7.3 Time-Dependent Density Functional Theory (TDDFT)7.3.8 Spin-Orbit Coupling 7.3.10 Examples

7.3.9 CIS-1D and TDDFT-1D

(May 21, 2025)

Propagating nonadiabatic systems forward in time remains challenging particularly at crossings between the ground state ( $S_{0}$ ) and the first excited state ( $S_{1}$ ). For $S_{1}/S_{0}$ conical intersections, conventional electronic structure methods (such as HF/CIS and DFT/TDDFT) recover the wrong dimensionality of the crossing seam. The TDDFT-1D method address this limitation by combining DFT and configuration interaction to achieve smooth crossings of $S_{0}$ and $S_{1}$ states ¹²⁹³ Teh H.-H., Subotnik J. E.
J. Phys. Chem. Lett.
(2019), 10, pp. 3426. Link while keeping the cost of the calculation remains low (comaprable to a standard TDDFT calculation).

The idea behind the CIS-1D and the TDDFT-1D method is to diagonalize a configuration interaction Hamiltonian whose basis includes the reference state, all singly excited configurations, and one unique doubly excited configuration. This unique double (the 1D in CIS-1D/TDDFT-1D) in the configuration interaction Hamiltonian gives rise to the CIS-1D method if one starts from a Hartree-Fock reference, and the TDDFT-1D method if one starts from a Kohn-Sham reference.

The doubly excited configuration is formed by exciting a pair of electrons from the HOMO ( $h$ ) to the LUMO ( $l$ ). To find the unique double, the cannonical molecular orbitals are rotated such that the energy of the doubly excited configuration ( $E_{d}$ ) is minimized ⁵⁵ Athavale V., Teh H.-H., Subotnik J. E.
J. Chem. Phys.
(2021), 155, pp. 154105. Link .

From these optimized orbitals, the following configuration interaction Hamiltonian is constructed, as explained previosly, in the basis of the reference state, all singles $\text{\textbar}{\psi_{i}^{a}}\rangle$ , and one unique double $\text{\textbar}{\psi_{h\bar{h}}^{l\bar{l}}}\rangle$ :

\bm{H}=\begin{pmatrix}\epsilon_{0}&0&\left\langle\psi_{0}|H|\psi_{h\bar{h}}^{l% \bar{l}}\right\rangle\\ 0&\left\langle\psi_{i}^{a}|H|\psi_{j}^{b}\right\rangle&\left\langle\psi_{i}^{a% }|H|\psi_{h\bar{h}}^{l\bar{l}}\right\rangle\\ \left\langle\psi_{0}|H|\psi_{h\bar{h}}^{l\bar{l}}\right\rangle&\left\langle% \psi_{h\bar{h}}^{l\bar{l}}|H|\psi_{i}^{a}\right\rangle&\left\langle\psi_{h\bar% {h}}^{l\bar{l}}|H|\psi_{h\bar{h}}^{l\bar{l}}\right\rangle\end{pmatrix}.

(7.47)

Diagonalizing the above Hamiltonian yields the CIS-1D or TDDFT-1D states. The excitation energies computed with TDDFT-1D are generally at the same level of accuracy as TDDFT with the Tamm-Dancoff approximation (TDA) at geometries far from crossing points. However, TDDFT-1D’s key advantage appears near ground and excited state crossing points, where it produces smooth potential energy surfaces.

To facilitate nonadiabatic dynamics simulations, analytical gradients and derivative couplings ⁵⁴ Athavale V. et al.
J. Chem. Phys.
(2022), 157, pp. 244110. Link are available for the TDDFT-1D method. Further, minimum energy crossing points (MECPs) can be located for TDDFT-1D states.

7.3.9.1 Job Control and examples for TDDFT-1D

The rem variables for TDDFT-1D calculations are similar to those for CIS calculations. The following rem variables are used to control the TDDFT-1D calculation:

CIS1D_N_ROOTS

CIS1D_N_ROOTS
       Sets the number of CIS-1D and TDDFT-1D states to calculate. The lowest CIS-1D/TDDFT-1D state is the ground state, which is followed by excited states.
TYPE:
       INTEGER
DEFAULT:
       0 Do not calculate any CIS-1D or TDDFT-1D states.
OPTIONS:
       $n$ $n>0$ , Calculate the lowest $n$ CIS-1D or TDDFT-1D states.
RECOMMENDATION:
       None

CIS1D_ED_CONVERGENCE

CIS1D_ED_CONVERGENCE
       The first step in TDDFT-1D is to find the optimized orbitals for the double by minimizing the energy of the doubly excited configuration, $E_{d}$ . This variable controls the convergence criterion for the minimization of $E_{d}$ . The orbitals are optimized when the error is lower than $10^{-\text{CIS1D\_ED\_CONVERGENCE}}$ .
TYPE:
       INTEGER
DEFAULT:
       7
OPTIONS:
       $n$ Convergence achieved when the error is lower than $10^{-n}$ .
RECOMMENDATION:
       The convergence criterion for the roots of the CIS-1D and TDDFT-1D calculations is set by the CIS_CONVERGENCE rem variable, which is set to 9 by default for a CIS-1D/TDDFT-1D calculation. If CIS_CONVERGENCE is increased, then CIS1D_ED_CONVERGENCE should also be increased.

