As Martin has said a practical way to decide if a molecule requires multi-reference treatment is to calculate the unrestricted Hartree Fock natural orbitals.[1] Significant fractional occupancy of a UHF natural orbital indicates that the orbital should be included in the active space.
For a nonorthogonal finite basis set, they are given as
$$\mathbf{S}(\mathbf{D_\alpha+\mathbf{D_\beta}})\mathbf{S}\mathbf{C} = \mathbf{S}\mathbf{C}\mathbf{n},$$
where $\mathbf{S}$ is the overlap matrix, $\mathbf{D}_\alpha$ and $\mathbf{D}_\beta$ are the reduced first order density matrices for the $\alpha$ and $\beta$ spins, respectively, the columns of $\mathbf{C}$ contain the coefficients of the natural orbitals, and the diagonal matrix $\mathbf{n}$ holds the occupation numbers.
Specifically wavefunction with significant fractional occupancy (roughly between 0.02 and 1.98) should constitute the active space for static (non-dynamical) correlation.
For example, this can be performed in Molpro as
***, HF
symmetry,nosym
memory,400,m
orient,noorient
basis=6-31G**
geometry = {
N 0 0 0
O 0 0 1.75
}
{UHF;save,2101.2}
{matrop
load,D,Den
load,S
natorb,Cnat,D
mult,SC,S,Cnat
tran,D_nat,D,SC ! = SC' D SC
prid,D_nat
save,Cnat,2150.2
}
{put,molden,orbitals.gmolden;orbital,2101.2}
{put,molden,check.gmolden;orbital,2150.2}
Molden will automatically print the occupations in the orbital dump (2101.2) but the matrop
can be also used to solve the eigenvalue problem manually
The result for example here is
1.99999995 1.99999988 1.99987846 1.99937413 1.99796035 1.96525179 1.96022720 1.00000000 0.03977280 0.03474821 0.00203965 0.00062587 0.00012154 0.00000012 0.00000005 0.00000000 -0.00000000 -0.00000000 -0.00000000 -0.00000000 -0.00000000 -0.00000000 -0.00000000 -0.00000000 -0.00000000 -0.00000000 0.00000000 0.00000000
Which is in agreement with reference [1].
(If you do plan to do multi-reference however, do never use Hartree-Fock orbitals. They are the worst! Use DFT orbitals as a starting place.)
References:
- Pulay, P.; Hamilton, T. P. UHF natural orbitals for defining and starting MC‐SCF calculations. J. Chem. Phys. 1988, 88 (8), 4926–4933. DOI: 10.1063/1.454704.