On Slater's transition state for ionization energies

chcmkal lllysics 7 (1975) 100-107 0 NOM-HollandPublishingCompany

ON SLATER’S TRANSITION STATE CONCEPT FOR IONIZATION ENERGIES Karlhcinz

l&rirur

SCHWARZ

@rTechnische

Eleknochemie.

Technische Hochschule Wien. A-1060

and IBM Research Lobomrory, fan lose,

Califomio

951

I 4.

Vienna, Ausnti

USA

Reccived4June1974

Revised manuscript received 7 October 1974

For the Xa- and the Xc&exchange approximation the transition ate concept for estimating ionization energies is analyzed mainly in atomic systems. Comparisons with HF results and experimental dab axe presented in order to put the different theoretical approaches in pro& perspective.

1. Introduction

tion is responsible for the relatively wide applicability of Koopmans’ theorem for valence orbitals.

Because of the rapid prcgress in experimental techniques such as ESCA [l] and photoelectron spectroscopy 121, interest has grown in theoretical models suitable for the calculation of ionization energies. All methods within the HF framework can make use of Koopmans’ theorem [3] in order to approximate the ionization potential 5 of the system by the orbital energy EjHF of the corresponding occupied SCF MO:

4 z -ejHF.

(1)

This follows from the fact that thejth HF energy eigenvalue equals the total energy difference between the ground state (atom, molecule) and the ion with the jth electron missing, when one assumes that the orbi. tals do not relax after the removal of thejth electron (indicated by the subscript 0): EjHF

EjHF

=

Em(nj

= njfJ)lo -EHF("j=

“jo

-

l)lo

(2)

Orbital relaxation effects can be taken into account by taking the total energy difference AE,, of two separate self-consistent calculations. This scheme of approximating Ij by UHF is superior to Koopmans’ values for many cases, in particular for core levels, where orbital relaxation exceeds the contributions to the ionization energy from ilectron correlation. In the case of valence levels, orbital relaxation and correlation effects are of timilar size but opposite sign [4]. This error compensa-

For many years Slater’s statistical approximation (Xol method [S]) to the electron exchange has been widely used in solid state calculations. More recently the Xor scheme has become one of the concepts upon which the SCF-a-scattered wave [6,7] and discrete variational [8-10) methods are based. These methods represent new techniques for calculating the electronic structure of polyatomic molecules. Another form of the statistical exchange approximation, namely the XF&3 method, has been discussed in a previous paper [ 1 I] and will also be used in the present work. It has been shown [ 5,12,13] that the eigenvalues in the statistical exchange approximations are partial derivatives of the statistical total energy (E) with respect to the orbital occupation number nj which is assumed to be non-integer [ 141:

~j. = aWanj.

(3)

This meaning of the one-electron energies Ej differs from HF theory. Koopmans’ theorem cannot be used in the Xa or XF@ method. Therefore an additional concept was necessary to enable the computation of ionization energies, or more generally, excitation energies. Slater’s transition state idea [S, 15,161 has proved to be very valuable in this context. It has been used already in atomic and molecular calculations and has been analyzed from different points of view [17-211.

101

R Sclr~c~or:/~ansirio,r slate 2. Motivation

of the transition

state

concept

The ionization in atoms is used in this section in order to demonstrate Slater’s transition state idea. Since in the statistical approximations (Xa or XF@3) the orbital occupation numbers nj are assumed to be non-integer [14] the ionization process becomes continuous. This is illustrated in fig. 1 fo: the 3plevel of the cNotine atom. Within the Xcz method the statistical total energy (EXJ is computed self-consistently for varying occupation number rr3,., and corresponding ionicity. The ground state is given for rrJp = 5 and the ion Cl+, with one 3p-electron missing, is represented by tr3p = 4. The energy eigenvalue ejp is, geometrically speaking, the tangent to (Ex,) according to (3). It can be seen from fig. 1 that, going from the atom to the ion. (EXJ becomes steeper, the eigenvalues mom negative, i.e., the electron is more tightly bound, as one would expect in this case. Using this picture a motivation for the transition state concept can be given by the following arguments: The ionization energy is represented by the total ener-

