Since the number of CSFs or Slater determinants and thus the computational
cost quickly increases with the number of active orbitals, it may
be desirable to use a smaller set of CSFs. One way to make a selection
is to restrict the number of electrons in certain subspaces. One could
for instance allow only single and double excitations from some
strongly-occupied subset of active orbitals,
or restrict the number of
electrons to at most 2 in another subset of active orbitals. In general,
such restrictions can be defined using the `restrict` directive:

`restrict`,*min, max, orbital list*

where *min* and *max* are the minimum and maximum number of electrons
in the given orbital subspace, as specified in the *orbital list*. Each
orbital is given in the form *number.symmetry*, e.g. 3.2 for the third
orbital in symmetry 2. The `restrict` directives (several can follow each
other) must be given after the `wf` card. As an example, consider
the formaldehyde example again, and assume that
only single and double excitations are allowed into the orbitals 6.1,
7.1, 2.2, 3.3,
which are unoccupied in the HF wavefunction. Then the input would be

{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 restrict,0,2, 6.1,7.1,2.2,3.3} !max 2 electrons in the given orbital list

One could further allow only double excitations from the orbitals 3.1, 4.1, but in this case this has no effect since no other excitations are possible anyway. In order to demonstrate such a case, we increase the number of occupied orbitals in symmetry 1 to 8, and remove the restriction for orbital 6.1.

{casscf closed,2 !2 inactive orbitals in Symmetry 1 (a1) occ,8,2,3 !7a1, 2b1, 3b2 occupied orbitals wf,16,1,0 !16 electrons, Symmetry 1 (A1), singlet restrict,0,2, 7.1,8.1,2.2,3.3 !max 2 electrons in the given orbital list restrict,2,4, 3.1,4.1} !at least 2 and max 4 electrons in the !given orbitals

It is found that this calculation is not convergent. The reason is that some orbital rotations are almost redundant with single excitations, i.e., the effect of an orbital transformation between the strongly and weakly occupied spaces can be expressed to second order by the single and double excitations. This makes the optimization problem very ill conditioned. This problem can be removed by eliminating the single excitations from / into the restricted orbital space as follows:

{casscf closed,2 !2 inactive orbitals in Symmetry 1 (a1) occ,8,2,3 !7a1, 2b1, 3b2 occupied orbitals wf,16,1,0 !16 electrons, Symmetry 1 (A1), singlet restrict,0,2, 7.1,8.1,2.2,3.3 !max 2 electrons in the given orbital list restrict,-1,0 7.1,8.1,2.2,3.3 !1 electron in given orbital space not !allowed (no singles) restrict,2,4, 3.1,4.1 !at least 2 and max 4 electrons in the !given orbitals restrict,-3,0,3.1,4.1} !3 electrons in given orbital space not !allowed (no singles)

and now the calculation converges smoothly.

Converging MCSCF calculations can sometimes be tricky and difficult. Generally, CASSCF calculations are easier to converge than restricted calculations, but even in CASSCF calculations problems can occur.

The reasons for slow or no convergence could be one or more of the following.

- -
- Near redundancies between orbital and CI coefficient changes as shown above.

Remedy: eliminate single excitations - -
- Two or more weakly occupied orbitals have almost the same effect on the energy, but
the active space allows the inclusion of only one of them.

Remedy: increase or reduce active space (`occ`card). - -
- An active orbital has an occupation number very close to two. The program may have
difficulties to decide which orbital is inactive.

Remedy: increase inactive space - -
- Correlation of an active orbital gives a smaller energy lowering than would be
obtained by correlating an inactive orbital. The program tries to swap active
and inactive orbitals.

Remedy: reduce (or possibly increase) active space. - -
- Another state of the same symmetry is energetically very close (nearly degenerate). The
program might oscillate between the states (root flipping).

Remedy: include all nearly degenerate states into the calculation, and optimize their average energy (see below).

As a rule of thumb, it can be said that if a CASSCF calculation does not converge or converges very slowly, the active or inactive space is not chosen well.

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molpro@molpro.net 2017-12-16