INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY SYMP. NO. 7, 353-367 (1973)

A Study of the Ground States of N,, O,, and F, and Their ESCA Spectra by the

Multiple Scattering Xa Method

PETER WEINBERGER* AND DANIEL D. KONOWALOW Department of Chemistry, State University of New York, Binghamton, New York 13901

Abstract MS-Xa-SCF calculations have been carried out over a wide range of separations for the lowest states

of N,, O,, and F,. These calculations give binding energies low by a factor of 3 to 5 and equilibrium internuclear separations too large by a factor of 1.6 to 2.0. “Transition-state” calculations of ionization excitation energies are shown for N, to give ionization potentials virtually identical to those calculated directly. Agreement with experimental ionization energies is adequate.

1. Introduction

Recently we reported a multiple-scattering self-consistent-field X a ( ~ s - X a ) calculation of the potential energy curve for the ground state of the Ne, molecule [l]. This paper presents a continuation of our investigations of the M S - X ~ scheme as applied to the study of molecular electronic structure of simple molecules. Our present object is to assess the potential and limitations of the method in calculating potential curves for the lowest states of N,, 02, and F,.

The MS-XH method was introduced by Slater and Johnson [2]. There have recently appeared extensive review articles [3,4] on the Xa exchange approximation and on the multiple-scattering method applied to clusters. Thus we mention here only those aspects of the method which are needed to establish notations in our tables and figures.

Molecular space is partitioned into three mutually exclusive regions of potential as shown in Figure 1. A sphere surrounding each atom comprises region I. A single outer sphere surrounds the cluster of atomic spheres. Region 111 lies outside the outer sphere. The remainder of space comprises region 11, the intersphere region. Inside a sphere of region I and for region 111, the molecular potential is averaged over angles to produce an approximate potential which is spherically symmetrical with respect to the center of the sphere under consideration. The molecular potential

Permanent address: Institut fiir Technische Elektrochemie der Technischen Hochschule Wien, A-1060 Vienna, Austria.

353 @ 1973 by John Wiley & Sons, Inc.

354 WEINBERGER AND KONOWALOW

Figure 1. Division of molecule space for two atoms interacting at a distance R into I-atomic, 11-interatomic, and III--extramolecular regions.

is averaged over the volume of region I1 to produce an approximate potential that is constant [2,4]. These combined processes yield a muffin-tin potential.

The atomic exchange parameter aM referred to as “virial theorem value” in the paper by Schwarz [5], namely, tl = 0.75118 for N2, a = 0.74367 for 02, and a = 0.73651 for F,, are used in our calculations. We use the same a value for all three regions.

The atomic sphere radii are chosen to be half of the intermolecular separation under consideration. This relative geometry is kept fixed for all separations and thus represents a scaling of the coordinate space.

The expansion of the molecular wave function in regions of spherical symmetrical potential into Im-like partial waves (Equation (11.7) of Reference [4]) was taken up to a standard I value of 2. Orbitals arising from the atomic 1s levels are treated as “semicore” levels, i.e., only 1s-like partial waves are considered for the above- mentioned expansion. This in fact is no restriction at all, since the splitting of the la, and la, orbitals, even for separations corresponding to the repulsive part of the total energy, was found to be less than 10- Ry. This is also the order of magnitude by which the la, and la, orbitals are altered in changing the maximum I value from 0 to 2.

In the calculations reported here, self-consistency of the molecular potential field was deemed to have been achieved when the potential changed by no more than lo-, % in successive iterations.

2. Total Energy and Orbital Energies

In Tables 1-111 the total energies are listed for N 2 (la2 10: 2a2 2a; ln: 303, of the internuclear separation. These energies were calculated according to the scheme outlined by Connolly and Sabin [6]. For each molecule, the statistical total energy at R = 10a, was calculated to be above (by an amount between 0.002 and 0.008 Ry) the m s - X a energy of the separated component atoms. We do not know whether this

O2 (la: la: 20,‘ 2a: In: 1.: 3a:), and F2 (la: la: 20,’ 2a: In, B 1 s 30,) 1 as a function

GROUND-STATE STUDY BY MS-XU METHOD 355

R,a,

Figure 2. Binding energy curves for N,, 0,, and F,. The zero of energy is taken to be the molecular energy at R = lOa,.

anomaly is due entirely to noise in our single-precision (32-bit word length) cal- culations, or whether this may be attributed in part to the muffin-tin approximation. Published accounts of MS-Xa potential energy curves for Li, [7] and CH, [S] make no mention of this question. Since a reputed advantage of the M S - X ~ method over ordinary SCF calculations is its ability to describe the long-range behavior more nearly correctly, this question deserves more study.

