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

(September 23, 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) afford incorrect dimensionality for the crossing seam. ¹⁴⁷⁶ Zhang X., Herbert J. M.
J. Chem. Phys.
(2015), 142, pp. 064109. Link ^, ¹⁴⁷⁸ Zhang X., Herbert J. M.
J. Chem. Phys.
(2021), 155, pp. 124111. Link The TDDFT-1D method address this limitation by combining DFT and CI to achieve smooth crossings of S ${}_{0}$ and S ${}_{1}$ states, ¹²⁹⁶ Teh H.-H., Subotnik J. E.
J. Phys. Chem. Lett.
(2019), 10, pp. 3426. Link while keeping the cost comparable to that of 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 CI Hamiltonian gives rise to the CIS-1D method if one starts from a Hartree-Fock reference, or TDDFT-1D 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 canonical 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 a CI Hamiltonian is constructed,

\mathbf{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)

in the basis of the reference state along with all singly substituted determinants ( $\text{\textbar}{\psi_{i}^{a}}\rangle$ ) and one unique doubly substituted determinant ( $\text{\textbar}{\psi_{h\bar{h}}^{l\bar{l}}}\rangle$ ). Diagonalizing $\mathbf{H}$ affords the CIS-1D or TDDFT-1D states. At geometries that are far from crossing points, excitation energies computed with TDDFT-1D are generally at the same level of accuracy as TDDFT with the Tamm-Dancoff approximation (TDA). 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 are available for the TDDFT-1D method. ⁵⁴ Athavale V. et al.
J. Chem. Phys.
(2022), 157, pp. 244110. Link This facilitates the optimization of minimum-energy crossing points (MECPs) between electronic states.

7.3.9.1 Job Control for TDDFT-1D

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

CIS1D_N_ROOTS

CIS1D_N_ROOTS
       Sets the number of CIS-1D and TDDFT-1D states to calculate. The lowest eigenstate of $\mathbf{H}$ is the ground state, 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
       Convergence criterion for the minimization of $E_{d}$ , used to optimize the orbitals. The stopping criterion is set to $10^{-\text{CIS1D\_ED\_CONVERGENCE}}$ .
TYPE:
       INTEGER
DEFAULT:
       7
OPTIONS:
       $n$ Convergence achieved when the error is lower than $10^{-n}$ .
RECOMMENDATION:
       Convergence on the roots of $\mathbf{H}$ is controlled by CIS_CONVERGENCE, which is set to 9 by default for a CIS-1D or TDDFT-1D calculation. If CIS_CONVERGENCE is increased then CIS1D_ED_CONVERGENCE should also be increased.

CIS1D_MAX_CYCLES

CIS1D_MAX_CYCLES
       Maximum number of Davidson iterations to diagonalize $\mathbf{H}$ .
TYPE:
       INTEGER
DEFAULT:
       60
OPTIONS:
       $n$ Use up to $n$ iterations.
RECOMMENDATION:
       The memory allocated for the Davidson iterations (used to diagonalize $\mathbf{H}$ ) is $(n_{\text{occ}}n_{\text{virt}}+\text{2})$ (CIS1D_MAX_CYCLES)(CIS1D_N_ROOTS)/2 double-precision numbers, where $n_{\text{occ}}$ and $n_{\text{virt}}$ are the number of occupied and virtual orbitals, respectively. Larger values of CIS1D_MAX_CYCLES may require increasing MEM_TOTAL.

CIS1D_SCALE_GD

CIS1D_SCALE_GD
       Scaling factor for the coupling matrix element $\langle\psi_{0}|\hat{H}|\psi_{h\bar{h}}^{l\bar{l}}\rangle$ between the reference state and the doubly substituted Slater determinant.
TYPE:
       INTEGER
DEFAULT:
       100
OPTIONS:
       $n$ $0\leq n\leq 100$ , The coupling element is scaled by $n/100$ .
RECOMMENDATION:
       There is no rigorous way to determine the coupling elements in $\mathbf{H}$ , since the Kohn-Sham determinant is not a true wave function, so the couplings are computed using a “pseudo-wave function” approach. ⁹⁸⁵ Ou Q., Alguire E. C., Subotnik J. E.
J. Phys. Chem. B
(2015), 119, pp. 7150. Link ^, ¹⁴⁷⁶ Zhang X., Herbert J. M.
J. Chem. Phys.
(2015), 142, pp. 064109. Link 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
       Scaling factor for the coupling matrix element $\langle\psi_{i}^{a}|H|\psi_{h\bar{h}}^{l\bar{l}}\rangle$ between the single excitations and the lone double excitation.
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

7.3.9.2 Examples of TDDFT-1D

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 Optimizing the 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 for TDDFT-1D

7.3.9.2 Examples of TDDFT-1D