State-averaged MCSCF#

CASCI response#

As for truncated CI, the exact state response formulas also apply to CASCI and other active space CIs. The method can thus create all excited states that arise from excitations within the active space. This can also be used as an approximation for a MCSCF response, neglecting the orbital part of the response.

Let us try it on our usual water case. We’ll use an active space including the 2 pairs of O-H \(\sigma\) and \(\sigma^*\) (4 electrons in 4 orbitals). However, since the main excitation also involves the oxygen lone pair, we add this orbital too, for a total CAS(6,5).

First, we try the intuitive concept of first optimizing the CASSCF wavefunction for the ground state, and then applying CASCI response with the 5 states.

import veloxchem as vlx
import multipsi as mtp
* Warning * Environment variable OMP_NUM_THREADS not set.
* Warning * Setting OMP_NUM_THREADS to 8.
* Warning * Setting MKL_THREADING_LAYER to "GNU".
# CAS(6,5) calculation of water

mol_str = """
O       0.0000000000     0.0000000000     0.1178336003
H      -0.7595754146    -0.0000000000    -0.4713344012
H       0.7595754146     0.0000000000    -0.4713344012
"""

molecule = vlx.Molecule.read_str(mol_str, units='angstrom')
basis = vlx.MolecularBasis.read(molecule, "6-31g")

scf_drv = vlx.ScfRestrictedDriver()
scf_drv.compute(molecule, basis)

space=mtp.OrbSpace(molecule,scf_drv.mol_orbs)
space.CAS(6,5) #3 O_2p and 2 H_1s
# CASSCF calculation
mcscf_drv=mtp.McscfDriver()
mcscf_drv.compute(molecule,basis,space)
                                                                                                                          
                                            Self Consistent Field Driver Setup                                            
                                           ====================================                                           
                                                                                                                          
                   Wave Function Model             : Spin-Restricted Hartree-Fock                                         
                   Initial Guess Model             : Superposition of Atomic Densities                                    
                   Convergence Accelerator         : Two Level Direct Inversion of Iterative Subspace                     
                   Max. Number of Iterations       : 50                                                                   
                   Max. Number of Error Vectors    : 10                                                                   
                   Convergence Threshold           : 1.0e-06                                                              
                   ERI Screening Scheme            : Cauchy Schwarz + Density                                             
                   ERI Screening Mode              : Dynamic                                                              
                   ERI Screening Threshold         : 1.0e-12                                                              
                   Linear Dependence Threshold     : 1.0e-06                                                              
                                                                                                                          
* Info * Nuclear repulsion energy: 9.1561447194 a.u.                                                                      
                                                                                                                          
* Info * Overlap matrix computed in 0.00 sec.                                                                             
                                                                                                                          
* Info * Kinetic energy matrix computed in 0.00 sec.                                                                      
                                                                                                                          
* Info * Nuclear potential matrix computed in 0.00 sec.                                                                   
                                                                                                                          
* Info * Orthogonalization matrix computed in 0.03 sec.                                                                   
                                                                                                                          
* Info * SAD initial guess computed in 0.00 sec.                                                                          
                                                                                                                          
* Info * Starting Reduced Basis SCF calculation...                                                                        
* Info * ...done. SCF energy in reduced basis set: -75.983870205311 a.u. Time: 0.07 sec.                                  
                                                                                                                          
* Info * Overlap matrix computed in 0.00 sec.                                                                             
                                                                                                                          
* Info * Kinetic energy matrix computed in 0.00 sec.                                                                      
                                                                                                                          
* Info * Nuclear potential matrix computed in 0.00 sec.                                                                   
                                                                                                                          
* Info * Orthogonalization matrix computed in 0.00 sec.                                                                   
                                                                                                                          
                                                                                                                          
               Iter. | Hartree-Fock Energy | Energy Change | Gradient Norm | Max. Gradient | Density Change               
               --------------------------------------------------------------------------------------------               
                  1       -75.983870373952    0.0000000000      0.00006826      0.00001638      0.00000000                
                  2       -75.983870375702   -0.0000000017      0.00002636      0.00000475      0.00006304                
                  3       -75.983870375765   -0.0000000001      0.00000396      0.00000061      0.00000524                
                  4       -75.983870375769   -0.0000000000      0.00000029      0.00000008      0.00000296                
                                                                                                                          
               *** SCF converged in 4 iterations. Time: 0.02 sec.                                                         
                                                                                                                          
               Spin-Restricted Hartree-Fock:                                                                              
               -----------------------------                                                                              
               Total Energy                       :      -75.9838703758 a.u.                                              
               Electronic Energy                  :      -85.1400150952 a.u.                                              
               Nuclear Repulsion Energy           :        9.1561447194 a.u.                                              
               ------------------------------------                                                                       
               Gradient Norm                      :        0.0000002874 a.u.                                              
                                                                                                                          
                                                                                                                          
               Ground State Information                                                                                   
               ------------------------                                                                                   
               Charge of Molecule            :  0.0                                                                       
               Multiplicity (2S+1)           :  1.0                                                                       
               Magnetic Quantum Number (M_S) :  0.0                                                                       
                                                                                                                          
                                                                                                                          
                                                 Spin Restricted Orbitals                                                 
                                                 ------------------------                                                 
                                                                                                                          
               Molecular Orbital No.   1:                                                                                 
               --------------------------                                                                                 
               Occupation: 2.000 Energy:  -20.56128 a.u.                                                                  
               (   1 O   1s  :    -1.00)                                                                                  
                                                                                                                          
               Molecular Orbital No.   2:                                                                                 
               --------------------------                                                                                 
               Occupation: 2.000 Energy:   -1.35434 a.u.                                                                  
               (   1 O   1s  :     0.21) (   1 O   2s  :    -0.47) (   1 O   3s  :    -0.48)                              
                                                                                                                          
               Molecular Orbital No.   3:                                                                                 
               --------------------------                                                                                 
               Occupation: 2.000 Energy:   -0.70774 a.u.                                                                  
               (   1 O   1p+1:     0.51) (   1 O   2p+1:     0.27) (   2 H   1s  :    -0.26)                              
               (   3 H   1s  :     0.26)                                                                                  
                                                                                                                          
