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7.3 Time-Dependent Density Functional Theory (TDDFT)

7.3.6 Calculations of Spin-Orbit Couplings Between TDDFT States

(July 14, 2022)

Several options for computing spin-orbit couplings (SOCs) between TDDFT states are available: (i) one-electron part of the Breit-Pauli Hamiltonian, (ii) one-electron SOC using scaled nuclear charges; (iii) full SOC using the mean-field treatment of the two-electron part. Options (i) and (ii) are are available for both TDA and RPA variants (including TDHF and CIS states), for restricted Kohn-Sham references only. Option (iii) is available for both restricted and unrestricted variants, but only within TDA. Calculations of SOC for SF-TDDFT are also possible within TDA using option (iii).

The implementation of one-electron SOC, options (i) and (ii), is based on the following. The SOCs are computed by evaluating matrix elements of the one-electron part of the Breit-Pauli Hamiltonian:

H^SO=-α022i,AZAriA3(𝐫iA×𝐩i)𝐬i (7.16)

where i denotes electrons, A denotes nuclei, α0=1/137.037 is the fine structure constant, and 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^SO,x =-α022pqL~x,pq2(a^pa^q¯+a^p¯a^q) (7.17a)
H^SO,y =-α022pqL~y,pq2i(a^pa^q¯-a^p¯a^q) (7.17b)
H^SO,z =-α022pqL~z,pq2(a^pa^q-a^p¯a^q¯) (7.17c)

where L~^α=L^αr-3 for α{x,y,z} and L~α,pq are the matrix elements of this operator. The single-reference ab initio excited states (within the TDA) are given by

|ΦsingletI =i,asiIa(a^aa^i+a^a¯a^i¯)|ΦHF (7.18a)
|ΦtripletI,Ms=0 =i,atiIa(a^aa^i-a^a¯a^i¯)|ΦHF (7.18b)
|ΦtripletI,Ms=1 =2i,atiIaa^aa^i¯|ΦHF (7.18c)
|ΦtripletI,Ms=-1 =2i,atiIaa^a¯a^i|ΦHF (7.18d)

where siIa and tiIa are singlet and triplet excitation coefficients of the Ith singlet or triplet state respectively, with the normalization

ia(siIa)2=ia(tiIa)2=12. (7.19)

The quantity |ΦHF refers to the Hartree-Fock ground state. Thus the SOC constant from the singlet state to different triplet manifolds are

ΦsingletI|H^SO|ΦtripletJ,Ms=0=α022(i,a,bL~z,absiIatiJb-i,j,aL~z,ijsiIatjJa) (7.20)


ΦsingletI|H^SO|ΦtripletJ,Ms=±1=α0222(i,a,bL~x,absiIatiJb-i,j,aL~x,ijsiIatjJa)+α0222i(i,a,bL~y,absiIatiJb-i,j,aL~y,ijsiIatjJa). (7.21)

The SOC constant between different triplet manifolds can be obtained as

ΦtripletI,Ms=0|H^SO|ΦtripletJ,Ms=±1=α0222(i,a,bL~x,abtiIatiJb+i,j,aL~x,ijtiIatjJa)α0222i(i,a,bL~y,abtiIatiJb+i,j,aL~y,ijtiIatjJa) (7.22)


ΦtripletI,Ms=±1|H^SO|ΦtripletJ,Ms=±1=±α022(i,a,bL~z,abtiIatiJb+i,j,aL~z,ijtiIatjJa). (7.23)

Note that

ΦtripletI,Ms=0|H^SO|ΦtripletJ,Ms=0=0=ΦtripletI,Ms=±1|H^SO|ΦtripletJ,Ms=1. (7.24)

The total (root-mean-square) spin-orbit coupling is

ΦsingletI|H^SO|ΦtripletJ =(Ms=0,±1ΦsingletI|H^SO|ΦtripletJ,Ms2)1/2 (7.25a)
ΦtripletI|H^SO|ΦtripletJ =(Ms=0,±1ΦtripletI,Ms|H^SO|ΦtripletJ,Ms2)1/2. (7.25b)

For RPA states, the SOC constant can simply be obtained by replacing siIatjJb with Xi,tripletIaXj,tripletJb+Yi,singletIaYj,tripletJb and tiIatjJb with Xi,tripletIaXj,tripletJb+Yi,tripletIaYj,tripletJb.

The calculation of SOCs using effective nuclear charges, option (ii), is described in Section The calculations of SOCs using option (iii)—with a mean-field treatment of the two-electron part—is implemented following the algorithm described in Ref.  962 Pokhilko P., Epifanovsky E., Krylov A. I.
J. Chem. Phys.
(2019), 151, pp. 034106.
and outlined in Section

The SOC calculation is activated by $rem variable CALC_SOC: CALC_SOC = 1 activates option (i), CALC_SOC = 4 activates option (ii), and CALC_SOC=2 activates option (iii).

Note:  Setting CALC_SOC = TRUE activates a one-electron calculation using old algorithm, i.e., option (i).


       Controls whether to calculate the SOC constants for EOM-CC, RAS-CI, ADC, CIS, TDDFT/TDA and TDDFT/RPA.
       FALSE Do not perform the SOC calculation. TRUE Perform the SOC calculation.
       Although TRUE and FALSE values will work, EOM-CC code has more variants of SOC evaluations. For details, consult with the EOM section. For TDDFT/CIS, one can use values 1, 2, and 4, as explained above.

Examples 7.3.6, 7.3.6, and 7.3.6 illustrate calculations of SOCs for (SF)-TDDFT states using the above features. These calculations can also be carried out for CIS states by modifying METHOD appropriately.

Example 7.10  Calculation of one-electron SOCs for water molecule using TDDFT/B3LYP 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

Example 7.11  Calculation of full SOCs for water molecule including mean-field treatment of the two-electron part of the Breit-Pauli Hamilton and UHF/TDDFT/B3LYP within the TDA.

Calculation of full SOCs for water molecule inlcuding mean-field treatment
of the two-electron part of the Breit-Pauli Hamiltonian and Wigner-Eckart theorem.
UHF/TDDFT/B3LYP/6-31G within the TDA.

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

jobtype                 sp
unrestricted            true
method                  b3lyp
basis                   6-31G
cis_n_roots             4
cis_convergence         8
cis_singlets            true
cis_triplets            true
calc_soc                2

Example 7.12  Calculation of SOCs for methylene using non-collinear SF-TDDFT/PBE0.

Calculation of SOCs for methylene using non-collinear SF-TDDFT/PBE0,
with tight convergence.

0 3
C  H1 1.0775
H2 C  1.0775 H1 133.29

method = pbe0
basis = cc-pvtz
scf_convergence = 12
cis_convergence = 12
cis_n_roots = 2
calc_soc = 2                    Compute full SOC with mean-field treatment of 2el part
WANG_ZIEGLER_KERNEL =  TRUE     Important for 1,1 diradicals
spin_flip = TRUE