THOMAS–FERMI MODEL ELECTRON DENSITY WITH CORRECT BOUNDARY CONDITIONS: APPLICATIONS TO ATOMS AND IONS

Atomic Data and Nuclear Data Tables 71, 41– 68 (1999) Article ID adnd.1998.0799, available online at http://www.idealibrary.com on

THOMAS–FERMI MODEL ELECTRON DENSITY WITH CORRECT BOUNDARY CONDITIONS: APPLICATIONS TO ATOMS AND IONS S. H. PATIL Department of Physics Indian Institute of Technology Bombay 400 076 India

We propose an electron density in atoms and ions, which has the Thomas–Fermi–Dirac form in the intermediate region of r, satisfies the Kato condition for small r, and has the correct asymptotic behavior at large values of r, where r is the distance from the nucleus. We also analyze the perturbation in the density produced by multipolar fields. We use these densities in the Poisson equation to deduce average values of r m , multipolar polarizabilities, and dispersion coefficients of atoms and ions. The predictions are in good agreement with experimental and other theoretical values, generally within about 20%. We tabulate here the coefficient A in the asymptotic density; radial expectation values ^r m & for m 5 2, 4, 6; multipolar polarizabilities a 1, a 2, a 3; expectation values ^r 0 & and ^r 2 & of the asymptotic electron density; and the van der Waals coefficient C 6 for atoms and ions with 2 # Z # 92. Many of our results, particularly the multipolar polarizabilities and the higher order dispersion coefficients, are the only ones available in the literature. The variation of these properties also provides interesting insight into the shell structure of atoms and ions. Overall, the Thomas–Fermi–Dirac model with the correct boundary conditions provides a good global description of atoms and ions. © 1999 Academic Press

0092-640X/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved.

41

Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999

S. H. PATIL

Thomas–Fermi Model

CONTENTS

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

2. DESCRIPTION OF THE MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Density in the Small-r Domain . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Density in the r 3 ` Region . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Modified Thomas–Fermi–Dirac Model . . . . . . . . . . . . . . . . . . . . 2.4. Modified TFD Model with a Perturbation . . . . . . . . . . . . . . . . .

43 43 43 44 45

3. RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Solutions for Unperturbed Atoms and Ions . . . . . . . . . . . . . . . . 3.2. Multipolar Polarizabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Dispersion Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46 46 47 48 50 52

APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52

EXPLANATION OF TABLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TABLE I. Coefficient of the Asymptotic Density, Radial Expectation Values, Polarizabilities, and Angle-Averaged van der Waals Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

54

55

1. INTRODUCTION The Thomas–Fermi (TF) model [1] provides a fairly useful description of the average properties of atoms and ions. Its importance lies in its great simplicity. It can be regarded as a statistical model with the local density deduced from phase space considerations. Alternatively, the local density can be obtained [2] from WKB radial wave functions in the classically allowed region. This approach makes it clear that the model is not accurate for atoms and ions, near the nucleus where the potential is expected to vary rapidly, and in the classically forbidden region far away from the nucleus. Many modifications to the Thomas–Fermi model have been introduced to improve its accuracy. Dirac introduced [3] a correction to the Thomas–Fermi density by including exchange properties of electron wave functions. The model based on the density including this exchange effect is known as the Thomas–Fermi–Dirac (TFD) model. The corrections to the total energy from the region near the nucleus have been considered by Scott [4]. More recently, Schwinger and his co-workers have proposed an approach [5] to understanding the outer regions of a Thomas–Fermi model, which includes exchange and first quantum kinetic energy corrections. These models provide a general descrip-

tion of some properties of atoms and ions, such as the total binding energy and effective potential. There have been some attempts to calculate diamagnetic susceptibilities [5, 6] and dipolar polarizabilities [7] of inert gas isoelectronic sequences, but with limited success. A point that needs to be stressed is that the Thomas– Fermi–Dirac density, or its improved version, provides a good description of the density at intermediate distances from the nucleus. However, we do have rigorous relations for the densities near the nucleus and far away from the nucleus. The angle-averaged density r (r), near the nucleus, satisfies the property

r 9~0! 5 22Z r ~0!,

(1.1)

where Z is the nuclear charge in atomic units. This is essentially a consequence of the Kato condition [8]. When one is far away from the nucleus, the form of the density is given by [9, 10]

r ~r! 3

42

A 22 b 22 a r r e , 4p

r 3 `,

(1.2)

Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999

S. H. PATIL

Thomas–Fermi Model

with

a 5 ~22E i! 1/ 2

the electron density for essentially all atoms and positive ions. This allows us to deduce average values of r m in general, and diamagnetic susceptibilities in particular. These predictions are in agreement with the experimental values within 15%. We have also analyzed changes in the density produced by multipolar fields. This allows us to obtain all multipolar polarizabilities and dispersion coefficients of atoms and ions. The agreement of the latter predictions with the experimental values is also generally good. The largest deviation is about 50% for three and four electron systems. Furthermore, the variation of these properties for different systems provides interesting insight into shell structure, number of outer electrons, etc. Overall, the Thomas–Fermi–Dirac model, modified in conformity with the Kato constraint and the asymptotic behavior of the density, provides a satisfactory description of the properties of atoms and ions. We use atomic units unless stated otherwise.

