INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL. 11, 663-685 (1968)

A Configuration Interaction Study of the Four Lowest ‘C+ States

of the LiH Molecule* RICHARD E. BROWN? AND HARRISON SHULL The Department of Chemistry, Indiana Unioersity, Bloomington, Indiana

Abstracts

A configuration interaction study was completed on the lC+ states of the LiH molecule using a nonorthogonal one-electron basis in elliptical coordinates. A few wave functions with highly optimized parameters were obtained for the X’C+ and AIC+ states and combined to construct a larger wave function which gave improved results for both states over a wide range of R values. The third and fourth roots are also reported since the wave function is extensive enough to give good upper bounds for the two states. Calculations were completed for 34 values of R in the range 1 Q R Q 10 bohr (b).

The calculated XIC+ curve has a minimum at Re = 3.060b (3.015b), with E(Re) = -8.0556 Hartree (H)( -8.0704),p(Re) = -5.89 debye (d) (-5.88d) andp/(Rdp/dR)IR, = 1.75 (1.80 f 0.3). FortheAlPstate, Re = 4.928b (4.906b), E(Re) = -7.9372H(-7.9496H), p(Re) = +3.96d and p/(R dp/dR)]~, = -0.471. The values in parentheses are experimen- tal results for comparison. The numerical vibrational and rotational analysis agreed well with experiment for both states. The A’C+ state exhibited a pronounced negative anharmonicity. Both states show strong interaction of three zeroth-order configurations, the nature of which changes considerably with R.

The thus far unobserved second excited state has two minima, a metastable one at R = 3.70b and another at R M 10.00b. The third excited state also appears to have a minimum at R = 7.00b.

Une etude a ttC faite pour les ttats lC+ de la moltcule LiH selon la mtthode d’interac- tions de configurations, a partir d’une base nonorthogonale de fonctions un Clectron en coordonntes tlliptiques. Pour les ttats XIC+ et AlC+ on a obtenu quelques fonctions d’onde avec des paramttres trts bien optimists. Ces fonctions-ci ont t t t combintes pour construire une fonction d’onde plus grande, ce qui a donnt des rtsultats amtliorts pour les deux &tats pour un grand nombre de valeurs de R. Comme la fonction d’onde est suffisamment vaste pour donner de bonnes bornes suptrieures pour le second et le troisitme ttat excite, on donne aussi la troisikme et la quatritme racine de l’tquation stculaire. Les calculs ont Ctt faits pour 34 valeurs de R dans I’intervalle 1 Q R Q 10 bohr (b).

La courbe XIC+calculCe a un minimum a Re = 3.060b (3.015b) avec E(Re) = -8.0556 Hartree (H)( -8.0704), p(Re) = -5.89 debye (d)( -5.88d) et p/(R dp/dR)I~, = +1.75 (1.80 f 0.3). Pour l’ttat AIC’+: Re = 4.928b (4.906b), E(Re) = -7.9372H (-7.9496H),

* This research was supported by grants from the Air Force Office of Scientific Research and

t Present address: The Department of Quantum Chemistry, The University of Uppsala, the National Science Foundation.

Uppsala, Sweden. 663

664 RICHARD E. BROWN AND HARRISON SHULL

p(Re) = +3.96d et p / ( R dp/dR)In, = -0.471. Les valeurs en parenthPses sont les resultats exptrimentaux. L’analyse vibrationelle et rotationnelle numtrique est en bon accord avec l’expkrience pour les deux Ptats. L’ttat AIC+ a une anharmonicit6 ntgative prononcee. Dans les deux ttats trois configurations d’ordre ztro interagissent fortement d’une faGon qui varie avec R.

Le second &at excite, qui n’a pas t t t observe jusqu’ici, a deux minima: l’un mttastable a R = 3.70b et l’autre R w 10.00b. Le troisieme Ptat excite parait avoir un minimum a R w 7.00b.

Die lC+ Zustande des LiH-Molekuls wurden mit der Konfigurationswechselwirkungs- methode studiert. Dabei wurde ein System von nicht-ortogonalen Einelektronfunktionen in elliptischen Koordinaten benutzt. Einige Wellenfunktion mit wohl optimisierten Para- metern wurden fur die XIC+ und AIC+ Zustande erhalten. Diese Funktionen wurden dann zu grosseren Wellenfunktionen kombiniert welche verbesserte Resultate fur beide Zustande uber eine weite Reihe von Kernabstanden gaben. Die dritte und vierte Wurzeln werden auch berichtet, da die Wellenfunktion geniigend umfassend ist um gute obere Grenzen fur die zwei Zustande zu geben. Die Berechnungen wurden fur 34 Kernabstande in der Reihe 1 < R Q 10 bohr (b) gemacht.

Die berechnete XIC+-Kurve hat ein Minimum fur Re = 3.060b (3.015b), mit E(Re) = -8.0556 Hartree (H)( -8.0704), !&(Re) = -5.89 debye (d) (-5.88d) und p / (Rdp /dR) ln , = +1.75 (1.80 f 0.3). Fur den A’C+-Zustand: Re = 4.92813 (4.906b), E(Re) = -7.9372H (-7.9496H), p(Re) = +3.96d und p / ( R dp/dR)]R, = -0.471. Die Werte in Klammern sind Experimentalwerte. Die numerische Schwingungs- und Rotations- analyse stimmt mit den Experimentalresultaten fur beide Zustande wohl iiberein. Der AIC+-Zustand hat eine ausgesprochene negative Anharmonizitat. Fur beide Zustande zeigen drei Konfigurationen nullter Ordnung starke Wechselwirkungen.

Der noch nicht beobachtete zweite angeregte Zustand hat zwei Minima, ein metastabiles Minimum fur R = 3.70b und ein anderes fur R M 10.00b. Der dritte angeregte Zustand erscheint auch ein Minimum fur R M 7.00b zu haben.

1. Introduction

Numerous calculations have been completed on the LiH molecule since it is a molecule of considerable theoretical interest.’ It offers many of the important features of larger molecules such as the effects of a closed inner shell, a pair of outer-shell “binding electrons”, different spin states with spin functions of varying importance, a number of stable states having markedly different properties and a number of strongly interacting zeroth-order configurations. Also a t least one state of 3X+ symmetry has never been observed. With only four electrons, many of the causes and effects in bonding may more easily be analysed. Quite accurate calculations have been completed on the united and separated atoms so that comparisons in these limiting cases can be made.

In light of these interesting aspects of LiH, the purpose of this research was primarily to investigate the nature of lX+ states of LiH, paying particular attention to the ground and first excited states and considering briefly the next two states of the same symmetry. The rearrangements in the electronic densities were noted

For a good review of previous calculations on LiH, consult Cade [ l ] .