gy difference AfZ between the atom and the ion, in this mode1 by M = A (I??,). From fig. I it can be seen that the slope of AE/Atr (secant line) is very nearly the same as that of the tangent in the midpoint (halfway between the atom and the ion) which corresponds to the transition state eigenvalue &. Thus one effectively mixes the initial states (atom) and the fiiial states (ion) in the transition state by removing half an electron from the shell under consideration.

of (El

3. Expansion

(E(Iz~)) = (E)o + aj(nj-tIjo) + ei(ni-niO)?

+ ___

(4)

According to (3) the eigenvalue eI as a function of Ri can be obtained from (4). Three different approximations to Ij can be derived: (i) the total energy difference between atom and ion

RY

1

series of ‘4

The statistical total energy of either the Xa or the XFctJ method can be expanded in a power-series of the occupation numbers ni [S. 13,161. A non-spinpolarized [S] ground state is taken to be the standard state with occupation numbers nio. For the ionization from orbital j a Taylor’s series expansion of(E) up to third power terms in short hand notation is

+ bj(nj-njo)’ E tata,.

in a power

TRANSITION

-918.0 -

Uj

=

=

‘E(flj

=

no))

-

=

tZjO_I

))

ai - 6, + cj - . . . ,

(ii) the transition

-919.5 -

‘E(nj

state eigenvalue

(5) e,E is given by (6)

~,~=a(E(nj=n~O-))~/anj=aj-bi+~Cj-....

(iii) the eigenvalue

$ which is obtained

as average

from the eigenvalues of the atom $‘lom and the ion e,?” [22], is

-919.0 L

45

’

“BP

5.0

Fig 1. Total energy Etotd of the chlorine atom (ion) as a function of the 3p orbital occupation number nsp. computed

self-consistently within the x0 method. The 3p-energy eigenvalues corresponding to the ion Cl+ kt”n), to the transition state (ES) end the atom (Atom) appey as tingents to Etow_ represents the ionization of one 3p

The secant line &/An

electron.

5 = f (+om

+~~)=a~--_~+;c~-..._

(7)

A comparison of eqs. (5). (6) and (7) shows that up to second order terms in the expansion of (13 in (4) the three approximations to fi yield the same result. Geometrically speaking, this means that 0 as a function of nr would be represented by a parabola (cf. fig. 1). however, if third order terms are important, then the three estimates for fi according to (5).

___-T ?

102

K. Schwan/lTonsifion

TJble 2 . Threeapproximations to the ionization energies for the N2 molecule with R = 2.068 au evaluated from results of an SCF-Xa-SW calculation reported in ref. 1201

cts

_L I -

_-

_-

hE

C

I

I- - --_

store

;i

Nz

lo

2ag

20”

1%

3Og

.= AE 7

29.9564 29.9641 29.9752

2.3157 2.3191 2.3272

1.3440 I .3469 1.3549

1.3398 I .3432 1.3512

1.0354 1.0402 1.0499

AU energies are negative and in rydbergs.

Fig. 2. Term scheme for the three approximations

to the ionization energy according to qs. (5). (6) and (7) up to QGrd order terms.

three in the expansion of (E) can be neglected [ 13,161. Furthermore it explains, in a formal way, why en differs from B. Since this difference is found to be small, the transition state eigenvalue e,F is a good approximation to the quantity one wants for computing the ionization energy 4. From a numerical point of view the transition state eigenvalue is easier to obtain than the difference in total energy, which would require extremely high precision of the total energy (especially in the cases of large systems) in order to compute an accurate small difference.