(1.3)

and

b512

Q , a

(1.4)

where 2E i is the separation energy of the last electron and Q is the charge of the remaining core seen by the electron when it is far away. All quantities are in atomic units. This behavior of the density can be generalized to include asymptotically nonleading terms. Here, we propose a density which has the TFD form in terms of the potential in the intermediate region, satisfies the Kato condition in Eq. (1.1), and has the correct asymptotic form at large values of r. Solving the Poisson equation with this density, we obtain

2. DESCRIPTION OF THE MODEL 2.2. Density in the r 3 ` Region

Here we briefly describe some general properties of the electron density in atoms and ions, and then a modified Thomas–Fermi–Dirac model which incorporates the boundary conditions for small and large values of r.

Many of the atomic and ionic properties depend on the density in the outer region. The behavior of the density in this region depends on the separation energy of the last electron. Consider an N-electron system governed by the Hamiltonian H (N) . When one of the electrons is far away from the nucleus, one has

2.1. Density in the Small-r Domain Consider the Schroedinger equation when one of the electrons, described by the coordinates r, is close to the nucleus: For the angle-averaged density, the main contribution is from the l 5 0 term of the wave function. Multiplying the Schroedinger equation by r, the radial part of the wave function, R 0 (r) satisfies the equation

r 1 3`

H ~N! ~r 1 , . . . , r N ! ™™™™3 H ~N21! ~r 2 , . . . , r N ! 1

2

2

1 d Z @rR 0 ~r!# 2 @rR 0 ~r!# 3 0 2 dr 2 r

for r 3 0.

for r 3 0

(2.5)

(2.1) where Q 5 Z 2 N 1 1 is the charge of the remaining core seen by the electron. Expanding the lowest energy eigenstate c 0(N) of H (N) in terms of eigenstates c n(N21) of H (N21) ,

Taking R 0 ~r! 3 a 0 1 a 1 r

1 2 Q p 2 , 2 1 r1

(2.2)

O f ~r !c `

c

we obtain from Eq. (2.1)

~N! 0

~r 1 , . . . , r N ! 5

n

1

~N21! n

~r 2 , . . . , r N !,

n50

a 1 5 2Za 0 .

(2.6)

(2.3) where n 5 0 corresponds to the ground state, we get

For the angle-averaged density, this leads to the relation

r ~r! 3 a 20 ~1 2 2Zr!

for r 3 0,

S

(2.4)

which is equivalent to the Kato condition in Eq. (1.1). It may be noted that l 5 1, . . . terms contribute terms of order r 2 and higher order.

2

D

1 2 Q ¹ 2 f ~r ! 1 ~E ~N21! 2 E ~N! 0 0 ! f 0 ~r 1 ! 2 1 r1 0 1 5O

43

SD

1 f ~r ! r3 0 1

for r 1 3 `.

(2.7)

Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999

S. H. PATIL

Thomas–Fermi Model

2.3. Modified Thomas–Fermi–Dirac Model

This leads to [9, 10]

c

~N! 0

S D

A ~r 1 , . . . , r N ! 3 4pN

S

b 3 r2 1 2

The Thomas–Fermi density is obtained from the local phase integral over the occupied states,

1/ 2

D

1 b r 2 b 21 e 2 a r 1 2 1 1

for r 1 3 `,

r TF~r! 5

(2.8)

with a and b given in Eqs. (1.3) and (1.4), where

5

2 E ~N! 2E i 5 E ~N21! 0 0

b 1 5 ~l 1 b !~ b 2 l 2 1!/ a ,

A ~r! 3 @r 22 b 2 b 1 r 22 b 21 #e 22 a r 4p

(2.10)

for r 3 `.

r TFDS~r! 5

O r P ~cos u !. N

L

(2.12)

i

1 @2~E i 1 f !#. p3

for E i . 2f ~r!.

The first-order change in the wave function, dc 0(N) , is obtained by solving the inhomogeneous equation

O r P ~cos u !c L i

L

i

~N! 0

.

S DS D 1 L11

1 L11 r P L ~cos u 1 ! c ~N! 0 a 1

(2.13)

for r 1 3 `.

E i 1 f ~r! 3

(2.14)

A @r L11 P L ~cos u !#r 22 b e 22 a r 2 pa ~L 1 1! for r 3 `.

(2.18)

S

D

r @E i 1 f ~r!#, r 1 c1

(2.19)

where c1 is a constant. We also add a term to the density which ensures the correct asymptotic behavior of the density at large r. Then the modified Thomas–Fermi equation is given by the Poisson equation

The corresponding first-order change in the density is

dr asym~rW ! 3

(2.17)

This density may be expected to provide a good description of the density in the intermediate r region, but not in the small r or large r domains. In particular, it may be noted that for r 3 0, the potential f (r) 3 Z/r and the density in Eq. (2.18) tends to infinity. To incorporate the correct behavior for small r and large r, we introduce the following modifications. We remove the infinity at the origin by the replacement

Using the asymptotic form of c 0(N) given in Eq. (2.8), one obtains [10]

dc ~N! 0 3

(2.16)

1 11 @2~E i 1 f !# 3/ 2 1 @2~E i 1 f !# 3p 2 9p 3

i51

~N! # dc ~N! @E ~N! 0 2 H 0 5 2

for E i . 2f ~r!,

There is an additional contribution of the same form, from quantum corrections, obtained by Schwinger [5], but smaller by a factor 92. We therefore write the Thomas– Fermi–Dirac–Schwinger (TFDS) density as

One can also deduce the asymptotic form of the atomic or ionic wave functions perturbed by an external multipolar potential

L i

1 @2~E i 1 f !# 3/ 2 3p 2

r ex~r! 5

(2.11)