               Molecular Orbital No.   4:                                                                                 
               --------------------------                                                                                 
               Occupation: 2.000 Energy:   -0.56024 a.u.                                                                  
               (   1 O   2s  :    -0.18) (   1 O   3s  :    -0.31) (   1 O   1p0 :    -0.55)                              
               (   1 O   2p0 :    -0.40)                                                                                  
                                                                                                                          
               Molecular Orbital No.   5:                                                                                 
               --------------------------                                                                                 
               Occupation: 2.000 Energy:   -0.50122 a.u.                                                                  
               (   1 O   1p-1:    -0.64) (   1 O   2p-1:    -0.51)                                                        
                                                                                                                          
               Molecular Orbital No.   6:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    0.20280 a.u.                                                                  
               (   1 O   3s  :    -1.17) (   1 O   1p0 :     0.23) (   1 O   2p0 :     0.48)                              
               (   2 H   2s  :     0.99) (   3 H   2s  :     0.99)                                                        
                                                                                                                          
               Molecular Orbital No.   7:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    0.29874 a.u.                                                                  
               (   1 O   1p+1:    -0.34) (   1 O   2p+1:    -0.82) (   2 H   2s  :    -1.38)                              
               (   3 H   2s  :     1.38)                                                                                  
                                                                                                                          
               Molecular Orbital No.   8:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    1.05548 a.u.                                                                  
               (   1 O   2p+1:     0.74) (   2 H   1s  :     0.97) (   2 H   2s  :    -0.50)                              
               (   3 H   1s  :    -0.97) (   3 H   2s  :     0.50)                                                        
                                                                                                                          
               Molecular Orbital No.   9:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    1.16442 a.u.                                                                  
               (   1 O   1p-1:    -0.96) (   1 O   2p-1:     1.04)                                                        
                                                                                                                          
               Molecular Orbital No.  10:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    1.18237 a.u.                                                                  
               (   1 O   2s  :     0.26) (   1 O   3s  :    -0.18) (   1 O   1p0 :    -0.67)                              
               (   1 O   2p0 :     0.22) (   2 H   1s  :    -0.84) (   2 H   2s  :     0.55)                              
               (   3 H   1s  :    -0.84) (   3 H   2s  :     0.55)                                                        
                                                                                                                          
          Active space definition:
          ------------------------
Number of inactive (occupied) orbitals: 2
Number of active orbitals:              5
Number of virtual orbitals:             6

    This is a CASSCF wavefunction: CAS(6,5)

          CI expansion:
          -------------
Number of determinants:      55


                                                                                                                          
        MCSCF Iterations
        ----------------
                                                                                                                          
     Iter. | Average Energy | E. Change | Grad. Norm | CI Iter. |   Time
     ---------------------------------------------------------------------
        1     -75.991312026     0.0e+00      3.6e-02          1   0:00:00
        2     -76.006166338    -1.5e-02      6.3e-02          1   0:00:00
        3     -76.032227092    -2.6e-02      3.2e-02          1   0:00:00
        4     -76.034907536    -2.7e-03      3.7e-02          1   0:00:00
        5     -76.037509034    -2.6e-03      1.5e-02          1   0:00:00
        6     -76.037708221    -2.0e-04      7.3e-03          1   0:00:00
        7     -76.037802925    -9.5e-05      4.9e-03          1   0:00:00
        8     -76.037839963    -3.7e-05      3.1e-03          1   0:00:00
        9     -76.037842633    -2.7e-06      1.4e-03          1   0:00:00
       10     -76.037843292    -6.6e-07      3.9e-04          1   0:00:00
       11     -76.037843330    -3.8e-08      2.0e-04          1   0:00:00
       12     -76.037843340    -9.4e-09      5.7e-05          1   0:00:00
                                                                                                                          
** Convergence reached in 12 iterations
                                                                                                                          
        Final results
        -------------
                                                                                                                          
* State 1
- Energy: -76.03784333970789
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.99931 1.97806 1.97495 0.02400 0.02368
                                                                                                                          
                                                 Spin Restricted Orbitals                                                 
                                                 ------------------------                                                 
                                                                                                                          
               Molecular Orbital No.   1:                                                                                 
               --------------------------                                                                                 
               Occupation: 2.000 Energy:  -20.55485 a.u.                                                                  
               (   1 O   1s  :    -1.00)                                                                                  
                                                                                                                          
               Molecular Orbital No.   2:                                                                                 
               --------------------------                                                                                 
               Occupation: 2.000 Energy:   -1.07299 a.u.                                                                  
               (   1 O   1s  :     0.22) (   1 O   2s  :    -0.48) (   1 O   3s  :    -0.57)                              
               (   1 O   1p0 :    -0.24) (   1 O   2p0 :    -0.19)                                                        
                                                                                                                          
               Molecular Orbital No.   3:                                                                                 
               --------------------------                                                                                 
               Occupation: 1.999 Energy:   -0.50217 a.u.                                                                  
               (   1 O   1p-1:    -0.64) (   1 O   2p-1:    -0.51)                                                        
                                                                                                                          
               Molecular Orbital No.   4:                                                                                 
               --------------------------                                                                                 
               Occupation: 1.978 Energy:   -0.83593 a.u.                                                                  
               (   1 O   1p0 :     0.51) (   1 O   2p0 :     0.35) (   2 H   1s  :    -0.21)                              
               (   3 H   1s  :    -0.21)                                                                                  
                                                                                                                          
               Molecular Orbital No.   5:                                                                                 
               --------------------------                                                                                 
               Occupation: 1.975 Energy:   -0.70221 a.u.                                                                  
               (   1 O   1p+1:     0.51) (   1 O   2p+1:     0.25) (   2 H   1s  :    -0.27)                              
               (   3 H   1s  :     0.27)                                                                                  
                                                                                                                          
               Molecular Orbital No.   6:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.024 Energy:    0.75288 a.u.                                                                  
               (   1 O   1p+1:     0.87) (   2 H   1s  :     0.46) (   2 H   2s  :     0.30)                              
               (   3 H   1s  :    -0.46) (   3 H   2s  :    -0.30)                                                        
                                                                                                                          
