Semiclassical theory of quantum defects: Alkali ...

Semiclassical theory of quantum defects: Alkali Rydberg states Charles Jaffe and William P. Reinhardt* Department of Chemistry. University of Colorado and Joint Institute for Laboratory Astrophysics. University of Colorado and National Bureau of Standards. Boulder. Colorado 80309 (Received 16 August 1976) A semiclassical theory of atomic quantum defects is given in terms of a "radial action defect" derived in the context of Hamilton-Jacobi theory. This derivation leads to the relation aleE) = 7TJLI(E) between the semiclassical (WKB) phase shift and the semiclassical quantum defect. a result identical in form to that of Seaton for the analogous fully quantum mechanical quantities. The semiclassical theory and is used to compute quantum defects for the I = 0-13 Rydberg states of Li and the I = 0-4 states of Na and K. with good results.

I. INTRODUCTION

There has recently been a resurgence of interest in the high Rydberg states of atoms 1 and molecules. 2 Since these states are often well parameterized via single,3 or many channe14 quantum defect theory, it is of interest to develop methods for direct computation of the quantum defects themselves, as their knowledge leads immediately to the Rydberg energy levels in onedimensional systems and through the theory of "frame transformation" 5 to many features of the spectra of molecular Rydberg levels. 6 That the high Rydberg states are characterized by very large quantum numbers makes variational quantum calculations tedious and often basis dependent. However, it has been shown by Percival and co-workers that in the limit as both l and n become large that semiclassical and classical correspondence principle arguments may be successfully used in discussions of the dynamics of highly excited atoms. 7 It is the purpose of this paper to begin development of a semiclassical theory of atomic and molecular quantum defects, taking advantage of two features of the problem. First, although in many observed Rydberg series l is not large, n is large, suggesting the possible utility of a semiclassical technique appropriate to the large n limit and, second, the asymptotic coordinate space potential seen by an atomic or molecular Rydberg electron is purely Coulombic, where it is well known that straightforward WKB quantization gives exact results. Specializing to atomic systems, we briefly review the quantum defect theory of Seaton, 3 pointing out that one need not explicitly perform a bound state quantization to obtain the quantum defect-it can be calculated directly as a function of energy in terms of the corresponding "negative energy" phase shift. In Sec. III an analogous development is made within the framework of classical Hamilton-Jacobi theory. Subsequent application of the generalized Bohr-Sommerfeld8 • 9 quantization rule 1==

;7r J Pdq==(n+*)n ,

(1)

being the Maslov index9 (see Appendix), gives a Rydberg_like lO formula and an identification of the semiclassical quantum defect as a difference between two actions. This difference is then shown to be equivalent Q!

to a "negative energy" WKB phase shift, giving rise to a semiclassical analog of the Seaton quantum defectphase shift relationship. This is followed (Sec. IV) by a discussion and classification of the classical orbits. In Sec. V semiclassical techniques are used to compute energy dependent quantum defects for the alkali Rydberg states for l == 0 - 4. The Hamilton-Jacobi technique and, for nonpenetrating orbits, a Born-Heisenberg 11 perturbative result are found to give good results when compared with experiment and with quantum calculations using the same model potentials. II. QUANTUM DEFECTS: QUANTUM MECHANICS

S eaton3 has discussed potentials of the form j(r) V(r)==- ,

(2)

r

wherej(r)-Ze 2 for r2':ro'

In the asymptotic region where r > ro the S chrodinger equation has solutions of the form (3)

where K==i(Ze 2 /k), k being the wave number, and where Yl and Y4 are appropriate regular and irregular Coulomb functions, analytic in 1/K 2 , /3(E) is determined by

the boundary conditions for matching the solution for r(rp) gives the usual separation of variables 1Z ,13: (9a) as) ( ae as) ( fJr

(2

r,4>

j32 )1/2 =Po= j38-~ ,

(9b)

=Pr =( 2m[E -

(9c)

V(r)]

j32 )1/2 -y! ,

8,4>

and the action integrals l3 ,14

=~{l 1:.