(6) and (7) differ as is illustrated schematically in fig. 2. For the Xa method fig. 2 represents the actual situation very well. as can be seen for the chlorine atom (table 1) and for the N2 molecule (table 2) since the difference between en and AE is about half the difference between AE and F_ The XFcr/3 results of table 1 will be discussed in the next section. This analysis shows that terms of order higher than Table 1 The three approximations eqs. (5). (6) and (7)

Mjv

cts. AE and bto the ionization energies for the chlorine atom according to

XFQ!3 i IS

2s

2P

3s

3p

aElani

=i

aE/anj

208.26115

208.51754 208.51558 208.51425

208.51490

et= AE F

208.26 124 208.25890 208.25339

cts AE F

19.67001 19.68033 19.70108

19.67000

15.5 1486

PE e

15.51485 15.52831 15.55527

ets AE E

I .89466 I.89937 1.90886

1.89466

ets

0.97943 0.9844 I 0.99442

0.97942

ets

AE c

208.25329

19.70106

15.55529

1.90883

0.99442

208.51574

19.58792 19.56303 19.61807

19.55298

15.32944 15.30522 15.36967

15.29179

1.85308 1.83309 1.86705

1.82850

0.93301 0.91215 0.94792

0.90721

19.58319

15.33214

1.84228

0.92205

ti = 0.72277 in Xu; XF@ refers to XFaallj) as dcfiicd in [ 111. y denotes the results obtained from eigenvalues and a&/an/ indicates the partial derivative resuk Ali energies are negative and in rydhergs.

103

4. The modifying function of the XF@3 method in connection with the transition state

” AE

The modifying function f which is used in the XFc$ method in order to keep the exchange-inhomogeneity correction (p-term) within certain limits, has been discussed in detail [ 1l] in connection with various total energy terms. It has been mentioned in [ 1 l] that eq. (3) holds rigorously for the XQ method. while for XFq3 it is satisfied only approximately when a modifying function is used. This fact plays an important role in connection with the transition state eigenvalue. The validity of (3) is checked numerically in order to estimate the effect of the modifying function on eigenvalues. For this purpose the ionization energies for all shells in the chlorine atom are computed using the three approximations according to (5). (6) and (7) within both, the Xa and the XFQ$ method (table 1). The quantities ets and Eare obtained once from the corresponding eigcnvalues (table 1, columns labeled by Ej), and once by numerical differentiation (table 1, columns labeled by aE/arri) according to (3). In the case of the Xormethod excellent agieement between the two techniques is found, as it should be, since (3) is obeyed rigorously. This agreement demonstrates that the numerical precision of our computer programs is sufficient for this kind of analysis. In the case of the XF@ method there are small deviations from (3) and therefore the two columns in table I labeled by pi and i3E/anj differ by a few hundredths of a rydberg. If all quantities are obtained from the total energy calculations only (i.e., E& and Care computed from partial derivatives instead of the corresponding eigenvalues), then the term scheme of fig. 2 is valid again, similar to the Xa results. For the 2p-state of the chlorine atom a typical situ. ation is illustrated in fig. 3. The three approximations ; es, AE and Pare compared between the xol and the XF@3 method, in which the two modifying functions f2 and fS are used as defined in [ I 1] _ The XCYresults are shown twice, once on the original energy scale and once shifted in energy such that A/Z matches the corresponding value from XFc@~~). These shifted XQ results agree well with the respective quantities of the XFOpcf,) method corresponding to the derivative values showing that the third order terms are very simila; in the two schemes.

t -15.45

PE

I

-15.55t

-

g I

/ I -15.50

Al

I

Fig. 3. Different approximations to the ionization energy cf Ihe Zpstate in the chlorine atom ax shown using the XQ method (a = 0.72777) and the xFa~3 method with the two modifying functions f2 and fs [ t I] _Tbere~titsobtained Gem eigenvalucs are labeled by ei, those from the pax&l derivative are marked by aElani.