               Molecular Orbital No.   7:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.024 Energy:    0.73040 a.u.                                                                  
               (   1 O   2s  :    -0.50) (   1 O   3s  :    -0.17) (   1 O   1p0 :     0.77)                              
               (   1 O   2p0 :    -0.18) (   2 H   1s  :     0.43) (   2 H   2s  :     0.20)                              
               (   3 H   1s  :     0.43) (   3 H   2s  :     0.20)                                                        
                                                                                                                          
               Molecular Orbital No.   8:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    0.68041 a.u.                                                                  
               (   1 O   2s  :    -0.19) (   1 O   3s  :     1.21) (   1 O   1p0 :     0.41)                              
               (   1 O   2p0 :    -0.76) (   2 H   1s  :     0.40) (   2 H   2s  :    -1.13)                              
               (   3 H   1s  :     0.40) (   3 H   2s  :    -1.13)                                                        
                                                                                                                          
               Molecular Orbital No.   9:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    0.78480 a.u.                                                                  
               (   1 O   1p+1:     0.28) (   1 O   2p+1:    -0.73) (   2 H   1s  :     0.62)                              
               (   2 H   2s  :    -1.60) (   3 H   1s  :    -0.62) (   3 H   2s  :     1.60)                              
                                                                                                                          
               Molecular Orbital No.  10:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    1.16356 a.u.                                                                  
               (   1 O   1p-1:     0.96) (   1 O   2p-1:    -1.04)                                                        
                                                                                                                          
                                                                                                                          
Total MCSCF time: 00:00:00
nstates=5

# CASCI excitation energies
cidrv=mtp.CIDriver()
cidrv.compute(molecule,basis,space,nstates)

# Transition properties
SI=mtp.InterState()
DipOsc=SI.diposc(molecule,basis,cidrv.vecs)
          Active space definition:
          ------------------------
Number of inactive (occupied) orbitals: 2
Number of active orbitals:              5
Number of virtual orbitals:             6

    This is a CASSCF wavefunction: CAS(6,5)

          CI expansion:
          -------------
Number of determinants:      55


                                                                                                                          
        CI Iterations
        -------------
                                                                                                                          
     Iter. | Average Energy | E. Change | Grad. Norm |   Time
     ----------------------------------------------------------
        1     -75.358106178     0.0e+00      2.7e-25   0:00:00
                                                                                                                          
** Convergence reached in 1 iterations
                                                                                                                          
        Final results
        -------------
                                                                                                                          
* State 1
- Energy: -76.03784333970788
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.99931 1.97806 1.97495 0.02400 0.02368
                                                                                                                          
* State 2
- Energy: -75.35358040236133
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.00000 1.97467 1.98958 1.00239 0.03337
                                                                                                                          
* State 3
- Energy: -75.34949654342167
- S^2   : -0.00  (multiplicity = 1.0 )
- Natural orbitals
1.00000 1.99202 1.97078 0.03398 1.00322
                                                                                                                          
* State 4
- Energy: -75.05973100146419
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.99901 1.91811 1.06476 0.08436 0.93377
                                                                                                                          
* State 5
- Energy: -74.98987960347861
- S^2   : -0.00  (multiplicity = 1.0 )
- Natural orbitals
1.97829 1.60769 1.38512 0.60865 0.42026
                                                                                                                          
List of oscillator strengths greather than 1e-10
                                                                                                                          
  From     to       Energy (eV)    Oscillator strength (length and velocity)
     1       3       18.73087         4.719252e-04    4.241774e-03
     1       4       26.61579         3.182229e-01    1.061922e-01
     1       5       28.51655         2.437476e-02    2.439845e-03
# Results computed in CI response
FCI_excitation_energies=[8.44992, 10.68451, 10.97654, 13.37692]
CIS_excitation_energies=[9.38059, 11.30739, 11.83172, 13.90237]
au2ev = 27.211386
print("Full CI Energies")
print(FCI_excitation_energies)
print("CIS Energies")
print(CIS_excitation_energies)
print("CASCI Energies")
print(au2ev*DipOsc['energies'])
Full CI Energies
[8.44992, 10.68451, 10.97654, 13.37692]
CIS Energies
[9.38059, 11.30739, 11.83172, 13.90237]
CASCI Energies
[18.61974291 18.73087038 26.61579239 28.51654574]

As we can see, the result is very disappointing, very far from the full CI result, and even from the CIS result. The reason for that is the orbitals. The orbitals were optimized for the ground state, and not for the excited states. Like for truncated CI, we thus have an imbalance between the description of the ground and excited states leading to (significantly) too high excitation energies. In particular, in this case, the \(\sigma^*\) orbitals are optimal to describe correlation in the O-H bonds, but not for the electronic excitation.

State-averaging#

Ideally, we would want each state to have its own orbitals, which a full MCSCF response would provide. However, there is a way to improve the result without including the orbital response. The concept is to use only one orbital set, but to make these orbitals a “compromise” between the needs of the ground state and that of the excited states. The description of each state would be worse but the hope is that this error would be consistent across all states and mostly cancel out in the excitation energies.

The way to do this is to construct a “state-average” MCSCF (SA-MCSCF), that is optimize the orbitals to minimize the average energy of all states. Doing this requires only a trivial modification of the code, namely the CI part of the MCSCF includes all states (here 5) and then the MCSCF energy is replaced by the average energy of all states and the CI densities are replaced by the average density of all states.

