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

(July 4, 2026)

The spin-flip approach provides a way to describe certain types of difficult multi-configurational states within a single-reference formalism. ⁷²⁶ Krylov A. I.
Chem. Phys. Lett.
(2001), 338, pp. 375. Link ^, ⁷²⁷ Krylov A. I.
Chem. Phys. Lett.
(2001), 350, pp. 522. Link ^, ⁷²⁸ Krylov A. I.
Acc. Chem. Res.
(2006), 39, pp. 83. Link ^, ²⁰⁵ 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) ²⁰⁵ Casanova D., Krylov A. I.
Phys. Chem. Chem. Phys.
(2020), 22, pp. 4326. Link as well as with DFT. ¹²³⁹ Shao Y., Head-Gordon M., Krylov A. I.
J. Chem. Phys.
(2003), 118, pp. 4807. Link ^, ¹¹⁴ 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, systems with small HOMO-LUMO gaps, and, in some cases, bond-breaking and conical intersections. ²⁰⁵ 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 to 1 (regular SF-TDDFT) or 2 (MRSF-TDDFT). 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, ¹²³⁹ Shao Y., Head-Gordon M., Krylov A. I.
J. Chem. Phys.
(2003), 118, pp. 4807. Link ^, ¹¹⁴ Bernard Y. A., Shao Y., Krylov A. I.
J. Chem. Phys.
(2012), 136, pp. 204103. Link behavior that was explained on theoretical grounds. ⁶⁰³ 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, ¹⁴²⁴ Wang F., Ziegler T.
J. Chem. Phys.
(2004), 121, pp. 12191. Link ^, ¹²³³ Seth M., Mazur G., Ziegler T.
Theor. Chem. Acc.
(2011), 129, pp. 331. Link has also been implemented. ¹¹⁴ 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 addressed by using a spin-adapted version of SF-TDDFT (Section 7.2.3.3), ¹⁵⁴⁹ Zhang X., Herbert J. M.
J. Chem. Phys.
(2015), 143, pp. 234107. Link ^, ¹⁰¹⁶ Ojha A. K., Herbert J. M.
J. Chem. Phys.
(2026), 164, pp. 174102. 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. SOCs and wave-function analysis (NTOs and exciton descriptors) are available for for standard collinear and non-collinear formulations of SF-TDDFT and for MR-SFTDDFT. Analytic gradients and NACs 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: 7.3.4.2, 7.2.3.3, 7.3.10.1, 7.3.4.2, 9.8.4, and 7.3.4.2. Other related methods include SF-XCIS (Section 7.2.3.2), spin-adapted SF-CIS (Section 7.2.3.3), EOM-SF-CC (Section 7.9.6), SF-ADC (Section 7.10.8), and SF-RASCI (Section 7.11).

7.3.4.1 Mixed-reference SF-TDDFT

In MRSF-TDDFT, ⁷⁸¹ Lee J. et al.
J. Chem. Phys.
(2018), 149, pp. 104101. Link ^, ⁷⁹³ 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 the singly occupied spin-orbitals. ⁷⁸¹ 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, MRSF-SFTDDFT 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. It also affords calculations of transition properties including matrix elements of the dipole and angular momentum operators, SOCs and wave-function analysis. 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 ⁷⁹³ 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 in which the singlet and triplet spaces are completely decoupled from each other, MRSF-TDDFT, in contrast to the original 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 STS_MOM=TRUE 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. To compute SOC, use CALC_SOC=2.

Example 7.3.4.2 illustrates MR-SF-TDDFT calculation for butadiene inlcuing state and transition properties and exciton analysis.

7.3.4.2 Examples

Example 7.8 SF-TDDFT SP calculation of the 6 lowest states of the TMM diradical using recommended 50-50 functional.

$molecule
   0 3
   C
   C  1  CC1
   C  1  CC2   2  A2
   C  1  CC2   2  A2     3  180.0
   H  2  C2H   1  C2CH   3    0.0
   H  2  C2H   1  C2CH   4    0.0
   H  3  C3Hu  1  C3CHu  2    0.0
   H  3  C3Hd  1  C3CHd  4    0.0
   H  4  C3Hu  1  C3CHu  2    0.0
   H  4  C3Hd  1  C3CHd  3    0.0

   CC1    = 1.35
   CC2    = 1.47
   C2H    = 1.083
   C3Hu   = 1.08
   C3Hd   = 1.08
   C2CH   = 121.2
   C3CHu  = 120.3
   C3CHd  = 121.3
   A2    = 121.0
$end

