10.3 Nonadiabatic couplings
Nonadiabatic (derivative) couplings are available for both CIS and TDDFT. The CIS nonadiabatic couplings can be obtained from direct differentiations of the wavefunctions with respect to nuclear positions [470, 422, 471]. For TDDFT, the same procedure can be carried out to calculate the approximate nonadiabatic couplings, in what has been termed the “pseudo-wavefunction” approach [422]. More rigorous TDDFT nonadiabatic couplings derived from quadratic response theory are also available, although they are subject to certain undesirable singularities that arise from the ubiquitous adiabatic approximation in TDDFT [472, 473].
In order to perform nonadiabatic coupling calculations, the $derivative_coupling section must be given:
$derivative_coupling
one line comment
$end
Here, are the states among which nonadiabatic couplings will be calculated and 
refers to the HF or DFT reference state.
10.3.1 Job Control
CIS_DER_COUPLE
Determines whether we are calculating nonadiabatic couplings.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Calculate nonadiabatic couplings.
FALSE
Don’t calculate nonadiabatic couplings.
RECOMMENDATION:
None.
CIS_DER_NUMSTATE
Determines among how many states we calculate nonadiabatic couplings.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
0
Don’t calculate nonadiabatic couplings.
Calculate
pairs of nonadiabatic couplings.
RECOMMENDATION:
None.
SET_QUADRATIC
Determines whether to include full quadratic reponse contributions for TDDFT.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
TRUE
Include full quadratic reponse contributions for TDDFT.
FALSE
Use pseudo-wavefunction approach.
RECOMMENDATION:
The pseudo-wavefunction approach is usually accurate enough. Consult Refs. Zhang:2015 and Ou:2015 for additional guidance.
10.3.2 Examples
Nonadiabatic couplings for TDDFT using pseudo-wavefunction approach:
Example 10.209 Nonadiabatic couplings among the lowest five singlet states of ethylene using B3LYP DFT/TDDFT
$molecule
0 1
C 1.85082356 -1.78953123 0.00000000
H 2.38603593 -2.71605577 0.00000000
H 0.78082359 -1.78977646 0.00000000
C 2.52815456 -0.61573833 0.00000000
H 1.99294220 0.31078621 0.00000000
H 3.59815453 -0.61549310 0.00000000
$end
$rem
jobtype sp
cis_n_roots 4
cis_triplets false
set_iter 50
CIS_DER_NUMSTATE 5
CIS_DER_COUPLE true
exchange b3lyp
basis 6-31G*
$end
$derivative_coupling
0 is the reference state
0 1 2 3 4
$end
Nonadiabatic couplings for spin-flip TDDFT:
Example 10.210 Nonadiabatic couplings between 
and 
states of ethylene using BH&HLYP spin-flip TDDFT
$molecule
0 3
C 1.85082356 -1.78953123 0.00000000
H 2.38603593 -2.71605577 0.00000000
H 0.78082359 -1.78977646 0.00000000
C 2.52815456 -0.61573833 0.00000000
H 1.99294220 0.31078621 0.00000000
H 3.59815453 -0.61549310 0.00000000
$end
$rem
jobtype sp
spin_flip true
unrestricted true
cis_n_roots 4
cis_triplets false
set_iter 50
CIS_DER_NUMSTATE 2
CIS_DER_COUPLE true
exchange bhhlyp
basis 6-31G*
$end
$derivative_coupling
comment
1 3
$end
Nonadiabatic couplings for TDDFT including full quadratic reponse:
Example 10.211 Nonadiabatic couplings between and 
states of ethylene using B3LYP TDDFT with full reponse
$molecule
0 1
C 1.85082356 -1.78953123 0.00000000
H 2.38603593 -2.71605577 0.00000000
H 0.78082359 -1.78977646 0.00000000
C 2.52815456 -0.61573833 0.00000000
H 1.99294220 0.31078621 0.00000000
H 3.59815453 -0.61549310 0.00000000
$end
$rem
jobtype sp
cis_n_roots 4
cis_triplets false
rpa true
set_iter 50
CIS_DER_NUMSTATE 2
CIS_DER_COUPLE true
exchange b3lyp
basis 6-31G*
set_quadratic true #include full quadratic response
$end
$derivative_coupling
comment
1 2
$end