J . Phys. Chem. 1984, 88, 2045-2048 plus damped dispersion model, it is interesting to compare the two models in greater detail in order to understand why the HTT surface is so much more repulsive. The SCF interaction energies utilized in the HTT surface are only slightly different from and, in fact, less repulsive than those employed in the HFD surface. Hence the HTT and HFD surfaces must differ appreciably in their modeling of the effects of electron correlation. In the HFD surfaces the correlation contribution, AECORPlto the interaction energy is given by the doubly damped dispersion series (2.2) whereas in the HTT surface it is given by the Drude correction denoted porn by Habitz the sum of prude, et and PIs,the attractive damped dispersion. The ratio of these correlation contributions to the HTT and H F D l surfaces is displayed in Figure 4 for the C, and C,, geometries. The two surfaces are seen to have very different correlation energy contributions for R 5 6 ~ 0 .These contributions differ even in sign for smaller R. Next, we examine whether the large differences in AEcoRR between the HTT and HFD surfaces are due to differing representations of the nonmultipole-expanded dispersion energy PIs. In the HTT model P I s is identical with PEwhereas in the HFD model P I s is given by the singly damped series PIS
=-
c ' g,(pR) C,R"
n=6
Figure 5 shows P I s for the HTT, HFD1, and HFDZ surfaces at two geometries. Note that PIS(y=O) and PIS(y=7r/2) cross for the H F D l and HFDZ surfaces but not for the HTT surface. This different behavior is due to the fact that the HFD damping functions g, are anisotropic as shown in Figure 6c-d whereas the HTT damping functionsf,, are isotropic as shown in Figure 6b. Figures 6b-d reveal that, in the region of the energy minimum (R= 7ao),the HFD dispersion is noticeably damped by the g,'s, especially for the C,, geometry, whereas the HTT dispersion is essentially undamped. Thus it is not surprising to see in Figure 5 that the HTT surface has a somewhat more attrnctiue P I s at the C,, geometry. However, the A E c o R R differences between the HTT and HFD surfaces are much too large to be due to the P I s
2045
differences shown in Figure 5. Hence the remaining contributions to AEcoRRmust be responsible for the HTT surface being so much more repulsive than the HFD surfaces. In the latter the remaining correlation effects are modeled by multiplying PIs with the damping function f shown in Figure 6a. However, this causes AECORR to be even less attractive than P I s in the HFD model. Therefore the Drude correction is the reason why the HTT surface is considerably more repulsive than any other He-N, is repulsive and surface currently available. For R > 2ao prude almost as large as the SCF interaction energy as shown by Figure 7 which compares the radial Legendre coefficients VnDrude(R) and VnSCF(R)for n = 0, 2, and 4. In order to decide which of the five surfaces discussed in this work is the most realistic and whether the electron correlation effects are better modeled by the HTT or HFD potentials, a detailed comparison must be made between the values of physical observables predicted by the various surfaces and experimental results. Such a comparison forms the subject of a later papers6 in which calculations are made for the HFD1, HFD2, HTT, KSK, and KKM3 potential surfaces of the second virial coefficient, the differential scattering cross section, and shear viscosity and diffusion coefficients. A major finding is that while the present HFDl surface gives overall the best agreement for the bulk data, viz., second virial coefficient, shear viscosity, and diffusion coefficients, its predicted behavior for the total differential scattering cross section is poor. Just the opposite situation occurs for the HTT surface which is in moderate agreement with the total differential scattering cross section but for which the calculated second virial coefficient is in severe disagreement with experimental values over the entire temperature range for which such measurements are available. Acknowledgment. This work was supported by the Natural Sciences and Engineering Research Council of Canada. Registry No. He, 7440-59-7;Nz, 7727-37-9. (56) McCourt, F. R.;Fuchs, R. R.; Thakkar, A. J. J . Chem. Phys., in press.
Numerical Multiconflguration Self-Consistent-Field Calculations on the First Excited State of LiHLudwik Adamowiczt and E. A. McCullough, Jr.* Department of Chemistry and Biochemistry, UMC 03, Utah State University, Logan, Utah 84322 (Received: October 20, 1983)
We have carried out numerical multiconfiguration self-consistent-field (MCSCF) calculations on the first excited state of LiH- in order to investigate the importance of electron correlation in excited states of anions of highly polar molecules. A discussion of the configuration selection and orbital optimization procedures is given. Comparison calculations were carried out on LiH so that an estimate of the electron binding energy could be made. Our best estimate, 95 bhartrees, is compared with the pseudopotential value obtained by Garrett. Some possible general conclusions regarding electron correlation in excited-state anions of this type are presented.
