CI response#

Full-CI response#

As we saw in the previous section, for an exact state, we have the formula:

\[\begin{equation*} \langle \! \langle \hat{\Omega}; \hat{V}^{\omega} \rangle \! \rangle = - \frac{1}{\hbar} \sum_{n>0} \left[ \frac{\langle 0 | \hat{\Omega} | n \rangle \langle n | \hat{V}^{\omega} | 0 \rangle } {\omega_{n0}-\omega} + \frac{\langle 0 | \hat{V}^{\omega} | n \rangle \langle n | \hat{\Omega} | 0 \rangle } {\omega_{n0}+\omega} \right] . \end{equation*}\]

For example, in the case of a molecule iradiated by light, we are interested in computing the polarizability \(\alpha_{\alpha\beta}(-\omega; \omega)\), corresponding to the response with \(\hat{\Omega} = \hat{\mu}_\alpha\) and \(\hat{V}^\omega = - \hat{\mu}_\beta\), where \(\alpha, \beta \in \{x,y,z\}\). Looking more precisely at the isotropic component, \(\bar\alpha (-\omega; \omega) = \frac{1}{3} \sum_\alpha \alpha_{\alpha\alpha} (-\omega; \omega)\), we obtain:

\[\begin{equation*} \bar\alpha (-\omega; \omega) = \frac{2}{3} \sum_{n>0} \sum_\alpha \frac{ \omega_{n0} | \langle 0 | \hat{\mu}_\alpha | n \rangle |^2 }{ \omega_{n0}^2 - \omega^2 } \end{equation*}\]

We now define the oscillator strength \(f_n\) for a transition \(0 \rightarrow n\) as:

\[\begin{equation*} f_n = \frac{2}{3} \sum_\alpha \omega_{n0} | \langle 0 | \hat{\mu}_\alpha | n \rangle |^2 \end{equation*}\]

such that

\[\begin{equation*} \bar\alpha (-\omega; \omega) = \sum_{n>0} \frac{ f_n }{ \omega_{n0}^2 - \omega^2 } \end{equation*}\]

Those formulas can be applied directly for a configuration interaction wavefunction, where it is easy to get the excited state energies simply by obtaining additional roots in the diagonalization of the Hamiltonian. We then only need to compute the matrix element \(\langle 0 | \hat{\mu}_\alpha | n \rangle\). In second quantization, the \(\alpha\) component of the dipole operator can be written as

\[\begin{equation*} \hat{\mu}_{\alpha} = \sum_{pq} \mu_{\alpha, pq} \hat{a}_p^\dagger \hat{a}_q . \end{equation*}\]

We thus need to compute the matrix \(\boldsymbol{\gamma}\), with elements \( \gamma_{pq} =\langle 0 |\hat{a}_p^\dagger \hat{a}_q | n \rangle\), which is called the transition density matrix, and contract it with the dipole moment integrals.

Let’s apply these for full CI (FCI) of a water molecule:

import veloxchem as vlx
import multipsi as mtp
import numpy as np
* Warning * Environment variable OMP_NUM_THREADS not set.
* Warning * Setting OMP_NUM_THREADS to 8.
* Warning * Setting MKL_THREADING_LAYER to "GNU".
# Compute the 5 lowest states of water using full CI

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.FCI(nfrozen=1) #Full CI with frozen 1s orbital

nstates=5
cidrv=mtp.CIDriver()
cidrv.compute(molecule,basis,space,nstates)

CIVecs=cidrv.vecs
                                                                                                                          
                                            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 0.01 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: 0.06 sec.                                  
                                                                                                                          
* Info * Overlap matrix computed in 0.01 sec.                                                                             
                                                                                                                          
* Info * Kinetic energy matrix computed in 0.02 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: 0.05 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: 1
Number of active orbitals:              12
Number of virtual orbitals:             0

    This is a CASSCF wavefunction: CAS(8,12)

          CI expansion:
          -------------
Number of determinants:      122760


                                                                                                                          
        CI Iterations
        -------------
                                                                                                                          
     Iter. | Average Energy | E. Change | Grad. Norm |   Time
     ----------------------------------------------------------
        1     -75.708185001     0.0e+00      3.6e-01   0:00:00
        2     -75.793127822    -8.1e-02      3.7e-02   0:00:00
        3     -75.799861180    -6.4e-03      3.6e-03   0:00:00
        4     -75.800508543    -4.7e-04      3.9e-04   0:00:00
        5     -75.800565380    -2.7e-05      7.3e-05   0:00:00
        6     -75.800574227    -2.5e-06      8.3e-06   0:00:00
        7     -75.800575131    -1.4e-07      1.2e-06   0:00:00
        8     -75.800575290    -1.1e-08      4.2e-07   0:00:00
        9     -75.800575329    -8.0e-10      1.7e-07   0:00:00
       10     -75.800575338    -6.6e-11      2.9e-08   0:00:00
       11     -75.800575340    -5.3e-12      3.9e-09   0:00:00
                                                                                                                          
