Sep 1, 1988 - Parker. Department ofPhysics, North Carolina State Uniuersity, Raleigh, North Carolina 27695 8202-. (Received 21 March 1988). Numerical ...
GENERAL PHYSICS
SEPTEMBER 1, 1988
THIRD SERIES, VOLUME 38, NUMBER 5
Numerical solution of the Thomas-Fermi equation for molecules
G. %. Parker Department
of Physics,
Raleigh, North Carolina 27695 (Received 21 March 1988)
North Carolina State Uniuersity,
8202-
Numerical solutions of the multicentered Thomas-Fermi equation for molecules are obtained by use of an efficient multigrid scheme based on the standard finite-difference discretization of the Laplacian in a modification of this nonlinear equation. It is found that the difference between the molecular potential and the sum of atomic potentials is bounded and varies smoothly through the nuclei, and is thus a good choice for the dependent variable to be discretized. This choice also results in a general program that is applicable to any molecule whose structure is known; the input data are the nuclear charges and their locations. The electron density and the difference between the electron density and the sum of atomic densities are obtained for N2 and compared with the results of molecular-orbital calculations. The method s applicability to larger molecules is illustrated by a calculation of the electron density of C6H6. Its extension to crystalline solids with cubic unit cells is straightforward.
where r, = r — R, ~, with r measured from an origin which is subsequently taken to be at the center of nuclear
INTRODUCTION
~
The Thomas-Fermi (TF) equation is the result of a simple and elegant formulation of the proble~ of calculating ground-state electron densities and effective potentials for The relation between atoms, molecules, and solids. ' the electron density n and one-electron potential energy V from the free-electron-gas model is extended to the case of a slowly varying potential energy V(r ) and combined with Poisson's equation to give a boundary value problem for V(r). A more general derivation uses an electron energy functional E(n) containing a Hartree-type electronelectron interaction, the interaction between electrons and nuclei, and a free-electron-gas kinetic energy term. Carrying out the variation of E subject to the constraint of normalization gives an algebraic relation between V and n which, for neutral atoms or molecules, is (atomic units are used throughout) n
(r) =(1/3n. )[ —2V(r)]
Combining this with Poisson s equation, one obtains the TF equation for a molecule,
V+
g Z, /r,
=(4/3~)( —2V} ~2
(2)
The nuclear charge density, produced by nuclei of charge Z, located at R„has been carried to the left-hand side of Eq. (2). Boundary conditions on this equation are
—Z,a /r,a (r, ~0) 0 (r~ao},
(3a) (3b) 38
charge. The several approximations leading to Eq. (2) require high electron densities for their validity in addition to This the constraints of the semiclassical approximation. latter approximation breaks down both near and far from the nuclei as reflected in the divergence of n as 1/r, ~ as r, goes to zero and its slow decrease as 1/r at large distances from a molecule. It is also known that TF theory does not yield bound rnolecules (no-binding theorem). '6 In spite of these limitations, the Thomas-Fermi approach is of interest if only as a starting point for a more accurate description of electron density. In particular, various corrections have been proposed for it that offer the to the accuracy comparable promise of obtaining Hartree-Fock method while still maintaining much of the computational simplicity of a Poisson-type equation. ' Because of this, I have investigated the possibility that the TF equation may be solved efficiently for an arbitrary collection of atoms. There are additional reasons for interest in the TF method. First, it becomes exact in the limit of large atomic numbers, which makes it a natural starting point for further developments. ' For example, the nonrelativistic to&al energy of a neutral atoms has been developed as a series involving different powers of Z with the leading term of this series the TF energy. ' Secondly, the TF density agrees with actual atomic densities in the core region of atoms' (even atoms with atomic numbers of order 10), and it has been used to predict results of x-ray and electron-scattering experiments. Furth-
"
2205
1988
The American Physical Society
G. W. PARKER
2206
ermore, the corresponding potential energy has provided a useful effective potential for the calculation of orbital energies. ' Finally, the TF equation predicts that neutral molecules will dissociate into neutral atoms in accord with experiment, and in contrast with the self-consistent' field linear combination of atomic orbitals method.
