Expanding Horizons in Quantum Chemical Studies: The Versatile Power of MRSF-TDDFT

Conspectus

While traditional quantum chemical theories have long been central to research, they encounter limitations when applied to complex situations. Two of the most widely used quantum chemical approaches, Density Functional Theory (DFT) and Time-Dependent Density Functional Theory (TDDFT), perform well in cases with relatively weak electron correlation, such as the ground-state minima of closed-shell systems (Franck–Condon region). However, their applicability diminishes in more demanding scenarios. These limitations arise from the reliance of DFT on a single-determinantal framework and the inability of TDDFT to capture double and higher excited configurations in its response space.

The recently developed Multi-Reference Spin-Flip Time-Dependent Density Functional Theory (MRSF-TDDFT) successfully overcomes these challenges, pushing the boundaries of DFT methods. MRSF-TDDFT is exceptionally versatile, making it suitable for various applications, including bond-breaking and bond-forming reactions, open-shell singlet systems such as diradicals, and a more accurate depiction of transition states. It also provides the correct topology for conical intersections (CoIns) and incorporates double excitations into the response space for a more precise description of excited states. With the help of its formal framework, core-hole relaxation for accurate X-ray absorption prediction can be also done readily. Notably, MRSF-TDDFT achieves an equal footing description of ground and excited states, with its dual-reference framework ensuring a balanced treatment of both dynamic and nondynamic electron correlations for high accuracy.

In predictive tasks, such as calculating adiabatic singlet–triplet gaps, MRSF-TDDFT achieves accuracy comparable to that of far more computationally expensive coupled-cluster methods. The missing doubly excited state of H₂ observed in TDDFT is accurately captured by MRSF-TDDFT, which also reproduces the correct asymptotic bond-breaking potential energy surface. Furthermore, the CoIns of butadiene, missed by both TDDFT and Complete-Active Space Self-Consistent Field (CASSCF) methods, are successfully recovered by MRSF-TDDFT, achieving results consistent with high-level theories, an important aspect for successful study of photochemical processes. Additionally, the common issue of CASSCF overestimating bright states (ionic states) due to the missing dynamic correlation is effectively resolved by MRSF-TDDFT.

Despite its numerous advancements, MRSF-TDDFT retains the computational efficiency of conventional TDDFT, making it a practical tool for routine calculations. In addition, it has been demonstrated that the prediction accuracy of MRSF-TDDFT can be further enhanced through the development of tailor-made exchange-correlation functionals, paving the way for the creation of new, specialized functionals. Consequently, with its remarkable versatility, high accuracy, and computational practicality, this innovative method significantly expands scientists’ ability to explore complex molecular behaviors and design advanced materials, including applications in photobiology, organic LEDs, photovoltaics, and spintronics, to name a few.

Key References

• Lee S.; Filatov M.; Lee S.; Choi C. H.. Eliminating spin-contamination of spin-flip time dependent density functional theory within linear response formalism by the use of zeroth-order mixed-reference (MR) reduced density matrix. J. Chem. Phys. 2018, 149, 104101. . The main derivations of MRSF-TDDFT are presented along with its hypothetical single-reference concept. It introduces the multireference concept into linear response theory for the first time.(1)

• Lee S.; Kim E. E.; Nakata H.; Lee S.; Choi C. H.. Efficient implementations of analytic energy gradient for mixed-reference spin-flip time-dependent density functional theory (MRSF-TDDFT). J. Chem. Phys. 2019, 150, 184111. . The derivations of MRSF-TDDFT gradient equations are presented.(2)

• Park W.; Komarov K.; Lee S.; Choi C. H.. Mixed-Reference Spin-Flip Time-Dependent Density Functional Theory: Multireference Advantages with the Practicality of Linear Response Theory. J. Phys. Chem. Lett. 2023, 14, 8896–8908. This perspective provides an in-depth theoretical analysis of MRSF-TDDFT. It highlights its potential to become a key tool for a wide range of routine tasks.³

• Shostak S.; Park W.; Oh J.; Kim J.; Lee S.; Nam H.; Filatov M.; Kim D.; Choi C. H.. Ultrafast excited state aromatization in dihydroazulene. J. Am. Chem. Soc. 2023, 145, 1638–1648. The dynamical aspects of the aromatization process in excited states have been demonstrated for the first time using MRSF-TDDFT in combination with NAMD. A competition between Bond-Length Alternation (BLA) and bond-equilization dynamics renders the system aromatic within 200 fs.⁴

