INTERATOMIC POTENTIALS BASED ON THOMAS–FERMI THEORY

CHAPTER

III

INTERATOMIC POTENTIALS BASED ON THOMAS-FERMI THEORY

3.1

INTRODUCTION

We have seen in Chapter II how the T F statistical theory of the atom produces an electronic screening function x(r/a) upon solution of the T F dimensionless equation. We also noted briefly that this function may be used in a simple-minded interaction potential between a pair of T F atoms (see Eq. (2.67)). In this chapter we shall examine more closely such a poten­ tial and those resulting from more elaborate treatments of the two-center situation on the T F and T F D models. It will become evident that care must be exercised in the interpretation of the physical significance of T F and T F D atoms, with reference to their two-body interaction. We remember that the electron density of the T F atom extends to infinity, so that any pair potential must be overestimated at large separations. The T F D atom, on the other hand, has a sharp electron density 35

ΠΙ

36

Thomas-Fermi Interatomic Potentials

cutoff at a finite radius, leading to the disappearance of the interaction potential for atom separations beyond twice this radius—again a rather artificial effect. Because of these long-range effects the T F and T F D inter­ atomic potentials are most reliable for small separations, typically less than 1 Â. 3.2

The who in general mation

THE FIRSOV THOMAS-FERMI PAIR POTENTIAL

earliest consideration of the two-center T F problem was by Firsov, two quite separate treatments evolved two-body potentials of the form of Eq. (2.67). His earlier derivation [1] assumed the approxi­ for the total electron energy in an atom [ l a ] : E =*= - 0 . 7 7 ( 4 π me*/h )Z 2

2

(3.1)

1/3

9l

Then the energy difference between two atoms entirely overlapped and entirely separate is given by the expression E.i(rUo

~0Jl(4n me lh ){(Z 2

4

+ Z ) ' - Ζψ

2

7

i

- Ζψ}

3

2

(3.2)

The term in brackets could be approximated by the formula ( Z + Z ) ' - Z ' - Ζψ^Ζ,Ζ (Ζ 7

T

3

7

+ Z y*

3

2

2

χ

(3.3)

2

Thus: J 5 ( r ) ^ -l.S(4n me /h )Z Z (Z 2

4

2

el

1

2

i

+ Z )

(3.4)

1 / 3

2

The interaction potential energy V{r) of the two atoms may be written as the sum of the nuclear and electronic energies: V(r) = Ζ Ζ γ

e fr + E (r) - AE (r)

(3.5)

2

2

9l

9l

Firsov applied first-order perturbation theory to calculate the change in energy AE (r) of the electrons when the two nuclei Z e and Z e draw apart by a distance r. This he postulated was equivalent in the first order to the change in energy of an atom whose nucleus is ( Z + Z )e when a charge + Z e is placed at r with —Z e at the center. This is the average potential difference of the electrons between zero and r: el

t

t

2

2

2

2

ΑΕ,Μ^ΖΛφ^-φ^))

(3.6)

where <^ (r) is the T F potential of the electrons, defined by Eq. (2.19), less the Coulomb potential of the nucleus: t

φ (τ) = {(Ζ, + Z )e/r}[ (r/ä) 1

2

X

- 1]

(3.7)

the screening radius a being defined by the equation a ^ 0.885Λ /4π W ( Z ! + Z ) 2

2

1 / 3

= 0.885α /(Ζ! + Z ) 0

2

1 / 3

(3.8)

3.3

The Firsov Variational Derivation of the Thomas-Fermi Potential

37

By expanding χ(τ/ά) about the origin and neglecting all but the first two terms, Firsov found that his E (r) of E<1- (3-5) was approximately canceled by the φι(0) term of A £ ( r ) , as would be expected from perturbation theory. His expression was asymmetric in Z and Z , a consequence of removing the charge Z e from the center of the atom and carrying the calculation to first order only. This could be easily remedied by adding or subtracting a secondorder term. This done, the Coulomb contributions in V(r) disappeared by subtraction, leaving el

el

i

2

2

V(r) = (Z Z e lr) (r/a)

(3.9)

2

i

2

X

Thus a rather crude first-order perturbation approach to the two-center problem nevertheless leads to an expression of the appropriate screened Coulombic form.