R,a, Figure 3. R dependence of virial coefficient calculated according to Equation (1).

356 WEINBERGER AND KONOWALOW

TABLE 1 . Ms-Xa total energies for N, in the configuration 10: lu~2a,22u: 1 z:3u: according to non-spin-polarized calculations with exchange parameter a = 0.751 18.

1.5

1.65

1-75

1.82

1.85

1.905

1.95

2.00

2.068

2.090

2.15

2.20

2.292

2.34

2.40

2.45

2.50

2.60

2.75

2.90

3.1

3.2

213.4020

U4.3040

214.7751

215.0618

215.1725

215.3655

215.5117

215.6589

215.8998

215.8438

216.0416

U6.1488

216.3268

216.4096

216.5041

216.5774

216.6435

216.7630

216.9054

217.0186

217.1273

217.1660

R ,a0

3.5

3.7

3.8

3.9

4.0

4.1

4.2

4.3

4.4

4.5

4.6

4.8

5.0

5.4

5.6

5.8

6.0

6.5

7.0

8.0

9.0

10.0

8

-Et 0ta19 Ry

217.2440

217.2711

217.2794

217.2844

217.2878

217.2882

217.2873

217.2859

217.2832

217.2791

217 2755

217.2655

217.2540

217.2339

217.2246

217.2162

217.2087

217.1926

217.1807

217.1667

217.1592

217.1568

217.1646

In order to estimate the M S - X ~ binding energies which are listed in Table IV we take the molecular total energy at R = 10ao to be identical with the dissociation limit. Thus the binding energy curves for N,, O,, and F, are seen in Figure 2 to coin- cide at R = 10~1,. These curves are qualitatively correct, but, as we show in more detail later, they have substantial quantitative shortcomings.

GROUND-STATE STUDY BY MS-xCr METHOD ' 351

TABLE 11. Ms-Xa total energies for 0, in the configuration la: lat2u:2a~ 1xt3a; 1x: according to non-spin- polarized calculations with exchange parameter a = 0.74367.

1.9

2.0

2.1

2.2

2.282

2.3

2.4

2.5

2.6

2.75

2.8

2.9

3.0

3.5

3.7

3.8

3.9

4.0

296.1509

296.6604

297.0881

297.4487

297 * 7009

297.7490

298.0020

298.2161

298.3904

298.5989

298.6553

298.7554

29s. 8369

299- 0557

299.0889

299.1008

299.1096

299.1130

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4'. 9

5.0

5.2

5.5

5.7

6.0

6.5

7.0

8.0

10.0

00

299.1167

299.1179

299.1174

299.1165

299.1152

299.1121

299.1084

299.1072

299.1035

299.1003

299.0942

299.0833

299.0767

299.0703

299.0601

299.0552

299.0461

299.0435

299.0454

A useful tool for examining the shape of potential curves is provided by the virial theorem [9],

(1) R dEldR = ('I - 1) ( T )

where E is the statistical total energy, R is the internuclear separation, ( T ) is the expectation value of the kinetic energy operator, and '1 = - E / ( T ) . The function 'I - 1 plotted in Figure 3 has a node precisely at the equilibrium internuclear separa- tion R = Re, where dEfdR = 0, and it approaches zero from above as R + 00.

We use this plot in two ways. First Figure 3 shows that at R = lOa, F, is nearly

358 WEINBERGER AND KONOWALOW

T~~~~ I ~ ~ . X U total energies for F, in the configuration 1.: 1u:2u:2oi 111t3u: ln: according to non-spin- polarized calculations with exchange parameter

Ra 0

1.5

1.75

2.00

2.20

2.40

2.60

2.71

2.80

3.0

3.2

3.4

3.6

3.8

4.0

4.1

4.2

4.3

4.4

u = 0.73651.

-Etotal*%

389.5144

392.7207

394.6931

395.7129

396.3945

396.8523

397.0154

397.1533

397.3491

397.4705

397.5474

397.5916

397.6172

397.6292

397.6326

397.6335

397.6355

397.6350

4.5

4.6

4.7

4.8

5.0

5.2

5.4

5.6

5.8

6.0

6.3

6.6

6.9

7.5

8.0

9.0

10.0

W

-Etotal’RY

397.6340

397.6323

397.6311

397.6289

397.6230

397.6211

397.672

397.6138

397.6111

397.6094

397.6079

397.6040

397.6011

397.5974

397.5942

397.5940

397.5938

397.5995

fully dissociated, while O2 and N2 are successively less so. This assessment is cor- roborated by the long-range behavior of the orbital energy versus R curves shown in Figures 4 and 5. All of this reinforces the need for a defined dissociation limit in order to estimate our Ms-xa binding energies. Second we use Figure 3 to fix the Ms-Xa values of Re by locating the nodes of the q - 1 curves. These are listed in Table IV.

