7.3 Time-Dependent Density Functional Theory (TDDFT)

7.3.6 Calculations of Spin-Orbit Couplings Between TDDFT States

Calculations of spin-orbit couplings (SOCs) for TDDFT states within the Tamm-Dancoff approximation or RPA (including TDHF and CIS states) is available. We employ the one-electron Breit Pauli Hamiltonian to calculate the SOC constant between TDDFT states.

H^SO = -α022i,AZAriA3(𝕣iA×𝕡i)𝕤i (7.16)

where i denotes electrons, A denotes nuclei, α0=137.037-1 is the fine structure constant. ZA is the bare positive charge on nucleus A. In the second quantization representation, the spin-orbit Hamiltonian in different directions can be expressed as

H^SOx = -α022pqLx~pq2(apaq¯+ap¯aq) (7.17)
H^SOy = -α022pqLy~pq2i(apaq¯-ap¯aq) (7.18)
H^SOz = -α022pqLz~pq2(apaq-ap¯aq¯) (7.19)

where Lα~=Lα/r3(α=x,y,z). The single-reference abinitio excited states (within the Tamm-Dancoff approximation) are given by

|ΦsingletI = i,asiIa(aaai+aa¯ai¯)|ΦHF (7.20)
|ΦtripletI,ms=0 = i,atiIa(aaai-aa¯ai¯)|ΦHF (7.21)
|ΦtripletI,ms=1 = i,a2tiIaaaai¯|ΦHF (7.22)
|ΦtripletI,ms=-1 = i,a2tiIaaa¯ai|ΦHF (7.23)

where siIa and tiIa are singlet and triplet excitation coefficients of the Ith singlet or triplet state respectively, with the normalization iasiIa2=iatiIa2=12; |ΦHF refers to the Hartree-Fock ground state. Thus the SOC constant from the singlet state to different triplet manifolds can be obtained as follows,

ΦsingletI|H^SO|ΦtripletJ,ms=0 = α022(i,a,bLz~absiIatiJb-i,j,aLz~ijsiIatjJa) (7.24)
ΦsingletI|H^SO|ΦtripletJ,ms=±1 = α0222(i,a,bLx~absiIatiJb-i,j,aLx~ijsiIatjJa) (7.25)

The SOC constant between different triplet manifolds can be obtained

ΦtripletI,ms=0|H^SO|ΦtripletJ,ms=±1 = α0222(i,a,bLx~abtiIatiJb+i,j,aLx~ijtiIatjJa) (7.26)
ΦtripletI,ms=±1|H^SO|ΦtripletJ,ms=±1 = ±α022(i,a,bLz~abtiIatiJb+i,j,aLz~ijtiIatjJa) (7.27)

Note that ΦtripletI,ms=0|H^SO|ΦtripletJ,ms=0=ΦtripletI,ms=±1|H^SO|ΦtripletJ,ms=1=0. The total (root-mean-square) spin-orbit coupling is given by

ΦsingletI|H^SO|ΦtripletJ = ms=0,±1ΦsingletI|H^SO|ΦtripletJ,ms2 (7.28)
ΦtripletI|H^SO|ΦtripletJ = ms=0,±1ΦtripletI,ms|H^SO|ΦtripletJ,ms2 (7.29)

For RPA states, the SOC constant can simply be obtained by replacing siIatjJb (tiIatjJb) with Xi,singletIaXj,tripletJb+Yi,singletIaYj,tripletJb (Xi,tripletIaXj,tripletJb+Yi,tripletIaYj,tripletJb) Setting the $rem variable CALC_SOC = TRUE will enable the SOC calculation for all calculated TDDFT states.

       Controls whether to calculate the SOC constants for EOM-CC, ADC, TDDFT/TDA and TDDFT.
       FALSE Do not perform the SOC calculation. TRUE Perform the SOC calculation.
       Although TRUE/FALSE values will work, EOM-CC code has more variants of SOC evaluations. For details, consult with EOM section.

Example 7.9  Calculation of SOCs for water molecule using TDDFT/B3LYP functional within the TDA.

   This sample input calculates the spin-orbit coupling constants for water
   between its ground state and its TDDFT/TDA excited triplets as well as the
   coupling between its TDDFT/TDA singlets and triplets.  Results are given in

   0 1
   H       0.000000    -0.115747     1.133769
   H       0.000000     1.109931    -0.113383
   O       0.000000     0.005817    -0.020386

   EXCHANGE             b3lyp
   BASIS                6-31G
   CIS_N_ROOTS          4
   MAX_SCF_CYCLES       600
   MAX_CIS_CYCLES       50
   SCF_ALGORITHM        diis
   MEM_STATIC           300
   MEM_TOTAL            2000
   SYMMETRY             false
   SYM_IGNORE           true
   CIS_SINGLETS         true
   CIS_TRIPLETS         true
   CALC_SOC             true
   SET_ITER             300