V52

d 3p

where f is the local electrostatic potential, E i is the highest energy of the occupied electron states, and [2(E i 1 f )] 1/ 2 is the corresponding momentum. The same result can be obtained [2] by using the WKB wave functions in the classically allowed region. The break in the density at f 5 2E i comes because of the inadequacy of the WKB wave functions near the turning points. Dirac [3] obtained the contribution coming from the exchange terms,

with l being the angular momentum quantum number of the last electron. As has been discussed earlier [10], l is well defined for most of the atoms and ions. The corresponding angle-averaged asymptotic density has the form

r

E

(2.9)

is the separation energy, and

asym

2 ~2 p ! 3

¹ 2 f 5 4 pr ~r!,

(2.15) 44

(2.20)

Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999

S. H. PATIL

Thomas–Fermi Model

which for the isotropic system reduces to 1 d2 @r f ~r!# 5 4 pr ~r!, r dr 2

density in Eq. (2.22), in powers of r, the Kato condition leads to the relation

FS D

(2.21)

3

2Z c1

1/ 2

1

22 3p

with

GF

E i 1 g9~0! 2

5 2~2Z! 2

r ~r! 5

F S

1 2r 2 3p r 1 c1

S S

D

3/ 2

D

G

11 2r ~E i 1 f ! u ~E i 1 f ! 9p 3 r 1 c1

1

A 4p

F

r r 1 1/ 2 a

3 12

S

D

n

c2 5

~r 1 1/ 2 a ! 22 b

1 1 c 2 2 r 1 1/ 2 a

DG

2

e 22 a r .

a 5 ~22E i!

1/ 2

1

G

11 , 3p

~l 1 b !~ b 2 l 2 1! ~n 1 2 b ! 2 , a 2a

(2.27)

(2.28)

where l is the angular momentum quantum number of the last electron. The remaining parameter A is determined by requiring that the Poisson equation in Eq. (2.21) be satisfied with the boundary condition

(2.22)

Here, 2E i is the separation energy of the last electron, 1/ 2

FS D 2Z c1

G

where g(r) 5 r f (r). The constant c 2 is determined by requiring that the asymptotic term in Eq. (2.22) conforms to the asymptotic behavior in Eq. (2.11). Expanding the asymptotic term in inverse powers of r, this leads to

~E i 1 f ! 3/ 2

1

1 Z c1

f ~r! 3 (2.23)

Z r

for r 3 0.

(2.29)

Thus the only input parameters for obtaining the complete solutions are Z, N, and the separation energy 2E i of the last electron.

and

b512

Q a

(2.24)

2.4. Modified TFD Model with a Perturbation When the system is perturbed by a multipolar potential given in Eq. (2.12), the change in the electron density for large r is given by Eq. (2.15). Then, substituting

as in Eqs. (1.3) and (1.4), Q 5 Z 2 N 1 1,

(2.25)

f 3 f 0 1 df

(2.30)

in Eq. (2.20), we obtain

Z is the nuclear charge, N is the number of electrons, c1 , c 2 , and A are constant parameters, u ( x) is the Heaviside step function, and n is taken to be 2 for smaller systems and 4 for larger systems as follows:

¹ 2 ~ df ! 5 4 p ~ dr !,

(2.31)

with n52 54

for 2 # N # 17 for N $ 18.

dr 5

(2.26)

F S

1 2r 2p 2 r 1 c1

1

This ensures that the density at smaller values of r is essentially unaffected by the asymptotic term. The larger value of n for larger systems reflects the fact that the corresponding outer electron wave functions have a larger number of zeros. The constant c1 is determined by requiring that the Kato condition in Eq. (2.4) be satisfied. Expanding the

S

D

3/ 2

~E i 1 f 0 ! 1/ 2 df

D G S

2r 11 A df u ~E i 1 f 0 ! 1 3 9p r 1 c1 2 pa ~L 1 1!

3 @r L11 P L ~cos u !#

F

3 12

45

S

r r 1 1/ 2 a

1 1 c3 2 r 1 1/ 2 a

DG

D

n

~r 1 1/ 2 a ! 22 b

2

e 22 a r .

(2.32)

Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999

S. H. PATIL

Thomas–Fermi Model

The constant c 3 is determined by requiring that the asymptotic density strictly conforms to the behavior in Eq. (2.15) and does not contain an additional r L22b term. This leads to

The equation is solved by starting with an asymptotic behavior,

h 3 rL 2

~n 1 2 b ! c3 5 2 . 2a

(2.33)

df 5 h ~r! P L ~cos u !,

(2.34)

aL 5 2

we get from Eqs. (2.31) and (2.32),

F S D S DG S D S DG

1 2r 1 d2 L~L 1 1! ~r h ! 2 h 5 4p r dr 2 r2 2p 2 r 1 c1

1

F

3 12

3/ 2

5

11 2r h u ~E i 1 f 0 ! 3 9p r 1 c1

r 4pA r L11 2 pa ~L 1 1! r 1 1/ 2 a

(2.36)

n

~r 1 1/ 2 a ! 22 b

e 22 a r .

E

E

~N! 3 L c ~N! 0 r P L ~cos u ! dc 0 d r,

r L P L ~cos u !~ dr !d 3 r,

(2.37)

where dr is given in Eq. (2.32). We have used Eq. (2.35) to obtain the values of a L though we did find that the values obtained from Eq. (2.37) are equal to those from Eq. (2.35), within the accuracy of the calculations. Here, a L is in atomic units. It may be multiplied by a 02L11 with a 0 being the Bohr radius (5 0.5292 3 10 28 cm in cgs units, for example), to get the value in other units.