Let’s see it in practice:

# CASSCF calculation
mcscf_drv.compute(molecule,basis,space,nstates) #5 states included already in the MCSCF.

# Transition properties
DipOsc=SI.diposc(molecule,basis,mcscf_drv.CIVecs)
          Active space definition:
          ------------------------
Number of inactive (occupied) orbitals: 2
Number of active orbitals:              5
Number of virtual orbitals:             6

    This is a CASSCF wavefunction: CAS(6,5)

          CI expansion:
          -------------
Number of determinants:      55


                                                                                                                          
        MCSCF Iterations
        ----------------
                                                                                                                          
     Iter. | Average Energy | E. Change | Grad. Norm | CI Iter. |   Time
     ---------------------------------------------------------------------
        1     -75.358106178     0.0e+00      3.2e-01          1   0:00:00
        2     -75.521955636    -1.6e-01      2.9e-01          1   0:00:00
        3     -75.654098795    -1.3e-01      2.0e-01          1   0:00:00
        4     -75.678346762    -2.4e-02      9.0e-02          1   0:00:00
        5     -75.684567371    -6.2e-03      7.8e-02          1   0:00:00
        6     -75.685966117    -1.4e-03      1.8e-02          1   0:00:00
        7     -75.686043611    -7.7e-05      3.7e-03          1   0:00:00
        8     -75.686051189    -7.6e-06      8.9e-04          1   0:00:00
        9     -75.686051430    -2.4e-07      2.3e-04          1   0:00:00
       10     -75.686051445    -1.5e-08      2.9e-05          1   0:00:00
       11     -75.686051445    -2.7e-10      7.9e-06          1   0:00:00
                                                                                                                          
** Convergence reached in 11 iterations
                                                                                                                          
        Final results
        -------------
                                                                                                                          
* State 1
- Energy: -75.98214412374172
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.99958 1.99145 1.98461 0.01560 0.00875
                                                                                                                          
* State 2
- Energy: -75.70439730725222
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.00000 1.99722 1.98783 0.01582 0.99913
                                                                                                                          
* State 3
- Energy: -75.62069020549886
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.00000 1.99411 1.99499 0.99764 0.01326
                                                                                                                          
* State 4
- Energy: -75.60257198343568
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.99968 0.77386 1.96066 0.04213 1.22366
                                                                                                                          
* State 5
- Energy: -75.52045360528247
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.99956 1.01895 1.97487 0.97610 0.03052
                                                                                                                          
                                                 Spin Restricted Orbitals                                                 
                                                 ------------------------                                                 
                                                                                                                          
               Molecular Orbital No.   1:                                                                                 
               --------------------------                                                                                 
               Occupation: 2.000 Energy:  -20.68941 a.u.                                                                  
               (   1 O   1s  :     1.00)                                                                                  
                                                                                                                          
               Molecular Orbital No.   2:                                                                                 
               --------------------------                                                                                 
               Occupation: 2.000 Energy:   -1.44678 a.u.                                                                  
               (   1 O   1s  :    -0.22) (   1 O   2s  :     0.49) (   1 O   3s  :     0.48)                              
                                                                                                                          
               Molecular Orbital No.   3:                                                                                 
               --------------------------                                                                                 
               Occupation: 1.976 Energy:   -0.80534 a.u.                                                                  
               (   1 O   1p+1:    -0.55) (   1 O   2p+1:    -0.28) (   2 H   1s  :     0.23)                              
               (   3 H   1s  :    -0.23)                                                                                  
                                                                                                                          
               Molecular Orbital No.   4:                                                                                 
               --------------------------                                                                                 
               Occupation: 1.604 Energy:   -0.58138 a.u.                                                                  
               (   1 O   2s  :     0.18) (   1 O   3s  :     0.24) (   1 O   1p0 :     0.60)                              
               (   1 O   2p0 :     0.40)                                                                                  
                                                                                                                          
               Molecular Orbital No.   5:                                                                                 
               --------------------------                                                                                 
               Occupation: 1.600 Energy:   -0.50270 a.u.                                                                  
               (   1 O   1p-1:    -0.67) (   1 O   2p-1:    -0.48)                                                        
                                                                                                                          
               Molecular Orbital No.   6:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.414 Energy:    0.21172 a.u.                                                                  
               (   1 O   1p+1:     0.43) (   1 O   2p+1:     0.69) (   2 H   2s  :     1.24)                              
               (   3 H   2s  :    -1.24)                                                                                  
                                                                                                                          
               Molecular Orbital No.   7:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.406 Energy:    0.11950 a.u.                                                                  
               (   1 O   2s  :     0.17) (   1 O   3s  :     1.09) (   1 O   1p0 :    -0.25)                              
               (   1 O   2p0 :    -0.41) (   2 H   2s  :    -0.93) (   3 H   2s  :    -0.93)                              
                                                                                                                          
               Molecular Orbital No.   8:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    1.00726 a.u.                                                                  
               (   1 O   2p+1:    -0.69) (   2 H   1s  :    -0.97) (   2 H   2s  :     0.59)                              
               (   3 H   1s  :     0.97) (   3 H   2s  :    -0.59)                                                        
                                                                                                                          
               Molecular Orbital No.   9:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    1.07263 a.u.                                                                  
               (   1 O   1p-1:    -0.94) (   1 O   2p-1:     1.05)                                                        
                                                                                                                          
               Molecular Orbital No.  10:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    1.09033 a.u.                                                                  
               (   1 O   2s  :     0.18) (   1 O   1p0 :     0.94) (   1 O   2p0 :    -1.01)                              
               (   2 H   1s  :     0.23) (   2 H   2s  :    -0.37) (   3 H   1s  :     0.23)                              
               (   3 H   2s  :    -0.37)                                                                                  
                                                                                                                          
                                                                                                                          
Total MCSCF time: 00:00:00
                                                                                                                          
List of oscillator strengths greather than 1e-10
                                                                                                                          
  From     to       Energy (eV)    Oscillator strength (length and velocity)
     1       2        7.55788         1.182610e-02    2.629897e-02
     1       4       10.32868         1.101141e-01    1.302112e-01
     1       5       12.56324         1.491877e-01    1.439100e-01
print("Full CI Energies")
print(FCI_excitation_energies)
print("CIS Energies")
print(CIS_excitation_energies)
print("SA-CASSCF Energies")
print(au2ev*DipOsc['energies'])
Full CI Energies
[8.44992, 10.68451, 10.97654, 13.37692]
CIS Energies
[9.38059, 11.30739, 11.83172, 13.90237]
SA-CASSCF Energies
[ 7.55787583  9.83566209 10.32868402 12.56323891]

The result is now significantly closer to the converged full CI result. In this particular case, the result is not better than CIS, but of course, one the main motivation to use MCSCF is for systems with strong correlation, which was not the case here.

