The calculations of first-order nonadiabatic couplings (NAC) between ground and excited-states (<0|Dx|Sn>), and those between the excited-states (<Sm|Dx|Sn> or <Tm|Dx|Tn>) at the TD-DFT/TDA level can be achieved by generalizing the standard linear and quadratic response theories, for details, see
1. Zhendong Li and Wenjian Liu, "First-order nonadiabatic coupling matrix elements between excited states: A Lagrangian formulation at the CIS, RPA, TD-HF, and TD-DFT levels", J. Chem. Phys. 141, 014110 (2014).
2. Zhendong Li, Bingbing Suo, and Wenjian Liu, "First-order nonadiabatic coupling matrix elements between excited states: II. Implementation and applications at the TD-DFT and pp-RPA levels", J. Chem. Phys. 141, 244105 (2014).
For convenience, however, in the input they are both specified by the QUAD keyword with single and double, respectively. Either analytic derivative or finite difference approach can be used. The latter is only allowed for C(1) symmetry, and for molecules without orbital degeneracy!
Compared with Hellman-Feynman results
By using the turn over rule, the NAC between two adiabatic states can be written as a integration of the gradients of V_nuc and the transition density scaled by the energy difference. We may call it Hellman-Feynman like expression.
However, for real application, neither the state is exact nor the basis is complete, so this form will not yield accurate results !!! As a simple illustration, we consider the NAC between two triplet states of MgH2 calculated at the pp-TDA level using several different basis sets.
pp-tda for MgH2:
$COMPASS Title nh3 Basis aug-cc-pvqz uncontracted # nac-mg #primitives Geometry Mg 0. 0. 5.0 H 0. 1.25 0. H 0. -1.25 0. End geometry units bohr skeleton $END $xuanyuan direct schwarz $end $scf RHF charge 2 spin 1 THRESHCONV 1.d-12 1.d-10 OPTSCR 1 $end $tddft imethod 4 isf 1 nexit 0 0 2 0 itda 1 ipprpa 1 idiag 1 istore 1 crit_e 1.d-12 crit_vec 1.d-10 $end $resp iprt 2 QUAD FNAC norder 1 method 2 nfiles 1 $end
BASIS SET 1: nac-mgh2 (only containing s/p)
No. Pair ExSym ExEnergies f D<S^2> Dominant Excitations IPA Ova En-E1
1 B1 1 B1 -18.6728 eV 0.0000 0.0000 92.8% VV(1): B1( 2 )-> A1( 5 ) -23.956 0.000 0.0000
2 B1 2 B1 -16.3909 eV 0.0000 0.0000 88.4% VV(1): B1( 3 )-> A1( 5 ) -21.065 0.000 2.2819
...
Gradient contribution from Final-NAC(R)-HFey
1 -0.0000000000 -0.0000000000 -0.7243310867
2 0.3451224889 0.0000000000 0.0561331731
3 -0.3451224889 -0.0000000000 0.0561331731
Sum of gradient contribution from Final-NAC(R)-HFey
-0.0000000000 -0.0000000000 -0.6120647404
...
Gradient contribution from Final-NAC(R)-Escaled
1 -0.0000000000 -0.0000000000 0.3549353015
2 0.7989929157 0.0000000000 -0.1473168767
3 -0.7989929157 -0.0000000000 -0.1473168774
Sum of gradient contribution from Final-NAC(R)-Escaled
-0.0000000000 -0.0000000000 0.0603015474BASIS SET 2: aug-cc-pVTZ
No. Pair ExSym ExEnergies f D<S^2> Dominant Excitations IPA Ova En-E1
1 B1 1 B1 -18.6736 eV 0.0000 0.0000 90.6% VV(1): B1( 2 )-> A1( 5 ) -23.766 0.000 0.0000
2 B1 2 B1 -16.9243 eV 0.0000 0.0000 87.8% VV(1): B1( 3 )-> A1( 5 ) -21.638 0.000 1.7493
...
Gradient contribution from Final-NAC(R)-HFey
1 -0.0000000000 -0.0000000000 -0.2135694894
2 0.3992751267 -0.0000000000 0.0624241280
3 -0.3992751267 -0.0000000000 0.0624241280
Sum of gradient contribution from Final-NAC(R)-HFey
-0.0000000000 -0.0000000000 -0.0887212334
...
Gradient contribution from Final-NAC(R)-Escaled
1 0.0000000000 -0.0000000000 0.3849500968
2 0.8010940332 0.0000000000 -0.1247569146
3 -0.8010940332 -0.0000000000 -0.1247570212
Sum of gradient contribution from Final-NAC(R)-Escaled
0.0000000000 -0.0000000000 0.1354361609BASIS SET 3: uncontracted aug-cc-pVQZ
No. Pair ExSym ExEnergies f D<S^2> Dominant Excitations IPA Ova En-E1
1 B1 1 B1 -18.6693 eV 0.0000 0.0000 90.6% VV(1): B1( 2 )-> A1( 5 ) -23.760 0.000 0.0000
2 B1 2 B1 -16.9256 eV 0.0000 0.0000 87.5% VV(1): B1( 3 )-> A1( 5 ) -21.636 0.000 1.7438
...
Gradient contribution from Final-NAC(R)-HFey
1 -0.0000000000 0.0000000000 0.1886204721
2 -0.4020811753 0.0000000000 -0.0612611462
3 0.4020811753 0.0000000000 -0.0612611462
Sum of gradient contribution from Final-NAC(R)-HFey
-0.0000000000 0.0000000000 0.0660981797
...
