Further,
if one is
interested in the d e p e n d e n c e of the energy on the p a r a m e t e r s that specify the c o n f i g u r a t i o n of the nuclei,
the e l e c t r o s t a t i c energy
due to the C o u l o m b r e p u l s i o n between the n u c l e i must be included. Effects of the finite nuclear masses are very small and c e r t a i n l y irrelevant at this level of accuracy. 41
As a matter of fact, what is d i s c u s s e d
in the text is not the
"usual argument" but a v a r i a n t of it. The d i s c u s s i o n is both simpler and more t r a n s p a r e n t this way.
~z
H a n d b o o k of C h e m i s t r y and Physics Ohio,
~3
(Chemical Rubber Co., Cleveland,
1979/80).
F. Hoare and G. Brindley,
Prec. Roy. Soc. London 159A, 395
(1937).
393
4~ S o m m e r f e l d ~ a p p r o x i m a t i o n to F(x), gives 8.71
that is F(x) ~ [1+(12-2/3x) Y] -3/Y,
for this integral. W h e n inserted after undoing the two-
fold partial integration,
it produces
8.67. Both numbers are in sa-
t i s f a c t o r y agreement w i t h the actual value.
4s From Ref.11
of Chapter Three.
46 In Ref.24 of C h a p t e r Two,
Levy and Perdew try to put the blame for
the d i s c r e p a n c y b e t w e e n e x p e r i m e n t and the HF predictions onto the experiments
(or the e x p e r i m e n t a l i s t s ) .
p l a u s i b l e to me. Also, differences
entirely
This does not seem
r e l a t i v i s t i c corrections cannot account for
this large.
47 B.-G. Englert and J. Schwinger,
Phys. Rev. A29,
2353
(1984).
48 A n o t h e r a p p l i c a t i o n is reported by A. Ma~anes and E. Santos, Rev. B 34,
5874
potentials
U and Ues
(1986). Unfortunately, (our denotation)
these authors confuse the with,
luckily,
no consequences
as far as the conclusions of the paper are concerned. however,
Phys.
The appendix,
and all related remarks in the text are erroneous.
4~ This m o d i f i e d TF d e n s i t y
(or its o n e - d i m e n s i o n a l analog)
has been
derived prior to the p u b l i c a t i o n of Ref.8 for the special situation of a linear potential,
when
(394) is the whole answer.
I am aware
of the f o l l o w i n g three papers: W. Kohn and L.J. Sham, Phys. A1697
(1965);
Nucl.
Phys. A349,
S.F. Timashev, 349
E l e k t r o k h i m i a 4_O0, 730
(1979);
Rev.13___~7, H. Gr~f,
(1980). All these authors squared the wave
f u n c t i o n in a linear p o t e n t i a l see Eq. (143)] to arrive,
[this is e s s e n t i a l l y an Airy function,
finally,
at
(394)
for that special poten-
tial.Our d e r i v a t i o n is m o r e general in not m a k i n g such assumptions about V. 5o The subscript
is non-standard; P any c o n f u s i o n w i t h ~ = 1.04018...
sl See R a d z i g a n d
Smirnov,
it is i n t r o d u c e d only to exclude of Eq. (2-313).
cited in F o o t n o t e 30.
s2 The e v a l u a t i o n of the i n t e g r a l
(140) and its y - d e r i v a t i v e for x=0
is a simple e x e r c i s e in p e r f o r m i n g complex contour integrals. worst,
At
the results can be looked up in Ref.6.
53 The HF d e n s i t y is c o m p i l e d from D.R.
Hartree,
Proc.Roy. Soc. London,
394
Ser. A 151, 96
s4 L.L.
DeRaad
(1935).
and J. Schwinger,
Phys. Rev. A 25,
2399
(1982).
Chapter Five. i
The first plot of this kind is c o n t a i n e d in the paper by D m i t r i e v a and Plindov,
cited in F o o t n o t e
10 to Chapter Three.
