7.3 Time-Dependent Density Functional Theory (TDDFT)7.3.3 TDDFT within a Reduced Single-Excitation Space 7.3.5 Electron-Affinity (EA-) TDDFT

7.3.4 Spin-Flip TDDFT

(December 11, 2025)

The spin-flip approach provides a way to describe certain types of difficult multi-configurational states within a single-reference formalism. ⁷¹⁰ Krylov A. I.
Chem. Phys. Lett.
(2001), 338, pp. 375. Link ^, ⁷¹¹ Krylov A. I.
Chem. Phys. Lett.
(2002), 350, pp. 522. Link ^, ⁷¹² Krylov A. I.
Acc. Chem. Res.
(2006), 39, pp. 83. Link ^, ²⁰⁰ Casanova D., Krylov A. I.
Phys. Chem. Chem. Phys.
(2020), 22, pp. 4326. Link SF is particularly suitable for states that can be described as two electrons in two orbitals [(2e,2o)] or (3e,3o). The idea is to describe such target states as spin-flipping excitations (e.g., $\alpha\rightarrow\beta$ ) from a high-spin reference determinant (triplet or quartet). SF treatment can be combined with different correlation treatments (e.g., EOM-CC, ADC, CI, RAS-CI) ²⁰⁰ Casanova D., Krylov A. I.
Phys. Chem. Chem. Phys.
(2020), 22, pp. 4326. Link as well as with DFT. ¹¹⁹⁸ Shao Y., Head-Gordon M., Krylov A. I.
J. Chem. Phys.
(2003), 118, pp. 4807. Link ^, ¹¹³ Bernard Y. A., Shao Y., Krylov A. I.
J. Chem. Phys.
(2012), 136, pp. 204103. Link SF-TDDFT can describe the ground state as well as a few low-lying excited states. It can be used to describe diradicals, triradicals, single-molecule magnets, and, in some cases, bond-breaking and conical intersections. ²⁰⁰ Casanova D., Krylov A. I.
Phys. Chem. Chem. Phys.
(2020), 22, pp. 4326. Link

SF-DFT calculations are deployed by choosing an appropriate multiplicity of the reference state and setting SPIN_FLIP or MR_SPIN_FLIP to TRUE. SF-DFT is only used within Tamm-Dancoff approximation (RPA must be set to FALSE).

The original SF-DFT, formulated using collinear kernel, requires the functionals with substantial fraction of Hartree–Fock exchange. Best results are obtained using functionals with $\approx 50$ % Hartree–Fock exchange, ¹¹⁹⁸ Shao Y., Head-Gordon M., Krylov A. I.
J. Chem. Phys.
(2003), 118, pp. 4807. Link ^, ¹¹³ Bernard Y. A., Shao Y., Krylov A. I.
J. Chem. Phys.
(2012), 136, pp. 204103. Link behavior that was explained on theoretical grounds. ⁵⁸⁷ Huix-Rotllant M. et al.
Phys. Chem. Chem. Phys.
(2010), 12, pp. 12811. Link Becke’s half-and-half functional BH&HLYP has become something of a standard approach when using standard SF-TDDFT.

A SF-TDDFT method with a non-collinear exchange-correlation potential, originally developed by Ziegler and co-workers, ¹³⁷⁶ Wang F., Ziegler T.
J. Chem. Phys.
(2004), 121, pp. 12191. Link ^, ¹¹⁹² Seth M., Mazur G., Ziegler T.
Theor. Chem. Acc.
(2011), 129, pp. 331. Link has also been implemented. ¹¹³ Bernard Y. A., Shao Y., Krylov A. I.
J. Chem. Phys.
(2012), 136, pp. 204103. Link This non-collinear version sometimes improves upon collinear SF-TDDFT for excitation energies but contains a factor of spin density ( $\rho_{\alpha}-\rho_{\beta}$ ) in the denominator that sometimes causes stability problems.

The SF-DFT states may suffer from spin-contamination. This problem can be mitigated by using a spin-adapted version of SF-TDDFT (Section 7.2.3.3), ¹⁴⁹¹ Zhang X., Herbert J. M.
J. Chem. Phys.
(2015), 143, pp. 234107. Link or by using mixed-reference SF-TDDFT formulation described below.

Calculations of permanent and transition dipole moments are available for all SF-TDDFT variants. Wave-function analysis, analytic gradients, NACs, and SOCs are available for non-spin-adapted collinear and non-collinear formulations of SF-TDDFT.

The following examples illustrate SF-DFT capabilities available in Q-Chem: 7.3.11, 7.2.3.3, 7.3.9.1, 7.3.11, 9.8.4, and 7.3.11. Other related methods include SF-XCIS (Section 7.2.3.2), spin-adapted SF-CIS (Section 7.2.3.3), EOM-SF-CC (Section 7.10.5), SF-ADC (Section 7.11.7), and SF-RASCI (Section 7.12).