Ii 27T jCl

Z (2m[E- V(r)]_Ii [l+(V 2 )]2)lIZ dr

- ;7T

{J2m (E+ Z;Z) _n [l+S/2)]T/Z dr} ,

(17) thus allowing 11-,(E) to be calculated directly, as a function of E, from classical trajectories. The fact that the separation action variable j30 is identified as ~ +(1/2)]n, I being an angular momentum quantum number, is discussed in the Appendix. Equation (17) is the semiclassical analog of the Seaton relationship between 11-,(E) and o,(E). This may be seen by noting that the WKB phase shift (relative to the WKB Coulomb phase shift) is, for positive energies, no,(E) =

f

r>ro

TV

(lOa)

1:.{ j382 -

1 18 =27T j 1 Ir= 27T

f{

j3~

sin2e

}1/2

2m[E - V(r)]

de ,

j32}I/Z -y! dr.

(lob) (10c)

The first two of these integrals are independent of V(r) and are easily integrated,13 giving I

,

10 = (j30 - j34» •

(lla) (llb)

Ir depends on the potential V(r), both through the ap-

pearance of the potential itself in the integrand and through the specific path C1 defined (in the one-dimensional case) by the appropriate periodic classical trajectory. As such, Ir is not usually analytically integrable. However for a pure Coulomb potential the integral

r

2

Pr(r) dr-

Jr>ro

(18)

pcouI(r)dr,

TCoul

where ry and rCoul are the inner turning points for the full and Coulomb potentials, respectively. For negative energies, assuming that ro is inside the outer turning point (i. e" that we are close to threshold), Eq. (18) may be extended to full periods, since Pr(r) and PCoul (r) are identical for r > rD. The result is

ol(E)=i(~ fc/r(r)dr-~ £/COUl(r)dr)

•

(19)

Equation (19) provides the analytic continuation of the WKB phase shift to negative energies, where we find, by comparison of Eqs. (17) and (19), the familiar result 7TI1-,(E) =15,(E), B. Secular perturbation theory In 1924 Born and Heisenberg (BH), l1,1Z treated the case of nonpenetrating orbits (high 1) by the method of secular perturbations using the polarization term as

J. Chern. Phys., Vol. 66, No.3, 1 February 1977

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1287

C. Jaffe and W. P. Reinhardt: Semiclassical theory of quantum defects

The depth of penetration into the core is greatest for 1=0 and decreases as 1 increases. From Eqs. (18), (21a), and (21b) it is immediately seen that the lower the angular momentum, the larger the quantum defect, the angle of preceSSion, and the time of precession. Based on this observation the orbits may be classified into three groups: nonpenetrating orbits (NPO's), shallowly penetrating orbits (SPO's), and deeply penetrating orbits (DPO's).

0.4 0.3 0.2 :i d

0.1 0

a: -0.1 >""-0.2 -0.3

Nonpenetrating orbits, which are high angular momentum states, precess slightly due to the polarization of the core by the Rydberg electron: The quantum defects of these orbits generally are less than 0.01, and are easily calculated via Hamilton-Jacobi theory, or by the Born-Heisenberg secular perturbation theory.

POTASSIUM .[ =2 MODEL POTENTIAL

-0.4 -0.5 -0.6 -0.7 0

R Bohr

FIG. 1. Potassium core model potential (see text and Ref. 19) for 1=2 illustrating barrier [near 2.5 a. u. (a o)] giving rise to quantum effects not included in the simple Hamilton-Jacobi semiclassical treatment.