Judging by the results presented in table 1 and fig. 3 one can see the close agreement between the transition state eigenvalue and AE. The various effects of the modifying functions in the XFOQ method on the transition state eigenvalues are all small in comparison to the difference which comes about by approximating exchange via the Xa or the X!%$ method, where even those differences are not very big. Fortunately the choice of modifying function is not too crucial in connection with transition state calculations.

5. Comparison

of different

theoretical

approaches

In order to stcdy the order of magnitude of sume effects which have been discussed in the previous sections, ionization energies for a number of selected atoms are evaluated within different theoretical approaches. All results summarized in table 3 are based on self-consistent XFJ calculations using the modify. ing function f2 as defined in [ 111. The HF results are obtained using the HF formulas but XF@3 orbit&.

R SchwanJlhnri~ion

104

Table 3 Comparison of different theoretical approaches for e&nating

stare

ionization potentials Ii of several xkcted

atanu

AlOlll

Orbit

He 2

Is

1.1063

1.8378

1.7233

1.9704

1.9215

C6

IS 22

20.1036 0.9584 0.3259

22.7346 1.4598 0.8249

21.8103 1.3205 0.73 18

22.4205 1.3892 0.7045

22.3582 1.3809 0.7199

61.0253 2.602: 0.9019

65.6361 3.9316 1.7682

63.8207 3.6122 1.4492

64.6349 3.2713 1.5307

64.6322 3.2885 1.5596

130.96f.3 10.1637 6.9575 0.7305 0.2477

137.6885 12.3610 8.5710 1.1009 0.5617

135.6078 11.8075 7.9488 1.0254 0.5217

136.5671 11.1424 8.0008 1.0226 0.4982

136.5369 11.1396 8.0017 1.0424 0.5091

228.4268 21.6195 lb.8152 1.6979 0.6859

237.2660 24.6740 19.1843 2.5739 1.1972

234.8312 23.8683 18.2843 2.4366 1.0833

235.8863 22.8653 18.2333 2.1373 1.0779

235.8807 22.9007 18.2601 2.1600 1.1048

355.6077 38.9869 32.5256 4.4868 2.7870 0.2557 0.2724

366.6141 42.9133 35.6600 5.8057 3.6452 0.7963 0.4533

363.8300 41.7622 34.3953 5.4486 3.3065 0.6145 0.4180

365.0346 40.4812 34.3104 4.9962 3.2680 0.6250 0.4485

364.9854 40.4866 34.2950 4.9826 3.2609 0.6405 0.4765

509.7279 51.0691 6.6952 4.3108 0.4939 0.3235

522.8935 64.0347 54.9991 8.4718 5.6147 1.2661 0.5354

519.6000 62.4005 53.2335 7.9197 5.0833 0.8746 0.4884

520.9732 60.8547 53.0017 7.2566 4.8551 0.9549 0.5262

520.9321 60.8657 53.0758 7.2540 4.8556 0.9703 0.5665

904.3191 114.7631 103.0020 15.0882 11.0485 3.5192 1.1556 0.4172

921.8350 121.44so 108.6543 17.9597 13.4202 5.3940 1.7073 0.7974

917.9145 119.3846 106.4133 17.2431 12.7075 4.6863 15999 0.7287

919-.4868 117.0495 105.8626 15.9019 11.8512 4.6834 1A901 0.7066

919.4633 117.0843 105.8761 15.9216 11.8720 4.7148 1.5178 0.7281

1420.6174 197.1959 1815716 33.3764 27.3776 16.4115

1442.5613 205.8596 189.0569 37.3155 30.7267 18.7286

1438.1704 203.4708 186.4498 36.2942 29.6854 17.7152

1439.8913 200.2710 185.4269 34.3641 28.3812 17.4741

1439.8570 200.301 I 185.4311 34.3883 28.4033 17.5025

2P Ne 10

1s 2s 2P

Si 14

IS 2s 2P 3s 3P

Ar

18

IS 2s 2P 3s 3P

Ti 22

IS 2s 2P 3s 3P 3d 4s

Fe 26

1s 23 2P 3S 3P 3d 4s

Se 34

1s 2s 2P 3s 3P 3d 4s 4P

Mo42 E 2P 3s 3P 3d

o# eigcnvalue

Koopmans theorem

Total energy difference (atom-ion)