1. Introductions

Quantum chemical theories are essential for advancing science and technology by providing a deep understanding of molecules and materials. However, they face limitations in certain scenarios-such as covalent bond breaking, diradicals, and conical intersections (CoIns)-which are challenging to address. Specifically, linear conjugated π-electron systems, such as polyenes and β-carotene, exemplified by trans-butadiene in the present account, play a critical role in photosynthesis as light-harvesting pigments as well as vision process. Similarly, the photodynamics of nucleobases, such as thymine, are of significant interest due to the potential solar UV damage inflicted on DNA and RNA. However, accurately describing the underlying potential energy surfaces of multiple electronic states remains a considerable challenge. Our interest in adopting these theories stems from their use in nonadiabatic molecular dynamics (NAMD) simulations, which deal with the dynamics of electronic state-to-state transitions. Surprisingly, we found that the major quantum chemical theories are not sufficiently robust for NAMD, which was one of the main motivations for developing a new quantum theory.

To be applicable to NAMD, quantum chemical theory should: (i) provide a balanced treatment of electron correlations in both dynamical and nondynamical characteristics (including critical double excitations) for an accurate and balanced description of excited states; (ii) yield the correct topology of CoIns between ground and excited states, an essential requirement for accurate description of internal conversions; and (iii) remain computationally feasible for statistical sampling over several picoseconds or longer. The most accurate methods, such as Multi-Reference (MR) coupled cluster (MRCC) and MR perturbation theory (MRPT), are prohibitively expensive and fail to meet criterion (iii). More economical methods like complete active space self-consistent field (CASSCF), second-order algebraic diagrammatic construction (ADC), and linear response time-dependent density function theory (LR-TDDFT) present significant limitations: for instance, CASSCF neglects dynamical electron correlation, while ADC(2) and TDDFT fail to capture the correct dimensionality of the S₁/S₀ CoIn seam, thereby reducing the reliability of their predictions.⁵

To account for these challenges, we have developed a new quantum chemical theory, Mixed-Reference Spin-Flip Time-Dependent Density Functional Theory (MRSF-TDDFT).¹⁻³ Additionally, we have also released OpenQP, a new quantum chemical software package^6,7 that includes MRSF-TDDFT, making this advanced methodology widely accessible to the scientific community. In the following sections, we will demonstrate how MRSF-TDDFT overcomes the limitations of existing quantum chemical theories by presenting studies that showcase the method’s ability to resolve complex problems and open new avenues for chemical research.

2. Nonconventional Idea in MRSF-TDDFT

The simplest wave function that satisfies the Pauli exclusion principle is the Slater determinant, which is constructed from a set of single-particle wave functions (referred to as orbitals) describing the electron configuration of a system.

As shown in Figure 1(a), traditional quantum chemical theories can be classified as either single-reference (single Slater determinant) or multireference (multi Slater determinant) methods. The former provides a more practical approach for obtaining molecular orbitals (MO), while the latter accounts for nondynamical (static) electron correlation from the outset. As a result, popular methods such as density functional theory (DFT), an example of the former, experience difficulties in recovering nondynamical correlation, which poses a major limitation to the theory. On the other hand, multireference theories, such as CASSCF, lack dynamical correlation and require additional, expensive, corrections. CAS-DFT was developed to overcome these limitations.⁸ Although it is not a black-box method, the flexibility in active space selection often provides significant advantages, including the ability to easily obtain various spin states. However, it still faces challenges, such as the double counting of dynamic correlation effects and the computational difficulties inherited from CASSCF. In addition, it has the drawback of not providing state-specific orbitals since it allows core-virtual orbital rotations based on an averaged state.

Figure 1.

(a) Conventional classification of quantum chemical theories. (b) The MRSF-TDDFT method employs a nonconventional approach, conducting a multireference response calculation based on single-reference orbital optimization. The wave functions Ψ_DFT(Ms=1) and Ψ_DFT(Ms=–1) represent the restricted open-shell Hartree–Fock (ROHF) determinants with M_s = +1 and M_s = −1, respectively. The former is obtained through the self-consistent field (SCF) cycle via the variational principle, while the latter can also be derived from the same SCF process, as both are identical in terms of spatial orbitals. The corresponding densities are added together yielding the term mixed-reference”.