the perturbation. However, at that time they were not aware that the angular momentum quantum numbers entered as half-integers. With this correction their expression for the quantum defect is

a

1

I1n,l

"'4

(

(l+ 1/2)5 3 -

(l+ 1/2)2)

n2

,

(20a)

which we have modified to read (20b) IV. CLASSICAL CHARACTERIZATION OF QUANTUM DEFECTS It is well-known that the Bohr orbits of hydrogen are ellipses. From this we expect the classical orbits to be elliptical in the asymptotic region, where the electron only feels the Coulomb field. When the electron penetrates into the core the orbit is no longer elliptical, resulting in precession of the asymptotic orbit. The angle of precession XI(E) (for a single passage through the core), can be related, in analogy to the relationship 15 between the WKB phase and the classical scattering angle, to the quantum defect by XI

(E)-2 -

7T

fJJ.lI(E) al

I

E'

-

fJJ.lI(E)1 ---aE I

•

Shallowly penetrating orbits, which are states of intermediate angular momentum, present a much more challenging situation. A variety of difficulties arises in their calculation, the first being that the electron samples the region of space where the matching of short and long range potentials is important, and the simple model potentials used in the present calculations are inadequate. In addition, quantum effects arise that do not occur in any of the other cases. These are related to barriers of the type shown in Fig. 1. V. NUMERICAL APPLICATIONS: ALKALI RYDBERG STATES

Since the alkali metals consist of a single electron outside a closed shell and exhibit unperturbed Rydberg series, they provide excellent systems for testing semiclassical ideas. The core model potentials were constructed from a static Hartree-Fock term, Riley and Truhlar's local semiclassical exchange approximation/ 7 TABLE I.

0

:3

(21b)

It should be noted that Eqs. (21a) and (21b) are valid

under the same conditions which allow us analytically to continue the WKB phase shift to negative energies.

Lithium quantum defects. "

Character of orbit

,J'H

I1"M

J.1 Exv t

DPO

10. :194 10. :195

4.62 4.61

0.40:1

0.404

0.400 0.400

- O. 001 - O. 01

SPO

10.0667 10.0659

0.0190 0.0187

0.0585 0.0581

0.0471 0.0467

- O. 001 - O. 01

NPO

I 0.001:17 10.00i35

0.00147 0.00144

0.00218 0.0021:1

0.00192

- O. 001 - O. 006

(21a)

Similarly, in analogy to the relationship16 between the collision time and the WKB phase shift, the "precession time" can be expressed in terms of the quantum defect as TI (E) - 7T

Deeply penetrating orbits, which are the low angular momentum states, precess by large amounts with correspondingly large quantum defects. The most important interaction in the DPO case is the static potential.

NPO

,J'J

0.00188

! 0.000265

0.00027:)

0,000295

0.000308

10.000254

0.000261

0.000283

0.000295

E

- O. 001 - O. 006

"As calculated using the Hamilton-Jacobi (HJ), Born-Heisenberg (BH), methods discussed in the text. The orbits are classified as DPO, SPO, and NPO following the discussion of Sec. IV. The quantum mechanical (QM) results were obtained solving the radial Schrodinger equation using the same model potential used to estimate JllJ. The experimental results are those of I. Johansson, Arkiv Fysik 15, 169 (1958), and P.d P. Risberg ibid, 10, 583 (1956). The energy E is measured in a. u. (Hartree units) from the threshold of ionization of the Rydberg electron.