** Convergence reached in 11 iterations
                                                                                                                          
        Final results
        -------------
                                                                                                                          
* State 1
- Energy: -76.12020540305305
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.98825 1.98065 1.97145 1.96801 0.02821 0.02666 0.01813 0.01218 0.00310 0.00222 0.00063 0.00050
                                                                                                                          
* State 2
- Energy: -75.80967663263323
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.98999 0.99416 1.97826 1.96500 0.03529 0.99612 0.00679 0.00969 0.00360 0.00263 0.00250 0.99612
                                                                                                                          
* State 3
- Energy: -75.72755706157494
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.98934 0.99423 1.96858 1.97697 0.99438 0.03496 0.00671 0.01152 0.00381 0.00220 0.01394 0.00335
                                                                                                                          
* State 4
- Energy: -75.71682482474833
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.98622 1.98084 0.84833 1.95656 0.04126 0.01163 0.01796 0.00636 0.00283 0.00240 0.00360 0.01163
                                                                                                                          
* State 5
- Energy: -75.62861277592373
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.98447 1.98207 1.01098 1.95661 0.97797 0.04117 0.01636 0.00643 0.00370 0.00204 0.01380 0.00441
au2ev = 27.211386

#Get the molecular orbitals within the active space
C=space.getMOAct()

#Compute dipole integrals in AO basis
dipole_drv = vlx.ElectricDipoleIntegralsDriver()
dipole_mats = dipole_drv.compute(molecule, basis)
dipole=[dipole_mats.x_to_numpy(),dipole_mats.y_to_numpy(),dipole_mats.z_to_numpy()]

#Transform to MO basis
dipole_mo=[]
for icomp in range(0,3):
    dipole_mo.append(np.einsum('pq,pt,qu->tu', dipole[icomp], C, C))
    

#Initialize the CIOperator class to compute the transition densities.
expansion=mtp.CIExpansion(space)
DenDriver=mtp.CIOperator(expansion)
        
#Compute all 0->n transitions
for n in range(1,nstates):
    dE=CIVecs[n].energy-CIVecs[0].energy
    tden=DenDriver.get1den(CIVecs[0],CIVecs[n]) #Transition density matrix
    dx=np.tensordot(tden,dipole_mo[0]) #<0|x|n>
    dy=np.tensordot(tden,dipole_mo[1]) #<0|y|n>
    dz=np.tensordot(tden,dipole_mo[2]) #<0|z|n>
    F=2/3*dE*(dx*dx+dy*dy+dz*dz)
    print("Excitation 0->",n," energy:",dE*au2ev,"eV oscillator strength:",F)
Excitation 0-> 1  energy: 8.449918235999155 eV oscillator strength: 0.013189367950823701
Excitation 0-> 2  energy: 10.684505582220593 eV oscillator strength: 1.0359199498717312e-26
Excitation 0-> 3  energy: 10.97654462115296 eV oscillator strength: 0.11535380248486023
Excitation 0-> 4  energy: 13.376916731570043 eV oscillator strength: 0.11498821812103205

We thus obtain 3 visible transitions and one dark state (near-zero dipole oscillator strength). We obtain the exact same result by using the built-in class of multipsi:

SI=mtp.InterState()
DipOsc=SI.diposc(molecule,basis,CIVecs)
                                                                                                                          
List of oscillator strengths greather than 1e-10
                                                                                                                          
  From     to       Energy (eV)    Oscillator strength (length and velocity)
     1       2        8.44992         1.318937e-02    4.625297e-02
     1       4       10.97654         1.153538e-01    1.749710e-01
     1       5       13.37692         1.149882e-01    1.223962e-01

Truncated CI response#

While they do not formally qualify as “exact state”, the response equations for truncated CI are the same as the exact case above. One can thus, in principle, obtain a hierarchy of method from CIS to full CI. However, the excitation energies do not necessarily improve from order to order. Let us demonstrate this on the water example.