$rem
   EXCHANGE          gen
   BASIS             6-31G*
   SCF_GUESS         core
   SCF_CONVERGENCE   10
   MAX_SCF_CYCLES    100
   SPIN_FLIP         1
   CIS_N_ROOTS       6
   CIS_CONVERGENCE   10
   MAX_CIS_CYCLES    100
$end

$xc_functional
   X  HF    0.50
   X  S     0.08
   X  B     0.42
   C  VWN   0.19
   C  LYP   0.81
$end

Example 7.9 SF-TDDFT with non-collinear exchange-correlation functional for low-lying states of $\rm CH_{2}$ .

$comment
  Non-collinear SF-DFT calculation for CH2 at 3B1 state geometry from
  EOM-CCSD(fT) calculation
$end

$molecule
   0 3
   C
   H  1 rCH
   H  1 rCH  2 HCH

   rCH = 1.0775
   HCH = 133.29
$end

$rem
   EXCHANGE             PBE0
   BASIS                cc-pVTZ
   SPIN_FLIP            1
   WANG_ZIEGLER_KERNEL  TRUE
   SCF_CONVERGENCE      10
   CIS_N_ROOTS          6
   CIS_CONVERGENCE      10
$end

Example 7.10 SF-TDDFT calculation with collinear B5050LYP for $p$ -benzyne with wave-function analysis (natural orbitals and NTOs) performed by libwfa.

$comment
Para-benzyne diradical
Equilibrium singlet state geom from:
J. Chem. Phys. 136, 204103 (2012)
Enu = 187.2138176166 hartree
$end

$molecule
   0 3
   H   2.145810  -1.225292   0.000000
   C   1.201382  -0.709285   0.000000
   C   1.201382   0.709285   0.000000
   H   2.145810   1.225292   0.000000
   C   0.000000   1.335664   0.000000
   C  -1.201382   0.709285   0.000000
   H  -2.145810   1.225291   0.000000
   C  -1.201382  -0.709285   0.000000
   H  -2.145810  -1.225291   0.000000
   C   0.000000  -1.335664   0.000000
$end

$rem
   METHOD          = b5050lyp
   BASIS           = 6-31G*
   CIS_N_ROOTS     = 4
   SPIN_FLIP       = true
   NEW_DFT         = true
   STATE_ANALYSIS  = true
   WFA_REF_STATE   = 1
   MOLDEN_FORMAT   = true
   NTO_PAIRS       = 4
$end

Example 7.11 Mixed-reference SF-TDDFT/BHHLYP calculation of butadiene including exciton analysis.

$comment
- Singlet states of butadiene using MRSF-TDDFT
- State-to-state transition dipole moments, and oscillator strengths
- Density matrix based analysis of the calculated MRSF states
$end

$molecule
0 3
C   -0.410990219   -1.798958603   0.000000000
C   -0.559475119   -0.470573512   0.000000000
C    0.410990219    1.798958603   0.000000000
C    0.559475119    0.470573512   0.000000000
H   -1.263114858   -2.463113554   0.000000000
H    0.571968115   -2.252448386   0.000000000
H    1.263114858    2.463113554   0.000000000
H   -1.554995711   -0.040726039   0.000000000
H   -0.571968115    2.252448386   0.000000000
H    1.554995711    0.040726039   0.000000000
$end

$rem
jobtype            SP
unrestricted       FALSE
basis              6-31G*
exchange           bhhlyp
correlation        none
scf_guess          core
SCF_CONVERGENCE    10
SCF_ALGORITHM      DIIS
MAX_SCF_CYCLES     100
PRINT_ORBITALS     false   ! do not print orbitals
SPIN_FLIP          2       ! keyword to perform MRSF-TDDFT calculation
CIS_N_ROOTS        4       ! 4 MRSF states requested
CIS_SINGLETS       TRUE    ! asking for singlet states
CIS_TRIPLETS       FALSE
CIS_CONVERGENCE    8
MAX_CIS_CYCLES     100
STS_MOM            TRUE    ! calculate state-to-state transition moments
STATE_ANALYSIS     TRUE    ! density matrix based analysis of MRSF states
NTO_PAIRS          2       ! write 2 NTO pairs per excited states
$end

Search Results

7.3.4 Spin-Flip TDDFT

7.3.4.1 Mixed-reference SF-TDDFT

7.3.4.2 Examples