FOUR LOWEST lC+ STATES OF THE LiH MOLECULE 665

for changes in the internuclear distance for each state and also in the transitions between states. Various physical properties including the vibrational-rotational energy levels were determined and tabulated. Particularly interesting is the first excited lC+ state since it exhibits a negative anharmonicity, as do all the alkali hydrides in this state.

2. Method of Calculation

The configuration-interaction method was employed here in probably its simplest form in which the approximate wave function is expanded in terms of a linearly independent set of configurations, {an}, of lX+ symmetry, such that,

Minimizing the energy with respect to the coefficients, C:, leads to the usual secular equation with the kth lowest root being an upper bound to the exact energy of the respective state.

The one-electron basis set consisted of the following nonorthogonai orbitals in elliptical coordinates,

(2) p = N(R/2)-3/2(2,rr)--1/25nrj[(EZ - 1) (1 - rz)] I~zI/2e-~5-BVe-zmb

with 5 = ( r A + rB)IR and 7 = ( r A - rB)/R. The #J coordinate is the azimuthal angle, R is the internuclear distance between the Li nucleus at point A and the H nucleus at point B and N is the normalization constant.

Using these orbitals, there result four types of unnormalized configurations of -1P symmetry:

(3.1) (1) R+ I Y ~ @ ~ Y ~ P ~ ~ I / ( ~ ! ) ” ~ (3-2) (2) R+{\Yi@i%@31 - 1pi951952p31>/2”2(4!)1’2

(3.3)

(3.4)

(3) R+{1Yi@2%g41 - 1pi@z@3T41 - I(?iPz%@41 -k 1(?1912@34)41}/2(4!)~/~

(4) R+(IYiY)2%(?41 - 6[1Y)i%P)3@4/ -k \‘?‘i@z(?3VaI -k l@lp?z%?4/

-k I(?~CPZ~~’P~I] -k 1~i(?2p3p41>/3”z(4!)1’2

The barred orbitals denote normalized spin orbitals with /3 spin and the unbarred orbitals denote those with u spin. R, is the C,, point group projection operator for C+ symmetry, &(E + 0,). Here E is the identity operator, and o, is the operator for reflection across the internuclear axis which simply changes the sign of the m quantum number in each orbital. When the configurations contain orbitals of only B symmetry, R+ reduces simply to just the identity operator. The upright bars denote the usual determinantal functions. Configurations (3.1)-(3.3) are singlet-coupled spin types with (3.3) reducing to (3.1) or (3.2) in closed-shell cases while configuration (3.4) is a triplet-coupled spin type.

666 RICHARD E. BROWN AND HARRISON SHULL

Once the wave functions are obtained, it proves useful to analyse them in terms of the following two functions, G ( z ) and G’(z) , which we designate respec- tively “electron count” and “planar density” :

The Li and H nuclei are situated on the 2 axis with the axis origin at the molecular midpoint. Here p(r) is the first-order density matrix and the n, are the occupation numbers of the respective NSO’S, x, [2]. G ( z ) simply denotes the number of electrons on the negative side of the X Y plane intersecting the 2 axis at the point 2 , and G‘(z ) is the density of electrons in this plane. Although these two functions are not a complete substitute for the detailed density maps, they do provide useful information on the electronic structure of the states.

3. Previous Results and Considerations

There have been many previous calculations on the LiH molecule, particularly on the ground lX+ state near the experimental minimum of Re = 3.015 bohrs. These begin with calculations of Knipp [3] published in 1936, and go all the way to the latest results of Bender and Davidson [4]. Karo and Olson [5] did numerous calculations on the ground state and the first excited states using wave functions of the VB and LCAO-SCF-MO types. Although their wave functions were limited, this investigation was probably the most thorough previous one since it covered both the ground and first excited IC.+ states for a wide range o f R values. Ebbing’s [6] calculation on the ground state a t R = 3.00b gave a notable improvement using ten SCF orbitals and 53 configurations to obtain 66 % of the binding energy. Browne and Matsen [ 7 ] demonstrated the utility of a mixed orbital basis set and reached 85% of the binding energy. Most recently, Bender and Davidson, generating the NSO’S directly, and using a considerably more extensive wave function, accounted for 89 % of the binding energy.

Previous calculations indicate that both the ground and first-excited states can be represented approximately as the sum of three zeroth-order singlet-coupled configurations [8, 91,

FOUR LOWEST lX+ STATES OF THE LiH MOLECULE 667

A, Boo and R, are the respective projection operators for the antisymmetry, singlet spin symmetry and X+ point group symmetry. The relative importance of each configuration changes considerably with R for each state. For R 5 6b, the energy of y3 alone is lower than that of either y1 or yz . For 6 < R lob, it gives an energy lower than yz . For the limiting case in which R approaches infinity, the dissociation products of y3 lie much higher than those of y1 or yz . Consequently as Mulliken and Rosenbaum have pointed out, these low-lying zeroth-order configurations should be strongly interacting for most R values of chemical interest.

These considerations suggest that a satisfactory wave function might be obtainable by combining functions which express all the zeroth-order configura- tions satisfactorily and thus we can avoid laborious reoptimization at each R value. First, highly optimized wave functions were obtained for the ground and first-excited states at R = 2.00, 3.00, 4.900 and 7.00b. These for the ground state at R = 3.00b and for the first excited states at R = 4.90b are listed in Tables I and 11, respectively. As expected, many of the orbitals and configurations were repetitious for the two states. Then a judicious choice was made from the orbitals and configurations of the optimized wave functions to construct a wave function with 18 orbitals and 69 configurations. The energy of this function was an improvement over that of each of the optimized wave functions mentioned above. In order to do a calculation at a particular R, the orbital parameters were simply scaled by a factor of R/3 from those parameters successfully used a t R = 3.00b. We judge from the improvements in the energy and the virial theorem that this wave function gave satisfactory results for 2 _< R 5 8b. In all cases, 0.98 < 1 T/El < 1.02, and close to the minimum for each state, 0.9996 < I TIE1 < 1.0006, even though an overall scale factor was not explicitly minimized. Other calcula- tions were completed at R = 1 .OO, 9.00 and 10.00b in order to obtain legitimate limits for the boundary conditions in the numerical calculation of the vibrational- rotational energy levels.

4. The Ground State of LiH

The calculations using the 69 configurational wave function, given in Table 111, were completed at 34 different values of R for 1.0 5 R 5 10.00b. For certain selected values of R, the eigenvector coefficients are given in [I91 and the energy and dipole moments at all 34 points are listed in Tables IV and V, respectively. Outside of the range, 2.00 5 R 8.00b, the exponential parameters are considered nonoptimum, and the discussion is limited primarily to this range. Also the orbitals, configurations and coefficients of a singlet-coupled 28 configurational wave function are listed in Table 1. For selected values of R, the NSO occupation numbers and the correlation energies are given in Tables VI and VII, respectively.