To avoid a possible m i s u n d e r s t a n d i n g :
it is not the n u m e r i c a l
a c c u r a c y of HF numbers that is q u e s t i o n e d here, but the p h y s i c a l r e l i a b i l i t y of the HF approximation.
C.E. Moore,
Atomic Energy Levels,
Ser., Natl.
Bur.
D.C.,
Stand.
Ref. Data
(U.S. GPO, Washington,
1970).
4
J.P. Desclaux,
s
J.M.S.
6
H.A. Kramers,
7
R.E. Langer,
Scott,
cited in Footnote
8 to Chapter Two.
ci£ed in F o o t n o t e I to Chapter Three.
Zschr.
f. Phys.
Phys. Rev. 51,
N.H. M a r c h and J.S. 419
Natl, Bur. Stand.
(U.S.) Circ. No. 3 5
3__9_9,836
669
Plaskett,
(1926).
(1937).
Proc.
R. Soc.
London,
Ser. A235,
(1956) a l r e a d y noticed that the s e m i c l a s s i c a l sum contains
the TF a p p r o x i m a t i o n in the c o n t i n u u m limit.
However,
their me-
thod is v e r y d i f f e r e n t from the one d i s c u s s e d in this text,
and
they did not d e v e l o p a s y s t e m a t i c way of a n a l y z i n g these q u a n t u m corrections.
s
i0
B.-G.
Englert and J. Schwinger,
Phys. Rev. A32, 26
The circular orbit is also the one which,
(1985).
for given angular momen-
tum, has least energy. ii
P r e s u m a b l y due to too crude an a p p r o x i m a t i o n for F(x), Fermi reported 3.2
(instead of 3.916)
for this integral in his c l a s s i c a l
paper on the systematics of the Periodic Table. The reference is:
395
E. Fermi,
12 13
B.-G.
Rend.
Lincei 7, 342
(1928).
Englert and J. Schwinger,
H. Hellmann,
Acta Physicochim.
to consider the £TF model.
Phys.
Rev. A 3 2 ,
URSS 4, 225
36
(1985)
(1936), was the first
Since he used the o r i g i n a l sum over Z,
not the Fourier formulation,
H e l l m a n n failed to r e c o g n i z e that one
can split the energy into ETF plus the q u a n t u m c o r r e c t i o n Equ.
l~
A simple c o u n t i n g of the r e s p e c t i v e points on a j,k lattice shows that this fraction equals
[ a r c t a n ( ~ ) - ~ / 4 ] / ~ = 0.098, a little bit
less than ten percent.
is
The factor k-j is m i s s i n g in Eqs. (I07) publication
(1986)]. Fortunately,
16
Phys. Rev. A3_~2, 47
in the H a n d b o o k of Mathe-
cited in F o o t n o t e 6 of Chapter Four.
Another way of p l o t t i n g the Fresnel functions, c h a r m i n g one,
gart,
and a p a r t i c u l a r l y
is p r e s e n t e d in Fig.16 of F. L~sch, T a f e l n h~herer
F u n k t i o n e n / T a b l e s of Higher Functions
18
(108) of the o r i g i n a l
this i n a d v e r t e n c e did not cause any harm.
A standard r e f e r e n c e is M. Abramowitz m a t i c a l Functions,
17
and
[B.-G. Englert and J. Schwinger,
(7-th edition,
Teubner,
Stutt-
1966).
See the paper cited in Footnote
15.
Chapter Six. z
Density. F u n c t i o n a l Methods in Physics
(Alcabideche/Portugal,
edited by R.M. D r e i z l e r and J. da P r o v i d ~ n c i a York,
1985);
France,
S e m i c l a s s i c a l Methods in N u c l e a r Physics
1984), edited by R.W. Hasse,
J. de Physique
3
f. Phys.
R. Arvieu,
(Grenoble/
and P. Schuck,
4_55, Coll. C 6.
M.S. V a l l a r t a and N. Rosen, Zschr.
1983),
(Plenum Press, New
82, 794
Phys. Rev.