Gradient contribution from Final-NAC(R)-Escaled
1 -0.0000000000 0.0000000000 -0.3856366870
2 -0.8006137635 0.0000000000 0.1251752431
3 0.8006137635 -0.0000000000 0.1251750107
Sum of gradient contribution from Final-NAC(R)-Escaled
0.0000000000 0.0000000000 -0.1352864331It is seen that the results [NAC-HFey] are quite differenct, while those [NAC(R)-Escaled] consider the finite basis set effects (Pulay terms) are similar. While for small molecules like H2, such difference seems to be not significant, however, for large molecules the difference can be very large. (This might be due to the fact that for small systems, the density is small such that the contraction with integrals is also not large).
The same conclusion can also be found for TD-DFT, as already demonstrated by the paper of Send and Furche (see TABLE IV), where u-aug-TZVPP basis set have been used to get qualitatively correct results for the coupling between the ground and first singlet excited state of Cinnoline (C8H6N2) using the PBE functional !
Analytic derivative approach
NAC between two 1A2 states of CH2O:
$COMPASS Title ch2o Basis 6-31GP Geometry C 0.00000000 -0.00000000 -0.53964037 O 0.00000000 0.00000000 0.68767663 H 0.00000000 0.93940400 -1.13178537 H 0.00000000 -0.93940400 -1.13178537 End geometry skeleton $END $xuanyuan direct schwarz $end $scf RHF charge 0 spin 1 THRESHCONV 1.d-10 1.d-8 OPTSCR 1 $end $tddft imethod 1 isf 0 nexit 0 2 0 0 itda 0 idiag 1 istore 1 crit_e 1.d-10 crit_vec 1.d-8 lefteig DirectGrid $end $resp iprt 1 QUAD FNAC double pairs 1 1 2 1 1 2 2 norder 1 method 2 nfiles 1 ignore 1 noresp $end
Finite difference approach
To use the finite difference approach, nosym must be used to avoid the rotation of molecules. Currently, only C(1) group is permitted. For the example considered above, it turns out that the first two 1A2 excited states are the 1st and 4th excited states. However, if only use iexit=4 with iterative diagonalization, the initialization based on orbital energy difference (IPA) will miss the 2rd excited states, as its IPA is very large, viz.,
No. Pair ExSym ExEnergies f D<S^2> Dominant Excitations IPA Ova En-E1
1 A 2 A 4.3504 eV 0.0000 0.0000 88.9% CV(0): A( 8 )-> A( 9 ) 15.558 0.476 0.0000
2 A 3 A 9.3394 eV 0.0008 0.0000 92.4% CV(0): A( 6 )-> A( 9 ) 21.045 0.546 4.9890
3 A 4 A 9.3480 eV 0.1782 0.0000 85.8% CV(0): A( 7 )-> A( 9 ) 17.814 0.822 4.9975
4 A 5 A 11.3372 eV 0.0000 0.0000 92.3% CV(0): A( 5 )-> A( 9 ) 22.197 0.535 6.9868
5 A 6 A 11.6654 eV 0.3266 0.0000 92.9% CV(0): A( 8 )-> A( 10 ) 18.527 0.461 7.3150Thus, iexit=5 is used in the following inputs:
$COMPASS Title ch2o Basis 6-31GP Geometry C 0.00000000 -0.00000000 -0.53964037 O 0.00000000 0.00000000 0.68767663 H 0.00000000 0.93940400 -1.13178537 H 0.00000000 -0.93940400 -1.13178537 End geometry skeleton group c(1) nosym $END $xuanyuan direct schwarz $end $scf RHF charge 0 spin 1 THRESHCONV 1.d-10 1.d-8 OPTSCR 1 $end $tddft imethod 1 isf 0 iexit 5 itda 0 idiag 1 istore 1 crit_e 1.d-10 crit_vec 1.d-8 lefteig DirectGrid $end $resp iprt 1 QUAD FNAC double pairs 1 1 1 1 1 1 4 norder 1 method 2 nfiles 1 ignore 1 noresp $end
The corresponding input for the finite difference approach is to add two keywords: FDIF for specification and step followed by a real number for step size in finite difference. If the unit is bohr in the COMPASS part, the 'BOHR' should be added in the RESP part also.
$COMPASS Title ch2o Basis 6-31GP Geometry C 0.00000000 -0.00000000 -0.53964037 O 0.00000000 0.00000000 0.68767663 H 0.00000000 0.93940400 -1.13178537 H 0.00000000 -0.93940400 -1.13178537 End geometry skeleton group c(1) nosym $END $xuanyuan direct schwarz $end $scf RHF charge 0 spin 1 THRESHCONV 1.d-10 1.d-8 OPTSCR 1 iaufbau 0 $end $tddft imethod 1 isf 0 iexit 5 itda 0 idiag 1 istore 1 crit_e 1.d-10 crit_vec 1.d-8 lefteig AOKXC DirectGrid $end $resp iprt 1 QUAD FNAC #single #states #1 #1 1 2 double pairs 1 1 1 1 1 1 4 norder 1 method 2 nfiles 1 FDIF step 0.001 ignore 1 noresp $end
To use finite-difference, a script fdiff.py along with fdmol.py in bdf-pkg/source/tools/fdiff should be used as
./fbdiff.py run.sh xxx.inp > log
After the calculation is done, an output file xxx.out will present in the current directory. The log file saves the information during the calculations.
Illustration
Visualisation of the results can be achieved via the tools in bdf-pkg/source/tools/fdiff/NACplot.nb.
Static: t1_nac.PNG
Dynamic: t1_nac.GIF