The energy and dipole moment curves were fitted with a sixth-order polyno- mial in R a t R = 2.80, 2.90, 3.00, 3.05, 3.10, 3.20 and 3.30b, and this polynomial

668 RICHARD E. BROWN AND HARRISON SHULL

TABLE I. 28 configurational wave function for the ground lZ+ state of LiH for R = 3.00b, E = -8.05240H.a.h*c

Basis orbitals

No. N J M Alpha Beta Norm.

1 2 3 4 5 6 7

9 10 11 12 13 14

a

0 0 1 0 0 0 1 0 0 0 0 0 0 0

0 0 0 0 1 0 1 2 0 1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 1 1 2 2

4.92

1.26

2.00

3.08

1.81

1.58 4.80 8.00 2.50 1.50 5.93 1.90

2.73 8.50

4.98 3.14

-0.22 -1.47

2.41

5.90 2.50

-0.10 - 1.50

5.93 -1.16

-1.23

8.50 -1.97

14.658295 1 7.29264313 1.60996034 4.26557669 3.84677532 3.44635068 5.405426 16

5392.18848 17.9422761 3.73497968

85.6321274 7.00236 1 81

730.954955 20.00 1241 2

Configurations

Configurations Coefficients Configurations Coefficients

1 2 3 4 1 2 5 6

1 1 1 1 3 4 1 1 1 1 5 6 13 1_3 3 4 13 13 5 6

I 2 14 1 3 9 9 1 2 1 2 4 9 1 2 6 9 1 2 3 1 0 1 2 5 1 0 1 2 9 1 0

1 2 1 2 g

0.567522 0.200839

-0.02763a -0.007155 -0.00472 1 -0.001223 -0.073319 -0.010398

-0.225787

- 0 . o ~ o a 4

0.128324

0.018325

0.155051

0.01 1740

7 7 3 4 7 7 5 6 1 7 3 4 1 7 5 6 2 7 3 4 2 7 5 6 8 8 3 4 8 8 5 6 1 8 3 4 1 8 5 6 2 8 3 4 2 8 5 6 7 8 3 4 7 8 5 6

-0.13 1757 -0.024218 - 0.02840 1 -o.ooi ooa -0.072852 -0.005686 -0.053400 -0.007945

0.004495 -0.00 1000

-0.035165 -0.09 12 17

-0.0152ao

-0.028687

a Any underlined orbitals, 4, denotes 4 = uu+, and all pairs, 44, are open shells. - - - Coefficients are for the symmetry adapted configurations defined in (1)-(3.3) using

,411 configurations are singlet coupled types. nonorthogonal but normalized orbitals.

FOUR LOWEST 'C+ STATES OF THE LiH MOLECULE 669

TABLE 11. 32 configurational wave function for the first excited lC+ state of LiH for R = 4.90b, and E = -7.933385H.a

Basis orbitals

NO. N J M Alpha Beta Norm.

1 2 3 4 5 6 7 8 9

10

0 0 0 1 0 0 1 1 0 0

0 0 0 0 1 0 1 1 0 0

0 0 0 0 0 0 0 0 1 1

8.12 5.13 1.95 1.93 0.96 2.53 2.70 6.00 9.70 2 9 0

8.12 5.05 1.95 1.93 0.96

-2.37 -2.10 10.00 9.70

-2.30

32.7226933 17.6053209 3.85094144 2.51864595 2.08156579 6.40035182 9.736001 11 0.502219690

293.04 1640 20.7963147

Configurations

Configurations Coefficients Configurations Coefficients

1 2 3 6 1 2 4 6 1 2 5 6 6 6 1 2 1 2 3 7 1 2 4 7 1 2 5 7 7 7 1 2 1 2 6 7 4 4 1 2 1 2 4 5 1 3 5 6 2 3 5 6 9 9 3 6 9 9 4 6 9 9 5 6

0.880853

0.72442 1 -0.659238

-0.378890 -0.046473 -0.048760 -0.097882 -0.265769

0.473549 -0.016796 - 0.0 140 13 -0.019778

0.0092 11

0.041964 -0.049259

-0.031061

6 6 9 9_ 7 7 9 9_ 9 9 6 7 1 2 1 o l ( J 1 8 3 6 2 8 3 6 8 8 3 6 1 8 4 6 2 8 4 6 8 8 4 6 1 8 5 6 2 8 5 6 8 8 5 6 6 6 1 8 6 6 2 8 8 8 6 6

0.013384 0.009561

-0.017324 -0.027 177 -0.195750 -0.196820 -0.270200

0.168925 0.170799 0.239977

-0.078955 - 0.086245 -0.132771 -0.006341 -0.009353 -0.007689

a Note the Footnotes (u)- (c) in Table I.

was used to obtain the values of some physical properties at R,: R, = 3.060b (3.015b)) E(R,) = -8.0556 Hartree (-8.0704H), p(Re) = -5.89 debye (-5.88d) and p / ( R dp1dR) I R e = 1.75( 1.80 f 0.30). The experimentally determined values are given in parentheses [lo-121. Most of the results agree favorably with experiment. The somewhat large value of Re agrees with the value obtained in most other computations of Re. Thus, Browne and Matsen obtained 3.046. The calculation reported here accounted for 84 % of the 0.092 1 Hartrees of binding energy. Also the first natural detor, x 3 , ~ 3 , , gave an energy of -7.9850H as compared to Ebbing's estimate of -7.98608 for the HF limit.

-7.5

-7.6

- 7.7

-7.9

-8.C

-8.1 2.0 4.0 6.0 8.0 10.0

R (bohrs)

Figure 1. Potential energy curves for the four lowest 'C+ states of LiH.

Figure 2. Dipole moment curves for the ground and first excited 'C+ states of LiH. 670

FOUR LOWEST lX+ STATES OF THE LiH MOLECULE 671

I t is of especial interest to compare the electron distribution along the axis with that of the isolated atoms placed at the same positions. It is clear from Figure 3 that this distribution in the ground XIE+ state is close to but not identical with the sum of the isolated Li+ + H- densities [ 141 .2 In the region of the Li nucleus, the electron count is the same within the width of the curve, but there is a definite deviation in the distribution around the H nucleus as the planar density plots in Figure 4 demonstrate quite markedly. This distribution could be explained largely by a shift of the broad H- distribution from the back side of the H nucleus

2 0x1s (bohrs)

Figure 3. Electron count curves for the ground and first excited 'C+ states of LiH at R = 3.00 and also for Li(?Y), Li+('S), H-('S), H(2S), Li(2S) + H(9S), and Li+('S) + H-('S). Where some curves approximately join into one, only one is

drawn.

into the internuclear region. Clearly there would be a very appreciable polariza- tion of the H- ion, but little polarization of Li+-a result to be expected from their relative polarizabilities and sizes.