3.3

T H E FIRSOV VARIATIONAL DERIVATION OF THE THOMAS-FERMI POTENTIAL

Firsov's second interatomic potential derivation [2] was a little more fundamental in its initial assumptions, setting out this time from the total Hamiltonian of a system of nuclei and electrons (see Eq. (2.27)). At any particular separation r of two atoms it may be reasonably assumed that the electron distributions will be distorted so that this total energy is a mini­ m u m for this separation. F o r two atoms,

where r are the distances from the origin to the nuclei. The procedure was to minimize with respect to the electron density p(r), which upon simpli­ fication leads to the expression for minimal energy: t

= (c /6) k

jp

5

/ 3 0

dr-ijZ

(

W

o

dx

(3.11)

This then provided a lower bound for the Hamiltonian of the system. In itself this is inadequate to define the interaction energy accurately, so Firsov next introduced a functional 3tf of a function / which was related to the electron density p. U p o n variation of J f with respect to / , was found to be maximum when / = f , given by l

t

l

0

Then the minimal of Jfx was identical to the expression (3.11) for 3tf0. Thus he established an upper and lower bound for 3tf, although we note

ΠΙ Thomas-Fermi Interatomic Potentials

38

that no physical reality may be attached to the functional except in the case where f = f 0 The procedure in the two-center case is to replace the overall electron density by the sum of extremals of the two densities p i ( i ) * Po2( 2)> found by minimizing Jtif for each atom separately: r

a n c

r

0

Poi(0 +

Pir) =

(3.13)

P02W

and by r e p l a c i n g / b y the sum of the extremals of / for each a t o m : / = / o i

+/02

(3.14)

These represent approximations with respect to the general treatment out­ lined above for a system of nuclei and electrons. Consequently the value of calculated using (3.13) will now differ from that of using (3.14). By comparison of and jfl9 Firsov was able to calculate the upper limit to the error made in assuming the extremal density summation of Eq. (3.12) and calculating on that basis. This was in the region of 8 %, which meant that if 34?ο were assumed to be the mean of 2tf and «?f then t

0

l 9

« # - i ( ^ + ^i)

(3.15)

0

The error in was less than \ | — |, in this case ~ 4 %. The maximum error occurred for atomic interactions where p and p were close in value, and the error dropped rapidly as the values of p and p differed. It is relatively simple to express the interatomic potential V(r) in terms of these expressions for the Hamiltonians. Thus t

0 1

0 2

0 1

0 2

V{r) = Ζ Ζ e /r + jf - J f ( c o )

(3.16)

2

λ

2

V (r) = Z Z x

X

e /r + ΜΤ - ^ ( o o )

(3.17)

2

2

±

where (00) and (οο) are the expressions for two entirely separate atoms. These expressions may be evaluated using the summation technique for ρ and / of Eqs. (3.13) and (3.14), and consequently the limits for V(r) may be obtained. Firsov, however, wished to express the potential if possible in simpler terms somewhat similar to those of his earlier treatment (see Eq. (3.9)) : λ

Voir) = (Z Z e /r)x(r/a )

(3.18)

2

i

where a

l2

2

12

is a screening radius symmetrical in Z and Z : l

a

l2

= 0M5a /&(Z

He tried different forms of ^ ( Z

o

1 ?

2

Z)

l3

(3.19)

2

Z ) , including the original one: 2

(z +z y> 1

3

2

Zf +Z| 3

/ 3

)

(Zj + ζ ψ γ ' /2

(3.20)

1 / 2

3

3.4

Solution of the Thomas-Fermi Equation for the Diatomic Molecule

39

F o r each of these and for different ratios Z / Z the V (r) was evaluated over a range of r and compared with the equivalent extremal values V(r) and V^r). Best agreement was found for the third form of &(Ζ Ζ ). Then V (r) differed from V(r) by less than 2 0 % over a range of variation of r/a from 0 to 10, which represents, depending on Z , a value of r in the region of 1 Â. At distances exceeding this, agreement is meaningless, since calculation on the basis of the statistical model loses its validity. Thus Firsov finally presented as his two-body interatomic potential based on T F theory, the form 2

l 5

0

ΐ9

2

0

12

V{r) = ( Z , Z e lr) {{Z\l 2

2

2

X

+ Z^ ) / (r/0.885a )} 2

2

3

0

(3.21)

valid in the range r < 1 Â. The Firsov potentials for copper and argon are traced in Fig. 3.1. The most notable feature is the very slow decay of these potentials as r increases, which limits their range of application.