1288


and a Buckingham polarization term with the parameters of Bardsley. 18 Potentials of this form were found previously to give excellent quantum defects when used in fully quantum mechanical calculations. 19 In the fully quantum calculations, quantum defects were obtained by numerical integration of the Schrodinger equation, matching to appropriate Coulomb functions in the asymptotic region. The Hamilton-Jacobi results were obtained by a GaUSS-Legendre quadrature of Eq. (18). The modified BH results were obtained from Eq. (20b). Semiclassical quantum defects for DPO's, NPO's, and SPO's for Li, Na, and K are presented in Tables I, II, and III, where they are compared with results of fully quantum mechanical calculations using the same model potentials and with experiment. In the case of NPO's the results are also compared with the modified BH quantum defects. As is evident from the tables the semiclassical quantum defects for the DPO's and NPO's are in excellent agreement with experiment, and with the fully quantum calculations using the same model potentials. The appropriately modified BH theory works well for the NPO's, where the assumptions embodied in its derivation are satisfied. For the SPO's the semiclassical results agree less well with the corresponding quantum results due to the complex turning points (see Fig. 1) lying near the real axis, which indicates the need for inclusion of more points of stationary phase in the semiclassical expansion of 0/(E). Such modifications were not made, as comparison of the quantum results for the SPO's with experiments indicates that the simple model potential is itself inaccurate. These quantum corrections could easily be made following the techniques of Ford et aZo 20 In conclusion we note that the semiclassical methods work well for l values down to l =0, as long as n is large. We also note that the Riley-Truhlar17 semiclassicallocal exchange approximation, which was designed for electron scattering at positive energies, provides an excellent ab initio way of estimating the "exchange" contribution to a model potential suitable for quantum calculation of quantum defects (see the discusof Ref. 19), and also that this same local exchange approximation appears to be perfectly suited to approxi-

Potassium quantum defects. a

TABLE III. Character of orbit

0

i!'1J

V'H

J,fIM

t-t

1:11. :1 1:11 . 1

2.175 2.178

2.180

DPO

2.18:)

-0.001 - O. 01

DPO

1 1. 712 11.717

O. '):19 0.5:12

1.712 1. 716

l. 17:1 I. 717

- O. 001 - O. 01

IO.52G

10.51:1

0.0418 0.040:1

0.420 0.:398

0.275 0.Z5G

- O. 001 - O. 01

o.OOn!)

0.00888

- O. 001 - O. 006

2

SPO

:1

NPO

10.00719 10.00691

0.00775 0.0074:)

0.00863 0.00825

4

NPO

I O. 00208 10.00192

0.00219 0.00204

0.002211 0.00203

Character of orbit

mate the exchange interaction between an atomic core and a distinguishable "classical" outer electron. ACKNOWLEDGMENTS

Helpful conversations with I. C. Percival, D. Truhlar, and W. C. Lineberger are gratefully acknowledged. This work was supported in part by National Science Foundation Grant CHE74-19605 and by ERDA. APPENDIX

In this appendix we outline the application of the Keller 21 _Maslov8 generalization of the Bohr-Sommerfeld quantization rule to the problem of the hydrogen atom. As these generalizations apply, following the work of Einstein,22 to the nonseparable case, their application to the separable Kepler problem is carried out to give a simple illustrative example of the method and to provide a framework for future extension to molecular Rydberg states. In the Hamilton-Jacobi formulation of the Kepler problem in spherical polar coordinates, 12,13 separation of variables and definition of the action integrals follows Eqs. (9) and (10), of the text. In particular (for the classical H atom), the action variables separate as 12 ,13 (Ala)

V'H

J,fIM

).,E%J)t

22.7 22.6

1.352 1. 354

1. 348 1. 349

- O. 001 - 0.01

11. 357 11.359

DPO

10.842 10.845

0.0932 0.0919

0.859 0.861

0.855 0.857

- O. 001 - 0.01

2

SPO

10.00686 10.00654

0.00723 0.00696

0.0150 0.0143

0.0148 0.0141

- O. 001 -0.01

3

NPO

10.00127 10.00122

0.00134 0.00128

0.00145 0.00139

0.00168 0.00159

- O. 001 - O. 006

4

NPO

\ O. 0003{j3 10.000335

0.000379 0.000353

0.000384 0.000354

"See Table I for explanation of the table.

(Alb)

f3 ,

e2j~~

,

(A1c)

E

1)1>0

0

= f3e -

Ir= - (I+le)+ !l1J

- O. 001 - 0.006

"See Table I for an explanation of the table.

