Given a set of occupied M.O.s,
ψ_{i }=Σ_{λ}c_{λi}φ_{λ},
the density matrix, P, is defined as:
Ψ_{λσ}
= 2Σ_{i}^{occ}
c_{λi}c_{σi}.
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_{λi}^{α}c_{σi}^{α} +
σ_{i}^{occ(β)}
c_{λi}^{β}c_{σi}^{β}.
If fractional occupancy of α
UHF M.O.'s is used, the "1"
would be replaced by the fractional occupancy.

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.

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_{λi}Ψ_{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_{λi}Ψ_{λ})
x Σ_{σ}^{l}
(c_{σi}Ψ_{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_{ki}O_{li }c_{ki}
x c_{li}, where O_{ki }and 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_{i }and ψ_{j }must 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_{λ}and 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_{i}Σ_{j}
c_{λi}Δ_{i,j}c_{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.