However, it is also imperative to consider the other major zeroth-order configuration, Li(2S)H(2S) [15]. Consequently also included in Figures 3 and 4 are the electron counts and planar densities of Li and H atoms and their sum. The Li atom distribution is too diffuse to represent well the back side of Li in LiH XIC+ and the H atom density is altogether too concentrated to represent well the H side for this state. There is therefore a very marked deviation of these zeroth-order curves from the observed LiH electron counts and planar densities which can be explained away only by the introduction of substantial amounts of polarization of Li, possibly via 2s-2p, hybridization. Thus these considerations clearly point out the ionic character of LiH XIC+ and that the most appropriate zeroth-order

The authors are indebted to A. Macias for supplying the NSO'S for the Weiss functions from yet unpublished data.

Z a x i s (bohrs)

Figure 4. Planar density curves for the same species as drawn in Figure 3. Where some curves approximately join into one, only one is drawn.

TABLE 111. The 69 configurational wave function.a.boG

Orbitals (R = 3.00b)

NO. N J M Alpha Beta Norm.

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19

0 0 1 0 0 0 1 0 1 0 0 1 0 0 1 0 0 0 0

0 0 1 2 1 0 0 1 0 0 0 1 1 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 2 2

4.94 3.1 1 3.90 8.00 2.00 1.04 1 .oo 0.58 1.26 2.50 1.55 1.65 1.50 5.93 7.00 1.90 2.50 8.50 2.73

4.94 3.1 1 6.40 2.50 2.41 1.20 1.37 0.51

-0.22 -0.10 -1.45 -1.29 - 1.50

5.93 7.00

-1.16 - 1 .oo

8.50 - 1.97

15.5276388 7.7563 1755 1.1 1063681

3.84677532 1.37838152 0.534966630 1.07873220 1.60996034

2.90032427 3.41 148023 3.73497968

5392.18848

17.9422761

85.632 1274 107.986999

7.00236181 20.6029825

20.0012412 730.954955

672 (Table continued)

TABLE 111.-continued.

Configurations

No. Orbitals Spin No. Orbitals Spin

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20

1 2 6 1 1 1 2 7 1 1 1 2 8 1 1

11 11 1 2 1 2 8 1 2 1 2 9 1 1 1 2 5 1 1 1 2 5 1 2 1 2 5 1 3 1 2 12 13 1 2 11 13 1 2 7 1 2

12 12 1 2 10 10 1 2 1 2 10 11 1 2 9 1 3

13 13 1 2 1 2 9 1 2 1 2 6 1 3 1 2 6 7

3 3 3 2 3 3 3 3 3 3 3 3 2 2 3 3 2 3 3 3

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

1 2 5 9 1 2 5 1 0 1 2 6 8 1 2 6 9 1 3 6 1 1 2 3 6 1 1 I 4 6 1 1 2 4 6 1 1 3 3 6 1 1 3 4 6 1 1 4 4 6 1 1 2 3 8 1 1 1 4 8 1 1 3 3 8 1 1 4 4 8 1 1 1 3 9 1 1 2 3 9 1 1 1 4 9 1 1 2 4 9 1 1 4 4 9 1 1

3 3 3 3 3 3 3 3 2 3 2 3 3 2 2 3 3 3 3 2

Configurations

No. Orbitals Spin No. Orbitals Spin

41 42 43 44 45 46 47 48 49 50 51 52 53 54 55

1 5 6 1 2 1 5 10 11 1 5 11 13 1 2 16 16 1 2 1 7 i - j 1 2 1 6 1 7

1 4 E 6 f i 14 14 8 11 14 14 9 11 14 14 11 13 11 fi 14 14 14 14 7 ii 15 15 6 11 15 8 11 15 13 9 11

3 3 3 3 3 3 3 3 3 3 2 3 3 3 3

56 57 58 59 60 61 62 63 64 65 66 67 68 69

15 15 11 13 14 15 6 11 14 15 8 11 14 15 9 11 14 15 11 13

18 6 11 18 18 8 11 18 18 9 11 18 2 11 13 18 18 7 11 1 5 6 1 1 1 5 7 1 2 1 5 6 1 2

1 2 1 9 1 _ 9

3 3 3 3 3 3 3 3 3 3 3 4 4 4

a Note Footnotes (u) and ( b ) in Table I. Spin type of configurations refer to the respective equations (3.1) to (3.4). The eigenvector coefficients of this wave function are listed in 1191 for all four states

and for various values of R.

673

6 74 RICHARD E. BROWN AND HARRISON SHULL

picture is Li+(lS)H-(lS). But in explaining the explicit electron distribution, substantial polarization is necessary in the LifjlS) H-(IS) and particularly the Li(2S)H(2S) zeroth-order configurations.

We can further explore the nature of the ground state by examining the natural orbital analysis and the correlation energies. At Re, the outer-shell NSO occupation numbers of xoz and xnl are 0.9716 and 0.0101, respectively, and the outer-shell 7r correlation energy is 0.01 16H. For Be the corresponding occupation numbers are 0.9103 and 0.0587. Even ifwe allow for the fact that there is spherical angular

TABLE IV. Energies from the 69 configuration function.

1 .oo 1.50 2.00 2.30 2.50 2.70 2.80 2.90 3.00 3.015 3.05 3.10 3.20 3.30 3.50 3.70 4.00 4.50 4.70 4.80 4.90 4.906 5.00 5.10 5.20 5.50 5.70 6.00 6.50 7.00 7.50 8.00 9.00

10.00

-7.32714 - 7 3043 7 -7.98124 -8.02563 -8.04 168 -8.05068 -8.05324 -8.05479 -8.05549 -8.05554 -8.05557 -8.05555 -8.05493 -8.05388 -8.05068 -8.04643 -8.03888 -8.02505 -8.01950 -8.01676 -8.0 1406 -8.01390 -8.01 140 -8.00879 -8.00623 -7.99890 -7.99436 -7.98809 -7.97925 -7.97247 -7.96762 - 7.96432 - 7.96074 -7.95924

-7.23927 -7.67122 - 7.83 169 -7.87664 -7.89557 -7.90874 -7.91 375 -7.91791 -7.92 138 -7.92185 -7.92289 - 7.9242 7 -7.92667 - 7.92866 -7.93 166 -7.93371 -7.93559 -7.93694 -7.93714 -7.93718 -7.93720 -7.93720 -7.93719 - 7.937 16 -7.93709 -7.93675 -7.93638 -7.93558 -7.93345 - 7.930 18 -7.92580 -7.92066 - 7.90975 -7.89958

-7.06054 -7.547 19 -7.72247 -7.76809 - 7.78573 -7.79662 -7.80030 -7.80292 -7.80486 - 7.80509 -7.80563 - 7.806 15 - 7.80682 -7.807 13 -7.80665 -7.80531 -7.802 19 -7.79609 -7.79376 -7.79272 -7.79175 -7.791 70 - 7.7908 1 -7.79005 - 7.78943 -7.78854 -7.789 12 - 7.792 18 -7.80 126 -7.81104 -7.8 1924 -7.82568 -7.83444 -7.83954

-6.96665 -7.29812 - 7.46306 -7.54009 - 7.57436 - 7.59777 - 7.60640 -7.61294 -7.61825 -7.61892 -7.62054 - 7.62235 - 7.62524 -7.62795 - 7.63346 -7.64214 -7.66023 -7.69325 -7.70552 -7.71 138 -7.7 1699 -7.7 1732 -7.72225 -7.72730 - 7.73204 -7.74407 -7.75004 -7.75561 - 7.75784 -7.75622 - 7.7 5344 - 7.75048 -7.745 13 -7.74080

a R is in bohrs and Ei is in Hartrees.