Introduction The class of negative ions formed by attaching an electron to a polar, closed-shell molecule are of widespread interest. To a first approximation, they may be understood as the binding of an electron by a dipole field, and it has been shown' that an infinite number of bound-electron states exist in the Born-Oppenheimer approximation for any system with a long-range dipole field ex-
'
Current Address: Quantum Theory Project, University of Florida, Gainesville, FL 3261 1.
0022-3654/84/2088-2045$01.50/0
ceeding 0.639 eao (1.625 D). The ground-state anions have received considerable study; a number of single-configuration calculations have been p e r f ~ r m e d , ~and - ~ there have also been ( 1 ) 0. H. Crawford, Mol. Phys., 20, 585 (1971); W.R.Garrett, Chem. Phys. Lett., 62, 325 (1979). ( 2 ) K. D . Jordan and W. Luken, J . Chem. Phys., 64, 2760 (1976). (3) K. D. Jordan, K. M. Griffing, J. Kenney, E. L. Anderson, and J. Simons, J . Chem. Phys., 64,4730 (1976). (4) B. Liu, K. 0-Ohata, and K. Kirby-Docken, J . Chem. Phys., 67, 1850
(1977).
0 1984 American Chemical Society
2046 The Journal of Physical Chemistry, Vol. 88, No. 10, 1984
several calculations which include the effects of electron correl a t i ~ n . ~ Less * ~ ~is~known ~ ' ~ about the excited states, however. Indeed, single-configuration results have been reported only for a few and even rudimentary information about the importance of electron correlation is available only for LiH-.I2 In addition to correlation error, basis set expansion calculations on the very diffuse excited state may be very susceptible to basis set error. Hence, we undertook a study of the first excited state of LiH- using our numerical muiticonfigurationself-consistent-field (MCSCF) method. This method can provide essentially exact results for a given configuration list, so that correlation error may be investigated independent of other errors. The quantity of principal interest is the vertical electron binding energy, tb, for the attached electron, which could be measured, say, by an electron-detachment experiment. The difficulty of computing tb as the energy difference between anion and neutral molecule may be readily understood through reference to pioneering work of Liu et al., on ground-state LiH-.4 They utilized large basis sets and extensive configuration interaction (CI) and argued convincingly that they had achieved absolute accuracies of the order of 0.5 mhartree in computing the valence-shell correlation energies of both LiH- and LiH. Since eb for ground-state LiH- is 10 mhartrees, computing Eb as the difference between their total energies is a meaningful procedure and should yield a reliable t b if correlation involving the l a orbital is indeed negligible, as they assumed. The straightforward procedure seems to be out of the question for the first excited state, where t b is -90 phartrees. Rather, it seems necessary to perform less accurate calculations which incorporate correlation effects common to the anion and neutral in precisely the same way. Such calculations must also account for all important orbital relaxation effects, as well as all important correlation effects unique to the anion. Our hope was that an MCSCF implementation of such an approach would lead to stable q, values which could be achieved with fairly short configuration lists.
-
Calculations and Results Our numerical MCSCF procedure has been described previously.12 All calculations were carried out on a VAX 11/780. The analogues of basis set error in our method are error in the numerical evaluation of matrix elements and error in the numerical solution of the Fock equations. To ensure that numerical errors from either source would be well below 90 phartrees, we performed two types of tests. First, we numerically evaluated numerous matrix elements involving diffuse sp Slater function hybrids and compared our results with the exact values. For appropriate choices of the Slater function exponents, such hybrids can be made to resemble the orbitals for the extra electron in these anions quite well. Second, we solved the electron-fixed dipole problem numerically for several values of the parameters and compared with published results13 for this exactly soluble model. These tests indicated that the numerical evaluation of matrix elements should contribute less than 0.5-phartree error, while the numerical solution of the orbital equations might introduce as much as 3 phartrees. The single-configuration description of the first excited state of LiH- is la22a24a. We ignored correlation involving l a , although this was done primarily for simplicity, and we have no direct evidence that the effects are truly negligible. All calculations were carried out at R = 3.015 a,, the equilibrium internuclear separation in LiH. Preliminary calculations indicated that the (5) K. D. Jordan and R. Seeger, Chem. Phys. Lett., 54, 320 (1978). (6) K. D. Jordan, Acc. Chem. Res., 12, 36 (1976). (7) W. J. Stevens, J. Chem. Phys., 72, 1536 (1980). (8) Y. Yoshioka and K. D. Jordan, J . Chem. Phys., 73, 5899 (1980). (9) A. U. Hazi, J . Chem. Phys., 75, 4586 (1981). (10) A. M. Karo, M. A. Gardner, and J. R. Hiskes, J . Chem. Phys., 68, 1942 (1978). (11) E. A. McCullough, Jr., J . Chem. Phys. 75, 1579 (1981). (12) E. A. McCullough, Jr., J . Phys. Chem., 86, 2178 (1982). (13) R. F. Wallis, R. Herman, and H . W. Milnes, J . Mol. Specfrosc., 4, 51 (1960).