2

1 1 c3 2 r 1 1/ 2 a

for r 3 `,

and then determining the multipolar polarizability a L by requiring that h have the behavior r L for r 3 0. We may also determine average polarizabilities a L from the relation

Writing

3 ~E i 1 f 0 ! 1/ 2 h 1

aL r L11

(2.35)

3. RESULTS We now describe the results obtained by solving Eq. (2.21) with the density given in Eq. (2.22) for the unperturbed atomic and ionic systems, and Eq. (2.31) with the change in density given in Eq. (2.32) for the system perturbed by a multipolar field of order L.

Eq. (2.21). The value of A is determined so as to satisfy the condition in Eq. (2.29): r f ~r! 3 Z

The Poisson equation in Eq. (2.21) is solved by starting with the solutions in the large r region:

r f ~r! 3 Z 2 N 1

E E

4 pr

dr9

r

c 1 5 0.95Z 21.05

~r0!r0dr0

r9

for r 3 `,

(3.1)

with

S DS

A r asym~r! 5 4p

r r 1 1/ 2 a

F

D S

^r m & 5

n

~r 1 1/ 2 a ! 22 b

1 1 3 1 2 c2 2 r 1 1/ 2 a

DG

E

r ~r!r m d 3r.

(3.5)

In particular, we obtain diamagnetic susceptibilities of atoms and ions,

2

e 22 a r .

(3.4)

for all Z and N. Some of the details of the calculations are given in the Appendix. The solutions for the densities and potentials provide us with very useful information about the properties of atoms and ions. They allow us to calculate expectation values ^r m & defined as

` asym

(3.3)

We also note that c1 which is determined iteratively by using Eq. (2.27), is given quite accurately, within a few percent, by the relation

3.1. Solutions for Unperturbed Atoms and Ions

`

for r 3 0.

(3.2)

x52

We then trace the solution to smaller values of r by using 46

1 ^r 2 &, 6c 2

(3.6)

Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999

S. H. PATIL

Thomas–Fermi Model

^r m16 & 5 ^r 6 &

S D 1 2a

m

G~m 1 9 2 2 b ! , G~9 2 2 b !

m $ 0,

(3.7)

where G( x) is the gamma function. The Table also includes the values of A, with A/4 p being the coefficient of the asymptotic density in Eq. (2.11). This coefficient plays an important role in the determination of many properties of atoms and ions, and their interactions with electrons and with each other. 3.2. Multipolar Polarizabilities The change in the density of a system perturbed by the multipolar potential in Eq. (2.12), is obtained by solving the Poisson equation in Eq. (2.35). One may start at large r with

h ~r! 5 r L 2

aL r L11

(3.8)

and trace the solution to smaller values of r by converting the second-order differential equation into a difference equation. The L-pole polarizability a L is then determined by requiring that h (r) vanish at the origin. Alternatively, we may start with

FIG. 1. Plot of the modified TFD (Eq. (2.22)) radial electron density, 4 p r 2 r (r), for neon, as a function of r 1/ 2 . The broken curve is the prediction of Roothan–Hartree–Fock calculation given in Ref. [16].

with c being the speed of light expressed in atomic units. The predictions for ^r m &, m 5 2, 4, 6 are given in Table I. For higher values of m, the dominant contribution is from the outer density, that is, the term proportional to A in Eq. (2.22). Using the large-r limit, we obtain the relation

h ~r! < a L r L

(3.9)

for small r and trace the solution to larger values of r by converting the second-order differential equation into a difference equa-

FIG. 2. Plot of number of outer electrons for neutral atoms, defined as the integral over the asymptotic density [Eq. (3.28)], as a function of n e (total) (5N), the total number of electrons.

47

Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999

S. H. PATIL

Thomas–Fermi Model

FIG. 3. Plot of ^r 2 & 1/ 2 for neutral atoms as a function of n e (total) (5N), the total number of electrons. The stars connected by the solid line represent the values from the present calculation and circles are the values from SCF calculations [17].

tion. The constant a L is determined so that the coefficient of r L in h(r) at large r is 1, and the corresponding coefficient of 21/r L11 at large r is the polarizability a L. We also verify the results by using Eq. (2.37) and dr in Eq. (2.32), for calculating a L. The calculated values of a 1, a 2, a3 for atoms, and singly and doubly ionized ions, are given in Table I. For higher order polarizabilities, we may use only the asymptotic part of dr to evaluate a L in Eq. (2.38). This leads to the relation

a L13 5 a 3

S D S DS 1 2a

2L

3

4 L14

7 2L 1 7

and more generally

ab

~l, n 2 l 2 1!,

(3.14)

l51

where C ab ~l, l9! 5

D

G~2L 1 10 2 2 b ! , G~10 2 2 b !

OC

n22

C ab ~2n! 5

~2l 1 2l9!! 2 p ~2l !!~2l9!!

E

`

a l~i v ! a 9l 9 ~i v !d v ,

0

(3.15) L . 0,

with a l and a 9l9 being the angle-averaged polarizabilities of the atoms or ions a and b,

(3.10)

which is quite reliable for L $ 3.

a l~i v ! 5 2

3.3. Dispersion Coefficients

i

(3.16)

The angle-averaged van der Waals coefficients for interaction between atoms or ions a and b are given by C ab ~6! 5 C ab ~1, 1!

(3.11)

C ab ~8! 5 C ab ~1, 2! 1 C ab ~2, 1!

