Interaction Energy Surfaces for Li(2²S) and Li (2²P) With H₂

Abstract

Interaction energy surfaces for the interaction of Li(2²S) and Li(2²P) with H₂ are calculated using approximate Hartree-Fock trial functions. The crossing of energy curves is observed for C_2v conformations if the H₂ internuclear distance is sufficiently large. No crossing is observed for colinear collissions for any H₂ distance.

The wave function of the strongly attractive state involved in the crossing is related to the metastable negative-ion states that are postulated to account for resonant electron-molecule scattering. Such a charge-transfer state can only be bound for C_2v conformations for the H₂ molecule. The likelihood and geometry of the crossing complexes for other molecules interacting with alkalis is discussed in terms of the formation of these resonance charge-transfer states.

1. Introduction

Fundamental to the understanding of electronic-vibrational energy transfer involving alkali atoms and diatomic molecules is an accurate representation of the energy of interaction. The qualitative features of these surfaces was first considered with semi-empirical techniques [1].¹ To this date there are no ab initio calculations. Accurate energy calculations utilizing correlated wave functions are not yet possible but these systems can be investigated immediately with available Hartree-Fock (H.F.) programs [2]. This is of value for two reasons. First, the approximate H.F. energy surface is considerably more reliable than the earlier results to the extent that the qualitative features of the surfaces can be fixed. In particular, the question of the angular distribution of the interaction energy can be settled and an estimate given given of the activation energy, if any, of the excitation of alkali D radiation by vibrationally excited diatomics. Second, the molecular orbital description is more natural to the discussion of these reactions in the light of recent studies of electron-diatomic molecule resonant scattering [3].

This note will examine the qualitative behavior of the Li(2²S) and Li(2²P) interaction with the H₂ molecule without chemical reaction as the simplest example of the alkali-diatomic system. There is a total of four curves, two repulsive ones, and one weakly, and one strongly attractive curve which are depicted in figure 1. Except for the strongly attractive state there is little overlap of the alkali and diatomic charge distributions. It is our contention that the H.F. surfaces are qualitatively correct and of semiquantitative accuracy for this type of surface. The near constancy of the correlation energy over a wide range of geometries is presumed here [4].

Figure 1a.

Interaction energies for Li(²S and ²P) with the ground state of H₂ for C_2v geometries as a function of the H₂ internuclear distance.Energies in a.u.

The correlation situation is quite different for the strongly attractive curve. This curve has long been characterized as a charge transfer state [1]. The H.F. function does, in fact, describe a charge transfer state and is related to the ionic valence bond configurations. The negative ion ${\text{H}}_2^{-}$ state formed here is also related to the lowest energy resonance state of ${\text{H}}_2^{-}$ considered responsible for enhanced vibrational excitation at low incident electron energies [3]. Three electrons are now interpenetrating and the correlation [5] is similar to that in ${\text{He}}_2^{+}$ and H₃. However, there are offsetting H.F. and correlation errors that are strongly dependent on the geometry. The depth of the H.F. attractive well should approximate the correlated one.

Since even an accurate H.F. calculation of the strongly attractive curve can only be considered qualitative, we have contented ourselves with self-consistent-field solutions far from the H.F. limit. More accurate solutions will be considered when they are needed as a base for a correlated calculation. The H.F. techniques are briefly described and the results for a rather small but meaningful basis are given in the next section. Similar calculations were completed for the isoelectronic LiHe system at the same level of accuracy. This permits some calibration of the approximate H.F. values since limited configuration interaction results [6] exist for LiHe.

In the final section the H.F. results are compared to the earlier semiempirical calculations. The angular distribution and the position of the crossing in the interaction curves are particularly examined and earlier deductions on the linear surface are questioned. Comparison of the H.F. wave functions for the charge-transfer state to the assumed resonance states in electron scattering is pursued to the point of speculation with regard to other systems.

2. Hartree-Fock Calculation

For doublet systems with one open-shell orbital outside of a closed shell core, the H.F. solutions satisfy a variant of the Brillouin theorem that was noted by Davidson [7]. If the vectors are designated C_i, _α where α designates the open-shell vector for which self-consistency is achieved then