0

1

2

3 Γ (Â)

Fig. 3.1. Firsov potential for copper and argon: ( 3.4

4

5

) copper; (

) argon.

SOLUTION OF THE THOMAS-FERMI EQUATION FOR THE DIATOMIC MOLECULE

Townsend and Handler [3] made a numerical study of the two-center T F problem in application to the neutral homonuclear diatomic molecule. They calculated the total energy of a molecule made u p of two atoms of atomic number Z , separated by a distance 2r, solving the T F equation by a finite difference relaxation procedure. In calculating this energy they included an electron exchange contribution. Subtracting the energy of two isolated atoms

ΠΙ

40

Thomas-Fermi Interatomic Potentials

from the total energy of the molecule, they gave the interaction energy or interatomic potential as V(r) = Ζ V / r + £ ( r ) - E (oo) T F

TF

+ E (r) a

E (œ) a

(3.22)

The results of Townsend and Handler minus the exchange contribution E agree well with the interatomic potential of Firsov for the same pair of atoms. In a sense this might be considered to be an improvement on the Firsov model, although its basis is not so simple and it requires numerical procedures to calculate the potential. a

VARIATIONAL DERIVATION OF THE

3.5

THOMAS-FERMI-DIRAC INTERATOMIC POTENTIAL

Abrahamson and co-workers attempted to extend the maximal-minimal principle used by Firsov to the T F D model of the atom, including the exchange contribution. As was indicated in Chapter II, in the T F D model the dimensionless equation was no longer universal, so that the interatomic potential had to be numerically calculated for each pair of atoms. After the manner of Firsov, Abrahamson et al. [4] wrote down the Hamiltonian for the system, including this time the exchange term in p and found by minimization. Thus was again a functional of a slightly modified function / , and again its maximum, together with the , limited the total energy. When applied to the two-center T F D problem, this treatment yielded an interatomic potential of the form 4 / 3

0

1

0

(3.23) where Ä is a function of p and p similar to that of Firsov, except that it includes terms in p to account for exchange. Here C is a parameter which depends on Z and Z and is constant for a given pair of ions and χ is the T F D screening function. Abrahamson et al. predicted that the potential represented by Eq. (3.23) would become somewhat inaccurate when r ex­ ceeded the smaller of the two T F D electron density cutoff radii, and would be inapplicable for r greater than the sum of the two radii, where, according to the T F D model, there is no further interaction. In later work, Abrahamson [5] used the above expression for V(r) slightly modified to calculate the interatomic potentials between homonuclear rare gas atom pairs at distances ranging from 0.01 Â to about 6 Â. His purpose was to compare these with other forms of interatomic potential and in par­ ticular with those derived from experimental a t o m - a t o m scattering results (see Chapter VIII). The C parameter of Eq. (3.23) was dropped in this later 0 1

02

4 / 3

z

t

2

z

3.5

Variational Derivation of the Thomas-Fermi-Dirac Interatomic Potential

41

work, since it was held formally responsible for maintaining the electron density unrealistically high near the T F D cutoff radius, and was at the same time negligibly small inside the atom. The interatomic potentials of Abrahamson for H e - H e , N e - N e and A r - A r interactions are shown in Fig. 3.2. They appeared to be in only order-of-magnitude agreement with experimental results. Values of the Abrahamson rare gas potentials over a range of r are reproduced in Table 3.1.

r (a.u.)

Fig. 3.2. TFD interatomic potentials of Abrahamson for He-He, Ne-Ne, and Ar-Ar interactions in atomic units (a.u.).