Achieving a balanced description of dynamical and nondynamical correlation effects in an efficient manner is one of the most challenging tasks in electronic structure theory, due to the formally nonintermixable nature of the two limiting approaches, as illustrated in Figure 1(a). MRSF-TDDFT aims to address this elusive objective^1,2 with the aid of spin-flip (SF)-TDDFT.⁹ In contrast to the α → α excitations of TDDFT from Restricted Hartree–Fock (RHF), SF-TDDFT is based on α → β excitations from Restricted Open-Shell Hartree–Fock (ROHF). This approach generates various important configurations, including doubly excited ones that are absent in conventional TDDFT. As a result, SF-TDDFT can effectively describe open-shell singlet systems, such as diradicals and systems undergoing bond-breaking. Although SF-TDDFT addresses the major limitations of conventional TDDFT, it is hindered by significant spin contamination due to the absence of key configurations. Traditionally, higher-rank excitations, such as double excitations (see Figure 2(a)) and beyond, are incorporated from a single reference to mitigate this issue, but at the cost of increased computational complexity.

Figure 2.

(a) The black configuration is obtained through a single spin-flip excitation from the ROHF reference with M_S = +1, while the red configuration requires a combination of both spin-flip and nonspin-flip excitations. (b) The red configuration can also be derived from the red ROHF reference with M_S = −1 by a single spin-flip excitation. The symbols C, O, and V represent the doubly occupied, singly occupied, and virtual orbitals in ROHF reference, respectively.

To escape from this vicious cycle between accuracy and computational expense, a second reference with M_S = −1 was introduced in MRSF-TDDFT (see Figure 2(b)), a novel approach that had never been attempted before.^1,2 Remarkably, this second reference generates the red configuration with a single spin-flip, eliminating the need for expensive double excitations, as shown in Figure 2(a), thus ensuring both accuracy and practicality. In terms of formal structures, MRSF-TDDFT takes advantage of both single-reference and multireference methods, as illustrated in Figure 1(b), where response excitations from two references (the M_S = ± 1 components of Ψ_ROHF) are recovered on the basis of single-reference ROHF (the M_S = +1 component of Ψ_ROHF) orbital optimizations. It is noted that a simple addition of the two corresponding densities (mixed-reference) introduces the issue of nonidempotent density, which was addressed by the ”hypothetical single reference” concept through an ingenious spin function transformation.¹

The resultant MRSF-TDDFT provides a more balanced treatment of dynamical and nondynamical electron correlations for both ground and excited states. Comprehensive benchmark calculations have been performed to demonstrate the robustness of MRSF-TDDFT for addressing commonly encountered challenges in the field.^5,10−24 Notably, the careful treatment of electron correlation in MRSF-TDDFT enables reliable NAMD simulations, as showcased in a series of recent studies.^4,19,25−29 A recent perspective further underscores the merits of MRSF-TDDFT in this context.³

3. Breaking Barriers in the Application Scope

As explained in the previous section, MRSF-TDDFT can overcome the limitations of conventional quantum chemical theories without the high cost of computational overhead. In the following, we will demonstrate how the new MRSF-TDDFT addresses challenging cases, where traditional theories either suffer from nonapplicability or incur excessive computational costs.

3.1. Describing Diradicals and Diradicaloids

Diradicals are among the most significant systems in chemistry, but they pose considerable challenges for traditional quantum chemical theories, as they cannot be adequately studied using single-reference methods such as HF and DFT. This is because diradicals contain two unpaired electrons occupying two (nearly) degenerate molecular orbitals, forming the open-shell singlet (OSS) state, which requires at least two configurations¹⁷ of L and R, as illustrated in Scheme 1a. The out-of-phase combination of L and R corresponds to the OSS state, while the in-phase combination gives the M_S = 0 triplet state. As the interaction between the pair of electrons increases in a diradical, the OSS (L and R) configurations begin to mix with the symmetric ionic closed-shell configurations (G) and (D), leading to a diradicaloid state. This diradical-ionic mixing correlates with the mixing of the ground-state (G) and the HOMO-to-LUMO doubly excited Slater Determinants (D), as depicted in Scheme 1b for p-benzyne. A pure G represents an equal mixture of the diradical and ionic configurations, while an equal mixture of G and D with the minus sign corresponds to a pure diradical.