Adamowicz and McCullough TABLE I: MCSCF Results for LiH with Configuration Lists of the Form lu2(2uz
+ x,' + x,Z + ...)
Xi 0
71
6
0 1 1 2 2 3
0 0 1 1
0 0 0 0 1 0
1 2
E , hartree
-7.987 -8.003 -8.016 -8.021 -8.021 -0.022
9,
352 735 043 267 725 403
ea,
-2.362 -2.235 -2.262 -2.306 -2.305 -2.300
TABLE 11: MCSCF Results for Excited-State LiH- with Configuration Lists of the Form lu'(20' + xi2 t x 1 2 t ...)40 u
0 1 1 2 2 3
Xi rn
0
6
0
0 1 1 1 2
8
0 0 0 1 0
Eb,
/&re,
phartree
ea,
6 45 47 59 64 61
-2.241 -2.266 -2.309 -2.310 -2.305
( Z ~ O ) , 10-4*,02),
ao2
a0
-63.0 -71.6 -76.6 -66.3 -68.4 -67.6
1.13 1.50 1.72 1.26 1.34 1.31
excited-state anion and neutral molecule ppential curves are nearly parallel in this region, so the Re for the two species should be almost identical. Correlation within the 20 shell of LiH has a pronounced effect on the dipole moment. Given the electron-bound-by-a-dipole nature of the anion, one might anticipate that 2a shell correlation would have a pronounced effect on 'b. This indeed appears to be the case. Tables I and I1 present MCSCF results for a sequence of configuration lists of the form
+ + + ...) la2(2a2+ x I 2 + x22 + ...) 4a
LiH: la2(2a2 x12 x22
(1)
LiH-: (2) The numbers of 2a shell correlation orbitals of each type ( a , P , 6) are tabulated; the first tabular entry is the single-configuration (restricted HartreeFock) result. For LiH, the total energy and dipole moment are shown. For LiH-, we show tb, along with the dipole moment of the neutral core, and expectation values of z and r2 for the 4a orbital. One sees that the LiH dipole moment is very sensitive to correlation until about the level of two a and one r correlating orbitals. Roughly speaking, tb follows this trend. Several features are worthy of note. First, the dipole moment of the neutral core in excited-state LiH- is nearly equal to the dipole moment of LiH at the same level of correlation. Second, the size of the 4a orbital fluctuates drastically in response to changing correlation within the 2a shell. Third, the changes in the total energies of anion and neutral as correlation is added are several orders of magnitude larger than the changes in tb. Fourth, even with the most extensive configuration lists that we used, a residual 2a shell correlation energy of 1500 phartrees remains unrecovered in both species (this is based on 2a shell correlation energy of Liu et aL4). Finally, within the numerical accuracy of our calculations, the last three tabular entries yield nearly the same t b . The requirement of orbital orthogonality in the MCSCF procedure introduces constraints which raise the energy of LiHrelative to LiH. For example, the requirement that 4a be orthogonal to the unoccupied orbitals in any configuration (or, equivalent, that 4a be precisely the same in every configuration) is such a constraint. It can be alleviated within the orthogonal orbital framework by introducing excitations of the form 4a nu, where n can be any of the 2a shell a orbitals including 20 itself. Another constraint is the requirement that any of the 2a shell orbitals be orthogonal to 4a, which is totally absent in LiH. This can be offset with excitations of the type nu240 na4a2. Configuration lists which include both types of corrections can be summarized symbolically as
-
-
-
lay282
+ x12 + x? + .,. + 4a2)(4u + 2a + x , + x* + ...) (3)
The Journal of Physical Chemistry, Vol. 88, No. 10, 1984 2047
First Excited State of LiHTABLE 111: MCSCF Results for Excited-State LiH- with Configuration Lists of the Form 102(2u2 + x,’ t x,’ + ... t 4u’)(40
A configuration list having the desired properties is now easily constructed. It is
+ 20 t x , + x, + ...)