nstates=5
Energies=np.empty((5,nstates-1)) #nstates-1 transitions for CIS, CISD, CISDT and CISDTQ and FCI
Energies[4,:]=au2ev*DipOsc['energies'] #Save the FCI result
#CIS to CISDTQ
space=mtp.OrbSpace(molecule,scf_drv.mol_orbs)
cidrv=mtp.CIDriver()
SI=mtp.InterState()
for exc in range(1,5):
    space.CI(exc,nfrozen=1)
    
    cidrv.compute(molecule,basis,space, nstates)

    DipOsc=SI.diposc(molecule,basis,cidrv.vecs)
    Energies[exc-1,:]=au2ev*DipOsc['energies']
          Active space definition:
          ------------------------
Number of inactive (occupied) orbitals: 1
Number of active orbitals:              12
Number of virtual orbitals:             0

    This is a GASSCF wavefunction
           
         Cumulated   Min cumulated    Max cumulated 
 Space    orbitals      occupation       occupation
     1           4               7                8
     2          12               8                8


          CI expansion:
          -------------
Number of determinants:      33


                                                                                                                          
        CI Iterations
        -------------
                                                                                                                          
     Iter. | Average Energy | E. Change | Grad. Norm |   Time
     ----------------------------------------------------------
        1     -75.642674442     0.0e+00      4.0e-27   0:00:00
                                                                                                                          
** Convergence reached in 1 iterations
                                                                                                                          
        Final results
        -------------
                                                                                                                          
* State 1
- Energy: -75.9838703757691
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
2.00000 2.00000 2.00000 2.00000 0.00000 0.00000 0.00000 0.00000 -0.00000 -0.00000 -0.00000 -0.00000
                                                                                                                          
* State 2
- Energy: -75.63914005847889
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.00003 2.00000 1.99997 2.00000 0.99997 0.00000 -0.00000 -0.00000 0.00003 0.00000 0.00000 0.00000
                                                                                                                          
* State 3
- Energy: -75.5683315663457
- S^2   : -0.00  (multiplicity = 1.0 )
- Natural orbitals
1.00003 2.00000 2.00000 1.99997 -0.00000 -0.00000 0.99997 0.00000 0.00003 0.00000 -0.00000 -0.00000
                                                                                                                          
* State 4
- Energy: -75.54906255700253
- S^2   : -0.00  (multiplicity = 1.0 )
- Natural orbitals
1.99696 1.99959 1.02031 1.98314 0.97969 0.00000 -0.00000 -0.00000 0.00304 -0.00000 -0.00000 0.00041
                                                                                                                          
* State 5
- Energy: -75.47296765017843
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
2.00000 1.99995 1.02876 1.97129 0.02871 -0.00000 0.97124 0.00005 0.00000 -0.00000 0.00000 0.00000
                                                                                                                          
List of oscillator strengths greather than 1e-10
                                                                                                                          
  From     to       Energy (eV)    Oscillator strength (length and velocity)
     1       2        9.38059         1.468873e-02    4.091187e-02
     1       4       11.83172         1.208787e-01    1.355741e-01
     1       5       13.90237         1.045445e-01    8.623014e-02
          Active space definition:
          ------------------------
Number of inactive (occupied) orbitals: 1
Number of active orbitals:              12
Number of virtual orbitals:             0

    This is a GASSCF wavefunction
           
         Cumulated   Min cumulated    Max cumulated 
 Space    orbitals      occupation       occupation
     1           4               6                8
     2          12               8                8


          CI expansion:
          -------------
Number of determinants:      729


                                                                                                                          
        CI Iterations
        -------------
                                                                                                                          
     Iter. | Average Energy | E. Change | Grad. Norm |   Time
     ----------------------------------------------------------
        1     -75.708067912     0.0e+00      3.5e-01   0:00:00
        2     -75.739528252    -1.4e-02      9.7e-03   0:00:00
        3     -75.740704223    -4.1e-04      2.2e-04   0:00:00
        4     -75.740732582    -6.7e-06      5.2e-06   0:00:00
        5     -75.740733616    -1.8e-07      1.8e-07   0:00:00
        6     -75.740733655    -3.9e-09      1.2e-08   0:00:00
        7     -75.740733657    -1.2e-10      6.1e-10   0:00:00
                                                                                                                          
** Convergence reached in 7 iterations
                                                                                                                          
        Final results
        -------------
                                                                                                                          
* State 1
- Energy: -76.1134021156927
- S^2   : -0.00  (multiplicity = 1.0 )
- Natural orbitals
1.98980 1.98356 1.97571 1.97263 0.02409 0.02279 0.01544 0.01054 0.00268 0.00195 0.00046 0.00034
                                                                                                                          
* State 2
- Energy: -75.7351931372965
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.99830 0.99907 1.99600 1.99110 0.01175 0.99382 0.00147 0.00171 0.00118 0.00082 0.00035 0.00444
                                                                                                                          
