For the CI eigenstate Ψ^[R,S,Sz], the total electron density function ρ^[R,S,Sz](r) is expressed in terms of the 2M occupied and virtual SOs χ of ψ_Ref by:

(11.16)

where γ_mi is the occupancy (0 or 1) of the i^th SO for the m^th microstate, s_z,i = 1/2 for alpha SOs and -1/2 for beta SOs. In terms of the corresponding M MOs, ρ^[R,S,Sz](r) is given by:

(11.17)

where γ_a,mi is the occupancy (0 or 1) of the alpha SO of the i^th MO for the m^th microstate and P_MO^[R,S,Sz] is the total one-electron density matrix in the MO basis, with alpha and beta contributions P_MO,α^[R,S,Sz] and P_MO,β^[R,S,Sz], respectively. In terms of a basis of L atomic orbitals (AOs) symbolized by ξ, ρ^[R,S,Sz](r) is given by:

(11.18)

where P_AO^[R,S,Sz] is the total one-electron density matrix in the AO basis, with alpha and beta contributions P_AO,α^[R,S,Sz] and P_AO,β^[R,S,Sz], respectively.

In AMPAC, when the keyword CIDIP is specified, the dipole moment and Mulliken atomic charges are calculated for both the primary and secondary CI eigenstates from the corresponding density matrices P_AO^[R,S,Sz]. In general, other one-electron properties which are also available without CI, such as ESP charges, are calculated in CI calculations from P_AO^[R,S,Sz], but only for the primary CI eigenstate.

The “electron spin density” ρ_σ^[R,S,Sz](r) corresponding to ρ^[R,S,Sz](r) is simply the alpha electron density ρ_α^[R,S,Sz](r) minus the beta electron density ρ_β^[R,S,Sz](r), which, along with the corresponding spin density matrices P_MO,α^[R,S,Sz] and P_AO,β^[R,S,Sz] is given by:

(11.19)

(11.20)

(11.21)

In AMPAC, when the ESR keyword is specified, the spin density matrices P_MO,α^[R,S,Sz] and P_AO,β^[R,S,Sz] are printed for the primary CI eigenstate along with the net Mulliken atomic electron spins for both primary and secondary CI eigenstates. The net Mulliken electron spin for the A^th atom, σ_A, is calculated like the corresponding Mulliken atomic electron population except that P_AO,σ^[R,S,Sz] is used instead of P_AO^[R,S,Sz]

(11.22)

The transition dipole moment μ^[R→n,S,Sz] between CI eigenstates Ψ^[R,S,Sz] and Ψ^[n,S,Sz] is an important result:

(11.23)

For example, contributions from all available transition dipole moments appear in the “sum-over-states” (SOS) expression for the dynamic polarizability tensor α^[R→n,S,Sz](ω), given by Equation (11.26). Individual transition dipole moments are also of interest because they yield information about the UV / visible spectrum of a molecule. The oscillator strength f^[R→n,S,Sz]between states ψ^[R,S,Sz] and ψ^[n,S,Sz] is proportional to the absorptivity of light at a wavelength λ^[R→n,S,Sz]:

(11.24)

(11.25)

where K is a constant. By default, AMPAC writes the transition dipole moments μ^[R→n,S,Sz], transition wavelengths λ^[R→n,S,Sz] and oscillator strengths f^[R→n,S,Sz] between the primary eigenstate Ψ^[R,S,Sz] and all of the secondary CI eigenstates Ψ^[n,S,Sz]. In AMPAC, the number of CI eigenstates to calculate, including the primary CI eigenstate, can be specified using the CISTATE=n keyword (some of these will have a different total spin quantum number S than the primary eigenstate and so their corresponding transition dipole moments vanish).

The “sum-over-states” (SOS) expression for the dynamic polarizability tensor α^[R→n,S,Sz](ω) for the CI eigenstate ψ^[R,S,Sz] is given by:

(11.26)

where ω is the external electric field frequency (in energy units) and the sum is over all possible CI eigenstates different from the primary eigenstate, but having the same S and S_z quantum numbers. In AMPAC, α^[R→n,S,Sz](ω) will be calculated and written to the AMPAC output file when the keywords DYNPOL or DYNPOL=n.nnnn are specified. Note that the keyword CISTATE=n has no influence on the calculation of dynamic polarizabilities, and vice versa, but the number of possible CI eigenstates (determined by the active space and hence the number of final microstates) does.