7.3.4.1 Mixed-reference SF-TDDFT

In MRSF-TDDFT, ⁷⁶⁰ Lee J. et al.
J. Chem. Phys.
(2018), 149, pp. 104101. Link ^, ⁷⁷² Lee S. et al.
J. Chem. Phys.
(2019), 150, pp. 184111. Link the reference state is constructed as an equal mixture of two Kohn–Sham determinants corresponding to the $M_{S}=+1$ and $M_{S}=-1$ components of the triplet state. The resulting mixed-reference reduced density matrix (RDM) defined as

\rho_{0}^{\rm MR}=\frac{1}{2}(\rho_{0}^{M_{S}=+1}+\rho_{0}^{M_{S}=-1})

(7.27)

is non-idempotent but the idempotency is restored through a complex transformation of spins of the singly occupied molecular orbitals. ⁷⁶⁰ Lee J. et al.
J. Chem. Phys.
(2018), 149, pp. 104101. Link By coupling the $\alpha\rightarrow\beta$ spin-flip excitations from the $M_{S}=+1$ component with the $\beta\rightarrow\alpha$ excitations from the $M_{S}=-1$ component, this approach recovers most of the determinants missing in conventional SF-TDDFT and by doing so restores spin-completeness.

The current implementation of MRSF-TDDFT is available for a ROHF reference (UNRESTRICTED = FALSE) and can describe both singlet and triplet states. MRSF-TDDFT employs a collinear exchange-correlation kernel; hence, hybrid functionals are recommended, particularly those with a larger fraction of Hartree-Fock exchange.

The orbital Hessian matrix is defined as ⁷⁷² Lee S. et al.
J. Chem. Phys.
(2019), 150, pp. 184111. Link

A_{pq,rs}^{(k)}=A_{pq,rs}^{(k)(0)}+A_{pq,rs}^{{}^{\prime}(k)}

(7.28)

where $k\in\{\text{S},\text{T}\}$ refers to a singlet or triplet state and

A_{pq,rs}^{(k)(0)}=U_{pq}^{k}\{\delta_{pr}F_{qs}^{\beta}-\delta_{qs}F_{pr}^{% \alpha}-c_{H}(pr|sq)\}U_{rs}^{(k)}.

(7.29)

The term $A_{pq,rs}^{{}^{\prime}(k)}$ takes care of the coupling between the determinants generated from the two triplet components and is given by

\displaystyle\begin{aligned} \displaystyle A_{pq,rs}^{{}^{\prime}(k)}&% \displaystyle=H_{p\underline{q},r\underline{s}}^{(k){\rm intra}}(U_{pq}^{\text% {CO1}}-U_{pq}^{\text{CO2}})(U_{rs}^{\text{CO1}}-U_{rs}^{\text{CO2}})\\ &\displaystyle\qquad+H_{\underline{p}q,\underline{r}s}^{(k){\rm intra}}(U_{pq}% ^{\text{O1V}}-U_{pq}^{\text{O2V}})(U_{rs}^{\text{O1V}}-U_{rs}^{\text{O2V}})\\ &\displaystyle\qquad+H_{pq,rs}^{(k){\rm inter}}(U_{pq}^{\text{CO1}}U_{rs}^{% \text{O2V}}+U_{pq}^{\text{CO2}}U_{rs}^{\text{O1V}}+U_{pq}^{O1V}U_{rs}^{\text{% CO2}}+U_{pq}^{\text{O2V}}U_{rs}^{\text{CO1}})\;.\end{aligned}

(7.30)

Here, $U_{pq}$ are dimensional transformation matrices and


$\displaystyle H_{pq,rs}^{(k)\text{intra}}$	$\displaystyle=\text{sgn}(k)c_{\text{H}}(ps\|rq)$	(7.31a)
$\displaystyle H_{pq,rs}^{(k)\text{inter}}$	$\displaystyle=\text{sgn}(k)c_{\text{H}}\big{[}(pq\|rs)-(pr\|sq)\big{]}$	(7.31b)

with

\text{sgn}(k)=\begin{cases}+1,&\text{if}~{}k=\text{S}\\ -1,&\text{if}~{}k=\text{T}\\ \end{cases}

(7.32)

and

\underline{p}=\begin{cases}\text{O2},&\text{if}~{}p=\text{O1}\\ \text{O1},&\text{if}~{}p=\text{O2}\\ \end{cases}

(7.33)

The dimensional transformation matrices ensure that the MRSF excitation subspace matches the dimensionality of the SF-TDDFT space for both $k=\text{S}$ and $k=\text{T}$ . Owing to this structure where the singlet and triplet spaces are completely decoupled from each other, MRSF-TDDFT, unlike the standard SF-TDDFT, solves separate eigenvalue problems for singlet and triplet manifolds.

By default, the calculation prints $\langle\hat{S}^{2}\rangle$ values for each state, TDDFT amplitudes, and transition dipole moments wrt the lowest SF state in each spin manifold.

Enabling keyword MRSF_DUMP = 1 prints additional information, including the Cartesian components of the permanent dipole moments (in a.u.) of each computed state, as well as the Cartesian components of the transition dipole moments (in a.u.), and the corresponding oscillator strengths between all computed SF states.

Example 7.3.11 illustrates MR-SF-TDDFT calculation for butadiene.

Note: MR-SF-TDDFT feature is being actively developed, so the keywords may change in the future release.

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7.3.4 Spin-Flip TDDFT

7.3.4.1 Mixed-reference SF-TDDFT