41, 708
(1932);
(1933).
F o o t n o t e 3 of Chapter Three applies here as well.
H. Jensen,
396
4
J. Schwinger,
cited in F o o t n o t e 2 of Chapter Three.
5
I.K. D m i t r i e v a and G.I.
Plindov,
cited in F o o t n o t e
10 of Chapter
Three.
6
From Ref.11 of Chapter Three.
7
J.M.S.
8
The number appearing here is
Scott, cited in F o o t n o t e
I of Chapter Three.
1 I 1T - 19 ~ ~ m1__~+112 ~ - - ~ - ~ - n~ 2.248 = 4 ~ n--I- + 4 ~ ' - ~'D-- _y n=1 n=1 n~ n=1 m=l n:l m=1
= 4 x 1.036928 - 19 x - g ~ + 4 x 1.265738 + 12 x 1.133479
9
W. Kohn and L.J. Sham,
10
D.A.
Kirzhnits and G.V. Shpatakovskaya,
Zh. Eksp. Teor. Fiz. 62,
2082
(1972)
(1972)] used methods re-
[Soy. Phys.
Phys.
Rev.
140, A1133
- JETP 35,
1088
(1965)
lated of those of Chapter Five to study shell effects in atomic densities.
11
Their results are encouraging,
E.P. Wigner,
but h a r d l y satisfactory.
cited in Footnote 5 of Chapter Four. We do not in-
clude the factors of 2~ into the d e f i n i t i o n of the Wigner function. - A recent review of this subject is M. Hillary, M.O.
Scully,
and E.P. Wigner,
Phys.
Rep.
106,
121
R.F. O'Connell, (1984).
397
INDEX A superscript
at the page number signifies
name or subject is not mentioned note to the indicated
that the respective
in the main text itself,
but in a foot-
page. Names
Abramowitz, M., 34816 Airy, G.B., 190 Amaldi, E., 22513 Antosiewicz, H.A., 1906 Arvieu, R., 3701 Avogadro, A., 257 Baker, E., 57,231 Baltin, R., 19912 Bander, M., 165 I° Bernoulli, J., 1364 Bhaduri, R.K., 1872'3 Bohr, N., 3,231 Brack, M., 1872,1884,19912 Brindley, G., 25743 Bush, V., 231 Caldwell, S.H., 231 Clausius, R.J.E., 263 Compton, A.H., 379 de Coulomb, C.-A., I Coulson, C.A., 6212 Darwin, C.G., 371 Desclaux, J.P., 568,2984 Dirac, P.A.M., 173,175,231f,245 Dmitrieva, I.K., 7418,8317,9818, 1651°,19611,231,2342~,24033, 2971,375f Donelly, R.A., 24134 Dreizler, R.M., 3701 Durand, M., 1884,19912 Emden, R., 230 Euler, L., 26,50 Fermi, E., 2,8317,104,22513, 230f,32411 Feynman, R.P., 235 Finkelstein, R., 1811 Fock, V., 2 Fourier, J., 24,312 Fraga, S., 16611,23528,24337, 25945,26445,3756 Fresnel, A.J., 348 Froese-Fischer, C., 24337 GauB, C.F.,
36
Gomb~s, P., 11,104,229,23932 Gr~f, H., 1872,26149 Grout, P.J., 23931 G~ttinger, P., 235 Hartree, D.R., 2,119,28953 Hasse, R.W., 3701 Heaviside, O., 7 Hellmann, H., 235,33813 Hill, S.H., 23931 Hillary, M., 380 T M Hille, E., 601°,6413 Hoare, F., 25743 Hohenberg, P., 10423 Jensen,
H., 231f,245,371
Karwowski, J., 16611,23528,2433v, 25945,26445,3756 Kirkwood, J.G., 189 s Kirzhnits, D., 19610,379 I° Kobayashi, S., 6514,231 Kohn, W., 10423,26149,378 Kompaneets, A., 1961° Kramers, H.A., 231,311 Kronecker, L., 58 Lamb, W.E., 241 Langer, R.E., 311 Levy, M., 112,115,24134,25946 Lieb, E.H., 43,1012°,124,1396 L~sch, F., 34917 Mananes, A., 26048 March, N.H., 11,6212,139,231,23931 3158 Matsukuma, T., 6514 Messer, J., 43 Milne, E.A., 230 Moore, C.E., 2983 Mosotti, O.F., 263 Nagai, S., 6514 O'Connell,