FOUR LOWEST lE+ STATES OF THE LiH MOLECULE 675

TABLE V. Dipole moments of the 69 configurational function for the lowest four lC+ states of LiH.

DZ

1 .oo 1.50 2.00 2.30 2.50 2.70 2.80 2.90 3.00 3.015 3.05 3.10 3.20 3.30 3.50 3.70 4.00 4.50 4.70 4.80 4.90 4.906 5.00 5.10 5.20 5.50 5.70 6.00 6.50 7.00 7.50 8.00 9.00

10.00

-4.788 -4.826 -4.952 -5.154 -5.323 -5.513 -5.615 -5.719 -5.827 -5.843 -5.881 -5.936 -6.048 -6.160 -6.387 -6.610 -6.931 -7.385 -7.521 -7.576 -7.621 -7.623 -7.655 -7.677 -7.686 -7.620 -7.487 -7.125 -6.041 -4.436 -2.660 -1.111 +0.839 + 1.719

+2.441 +4.394 +5.016 +5.204 +5.284 +5.327 +5.339 +5.340 +5.338 +5.337 $5.337 $5.329 +5.310 +5.288 $5.221 +5.133 +4.949 +4.504 +4.270 +4.140 +4.000 +3.991 +3.845 +3.682 $3.505 +2.882 +2.378 + 1.468 -0.506 -2.957 -5.527 -7.804 - 10.954 - 12.605

- 10.449 -9.519 -8.224 -7.375 -6.790 -6.187 -5.888 -5.580 -5.278 -5.232 -5.121 -4.976 -4.666 -4.361 -3.742 -3.122 -2.187 -0.696 -0.166 +0.071 +0.285 +0.297 +0.470 +0.614 +0.710 +0.562 -0.036 - 1.605 -4.059 -5.348 -6.090 -6.642 -7.665 -8.91 7

+3.898 +2.268 -1.741 -1.279 -0.963 -0.808 -0.789 -0.836 -0.923 -0.943 -0.987 - 1.076 -1.317 -1.633 -2.544 -3.565 -4.652 -5.477 -5.629 -5.666 -5.677 -5.677 -5.666 -5.608 -5.504 -4.789 --?.846 -1.817 +1.218 +2.848 +3.688 $4.127 +4.392 +4.269

a R is in bohrs and dipole moments, Di , are in debyes.

correlation in Be and only cylindrical angular correlation in LiH, both the T

occupation numbers and n- correlation energy are much greater in Be than LiH. Thus there is no strong resemblance between LiH and the united atom. However, in this respect the LiH outer shell does more nearly resemble the H- ion. The corresponding occupation numbers of H- are 0.9439 and 0.0082, respectively, and the n- correlation energy of H- is 0.0082H [ 13, 141. The comparison with the separated atoms, Li(2S) + H(%), as a zeroth-order approximation is not at all realistic since the corresponding occupation numbers in this case are about 0.5

TABLE VI. Correlation energies of LiH in Hartrees.

Ground state

R = 2.00 3.00 4.90 7.00 8.00

Total x + 6 0.0303 0.0283 0.0237 0.0181 0.0160 Total x 0.0281 0.0263 0.022 1 0.0170 0.0154 Inner v 0.0 145 0.0146 0.0147 0.0147 0.0146 Outer x 0.0135 0.0116 0.0073 0.0023 0.0008 Inner 6 0.0009 0.0009 0.0009 0.0008 0.0004 Outer 6 0.0006 0.0006 0.0003 0.0001 0.0000

First excited state

R = 2.00 3.00 4.90 7.00 8.00

Total x + 6 0.0154 0.0164 0.0176 0.0204 0.0205 Total x 0.0143 0.0152 0.0164 0.0191 0.0196 Inner ,r 0.0139 0.0147 0.0149 0.0148 0.0146 Outer x 0.0004 0.0005 0.0015 0.0043 0.0049 Inner 6 0.0009 0.0009 0.0009 0.0008 0.0004 Outer 6 0.0000 0.0000 0.0001 0.0002 0.0002

TABLE VII. NSO analysis occupation numbers.=

Ground state

1 .o 2.0 3.0 4.0 4.90 6.0 8.0

10.0

0.947420 0.972083 0.97 1587 0.96 1209 0.941318 0.893 128 0.700717 0.575598

0.028567 0.012420 0.014180 0.026747 0.048810 0.099973 0.29 7646 0.424122

0.002816 0.002940 0.003693 0.003244 0.002618 0.00 1837 0.001327 0.0013 19

0.02 104 0.01214 0.0 1010 0.00833 0.00680 0.00472 0.00098 0.000 12

0.997077 0.997072 0.996983 0.996901 0.996868 0.996860 0.996876 0.996890

0.00100 0.00103 0.00112 0.001 14 0.001 16 0.001 18 0.001 18 0.001 18

First excited state

1 .o 2.0 3.0 4.0 4.90 6.0 8.0

10.0

0.747874 0.640662 0.639166 0.670960 0.7 1749 1 0.791999 0.883383 0.864553

0.2 4667 5 0.3 5 7589 0.359504 0.327285 0.280041 0.203995 0.109705 0.129349

0.002935 0.001 762 0.001632 0.001562 0.001522 0.001 5 18 0.001630 0.001501

0.000206 0.000160 0.000360 0.0008 18 0.001424 0.002604 0.004940 0.004450

0.997000 0.996961 0.996756 0.996717 0.9967 15 0.996720 0.996786 0.996885

0.00090 0.00090 0.001 16 0.00118 0.001 18 0.001 18 0.001 18 0.001 18

a NSO'S xu, and xn, are localized in the "inner shell" and xu, , ,yo, , and ,yno are localized in the "outer shell".

R is given in bohrs. 676

FOUR LOWEST lC+ STATES OF THE LiH MOLECULE 677

for each of the first two outer CI orbitals and essentially zero for the outer 7~ orbital. From the occupation number standpoint, LiH is closest to Li+H-, but in any event, it is in more of a closed-shell configuration than either Be or Li+H-.