The set of occupied and virtual MOs whose corresponding SOs are allowed to exchange electrons in Ψ_Ref to form new microstates ψ are called the CI-active MOs or the “active space”. The choice of active space is one of the most crucial, and sometimes difficult, steps in a CI calculation, both computationally and in terms of physical results. Given this importance, the CI-active MOs are usually specified along with the keywords which invoke CI, possibly together with the RECLAS(n,m) keyword and its associated MO permutation data. For example, C.I.(5,8) means “do a CAS-CI using MOs 5,6,7 and 8 as the CI-active MOs”. It is essential that all or none of the members of a degenerate set of MOs be included in the active space. By default, AMPAC will abort if this is not the case. The keywords, CIGAP=n.nnnn and CI-OK can be used to alter the definition of MO degeneracy and to allow the active space to contain an incomplete set of degenerate MOs. By default, all of the MO energies are printed to the AMPAC output file. The keywords VECTORS and ALLVEC can be used to print both the MO energies and AO coefficients to the AMPAC output file for inspection. This information is also present in an AMPAC visualization file so that MOs can be visualized with AMPAC’s GUI. It is important to know the order in which the SCF MOs occur and their corresponding labeling. For a system with M MOs, the MOs are ordered from 1 to M by increasing occupancy, i.e., first doubly-occupied MOs, then partially occupied MOs and finally unoccupied (virtual) MOs. This order usually coincides with increasing MO energy for the entire list from 1 to M, but within each subset of the same occupancy the order always coincides with increasing energy.

For RHF open-shell calculations, the SCF calculations in AMPAC are done using the “Half-Electron” method instead of the ROHF (Restricted Open-Shell Hartree-Fock) method used by others. In the “Half-Electron” method, the usual “spin-less” closed-shell RHF SCF formalism is used to calculate Ψ_SCF, except that instead of N / 2 doubly occupied spatial MOs there are assumed to be N / 2 - n (N even) or N / 2 + 1 - n (N odd) doubly occupied MOs and m MOs with fractional occupancies which sum to n, where n is the number of open-shell electrons . When the OPEN(n,m) keyword is specified, the m open MOs have an equal occupancy of (n/m). When the SCFCI(n,m1,m2,r) keyword is specified, the set of m = m₁ + m₂ open MOs consists of a group of m₁ MOs each with an occupancy of (nm₁)/(m₁+rm₂) and a group of m₂ MOs each with an occupancy of (nrm₂)/(m₁+rm₂). A fractionally occupied MO in the “Half-Electron” method may be thought of as being occupied by two “half-electrons” of opposite spin and with a charge equal to half the occupancy of the MO, e.g., (n / 2m) when OPEN(n,m) is used. This leads to an energy expression which is similar to Roothan’s multiconfiguration open-shell SCF energy expression after spurious coulomb and exchange energies arising from the interaction between “half-electrons” are subtracted out. In AMPAC, however, the energy calculated using the “Half-Electron” method is never used, since it is non-variational, but the corresponding set of SCF MOs are, either in a “minimal” CAS-CI calculation involving all of the partially occupied MOs of Ψ_SCF as the active space if CI is not otherwise invoked, or more generally in any specified type of CI calculation. While the fractionally occupied SOs of Ψ_SCF determine the active space of corresponding MOs, the reference wavefunction Ψ_Ref used from an open-shell RHF calculation is not Ψ_SCF but rather a determinant obtained by filling the SOs of Ψ_SCF in the standard way using the Aufbau principle. It is important to note that, in general, the number of open-shell electrons to assume for the SCF should be specified explicitly using one of the keywords OPEN(n,m), BIRADICAL, EXCITED or SCFCI, otherwise AMPAC will assume the minimum number of open-shell electrons (0 for even-electron systems and 1 for odd-electron systems) for the SCF. The spin-multiplicity keywords (e.g., SINGLET, DOUBLET, TRIPLET, etc.) are not used in RHF until the CI portion of the calculation. Thus, for the oxygen molecule, OPEN(2,2) should be specified even if TRIPLET is also specified.