(3.12)

C ab ~10! 5 C ab ~2, 2! 1 C ab ~1, 3! 1 C ab ~3, 1!

O

l 2 9 u^0u ¥ j r j P l ~cos u j !ui&u ~E i 2 E 0 ! . ~E i 2 E 0 ! 2 1 v 2

l

Here a (0) is the usual multipolar polarizability. For v 3 `, we have v3`

a l ~i v ! O ¡ D l/ v 2,

(3.13) 48

(3.17)

Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999

S. H. PATIL

Thomas–Fermi Model

FIG. 4. Plot of ln(a 1) for neutral atoms as a function of n e (total) (5N), the total number of electrons.

with Dl 5 2

O9 ~E 2 E !u^0u O r P ~cos u !ui&u . i

l j

0

i

2

l

j

Since this term represents the contribution of the asymptotic electrons, we have from Eq. (3.19), (3.18)

d l 5 l^r 2l22 & asym ,

j

where

It is well known [11] that for angle-averaged states, D l 5 l^0u

Or

2l22 j

u0&

^r 2l & asym 5

j

5 l^r 2l22 &.

(3.23)

(3.19)

E

r asym~r!r 2l d 3 r.

(3.24)

Substituting Eq. (3.21) into Eq. (3.15), we obtain Now consider a representation dl d˜ l a l ~i v ! 5 2 , 2 1 2 v 1 v l v 1 v˜ 2l

C ab ~l, l9! 5 (3.20)

dl , v 1 v 2l

(3.21)

d l / v 2l 5 a l ~0! 5 a l .

(3.22)

2

F

G

~d l d9l 9 ! 1/ 2 a l a 9l 9 , ~d l a 9l 9 ! 1/ 2 1 ~d9l 9 a l ! 1/ 2 (3.25)

where a l , a 9l9 are the multipolar polarizabilities of the atoms/ions a and b, and d l , d9l9 are given by Eq. (3.23), for the atoms/ions a and b. Table I contains the values of a 1, a 2, a 3. Higher order polarizabilities can be obtained from the recursion relation in Eq. (3.10). Table I also contains ^r 0& asym and ^r 2& asym. Average values ^r 4& asym and ^r 6& asym are approximately equal to ^r 4& and ^r 6& for the whole atom/ion, given in Table I. We can evaluate the average values of still higher powers of r by using the recursion relation in Eq. (3.7). Thus, we can use Eq. (3.25) to obtain all the dispersion coefficients for pairs of atoms/ions. Though C ab(l, l9) are easy to evaluate by using Eq. (3.25), we have given C 6 5 C(1, 1) for all like atoms and ions, in Table I, for ready reference.

where the first term represents the contribution of the outer electrons described by the asymptotic part of the density in Eq. (2.22), and the second term represents the contribution from the inner electrons. It is found from Eqs. (2.37) and (2.35) that the contribution of the asymptotic term to a l is dominant. We therefore take

a l~i v ! <

~2l 1 2l9!! 4~2l !!~2l9!!

with

49

Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999

S. H. PATIL

Thomas–Fermi Model

FIG. 5. Plot of ln(C 6 ) for neutral atoms as a function of n e (total) (5N), the total number of electrons.

We can also calculate the three-body dispersion coefficients Z(l, l9, l0) which allows us to obtain [12] the three-body potential: Z~l, l9, l0! 5

1 p

E

a l ~i v ! a 9l 9 ~i v ! a 0l 0 ~i v !d v .

are not sensitive to the choice of l, but the polarizabilities are smaller for the choice of larger values of l. The solutions to Eq. (2.21) provide us with densities for all atoms and ions. As a typical case we have plotted in Fig. 1 the predicted density for neon. This figure also contains the density obtained from Roothan–Hartree–Fock calculations [16]. The comparison shows that the present model provides a smoothed-out density in the interior. Since it contains the correct asymptotic behavior, one may expect its predictions for susceptibilities and polarizabilities, which are more sensitive to the outer domains, to be quite reliable. Since our total density contains an explicit term in Eq. (2.22) to describe the asymptotic density, it is interesting to analyze this term. In Fig. 2 we have plotted the integral of the asymptotic density which may be regarded as the number of outer electrons:

(3.26)

Using Eq. (3.21) we obtain Z~l, l9, l0! 5 3

1 d d9 d 0 2 l l9 l0

~ v l 1 v 9l 9 1 v 0l 0 ! , v l v 9l 9 v 0l 0 ~ v l 1 v 9l 9 !~ v 9l 9 1 v 0l 0 !~ v 0l 0 1 v l !

(3.27)

where v l 5 (d l / a l ) 1/ 2 , and d l is given in Eq. (3.23). 3.4. Discussion We now analyze the qualitative and quantitative aspects of the predictions of the Thomas–Fermi model with the modified density. We emphasize that the only inputs in these calculations are the separation energy 2E and the angular momentum quantum number l of the last electron. The separation energy is taken from Moore’s tables [13] and the CRC handbook [14], and the angular momentum quantum number l is taken from the electron configurations given by Radzig and Smirnov [15] and from Moore’s tables [13]. It may be noted that there are differences in the angular momentum quantum number l suggested by different authors for some positively charged ions. Our predictions for ^r 2 &, ^r 4 &, etc.,

n e ~outer! 5 A

E

`

drr n12 ~r 1 1/ 2 a ! 2n22 b

0

F

3 12

S

1 1 c 2 2 r 1 1/ 2 a

DG

2

e 22 a r .