Figure 4 shows that the so-called “frozen-core’’ approximation is unrealistic, especially at small separations, for the molecules we treat. It appears that the core orbitals may be active participants in the molecular bonding. Similar behavior has been observed for the core orbitals of S , [lo].

GROUND-STATE STUDY BY MS-XCI METHOD

-28.3-

+.-28.5

P 2 0

w' -378

6 -38.0 I c - z

b - 48.6

359

I I I I I I I I - /

N2 - -

4 > - - - 02

- - > <

F2 - -

TABLE IV. Dissociation energies and equilibrium separations for ground state N,, 02, and F,.

Substance Data Source Re(ag) De(eV)

n2 experiment ' 2.07 9.91 E-XCJJ 4.1 1-79 Hartree-Fock' 2.01 5.27 MC-SCFd 8.02

02 experiment' 2.28 5.21 4.2 1.01 E-Xu

Hartree-Fock' 2.18 1.43

MC-SCF~ 2.31 4.72

b

F2 exper iment' 2.68 1.65 MS-Xab 4.3 0.57

Hartree-Fockl! 2.51 -1.63 MC-SCF~ 2.67 1.67

~~

"Reference [ 11 1. *This work. 'Reference [12] dMulticonfiguration SCF results, Reference [ 131. 'Reference [ 141. lReference [15].

Figure 4. R dependence of lo core orbital energies.

360 WEINBERGER AND KONOWALOW

Figure 5. R dependence of valence shell molecular orbitals for N,, 0,, and F,. Reading from top to bottom at R = 4.0a0 are the following orbitals: In, and 3u0 for N,; lno, In,, and 3ug for 0,; In,, In,, and 3ug for F,; and then 2u, and 2a, in order for each of N,, 0,, and F,.

3. Discussion of Results

A. Potential Curves

The M S - X a potential curves for N,, 0,, and F, shown in Figure 2 are qualita- tively correct. That is, they all exhibit a steep repulsion, pass through a minimum, and approach the large-separation limit of dissociated atoms essentially correctly. However, the quantitative shortcomings of these curves is also evident from that figure and the data [ll-151 in Table IV. There it is seen that our calculated potential curves correspond to dissociation energies D, which are substantially too small, and which correspond to equilibrium separations Re which are much too large, compared to the experimental values [ 111. For N, , 0, , and F, the Ms-Xa values of the calculated equilibrium separations are respectively a factor of 2.0, 1.8, and 1.6 times larger than the experimental ones. Concomitantly, the calculated D, values are, respectively, only 18 %, 19 %, and 35 % of the experimental values.

Our M S - X ~ results provide an interesting contrast with those obtained from re- stricted Hartree-Fock (RHF) calculations [12-141. The RHE Re values are roughly 3-5% smaller than the experimental ones. The RHF D, values get progressively worse as one considers N,, 0,, and F, in turn. The negative binding energy of F, according to RHF calculations [14] is a well-known result. Painstaking techniques

GROUND-STATE STUDY BY MS-XCI METHOD 361

[15] have been developed to correct the faulty RHF potential curves. It is noteworthy that the MS-XCI potential curve for F, does predict a bound molecule.

It should be noted, by contrast, that the MS-XCI calculations reported for Li, [7] and CH4 [8] give appreciably more realistic potential curves than the ones reported here. For example, the Li, MS-XU curve has an equilibrium internuclear distance only slightly larger than the experimental value, and corresponds to a dissociation energy approximately 75 % that of the experimental value [7,16]. We need to learn why the MS-XCI results for Li, are so much more realistic than those for F,, O,, or N,. This question is currently being investigated in this laboratory. As test examples we treat several alkali metal diatomic molecules. So far we can only suggest that the fraction of charge in the intersphere region I1 must be relatively low before the corresponding calculations give good results.