0&6, V&S&/2 .
bn~0(r,
My method is based on a transformation to a depenthe dent variable which automatically incorporates ' This is boundary condition at the nuclei, Eq. (3a). achieved by utilizing the solutions V, for isolated atoms. These atomic solutions have the form
= —Z, 4(r, )/r, , 4(0) =1 is the atomic
(4)
equivalent of Eq. (3a). For reference, I note here that at large distances where
V.
k/— r.' (r. »1),
~ (
V, +Z,
/r, ) =(4/3n
)(
—2V, )
(5)
d, =(g/15~)(2Z, )'
Defining the sums
V'
g —2V,
(b, V) =(4/3m)[(
(
)
—2b, V+Sl } —Sq]
Here b, n = n — n, is the difference between the molecular electron density and the sum of TF atomic densities. Some insight into the solution of (7) can be obtained from the potential hVN, , which makes the density difference "noninteracting" hn =0. This potential energy difference is given by
g,
( I/2)[Si VNt ——
—(S
)
],
(g)
and although not, of course, a solution to the TF equation, it is expected to give a rough approximation to the true solution owing to the no-binding theorem. In particular, we see that the relation of the true solution to the noninteracting solution determines whether the electron density increases or decreases on bonding TF atoms into a molecule. The following two inequalities, with their implications in parentheses, state this connection: b V&hVN,
(bn &0), hV&hVN,
(bn &0) .
(9)
Bounds for 6V follow from the fact that V(0 and the evident fact that the molecular electron density will be contracted, relative to the sum of atomic densities, so that V & V, and thus
g,
g
i2V (a)
i
—25V(a)
',
(b, W) =
4'(b, n —b, n,
—
),
'
(r,
&&Z
'
)
.
(12)
',
Equation (11) is applied near nuclei where r, is small compared to Z, the latter factor being the basic length scale introduced by the TF equation. The asymptotic solution at large distances from the molecule is obtained from Eq. (2) by expanding the nuclear potential about the center of nuclear charge, giving
gZ, /rg —gZ, /r+O(1/r
)
.
This shows that the molecular TF equation reduces to the equation for an atom of charge Z equal to the total nuclear charge. In practice, this asymptotic solution V„ is used to provide the boundary condition on Eq. (7). To summarize, I have reduced the TF equation for molecules to Eqs. (7) and (11), together with the boundary condition at large distances that 6 V approach V„— V„ the TF solution for an atom located at the center of nuclear charge with Z, minus the sum of TF atomic potentials. Following a description of the dissociation limit, I will describe brieAy how these equations, as well as Eq. (5), have been solved numerically.
TF
=4mhn .
b
.
where the density difference near the nucleus of charge Z„which turns out to be positive, is
and combining Eq. (2) with Eqs. (5) and (6}, a new form of the TF equation for molecules is obtained,
—V
'
bn, =(15d, /16~)r, (6)
Sq ——
EV~b V(a)+O(r, i2)
A8'=hV+d, r, i
of nuclear charge. The
V, .
S, =g( —2V, ),
),
The TF density difference diverges at nuclei but less strongly than n itself, while the new variable hV approaches a constant value b, V(a). This behavior near nuclei suggests a transformation of Eq. (7) to a form more suitable for relaxation there. I put
The new dependent variable is then taken to be b, V=V —g
'
which gives
where k is a constant independent atomic potential energy satisfies
—7
(10)
Equation (8) is consistent with these bounds. It may also be noted that 5 VN& approaches a finite value at each nucleus, a behavior which is also found to hold for b V. Equation (7), examined in the limit r, ~0, yields the following results
REFORMULATION OF THE TF EQUATION FOR MOLECULES
V,
38
g,
Z=g,
DISSOCIATION LIMIT Neutral molecules dissociate into neutral atoms, and as shown by Yonei for diatomics, ' the TF method (including some corrections) contains this fact. I will show how this limit is obtained from Eq. (7) for an arbitrary mole-
cule. Surround each atom with a sphere of radius R ~ centered on the atom. A typical interatomic distance is R,b, and I consider the case where
R
b
))R g ))
1
NUMERICAL SOLUTION OF THE THOMAS-FERMI EQUATION. . .