R.F.,
380
Palke, W.E., 24134 Parr, R.G., 24134 Pauli, W., 235
TM
398
Scott, J.M.S., 130f,138f,173,231,298, 375f Scully, M.O., 380 T M Shakeshaft, R., 124 Sham, L.J., 26149,378 Shpatakovskaya, G.V., 3791° Simon, B., I012° Slater, J.C., 223 Smirnov, B.M., 23830,26451 Sommerfeld, A., 6111,124,25944 Spruch, L., 124 S~Bmann, G., 1364
Pavlovskii, E., 196 I° Perdew, J.P., 112,115,25946 Plaskett, J.S., 3158 Plindov, G.I. 741s,831~,8918 16510,1961~,231,23417,2403~, 2971,375f Poisson, S.D., 27,173 da Provid~ncia, J., 3701 DeRaad, L.L., 29454 Radzig, A.A., 23830,2645~ Rasetti, F. 2311~ Riemann, B., 165 Rosen, N., 371 Santos, E., 26048 Saxena, K.M.S., 16611,23528, 2433~,2594s,26445,3756 Schr~dinger, E., 105 Schuck, P., 1872,1884,19912,3701 Schwinger, J., 415,130,139,143 v, 193,1935,196,2178,231,24938 , 2528,25438,25638,2604~,27847 , 2814~,8,28947,29454,3179 , 3249,33512,34615,35815,375
Thirring, W., 43 Thomas, L.H., 2,104,230f Tietz, T., 12628 Timashev, S.F., 26149 Umeda,
K., 6514
Vallarta, ~.S., 371 Van Vleck, J.H., 23 s von Weizs~cker, C.F., 232 Wigner, E., 1893,380
175,193f,196,
Subjects a, 36 actinides, 329 Ai(x), 190f,197,201,203ff,244, 260,270,281 Airy average(s), 191,199ff,311, 381 Airy's function ~6£ Ai(x) alkaline metals, 238 Amaldi correction, 22513 argon, 260 astatine, 2383o Avogadro's number, 257 B, 38,49f,53f,65 Baker's constant dee B Baker's series, 57,60,60 I° Bernoulli polynomials, 136~368 binding energy(-ies) of Bohr atoms, 6,300 of Bohr atoms with shielding, 13,21 of neutral HF atoms, 56 of neutral TF atoms, 38,56,66 of HF ions, 234 of TF ions, 78,81,83,96 of neutral TFS atoms, 138 of neutral SM atoms, 230 of SM ions, 232ff experimental, 297f relativistic correction, 374,3"26 binding-energy oscillations, 297ff double-peak structure, 297,341,
361ff Bohr atoms, 4ff,132ff,300ff,335 with shielding, 11ff size, 21ff,26 Bohr radius, 3,256 Bohr shell(s), 4ff,132,138ff,169, 220,287,304,371 Bohr-shell artifacts, 136,143,336, 210 de Broglie wavelength, 11,100,182,187 Ck, 63,88f,92,235 Ok, 356ff centrifugal barrier, 311 cesium, 288f chemical potential, 239ff Clausius-Mosotti formula, 263 closed-shell atoms, 256,286f,299 Compton profile, 379 correlation energy, 225,298,361 Coulson-March expansion, 6212,64,6413 count of states S6£ N(E) CSBE, 245 dk, 89f,93f Darwin term, 371ff degeneracy, 299f in the TF potential, 323ff linear ~, 335f lines of ~, 313,315ff,335,338 density (of electrons), 27,29ff,I05, 121 momentum space ~, 379f
399