Scheme 1. Major Electronic Configurations of (a) Diradical and (b) Diradicaloid Molecule.

(a) Two electronic configurations L and R can be combined as L-R giving open-shell singlet (OSS) or as L+R yielding the M_S = 0 triplet component. (b) Ionic and covalent resonance forms of p-benzyne that contribute to a diradicaloid. The negative combination of G-D with the same magnitude represents a pure diradical, and the minus combination with more G and less D represents a diradicaloid. SOMO = singly occupied molecular orbital, HOMO = highest occupied molecular orbital, LUMO = lowest unoccupied molecular orbital. Reproduced from ref (17). Copyright 2021 American Chemical Society.

To address these challenges,³⁵ the broken-symmetry spin-unrestricted formalism, as well as SF-TDDFT,³⁶ have been used to describe multiconfigurational states. However, these methods suffer from notorious spin contamination,³⁷ and they cannot produce all the necessary configurations in a spin-adapted, balanced way. This challenge has been successfully addressed by MRSF-TDDFT, as demonstrated in the calculations of adiabatic singlet–triplet (ST) gaps, ΔE_ST = E_S – E_T, reported by Horbatenko et al.¹⁷ The MRSF-TDDFT results, compared with experimental and other computational results in Table 1, show a sufficiently small mean absolute error (MAE) of 0.14 eV without any spin contamination. This is comparable to the MAE (0.09 eV) of the equation-of-motion spin-flip coupled cluster (EOM-SF-CCSD(dT)) method. Evidently, for open-shell systems, MRSF-TDDFT demonstrates reliability comparable to the EOMCC class of methods, but with significantly reduced computational overhead. This is because, even though MRSF-TDDFT is based on single-reference ROHF, its responses from two configurations sufficiently reproduce most of the necessary configurations for the description of diradicals and diradicaloids.

Table 1. Adiabatic Singlet-Triplet Gaps (ΔE_ST, in eV) for MRSF-TDDFT with BH&HLYP Functional and cc-pVTZ Basis Set^a.

	MRSF-BH&HLYP	EOM-SF-CCSD(dT)b	Exp
3m-Xy	0.46	0.45	0.42c
32,7-NQ	0.68		0.55c
3TMM (1A1)	0.91	0.85	0.79d
3TMM (1B2)	0.73	0.70
3α,2-DHT	0.21	0.29
3α,4-DHT	0.15	0.25
1α,3-DHT	–0.01	–0.06
1o-By	–1.86	–1.62	–1.66e
1m-By	–0.96	–0.89	–0.91e
11,8-NQ	–0.13		–0.03f
1p-By	–0.16	–0.17	–0.16e
MAE	0.091	0.032

^aTaken from ref (17). The EOM-SF-CCSD(dt) calculations from ref (30) and experimental available values are given for comparison. The mean absolute errors (MAEs) (in eV) compared to previous experimental values are also given. The negative values of the gap indicate the singlet state is more stable than the triplet state. The numeric superscript in front of each compound’s abbreviation indicates the ground state spin multiplicity. Reproduced from ref (17). Copyright 2021 American Chemical Society.

^bRef (30).

^cRef (31).

^dRef (32).

^eRef (33).

^fRef (34).

3.2. Doubly Excited State and Bond Breaking of H₂

The OSS configurations of L and R, as well as the ionic closed-shell configurations of G and D, shown in Scheme 1, are responsible for the correct ground-state bond-breaking. Since DFT only utilizes G, it tends to overestimate bond dissociation. The doubly excited D configuration is also missing in TDDFT, which limits its applicability to certain cases.

In this section, we demonstrate how MRSF-TDDFT overcomes these limitations and successfully recovers the correct chemical picture in the bond-breaking of the H₂ molecule. The four lowest singlet states, 1¹Σ_g⁺, 2¹Σ_g⁺, 3¹Σ_g⁺, and 1¹Σ_u⁺, were considered, and their potential energy surfaces (PESs) along the entire bond dissociation path were calculated using MRSF-TDDFT, as presented in Figure 3. The results are compared with those obtained using Full Configuration Interaction (FCI) and LR-TDDFT in Figure 3.

Figure 3.