In Figure 6 the percentages of the electronic charge which lies in regions I1 and I11 for each of N2, O,, and F, are plotted as a function of the internuclear distance. A logarithmic scale for the ordinate is chosen, mainly to demonstrate that these charges decay exponentially over a rather wide range of R values. In order to point out the analogy with a previous calculation for the ground state of the Ne, molecule [I], the corresponding values for Ne, are also shown in this figure. We have already suggested previously that the muffin-tin approximation for the potential was probably

Figure 6. R dependence of gross distribution of electron probability among regions of molecule space. Results for Ne, from Reference [l].

362 WEINBERGER AND KONOWALOW

the main drawback of the current version of the MS-XCI scheme. Figure 6 serves to emphasize this point. The fraction of charge in region I1 (where the muffin-tin potential is constant) increases substantially with each unit increase in the molecular bond order. (A similar pattern is exhibited by the fraction of charge in region 111.) For N,, for example, we find that between 20 and 25% of the electronic charge lies in region I1 for all separations 1.5 5 R/u, 5 2.6. Note that these separations span the experimental equilibrium value (Re = 2 . 0 7 ~ ~ ) . With so large a proportion of the charge in the region where the potential is probably least realistic, it should be no surprise that we have obtained unrealistic binding energies for N,. Similar state- ments, appropriately modulated, could be made for 0, and F,. It is suggestive, however, that better agreement with experiment accompanies a diminished p r e portion of charge in region 11. It remains to be seen whether this relation is a causal one.

B. Ionization Potentials

The kth ionization potential Ik(R) is defined by

for the ionization of N, by the removal of an electron from the kth molecular orbital of N, to form N: . (Similarly, one could define ionization potentials corresponding to multiple ionization excitations, etc., but the above definition will be sufficient for our purposes.) This definition corresponds to a so-called “vertical” ionization potential. Values of I k (R) calculated according to Equation (2) are not precisely comparable to experimental ionization potentials [ 17,181 without taking into account the vibrational structure of the molecules involved. (We shall ignore this effect in our subsequent discussions.)

According to Slater and coworkers [3,19] excitation energies between two states of a molecule can be calculated within the MS-XC~ method from so-called “transition-state’’ formulas. There the kth ionization potential is given by the kth orbital energy in a hypothetical transition state where the kth orbital occupancy nk of the ground state has been reduced by one half an electron, with all other orbital occupation numbers remaining the same as in the reference unionized state. That is, (3) I F ( R ) = - &k(R, nFm = nk - 3) The definition of the transition-state ionization potentials given here allows for orbital relaxation effects, at least to first order [3, 19,201. Clearly, the ionization potential as ordinarily defined, as in Equation (2), contains orbital relaxation effects to all orders.

So far, the transition state concept has been verified by actual computations only for the atoms Cr and Ne [19,20]. We show in Table V that the several ionization potentials of N2 calculated according to Equations (2) and (3) agree very nicely with each other. The discrepancies are about 0.5% at worst. It is curious to note that the overall agreement afforded by these two modes of calculation is slightly better

GROUND-STATE STUDY BY MS-XCL METHOD 363

TABLE V. Comparison of direct and transition-state calculation of ionization excitation energies of ground state of N, into various states of Ni. For all calculations, a = 0.75118.

Direct calculation. Ea “2) “Transition State” calculation. Ea ( 3 )

R Configurationa -qotsl Ionization Energy a0 RY RY

2.068 (laI3G1 185.8797 29.9641

(20g)’G2 213.5247 2.3191

(20,)’C3 214.4969 1.3469

( l ~ r , ) ~ G , 214.5006 1.3432

(3Og)’G5 214.8036. 1.0402

4.1 (10 1 3 ~ 1 187.5158 29.7724

(2Ug)‘G2 215.4730 1.8152

(2oU)’G3 215.5707 1.7175

(ln,J3G,, 216.3417 0.9465

(30,)’Cs 216.2887 0.9995

Configurationa Ionization Energy RY

(10 I 3.5G‘ 29.9564

(2o,) 1. 5G2 2.3157 (~o,)~.~G, 1.3440

( IT,) ’ 5G 4 1.3398

(30g) 1. 5G5 1.0354

(10) ’. 29.7740

( 2 0 ~ ) ~ ’ 5G2 1.8148 ( 2ou) 1. 5c3 1.7177 (IT,) 3 . 5G4 0.9454

(30g) ’’ 5G5 0.9990

“The G, are defined as follows: G , = (2~,)~(2u,)~(ln,)~(3a,)Z; G, = (la)4(2a,)2(1nu)4(3a~)2; etc., is., G, represents all the ground-state configuration except for the kth molecular orbital.

for R = 4.1a0 than for R = 2.0680,. It appears that we could use either of Equations (2) or (3 ) interchangeably to calculate ionization potentials. Since we had already calculated the ground-state potential curves, it is no more convenient to use Equation (2) than to use Equation (3) . To gain experience, we used the transition state formula- tion of Equation (3) in all the ionization potential studies we discuss subsequently.