38
It then follows from the definitions within the sphere about atom a,
above that
given
S, =S, +O(s'), S, =S.' '+O(s'), where c. =R&/R, b and S, = 2V, ~. Substituting ~
sums into Eq. (7) and taking the limit as a
~0 gives
these
'"],
V'(6 V) = —(4/3~)[( —2h V+S. )'"—S.
which has the solution AY=0 corresponding to n =n, for r, ~R~. Now R~ may be increased without limit, effectively enclosing all electrons in their respective neutral spheres.
NUMERICAL METHOD
Previous workers have mostly concentrated on solving the TF or TF-Dirac equation for diatomic rnoleThis has typically involved rewriting the TF cules. ' in a special coordinate system following the inequation troduction of a molecular screening function 4, V
= —(Z, /r, + Z2 /r
&
)P
.
The equation in these special coordinates is then discretized, taking care to handle special regions near nuclei and other locations at which singular behavior might be introduced by the special coordinates chosen. In particular, this approach requires nuclei to be located at points of the grid. By introducing Eq. (6), I have avoided having to satisfy any special boundary conditions at the nuclei. Consequently, I discretize Eq. (7) in ordinary Cartesian coordinates, using the standard seven-point O(h ) replacement for the Laplacian on a cubic grid of spacing h, which makes our program applicable to any number of atoms without change. The discretized Eq. (7) has been ' which I will now solved by two multigrid algorithms, outline. The molecule is embedded near the center of a finite simple-cubic-lattice grid Gh of minimum spacing h having Nz grid points. The distance from the center of nuclear charge to outer grid boundaries, where the asymptotic solution takes over, is chosen large enough so that no significant change occurs in the solution over the region of interest (specifically, when the boundary error falls below the truncation error due to discretization, the boundary need not be further which is of order enlarged). The resulting equation on Gz are relaxed by two or three Gauss-Seidel sweeps where one iteration of Newton's method is sufficient to solve the nonlinear equation at each grid point. This effectively eliminates significant errors whose wavelengths are of order 2h, the "high-frequency" error components on G& (Ref. 26). The solution is then transferred to a coarser grid of spacing H =2h, followed by relaxation on that level to smooth out most of the error for wavelengths of order 2H, and so on down to some lowest level which is effective in reducerrors. Then the solution on ing the longest-wavelength the lowest level is interpolated back to the next higher level, followed by relaxation there to reduce errors introduced by interpolation, and so on, back to the highest level Gj, . This constitutes one cycle of the first solution
h,
2207
algorithm. Only two or three such cycles are usually needed to reduce the algebraic error (the difference between the current approximation and the exact solution of the discrete equations) below the truncation error, at which point the cycling process is automatically stopped. The algebraic error is reduced at an exponential rate, so that the time for the cycling process to be completed is essentially proportional to the number of grid points on the finest grid. Since all atoms are of about the same size, this time increases asymptotically no faster than N„, the number of atoms in the molecule. In setting up the cycling procedure S& and S2 are evaluated and stored on all grids, which requires a time essentially proportional to N„)& Nh, which dominates the cycling time. A second algorithm with more efficiency has also been used. This method starts on the coarsest grid of the sequence of grids described above. Relaxation on such a coarse grid, H =2 a.u. , rapidly produces convergence to an initial approximate solution. This solution is then interpolated to the next finer grid GH&2, with an interpolation formula which uses the differential equation itself to evaluate higher-order terms resulting in errors formally of order (8/2) . Relaxation on GH&2 is now performed to begin one cycle like those described in the previous paragraph but now operating on only two grids. At the end of one such cycle the high-order interpolation is performed again to raise the solution to the next finer grid followed by another cycle involving all the grids used thus far. This process is repeated until terminating on the highest level as in the first method. This second algorithm uses about one-third the relaxation work of the