CIS1D_SCALE_GD

CIS1D_SCALE_GD
       The coupling element between the ground and doubly excited configuration in the Hamiltonian in Eq. (7.47), ( $\langle\psi_{0}|H|\psi_{h\bar{h}}^{l\bar{l}}\rangle$ ) are scaled by this factor.
TYPE:
       INTEGER
DEFAULT:
       100
OPTIONS:
       n $0\leq n\leq 100$ , The coupling element is scaled by $n/100$ .
RECOMMENDATION:
       Since the KS reference is not a true wave function, there is no rigorous way to determine the coupling elements in the configuration interaction Hamiltonian. It sometimes becomes necessary to scale the coupling elements to obtain accurate potential energy surfaces, and the scaling factor has to be determined with some benchmarking ⁵⁵ Athavale V., Teh H.-H., Subotnik J. E.
J. Chem. Phys.
(2021), 155, pp. 154105. Link .

CIS1D_SCALE_SD

CIS1D_SCALE_SD
       The coupling element between the singles and the double in the Hamiltonian in Eq. (7.47), ( $\langle\psi_{i}^{a}|H|\psi_{h\bar{h}}^{l\bar{l}}\rangle$ ) are scaled by this factor.
TYPE:
       INTEGER
DEFAULT:
       100
OPTIONS:
       n $0\leq n\leq 100$ , The coupling element is scaled by $n/100$ .
RECOMMENDATION:
       Same as CIS1D_SCALE_GD.

CIS1D_STATE_DERIV

CIS1D_STATE_DERIV
       Selects the CIS-1D/TDDFT-1D state for which gradients are calculated. This is useful for jobs such as geometry optimizations.
TYPE:
       INTEGER
DEFAULT:
       -1 Does not select any state
OPTIONS:
       $n$ $n\geq 0$ , Selects the $n$ th CIS-1D/TDDFT-1D state.
RECOMMENDATION:
       None

CIS1D_DER_NUMSTATE

CIS1D_DER_NUMSTATE
       Determines the number of states for which derivative couplings are to be calculated. The states are specified in the $derivative_coupling section
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       0 Do not calculate derivative couplings. $n$ Calculate $n(n-1)/2$ pairs of derivative couplings.
RECOMMENDATION:
       None

Example 7.23 TDDFT-1D calculation of water using the B3LYP functional for the lowest 5 states.

$molecule
0 1
H 0.96 0 0
O 0.000000000000000 0.000000000000000 0.000000000000000
H -0.240364803892264 0.929421734762983 0
$end

$rem
METHOD    b3lyp
BASIS     cc-pvdz
SYM_IGNORE  true
cis1d_n_roots 5
$end

Example 7.24 TDDFT-1D calculation of the gradient of the 1st excited state of water in the presence of external charges

$molecule
0 1
H 0.96 0 0
O 0.000000000000000 0.000000000000000 0.000000000000000
H -0.240364803892264 0.929421734762983 0
$end

$rem
METHOD    b3lyp
BASIS     cc-pvdz
SYM_IGNORE  true
cis1d_n_roots 4
cis1d_state_deriv 1
jobtype force
$end

$external_charges
0.0 0.0 -1.0 -1.0
0.0 0.0 1.2 1.0
0.0 0.0 -2.0 -1.0
0.0 0.0 2.2 1.0
$end

Example 7.25 Minimum energy crossing point (MECP) between the ground and the 1st excited state of ethylene using the $\omega$ B97X functional.

$rem
geom_opt_driver optimize
scf_convergence 10
thresh_diis_switch 9
cis_convergence 8
jobtype opt
mecp_opt      true
mecp_methods  branching_plane
MECP_PROJ_HESS true
mecp_state1  [0,0]
mecp_state2  [0,1]
unrestricted false
calc_nac 1
scf_algorithm diis_gdm
cis1d_n_roots 5
exchange wb97x
basis 6-31G*
symmetry_ignore true
$end

$molecule
0 1
C         0.0188526101    0.0181382820   -0.3533168752
C         1.3813178426    0.0038321438   -0.0594695222
H         2.0062266305   -0.8967843506    0.0891927721
H        -0.3788613100   -0.7769905091   -1.0012547284
H        -0.0243170648   -0.6123230262    0.6288689086
H         1.9213240937    0.9262667800    0.1943673040
$end

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7.3.9 CIS-1D and TDDFT-1D

7.3.9.1 Job Control and examples for TDDFT-1D