The ROHF of GAMESS for TDDFT calculation


Unfortunately, there are many definitions of ROHF wavefunction. The Fock matrix in the MO basis has the form

                                              closed       open        virtual
        closed      F2      |     Fb     | (Fa+Fb)/2
                 -----------------------------------
        open        Fb      |     F1     |    Fa
                 -----------------------------------
        virtual   (Fa+Fb)/2 |     Fa     |    F0

 where Fa and Fb are the usual alpha and beta Fock matrices any UHF program produces.  All ROHF methods agree on these, as they are the variational conditions that separate the doubly occupied, alpha occupied, and empty orbital spaces.

The diagonal blocks can be written
               F2 = Acc*Fa + Bcc*Fb
               F1 = Aoo*Fa + Boo*Fb
               F0 = Avv*Fa + Bvv*Fb
Some choices for the canonicalization coefficients to define the diagonal blocks are
                          Acc  Bcc  Aoo Boo  Avv  Bvv
 Guest and Saunders       1/2  1/2  1/2 1/2  1/2  1/2
 Roothaan single matrix  -1/2  3/2  1/2 1/2  3/2 -1/2
 Davidson/1988            1/2  1/2   1   0    1    0
 Binkley, Pople, Dobosh   1/2  1/2   1   0    0    1
 McWeeny and Diercksen    1/3  2/3  1/3 1/3  2/3  1/3
 Faegri and Manne         1/2  1/2   1   0   1/2  1/2
 GVB program/Euler        1/2  1/2  1/2  0   1/2  1/2
 "canonical 1"            0    1    1    0    1    0
 "canonical 2"     (2S+1)/2S -1/2S  0    1  -1/2S (2S+1)/2S


 The choice of the diagonal blocks is arbitrary, as ROHF is converged when the off diagonal blocks go to zero.  The exact choice for these blocks can however have an effect on the convergence rate. This choice also affects the MO coefficients, and orbital energies, as the different choices produce different canonical orbitals within the three subspaces.  All methods, however, will give identical total wavefunctions, and hence identical properties such as nuclear gradients and hessians.  Some of the perturbation theories for open shell cases are defined in terms of a particular canonicalization, if so, GAMESS automatically canonicalizes after convergence so the desired orbitals and energies are given to the perturbation theory codes.

The critical problem occurs when you do TDDFT (including SF-TDDFT and MRSF-TDDFT) calculation. The GAMESS utilizes the Roothaan single matrix scheme for SCF part of DFT. However, it takes the Guest and Saunders' scheme for TDDFT (including SF-TDDFT, MRSF-TDDFT). This is clearly a BIG problem, since the molecular orbitals by SCF are utilized in the CPHF  as well as gradients of excited states. To sort things out, the same Guest and Saunders' scheme MUST be used for SCF in the calculation of TDDFT. This can be done by including

couple=.true. alpha(1)=0.5,0.5,0.5 beta(1)=0.5,0.5,0.5

in $SCF group.

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