I have two questions about the orbitals that we get after the CASSCF calculations.

First, what are the CASSCF orbitals that we get from these calculations? I mean after optimization, in the CASSCF wavefunction we have various configurations (CSFs) with different weights and in each of these CSFs we have MO orbitals with different MO coefficients. So my question is why CASSCF software packages (in particular Gaussian) display MO orbitals after the CASSCF calculations? These orbitals belong to which CSFs? The CSF with the highest weight/coefficient?

Second, if the orbitals after the CASSCF calculations are valid, should they approximately the same as the initial orbitals (guess orbitals)? If they are not, it means that the initial guess should be changed?

  • $\begingroup$ Could you post an example input to generate this? MOs are not printed by default I think for any package, just the density matrix for each state. $\endgroup$ – pentavalentcarbon Jun 23 '18 at 23:33
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    $\begingroup$ The MOs will not be printed unless you request them. My question is not why they are printed (upon request), but how it is possible that we have orbitals when we have many configurations in the CASSCF wavefunction. Let’s say we do a CASSCF calculations. After SCF convergence we get 20 different configurations each with different MO coefficients. So how can that be displayed? $\endgroup$ – Gilli Jun 24 '18 at 0:46

The orbitals in CASSCF (or MCSCF in general) are optimized for a specific electronic state (= linear combination of CSFs) or a user-defined (weighted) state-average.

If they would belong just to the largest CSF, they would be equal to the HF orbitals.

Also, there is no unique set of MCSCF orbitals, but they depend on how you set up the MCSCF calculation: if you are interested in multiple electronic states, you can optimize orbitals for each state individually by separate calculations, or find one set of orbitals describing all states in a balanced way for a state-average (yielding larger energies).

The MCSCF orbitals usually look very similar to the canonical orbitals, and have similar eigenvalues. Also occupation numbers should be somewhat similar, although they are no longer restricted to being integer values. Basically, the better the HF approximation for a specific system, the larger the HF configuration in the CI vector, and the closer should the MCSCF orbitals be to the HF orbitals.

If you have large differences, then you probably have a multi-reference problem. The whole point of MCSCF calculations is to find improved orbitals for such cases.

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    $\begingroup$ My question is just about the ground state. Let's say we select four orbitals to be in the active space and we do a CAS(4,4) calculations. After the calculation we get 21 CSFs each with different CI coefficients. So in total I think we should have 84 orbitals (21 CSFs * 4 orbitals). However, the visualization package just show me four orbitals instead of 84. So my question is why only four orbitals are shown and what are these four orbitals? Are they just belong to one CSF? $\endgroup$ – Gilli Jun 24 '18 at 16:14
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    $\begingroup$ No, please read my first sentence again. The orbitals do NOT belong to any CSF. They belong to a certain linear combination of CSFs: the electronic (ground) state. Each CSF is a certain vector in the CI space, the ground state is a vector in CI space as well, just developed in terms of CSFs. The orbitals may be optimized for any arbitrary CI vector, but you want them to be optimized for the ground state (not CSF). $\endgroup$ – Feodoran Jun 24 '18 at 21:31

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