Transition Moments and Spin-Orbit Coupling

GAMESS can compute transition moments and oscillator strengths for the radiative transition between two CI wavefunctions. The moments are computed using both the "length (dipole) form" and "velocity form". GAMESS can also compute the one electron portion of the "microscopic Breit-Pauli spin orbit operator". The two electron terms can be approximately accounted for by means of effective nuclear charges.

The orbitals for the CI can be one common set of orbitals used by both CI states. If one set of orbitals is used, the transition moment or spin-orbit coupling can be found for any type of CI wavefunction GAMESS can compute.

Alternatively, two sets of orbitals (obtained from separate MCSCF orbital optimizations) can be used. Two separate CIs will be carried out (in 1 run). The two MO sets must share a common set of frozen core orbitals, and the CI -must- be of the complete active space type. These restrictions are needed to leave the CI wavefunctions invariant under the necessary transformation to corresponding orbitals. The non-orthogonal procedure implemented in GAMESS is a GUGA driven equivalent to the method of Lengsfield, et al. Note that the FOCI and SOCI methods described by these workers are not available in GAMESS.

If you would like to use separate orbitals for the states, use the FCORE option in $MCSCF in a SCFTYP=MCSCF optimization of the orbitals. Typically you would optimize the ground state completely, and use these MCSCF orbitals in an optimization of the excited state, under the constraint of FCORE=.TRUE. The core orbitals of such a MCSCF optimization should be declared MCC, rather than FZC, even though they are frozen.

In the case of transition moments either one or two CI calculations are performed, necessarily on states of the same multiplicity. Thus, only a $CIDRT1 is read. A spin- orbit coupling run almost always does two CI calculations, as the states are usually of different multiplicities. So, spin-orbit runs might read only $CIDRT1, but normally read both $CIDRT1 and $CIDRT2. The first CI calculation, defined by $CIDRT1, must be for the state of lower spin multiplicity, with $CIDRT2 defining the state of higher multiplicity.

You will probably have to lower the symmetry in $CIDRT1 and $CIDRT2 to C1. You may use full spatial symmetry in the CIDRT groups only if the two states happen to have the same total spatial symmetry.

The transition moment and spin orbit coupling driver is a rather restricted path through GAMESS, in that

  1. Give SCFTYP=NONE. $GUESS is not read, as the program expects to MOREAD the orbitals $VEC1 group, and perhaps a $VEC2 group. It is not possible to reorder orbitals.
  2. $CIINP is not read. The CI is hardwired to consist of CIDRT generation, integral transformation/sorting, Hamiltonian generation, and diagonalization. This means $CIDRT1 (and maybe $CIDRT2), $TRANS, $CISORT, $GUGEM, and $GUGDIA input is read, and acted upon.
  3. The density matrices are not generated, and so no properties (other than the transition moment or the spin-orbit coupling) are computed.
  4. There is no restart capability provided.
  5. CIDRT input is given in $CIDRT1 and maybe $CIDRT2.
  6. IROOTS will determine the number of eigenvectors found in each CI calculation, except NSTATE in $GUGDIA will override IROOTS if NSTATE is larger.

References:

F.Weinhold, J.Chem.Phys. 54,1874-1881(1970)

B.H.Lengsfield, III, J.A.Jafri, D.H.Phillips, C.W.Bauschlicher, Jr. J.Chem.Phys. 74,6849-6856(1981)

"Intramediate Quantum Mechanics, 3rd Ed." Hans A. Bethe, Roman Jackiw Benjamin/Cummings Publishing, Menlo Park, CA (1986), chapters 10 and 11.

For an application of the transition moment code: S.Koseki, M.S.Gordon J.Mol.Spectrosc. 123, 392-404(1987)

For more information on effective nuclear charges:
S.Koseki, M.W.Schmidt, M.S.Gordon J.Phys.Chem. 96, 10768-10772 (1992)
S.Koseki, M.S.Gordon, M.W.Schmidt, N.Matsunaga J.Phys.Chem. 99, 12764-12772 (1995)


Special thanks to Bob Cave and Dave Feller for their assistance in performing check spin-orbit coupling runs with the MELDF programs.

Here is an example. Note that you must know what you are doing with term symbols, J quantum numbers, point group symmetry, and so on in order to make skillful use of this part of the program.