first method, which results in a net gain in efficiency. One obvious improvement in our program would be to use only coarser grids at larger distances. This would increase the efficiency by reducing the number of grid point used for a given boundary size. Our method requires the atomic solution to be available during the solution process to provide the asymptotic molecular solution and the sums of atomic potential energies. I have therefore solved the atomic problem as oneboundary-value problem using a two-point dimensional (1D) form of the code. This code employs an accurate, multitermed, asymptotic TF atomic solution for and results in a uniform acits boundary condition the whole of space 0. or better 02% throughout of curacy for a 1D grid with h =2 =0.007815 and the boundary 20 a.u. from the origin. This solution is then extended to arbitrary points in space using a cubic spline interpola-
tion. RESULTS AND DISCUSSION
The nitrogen molecule has become a standard testing ground for solutions of the TF and related equations. I have likewise chosen it for a first case to study. The equilibrium internuclear distance in N2 is 2.068 a.u. Initial runs used a computational cell in the form of a cube, 9.6 a.u. , on a side containing 35 937 points with h =0. 30 a.u. This case took about 10 s of CPU time to run on an IBM 3081. As a check, a run with h =0. 15 a.u. was made and contour plots were made from its output. As expected
G. W. PARKER
2208
38
0.7
10'
0.6
10':.
;.
0.5 10'
0.4 Cg
0.3
10
0.2 10
0. 1
0.0—
10
3 x
FIG. 1. 4 V and
'
I
—3
0
(a.u. )
x
axis of N2. The VN& along the internuclear x & 2 a.u. is the noninteracting potential enb
(a.u. )
FIG. 3. Electron densities from the TF
model (solid curve), (open circles) along
upper curve for ergy difference. The origin is at the center of nuclear charge, and the nuclei are located 1.034 a. u. from this point.
and a molecular-orbital (MO) calculation the internuclear axis of N2.
from the smoothness in 6 V the smaller grid size was unnecessary; interpolated values of this variable were used to generate more detailed line plots of the densities and related quantities. The divergence of the TF densities was truncated by restricting r, to be no smaller than 0.0005 a.u. Figure 1 shows a plot of b, V and 6 VN& along the internuclear axis. The origin is taken at the center of nuclear charge. Both these quantities are smooth through the nuclei (where they are both finite), with b VNI & 6 V in the x &2. This behavior confirms the appropriateregion ness of our choice of discretization variable; hV varies even more smoothly than the noninteracting difference. It is, in fact, equivalent to the electrostatic potential produced by a charge density b, n. Given that this neutral density is peaked positive near each nucleus, it follows that b, V should increase along the internuclear axis as along the shown. Figure 2 shows a plot of b, V/V same axis. It shows that in the region containing most of the electronic charge, b, V is no more than about 10% of the magnitude of V. This percentage increases to 100% b, V/V ~(N, — 1), in the asymptotic regime, where with N, =2. In Fig. 3 I have plotted electron densities
from our solution and from a molecular-orbital (MO) calculation on a logarithmic scale. ' The TF density, as noted before, diverges at the nuclei (which was truncated here), while this MO density peaks at about 198 a.u. Moving away from these nuclei there is a core region where reasonably good agreement is obtained between these densities. This soon breaks down between nuclei and outside them. The density error is emphasized by plotting density differences in Fig. 4. The TF difference peaks positively at each nucleus and is somewhat larger between nuclei than just outside their locations. The MO density, on contrast, shows considerable structure. In particular, it peaks on both sides of the nuclei. More accurate MO densities show an additional peak at the center of the bond. Clearly, the TF equation does not provide the flexibility needed to generate such behavior. Near each nucleus An, can only diverge positively or negatively and lower energy dictates the positive divergence. A final comparison is provided in Figs. 5(a) and 5(b), which show contour plots of the electron density in a plane containing the nuclei. Contour intervals are at 0.01, 0.03, 0.05, 0. 10, 0.20, 0.40, 0.60, and 1.20 a.u. The TF density more nearly resembles the overlap of spheri-
I
~
I
~
~
~
~
~
0.4
12
0.2
10 8
0.0 CI5
)
—0.2
CI
O O
—0.4 2
—0.6 —
0
3
—3
x (a.u. )
x (a.u. )
10O
Fig. 1.