(3.28)

This shows a clear increase as the shells and subshells get filled, and a clear decrease when new shells and subshells begin. What needs to be emphasized is that the n e(outer) is around 2 or less when a new shell starts and has a value of about 8 when 50

Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999

S. H. PATIL

Thomas–Fermi Model

TABLE A Comparison of Our Predictions for ^r 2 &, Dipolar Polarizability a 1, and Dispersion Coefficient C 6 , (in Atomic Units) with Some Experimental Values and Values from Other Calculations

Ne Na 1

a

^r 2 & (others)

10.9

9.36 a 9.37 b 6.40 a 6.42 b 26.0 a 26.0 b 19.5 a 19.6 b 39.1 a 39.6 b 31.4 a 31.8 b 61.1 a 62.3 b 51.2 a 52.6 b 27.2 b 51.3 b 68.2 b 98.0 b 18.6 a 18.5 b 42.9 a 57.6 b

6.68

Ar

24.5

K1

17.2

Kr

40.6

Rb 1

31.1

Xe

55.6

Cs 1

44.7

Na K Rb Cs Cu 1

40.8 57.1 77.2 93.9 29.4

Pd

58.5

Reference Reference c Reference d Reference e Reference b

^r 2 &

[21] [17] [22] [23] [24]

a1

a 1 (others)

C6

120 198 250 317 6.77

2.38 a 2.67 c 0.946 a 1.01 b 10.8 a 11.1 c 5.46 a 6.4 b 16.5 a 16.8 c 9.08 a 11.1 b 27.0 a 27.3 c 15.8 a 20.8 b 159 c 293 c 320 c 403 c 5.36 a

1350 2740 4120 5800 32.7

22.8

21.2 a

208

2.92 0.98 9.8 4.3 16.6 8.3 24.8 13.5

9.77

C 6 (others) 6.88 d

1.91 62.1

67.2 d

18.0 156

133 d

55.5 290

299 d

115 1390 e 3810 e 4430 e 6330 e

(Johnson et al.). (Fraga et al.). (Miller and Bederson). (Standard and Certain). (Marinescu et al.).

al. [21], and the (mostly experimental) recommended values of Miller and Bederson [22]. For alkali atoms, our values are somewhat smaller than the values recommended by Miller and Bederson [22]. Considering the simplicity of the present approach, and the wide spread in the values, the agreement seen here can be regarded as very encouraging. Our predictions for C 6 also are close to the average values of the lower and upper bounds obtained by Standard and Certain [23], and the values obtained by Marinescu et al. [24] using model potentials. It may be further noted that (61)^r 2& can be deduced from the experimental values of diamagnetic susceptibilities. With the correction by a factor of 1.07 suggested recently [18, 19], the experimental values [20] of (61)^r 2& for Ne, Ar, Kr, Xe are 1.60, 4.28, 6.52, 10.2 in atomic units which may be compared with our values of 1.82, 4.08, 6.76, 9.27, respectively. Our approach allows us to obtain all the dispersion coefficients for all the atoms and ions, in terms of Eq. (3.25), which depends on the polarizabilities a l and d l which are related to ^r 2l22 & asym as in Eq. (3.23). To illustrate the point, we note that for Kr, a 1 5 16.6, a 2 5 74.79, a 3 5

the shell closes, which is in conformity with the usual description in terms of hydrogenic states. The plot also shows the effect of subshells; for example, n e(outer) decreases when n e(total) increases from 12 to 13, or increases only slightly when n e(total) (5N in Eq. (2.25)) increases from 15 to 16. We have plotted in Fig. 3 the rms radius of atoms, along with those from the results of self-consistent-field calculations in Ref. [17]. There is a general agreement between the two, though the present results show a clear indication of the subshell structures. In Fig. 4, we have plotted the dipolar polarizabilities as a function of the total number of electrons, and in Fig. 5, we have plotted the dispersion coefficient C 6 for like atoms as a function of the total number of electrons. We can compare our predictions for the values of ^r 2&, dipolar polarizabilities, and dispersion coefficients with some experimental values and the results of other calculations. In Table A, we have presented these for inert gas atoms and their isoelectronic singly charged ions, for alkali atoms, and for Cu 1 and Pd. Our predictions for the dipolar polarizabilities of inert gases are close to the accurate calculated values of Johnson et 51

Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999

S. H. PATIL

Thomas–Fermi Model

710.9, d 1 5 9.444, d 2 5 2 3 31.75, d 3 5 3 3 241.7, and for Cs, a 1 5 316.9, a 2 5 8.00 3 10 3, a 3 5 3.93 3 10 5, d 1 5 1.88, d 2 5 2 3 54.17, d 3 5 3 3 2.88 3 10 3 , with d 3 obtained from ^r 4 & as described in Section 3.3. On using Eqs. (3.25) and (3.11)–(3.13), these lead to C 6 5 551.5,

C 8 5 5.65 3 10 4 ,

which is within the bounds of 1.79 3 10 3 and 1.91 3 10 3 given by Standard and Certain [23]. 3.5. Conclusions

C 10 5 6.85 3 10 6

With the modifications in the Thomas–Fermi–Dirac– Schwinger density to incorporate the Kato condition for small r, and the correct asymptotic behavior at large r, the Thomas–Fermi model becomes a powerful tool in calculating many atomic and ionic properties, such as densities, diamagnetic susceptibilities, and multipolar polarizabilities. In this, the asymptotic density governed by the separation energy of the last electron plays a particularly dominant role. We have presented here a set of such results for a broad range of atoms and ions. Many of our results, particularly the multipolar polarizabilities and the higher order dispersion coefficients, are the only ones available in the literature.