Tables VI-VIII list the ionization potentials for N,, O,, and F, corresponding

TABLE VI. Comparison of experimental” and calculatedb*‘ orbital binding energiesd in lowest Z, state of N,.

Orbital Experimental‘ SCF-Xcr Calculations’ Hartree-Fock‘

R = 2.068a0 R = 4.lao R = 2.068ao

lo 409.9 407.6 405.1 426.7

2% 37.3 31.5 24.7 40.1

2% 25.OSe 18.6 18.3 23.4 21.2

1% 16.8 18.2 12.9 16.8

3% 15.5 14.1 13.6 17.3

”Reference [17], Table 5.2.1. bCalculated according to transition-state formula, Equation (3). ‘Calculated according to Koopman’s theorem from orbital energies

dEnergies in eV. ‘This ESCA line is attributed to “shake up,” see Reference [17].

listed in Reference [12(a)], Table IV.

364 WEINBERGER AND KONOWALOW

TABLE V11. Comparison of experimental and calculated orbital binding energies for lowest Eg state of 0,.

Orb i t a l Orb i t a l binding energies‘ experimentalb calculated‘

R = 2 . 2 8 2 ~ ~ R = 4.2a0

l o 543.1, 544.2 537.0 536.3

2% 39.6, 41.6 35.8 29.9

2% 25-39 27.9 26.0 29.2

18.8, 21 .1 18.1 15.6 3Og 1% 17.0, (17.8) 18.9 14.9

lT!3 13.1 12.6 14 .5

“Energies in eV. bReference [17], Table 5.2.3. The entries to the left

correspond to cxcitation into a ‘Z state of 0; while those on the right correspond to doublets.

‘This work, Equation (3).

TABLE VIII. Comparison of experimental and calculated orbital binding energies for lowest Z, state or F,.

b experimental‘ SCF-Xu Calculations Orbit 81 R = 2.68ao R = 4.3a0

l o 686.5 686.2

2og 39.3 35.8

2%

3%

34.7 35.5

20.8 17.6

1% 19.5 17.1

1% 15.7 16.8 16.9

“Reference [ 181. bThis work, Equation (3).

to excitation from each of the molecular orbitals occupied in the ground state. We have performed two sets of calculations for each molecule, one set corresponding to the experimentally observed Re, and the other to the equilibrium internuclear separation calculated according to the MS- Xa method. (Recall from the previous section that the two sets of separations are substantially different according to our calculations.) Let us discuss each molecule in turn.

GROUND-STATE STUDY BY MS-xa METHOD 365

In Table VI we compare the experimental orbital binding energies [17] for N, with our two sets of MS-XU calculations and with the RHF calculations of Cade and coworkers [12(a)]. Our results for R = 2 . 0 6 8 ~ ~ are in essential agreement with those reported for the separation R = 2 . 0 7 4 ~ ~ by Connolly and coworkers [20]. They have already pointed out the good agreement between the experimental orbital binding energies and the MS-XU values calculated at the experimental equilibrium separation. Clearly those M S - X ~ calculations correspond to the experimental ordering of the energy levels (20,, 1n,,3ag) while the RHF results predict the order to be (2au, 3ag, ln,). It would seem more appropriate to carry out the MS-XE calculations at a value of the internuclear separation which corresponds to the calculated rather than the experimental equilibrium separation. Our vertical ionization energies for R = 4 . 1 ~ ~ agree with the ordering (20,, 3ag, In,) predicted by the RHF calculations [12(a)] but disagree with the experimental assignment [17] and the M S - X ~ results for R = 2 . 0 6 8 ~ ~ .

This difficulty can be understood by a detailed examination of the R dependence of the orbital energies shown in Figure 7. It is seen that our calculated orbital energies for the unionized N, molecule lie in the order (2ag, In,, 2au, 3ag) for internuclear separations Ria, 7 2.05, in the order (20g, 2a,, ln,, 30,) for 2.05 7 R/ao 7 2.95, and in the order (2a,, 20,, 3ag, ln,) for 2.95 7 R/a,. Let us assume that the ionization energies calculated according to the transition approximation do not alter the order of energies exhibited by the ground-state orbitals. (The limited experience available to date here and elsewhere [20] suggests this assumption to be a safe one.) Then it appears that one could obtain the same ionization energy ordering as in- dicated by experiment so long as the M S - X ~ calculation is carried out for any separation in the range 2.05 7 R/a, 7 2.95. It hardly seems necessary to point out the near coincidence between the lower limit and the experimental equilibrium separation. Rather, it seems that agreement with the experimental ionization energy order is not necessarily so fine a test of the appropriateness and efficacy of these calculations as we might at first have supposed.