From the NSO occupation numbers given in Table VII, the wave functions can be written quite accurately as the sum of two closed-shell configurations,

Y x:,[0.9928~;, - 0.1 199xz3].

Both xs2 and xo3 are plotted in Figure 5. Then this function can be transformed

Figure 5 . Plots of the “outer shell” NSO, U, and u3 for the ground and first excited 1Z+ states of LiH at R = 3.00b.

exactly to the usual (u, u ) form such that,

Y E 0.5563~:,[~(3)u(4) + u(3)u(4)]

with the normalized u, u = 0 . 9 4 4 9 ~ ~ ~ f 0.3286~~~ which are plotted in Figure 6. Inspection of the plots indicate that the u orbital resembles very much a hydrogenic 1s orbital on the H nucleus with a small cusp at the Li nucleus, and the u orbital resembles a 2s-2pU hybrid on the Li atom with apparently almost no cusp a t the H nucleus. Such hybridization could account for the dipole moment and the ionic electron distribution as observed in the electron count and planar density plots [5, 181.

Possibly the occupation numbers relative to Li+H- can be rationalized by assuming that extensive polarization of H- by Li+ decreases the effectiveness of

67% RICHARD E. BROWN AND HARRISON SHULL

Figure 6. Plots of the u, u orbitals for the ground and first excited 'C+ states of LiH at R = 3.00b.

the radial correlation in H- (the most important type here). The angular correlation becomes correspondingly more important in the polarized atom for it now assumes some of the correlation previously contributed by the radial term. The trend is, incidentally, precisely parallel to the trend in the occupation numbers in the He isoelectronic series as a function of 2. As 2 increases, the corre- lation energy stays constant, but the occupation numbers of the first orbital tends toward unity. Thus another rationalization of the situation is to say that we have a natural trend toward a modified H- ion with, however, a somewhat larger effective nuclear charge than in the isolated atom.

Both Browne and Matsen and Bender and Davidson have published functions with energy lower than that reported here. Apparently the improvement of Browne and Matsen is largely in the inner shell where the STO basis they used is more satisfactory. Bender and Davidson's result was significantly better (0.005 1 H) . Judging from the correlation energies, their function is an improvement over the one reported here in all types of terms in both shells. Since there is no major deficiency in our function in any one category, we feel it is a reasonably good representation of the distance variation of the function with relatively small additional effort because of the scaling properties of elliptical functions.

At some small distance, one can expect LiH to become more nearly like the united atom, Be. This has still not occurred at R = 2.00b where the occupation numbers show that ng2 is still higher than that for R = 3.00b whereas the Be ncL is lower. But by the time R = 1.00b is reached, the data in Table VII show that the occupation numbers have become quite similar to those found in Be.

For very large distances the molecule dissociates into Li + H, and we can expect in this limit that ng2 and nn3 become equal to each other and to 0.5. The

FOUR LOWEST lZ+ STATES OF THE LiH MOLECULE 6 79

data of Table VII show a trend in this direction for all distances beyond R = 2.00b with the most marked movement occurring beyond R = 6.0b.

The variation of the dipole moment with distance (Table V) is also instructive. At a distance as low as R = 1.00b where united atom occupation numbers are already in effect the dipole moment is still very large, in fact considerably more than one electronic charge placed at this internuclear distance. The magnitude of the dipole moment rises continuously until about R = 5.2b, but not as rapidly as it would if it represented one charge separated by the internuclear distance. This latter situation is reached just short of R = 2.0b. The continuous rise of dipole moment from R = 2 to R = 5.2b corresponds to the ascendancy of the configuration Li+H-. The sharp drop after R = 6b corresponds to the avoided crossing of the Li+ + H- and Li + H potential curves in that region (see Figure 1). The subsequent change of sign of the dipole moment computed for R = 9 and R = 10b of Table V is somewhat surprising, but as has been stated earlier, these R values are beyond the range of well-optimized wave functions, and the results may merely be a function of inaccurate computations and inadequacies of elliptic basis orbitals for large R values.

5. The First Excited State of LiH

The 32 configurational wave function utilizing ten orbitals reported in Table I1 gives an energy for the first excited state of - 7.93385H a t R = 4.90b. Also the energy of this state was calculated a t 34 different R values using the second root of the 69 configurational function given in Table 111. For R = 4.906b, the experi- mental Re, the energy obtained, -7.93720H as compared to the experimental value of -7.94956H, accounts for 65 % of the binding energy. Fitting the potential energy and dipole moment curves at eight points, R = 4.50,4.70,4.80,4.90, 5.00, 5.10, 5.20 and 5.50b, with a sixth-order polynomial in R gave Re = 4.928b) E(Re) = -7.9372H, p(Re) = +3.957d and p/(Rdp/dR)J,, = -0.471. For the latter two quantities, there are no experimental data a t present for comparison. Between R = 4.84 and 5.01b, the curve is extremely flat since the energy changes by only 0.0000lH. As with the ground state, the wave functions are considered reliable between 2 Q R < 8b.

For small R, the NSO occupation numbers indicate that the first excited state exists in more of an open-shell configuration than the ground state. Of particular interest are the charge rearrangements in the transition from the ground to the first excited state at R = 3.00b, the equilibrium distance for LiH X l P . Here the occupation numbers for the first excited state NSO’S, xoz and x o 3 , are 0.6392 and 0.3595, respectively, as opposed to the respective values of 0.9716 and 0.0142 for the ground state. The electron count and planar density plots in Figures 3 and 4 demonstrate an unusually large charge accumulation behind the Li nucleus while the electron density around H is not too unlike that of the simple H ( 3 ) atom.

680 RICHARD E. BROWN AND HARRISON SHULL

Obviously at this distance this large unbalance of charge is the major force towards a larger internuclear distance at equilibrium.

As with the ground state, the occupation numbers indicate that the excited state wave functions at R = 3.00b can accurately be written as,

Y x:,[0.7996~:, - 0.5997~%,]

This can be transformed to the (u, v) form such that,

'Y 0.6997xz1[u(3)v(4) + v(3)u(4)]

with the normalized u, v = 0 . 6 3 2 ~ ~ ~ f 0 . 5 4 8 ~ ~ ~ . The functions, u, u, are plotted in Figure 6. The u orbital resembles a hydrogenic Is orbital with a small cusp a t the Li nucleus, and the u orbital resembles a Zs-Zp, hybrid on Li with the proper phase to transfer charge to the back side of Li. Interestingly enough, there is a strong resemblance of the u orbitals for the two states while the u orbitals can approximately be considered the 2s-2pC hybrids of opposite phase.

At R = 4.90b, the equilibrium R distance for the first-excited state, the dipole moment has decreased to +3.991d, and the electron count and planar density calculations here confirm, as expected, a similar but somewhat less polarized electron distribution than for LiH AIZ+ at R = 3.00b. The electron count has only decreased from 1.90 to 1.80 at the Li nucleus, from 2.97 to 2.88 a t 2 = 0.0 and from 3.50 to 3.45 at the H nucleus. Similarly, the planar densities show a small shift of charge from behind the Li to the H.