12(20.2 + x,2
Xi
Eb, phartree
U
71
1 2
1
55
1
12
in the sense that, if this expression is formally multiplied out taking account of spin and spatial symmetry, all the necessary configurations will be generated. The results of calculations incorporating these “orthogonality correcting” configurations are shown in Table 111. In these calculations, all orbitals except l a were reoptimized with the extended configuration lists. One sees that the effect of these corrections on tb is certainly not negligible, although the effect on the total energy of LiH- is, of course, very small. Because the optimum orbitals are determined by minimizing the total energy, it would probably suffice to omit the na24u na4uz excitations during the optimization step and add them in afterward as a small perturbation by CI. All calculations reported in Tables I1 and 111 were performed directly on the excited state. That is, no configurations approximating the ground state of LiH- were included, and the minimization was done for thefirst eigenvalue of the Hamiltonian. This was possible because (a) no linear combination of the orbitals inside the second set of parentheses in (3) gives an approximation to the ground-state 3u orbital yielding a lower energy than the excited state, and (b) good starting guesses for the excited-state orbitals could be generated. We can demonstrate that these calculations do produce valid approximations to the excited state rather than very poor approximations to the ground state in the following manner. Suppose the Hamiltonian is diagonalized for an extended configuration list composed of (3) plus some additional configurations, possibly involving additional orbitals. Suppose the result is that the first eigenvalue obtained with (3) is exactly equal to the second eigenvalue obtained with the extended list. Then, the first eigenvalue given by (3) is a variational upper bound to the first-excited-state energy. The question is whether an extended configuration list with this property can be constructed for the case of interest? It can, although the procedure is somewhat involved. The configuration list 3 can be reexpressed as -+
+
+
+
lU*(20* x12 x22 + ...)4u laz(xl2 x2’ +
+
,..
+ 4u2)2a + ... (4)
where the first term represents a 40 electron outside a correlated LiH core, and the additional terms represent the various corrections discussed previously. Freeze the LiH core, keeping both the orbitals and the relative C I coefficients fixed at the values determined for the excited state. Next, construct the configuration list
l02[2uZ
+ x12 + x22 + ...I( 3u’ + 4a + 2a + x , + x2 + ...) (5a)
or equivalently
lu2[2u2
+ x12+ x22 + ...I 3a’ + la2[2a2+ x I z+ x2’ + ...I 4a + ... (5b)
where the brackets denote the frozen core, and optimize only orbital 3d for the LiH- ground state. In effect, this produces an approximate ground-state LiH- wave function for a LiH core appropriate for the excited state. We call the extra orbital 3a’ because it is not the same as the 3u orbital that would be obtained if the wave function were completely optimized for the ground nu included in (5) are state. Furthermore, the excitations 3d not small corrections for the ground state since the other orbitals and the relative CI coefficients in the frozen core are far from optimum for the ground state. The net result of this calculation is a ground-state energy below the excited-state energy obtained with configuration list 3.
