Mixed-Reference Spin-Flip Time-Dependent Density Functional Theory: Multireference Advantages with the Practicality of Linear Response Theory

Abstract

The density functional theory (DFT) and linear response (LR) time-dependent (TD)-DFT are of the utmost importance for routine computations. However, the single reference formulation of DFT suffers in the description of open-shell singlet systems such as diradicals and bond-breaking. LR-TDDFT, on the other hand, finds difficulties in the modeling of conical intersections, doubly excited states, and core-level excitations. In this Perspective, we demonstrate that many of these limitations can be overcome by recently developed mixed-reference (MR) spin-flip (SF)-TDDFT, providing an alternative yet accurate route for such challenging situations. Empowered by the practicality of the LR formalism, it is anticipated that MRSF-TDDFT can become one of the major workhorses for general routine tasks.

Scheme 1. Challenging Cases for DFT and LR-TDDFT

Figure 1

Upper panel shows the two references of MRSF-TDDFT denoted by black and red arrows. The zeroth-order MR-RDM which combines M_S = +1 and −1 RDMs is used in MRSF-TDDFT, while only the M_S = +1 RDM are used in SF-TDDFT. In the lower panel, the electronic configurations that can be generated by spin-flip linear responses from the MR-RDM are given by blue, black, and red arrows. The blue ones are generated from both references, which require a symmetrization procedure to eliminate OO-type spin contamination. The black and red ones are generated from M_S = +1 and −1 references, respectively. By contrast, those of SF-TDDFT are only the blue and black ones. Configurations that cannot be obtained even in MRSF-TDDFT are denoted by gray arrows.

Figure 2

(a) The concept of spinor transformation, which combines two references into a single hypothetical reference. (b) The three axes of response theory, spin-flip idea, and multireference components, where the shaded area represents the MRSF-TDDFT.

Figure 3

(a) The characteristic topology of the conical intersection (left) and linear intersection (right) between two potential energy surfaces of the S₁ and S₀ states. While MRSF-TDDFT can calculate both states, the combination of LR-TDDFT and DFT is needed for them. (b) The S₁ (red) and S₀ (blue) energies of CI_tw–BLA of trans-penta-2,4-dieniminium cation (PSB3) were calculated around a loop with the radius of 0.01 Å. The branching plane vectors were calculated using the algorithm by Maeda et al., which yields orthogonal GDV and DCV. (c) NAC vectors (or DCVs) at the MECI geometries of PSB3, where the left and right images are obtained by MRCISD and MRSF-TDDFT/BH&HLYP, respectively. The MECI geometries were optimized by MRCISD in ref (48). The numbers in parentheses display the inner products between the MRSF and the MRCISD NAC vectors and the ratio |NAC_MRSF|/|NAC_MRCISD| of the norms of the NAC vectors, respectively.

Figure 4

1¹B_u⁺ (red) and 2¹A_g^– (black) energy profiles along the minimum-energy paths (MEPs) of trans-butadiene by (a) SA-CASSCF(4,4) (solid lines), δ-CR-EOMCC(2,3) (dashed lines), and XMS-CASPT2(4,4) (dotted lines). (b) The MEPs by MRSF/BH&HLYP (solid lines) and TDDFT/B3LYP (dashed lines). All calculations taken from ref (42) were done with cc-pVTZ. The BLA coordinate is defined as the difference between the average length of single bonds and the average length of double bonds.

Figure 5

Time evolution of the adiabatic S₀ (black), S₁ (red), and S₂ (blue) populations for (a) the first 100 fs and (b) the entire 2 ps duration of the NAMD simulations taken from ref (86). The light blue curve in panel a and the green curve in panel b represent fittings of the S₂ and S₁ populations by a monoexponential function, respectively. Panel c compares the S₂ (blue) and S₁ (red) PES profiles along the MRSF MEPs (solid lines) with the EOM-CCSD curves (dashed lines). The molecular structure with atom numbers is given in the inset. Panel d shows the S₂ and S₁ PES profiles obtained with the 3SA-CASSCF(10,8) (solid lines) and the eXtended Multi-State Complete Active Space second-order Perturbation Theory (XMS-CASPT2, dashed lines). MEPs on the S₂ (blue) and S₁ (red) PESs optimized using the nudged elastic band (NEB)^, method in connection with MRSF-TDDFT and connecting the FC region, the CI_21,BLA, and the S_1, min geometries; the respective BLA values are given parenthetically. The BLA coordinate is defined as the difference between the average increments of the lengths of the double bonds and the decrease of the single bond, BLA = , where ΔR’s are displacements with respect to the S₀ equilibrium geometry. For all other electronic structure methods, the MRSF-TDDFT MEP geometries are utilized by employing a 6-31G* basis set with Cs symmetry restriction. Adopted with permission from ref (84). Copyright 2009 American Chemical.

Figure 6

(a) The core-hole relaxation is accomplished by replacing the two singly occupied open (O1 and O2) orbitals of the ROHF reference with other ones. While the O1 is replaced with 1s, the O2 can be (a) the original LUMO orbitals. (b) The simulated core to valence hole spectra without an empirical shift and corresponding orbital transition diagrams of valence excited states of thymine by ΔCHP-MRSF(R)/BH&HLYP with aug-pcX-2/aug-pcseg-1, where the π and n holes are represented by blank circles. Oscillator strengths for ΔCHP-MRSF(R) are taken from CHP-MRSF(R) results. The core to π, n, and π* + 1 holes are represented by red, blue, and green colors. The solid and dotted lines represent the excitations from 1s(O8) and 1s(O7) core, respectively (see Figure 5 for atom numbering). Here, we introduce a double hole particle relaxation, which relaxes a core and a valence hole at the same time. For example, the final configuration of 1s¹n²π*¹, which is 1s → n core excitation of n_O8π* state can be accomplished by the π* → n response excitation from the reference double hole particle CHP configuration of 1s¹n¹π*². Adopted from with permission from ref (43). Copyright 2022 American Chemical Society.

Figure 7

3D potential energy surfaces of S₁ (red) and S₂ (blue) states around conical intersection (in D_3h system) between them. The energy paths that connect the stationary points on the lowest-energy triplet (blue) and two lowest-energy singlet (black and red) are shown in line. The triplet energy paths are projected on the bottom. The energies of MRSF-TDDFT and XMS-CASPT2 (in parentheses) are given next to the stationary points. The point with D_3h symmetry is marked as D_3h. The rest of the points have C_2v symmetry. ϕ is defined as the torsion angle of CH₂ while, δ is defined as the difference between two identical C–C bonds out of three (denoted as A) and the other bond (denoted as B). All data were taken from ref (40).