The work was subsequently extended to a number of heteronuclear pairs of the rare gas type [6], and a recent publication of Abrahamson [7] has attempted to overcome the nonuniversal aspect of the T F D pair interaction by fitting a Born-Mayer type of analytical potential: V{r) = Ae~

Br

(3.24)

42

ΠΙ Thomas-Fermi Interatomic Potentials

TABLE

3.1

T F D INTERATOMIC POTENTIALS OF ABRAHAMSON FOR RARE GAS ATOMS (IN ATOMIC UNITS) '" 0

10

2 K*o) 0.01 0.03 0.06 0.1 0.3 0.6 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 8.0

36

18

54

86

276620 83813 37 001 19105 3 535.6 897.32 259.12 79.906 29.594 11.863 4.8152 1.8505 (0.76) (0.30) (0.12) (0.044) (0.018) (0.00044) 4.657 7

697150 208450 90535 46067 8 001.8 1 922.8 533.61 159.95 58.428 23.438 9.5162 3.7429 1.325 7 (0.57) (0.22) (0.083) (0.032)

\ 9690.0 31206 9 690.2 3 039.3 4407.9 1 362.9 2 360.7 747.48 491.01 166.40 49.404 138.75 43.807 16.229 5.407 8 14.298 2.053 5 5.4115 0.79962 2.145 9 0.297 11 0.836 33 0.10078 0.298 79 0.040 756 0.10195 (0.015) (0.038) (0.005 6) (0.014) (0.0021) (0.005 2) (0.001 9) —

391.91 125.31 59.155 33.073 7.999 5 2.5997 0.908 24 0.303 89 0.10711 0.035 695 0.011982 (0.0039)* (0.001 2) (0.00042) (0.000 14) —

—

123 720 37 908 16921. 8881.1 1714.1 453.15 136.51 42.843 16.081 6.4522 2.5829 0.970 50 0.327 82 0.14192 (0.055) (0.022) (0.008 3)

—

—

—

—

3.321 0

4.0507

4.281 8

4.527 5

—

4.795 2

After Abrahamson [5]. * Extrapolated values are enclosed in parentheses. β

This is feasible because, as we note from Fig. 3.2, the T F D potential is quite linear on a semilogarithmic plot over a considerable range of r, typically from about \.5a to 3 . 5 # , although extrapolation of the analytical curve to ~6a still produced better than order-of-magnitude agreement with the full T F D potential. In this way Abrahamson was able to list the constants A and Β for homonuclear pair interactions of all atoms from Ζ = 2 to Ζ = 102. This list is reproduced in Table 3.2. Using these values, a combination rule could be used to calculate all possible heteronuclear pair interactions: 0

0

0

V **(V V ) '

(3.25)

1 2

l2

11

22

The theoretical basis for this combination rule in a more general form has been discussed recently by Smith [7a]. Using an atomic distortion model to describe the atoms during the collision, he finds that the mean BM para­ meters A and Β are given by {A B ) ' » 2 B

12

l2

= (A B y » /B

n

n

+(A B y> » B

22

22

(3.25a)

3.5

Variational Derivation of the Thomas-Fermi-Dirac Interatomic Potential

43

This is equivalent t o Eq. ( 3 . 2 5 ) only in the case where £ = B · However, since the variation of the BM exponential parameter Β is usually quite small (see Table 3 . 2 ) , the use of Eq. ( 3 . 2 5 ) as a combination rule introduces very little error in comparison with other sources of uncertainty. Thus a simple analytical expression has been made available for over 5000 a t o m - a t o m interactions. This expression is equivalent over a reasonable range of r to that arising from the two-center solution of the T F D statistical model of the atom. Such an expression would be extremely desirable for many applications, were it not for the fact that its reliability, and indeed that of the whole Abrahamson variational treatment of the T F D problem, has been questioned by Günther [ 8 ] . Abrahamson a n d co-workers, in their formulation of the maximum principle, did not allow the finite boundary of the T F D electron density to be varied in the maximization. Their atomic radius was assumed to remain constant during the interaction, equal t o the normal T F D radius. Günther included this condition in a reformulation of the maximal principle, a n d found that the extremal of the modified Abrahamson functional did not represent a maximum with respect to surface variation of the two atoms, but rather had a cubic form of behavior. Thus the maximal principle could not be used with surface variation to obtain a limit to the total energy. Günther showed that if the correct T F D radius were known from the beginning of the calculation and retained in the variation, the extremal value of the maximal principal did in fact prove t o be a maximum. It would not however be known in most cases of practical interest. T h e unjustified constant T F D radius assumption was further shown by Günther t o lead in the two-center case to an error in the development of adding to the discrepancy in the final result. In a convincing comparison of T F D interaction energies for two argon atoms, Günther demonstrated that the Abrahamson potential overestimated the interatomic potential by quite a serious margin. In the absence of a reliable method of calculating the exact T F D interaction, he obtained an upper limit using the results of Townsend and Handler [ 3 ] (see Section 3 . 4 ) , whose T F potential without exchange agreed well with the work of Firsov. This upper limit Vj (r) was realized using the exact T F D energy a n d exchange energy of the isolated atoms : n