This method is of course not without issues. In particular, unlike standard responses calculation, the introduction of state-averaging means that the excitation energies depend on the total number of states, since the state-averaged orbitals depend on all states. In particular, we can expect the results to deteriorate with increasing number of states as the orbitals begin to compromise many states of widely different natures:

first_exc_energy=[]
for nstates in range(2,6):
    
    # CASSCF calculation
    mcscf_drv.compute(molecule,basis,space,nstates) #5 states included already in the MCSCF.

    # Transition properties
    DipOsc=SI.diposc(molecule,basis,mcscf_drv.CIVecs)
    
    first_exc_energy.append(au2ev*DipOsc['energies'][0])
          Active space definition:
          ------------------------
Number of inactive (occupied) orbitals: 2
Number of active orbitals:              5
Number of virtual orbitals:             6

    This is a CASSCF wavefunction: CAS(6,5)

          CI expansion:
          -------------
Number of determinants:      55


                                                                                                                          
        MCSCF Iterations
        ----------------
                                                                                                                          
     Iter. | Average Energy | E. Change | Grad. Norm | CI Iter. |   Time
     ---------------------------------------------------------------------
        1     -75.843270715     0.0e+00      1.1e-01          1   0:00:00
        2     -75.858829274    -1.6e-02      6.9e-02          1   0:00:00
        3     -75.864455907    -5.6e-03      2.8e-02          1   0:00:00
        4     -75.864858794    -4.0e-04      9.1e-03          1   0:00:00
        5     -75.864894174    -3.5e-05      3.0e-03          1   0:00:00
        6     -75.864897105    -2.9e-06      7.1e-04          1   0:00:00
        7     -75.864897238    -1.3e-07      1.2e-04          1   0:00:00
        8     -75.864897243    -4.9e-09      2.3e-05          1   0:00:00
                                                                                                                          
** Convergence reached in 8 iterations
                                                                                                                          
        Final results
        -------------
                                                                                                                          
* State 1
- Energy: -76.01351196497126
- S^2   : -0.00  (multiplicity = 1.0 )
- Natural orbitals
1.99925 1.98571 1.97737 0.02254 0.01513
                                                                                                                          
* State 2
- Energy: -75.71628252119797
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.00000 1.99623 1.97627 0.02604 1.00146
                                                                                                                          
                                                 Spin Restricted Orbitals                                                 
                                                 ------------------------                                                 
                                                                                                                          
               Molecular Orbital No.   1:                                                                                 
               --------------------------                                                                                 
               Occupation: 2.000 Energy:  -20.64857 a.u.                                                                  
               (   1 O   1s  :     1.00)                                                                                  
                                                                                                                          
               Molecular Orbital No.   2:                                                                                 
               --------------------------                                                                                 
               Occupation: 2.000 Energy:   -1.19270 a.u.                                                                  
               (   1 O   1s  :    -0.22) (   1 O   2s  :     0.51) (   1 O   3s  :     0.56)                              
               (   1 O   1p0 :     0.22) (   1 O   2p0 :     0.16)                                                        
                                                                                                                          
               Molecular Orbital No.   3:                                                                                 
               --------------------------                                                                                 
               Occupation: 1.985 Energy:   -0.86833 a.u.                                                                  
               (   1 O   1p0 :    -0.55) (   1 O   2p0 :    -0.35) (   2 H   1s  :     0.19)                              
               (   3 H   1s  :     0.19)                                                                                  
                                                                                                                          
               Molecular Orbital No.   4:                                                                                 
               --------------------------                                                                                 
               Occupation: 1.973 Energy:   -0.76522 a.u.                                                                  
               (   1 O   1p+1:     0.54) (   1 O   2p+1:     0.27) (   2 H   1s  :    -0.25)                              
               (   3 H   1s  :     0.25)                                                                                  
                                                                                                                          
               Molecular Orbital No.   5:                                                                                 
               --------------------------                                                                                 
               Occupation: 1.500 Energy:   -0.43591 a.u.                                                                  
               (   1 O   1p-1:    -0.66) (   1 O   2p-1:    -0.49)                                                        
                                                                                                                          
               Molecular Orbital No.   6:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.515 Energy:    0.12362 a.u.                                                                  
               (   1 O   2s  :     0.18) (   1 O   3s  :     1.04) (   1 O   1p0 :    -0.33)                              
               (   1 O   2p0 :    -0.38) (   2 H   2s  :    -0.89) (   3 H   2s  :    -0.89)                              
                                                                                                                          
               Molecular Orbital No.   7:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.028 Energy:    0.60893 a.u.                                                                  
               (   1 O   1p+1:    -0.82) (   2 H   1s  :    -0.43) (   2 H   2s  :    -0.44)                              
               (   3 H   1s  :     0.43) (   3 H   2s  :     0.44)                                                        
                                                                                                                          
               Molecular Orbital No.   8:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    0.84499 a.u.                                                                  
               (   1 O   1p+1:    -0.29) (   1 O   2p+1:     0.56) (   2 H   1s  :    -0.70)                              
               (   2 H   2s  :     1.51) (   3 H   1s  :     0.70) (   3 H   2s  :    -1.51)                              
                                                                                                                          
               Molecular Orbital No.   9:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    1.10548 a.u.                                                                  
               (   1 O   1p-1:    -0.95) (   1 O   2p-1:     1.05)                                                        
                                                                                                                          
               Molecular Orbital No.  10:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    1.11162 a.u.                                                                  
               (   1 O   3s  :     0.24) (   1 O   1p0 :     0.90) (   1 O   2p0 :    -0.88)                              
               (   2 H   1s  :     0.42) (   2 H   2s  :    -0.54) (   3 H   1s  :     0.42)                              
               (   3 H   2s  :    -0.54)                                                                                  
                                                                                                                          