Given ψ_Ref and a corresponding active space, a definition of which microstates to generate and potentially use for the expansion of the CI eigenstates is necessary. In the “Complete Active Space” method (CAS-CI), specified by C.I.=n or C.I.(n,m) and the default when CI is only implied by OPEN(n,m), all possible microstates which can be generated by permutations of the electrons among the SOs within the active space are potentially used. In the “CI Singles” method (S-CI), specified by SC.I.=n or SC.I.(n.m), all possible singly-excited microstates ψ^ia are potentially used. In the “CI Singles and Doubles” method (SD-CI), specified by SDC.I.=n or SDC.I.(n,m), all possible singly-excited microstates ψ^ia and doubly-excited microstates ψ^ia,jb are potentially used. In the “CI Singles, Doubles and Triples” method (SDT-CI), specified by SDTC.I.=n or SDTC.I.(n,m), all possible singly-excited microstates ψ^ia, doubly-excited microstates ψ^ia,jb and triply-excited microstates ψ^ia,jb,kc are potentially used. The initial set of microstates is referred to here as {I}_MS.

The size of {I}_MS grows very rapidly (combinatorially) as the size of the active space increases, especially when CAS-CI is used. (For a CAS-CI involving 10 electrons and 10 CI-active MOs, the number of possible microstates is over 60000, after spin degeneracies are excluded.) In some cases, all of {I}_MS should be used, if possible. If this is not the case, whether due to resource limitations and / or to avoid “over-correlating” the already partially correlated, semi-empirically calculated ground state energy, then some means of efficiently selecting the most important “final” set of microstates, referred to here as {F}_MS, from {I}_MS is necessary. Typically, only a relatively small “target” set of the possible CI eigenstates, {R}_ES, are of interest. For example, {R}_ES might be composed of the singlet ground CI eigenstate and the first excited singlet and triplet CI eigenstates. {R}_ES can usually be characterized in terms of relatively large contributions from a small subset of “germ” microstates {G}_MS = {G₀, G₁, G₂, ?}_MS, where G ₀≡ ψ_Ref roughly corresponds to the ground CI eigenstate R0, G1 to a first excited CI eigenstate R1, etc. While, much of the information relevant to {R}_ES is included in {G}_MS, the CI eigenstates of {R}_ES constructed from {G}_MS alone would generally have two significant deficiencies. First, there is generally a lack of specific correlation within the set {G}_MS. Second, the excited members {G}_MS are lacking in “repolarization” because the SCF orbitals from which {G}_MS is generated are obtained from a ground state wavefunction optimization. The objective of the microstate selection procedure used in AMPAC to produce {F}_MS is to extract from the enormous list of initial microstates in {I}_MS and not in {G}_MS, the ones which should contribute most to specific correlation and repolarization. This microstate selection consists of four major steps:

At each stage of this microstate selection procedure, the sets of microstates selected are required to preserve spatial degeneracy, i.e., all members of a degenerate set of microstates are kept if there is space available in the target list, or not kept if there is not space available in the target list. This is achieved by simple inspection of the Møller-Plesset zero-order energies, using a degeneracy threshold of 1.0 × 10^-4 eV, which is adjustable by the keyword CIGAP=n,n. Of course, this procedure will not cover the case of an active space containing only a partial set of degenerate MOs. It is important to remember that either all or none of the members of a degenerate set of MOs should be included in the active space.

In AMPAC, the above microstate selection procedure can be partially customized by specifying the parameters J₂, J₃ and J₄ using the keywords CIMAX=J₄, PERTU=J₂ and PERTU(J₂,J₃).

The following keywords will cause a CI calculation to be done following an RHF calculation:

The following keywords will cause at least a minimum CAS-CI calculation to be done following an RHF calculation. These may also be combined with a keyword which explicitly invokes a larger CI calculation:

	Note
	An odd electron system without UHF specified and without an explicit C.I. keyword will implicitly invoke OPEN(1,1), which in turn invokes a minimum CAS-CI calculation.

The CI-active MOs are always at least least initially defined by the keywords which invoke CI, possibly together with the RECLAS(n,m) keyword. The following keywords modify this definition or have an important bearing on it:

Given an initial set of microstates {I}_MS defined by the CI-active MOs, the following keywords help determine the number and nature of the final microstates {F}_MS actually used in the CI eigenstate expansions:

The following keywords determine the spin multiplicity SM, CI root and handling of degeneracies (EPS) for the primary CI eigenstate, as well as the number and kind of secondary CI eigenstates calculated:

The following keywords control what kind of extra information is printed to the AMPAC output file or other result files (CIOUT) following any CI calculation:

The following keywords can be used to calculate some extra molecular properties which are specific to a CI calculation:

Specifying any of the following keywords together with a keyword which either explicitly or implicitly invokes CI is an error, and will result in AMPAC aborting the job.