22

FD

VTFDW

= V (r) m

+ £ (oo) + £ (oo) - £ XF

fl

TFD

(oo)

(3.26)

where V (r) is the Townsend a n d Handler interaction potential given by TH

Eq. ( 3 . 2 2 ) . We have effectively

F

T F D

( r ) = Ζ V / r + E (r) TF

+ E (r) a

E (œ) TFD

(3.27)

ΠΙ Thomas-Fermi Interatomic Potentials

44

TABLE 3.2 NUMERICAL VALUES OF BORN-MAYER PARAMETERS A AND Β FOR NEUTRAL GROUND-STATE TFD

Atomic number, Ζ

ATOMS WITH Ζ = 2 το Ζ =

A Chemical symbol

105

e

Β ε"

(e /a ) 2

Q

(eV)

1

(A" )

(%)

(at )

1

2 3 4 5 6 7 8 9 10

He Li Be Β C Ν Ο F Ne

8.6047 16.109 24.599 35.606 48.367 62.840 78.771 96.267 114.72

234.13 438.33 699.34 968.84 1316.1 1709.9 2143.4 2619.4 3121.5

2.207 79 2.12081 2.05904 2.027 71 2.015 92 2.00900 2.00474 2.00229 1.999 54

4.17217 4.007 80 3.891 07 3.831 87 3.809 59 3.79651 3.78846 3.783 83 3.778 63

4.1 4.2 4.7 4.1 4.5 5.1 5.0 5.1 5.6

11 12 13 14 15 16 17 18 19 20

Na Mg Al " Si Ρ S Cl Ar Κ Ca

134.56 140.72 157.85 186.39 205.67 222.99 235.64 255.82 277.95 298.57

3661.4 3829.0 4295.1 5071.7 5596.3 6067.6 6411.8 6960.9 7563.0 8124.1

1.99899 1.95694 1.94681 1.958 88 1.95137 1.94026 1.92449 1.91901 1.91634 1.91044

3.777 59 3.69813 3.67899 3.701 80 3.68760 3.66661 3.636 81 3.62645 3.621 41 3.61026

5.8 4.9 4.2 4.3 3.9 3.7 3.6 3.6 3.6 3.4

21 22 23 24 25 26 27 28 29 30

Sc Ti V Cr Mn Fe Co Ni Cu Zn

319.62 343.72 366.70 389.77 414.53 438.46 463.35 487.72 511.53 539.78

8696.9 9352.6 9977.9 10606 11279 11931 12608 13271 13919 14687

1.90544 1.90402 1.90077 1.89794 1.89607 1.893 00 1.891 16 1.88818 1.884 57 1.88424

3.60081 3.598 12 3.591 98 3.58663 3.583 10 3.577 30 3.573 82 3.56819 3.561 37 3.56074

3.2 3.2 3.2 3.1 2.9 3.0 3.0 2.7 2.7 2.8

31 32 33 34 35 36 37 38 39 40

Ga Ge As Se Br Kr Rb Sr Y Zr

564.68 590.33 618.26 645.35 672.43 703.70 731.90 763.87 792.10 819.91

15 365 16063 16823 17 560 18 297 19148 19915 20785 21553 22310

1.88099 1.87903 1.878 33 1.87688 1.87495 1.875 20 1.873 73 1.873 95 1.87249 1.87042

3.55460 3.55090 3.549 58 3.54684 3.543 19 3.543 66 3.54088 3.541 30 3.538 54 3.53463

2.8 2.8 2.7 2.8 2.9 2.8 2.9 2.8 2.8 2.9

3.5

Variational Derivation of the Thomas-Fermi-Dirac Interatomic Potential

TABLE 3.2

{continued)

