7.1 Complete Active Space Self-Consistent Field, CASSCF

In a CASSCF wavefunction the occupied orbital space is divided into a set of inactive or closed-shell orbitals and a set of active orbitals. All inactive orbitals are doubly occupied in each Slater determinant. On the other hand, the active orbitals have varying occupations, and all possible Slater determinants (or CSFs) are taken into account which can be generated by distributing the $N_{act}=N_{el}-2 m_{closed}$ electrons in all possible ways among the active orbitals, where $m_{closed}$ is the number of closed-shell (inactive) orbitals, and $N_{el}$ is the total number of electrons. Thus, it corresponds to a full CI in the active space.

The CASSCF program is invoked using the

casscf

Aliases for this command are mcscf or multi. This command is optionally followed by further input defining the wavefunction. The inactive orbital space is defined using the closed directive.

closed, $n_1,n_2,\ldots,n_d$

where $n_i$ is the number of doubly occupied (inactive) orbitals in irreducible representation $i$. The total number of occupied orbitals is specified using the occ directive, as in Hartree-Fock.

occ, $m_1,m_2,\ldots,m_d$

where $m_i \ge n_i$. The number of active orbitals in irreducible representations $i$ is then $m_i - n_i$. Note that the inactive orbitals are always the first in each symmetry, i.e., inactive and active spaces cannot be mixed. The number of electrons, as well as the symmetry of the wavefunction and the total spin is specified using the wf directive, as explained already for open-shell HF:

wf, $N_{el},isym,ms2$

where $isym$ is the symmetry of the total wavefunction (the direct product of the symmetries of all occupied spin orbitals), and $ms2=2S$ defines the spin (0=singlet, 1=doublet, 2=triplet etc).

From the above it follows that the number of active electrons is

$$N_{act} = N_{el} - 2 \sum_i n_i$$ (1)

By default, the inactive space consists of all inner-shell orbitals, and the active space of all valence orbitals which are obtained from the atomic valence orbitals (full valence active space). The default number of electrons equals the sum of nuclear charges, the default wavefunction symmetry is 1 and singlet. The default starting guess for the orbitals is taken from the most recent orbital optimization, e.g., Hartree-Fock. The simplest input for a CASSCF calculation for formaldehyde is therefore

***,formaldehyde
print,orbitals,civector   !this is optional: print the occupied orbitals
                          !and the CI vector
                          !by default, only coefficients larger than 0.05
                          !are printed.
geometry                  !define the nuclear coordinates
ang
C
O  , C , rco
H1 , C , rch , O , hco
H2 , C , rch , O , hco , H1 , 180
end

rco=1.182 Ang
rch=1.102 Ang
hco=122.1789 Degree

basis=vdz              !Select basis set
hf                     !Perform HF calculation
casscf                 !Perform CASSCF calculation,
                       !using the HF orbitals as starting guess.

In this case, the carbon and oxygen $1s$ orbitals are inactive, and the carbon and oxygen $2s$, $2p$ as well as the hydrogen $1s$ orbitals are active. This corresponds to the following explicit input, which could be given after the casscf directive:

{casscf
closed,2               !2 inactive orbitals in Symmetry 1 (a1)
occ,7,2,3              !7a1, 2b1, 3b2 occupied orbitals
wf,16,1,0}             !16 electrons, Symmetry 1 (A1), singlet

Thus, there are five $a_1$, two $b_1$, and three $b_2$ active orbitals. This yields 3644 CSFs or 11148 Slater determinants. Note that the wf directive must be given after the occ and closed ones. A shorter expansion results if the $2s$ orbital of oxygen is made inactive. In this case the input would be

{casscf
closed,3               !3 inactive orbitals in Symmetry 1 (a1)
occ,7,2,3              !7a1, 2b1, 3b2 occupied orbitals
wf,16,1,0}             !16 electrons, Symmetry 1 (A1), singlet

and now only 1408 CSFs or 4036 Slater determinants are generated.