PESs for ¹H₂ dissociation calculated at the level of (a) FCI, (b) MRSF-TDDFT/STG1X, (c) MRSF-TDDFT/BH&HLYP, and (d) LR-TDDFT/PBE0, methods. The cc-pVTZ basis set is employed in all these calculations. The black, red, blue, and green curves correspond to the S₀, S₁, S₂, and S₃ states, respectively. The leading configurations in the states at the Franck–Condon (FC) geometry (R = 1.40 au) are shown as insets with the corresponding color and distributed in the four panels. Reproduced from ref (13). Copyright 2021 American Chemical Society.

At the bonding limit, the ground electronic state 1¹Σ_g⁺ and the third excited state 3¹Σ_g⁺ are predominantly described by a single Slater determinant with doubly occupied 1σ_g and 1σ_u orbitals, respectively. The latter represents a double excitation (D in Figure 1). The other two states, 1¹Σ_u⁺ and 2¹Σ_g⁺, are primarily characterized by one-electron excitations, resulting in the OSS configurations of L and R (see insets in Figure 3).

The FCI calculation shows that the doubly excited 3¹Σ_g⁺ state has an avoided crossing with the second 2¹Σ_g⁺ state at around 2.5 au (Figure 3(a)). As a result, the doubly excited character is transferred to the second 2¹Σ_g⁺ state, which subsequently undergoes another avoided crossing with the ground 1¹Σ_g⁺ state. This illustrates that the character of the third excited state becomes an important ingredient in the ground-state bond-breaking process. The MRSF-TDDFT method, combined with two density functionals, STG1X¹² and BH&HLYP, accurately captures the characteristics of the FCI results, as shown in Figure 3(b) and (c). This is because MRSF-TDDFT includes the essential doubly excited D configuration in Scheme 1(b) as a response state. On the other hand, due to the absence of this configuration, TDDFT fails to capture the 3¹Σ_g⁺ state and the corresponding avoided crossings, resulting in unphysical PESs as shown in Figure 3(d).

Overall, it should be noted that TDDFT can potentially miss important states, and their remarkable recovery by MRSF-TDDFT fundamentally restores the correct chemical picture.

3.3. Topology of Conical Intersections

The correct description of CoIns represents a hallmark challenge in electronic structure theories, which is of utmost importance for accurately describing nonadiabatic processes between electronic states (see the left panel of Figure 4). CoIns are special geometries where two (or more) adiabatic electronic states become degenerate.³⁸⁻⁴⁸ Surprisingly, all single-reference theories, such as DFT and TDDFT,⁴⁹ fail to describe them. This is because the ground and excited states must be obtained separately using DFT and TDDFT, respectively,⁵⁰ where state coupling between the two quantum methods cannot be introduced leading to unphysical linear crossing shown in the right panel of Figure 4. Although multistate multireference computational methods are capable of producing the correct CoIn topology, they are not practical due to the significant computational overhead.

Figure 4.

Characteristic topology of conical intersection (left) and linear intersection (right) between two potential energy surfaces of S₁ and S₀ states. While MRSF-TDDFT can calculate both states producing correct topology, DFT and LR-TDDFT are performed for respective states leading to unphysical linear crossing. Reproduced from ref (3). Copyright 2023 American Chemical Society.

In this challenging situation, it is remarkable that MRSF-TDDFT can produce the correct topology, as it equally generates both S₁ and S₀ as response states (left panel of Figure 4). In a benchmark study of 12 CoIns, MRSF-TDDFT shows excellent agreement with MRCISD for geometries (RMSD of 0.067 Å), while the relative energies (RMSD of 0.41 eV) deviate slightly but still capture the overall topological features of the potential energy surfaces.^10,14 Since MRSF-TDDFT can produce the correct topology and accurate energies, it is a promising new tool that can significantly facilitate future explorations of excited-state chemistry.

3.4. Dynamical vs Nondynamical Excited States

In the case of all-trans polyenes, such as trans-butadiene, the bright π → π* transition promotes one electron from the bonding HOMO to the antibonding LUMO. In terms of bond order, this transition removes one bonding orbital due to the singly occupied bonding and antibonding combination, significantly affecting the electron distribution. Consequently, the system introduces zwitterionic charge separation to achieve stabilization, as shown in Figure 5(a). In linear all-trans polyenes, this corresponds to the bright 1¹B_u⁺ state. Additionally, there is an optically dark, multiconfigurational 2¹A_g^– state characterized by prominent π → π* double excitations and distinctive diradical features, as illustrated in the right panel of Figure 5(a). From a purely theoretical perspective, these two distinctly different characteristics necessitate the corresponding dynamical and nondynamical electron correlations, respectively. In other words, their correct relative ordering in terms of state energy is highly dependent on a balanced treatment of electron correlations, which remains one of the most challenging tasks in quantum chemistry. High-level theoretical calculations predict that, at the Franck–Condon (FC) geometry, the 1¹B_u⁺ state is slightly more stable than the 2¹A_g^– state, as depicted in Figure 5(b).⁵¹

Figure 5.