$$C_{i,\alpha }S{C}_{j,\alpha }={\delta }_{ij}$$

and

$$\langle {\Psi }_{i,\alpha }H{\Psi }_{j,\alpha }\rangle =E_i{\delta }_{ij}$$

where S is the basis function overlap matrix and H is the total electronic Hamiltonian. Ψ_{i, α} is the H.F. function where the excited ϕ_i orbital replaces the ϕ_α orbital in Ψ_{α, α}. Calculation of any one state will yield upper bounds to higher Rydberg states. The accuracy of the bounds depends upon the size of the basis set and its ability to fit the higher Rydbergs and the extent to which the core orbitals relax for different Ψ_{α, α}. As noted in the introduction no special effort will be made to achieve the H.F. limit. But a reasonably large set must be used to insure correct qualitative behavior. Both the Li(²S) and H(²S) bases are chosen from the smallest of the accurate Gaussian-type functions (GTF) sets given by Huzinaga [8]. The 2s_H, 2p_Li, 2p_H, 3s_Li, and 3p_Li orbitals are approximated by choosing effective Slater exponents for the appropriate atomic Rydberg orbitals and using the large coefficient Gaussian fits to the Slater orbitals [9]. This procedure is very rough since not all the GTF required can be used because of computer time limitations so that compromise exponents must be chosen. This set has been contracted severely with the resultant H₂ and Li energies in table 1. The Li(²S-²P) interval is 0.11 eV larger than the H.F. value and 0.12 eV larger than the experimental value [10]. The error is almost entirely in the 2s_Li due to the use of 1s_Li coefficients to severely contract the s_Li set. For H₂ the error in the H.F. energy is about 0.6 eV at 1.5 a.u. compared to an accurate H.F. energy. This large discrepancy is due to the contraction of the H₂ basis using the Huzinaga expansion coefficients without scaling the orbital exponents. Since the contracted function is equivalent to a single Slater-type function (STF) the 1s function exponents determined by Fraga and Ransil [11] can be used as a guide. The larger exponent would improve the energy significantly but would effect improvements mainly in the H₂ binding and not in the interaction region between the Li and H₂ which is the main concern here. More flexible bases will be used when the correlation is calculated in subsequent work. The errors due to the small basis considerably exceed those that arise from the use of orbital energies to compute the total energies. The general shapes of all curves can be found in figure 1 for the neutral states and in figure 2 for the Li⁺ + H₂ interaction.

Table 1.

Energies of asymptotic atoms and molecule

	−Ea
Atom/molecule	Present calculation	Accurate H.F.
	He	2.86113	b2.86168
Li+	7.23597	b7.23641
Li(2S)	7.42714	c7.432724
Li(2P)	7.36342	c7.365068
H2(1.0)d	1.02984
H2(1.5)	1.11010
H2(2.0)	1.08197
H2(2.5)	1.03299
H2(1.4)		e1.13354

^aEnergy in atomic units: a.u. of energy = e²/a₀ = 27.210 eV.

^bT. Gilbert and P. Bagus, unpublished results.

^cA. W. Weiss, Astrophys. J. 138, 1262 (1963).

^dH₂(R) where the distance R is in atomic units: 1 a.u. of length = a₀ = 0.52793 10⁻⁸ cm.

^eW. Kolos and C. C. J. Roothan. Revs. Modern Phys. 32, 205 (1960).

Figure 2.

Interaction energies for Li ⁺(¹S) with the ground state of H₂ for (a) C_2ν and (b) C_∞ν geometries as a function of the H₂ internuclear distance.Energies in a.u.

The quantitative value of the B₂ curves is limited by an uncertain correlation error. Correlation effects will be considered in two categories. The first type are terms which correlate the asymptotic fragments. The K shell of the Li atom is, of course, essentially unaffected by the interaction. It is presumed that for all but the B₂ state the correlation of the H₂ bond is only slightly changed for moderate interaction energies. The correlation energy curve for H₂ is quite constant to about 1.5 a.u. and then starts to rise rapidly [14]. For the B₂ state the separation of the two hydrogen atoms in the presence of the Li atom does not lead to a large increase in the correlation energy as R(H − H) exceeds 1.5 a.u. Since a b₂(1σ_u) orbital is occupied, the asymptotic fragments would be H+ H⁻ with Li⁺ for electrical neutrality. The correlation energy [15] for H⁻ is only a little less than that for H₂ and the usual large asymptotic correlation error is avoided. For H₂ separations less than about 1.5 a.u. the correlation energy in the H₂ pair should be comparable for the A₁, B₁, and B₂ states.

At R(H − H) = 2.5 a.u. the B₂ state would be about 0.01 a.u. lower in energy than the A₁ and B₁ states due to the difference in correlation energy in the H₂ electron-pair for the two categories of states. Off-setting this error would be the increase in the second category of the correlation energy due to the three interpenetrating electrons in the B₂ state. For ${\text{He}}_2^{+}$ the correlation energy at equilibrium exceeds that for the He atom by about 0.025 a.u. It is not expected that the additional correlation energy in LiH₂ ²B₂ will be that large since the charge is not completely transferred from the Li atom in the range of conformations of interest. Assuming a direct proportionality of the excess correlation energy to the amount of charge transferred, the value is 0.008 a.u. for the curve where R(H − H) is 1.5 a.u. On this basis it is felt that the calculated surfaces are semiquanti-tatively correct. For these surfaces the zero activation energy crossing for activation of the ²P state by colliding the ²S atom with a vibrationally excited molecule occurs at a conformation where R(H − H) is a little less than 2.0 a.u.