Vos

?HF

?iHF

59.2268

Trandtion State eigenvalue

qo$

G&

K. SehwarzflFansirion

LOS

slate

Table 3 (continued) Atom

Orbit

4s 4P 4d 5s

Sn 50

IS

2s 2p 3s 3P 3d 4s 4P 4d 5s 5P Ce 58

1s 2s 2P 3s 3P 3d 4s 4P 4d 5s 5P

4r Sd 6s

ag3eigenvalue

Koopmans’ theorem

Total energy difference (atom-ion)

Transition stale eigenvalue

%xB

‘IHF

+HF

5%

4.3961 2.7027 0.2276 0.2260

5.6546 3.5667 0.6497 0.4259

2056.1156 303.3009 283.6557 58.2663 50.1783 35.2122 9.0434 6.3580 1.9277 0.6521 0.2208

2082.5097 314.0159 293.0498 63.2441 54.4719 38.3876 11.0688 7.9788 2.1116 0.9750 0.4990

2815.3293 437.6962 413.8983 94.0916 83.8213 64.7344 18.8517 15.0191 8.3069 2.6624 1.6281 0.5731 0.2113 0.2092

2846.2144 450.5039 425.2460 100.1408 89.095 1 68.7691 21.6090 Ii.2983 9.6299 3.5047 2.1484 1.3475 0.5 154 0.3451

A% 4.7821 3.0720 0.5111 0.355 1

4.8056 3.0904 05343 0.4219

2017.7750 311.4031 290.187 1 62.0649 53.2565 37.0560 10.6323 7.5686 2.4379 0.9100 0.4107

2079.6175 301.213 1 288.6181 59.5623 51.5218 36.7072 9.6178 6.9101 2.4065 0.8964 0.4172

20195613 307.2924 288.6079 59.S829 5LJ370 36.7255 9.6285 6.9202 2.4224 0.9269 0.4452

2841.2443 447.7500 422.2196 93.8314 87.1440 67.2732 21.0628 16.7650 9.1277 3.3491 2.0103 1.0252 0.4678 0.3296

2843.1551 442.6336 420.0472 95.1205 85.5319 66.6946 19.5756 15.7243 8.9644 3.0194 1.9573 1.0935 0.4353 0.3362

2843.0191 442.6349 420.0149 95.7259 85.5292 66.6937 19.5715 15.7208 8.9670 3.0194 1.9562 1.0984 0.4536 0.3677

5.2834 3.2379 0.4797 0.3823

Q@refers to XFo$cfz) as detined in [ 111. lhc HF results are obtalned using the HF expressions negative and in rydbcrgs.

but @3 orbitals. AU energies are

For every ionization energy I’ five estimates are presented in table 3: first, the XFc$ eigenvalue eias which would be a poor approximation to 5; second, the HF one-electron energy eiHF which corresponds to Koopmans’ theorem [3]; next, the total energy difference AE taken between atom and ion of two separate self-consistent calculations once computed with the HF and once with the statistical ((EXF,&) total energy E,,g and last, the transition.state eigenvalue

table 3 exhibiting the significant numerical difference between these two quantities. Orbital relaxation effects are responsible for the difference between eiHF and The deviations of AL$HF from tiiGJ are due lo the different treatment of exchange in the total energy in HF and XF&. respectively. The relatively close agreement between AEjmB and E& has been discussed in section 4 in more detail.

e,&?That ejas and e,xF have a different meaning according to (2) and (3) has been mentioned in the introduction and becomes apparent from the results of

6. Comparison with experiment

tijHF.