For the (u, v) form a t R = 4.90b,

Y ~;~[0.849~: , - 0.530~:,]

'3! 0.6903~:,[~(3)~(4) + v(3)u(4)]

with u, v = 0 . 7 8 5 ~ ~ ~ f 0 . 6 2 0 ~ ~ ~ . These are plotted in Figure 7. Inspection of Figure 7 indicates this bond is fairly closely a classical valence bond between neutral Li and neutral H as was observed a t smaller R.

As we follow the nonequilibrium wave functions a t larger R, the most significant change occurs again in the region of the avoided potential crossing of the unper- turbed Li+H- and LiH curves beyond R = 5.5b. The dipolemoment ofthe excited state suddenly becomes large and negative just as the curve tends towards the ionic limit. Since the actual dissociation limit at infinite distance for this state is not, however, Li+ + H-, but rather Li(,P) + H(2S), we can expect that at even larger R there will again be an avoided potential crossing, and the dipole moment will tend to zero after this occurs. We should point out once again that the precise data computed for R > 8 cannot be relied upon using the present wave function which is not adapted for these large R values. But these results can be expected to be qualitatively correct.

FOUR LOWEST lE+ STATES OF THE LiH MOLECULE 68 1

0.6 r

0.5 -

0.4 -

0.3

0.2

0.1

-

-

-

Z (bohrs)

-0.3 -02 t Figure 7. Plots of the u, u orbitals for the first excited lZ+ state of LiH at R =

4.90b.

Most of these results agree favorably with those of Karo who used a much simpler wave function, except Karo’s function did not emphasize the importance of the ionic configuration enough. The dipole moments obtained by Karo were not nearly so negative at large R as obtained here. Karo used closed shells whenever possible and included no angular correlation terms in his function.

6. The Second and Third Excited States of LiH

Since there are an ample number of configurations and orbitals in the 69 configurational wave function, the third and fourth roots of the secular equation should give informative upper bounds for the respective states. No attempt was made here to reoptimize the orbitals and undoubtedly more lithium 3s and 3p character is needed in the basis set. The results, however, seem to give fairly good energies, particularly for the second excited state, and so are considered reliable enough to be at least qualitatively correct. The eigenvector coefficients for various R values are given in [ 191.

Neither of these states has hitherto been observed experimentally, and no minimum was necessarily expected for either, but as Figure 2 illustrates, both possess a t least one minimum. The second excited state has a metastable minimum ofabout 0.015H close to R = 3.70b and presumably another one near R = 10.00b. The reason for this latter conjecture is that the energy obtained at R = 10.00b is only 0.018H above the sum of the experimental energies of the dissociation products, while the ground-state error at this distance was 0.0228H even when the orbitals were explicitly tailored for that state. This second minimum probably results from the stability of the Li+H- configurations which should become important for larger R. Csizimadia, Barnett and Sutcliffe investigated the second excited state with a much more restricted wave function and noticed only a flattening of the curve close to R = 3.50b [16].

682 RICHARD E. BROWN AND HARRISON SHULL

The dipole moment function for this second excited state has even more interesting variation with R than exhibited by the lower states. At the smallest R computed, the result is extremely negative, corresponding to a large electron displacement behind the hydrogen atom. As R increases, the moment steadily decreases in magnitude, finally becoming positive in the region near 5-6b. The ionic curve Li+H- now again exerts a large influence, and the moment becomes once again large and negative. Even further out, it must turn around toward zero.

The third-excited state also possesses a minimum around R = 7.50b where the dipole moment is positive. Since the bask set was optimized for the ground and first excited states, possibly reoptimizing the basis set to include more 3sL1 and 3pLi orbitals may alter the position and depth of the minimum, and materially change the electron distribution. Examination of the relative positions of the levels of the dissociation products suggests, however, that avoided crossings are the rule rather than the exception in these states, and the properly detailed potential surfaces and wave functions can be expected not infrequently to have somewhat bizarre appearances.

7. The Vibrational and Rotational Analysis The vibrational and rotational energy levels were determined for both the

ground and first excited states by a numerical method introduced by Davidson [17]. To obtain the additional points needed on the electronic energy curve, a four point gaussian interpolation was used on points , Ri , Ri+l and Rz+.2 to obtain additional points between Ri and R,+l. Since these curves were quite smooth and have little 1/R dependence like a Morse-type curve, the interpolated points were sufficiently accurate. For a step function approximation to the potential curve, E = 0.0006H which was somewhat smaller than necessary. All energy levels were determined to within -0.2 cm-l of the exact eigenvalues of the interpolated curve. For both the XIZ+ and AlCf states of LiH and LID, the vibrational and rotational energy levels were determined up to u = 18 and J = 4. The resulting energies are listed in Table VIII.

For the ground state of LiH, these energy levels give w, = 1374.3 cm-l (1405.65), w,x, = 22.5 cm-I (23.20) and B, = 7.2 cm-1 (7.51) where the experi- mental results are given in parentheses [lo, 121. Likewise for LID XIZ+, w, = 1033.3 cm-l (1055.12), w,x, = 13.5 cm-l (13.23) and Be = 4.0 cm-l (4.23).

Among all the diatomic states, the first-excited lC+ states of the alkali hydrides are particularly interesting in that they uniquely have a negative anharmonicity. The AG,,, values increase at first with higher u and then decrease normally thereafter. For the A1Z+ states of LiH and LiD, the experimentally determined AG values reach a maximum a t the 9-10 and 13-14 transitions, respectively, while here the calculated AG values reach a maximum at the 11-12 and 15-16 transitions, respectively. For LiH AIC+, we = 291.5 cm-I (234.41), w , ~ , = -12.6 cm-I (-28.95) and B, = 2.8 cm-I (2.82). Also for LiD AIC+, we = 217.4 cm-' (180.71), w,x, = -7.6 cm-l (-13.99) and Be = 1.6 cm-1 (1.61).