(3.29) for Kr–Cs coefficients, which may be compared with the lower and upper bounds suggested by Standard and Certain [23], (515, 561), (4.26 3 10 4, 5.19 3 10 4), (4.81 3 10 6, 5.56 3 10 6), respectively. We are also able to calculate all the three-body coefficients Z(l, l9, l0) in terms of Eq. (3.27). As an example, we obtain for Xe–Xe–Xe, with d 1 5 9.77, a 1 5 24.81, v 1 5 (9.77/ 24.81) 1/ 2 , Z~1, 1, 1! 5 1.797 3 10 3 ,

(3.30)

APPENDIX The solution for r f (r) in the asymptotic region can be written in the form

F

we get g~r 2 D! 2 2g~r! 1 g~r 1 D! 5 4 p r r ~r!, D2

r f ~r! 3 Z 2 N 1 A f ~n 1 2 b ! 2 c 2 f ~n 1 2 b 1 1! 1

1 2 c f ~n 1 2 b 1 2! 4 2

G

for r 3 `,

where r (r) is given in Eq. (2.22). Using Eq. (A.5), we trace the solution to smaller values of r. The constant A is varied till we get

(A.1)

where

E E `

f ~a! 5

g~0! 5 Z. `

dr9

r

dr0

r9

~r0! n11 e 22 a r0 . ~r0 1 1/ 2 a ! a

F

S

1 2 n11 a 2 2 1 3 ~2 a ! ~2 a ! r r 1 1/ 2 a

1

S

D

n~n 1 1! 3 a~a 1 1! 1 ~2 a ! 4 r2 ~r 1 1/ 2 a ! 2

D

2a~n 1 1! 2 1··· r~r 1 1/ 2 a !

G

References

r n11 e 22 a r . ~r 1 1/ 2 a ! a (A.3)

1. L. H. Thomas, Proc. Cambridge Philos. Soc. 23, 542 (1926); E. Fermi, Z. Phys. 48, 73 (1928) 2. D. A. Kirznitz, Yu. E. Lozovik, and G. V. Shpatakovskaya, Usp. Fiz. Nauk 117, 3 (1975) [Sov. Phys. Usp. 18, 649 (1976)]

To continue the solution to smaller values of r, we convert the differential equation in Eq. (2.21) into a difference equation. In terms of g(r), g~r! 5 r f ~r!,

(A.6)

The value of c1 is determined iteratively. We start with, for example, c1 5 1/Z. Then, we solve the differential equation (2.21) for a given value of A, use the value of g9(0) of this solution to obtain c1 from Eq. (2.27), and repeat the procedure till the input and output values are equal. We find that c1 is given quite accurately by the expression in Eq. (3.3), for Z . 2. For Z 5 2, the solution is insensitive to the value of c1 , and one may still use the expression in Eq. (3.3).

(A.2)

Integrating by parts repeatedly, we get f ~a! 3

(A.5)

3. P. A. M. Dirac, Proc. Cambridge Philos. Soc. 26, 376 (1930)

(A.4) 52

Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999

S. H. PATIL

Thomas–Fermi Model

4. J. Scott, Philos. Mag. 43, 859 (1952)

13. C. E. Moore, Natl. Bur. Stand. (U.S.), Circ. 467 (1949 – 1958)

5. J. Schwinger, Phys. Rev. A 24, 2353 (1981); L. DeRaad and J. Schwinger, Phys. Rev. A 25, 2399 (1982); B.-G. Englert and J. Schwinger, Phys. Rev. A 26, 2322 (1982)

14. CRC Handbook of Chemistry and Physics, 73rd ed. (CRC Press, Boca Raton, 1993)

6. S. H. Patil, J. Chem. Phys. 80, 5073 (1984)

15. A. A. Radzig and B. M. Smirnov, Reference Data on Atoms, Molecules, and Ions (Springer-Verlag, Berlin, 1985)

7. L. W. Bruch and A. P. Lehnen, J. Chem. Phys. 64, 2065 (1976)

16. E. Clementi and C. Roetti, ATOMIC DATA DATA TABLES 14, 177 (1974)

8. T. Kato, Commun. Pure Appl. Math. 10, 151 (1957)

17. S. Fraga, J. Karwowski, and K. Saxena, Handbook of Atomic Data (Elsevier, Amsterdam, 1976)

9. E. N. Lassettre, J. Chem. Phys. 43, 4475 (1965); J. Katriel and E. R. Davidson, Proc. Natl. Acad. Sci. U.S.A. 77, 4403 (1980); A. Ahlrichs, M. HoffmannOstenhoff, T. Hoffmann-Ostenhoff, and J. D. Morgan, Phys. Rev. A 23, 2106 (1982); D. Berdichevsky and U. Mosel, Nucl. Phys. A 388, 205 (1982); A. A. Radzig and B. M. Smirnov, Reference Data on Atoms, Molecules, and Ions (Springer-Verlag, Berlin, 1985)

AND

NUCLEAR

18. M. Levy and J. P. Perdew, Phys. Rev. 32, 2011 (1985) 19. Y. Zhang and M. Fink, Phys. Rev. A 35, 1943 (1987); S. H. Patil, J. Phys. B 27, 1823 (1994) 20. C. Barter, R. G. Meisenheimer, and D. P. Stevenson, J. Phys. Chem. 64, 1312 (1960) 21. W. R. Johnson, D. Kolb, and K.-N. Huang, ATOMIC DATA AND NUCLEAR DATA TABLES 28, 333 (1983)

10. S. H. Patil, J. Phys. B 22, 2051 (1989)

22. T. M. Miller and B. Bederson, Adv. At. Mol. Phys. 13, 1 (1977)

11. H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One and Two Electron Atoms (Springer, Berlin, 1957), sec 62, 61a

23. J. M. Standard and P. R. Certain, J. Chem. Phys. 83, 3002 (1983)

12. H. B. G. Casimir and D. Polder, Phys. Rev. 73, 360 (1948)

24. M. Marinescu, H. R. Sadeghpour, and A. Dalgarno, Phys. Rev. A 49, 982 (1994)

53

Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999

S. H. PATIL

Thomas–Fermi Model

EXPLANATION OF TABLE TABLE I.