Ie

Sodium quantum defects. a

+le+ 1,)2 •

(A2)

One obtains the Bohr formula by writing 13 (I + Ie + I,) This is clearly the "correct" result 23; however, it seems to be in conflict with the fact that the quantization rule 24

=nn.

1= ;1T

f

p dq =(n+ 1/2)n

(A3)

must be used to obtain correct results for the oscillaJ. Chern. Phys., Vol. 66, No.3, 1 February 1977



tor. The question is, when does one quantize to an integral and when toa half-integral multiple of no Resolution of this problem for nonseparable systems follows from the work of Maslov8 generalizing the ideas of Keller. 2o Percival9 has recently reviewed the situation, both with respect to quantization, and to the form of semiclassical wave function. For the separable case at hand the proper quantization rule is 9

where Cl!k is the topological "Maslov index" which has the value "2" for librations and "0" for rotations. We refer to this as the generalized Bohr-Sommerfeld quantization rule. Since e and rare libration variables and ¢ is a rotation variable, we have nq,

=0,1,2 •.•

(A5a)

,

(A5b) Ir=(n r + 1/2)n,

(A5c)

n r =0,1,2 ••• ,

giving (A6)

[q,+le+lr=(nq,+ne+nr+ 1)n ,

namely that the total action is quantized in integral multiples of n, beginning at 1. Completing the argument, we note that

=~ 21T

I Coul r

f

C2

[2m(E+~) r

I/2 (1;+[6)2J

r2

dr.

(A7)

Quantization of I q, and 10 gives Iq, + 16

=/36 =(nq, + n6 + 1/2)1/,

n e , nq, =0,1,2 .••• (AS)

However, since I q, + Ie corresponds to the total angular momentum, 13 we can write

f3e = (1 + 1/2)1/,

(A9)

1=0,1,2 •••

and interpret [=nq,+n e as an "angular momentum" quantum number giving [Caul r

=~ 21T

f

[2mIE+~)-1/2 (l+ 1/2)2J1/2 dr C2

\'

r

r2

(A10)

and E

=-

(AHa)

21/2(nr + 1/2 + 1+ 1/2)2 me 4

nr' l=0,1,2 ... ,

(Allb)

which is formally identical with the fully quantum result,25 giving the identification of n r + 1+ 1 as the "principal" quantum number n = 1,2,3 ••• 0

We thus see that the use of the 1+ i in Eqo (17) and the integral quantization of Ir+ 16 + Iq, in Eq. (15) both follow at once from the generalized quantization procedure. The results of this Appendix as applied to the separable Kepler problem are, of course, not new,28 but indicate how one will proceed in the nonseparable case, where the Keller-Maslov generalizations give the correct quantization procedure.