TABLE VIII. The vibrational-rotational energy levels for the ground and first excited lC+ states of LiH and LiD.a*b

LiH XIC+ p = 0.88151 E,, = 683.1 cm-l

J

U 0 1 2 3 4

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18

0.0 1329.3 2613.6 3857.8 5061.2 6224.4 7348.6 8435.2 9483.9

10496.1 11471.9 12410.9 13312.6 141 76.7 15002.6 15788.3 16531.7 17230.4 17880.9

14.3 1343.3 2627.2 3871.1 5074.0 6236.7 7360.6 8446.8 9495.2

10507.1 11482.5 12421.2 13322.5 14186.1 1501 1.6 15797.1 16540.1 17238.3 17888.4

42.9 1371.1 2654.2 3897.3 5099.5 6261.4 7384.5 8470.1 9517.7

10528.9 11503.6 12441.6 13342.1 14205.2 15030.0 15814.6 16556.7 17254.2 17903.3

86.0 1412.9 2694.7 3936.6 5137.7 6298.6 7420.5 8504.9 955 1.4

10561.6 11535.3 12472.0 1337 1.7 14233.7 15057.2 15840.8 16581.7 17277.7 17925.5

143.2 1468.4 2748.7 3989.0 5188.6 6347.9 7468.3 855 1.3 9596.5

10605.1 11577.3 12512.9 13410.9 1427 1.5 15093.7 15875.6 16614.9 17309.3 17955.1

LiD XIC+ p = 1.56535 E, = 514.6 cm-l

J

U 0 1 2 3 4

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18

0.0 1006.3 1985.6 2943.0 3875.7 4786.2 5674.8 6539.7 7383.1 8205.7 9006.7 9786.6

10546.2 11285.1 12003.4 12700.8 13377.3 14032.7 14666.7

8.0 1014.2 1993.4 2950.5 3883.2 4793.5 5681.9 6546.6 7389.9 8212.2 9013.2 9793.0

10552.4 11291.2 12009.4 12706.5 13382.8 14038.2 14672.0

24.2 1030.2 2008.9 2965.6 3898.1 4808.0 5696.1 6560.6 7403.4 8225.5 9026.1 9805.7

10564.8 11303.3 1202 I . 1 12718.0 13394.0 14048.9 14682.4

48.6 1053.9 2032.2 2988.4 3920.4 4829.8 5717.4 658 1.4 7423.8 8245.4 9045.6 9824.6

10583.2 11321.3 12038.8 12735.2 13410.8 14065.2 14698.2

81.0 1085.7 2063.2 3018.7 3950.0 4858.8 5745.7 6609.1 7450.8 8271.8 9071.4 9849.8

10607.9 11345.4 12062.2 12758.0 13433.0 14086.9 14719.3

683 (Table continued)

TABLE VIII.-continued,

LiH AIC+ p = 0.88151 E,, = 151.8cm-l

J

U 0 1 2 3 4

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18

0.0 316.7 658.6

1020.7 1399.0 1790.6 2 192.0 2601.8 3018.1 3438.2 3861.6 4286.2 4710.9 5 134.9 5557.2 5977.3 6394.2 6807.5 7216.4

5.6 322.3 664.4

1026.4 1404.8 1796.3 2197.8 2607.5 3023.8 3443.9 3867.2 429 1.7 4716.4 5140.4 5562.7 5982.5 6399.4 6812.7 7221.5

16.9 333.7 675.7

1037.8 1416.2 1807.7 2209.2 2619.0 3035.2 3455.1 3878.5 4302.7 4727.4 5151.2 5573.4 5993.1 6409.8 6822.9 7231.7

33.6 350.6 692.7

1055.0 1433.4 1824.9 2226.3 2636.0 3052.1 3472.1 3895.1 43 19.3 4743.8 5167.6 5589.4 6009.0 6425.4 6838.4 7246.9

55.9 373.1 715.5

1077.8 1456.3 1847.8 2249.0 2658.8 3074.8 3494.5 3917.5 4341.5 4765.7 5189.1 5610.8 6030.1 6446.2 6858.9 7267.0

LiD AIC+ p = 1.56535 E,, = 112.3 cm-I

J

U 0 1 2 3 4

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18

0.0 232.7 480.7 741.6

1013.6 1295.2 1584.2 1880.1 2181.6 2488.1 2798.3 3111.2 3426.9 3744.3 4062.5 4381.1 4699.9 5018.3 5336.0

3.2 235.9 483.9 744.8

1016.9 1298.5 1587.5 1883.3 2 184.8 249 1.4 2801.3 31 14.4 3430.2 3747.4 4065.5 4384.3 4703.0 502 1.4 5339.0

9.6 242.2 490.2 75 1.3

1023.2 1304.9 1594.0 1889.7 2191.3 2497.7 2807.9 3120.9 3436.6 3753.7 407 1.8 4390.5 4709.2 5027.6 5345.1

19.0 25 1.8 499.8 760.9

1032.9 1314.6 1603.7 1899.4 2200.9 2507.4 2817.5 3130.3 3446.0 3763.2 4081.2 4399.9 4718.5 5036.7 5354.2

31.5 264.5 512.6 773.6

1045.8 1327.5 1616.6 1912.3 2213.7 2520.3 2830.3 3 143.1 3458.7 3775.8 4093.8 4412.3 4730.7 5049.0 5366.4

a The energy levels are given relative to the (0, 0) level and E, is the energy of the (0, 0) level relative to the absolute minimum of the potential curve. p = reduced mass.

The energy is given in cm-l units where 1 Hartree = 219355.152 cm-l. 684

FOUR LOWEST %+ STATES OF THE LiH MOLECULE 685

Acknowledgements

The authors are particularly indebted to Professor S. Hagstrom, Dr. F. Prosser, Dr. G. P. Barnett and Dr. A. Macias for the use of some indispensable computer programs and for helpful consultations on matters of research.

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[6] D. D. Ebbing, J. Chem. Phys. 36, 1361 (1962). [7] J. C . Browne and F. A. Matsen, Phys. Rev. 135, A1227 (1964). [8] R. S. Mulliken, Phys. Rev. 50, 1028 (1936). [9] E. J. Rosenbaum, J. Chem. Phys. 6, 16 (1938).

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[lo] F. H. Crawford and T. Jorgenson, Phys. Rev. 47, 358,932 (1935). F. H. Crawford and T. Jorgenson, Phys. Rev. 49, 745 (1936).

[11] L. Wharton, L. P. Gold, and W. Klemperer, J. Chem. Phys. 37, 2149 (1962). [12] G. H. Herzberg, Spectra of Diatomic Molecules (D. Van Nostrand Co., Inc., Princeton, New

Jersey, 1950), pp. 328, 346, Table 39. [13] G . P. Barnett, J. Linderberg, and H. Shull, J. Chem. Phys. 43, 580 (1965). [14] A. W. Weiss, Phy. Rev. 122, 1826 (1961). [15] C. C. J. Roothaan, L. M. Sachs, and A. W. Weiss, Rev. Mod. Phys., 32, 186 (1960). [16] I. G. Csizimadia, B. T. Sutcliffe, and M. P. Barnett, Can. J. Chem. 42, 1645 (1964). [17] E. R. Davidson, J. Chem. Phys. 34, 1240 (1961). [18] P. Politzer and R. E. Brown, J. Chem. Phys. 45, 451 (1966). [19] R. E. Brown, A Configuration Interaction Study of the lC+ States of the LiH Molecule, Ph.D. Thesis,

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Received March 21, 1968.