Coefficient of the Asymptotic Density, Radial Expectation Values, Polarizabilities, and Angle-Averaged van der Waals Coefficients N

Total number of the electrons in the atom or the ion. Below each atom or ion is listed the following:

l 2E(eV) A ^r m & aL ^r m & asym C6

Angular momentum quantum number l of the last electron taken from Refs. [13, 15]. Ionization energy of the last electron in eV from Refs. [13, 14]. Parameter A with A/4 p being the coefficient of the asymptotic density in Eqs. (1.2) and (2.22). Radial expectation value from Eq. (3.5), for m 5 2, 4, 6, in units of a 0m with a 0 being the Bohr radius. Multipolar polarizabilities from Eq. (2.37), for L 5 1, 2, 3, in units of a 02L11 with a 0 being the Bohr radius. Radial expectation value for the asymptotic electron density as in Eq. (3.24), for m 5 0, 2, in units of a 0m with a 0 being the Bohr radius. ^r 0 & asym is equal to n e (outer) of Eq. (3.28). Angle-averaged van der Waals coefficient as in Eq. (3.11) and Eq. (3.25), in units of a 05 with a 0 being the Bohr radius.

54

Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999

S. H. PATIL

Thomas–Fermi Model

TABLE I. Coefficient of the Asymptotic Density, Radial Expectation Values, Polarizabilities, and Angle-Averaged van der Waals Coefficients See page 54 for Explanation of Table

55

Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999

S. H. PATIL

Thomas–Fermi Model

TABLE I. Coefficient of the Asymptotic Density, Radial Expectation Values, Polarizabilities, and Angle-Averaged van der Waals Coefficients See page 54 for Explanation of Table

56

Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999

S. H. PATIL

Thomas–Fermi Model

TABLE I. Coefficient of the Asymptotic Density, Radial Expectation Values, Polarizabilities, and Angle-Averaged van der Waals Coefficients See page 54 for Explanation of Table

57

Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999

S. H. PATIL

Thomas–Fermi Model

TABLE I. Coefficient of the Asymptotic Density, Radial Expectation Values, Polarizabilities, and Angle-Averaged van der Waals Coefficients See page 54 for Explanation of Table

58

Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999

S. H. PATIL

Thomas–Fermi Model

TABLE I. Coefficient of the Asymptotic Density, Radial Expectation Values, Polarizabilities, and Angle-Averaged van der Waals Coefficients See page 54 for Explanation of Table

59

Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999

S. H. PATIL

Thomas–Fermi Model

TABLE I. Coefficient of the Asymptotic Density, Radial Expectation Values, Polarizabilities, and Angle-Averaged van der Waals Coefficients See page 54 for Explanation of Table

60

Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999

S. H. PATIL

Thomas–Fermi Model

TABLE I. Coefficient of the Asymptotic Density, Radial Expectation Values, Polarizabilities, and Angle-Averaged van der Waals Coefficients See page 54 for Explanation of Table

61

Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999

S. H. PATIL

Thomas–Fermi Model

TABLE I. Coefficient of the Asymptotic Density, Radial Expectation Values, Polarizabilities, and Angle-Averaged van der Waals Coefficients See page 54 for Explanation of Table

62

Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999

S. H. PATIL

Thomas–Fermi Model

TABLE I. Coefficient of the Asymptotic Density, Radial Expectation Values, Polarizabilities, and Angle-Averaged van der Waals Coefficients See page 54 for Explanation of Table

63

Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999

S. H. PATIL

Thomas–Fermi Model

TABLE I. Coefficient of the Asymptotic Density, Radial Expectation Values, Polarizabilities, and Angle-Averaged van der Waals Coefficients See page 54 for Explanation of Table

64

Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999

S. H. PATIL

Thomas–Fermi Model

TABLE I. Coefficient of the Asymptotic Density, Radial Expectation Values, Polarizabilities, and Angle-Averaged van der Waals Coefficients See page 54 for Explanation of Table

65

Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999

S. H. PATIL

Thomas–Fermi Model

TABLE I. Coefficient of the Asymptotic Density, Radial Expectation Values, Polarizabilities, and Angle-Averaged van der Waals Coefficients See page 54 for Explanation of Table

66

Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999

S. H. PATIL

Thomas–Fermi Model

TABLE I. Coefficient of the Asymptotic Density, Radial Expectation Values, Polarizabilities, and Angle-Averaged van der Waals Coefficients See page 54 for Explanation of Table

67

Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999

S. H. PATIL

Thomas–Fermi Model

TABLE I. Coefficient of the Asymptotic Density, Radial Expectation Values, Polarizabilities, and Angle-Averaged van der Waals Coefficients See page 54 for Explanation of Table

68

Atomic Data and Nuclear Data Tables, Vol. 71, No. 1, January 1999