* State 3
- Energy: -75.65407733183052
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.99830 0.99900 1.99144 1.99450 0.99166 0.01390 0.00153 0.00149 0.00117 0.00047 0.00575 0.00080
                                                                                                                          
* State 4
- Energy: -75.64356171507323
- S^2   : -0.00  (multiplicity = 1.0 )
- Natural orbitals
1.99571 1.99582 1.12840 1.97554 0.02646 0.00507 0.00379 0.00180 0.00142 0.00091 0.00051 0.00507
                                                                                                                          
* State 5
- Energy: -75.55743398573284
- S^2   : -0.00  (multiplicity = 1.0 )
- Natural orbitals
1.99559 1.99775 1.02434 1.96489 0.96813 0.03602 0.00191 0.00164 0.00182 0.00039 0.00618 0.00134
                                                                                                                          
List of oscillator strengths greather than 1e-10
                                                                                                                          
  From     to       Energy (eV)    Oscillator strength (length and velocity)
     1       2       10.29159         1.548866e-02    3.658376e-02
     1       4       12.78501         1.247451e-01    1.355482e-01
     1       5       15.12866         1.136810e-01    9.274229e-02
          Active space definition:
          ------------------------
Number of inactive (occupied) orbitals: 1
Number of active orbitals:              12
Number of virtual orbitals:             0

    This is a GASSCF wavefunction
           
         Cumulated   Min cumulated    Max cumulated 
 Space    orbitals      occupation       occupation
     1           4               5                8
     2          12               8                8


          CI expansion:
          -------------
Number of determinants:      6329


                                                                                                                          
        CI Iterations
        -------------
                                                                                                                          
     Iter. | Average Energy | E. Change | Grad. Norm |   Time
     ----------------------------------------------------------
        1     -75.708185001     0.0e+00      3.5e-01   0:00:00
        2     -75.787071882    -7.3e-02      1.4e-02   0:00:00
        3     -75.791912542    -4.3e-03      1.2e-03   0:00:00
        4     -75.792184181    -1.7e-04      8.9e-05   0:00:00
        5     -75.792206026    -8.9e-06      1.1e-05   0:00:00
        6     -75.792207797    -4.6e-07      8.3e-07   0:00:00
        7     -75.792207936    -2.0e-08      7.9e-08   0:00:00
        8     -75.792207948    -9.5e-10      7.6e-09   0:00:00
                                                                                                                          
** Convergence reached in 8 iterations
                                                                                                                          
        Final results
        -------------
                                                                                                                          
* State 1
- Energy: -76.11438963653255
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.98960 1.98327 1.97493 1.97168 0.02478 0.02338 0.01562 0.01071 0.00289 0.00208 0.00060 0.00047
                                                                                                                          
* State 2
- Energy: -75.80003060650523
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.99159 0.99537 1.98187 1.97002 0.03029 0.99592 0.00554 0.00820 0.00318 0.00237 0.00236 0.99592
                                                                                                                          
* State 3
- Energy: -75.71893849479284
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.99111 0.99537 1.97319 1.97998 0.99427 0.03033 0.00552 0.00953 0.00339 0.00197 0.01223 0.00311
                                                                                                                          
* State 4
- Energy: -75.70731803768399
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.98804 1.98430 0.86217 1.96044 0.03726 0.01016 0.01467 0.00542 0.00263 0.00217 0.00348 0.01016
                                                                                                                          
* State 5
- Energy: -75.6203629649589
- S^2   : 0.00  (multiplicity = 1.0 )
- Natural orbitals
1.98665 1.98540 1.01392 1.95745 0.97598 0.04022 0.01331 0.00547 0.00341 0.00187 0.01214 0.00418
                                                                                                                          
List of oscillator strengths greather than 1e-10
                                                                                                                          
  From     to       Energy (eV)    Oscillator strength (length and velocity)
     1       2        8.55414         1.328950e-02    4.580294e-02
     1       4       11.07698         1.154227e-01    1.705637e-01
     1       5       13.44315         1.118511e-01    1.179731e-01
          Active space definition:
          ------------------------
Number of inactive (occupied) orbitals: 1
Number of active orbitals:              12
Number of virtual orbitals:             0

    This is a GASSCF wavefunction
           
         Cumulated   Min cumulated    Max cumulated 
 Space    orbitals      occupation       occupation
     1           4               4                8
     2          12               8                8