An oblique check on tnese calculations can also be made by considering the isoelectronic diatomic interaction Li + He for which accurate calculations exist. The asymptotic error is still due to the poor Li(²S) function. Only the ²Π state was calculated directly and the well depth and spectroscopic constants for the ²Π states are compared in table 2. All the LiHe interaction energy curves are exhibited in figure 3.

Table 2.

Spectroscopic constants and dissociation energy, LiHe

Molecular state	2Π
	This work	Ref. [6]
De(eV)	0.037.	0.06267
re(10−8cm)	2.04	1.964
ωe(cm−1)	261.9	285.7
ωeXe(cm−1)	28.66	39.68
αe(cm−1)	0.144	0.183

Figure 3.

Interaction energies for Li(²S and ²P) with the ground state of He.Energies in a.u.

3. Discussion

The present results confirm the early Heitler-London predictions of crossings in the energy curves for triangular geometries [1a]. There are no crossings in the colinear cases. In fact the linear curves strongly resemble the LiHe interaction energy curve. Laidler [1b] has discussed the Na(²P) quenching in terms of the linear complex and has argued that the conclusions would not be affected by the shape of the complex. For the Li reactions the portion of the surface calculated here would argue quite the opposite. It is improbable that colinear collissions of vibrationally excited or dissociated H₂ with Li(²S) would lead to excitation to Li(²P) or the converse. But crossings are available in the C_2v configuration and there is no classical activation energy. The curves indicate a very rapid change in crossing position and well depth as a function of the internuclear H₂ distance. This would permit the strong coupling of the electronic and vibrational degrees of freedom which is in accord with both past [1b] and present [12] interpretations of these reactions. In particular deactivation of alkali resonance radiation will occur predominantly by vibrational excitation.

Two features of the curves are worth specia, mention. First, the crossing in the Hartree-Fock approximation is a kind of conical intersection [13] of energy curves. The doublet open-shell configurations have all off-diagonal energy matrix elements zero even if the symmetries are identical. ²A′ states would exhibit a conical intersection in the H.F. approximation. However, it is difficult to see how the degeneracy could be maintained when the full correlation interaction is introduced although the splitting is probably small. The second feature is the almost congruence of the H.F. ²B₁ or ²Π neutral curves and the ion curve for the same geometry. The well depth in the Π-type states is not due, in this approximation, to any slight binding in the π orbital but predominately to the ion core interaction with a polarizable, unshielded atom. Similar results are found for LiHe. Although the well depths are about 40 percent smaller than those of Schneiderman and Michels or an accurate H.F. result, the qualitative behavior is similar. The ²Π and¹Σ⁺ well depths are quite comparable with the ²Π dissociation energy greater by 0.01 eV.

The large H₂ separations required for the crossing were previously linked to the ${}^2{\sum }_u^{+}$ state [1] of ${\text{H}}_2^{-}$. H.F. wave functions can be similarly interpreted. For the B₂ state the p orbital is oriented along the H₂ molecule. As the Li(²P) approaches H₂, a fraction of the charge transfers to the H₂ in a charge distribution very much like that of an isolated 1σ_uH₂ orbital. This is a penetrating valence orbital with strong antibonding characteristics which determines both the deep well and the large H₂ equilibrium separations. The picture that develops is that of a charge-transfer state arising from a low ionization potential atom and a diatomic molecule that can support a valencelike negative-ion resonance state. Those molecules which have no available low-lying valence-type orbitals will not have deep attractive interactions with excited alkali atoms. In this category would fall the Ne-isoelectronic molecules and saturated hydrocarbons. Quenching cross-sections are small for such molecules [16]. On the other hand the diatomics like N₂ have the π_g and σ_u type valence orbitals available. The Π_g state can arise from a p orbital in a co-linear collision for an orientation perpendicular to the line of centers. The C_2v geometry requires a d orbital on the alkali since the π_g orbital correlates with an A₂ representation. But the σ_u antibonding valence-type orbital plays the same role in N₂ as it does in H₂. The relative well depths with angular orientation should provide another measure of the kind of one-particle resonance involved in low energy electron excitation of vibrations. Contrary to H₂ there will probably be a crossing for colinear as well as C_2v collisions which could, in part, account for the higher N₂ cross section [17]. The character of the negative-ion should change as a function of angle and as a result so should the vibrational-electronic coupling.

Figure 1b.

Interaction energies for Li(²S and ²P) with the ground state of H₂ for C_∞v geometries as a function of the H₂ internuclear distance Energies in a.u.