I
6 V/ V
I
along the internuclear
0
axis of N2, as in
FIG. 4. Electron-density differences from the TF model and a MO calculation along the internuclear axis of N2.
..
NUMERICAL SOLUTION OF THE THOMAS-FERMI EQUATION.
38
2209
2. 4
3. 1.2 D
&
no
0. 0
-3
-1.2
-6, -2. 4
I
I
I
-1.2
1.2
0.0 X (a. u. )
I
I
l
I
I
I
I
I
-3
-6 2.4
X
I
(
I
I
I
0 (a. u. )
I
1
I
I
I
I
3
I
6
FIG. 6. Contours of the electron density in benzene (C6H6) in the plane containing the nuclei. Contour levels are the same as those in Fig. 5 There is a distortion of the 1.20 level around the hydrogen nuclei.
2. 4
~
1.2
&
0.0
-1.2
-2. 4 -2. 4
I
I
I
f
1
1
-1.2
I
I
I
I
I
1
I
0.0 X (a.u. )
I
I
I
I
1
1.2
l
I
I
I
2.4
FIG. 5. (a) Contours of the electron density in N& in the plane containing the nuclei from the TF model. Contours levels are 0.01, 0.03, 0.05, 0. 10, 0.20, 0.40, 0.60, and 1.20 a.u. (b) Contours of the electron density in Nz in the plane containing the nuclei from a MO calculation. Contour levels are the same as (a).
cally symmetric densities, which was the basis for the first approximate solution of the TF equation for molecules by Hund's trial function approach has been used in Hund. a number of cases and it has recently been extended to obtain more detailed fits to molecular solutions with and without additional correction terms. This latter approach has been applied by Toepfer, Gross, and
L. H. Thomas, Proc. Cambridge Philos. Soc. 23, 542 (1926). ~E. Fermi, Z. Phys. 48, 73 (1928). N. H. March, Adv. Phys. 6, 1 (1957). D. A. Kirzhnitz, Yu. E. Lozovik, and G. V. Shpatakovskaya,
Dreizler to Nz at its equilibrium separation and a density plot like Fig. 5 has been given. ' Although a detailed comparison has not been made, it is evident that our plot agrees well with theirs, as their work suggests it should; the only noticeable difference is that our 0.40 contour is more strongly necked between the atoms. Another molecule treated by Hund's procedure is benzene (CsH6). Figure 6 shows our plot of the TF electron density in the plane containing the nuclei at their equilibrium separations. The computational cell was 16' 16)& 8 a.u. with h =0.25 a.u. and it contained 139 425 points on the finest grid. Again it is found that hV is less than hVN, in the region containing most of the charge. The values of the density at the center and at the midpoint of a C C bond are in agreement with those obtained by Hund's method. As these examples have shown, it is possible to efficiently solve the TF equation for molecules; the extension to crystalline solids is most straightforward for solids with cubic unit cells. The two key elements in achieving efficiency and generality were (I) the transformation to a dependent variable that is approximated well on relatively coarse grids, and (2) the use of the multigrid scheme to obtain solutions in a time essentially proportional to the number of grid points used. It should be possible to apof the TF ply this approach to some modifications method that include corrections for effects originally neglected. This could lead to predictions of binding energies and equilibrium structures.
—
Usp. Fiz. Nauk. 117, 3 (1975) [Sov. Phys.