at the nucleus, 147,164ff,363f pseudo ~, 246 quantum corrected ~, 216 radial %, 101,220f,288ff shells effects in ~, 379 TF ~, 35,102ff TFS ~, 147 density functional(s), I04ff,172, 196 of the energy (general), 106 of the exchange energy, 227f of the interaction energy, I06,221ff,245 of the kinetic energy, 106, 129,194 minimum property, 46,106,112 of the TF energy, 44 density matrix one-particle ~, I07,222ff,379 two-particle ~, 222ff density operator, 108,222 diamagnetic susceptibilities, 225f,281,379 dielectric constant, 263f dipole moment, induced electric %, 263 double counting of pairs, 17,31
F(x), 38,57,63ff,126,259,323ff Fro(y), 199ff,206f,282,293f Fm(V , I~V[), 211ff,278f,282 f(z),f(z), 349f,354,356 fk(x), 83,88 f~(x), 37,83,85,90,232f F~rmi momentum, 34 Fermi statistics, 3 field, electrostatic, 33,36,263ff fine structure constant, 124,242,370 francium, 238 Fresnel integral, 348ff G(y), 61ff g(z), ~(z), 349f,354 Gauss's law, 36 Gaussian notation, 5 gradient corrections,176 90ff,235 ~q,"expanding
in powers of ~," 4,188 Hamilton operator effective one-dimensional ~, 309 independent-particle ~, 27,241, 370 many-particle ~, 2,104,119 single-particle ~, 7,123 Hartree energy, 122 E, 228,232 Hartree equations, 124 Eosc, 296,301f,304f,334,338 Hartree-Fock ~ s e e HF iTF prediction, 340ff,345ff Hartree's method, relation to TF l,v contribution to ~, 346f approximation, 119ff I contribution to ~, 354,356 Hellmann-Feynman theorem, 23 s semiclassical prediction for ~, 360ff HF, 1,2,119 HF predictions Es, 131ff,211ff,371ff binding energy, 55f, 234f density at the nucleus, 166 mean square radius, 259 polarization radius, 264 192f,198,210ff,336,372f relativistic energy correction, El, 28ff,108ff 298,375 discrete sums for ~, 306 shielding of nuclear magnetic quantum corrections to %, moment, 244 193,210,335 Hohenberg-Kohn theorem, 104,106, TF version, 34 TFS version, 130,141 107,222 hydrogen, 297 e(q), 71,78,80,86,97,98 (q), 232ff independent-particle energy, 16ff,28 edge of the atom, 36,251f,260ff, inert gases s e e closed-shell atoms 268,270,274f,280,283,291f inhomogeneity corrections, 176,194 electric polarizability, 263f, interaction energy, 31,104,106,111ff, 266ff,275,294 120,221ff,245 electronegativity, 241 ionization energy, 162,236ff,298,364 ES model, 249ff,270,292,378 exchange energy, 16,162,167,175f, J, 144,168ff,214ff 193,221ff,245,262,270,338 kinetic energy, 44,104,106,111ff, Dirac-Jensen approximation, 117,119,129,222,379 228,261,276,378 Kohn-Sham equations, 377ff self ~, 16,225 krypton, 221 exchange interaction, 16,32,119, Lagrangian multiplier, 44,107,121 338 expectation value of I/r, 22f, lanthanides, 329 Lieb-Simon theorem, 1012° 241ff,365ff
E~j, E~s, 131,134,144,157,177f,
400 local oscillator approximation, 183 ZTF model, .338ff MES model, 272ff mercury, 164,170ff,243 MIT Differential Analyzer,
231