With all the studies made in the previous sections we now can compare experimental data with the dif-

106

K. Schwarz!~ansition

ferent models for computing the ionization potentials

slate

atom that in the Xa method a Taylor’s

tion state eigenvalues eErr and e,” in both the XF@ and the Xn method, respectively, are given in table 4. Since HF theory is the theoretical counterpart to the statistical methods, its estimates to Zi are also pre-

sion of the ionization energy agrees very well With the transition state energy which’is the first tern in the

scnted. According to Koopmans’ theorem the HF oneelectron eIlergieS E~HF are approximations to fj. Orbi-

series. Using his result we have completely neglected

tal relaxation effects are accounted for by taking the HF-total energy difference A,!?,, (table 3). The quantities ~~~(43) and AEHF(Qp) are again obtained using the HF formulas but XF& orbitals. The energies EHF(4) are Seen to agree wc!g With true HF values eHF obtained by Mann [25) showing the close agreement between XFc@ and HF orbit& in accord with the situation which has been discussed in [ 1 l] in connection with the HF-total energy.

!i*

Recently

Beebe

[21]

higher order corrections

has shown

for the chlorine series expyl-

in the present

paper.

The chlorine atom is also chosen for our comparison with experiment. The multiplet splitting of each configuration has not been taken into account in the present study. We consistently use the average of the configuration. Based on experimentll data these averages have been estimated by Slater [23]. Since all approximations to Ii which we want to discuss here are nonrelativistic, the experimental data are corrected to nonrelativistic values (table 4) where the relativistic corrections are obtained within the XF@ method by first order perturbation theory as discussed in [24]. Six theoretical estimates for Ii are presented in table 4 for a comparison with the non-relativistic experimental data. As has been mentioned before, the one-electron energy eigenvalues eipa in the XFoIp method would be a poor approximation to Ii, a fact which is now well understood [5,13]. From the three quantities E,~. aEj and 5 which are the proper estimates for Ii within the statistical methods and which have been compared in the previous section, only the transi-

Table 4 EStimateS of ionization potentis xF0gf.f~) method [ 1 I ]

Zj for the chlorine atom, where a=

7. Conclusion Judging by the agreement

with experiment

The transition state which effectively takes orbital relaxation into account, is a very useful concept. It has

computational advantages, and leads to close agreement with experimental data. The modifying function

0.72277

in the Xa

method

and 4

refers

IS

2s

2P

3s

3P

experiment [ 23) r&tilJiStiC corrections non-relativistic experiment

208.4

20.3

15.3

1.86

1.01

0.02

0.01

g eHF(@) -F-

(2511

UHF(&) All energies are in rydbergs.

we can

conclude that in the statistical methods, Xa or XFc@, we are forced to go beyond the one-electron energy cigenvahres, if we are interested in ionization potentials, because Koopmans’ theorem cunnof be used.

to the

1.0

0.2

207.4

20.1

15.2

1.84

1.00

208.2612 21)8.517.5 201.5285

18.3819 19.6700 19.5819

15.5148 13.9829 15.3294

1.8531 1.4239 1.8947

0.9194 0.5488 0.9330

0.01

209.7845

21.2231

16.1637

2.1495

1.0146

209.7678 207.4722

21.2148 20.5036

16.1444 15.3568

2.1458

1.0128

2.0436

0.9307

K. Schwarz/Ttansition

in the XFcq3 methd is no severe complication, although it is responsible for small deviations from (3) (section 4). Differences between XQ and XF@ results exist. but based on the results of the present does not seem that one method is sipifi-tly

work

it

super-

ior to the other.