                                                                                                                          
Total MCSCF time: 00:00:00
                                                                                                                          
List of oscillator strengths greather than 1e-10
                                                                                                                          
  From     to       Energy (eV)    Oscillator strength (length and velocity)
     1       2        8.08803         8.153140e-03    3.791438e-02
          Active space definition:
          ------------------------
Number of inactive (occupied) orbitals: 2
Number of active orbitals:              5
Number of virtual orbitals:             6

    This is a CASSCF wavefunction: CAS(6,5)

          CI expansion:
          -------------
Number of determinants:      55


                                                                                                                          
        MCSCF Iterations
        ----------------
                                                                                                                          
     Iter. | Average Energy | E. Change | Grad. Norm | CI Iter. |   Time
     ---------------------------------------------------------------------
        1     -75.724068112     0.0e+00      4.6e-02          1   0:00:00
        2     -75.725200061    -1.1e-03      1.2e-02          1   0:00:00
        3     -75.725331152    -1.3e-04      2.9e-03          1   0:00:00
        4     -75.725334435    -3.3e-06      8.8e-04          1   0:00:00
        5     -75.725334998    -5.6e-07      3.2e-04          1   0:00:00
        6     -75.725335073    -7.5e-08      1.4e-04          1   0:00:00
        7     -75.725335090    -1.7e-08      2.4e-05          1   0:00:00
        8     -75.725335090    -5.3e-10      9.8e-06          1   0:00:00
                                                                                                                          
** Convergence reached in 8 iterations
                                                                                                                          
        Final results
        -------------
                                                                                                                          
* State 1
- Energy: -76.00986345021587
- S^2   : -0.00  (multiplicity = 1.0 )
- Natural orbitals
1.99931 1.98623 1.97762 0.02272 0.01412
                                                                                                                          
* State 2
- Energy: -75.71760512866494
- S^2   : -0.00  (multiplicity = 1.0 )
- Natural orbitals
1.00000 1.99620 1.97731 0.02548 1.00100
                                                                                                                          
* State 3
- Energy: -75.4485366920498
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.99771 1.99442 1.00266 0.00751 0.99771
                                                                                                                          
                                                 Spin Restricted Orbitals                                                 
                                                 ------------------------                                                 
                                                                                                                          
               Molecular Orbital No.   1:                                                                                 
               --------------------------                                                                                 
               Occupation: 2.000 Energy:  -20.66135 a.u.                                                                  
               (   1 O   1s  :     1.00)                                                                                  
                                                                                                                          
               Molecular Orbital No.   2:                                                                                 
               --------------------------                                                                                 
               Occupation: 2.000 Energy:   -1.25118 a.u.                                                                  
               (   1 O   1s  :    -0.22) (   1 O   2s  :     0.52) (   1 O   3s  :     0.56)                              
               (   1 O   1p0 :     0.18)                                                                                  
                                                                                                                          
               Molecular Orbital No.   3:                                                                                 
               --------------------------                                                                                 
               Occupation: 1.988 Energy:   -0.84539 a.u.                                                                  
               (   1 O   1p0 :    -0.56) (   1 O   2p0 :    -0.36) (   2 H   1s  :     0.19)                              
               (   3 H   1s  :     0.19)                                                                                  
                                                                                                                          
               Molecular Orbital No.   4:                                                                                 
               --------------------------                                                                                 
               Occupation: 1.666 Energy:   -0.50141 a.u.                                                                  
               (   1 O   1p-1:     0.66) (   1 O   2p-1:     0.48)                                                        
                                                                                                                          
               Molecular Orbital No.   5:                                                                                 
               --------------------------                                                                                 
               Occupation: 1.648 Energy:   -0.70435 a.u.                                                                  
               (   1 O   1p+1:    -0.55) (   1 O   2p+1:    -0.28) (   2 H   1s  :     0.25)                              
               (   3 H   1s  :    -0.25)                                                                                  
                                                                                                                          
               Molecular Orbital No.   6:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.675 Energy:    0.08480 a.u.                                                                  
               (   1 O   2s  :     0.16) (   1 O   3s  :     1.07) (   1 O   1p0 :    -0.31)                              
               (   1 O   2p0 :    -0.39) (   2 H   2s  :    -0.92) (   3 H   2s  :    -0.92)                              
                                                                                                                          
               Molecular Orbital No.   7:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.023 Energy:    0.55592 a.u.                                                                  
               (   1 O   1p+1:    -0.77) (   1 O   2p+1:    -0.18) (   2 H   1s  :    -0.44)                              
               (   2 H   2s  :    -0.51) (   3 H   1s  :     0.44) (   3 H   2s  :     0.51)                              
                                                                                                                          
               Molecular Orbital No.   8:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    0.84549 a.u.                                                                  
               (   1 O   1p+1:    -0.26) (   1 O   2p+1:     0.40) (   2 H   1s  :    -0.75)                              
               (   2 H   2s  :     1.44) (   3 H   1s  :     0.75) (   3 H   2s  :    -1.44)                              
                                                                                                                          
               Molecular Orbital No.   9:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    1.08882 a.u.                                                                  
               (   1 O   1p-1:     0.95) (   1 O   2p-1:    -1.05)                                                        
                                                                                                                          
               Molecular Orbital No.  10:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    1.09854 a.u.                                                                  
               (   1 O   3s  :     0.23) (   1 O   1p0 :     0.89) (   1 O   2p0 :    -0.79)                              
               (   2 H   1s  :     0.50) (   2 H   2s  :    -0.54) (   3 H   1s  :     0.50)                              
               (   3 H   2s  :    -0.54)                                                                                  
                                                                                                                          
                                                                                                                          
Total MCSCF time: 00:00:00
                                                                                                                          
List of oscillator strengths greather than 1e-10
                                                                                                                          
  From     to       Energy (eV)    Oscillator strength (length and velocity)
     1       2        7.95275         8.899154e-03    4.003592e-02
     1       3       15.27448         3.502949e-01    3.169114e-01
          Active space definition:
          ------------------------
Number of inactive (occupied) orbitals: 2
Number of active orbitals:              5
Number of virtual orbitals:             6

    This is a CASSCF wavefunction: CAS(6,5)