          CI expansion:
          -------------
Number of determinants:      27763


                                                                                                                          
        CI Iterations
        -------------
                                                                                                                          
     Iter. | Average Energy | E. Change | Grad. Norm |   Time
     ----------------------------------------------------------
        1     -75.708185001     0.0e+00      3.6e-01   0:00:00
        2     -75.793118310    -8.1e-02      3.6e-02   0:00:00
        3     -75.798908867    -5.2e-03      2.0e-03   0:00:00
        4     -75.799401560    -3.9e-04      2.2e-04   0:00:00
        5     -75.799440899    -2.2e-05      2.9e-05   0:00:00
        6     -75.799446125    -1.5e-06      3.3e-06   0:00:00
        7     -75.799446623    -9.5e-08      5.5e-07   0:00:00
        8     -75.799446711    -5.1e-09      2.1e-07   0:00:00
        9     -75.799446734    -3.7e-10      6.1e-08   0:00:00
       10     -75.799446738    -2.9e-11      7.4e-09   0:00:00
                                                                                                                          
** Convergence reached in 10 iterations
                                                                                                                          
        Final results
        -------------
                                                                                                                          
* State 1
- Energy: -76.12003281964283
- S^2   : -0.02  (multiplicity = 1.0 )
- Natural orbitals
1.98832 1.98078 1.97163 1.96820 0.02805 0.02650 0.01801 0.01210 0.00308 0.00221 0.00062 0.00049
                                                                                                                          
* State 2
- Energy: -75.80828932731461
- S^2   : -0.03  (multiplicity = 0.9 )
- Natural orbitals
1.99041 0.99448 1.97922 1.96630 0.03407 0.99598 0.00646 0.00935 0.00350 0.00257 0.00243 0.99598
                                                                                                                          
* State 3
- Energy: -75.72622990152138
- S^2   : -0.03  (multiplicity = 0.9 )
- Natural orbitals
1.98982 0.99453 1.96971 1.97785 0.99421 0.03391 0.00639 0.01098 0.00372 0.00215 0.01350 0.00324
                                                                                                                          
* State 4
- Energy: -75.71537749255114
- S^2   : -0.03  (multiplicity = 0.9 )
- Natural orbitals
1.98672 1.98172 0.84888 1.95763 0.04031 0.01128 0.01710 0.00608 0.00280 0.00235 0.00347 0.01128
                                                                                                                          
* State 5
- Energy: -75.62730415146522
- S^2   : -0.03  (multiplicity = 0.9 )
- Natural orbitals
1.98504 1.98291 1.01141 1.95734 0.97774 0.04059 0.01556 0.00615 0.00364 0.00200 0.01336 0.00427
                                                                                                                          
List of oscillator strengths greather than 1e-10
                                                                                                                          
  From     to       Energy (eV)    Oscillator strength (length and velocity)
     1       2        8.48297         1.323123e-02    4.607715e-02
     1       4       11.01123         1.154557e-01    1.737581e-01
     1       5       13.40783         1.150427e-01    1.217901e-01
import matplotlib.pyplot as plt

plt.figure(figsize=(6,4))
x = np.array(range(1,6))
plt.plot(x,Energies[:,0], label='0->1')
plt.plot(x,Energies[:,1], label='0->2')
plt.plot(x,Energies[:,2], label='0->3')
plt.plot(x,Energies[:,3], label='0->4')
plt.title('Convergence of excitation energies')
plt.xticks([1, 2, 3, 4, 5], ['CIS', 'CISD', 'CISDT', 'CISDTQ', 'FCI'])
plt.ylabel("Energies (eV)")
plt.legend()
plt.tight_layout(); plt.show()
../../_images/ci_response_8_0.png
Energies[0,:]
array([ 9.38058973, 11.30738694, 11.83172339, 13.90237127])

As we can see, we do not have a monotonous convergence towards the full CI result. In particular, CISD is actually worse than CIS. The reason for this is that CIS has a more even-handed treatment of the ground and excited states than CISD. In CIS, due to the Brillouin theorem, the ground state does not improve with the inclusion of single excitations, but the single excitations are needed to generate the dominant configuration for the (singly) excited states. Both ground and excited states are thus treated at a mean-field (HF-like) level.

However, in CISD, the ground state contains double excitation, and thus dynamical correlation, while the excited states would need up to triple excitations to both generate the singly-excited dominant configuration and correlating double excitations on top. There is an unbalance, the ground state being described more accurately than the excited states, leading to too high excitation energies. Adding the triple excitations bring correlation also for excited states, which brings the result closer to the final FCI result, and in this simple case is essentially converged.

For this reason, while CIS is sometimes used to compute excitation energies (and as we can see in the next section, is actually related to HF response), the other truncated CIs are not.