Atomic

Β symbol

(e M

41 42 43 44 45 46 47 48 49 50

Nb Mo Tc Ru Rh Pd Ag Cd In Sn

850.89 881.65 911.85 941.72 972.66 1005.2 1040.7 1071.1 1102.6 1140.2

23153 23 990 24811 25 624 26466 27 352 28318 29145 30002 31025

51 52 53 54 55 56 57 58 59 60

Sb Te I Xe Cs Ba La Ce Pr Nd

1171.4 1199.4 1231.2 1265.2 1298.9 1336.4 1370.2 1403.3 1436.0 1471.5

61 62 63 64 65 66 67 68 69 70

Pm Sm Eu Gd Tb Dy Ho Er Tm Yb

71 72 73 74 75 76 77 78 79 80

Lu Hf Ta W Re Os Ir Pt Au Hg

ζ

45

2

(eV)

(A- )

(%)

1.86990 1.86926 1.86489 1.86723 1.86643 1.865 99 1.866 57 1.86521 1.86416 1.865 10

3.533 65 3.53244 3.52418 3.528 60 3.52709 3.52626 3.527 35 3.52478 3.52280 3.52458

2.9 3.0 3.1 3.1 3.1 3.0 3.1 3.2 3.3 3.4

31874 32 636 33501 34426 35 343 36363 37283 38184 39074 40040

1.863 82 1.86136 1.86020 1.859 80 1.85901 1.859 56 1.85891 1.85796 1.85679 1.85664

3.52216 3.51751 3.515 32 3.51456 3.51307 3.51411 3.51288 3.51108 3.508 87 3.508 59

3.5 3.5 3.6 3.5 3.4 3.5 3.5 3.5 3.6 3.6

1508.4 1544.3 1580.5 1615.9 1651.4 1688.7 1724.9 1765.1 1801.4 1842.3

41044 42020 43005 43969 44935 45 950 46935 48028 49016 50129

1.856 51 1.85621 1.85606 1.855 71 1.855 12 1.855 13 1.85465 1.85493 1.85449 1.85489

3.508 34 3.507 78 3.50749 3.50683 3.505 72 3.505 73 3.50483 3.505 36 3.504 52 3.505 28

3.7 3.7 3.8 3.7 3.8 3.8 3.8 3.6 3.7 3.8

1879.8 1914.1 1952.7 1991.2 2028.1 2067.9 2104.6 2144.2 2185.7 2230.0

51 149 52083 53133 54181 55185 56268 57266 58 344 59473 60678

1.85459 1.85344 1.853 46 1.853 00 1.85231 1.85240 1.85190 1.85191 1.85204 1.852 54

3.50471 3.502 54 3.502 58 3.501 71 3.50041 3.50058 3.499 63 3.49965 3.499 89 3.50084

3.8 3.8 3.9 4.0 4.0 4.0 4.0 4.0 4.1 4.1

1

ΠΙ Thomas-Fermi Interatomic Potentials

46 TABLE 3.2.

(continued)

Atomic number, Ζ

Chemical symbol

(e la )

81 82 83 84 85 86 87 88 89 90

Tl Pb Bi Po At Rn Fr Ra Ac Th

2270.7 2311.8 2349.2 2395.9 2434.6 2476.5 2517.4 2623.0 2665.6 2716.9

91 92 93 94 95 96 97 98 99 100

Pa U Np Pu Am Cm Bk Cf Es Fm

101 102 103 104 105

Md No Lw

a b

Α

— —

Β e (%) b

(aô )

(Â" )

61 786 62904 63 922 65192 66246 67 386 68499 71372 72531 73927

1.85246 1.85258 1.85164 1.85224 1.85173 1.85163 1.85137 1.863 68 1.863 60 1.86470

3.50069 3.50092 3.49914 1.50027 3.499 31 3.49912 3.49863 3.521 89 3.521 74 3.523 82

4.1 4.2 4.2 4.1 4.1 4.2 4.2 5.7 5.6 5.3

2761.1 2806.6 2850.8 2895.1 2940.2 2981.6 3027.6 3072.8 3115.2 3160.0

75130 76368 77 570 78 776 80003 81129 82381 83 611 84765 85 984

1.86480 1.86519 1.86518 1.865 32 1.86544 1.86487 1.86511 1.86506 1.86456 1.86461

3.52401 3.52475 3.52473 3.524 99 3.525 22 3.524 14 3.52459 3.52450 3.523 55 3.523 65