N, 2 N(~), N(E), 28,35,312ff,329ff N (E), 339,344,351ff,358f osc N (E), 314,329,332ff,337ff, qu343ff,348,350 Ns, 133,211 n(r) s e e density ~, 147,161,215ff nIME, 147,214,281 ns,
of the TFD model, ~j,146,168ff,213ff,28137
quantum corrections, 11,175ff,209, 314,335 to the count of states, 314 to exchange, 275 to the time transformation function, 177ff to traces, 191f quantum number(s), 30,299,378 angular %, 25,140,299,306,312, 327ff,371,381 maanetic ~, 140,306 principal ~, 5,25,132,140,169, 351 rE,ro,
132,211,372
2.49
315ff,319ff,325ff
NIE, 4ff,11,379 noninteracting electrons s e e NIE nuclear magnetic moment, shielding of ~, 241ff, (365ff)
radium, 328 radius, mean-square ~, 256ff relativistic corrections, 25946,298, 370ff rubidium, 288ff Rydberg energy, 3
orbits, classical ordered operator,
s, 181,183ff Sk, 57ff
no, 164ff,282f
~, 320ff 10f,382
Periodic Table, 2,298f,327ff phase, 180,183ff,226,380 Poisson equation, 27,31,35,36, 168,193,265,273,275,278 Poisson's identity, 173,313 polarization radius, 264 potassium, 288f potential(s) atomic ~, 317f effective ~, 14,16,27f,31,104, 108,122,149,280f,285f,306 effective vector ~, 124 electrostatic ~, 27,31,337 electrostatic pseudo ~, 250, 268ff exchange ~, 245,263,275ff, 286,291f,337 external ~, 104ff,116,149, 153,265 interaction ~, 31 pseudo ~, 247 pseudo exchange ~, 247,273 potential functional(s), 104ff, 196,337 in electrostatics, 41 of the ES energy, 253 maximum property, 40 of the TF energy, 34 of the TFS energy, 145 potential-density functional(s), 110 of the ES model, 248 of the MES model, 272 of the TF model, 110
Sk, 354ff,369 scale atomic ~, 3,371 nuclear ~, 371 TF ~, 131,153 scaling properties of the external potential, 116 of the interaction-energy density functional, 111ff of the kinetic-energy density functional, 111f of the E I potential functional, 115ff of the TF potential functional, 66ff of the TFS potential functional, 155ff Schr~dinger equation, 105,112,122, 181,306,308,377 Scott correction, 131,136,139,143, 145,153,155,158,162,163,172,186, 209,22513,240,277,335f screening, inner-shell ~, 12,13 self energy, 14ff,20,225 shell(s) closed %, 299 closed ~ of NIE, 7,305 filling of ~, 7,299,302,329 partly filled ~, 8,304 shell effects, 4,7,135,167,238,258, 295fff,379 size, atomic ~, 21ff,26 Slater determinant, 223,225 SM s e e statistical model
401
spectrum s e m i c l a s s i c a l ~, 189 quantum corrected ~, 190f statistical model, 228 statistical model predictions, 287ff binding energy, 230,234 d e n s i t y at the nucleus, 283 exchange potential, 291 ionization energy, 237 mean square radius, 260 radial densities, 288ff shielding of nuclear magnetic moment, 244 strongly bound electrons, 124, 130-172,186f,194,196,278, 292,299,311,316,326,335f, 371ff,376f,381 