1289

*Camille and Henry Dreyfus Teacher-Scholar. IFor example: W. H. Wing and W. E. Lamb, Jr., Phys. Rev. Lett. 28, 265 (1972); T. F. Gallagher, S. A. Edelstein, and R. M. Hill, Phys. Rev. A 11, 1504 (1975); Phys. Rev. Lett. 35, 644 (1975); P. M. Koch and J. E. Bayfield, ibid., 34, 448 (1975); T. W. Ducas, M. G. Littman, R. R. Freeman, and D. Kleppner, ibid. 35, 366 (1975); K. B. McAdam and W. H. Wing, Phys. Rev. A 12, 1464 (1975); H. J. Beyer and K. J. Kollath, J. Phys. B 9, L185 (1976); P. Esherick, J. A. Armstrong, R. W. Dreyfus, and J. J. Wynne, Phys. Rev. Lett. 36, 1296 (1976); M, G. Littman, M. L. Zimmerman, T. W. Ducas, R. R. Freeman, and D. Kleppner, ibid. 36, 788 (1976). 2For example: W. A. Chupka, P. M. Dehmer and W. T. Jivery, J. Chem. Phys. 63, 3929 (1975); J. A. Schlavone, K. C. Smith, and R. S. Freund, J. Chern. Phys. 63, 1043 (1975); T. G. Finn, B. L. Carnahan, W. C. Wells, and E. C. Zipf, ibid. 63, 1596 (1975); W. P. West, G. W. Foltz, F. B. Dunning, C. J. Latimer, and R. F. Stebbings, Phys. Rev. Lett. 36, 854 (1976); see also Ref. 6. 3M. J. Seaton, Mon. Not. R. Astron. Soc. 118, 504 (1958). At alternate derivation based on the phase amplitude method appears in J. L. Dehmer and U. Fano, Phys. Rev. A 2, 304 (1970), 4M. J. Seaton, Proc. Phys. Soc. 88, 801 (1966); 88, 815 (1966); K. T. Lu, Phys. Rev. A 4, 579 (1971); C. M. Lee and K. T. Lu, ibid. 8, 1241 (1973); C. M. Lee, ibid. 10, 584 (1974). 5See , for example: U. Fano, Phys. Rev. A 2, 353 (1970); E. S. Chang and U. Fano, ibid. 6, 173 (1972); u. Fano, J. Opt. Soc. Am. 65, 979 (1975), 6G. Herzberg and Ch. Jungen, J. Mol. Spec. 41, 425 (1972); O. Atabek, D. Dill, and Ch. Jungen, Phys. Rev. Lett. 33, 123 (1974), 71. C. Percival and D. Richards, Adv. Atom. Mol. Phys. 11, 1 (1975). By. P. Maslov, Theorie des Perturbations et Methods Asymptotiques (Dunod, Gauthier-Yillars, Paris, 1972), 91. C. Percival, Adv. Chern. Phys. (to be published), IOJ. R. Rydberg, Phil. Mag. 29, 331 (1890), A good discussion appears in M. Jammer, Conceptual Development of Quantum Mechanics (McGraw-Hill, New York, 1966), Chap.2. "M. Born and W. Heisenberg, Z. Phys. 23, 388 (1924); M. Born, Ref. 12, p. 165. 12M. Born, The Mechanics of the Atom (Ungar, New York, 1960). 13H. Goldstein, Classical Mechanics (Addison Wesley, Reading, Mass., 1950). 14Ik=Jk/21'f, where the J k are those of Goldstein, Ref. 13. 15K. W. Ford and J. A. Wheeler, Ann. Phys. (N. Y.)7, 259(1959). 16See, for example, J. R. Taylor, Scattering Theory (John V-iley, New York, 1972), p. 251. 17M. E. Riley and D. G. Truhiar, J. Chern. Phys. 63, 2182 (1975); 65, 792 (1976). 1BJ. N. Bardsley, Case Stud. At. Phys. 4, 301 (1974). 19c. Jaffe andW. P. Reinhardt, J. Chem. Phys. 65, 4321 (1976). 2o K . W. Ford, D. L. Hill, M. Wakano, and J. A. Wheeler, Ann. Phys. (N.Y.) 7,239 (1959); W. H. Miller, J. Chem. Phys. 48, 1651 (1968). 21 J . B. Keller, Ann. Phys. (N.Y.) 4, 180(1958). 22 A. Einstein, Yerhand. Deut. Phys. Ges. 19, 82 (1917). 23See for example: L. Pauling and E. B. Wilson, Jr., Introduction to Quantum Mechanics (McGraw-Hill, New York, 1935), p. 39, where the Wilson-Sommerfeld rules are quoted as Ir=n,n, I9=n/i, Iq,=n,/f. 24See, for example, L. D. Landau and E. M. Lifshitz, Quantum Mechanics, 2nd ed. (Pergamon-Addison-Wesley, Reading, Mass., 1965), p. 163. 25Reference 23, p. 125. 26H. A. Kramers, Z. Phys. 39, 828 (1926): See also R. A. Marcus, J. Chern. Phys. 54, 3965 (1971) [Eq. (3.4»).

