[molpro-user] CASSCF energy is significantly changed by state-averaging

Lorenzo Lodi l.lodi at ucl.ac.uk
Fri Nov 22 13:38:43 GMT 2013

On 13/11/13 15:58, Qinghua Cao wrote:
> Hey, everyone,
> I'm working on the vertical transition energy in two ways:
> 1. a state-averaged CASSCF calculation of all states in one
> calculation (i also tried to include different numbers of states, and
> the result may vary significantly with the number of state!!)
> 2. CASSCF calculations of ground state and excited states individually
> it turns out that the transition energies are very different from the
> above ways, and they could differ as large as 1 eV. Moreover,
> including 10 states and 20 states in a state-averaged CASSCF
> calculation also lead to very different results.
> I don't know whether this ever occurs to any of you. If so, are there
> any reasons or explanations for this? Which is the most reliable way
> to calculate the energy?
> Thanks in advance!
> Qinghua

Dear Qinghua,
as no one more competent than me has yet commented on your question,
I'll give you my opinion on the matter.
It is to be expected that vertical excitation energies of state-averaged
CASSCF differ depending on the number of states included (and on the
active space used); that said, a difference of 1 eV looks indeed quite
large, but not knowing which system you are studying it is difficult to
comment on that. I also think it is impossible to blindly say which
strategy gives the `best' (=closest to experiment) excitation energies.

Leaving aside numerical problems of incorrect convergence to local
minima, the reason for these difference is due to the orbital
optimization. It seems to be the case that for your system and for the
active space you are using (for example, the `full valence' one used by
molpro by default) the electronic states you are considering have rather
different optimal orbitals.

As an idealized example, suppose there are three electronic states, a
ground state G and two excited ones A and B, and that when you do CASSCF
for each state one state at a time you get orbitals O1 for G, again O1
for A and O2 for B. In other words G and A have the same optimal CASSCF
orbitals while B's optimal orbitals are different.
When you do a state-average CASSCF calculation with all three states you
get orbitals which are intermediate between O1 and O2 (closer to O1 if
you use standard weights as two states `want' O1 and only one O2); the
absolute energies of all states will be increased with respect to the
one-at-a-time values, but presumably the energy of B will be increased
more than G and A because the intermediate orbitals are less optimal for
that state.
If you do a calculation with G+B the orbitals will be `half way' between
O1 and O2, and with respect to the three-state-averaged calculation G
will be higher in energy and B lower, and as a consequence E(B)-E(A)
[G+A+B state-averaged] >  E(B)-E(A) [G+B state-averaged].

In your case where you have 20 states (each with different optimal
orbitals) the situation is more complicated but the idea is the same. If
the optimal orbitals are very different for different states you'll see
large differences depending on which states are included in the
A possibles, straightforward suggestion is to try a different reference
space (e.g., larger) to see if it reduces the differences between
state-averaged and two-states-at-a-time excitation energies.
On a more complicated level one could also try to group together states
which have similar optimal orbitals and get the relative energies within
each group, and then carefully perform two-state-averaged calculations
to link the networks (I don't know if this strategy has been used in the

In any case one should not expect CASSCF energies to give very accurate
(e.g., accurate to ~0.25 eV or so) excitation energies because of the
lack of dynamical electron correlation.  You should also try a
post-CASSCF method (RS2, MRCI, CIPT2) and see what difference it makes
to your results.

I hope it helps.

Kind regards,
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