          !  Compute the splitting of the famous sodium D line.
          !
          !  The two SCF energies below give an excitation energy
          !  of 16,044 cm-1 to the 2-P term.  The computed spin-orbit
          !  levels are at RELATIVE E=-10.269 and 5.135 cm-1, which
          !  means the 2-P level interval is 15.404 cm-1.
          !
          !  Charlotte Moore's Atomic Energy Levels, volume 1, gives
          !  the experimental 2-P interval as 17.1963, the levels are
          !  at 2-S-1/2=0.0, 2-P-1/2=16,956.183, 2-P-3/2=16,973.379
          !
          
          1. generate ground state 2-S orbitals by conventional ROHF.
             the energy of the ground state is -161.8413919816
          
          --- $contrl scftyp=rohf mult=2 $end
          --- $system kdiag=3 memory=300000 $end
          --- $guess  guess=huckel $end
          --- $basis  gbasis=n31 ngauss=6 $end
          
          2. generate excited state 2-P orbitals, using a state-averaged
             SCF wavefunction to ensure radial degeneracy of the 3p shell
             is preserved.  The open shell SCF energy is -161.7682895801.
             The computation is both spin and space restricted open shell
             SCF on the 2-P Russell-Saunders term.  Starting orbitals are
             reordered orbitals from step 1.
          
          --- $contrl scftyp=gvb mult=2 $end
          --- $system kdiag=3 memory=300000 $end
          --- $guess  guess=moread norb=13 norder=1 iorder(6)=7,8,9,6 $end
          --- $basis  gbasis=n31 ngauss=6 $end
          --- $scf    nco=5 nseto=1 no(1)=3 rstrct=.true. couple=.true.
          ---             f(1)=  1.0  0.16666666666667
          ---         alpha(1)=  2.0  0.33333333333333  0.0
          ---          beta(1)= -1.0 -0.16666666666667  0.0 $end
          
          3. compute spin orbit coupling in the 2-P term.  The use of
             C1 symmetry in $CIDRT ensures that all three spatial CSFs
             are kept in the CI function.  In the preliminary CI, the 
             spin function is just the alpha spin doublet, and all three 
             roots should be degenerate, and furthermore equal to the 
             GVB energy at step 2.  The spin-orbit coupling code uses
             both doublet spin functions with each of the three spatial
             wavefunctions, so the spin-orbit Hamiltonian is a 6x6 matrix.  
             The two lowest roots of the full 6x6 spin-orbit Hamiltonian 
             are the doubly degenerate 2-P-1/2 level, while the other 
             four roots are the degenerate 2-P-3/2 level.
             
           $contrl scftyp=none cityp=guga runtyp=spinorbt mult=2 $end
           $system memory=500000 $end
           $gugdia nstate=3 $end
           $transt numvec=1 numci=1 nfzc=5 nocc=8 iroots=3 zeff=10.04 $end
           $cidrt1 group=c1 fors=.true. nfzc=5 nalp=1 nval=2 $end
          
           $data
          Na atom...2-P excited state...6-31G basis, typed w/o L shells.
          Dnh 2
          
          Na 11.0
            s 6 1
              1 9993.2 0.00193766
              2 1499.89 0.0148070
              3 341.951 0.0727055
              4 94.6796 0.252629
              5 29.7345 0.493242
              6 10.0063 0.313169
            s 6 1
              1 150.963 -0.00354208
              2 35.5878 -0.0439588
              3 11.1683 -0.109752
              4 3.90201 0.187398
              5 1.38177 0.646699
              6 0.466382 0.306058
            p 6 1
              1 150.963 0.00500166
              2 35.5878 0.0355109
              3 11.1683 0.142825
              4 3.90201 0.338620
              5 1.38177 0.451579
              6 0.466382 0.273271
            s 3 1
              1 0.497966 -0.248503
              2 0.0843529 -0.131704
              3 0.0666350 1.233520
            p 3 1
              1 0.497966 -0.0230225
              2 0.0843529 0.950359
              3 0.0666350 0.0598579
            s 1 1
              1 0.0259544 1.0
            p 1 1
              1 0.0259544 1.0
            
           $end
                 
          --- GVB ORBITALS --- GENERATED AT  7:46:08 CST 30-MAY-1996
          Na atom...2-P excited state
          E(GVB)=     -161.7682895801, 5 ITERS
           $VEC1
          ...orbitals from step 2 go here...
           $END

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