5.2 5.0 4.7 4.5 4.3 4.2 4.2 4.1 4.2 4.2

3207.8 3253.6 3300.2 3346.4 3393.9

87284 88 531 89 798 91056 92348

1.86485 1.86495 1.86511 1.86499 1.865 25

3.52410 3.52429 3.524 59 3.52437 3.52486

4.2 4.2 4.2 4.3 4.3

2

0

(eV)

1

1

After Abrahamson [7]. ε is the magnitude of the maximum percent error for each fit.

This represents an upper bound since E (r) and E (r) will be greater than the exact E (r) found by minimization. It may not be considered a good potential since Vj (co) is nonzero, but it is nevertheless useful as a limiting potential for the purposes of comparison. In Table 3.3, reproduced from Giinther's work, the different potentials are compared for four values of inter­ atomic separation r. There are the T F potentials of Townsend and Handler omitting and including exchange, the upper bound for the T F D potential, and the T F D potential according to Abrahamson, interpolated from Table 3.1. All potentials are for A r - A r interactions. TF

TFD

FD

a

3.6

Analytical Thomas-Fermi Two-Body Potentials

47

TABLE 3.3 ARGON-ARGON POTENTIAL IN ATOMIC UNITS

Φα y TFD V — upper bound for exact TFD potential

0

0.076314

0.76314

1.52629

3.81572

3062.3 3037.8

59.730 53.883

10.321 8.315

0.605 0.353

3039.3 3222

55.4 80.57

9.8 13.54

1.8 0.152

31

28

Minimum error in Abrahamson potential in %

5.7

—

° After Gunther [8]. After Townsend and Handler [3]. After Abrahamson [5]. b

c

It may be seen from this table that in the intermediate range of r, the Abrahamson potential is considerably higher than that calculated from the Townsend-Handler work. At small r the nuclear repulsion becomes pre­ dominant and the difference is less noticeable. The table also shows that the true T F D potential lies below the T F potential over the whole range of r, instead of being above it at intermediate separations as found by Abrahamson. This is in accord with the results of Townsend and Handler including ex­ change, and with those of a separate study by Günther [9] on the basis of mutual penetration of the shell structure of the two interacting atoms.

3.6

ANALYTICAL THOMAS-FERMI TWO-BODY POTENTIALS

T o complete the present chapter we would draw attention to the two-body potentials which make use of analytical approximations to the accurate T F screening function χ(χ). These are simplifications of the Firsov potential and can use the same screening length a. They belong more appropriately to the next chapter, where we shall discuss them in relation to other forms of empirical interatomic potential. In this chapter some rather complicated mathematical arguments have been radically pruned, and the methods have been presented only in outline. The author apologizes for any lack of clarity which results from this type of presentation. The alternative was to reproduce in toto the mathematical development of the authors in question, which is inappropriate when the original article is available for consultation by the interested reader.

48

ΠΙ Thomas-Fermi Interatomic Potentials REFERENCES

1. O. B. Firsov, Dokl. Akad. Nauk USSR 91, 515 (1953). la. P. Gombas, "Die Statische Theorie des Atoms und ihre Anwendungen." SpringerVerlag, London and New York, 1952. 2. Ο. B. Firsov, Zh. Eksperim. Teor. Fyz. 33, 696 (1957) [English transi.: Sov.Phys.JETP 6, 534 (1958)]. 3. J. R. Townsend and G. S. Handler, Chem. Phys. 36, 3325 (1962). 4. Α. A. Abrahamson, R. D. Hatcher, and G. H. Vineyard, Phys. Rev. Iii, 159 (1961). 5. Α. A. Abrahamson, Phys. Rev. 130, 693 (1963). 6. Α. Α. Abrahamson, Phys. Rev. 133, A990 (1964). 7. A. A. Abrahamson, Phys. Rev. 178, 76 (1969). 7a. F. T. Smith, Phys. Rev. A 5, 1708 (1972). 8. K. Günther, Ann. Phys. 14, 296 (1964). 9. K. Günther, Kernenergie 7, 443 (1964).