## 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, ψi λcλiφλ, the density matrix, P, is defined as:  Ψλσ = iocc cλicσi.  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: Ψλσ = 1σiocc(α) cλiαcσiα + σiocc(β) cλiβcσiβIf 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 Φi can be defined as Φi = Σλl (cλiΨλ), then the contribution to the electron population can be expressed in terms of active-space M.O.s ψk  and ψl  as Δkl.

Calculating Δkl requires evaluating the product Φi x Φi, i.e., evaluating the set: Σλl (cλiΨλ) x Σσl (cσiΨσ). This operation involves evaluating three types of integrals:

(A) Ψλ= Ψσ

The contribution to the active-space M.O.s ψk  and ψl is Δkl = OkiOli cki x cli, where Oki and Oli are 1 (unity) if the M.O. is occupied, zero otherwise.

(B) Except for M.O. ψ in Ψλ and ψ in Ψσ,  Ψλ= Ψσ, for example, if ψλ has active-space occupancy [000101] and Ψσ, has occupancy [000011] so  ψ in Ψλ  would be 4 and ψ in Ψσ would be 5.  Both M.O.s ψand ψmust have the same spin.

The contribution to the active-space M.O.s ψi  and ψj is Δij = (-1)p.cki.cli, where "p" is the number of permutations needed to put ψ in Ψλ  into coincidence with ψ in Ψσ .  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 Ψλand Ψσ, 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:

Pλσ = P'λσ + ΣiΣj cλiΔi,jcσj

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.