Calculation of the Density Matrix

Normal Semiempirical Density Matrix and the Density Matrix from a C.I. calculation

All the semiempirical methods in MOPAC use the same method for constructing the density matrix. The calculation of the RHF density matric is straightforward, for UHF, it's a bit more complicated, and for configuration interaction systems the calculation is quite difficult.

Normal Density Matrix

Given a set of occupied M.O.s, ψ_i=Σ_λc_λiφ_λ, the density matrix, P, is defined as: Ψ_λσ = 2Σ_i^occ c_λic_{σ_i}. If ROUHF is used, the "2" would be replaced by the fractional occupancy.

For UHF, the density matrix is given by the sum of the alpha and beta density matrices: Ψ_λσ = 1σ_i^occ(α) c_{λ_i}^αc_{σ_i}^α + σ_i^occ(β) c_{λ_i}^βc_{σ_i}^β. If fractional occupancy of α UHF M.O.'s is used, the "1" would be replaced by the fractional occupancy.

Density Matrix from a C.I. calculation

When configuration interaction is used, the total density matrix, P, is the sum of the density matrix for the doubly-occupied M.O.s plus the density matrix for the active space M.O.s.

Contributions to the density matrix arising from the M.O.s in the active space.

Given n M.O.s in the active space, and l microstates, Ψ, in each state, so that a State Φ_i can be defined as Φ_i = Σ_λ^l (c_λiΨ_λ), then the contribution to the electron population can be expressed in terms of active-space M.O.s ψ_k and ψ_l as Δ_kl.

Calculating Δ_kl requires evaluating the product Φ_i x Φ_i, i.e., evaluating the set: Σ_λ^l (c_λiΨ_λ) x Σ_σ^l (c_σiΨ_σ). This operation involves evaluating three types of integrals:

(A) Ψ_λ= Ψ_σ

The contribution to the active-space M.O.s ψ_k and ψ_l is Δ_kl= O_kiO_lic_ki x c_li, where O_kiand O_{_li} are 1 (unity) if the M.O. is occupied, zero otherwise.

(B) Except for M.O. ψ_i in Ψ_λ and ψ_j in Ψ_σ, Ψ_λ= Ψ_σ, for example, if ψ_λ has active-space occupancy [000101] and Ψ_σ, has occupancy [000011] so ψ_i in Ψ_λ would be 4 and ψ_j in Ψ_σ would be 5. Both M.O.s ψ_iand ψ_jmust have the same spin.

The contribution to the active-space M.O.s ψ_i and ψ_j is Δ_ij= (-1)^p.c_ki.c_li, where "p" is the number of permutations needed to put ψ_i in Ψ_λ into coincidence with ψ_j in Ψ_σ . In this example, "p" would be zero. If the microstates were [000110] and [000011] the number of permutations would be one.

(C) If more than one difference exists between Ψ_λand Ψ_σ, or if the number of α electrons in the two microstates is different, then the integral is zero.

Using the newly-formed matrix Δ, and the density matrix over M.O.s below the active space, P', the change to the density matrix is given by:

P_λσ = P'_λσ + Σ_iΣ_j c_λiΔ_i,jc_σj

where i and j run over the active-space M.O.s

When degenerate C.I. States are present, Δ is calculated using the weighted average of all components of the degenerate manifold. This is necessary in order to preserve symmetry.