Within HF theory

the one-electron

energies cdn

be

used to approximate ionization energies according

to Koopmans’ theorem. As consequence of a compensation of errors due to orbital relaxation and correlation effects, Koopmans’ values are often relatively good estimates. In particular for core levels, however, the total energy differences yield better results and are in general of quality comparable to the transition state estimates (table 4). When comparing statistical methods with HF. for example in connection with molecular calculations performed with the SCF-X&W method [6,7], one should keep in mind the level of sophistication of computing ionization energies. Thus for example, the question whether or not orbital relaxation is accounted for in both schemes, should be answered for a fair comparison.

slate

107

References

[ 11 K. Siegbshn, in: The As&mar conference.: EIectcon D.A. Shirley (North-Holland, Amstecdzn. 1972). and references therein. [Z] D.W. Turner, C. Baker, A.D. Baker and C.R. Brun~lc, Molecule photo&&on spectroscopy (Wiley-interscience. London, 1970). 131 T. Koopmans. Physica l(1933) 104. 141 W. Meyer. 1nte.m. J. QuantumChcm.5 (1971) 341. [S] J.C. Slater. in: Advances in quantum chemistry. Vol. 6 (Academic Press. New York. 1972) p. I. 161 K.H. Johnson, in: Advances in quantam chemistry. Vol. 7 (Aademic Press, New Y&k, 1973) P. 143. I71 J.C. Slater and K.H. Johnson, Phys. Rev. 65 (1972) 844_ [S] D.E. Ellis and C.S. Painter, Phys Rev. B2 (1970) 2887. 191 E.J. Eiacrends, D.E. Ellis and P. Ros, Chem. Phys. 2 ‘(1973)41. [IO] E.J. Baezcnds and P. Ros. Chcm. F’hys. 2 (1973) 52. [ 111 K. Schwarz, Chcm. Phys. 7 (1975) 94. [ 121 J.C. Slater, TM Wilson and J.H. Wood, Phys. Rev. 179 (1969) 28. [ 131 J.C. Shter and J.H. Wood, Intern. J. Quantum Chem. 4s (1971) 3. [ 141 J.C. Slater. J.B. hlann. T.M. Wilson and J.H. Wood, phys. Rev. 184 (1969)672. 1151 J.C. Shter. Inwrn. I. Quantum Chcm. 3s (1970) 727. [ 16) J.C. Slater. in: CompulationaJ methods in b3nd theory. eds. P.M. hkucus, J.F. Janak and A.R. Wtiiams. (Plenum

specl~oscopy. ed.

Acknowledgement

Ress. New York-London, 1971) P. M7. [ 17) 0. Coscinski, B.T. Pickup and G. Purvis. Chem. Phyr.

The author is pleased to acknowledge the encouragement and many helpful discussions with professor A. Neckel. It is a pleasure to thank Dr. F. Herman for his interest and critical comments. The author wishes to express his gratitude to IBM World Trade Corporation and IBM Austria for enabling him to spend a period of time at the IBM San Jose Research Laboratory as a World Trade Fellow. He is also grateful to IBM San Jose Research Laboratory for the kind hospitality he enjoyed.

Letters 22 (1973) 167. [18] S.B. Trickey. Chem. Phys. Letters 21 (1973) 581. [ 191 P. Jbrgensen and Y. ehm. Phyn Rev. A8 (1974) 112. [20] P. Weinberger and D.D. Konowalow. Intern. J. Quantum Chem. 7s (1974) 353. (211 N.H.F. Beebe. Chem. Phys. Letters 19 (1973) 290. (221 D.A. Libennan. Boll Amer. Phys. Sot. 9 (1964) 731. 1231 J.C. Slatcr. Phys. Rev. 98 (1955) 1040. [241 F. Herman and S. Skillman. Atomic Structure CslcuIalions (Prentice-Hall, Englewood Cliffs, 1963). 1251 J.B. Mann, Atomic Structure Calculations I. HarUeeFock Energy Results for the Elements Hydrogen to Lawrencium. Los Alamos Scientific Laboratory Report LA-3690 (1967). unpublished.