          CI expansion:
          -------------
Number of determinants:      55


                                                                                                                          
        MCSCF Iterations
        ----------------
                                                                                                                          
     Iter. | Average Energy | E. Change | Grad. Norm | CI Iter. |   Time
     ---------------------------------------------------------------------
        1     -75.650640385     0.0e+00      1.1e-01          1   0:00:00
        2     -75.692220054    -4.2e-02      5.0e-02          1   0:00:00
        3     -75.694350434    -2.1e-03      1.3e-02          1   0:00:00
        4     -75.694439713    -8.9e-05      3.7e-03          1   0:00:00
        5     -75.694445644    -5.9e-06      1.2e-03          1   0:00:00
        6     -75.694446718    -1.1e-06      4.2e-04          1   0:00:00
        7     -75.694446789    -7.2e-08      1.9e-04          1   0:00:00
        8     -75.694446804    -1.5e-08      2.5e-05          1   0:00:00
        9     -75.694446804    -2.8e-10      5.8e-06          1   0:00:00
                                                                                                                          
** Convergence reached in 9 iterations
                                                                                                                          
        Final results
        -------------
                                                                                                                          
* State 1
- Energy: -75.9969810636317
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.99942 1.98702 1.98210 0.01731 0.01415
                                                                                                                          
* State 2
- Energy: -75.70831737893653
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.00000 1.99704 1.98460 0.01812 1.00024
                                                                                                                          
* State 3
- Energy: -75.62405350917156
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.00000 1.98743 1.99379 0.99822 0.02056
                                                                                                                          
* State 4
- Energy: -75.44843526583905
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.99835 1.99702 1.00111 0.00526 0.99825
                                                                                                                          
                                                 Spin Restricted Orbitals                                                 
                                                 ------------------------                                                 
                                                                                                                          
               Molecular Orbital No.   1:                                                                                 
               --------------------------                                                                                 
               Occupation: 2.000 Energy:  -20.67802 a.u.                                                                  
               (   1 O   1s  :    -1.00)                                                                                  
                                                                                                                          
               Molecular Orbital No.   2:                                                                                 
               --------------------------                                                                                 
               Occupation: 2.000 Energy:   -1.24525 a.u.                                                                  
               (   1 O   1s  :     0.22) (   1 O   2s  :    -0.52) (   1 O   3s  :    -0.55)                              
               (   1 O   1p0 :    -0.20)                                                                                  
                                                                                                                          
               Molecular Orbital No.   3:                                                                                 
               --------------------------                                                                                 
               Occupation: 1.988 Energy:   -0.87394 a.u.                                                                  
               (   1 O   1p0 :     0.56) (   1 O   2p0 :     0.36) (   2 H   1s  :    -0.18)                              
               (   3 H   1s  :    -0.18)                                                                                  
                                                                                                                          
               Molecular Orbital No.   4:                                                                                 
               --------------------------                                                                                 
               Occupation: 1.734 Energy:   -0.74018 a.u.                                                                  
               (   1 O   1p+1:    -0.56) (   1 O   2p+1:    -0.28) (   2 H   1s  :     0.24)                              
               (   3 H   1s  :    -0.24)                                                                                  
                                                                                                                          
               Molecular Orbital No.   5:                                                                                 
               --------------------------                                                                                 
               Occupation: 1.499 Energy:   -0.46605 a.u.                                                                  
               (   1 O   1p-1:     0.67) (   1 O   2p-1:     0.48)                                                        
                                                                                                                          
               Molecular Orbital No.   6:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.512 Energy:    0.10271 a.u.                                                                  
               (   1 O   2s  :    -0.16) (   1 O   3s  :    -1.07) (   1 O   1p0 :     0.30)                              
               (   1 O   2p0 :     0.39) (   2 H   2s  :     0.92) (   3 H   2s  :     0.92)                              
                                                                                                                          
               Molecular Orbital No.   7:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.267 Energy:    0.23802 a.u.                                                                  
               (   1 O   1p+1:     0.45) (   1 O   2p+1:     0.67) (   2 H   1s  :     0.18)                              
               (   2 H   2s  :     1.19) (   3 H   1s  :    -0.18) (   3 H   2s  :    -1.19)                              
                                                                                                                          
               Molecular Orbital No.   8:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    0.99722 a.u.                                                                  
               (   1 O   2p+1:    -0.62) (   2 H   1s  :    -0.97) (   2 H   2s  :     0.66)                              
               (   3 H   1s  :     0.97) (   3 H   2s  :    -0.66)                                                        
                                                                                                                          
               Molecular Orbital No.   9:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    1.08042 a.u.                                                                  
               (   1 O   1p-1:     0.94) (   1 O   2p-1:    -1.05)                                                        
                                                                                                                          
               Molecular Orbital No.  10:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    1.09191 a.u.                                                                  
               (   1 O   3s  :     0.21) (   1 O   1p0 :     0.91) (   1 O   2p0 :    -0.87)                              
               (   2 H   1s  :     0.42) (   2 H   2s  :    -0.51) (   3 H   1s  :     0.42)                              
               (   3 H   2s  :    -0.51)                                                                                  
                                                                                                                          
                                                                                                                          
Total MCSCF time: 00:00:00
                                                                                                                          
List of oscillator strengths greather than 1e-10
                                                                                                                          
  From     to       Energy (eV)    Oscillator strength (length and velocity)
     1       2        7.85494         7.812110e-03    4.149883e-02
     1       4       14.92669         4.583994e-01    3.096527e-01
          Active space definition:
          ------------------------
Number of inactive (occupied) orbitals: 2
Number of active orbitals:              5
Number of virtual orbitals:             6

    This is a CASSCF wavefunction: CAS(6,5)

          CI expansion:
          -------------
Number of determinants:      55


                                                                                                                          
        MCSCF Iterations
        ----------------
                                                                                                                          
     Iter. | Average Energy | E. Change | Grad. Norm | CI Iter. |   Time
     ---------------------------------------------------------------------
        1     -75.630250938     0.0e+00      7.0e-02          1   0:00:00
        2     -75.672532201    -4.2e-02      9.5e-02          1   0:00:00
        3     -75.683260759    -1.1e-02      4.7e-02          1   0:00:00
        4     -75.686039130    -2.8e-03      5.0e-03          1   0:00:00
        5     -75.686051087    -1.2e-05      1.1e-03          1   0:00:00
        6     -75.686051424    -3.4e-07      2.2e-04          1   0:00:00
        7     -75.686051445    -2.1e-08      2.9e-05          1   0:00:00
        8     -75.686051445    -2.8e-10      8.7e-06          1   0:00:00
                                                                                                                          