-
+ x? + ... + 4u2)(4u + 2a + x , + x2 + ...) + lu2[2a2+ x12+ xz2+ ...I 3u’ ( 6 )
Diagonalization of the Hamiltonian for this configuration list is certain to give a ground-state energy no higher than that obtained with (5) because (6) is more flexible. Comparing (6) with (4), one sees that the additional term in (6) has the effect of allowing a change in orbital 40 with a fixed LiH core. But orbital 4u is already optimum for the excited state with that core, so no change in the excited-state enegy is observed. This illustrates the point which has been made that the presence of the lower eigenvalues of the same symmetry is not a necessary condition for a MCSCF calculation on an excited state.I4 On the basis of the results presented in Table 11, we concluded that a configuration list containing two a and one R correlating orbitals was about the shortest that could adequately describe correlation within the 2a shell. The next step was to include additional configurations accounting for correlation between the 2a and 4u shells. Since the configuration 1 a22u240dominates the wave function near Re, the most important excitations are 2a240 2uyizi, where yi and zi must be of the same spatial symmetry to yield a 2 molecular state, but may be singlet or triplet spin coupled. We first determined a set of approximate yi and ziorbitals using fewer than the full number of 2a shell correlating orbitals. We then checked the overlap integrals betwen these yi or ziorbitals and the 2u shell xi orbitals obtained with extended configuration lists of type 3. Several of the overlaps were substantial, which implies that the requirement of orthogonality between yi or zi orbitals and xi orbitals would constitute a severe constraint. Alleviating these constraints was more difficult than previously. For example, attempts to optimize orbitals in configuration lists of the form
-
1u2(2a2
+ x12+ x22 + ...)(26 + 4a + x , + x2 + ... + y , + + ...) + l U 2 2 U ( y l Z , + y2z2 + ...) (7) 2,
failed. The forms of the yi and zi orbitals turn out to be such that the linear combination inside the second set of parentheses can generate an approximation to the ground-state 3u orbital yielding a lower energy than the excited state. When minimization of the first eigenvalue of the Hamiltonian was attempted, variational collapse to the ground state occurred. It appeared that optimization for the second eigenvalue was now essential. The problem was how to guarantee a stable first eigenvalue in order to avoid the other well-known problem with excited-state MCSCF calculations: root switching. State average procedure^'^-'^ seemed undesirable. The optimum 2a shell orbitals for the ground-state and excited-state anions are clearly rather different. Furthermore, even a slight imbalance in the treatment of 2u shell correlation in excited-state LiH- relative to LiH might introduce substantial errors, given the very small magnitude of tb and the very large magnitude of the 2a shell correlation energy. The procedure that we adopted was to include a set of configurations incorporating an extra orbital chosen to guarantee an approximation to the ground state. If this extra orbital makes any contribution to the excited-state energy, minimization of that energy will, of course, change that orbital and may result in root switching. Consequently, we abandoned the idea of solving a Fock equation for the extra orbital. Instead, we minimized the excited-state energy only with respect to a unitary transformation between this orbital and all the others. The configuration lists were chosen so that, to a first approximation, the unitary trans(14) J. Olsen, J. Jmrgensen, and D. L. Yeager, J. Chem. Phys., 76, 527 (1982). (15) (16) Chem., (17)
K. K. Docken and J. Hinze, J . Chern. Phys., 57,4928,4936 (1972). L. M. Cheung, S. T. Elbert, and K. Ruedenberg, In?. J. Quantum
16, 1069 (1979). H . J. Werner and W. Meyer. J . Chem. Phys., 74,5794,5802 (1981). (18) R. N. Diffenderfer and D. R. Yarkony, J . Phys. Chem., 86, 5098 (1982).
2048
The Journal of Physical Chemistry, Vol. 88, No. 10, 1984
TABLE IV: Results for Excited-StateL W with Configuration Lists Incorporating 20-40Correlationa fb, phartree
33-configMCSCF (70, 80, 2n, 3n only) 33-configMCSCF ( ~ u - ~ cln-3n) J, 53-configCI a See text.
82 82 95
formation could not change the ground-state energy. The extra orbital 3a’ was generated in the same manner as previously described. To minimize the lengths of our configuration lists, some of the least important orthogonality correcting configurations were dropped from the MCSCF, including all of type na24a na4a2. The final configuration list had the form (all orbitals are now explicity shown)
-
+
+
la2(2a2+ 5a2 6a2 1a2)(4a 8a 3a’) la22a(7080
+
+
+ 2a + 5a + 6a + 70 + + 2 a 3 a + 5a70 + l a 3 n )
(8)
which generates 33 spin and spatial symmetry adapted configurations. Taking the la-60 and l a orbitals from a previous calculation (line 2 in Table 111), we first optimized only the 2u-4u correlating orbitals 717, 8a, 27, and 3a. The results are shown as the first entry in Table IV. Next, optimization of all orbitals except 1a was performed. This did not improve the energy within the accuracy of our calculation, as can be seen from line 2 of Table IV. We emphasize that these optimizations were preformed on the second eigevalue and that orbital 3a’ was only rotated with the other orbitals. As one sees from the form of (S), the rotation cannot alter the space spanned by the linear combination inside the second set of parentheses. In fact, this space would remain invariant except that the other orbitals do, of course, change somewhat during the MCSCF iteration. This guarantees a stable approximation to the ground-state energy, at the expense of having one orbital which cannot be fully optimized for the excited state.19 Our final calculation was a 53-configuration CI using the optimum orbitals for configuration list 8 in the configuration list
+
lo2(2a2+ 5a2 + 6a2 + l a 2 + 4a2)(4a + 2a + 5 a + 7u 8 a + 30’) lu22a(7u8a 2 a 3 a 507a 5a8a 6a7o 6o8a + 5a6a + 1a2a l a 3 n 7a2 8a2 2n2 3s2) (9)
+
+
+
+
+
+ + +
+ +
+
which includes a number of relatively important configurations omitted in the MCSCF step. The cb obtained, 95 phartrees, is given in the last line of Table IV.
