In the RHF formalism a MECI calculation is performed.
In general, `ESR` can be used for *any* system for which M.
If the C.I. calculation
results in many states, then the spin density for the state requested
and the next few states will be printed. For example, if benzene cation, D_{6h},
is calculated, using `ESR OPEN(3,2)`, then the spin density for
the two degenerate components of E_{1g} will be printed. For this system, the
total spin density on any given atomic orbital or any atom is given
by the *average* of the spin densities for the two components. For
benzene^{+}, this would be 1/6 electrons.

Thus, for example, for ethylene, `ESR TRIPLET
C.I.=2` would
give meaningful results, as would
`ESR MS=1 C.I.=2`. However, `ESR
ROOT=2 C.I.=2` would not, as this
would be used to calculate the spin density arising from the M_{S} = 0component of a triplet
state, which will have a zero spin density.

Spin density for state **Ψ**_{j},
calculated as spin, S_{A}, on atom A, is given in terms of contributions
from the M.O.s of the active space *ψ*_{i}, *
ψ*_{i} = Σ_{λi}*φ*
as:

S_{A} = Σ_{λ}^{A}*φ*_{λ}_{i}^{2}Σ_{i}S_{i},

where the S_{i} are the contributions of
spin from each M.O., expressed as a sum over the microstates of the C.I.,
**Ψ**_{j}
= Σc_{kj}Ψ_{j},
:

S_{i} = Σ_{k}(O_{i}^{α}^{k}-O_{i}^{β}^{k})c_{kj}_{,}

Where O_{i}^{αj}
is the alpha or "spin up" occupancy of M.O. *ψ*_{i},
in microstate Ψj.
Although φ_{λ}_{i}^{2
}is obligate positive, S_{i}
can be positive or negative, therefore S_{A} can be positive or
negative, although it is unlikely to be very negative, and the sum over all
atoms must equal 2*M*_{S}(Z)

If the keywords `OPEN` and `C.I.=` are both
absent, then only a single state is
calculated. The spin density is then calculated from the state
function. In order to have spin density on the hydrogens in,
for example, the phenoxy radical, several states should be mixed.