Open-Shell Energy Coefficients

The coefficients describing the electron repulsion interactions within and between open shells must be consistent with the number of electrons involved. Within a shell of occupation number there are pairs of electrons, and between shells with occupation numbers and there are pairs of electrons. Classically, one would expect an electrostatic integral involving the charge distributions of two orbitals, , for each pair of electrons. The same result can be obtained quantum mechanically by expressing the wavefunction in terms of Slater determinants and noting that integrals of this type can only be obtained in diagonal determinantal matrix elements, and that they occur there once for each pair of electrons.

All of the integrals in averaged coulomb integral summations are of the classical electrostatic type, while in averaged exchange integral summations only the intrashell terms with are of this type. Thus, if the energy can be expressed in terms of the averaged integrals, the intrashell coefficients must satisfy

or

and the intershell coefficients must satisfy

or

Once a state has been found whose energy can be expressed in terms of the averaged integrals, then other states with energies of this form can often be found by a general procedure based on the fact that the two-electron spatial function is both totally symmetric and invariant to the interchange of the coordinates of the two electrons. An -electron wavefunction of the desired form for the shell is multiplied both by this spatial factor and by a two-electron singlet spin factor, the electrons in both factors being numbered , . Provided that it does not vanish upon antisymmetrization, the new wavefunction will have the same space and spin symmetry as the original wavefunction, but will contain two more electrons. Due to the form of the two-electron factors, the additional electron repulsion terms can be expressed in terms of the averaged integrals. Series of wavefunctions related in this manner are said to have the same seniority [16], which is defined as the lowest value of for which such a wavefunction exists. The seniority is a useful label when more than one term of a particular type arises from an electron configuration. For example, from there are two wavefunctions, one of seniority 1 and one of seniority 3.

If is zero or one, there is no electron repulsion energy, so two series of states can be generated from these very simple starting points. The first series (case I in Table 1) is a set of totally symmetric singlets with seniority zero. They start at the empty shell ( ) and end at the closed shell ( ). The second series (case II in Table 1), is a set of doublets of symmetry and seniority one. They start at the particle state ( ) and end at the hole state ( ). The general formulas given for these two cases in Table 1 have been proven for most cases of interest, but not generally. Wavefunctions which are general in were written down for . The resulting energy expressions were fit to quadratic forms in and were found to satisfy the general relationship between the energies of corresponding states from shells of and electrons, which in terms of averaged integrals is

Thus the formulas are proven for all when , and are probably valid in general.

Other types of states whose energy can be expressed in terms of averaged integrals are the half-filled shell ( ) with all spins parallel, and the states obtained by adding or removing an electron ( ) from this state. The latter states (case III in Table 1) have spin ) and the former state (case IV in Table 1) has spin . These three states are the highest spin states.

It is widely known that the average energy of all the states of a configuration, and therefore the total energy also, have the form of the closed-shell energy, but reduced by a multiplicative constant.

It is also known, but not as widely, that the average and total energies of all the states of a configuration with a given spin are also expressible in terms of the averaged integrals, although not in the closed-shell form. Thus if all the states of a configuration except one, or even all those of a given spin except one, fall into the four cases treated already, the energy of the remaining state must be expressible in terms of the averaged integrals also. There are only a few instances (case V in Tables 2, 3) where this occurs. For , , there are case I and case IV states and, except in symmetry groups where the principal axis is a fourfold proper or improper rotation axis, only one other state, (see in Table 3). For in spherical or icosahedral symmetry, and for , there are case I and case III states and only one other state, (see , in Table 2). In the same situation except that , there are case II and case IV states and only one other state (see in Table 2).

There can be two shells of the same symmetry only if the same coefficients can serve both as intrashell and intershell coefficients. This requires that the intrashell coulomb coefficient be zero, which is only possible if it is half-filled with all spins parallel (case IV). Thus the coupling between shells must also be of the all-spins-parallel type and the only possibilities are , , etc. In improved virtual orbital calculations [17] only the intershell coefficients play a role in determining the virtual orbitals, so it is possible to proceed whenever they can be specified correctly.

When the open-shell orbital is not degenerate ( , ), the coefficients are not unique because . There is no electron repulsion so the coefficients need satisfy only

or

The choice would be appropriate if it is desired that the virtual orbitals approximate Rydberg orbitals as closely as possible. If there is more than one open shell of symmetry, the choice must be , .

If the energy of the state of interest cannot be expressed in terms of the averaged integrals, the only alternative within the present formulation is to assume that the orbitals in one or more other states are the same, and to average the energies of these states to obtain the desired form for the energy expression. This may involve all the states of a given configuration, all of the states with a particular spin from the configuration, or sometimes a still smaller number of states. This energy value, when computed, is useful when the splittings are small, but not otherwise. For example, of the six , states of benzene (the coefficients are the same as for the case in Table 6), only the can be done individually. The singlets must be averaged together and the value of this average energy is not particularly useful because the splittings are large and the orbitals for each state probably differ significantly in diffuseness. Rydberg states often have small splittings and for them an average energy can be quite useful. Sometimes it is the orbitals rather than the energy which are the quantities of principal interest. The averaging often has little effect on the orbitals.

General formulas for intrashell coefficients are given in Table 1. Values of intrashell coefficients are given for atoms in Table 2 and for linear molecules in Table 3. Values of intershell coefficients are given for atoms in Table 4, for linear molecules in Table 5, and for linear molecules in Table 6, which contains several examples of averages of states. Values of intrashell coefficients are given for octahedral and tetrahedral molecules in Table 7, including all strong-field states. Values of intershell coefficients are given for weak-field states of octahedral and tetrahedral molecules in Table 8. Values of intrashell coefficients to accompany intershell coefficients can be obtained by simple angular momentum or symmetry coupling. Values of some tabulated coefficients were obtained from references already cited and others were worked out for this review.