(1976)]. 5E. Teller, Rev. Mod. Phys. 34, 627 (1962). N. Balazs, Phys. Rev. j.56, 42 (1967).
—Usp.
18, 649
2210
G. W. PARKER
7B.-G. Englert and J. Schwinger, Phys. Rev. A 29, 2331; 29, 2339 (1984); 29, 2353 (1984). 8E. H. Lieb and B. Simon, Phys. Rev. Lett. 31, 681 (1973). E. H. Lieb, Rev. Mod. Phys. 48, 553 (1976); 53, 603 (1981). J. Schwinger, Phys. Rev. A 22, 1827 (1980); 24, 2353 (1981); B.-G. Englert and J. Schwinger, ibid. 26, 2322 (1982). "R. Shakeshaft, L. Spruch, and J. Mann, J. Phys. B 14, L121
(1981).
P. Politzer, J. Chem. Phys. 72, 3027 (1980). R. Latter, Phys. Rev. 99, 510 (1955). K. Yonei, J. Phys. Soc. Jpn. 31, 882 (1971). ' G. W. Parker, Bull. Am. Phys. Soc. 32, 1084 (1987). ' Leading terms in the expansions
are spherically
symmetric
about each nucleus. H. Glazer and H. Reiss,
J. Chem. Phys. 21, 903 (1953); 23, 937 (1955). J. W. Sheldon, Phys. Rev. 99, 1291 (1955). J. R. Townsend and G. S. Handler, J. Chem. Phys. 36, 3325 (1962); 38, 2499 (1963). J. Goodisrnan, Phys. Rev. A 3, 1819 (1971). E. K. U. Gross and R. M. Dreizler, Phys. Lett. 57A, 131 (1976). 22K. Yonei and J. Goodisman, Int. J. Quantum Chem. 11, 163 (1977). K. Yonei and J. Goodisman, J. Chem. Phys. 66, 4551 (1977). ~4B. Jacobs, E. K. U. Gross, and R. M. Dreizler, J. Phys. B 11, 3795 (1978). The largest molecules for which the TF equation ously been solved are H20 and SO2', see Ref. 24.
has previ-
38
A. Brandt, Multigrid Methods, Vol. 960 of Lecture Notes in Mathematics, edited by W. Hackbush and U. Trottenberg (Springer, Berlin, 1982), pp. 221 —312. A. Brandt, Institute for Compute Applications in Science and Engineering Report No. 78-18 (NASA Langley Research Center, Hampton, VA, 1978), p. 20. C. A. Coulson and N. H. March, Proc. Phys. Soc. London, Sect. A 63, 367 (1950). N. H. March, Proc. Cambridge Philos. Soc. 48, 665 (1952). S. Kobayashi, T. Matsukuma, S. Nagai, and K. Umeda, J. Phys. Soc. Jpn. 10, 759 (1955). 'B. J. Ransil, Rev. Mod. Phys. 32, 245 (1960). R. F. W. Bader, W. H. Henneker, and P. E. Cade, J. Chem. Phys. 46, 3341 (1967). F. Hund, Z. Phys. 77, 12 (1932). C. A. Coulson, N. H. March, and S. Altmann, Proc. Natl. Acad. Sci. U. S.A. 38, 372 (1952). N. H. March, Acta Crystallogr. 5, 187 (1952). P. Gombas, Acta Phys. Acad. Sci. Hung. 9, 461 (1959). J. Eichler and U. Wille, Phys. Rev. Lett. 33, 56 (1974); Phys. Rev. A 11, 1973 (1975). R. M. Dreizler, E. K. U. Gross, and A. Toepfer, Phys. Lett. 71A, 49 (1979). E. K. U. Gross and R. M. Dreizler, Phys. Rev. A 20, 1798 (1979). ~A. Toepfer, E. K. U. Gross, and R. M. Dreizler, Z. Phys. A 298, 167 (1980). A. Toepfer, E. K. U. Gross, and R. M. Driezler, Phys. Rev. A 20, 1808 (1979).