T(r',s,t) s e e tyme TF, 1,2,7 TF degeneracy, 323ff density, 35,102ff,193 equation~ 35,37 function see F(x) and f (x) model, 33ff,99ff,207ff q quantization, 308,311,322 variables, 46 TF predictions binding energy, 55,66,78,96ff mean square radius, 259 TFD model, 175,249,251f,255,292 TFS, 130 TFS density, 147 model, 130ff TFS predictions binding energy, 138 density at the nucleus, 165 Thomas-Fermi ~ s e e TF T h o m a s - F e r m i - D i r a c ~ s e e TFD T h o m a s - F e r m i - S c o t t ~ s e e TFS Thomas-Fermi-Scott-Weizs~ckerDirac model, 228 T h o m a s - F e r m i - y o n W e i z s ~ c k e r model, 175 Tietz potential, 12628,220,368 time t r a n s f o r m a t i o n function, 177ff,225,293,380 traces and phase-space integrals, 10f quantum corrected ~, 191,198 s e m i c l a s s i c a l evaluation of ~, 9,11,178 v a r i a t i o n of ~, 29 tyme, 180,183ff,226,380 units, atomic, uranium, 299
2,3,4,9,124
V(~) See potential(s), effective virial theorem(s), 68,70,112,140, 163
wj, 144,168ff,214ff von W e i z s ~ c k e r term, 194ff,196 I° Wigner functions, 379ff W i g n e r - K i r k w o o d expansion, 189 WKB q u a n t i z a t i o n s e e TF q u a n t i z a t i o n Xo (q) , 37,78,86,98 Y' Ys' Yj' y (x) , 62
199ff,210ff,282,284,292
Z, 2 z,z s , 197ff ~, 82 B, 62,65 y, 61 ,62 ~, 28,239ff, 378 ~j' ~S' 131,142,168ff,210ff,281, 335f,372f q(X),
7,304
A, 8Off, 127ff I, 1 E, 1 o, 303,312,314ff,323,325ff, 329,340 l(q), 74f,78,87 I oscillations, 35Off l,v oscillations, 344,351 V,VE(1),Vo(1) , 307,312f,316,318ff, 322ff, 329,332,344 ' " 317,320,325f, 343, 'v E, 'v O, 318f,324,343,345 v s, 133ff,372 v oscillations, 351,358ff Ps'
140,150,156,211
o, 82 ~(t), 81f,89,236 ¢(~',~,t) s e e phase ~k(t) , 77,127,233 ~i (t) , 74ff,232f ~m(t), l~nl a2v
90,236 '
142,148,150,156,214,381
cOE, e o, 316f,319f,325ff [y], , 5 °, 191,197ff
Semiclassical Theory of Atoms (Lecture Notes in Physics, Vol. 300, Springer 1988) by B.-G. Englert List of typographical errors (updated February 1996) 1. On p. 3, the second line after Eq. (1–8), it should be ‘macroscopic’, rather than ‘macroscopics’. 2. In the text between Eqs. (1–17) and (1–18), p. 6, the reference should be to Eq. (16), not to Eq. (15). 3. The right–hand side of the second equation in Problem 1–3, p. 25, should read p+Ft, rather than p+xFt. 4. In Eq. (2–94), p. 44, replace ζ by γ
Z . r γ
5. Read β˜ (x2 /˜ x2 ) rather than β (x2 /˜ x2 ) in Eq. (2–202) on p. 65. 6. The first line of Eq. (2–238), p. 70, should end with (− 73 ETF − ζN), not with (− 37 ETF + ζN). 7. On p. 75, read t=1 instead of t>1 in the line preceding Eq. (2–266). 8. The numerator in the integrand of Eq. (2–268), p. 75, should be q0 4/3 , rather than q0 7/3 . 9. In the first line on p. 79 replace ‘(287)’ by ‘(286)’.