N2 ORBITAL ENERGIES

R, a 0

Figure 7. R dependence of three highest orbital energies for N2. Note level crossings at about R = 2 . 0 5 ~ ~ and 2.95a0.

366 WEINBERGER A N D KONOWALOW

In Table VII we compare the ESCA experimental ionization energies [17] with our calculations for 0, corresponding to the experimental equilibrium separation and our calculated one. Here the comparison with the ESCA results presents an additional difficulty. The ESCA spectrum is resolved into one set of components cor- responding to excitation into spin doublets and another set'corresponding to excita- tion into spin quartets. Obviously, our non-spin-polarized calculations cannot reproduce such structure. Nevertheless, one might seek to use a non-spin-polarized calculation such as ours to predict (or corroborate) the experimental order of energy levels. It turns out that our calculation of the vertical ionization potentials for R = 4.2a0 (the calculated R e ) corresponds to the experimentally observed order (2a,, 3ag, ln,, lng), while similar calculations at R = 2 . 2 8 2 ~ ~ (the experimental R e ) give the order (2a,, In,, 3ag, ln,) which is in disagreement with experimental observations. It is evident from Figure 8 that our non-spin-polarized calculations for 0, afford agreement with the experimentally determined ordering of orbital energies for all internuclear separations greater than about 2 . 5 ~ ~ 0 . Bagus and Schaeffer [21] obtain excellent agreement with experiment with their direct hole-state MCSCF calculation of the 1s hole state of 0,. To obtain this good agreement, however, they had to relax the restriction that each molecular orbital be of g or u symmetry.

In Table VIII we present the ionization potentials calculated for F, at the inter- nuclear separations 2 . 6 8 ~ ~ and 4.34, which correspond to the experimental and our calculated values of Re. In this case both calculations suggest the same order of energy levels, namely, (la, 2a,, 2au, 3ag, lz,, lng). Both calculations give fairly good agreement with the single experimental level of which we are aware [ 181.

4. Summary

In summary, then, the MS- Xcr method gives qualitatively correct potential energy curves for the diatomic molecules N,, O,, and F,. Substantial improvements in the method are needed to attain quantitatively reliable potential curves.

02 ORBITAL ENERGIES I I I I

-0.6 -

w

2 3 4 5 R.ao

Figure 8. R dependence of three highest orbital energies for 02.

GROUND-STATE STUDY BY MS-XO! METHOD 367

The transition-state calculations of ionization potentials are in excellent agree- ment with those calculated conventionally in the M s - x a approximation. Moderately good agreement with experimental ionization potentials has been obtained.

Acknowledgment

Thanks go to John Connolly for providing the original versions of the programs used here, and for continued correspondence on the MS-XU method. One of us (P. W.) was supported by the Austrian Studies Program of the State University of New York at Binghamton.

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[ I 31 H. F. Schaefer, 111, The Electronic Structure of Atoms and Molecules (Addison-Wesley, Reading,

[I41 A. C. Wahl, J. Chem. Phys. 41, 2600 (1964). [15] G. Das and A. C. Wahl, J. Chem. Phys. 56,3532 (1972). [I61 M. R. Mattox and D. D. Konowalow, unpublished calculations. They find for Li,, D, : 0.82 eV

(171 K. Siegbahn et al., ESCA Applied to Free Molecules (North Holland Publ. Co., Amsterdam, 1969). [I81 R. P. Iczkowski and J. L. Margrave, J. Chem. Phys. 30,403 (1959). [19] J. C. Slater, J. B. Mann, T. M. Wilson, and J. H. Wood, Phys. Rev. 84, 672 (1969). [20] J. W. D. Connolly, H. Siegbahn, U. Gelius, and C. Nordling, J. Chem. Phys. 58, 4265 (1973); see

[21] P. Bagus and H. F. Schaefier, 111, J. Chem. Phys. 56,224 (1972).

Malli, and H. Popkie, unpublished results.

Mass., 1972).

at R , = 5.25a,.

also J. W. D. Connolly, Internat. J. Quant. Chem. 6S, 201 (1972).