Discussion Several conclusions can be drawn from this study which may be of general applicability in calculations on excited-state anions of this type. First, the importance of a correct description of the dipole field in which the attached electron moves has been emphasized by others.6,8,20 Our results tend to support this view. A corollary is that, if correlation within the neutral core of the anion changes the dipole field significantly, uncorrelated calcu(19) Frozen orbital methods have also been applied to excited-state MCSCF calculations by C. W. Bauschlicher, Jr., and D. R. Yarkony, J . Chem. Phys, 72, 1138 (1980). See also ref 17. (20) W. R. Garrett, J. Chem. Phys., 71, 651 (1979).
Adamowicz and McCullough lations on an excited state cannot be expected to yield quantitatively reliable results, except accidentzlly. In LiH-, correlation within the 2u shell decreases the core dipole moment and also cb, while 2o-4a correlation, of course, increases eb relative to the single-configuration value. The final q, is thus closer to the single-configuration prediction than would be the case if correlation within the 2a shell increased q,. Since, for the excited states, the attached electron appears to perturb the core hardly at all, the effect of correlation on the dipole moment of the neutral molecule probably can be used as a reliable guide for the anion as well. Our correlated LiH dipole moments stabilize at a value that is in good agreement with the experimental (vibrationally averaged) ground vibrational state value of -2.32 eao.21 Correlation between the LiH core and the attached electron appears to be no less important for the first excited state of LiHthan for the ground state. This is somewhat surprising, for the attached electron has a much lower probability of being close to the core electrons in the excited state. Nevertheless, 2a-4u correlation more than offsets the drastic decrease in eb arising from correlation within the 2u shell, and our final correlated t b is about 10%greater than the single-configuration value. Liu et al! found about a 20% increase for the ground state, and their treatment was much more exhaustive than ours. Although it is not possible to partition the contributions to their ground-state t b as we have tried to do here, it is interesting that, as percentages of the total cb, correlation appears to have roughly the same effect for both ground- and excited-state LiH-. Our results also underscore the need to allow for substantial relaxation in the orbitals involving the attached electron. For example, Table I1 shows that a 4 a orbital determined in a single-configuration calculation is hardly optimal for a highly correlated LiH core, so that if a MCSCF procedure is not employed, single excitations involving the attached electron orbitals can be expected to be very important. In a basis set expansion calculation, sufficient flexibility must also be present in the subset of diffuse basis functions to accomodate such relaxation effects. A point not considered at all in this paper is corrections to the Born-Oppenhemier approximation. These may be substantial when molecular rotation is introduced, as Garrett has demonstrated.22 Indeed, for a rotating molecule only a finite number of bound electron states will exist, and those that do may not follow a normal rotational progression. Using a pseudopotential technique, Garrett has established that the first excited state of LiHprobably remains bound through the first three rotational j states. He predicts a binding energy of 96 khartrees for t h e j = 0 level, which is in excellent agreement with our final Born-Oppenheimer estimate of 95 phartrees. Of course, such spectacular agreement is undoubtedly fortuitous. For example, our calculation might easily underestimate the true Born-Oppenheimer eb by 5-10 phartrees. Nevertheless, the general agreement between the results of these two distinct approaches does give some reason for optimism regarding the theoretical treatment of anions of this type. Acknowledgment. We thank the National Science Foundation for partial support of this research. Registry No. LiH-, 14808-05-0; LiH, 7580-67-8. (21) L. Wharton, L. P. Gold, and W. Klemperer, J. Chem. Phys., 37, 2149
(1 962).
(22) W. R. Garrett, J. Chem. Phys, 77, 3666 (1982).