1
10. In Eq. (2–349), p. 88, read [y(x)]k instead of [y(x)k ]. 11. On p. 93, the first line after Eq. (2–379), replace ‘statet’ by ‘stated’. 12. In the line preceding Eq. (2–436), p. 110, read ‘functional’ instead of ‘funcitonal’. 13. On p. 134, the last term in Eq. (3–15) should be (ζs − ζ) tr η(−H − ζs ) rather than (ζs − ζ) tr (−H − ζs ). 14. On p. 138, the second line in Section Scott’s original argument, replace ‘as as’ by ‘as’. 15. On p. 145, the fourth line after Eq. (3–64) should begin with ‘understood’, rather than ‘unterstood’. 16. The fourth line after Eq. (4–67), p. 186, should end with ‘of’, not with ‘or’. 17. The line after Eq. (4–91), p. 191, should begin with ‘Consequently’, not with ‘Concequently’. 18. The second line after Eq. (4–99), p. 192, should begin with ‘Delta’, not with ‘delta’. 19. In Eq. (4–129), p. 198, an integral sign is missing in front (d~r) (d~p) of on the right-hand side. (2π)3 20. The second line after Eq. (4–157), p. 203, should end with ‘ for y → ∞ ’, rather than ‘ for z → ∞ ’. 2
21. For ‘contruction’ in the first line before Eq. (4–236), p. 223, read ‘construction’. 22. On p. 230, the third line in Section History, replace ‘remakred’ by ‘remarked’. 23. On p. 232, the sixth line from the top, it should be ‘performed’, rather than ‘preformed’. 24. On p. 239, the fourth line before Eq. (4–299), insert ‘electron from the’ after ‘removal of one’. 25. In Eq. (4–320), p. 244, the power 5/3 should be replaced by 5/2. 26. On p. 246, the integral in the second line of Eq. (4–332) should also be performed over (d~r 0 ). 27. In the last line on p. 263 read ‘Clausius–Mossotti’ instead of ‘Clausius–Mosotti’. The same applies to the corresponding entries in the Index, pp. 397 and 398. 28. In Eq. (4–485), p. 276, replace [n(~r)]4/3 by [3π 2 n(~r)]4/3 . 29. In the seventh line of Problem 4–10, p. 294, read F21 (y) instead of F2n+1 (y). 1/3 and 30. In Eq. (5–27), p. 305, replace 32 Z 1/3 −1/3 3 3 and 2 Z , respectively. 2Z
−1/3 2 3Z
by
31. On p. 309, the fourth line after Eq. (5–41), replace ‘do’ by ‘to’. 3
32. On p. 309, the third line before Eq. (5–42), replace r0 (x0 → ∞) = 0 by r0 (x0 → −∞) = 0. 33. Equation (5–45), on p. 309, should be numbered correctly. 34. In Eq. (5-47), p. 309, it should read 4 s3 (x) square brackets, rather than 4 s(x)
d2 s(x) . dx2
d2 s(x) in the dx2
35. In the line after Eq. (5–58), p. 312, replace ‘Eq. (209)’ by ‘Eq. (2–9)’. 36. For ‘centripal’ read ‘centripetal’ in the first line before Eq. (5–111), p. 320. 37. The abscissa of Fig. 5–9, p. 323, should be labeled λ/Z1/3 , rather than λ/Z4/3 . 38. Read (Z/Z1s )1/3 instead of (Z/Z1s )1)3 in the top line of Eq. (5-145) on p. 330. 39. The second term on the right–hand side of Eq. (5–149), p. 331, should be subtracted rather than added. 40. On p. 345, the first line after Eq. (5–191), replace ‘insteat’ by ‘instead’. 41. On p. 355, the second sentence after Eq. (5–232), exchange the words ‘even’ and ‘odd’ with one another.
4
42. In the first line of Eq. (5–236), p. 356, read ‘odd’ rather than ‘even’; in the second line read ‘even’ rather than ‘odd’. 43. On p. 361, the sixth line from the bottom, read ‘Fig. 18’ instead of ‘Ref. 18’. 44. The abscissa of Fig. 5–22, p. 366, should be labeled Z1/3 , rather than Z. 45. The last line in Problem 5–1, p. 367, should begin with ‘Observe’, not with ‘observe’. 46. On p. 370, the second line before Eq. (6–1), replace ‘(2–1)’ by ‘(2–3)’. 47. On p. 378, the sixth line from the bottom, insert ‘done’ between ‘must not be’ and ‘more accurately’. 48. In the third line of Eq. (6–51), p. 381, replace
1 2
by 2t .
49. On p. 385, the last equation in Footnote 7 should end with d [ dr (rV)]2 , rather than [(rV)]2 . 50. On p. 385, the fourth line of Footnote 14 read ‘observe’ rather than ‘ovserve’. 51. In the last sentence of Footnote 14, p. 385, read ‘decimals’ instead of ‘decimales’. 52. In Footnote 15, p. 395, the year of publication should be 1985, not 1986.
5