** Convergence reached in 8 iterations
                                                                                                                          
        Final results
        -------------
                                                                                                                          
* State 1
- Energy: -75.98214464848317
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.99958 1.99145 1.98461 0.01560 0.00875
                                                                                                                          
* State 2
- Energy: -75.70439712311855
- S^2   : -0.00  (multiplicity = 1.0 )
- Natural orbitals
1.00000 1.99722 1.98783 0.01582 0.99913
                                                                                                                          
* State 3
- Energy: -75.620690090302
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.00000 1.99411 1.99499 0.99764 0.01326
                                                                                                                          
* State 4
- Energy: -75.60257199547213
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.99968 0.77386 1.96066 0.04213 1.22366
                                                                                                                          
* State 5
- Energy: -75.52045336778448
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.99956 1.01895 1.97487 0.97610 0.03052
                                                                                                                          
                                                 Spin Restricted Orbitals                                                 
                                                 ------------------------                                                 
                                                                                                                          
               Molecular Orbital No.   1:                                                                                 
               --------------------------                                                                                 
               Occupation: 2.000 Energy:  -20.68941 a.u.                                                                  
               (   1 O   1s  :    -1.00)                                                                                  
                                                                                                                          
               Molecular Orbital No.   2:                                                                                 
               --------------------------                                                                                 
               Occupation: 2.000 Energy:   -1.44678 a.u.                                                                  
               (   1 O   1s  :     0.22) (   1 O   2s  :    -0.49) (   1 O   3s  :    -0.48)                              
                                                                                                                          
               Molecular Orbital No.   3:                                                                                 
               --------------------------                                                                                 
               Occupation: 1.976 Energy:   -0.80534 a.u.                                                                  
               (   1 O   1p+1:    -0.55) (   1 O   2p+1:    -0.28) (   2 H   1s  :     0.23)                              
               (   3 H   1s  :    -0.23)                                                                                  
                                                                                                                          
               Molecular Orbital No.   4:                                                                                 
               --------------------------                                                                                 
               Occupation: 1.604 Energy:   -0.58138 a.u.                                                                  
               (   1 O   2s  :    -0.18) (   1 O   3s  :    -0.24) (   1 O   1p0 :    -0.60)                              
               (   1 O   2p0 :    -0.40)                                                                                  
                                                                                                                          
               Molecular Orbital No.   5:                                                                                 
               --------------------------                                                                                 
               Occupation: 1.600 Energy:   -0.50270 a.u.                                                                  
               (   1 O   1p-1:     0.67) (   1 O   2p-1:     0.48)                                                        
                                                                                                                          
               Molecular Orbital No.   6:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.414 Energy:    0.21172 a.u.                                                                  
               (   1 O   1p+1:     0.43) (   1 O   2p+1:     0.69) (   2 H   2s  :     1.24)                              
               (   3 H   2s  :    -1.24)                                                                                  
                                                                                                                          
               Molecular Orbital No.   7:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.406 Energy:    0.11950 a.u.                                                                  
               (   1 O   2s  :    -0.17) (   1 O   3s  :    -1.09) (   1 O   1p0 :     0.25)                              
               (   1 O   2p0 :     0.41) (   2 H   2s  :     0.93) (   3 H   2s  :     0.93)                              
                                                                                                                          
               Molecular Orbital No.   8:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    1.00726 a.u.                                                                  
               (   1 O   2p+1:    -0.69) (   2 H   1s  :    -0.97) (   2 H   2s  :     0.59)                              
               (   3 H   1s  :     0.97) (   3 H   2s  :    -0.59)                                                        
                                                                                                                          
               Molecular Orbital No.   9:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    1.07263 a.u.                                                                  
               (   1 O   1p-1:     0.94) (   1 O   2p-1:    -1.05)                                                        
                                                                                                                          
               Molecular Orbital No.  10:                                                                                 
               --------------------------                                                                                 
               Occupation: 0.000 Energy:    1.09033 a.u.                                                                  
               (   1 O   2s  :    -0.18) (   1 O   1p0 :    -0.94) (   1 O   2p0 :     1.01)                              
               (   2 H   1s  :    -0.23) (   2 H   2s  :     0.37) (   3 H   1s  :    -0.23)                              
               (   3 H   2s  :     0.37)                                                                                  
                                                                                                                          
                                                                                                                          
Total MCSCF time: 00:00:00
                                                                                                                          
List of oscillator strengths greather than 1e-10
                                                                                                                          
  From     to       Energy (eV)    Oscillator strength (length and velocity)
     1       2        7.55790         1.182597e-02    2.629915e-02
     1       4       10.32870         1.101138e-01    1.302110e-01
     1       5       12.56326         1.491866e-01    1.439092e-01
import matplotlib.pyplot as plt
import numpy as np

plt.figure(figsize=(6,4))
x = np.array(range(2,6))
plt.plot(x,np.array(first_exc_energy), label='First SA-CASSCF excitation energy')
plt.title('Deteriotation of state-averaged CASSCF excitation energies')
plt.ylabel("Energies (eV)")
plt.xlabel("Number of states")
plt.axhline(y=FCI_excitation_energies[0],label ='FCI',color='red')
plt.xticks(x)
plt.legend()
plt.tight_layout(); plt.show()
../../_images/sa_mcscf_10_0.png

While for 2 states, the first excitation energy was relatively close to the fullCI, it degrades significantly as more states are added, with a difference of more than 0.5 eV going from 2 to 5 states. Adding dynamical correlation on-top of the CASSCF typically reduces this error, but it is still an undesirable behaviour.

Still, because of its simplicity and its reasonable accuracy, SA-MCSCF is one of the most frequently used technique to compute properties at the MCSCF level.