(a) Lewis diagrams showing the ionic characteristics of the 1¹B_u⁺ state and the diradical characteristics of the 2¹A_g^– state of butadiene. (b) Illustration of the possible ordering of the PESs of butadiene. FC represents Franck–Condon and IC represents internal conversion. Reproduced from ref (5). Copyright 2023 American Chemical Society.

Beyond the FC region, internal conversion (IC) between the 1¹B_u⁺ and 2¹A_g^– states has been proposed⁵² to occur through a true crossing between their PESs, as depicted in Figure 5(b) along the bond length alternation (BLA = difference between average of single and double bond lengths) vibrational mode.⁵³ Correctly predicting the IC can be a good indicator of the quality of a quantum theory, as the location of the IC in terms of molecular geometry is highly sensitive to the electron correlation balance. In fact, the two high-level theories, completely renormalized EOMCC denoted as δ-CR-EOMCC(2,3) (dashed lines) and extended multistate complete active-space second-order perturbation theory denoted as XMS-CASPT2 (dotted lines) in Figure 6(a), recover the correct state ordering at the FC region and remarkably predict the curve crossing between the 1¹B_u⁺ and 2¹A_g^– states of butadiene, with the corresponding BLA values of −0.03 and −0.06 Å, respectively.⁵

Figure 6.

1¹B_u⁺ (red) and 2¹A_g^– (blue) energy profiles along the MEPs of trans-butadiene. Panel (a) shows the MEPs (minimum energy paths) by SA-CASSCF (solid lines), δ-CR-EOMCC(2,3) (dashed lines), and XMS-CASPT2 (dotted lines). Both SA-CASSCF and XMS-CASPT2 utilized 4 electrons and 4 orbitals in the active space. Panel (b) shows the MEPs by MRSF-TDDFT/BH&HLYP (solid lines) and TDDFT/B3LYP (dashed lines). All calculations were done with cc-pVTZ basis sets. Reproduced from ref (5). Copyright 2023 American Chemical Society.

The state-averaged complete active-space self-consistent field (SA-CASSCF(4e,4o)) calculation, one of the popular theories for excited states, significantly overestimates the energy of the ionic 1¹B_u⁺ state by ca. 2 eV (red solid lines) in Figure 6(a)) due to the absence of dynamical correlation.⁵¹ Although TDDFT (dashed lines in Figure 6(b)), another popular theory for excited states, produces the correct 1¹B_u⁺ lt 2¹A_g^– ordering at the FC structure, it dramatically misses the curve crossing due to the absence of the D configuration.

In contrast to these two popular quantum theories, MRSF-TDDFT remarkably reproduces both the correct ordering at the FC and the curve crossing (solid lines) in Figure 6(b), with the crossing position (BLA = −0.04 Å) falling between the predictions of δ-CR-EOMCC(2,3) and XMS-CASPT2. Accurately recovering the conical intersection by MRSF-TDDFT underscores its superior performance in handling electron correlations, while significantly reducing computational cost.

3.5. Customized XC Functionals for MRSF-TDDFT

Thus, far, we have demonstrated that MRSF-TDDFT has the potential to overcome the major limitations of DFT, TDDFT, SF-TDDFT, and CASSCF, while maintaining significantly lower computational overhead compared to high-level theories. Although MRSF-TDDFT offers formal advantages, its prediction accuracy still depends on the choice of exchange-correlation (XC) functionals.

We showed⁵⁴ that by individually tuning XC functionals for the mean-field ground-state SCF and response calculations-referred to as “double tuning”-better XC functionals for response theories can be achieved. As illustrated in Figure 7, this method notably reduces prediction errors, cutting MAEs for vertical excitation energy (VEE) calculations by a factor of 2 across all XC functionals (see the bottom of Figure 7). This “double tuning” approach led to the development of two new functionals: the doubly tuned Coulomb attenuated method (DTCAM)-VEE and (adaptive exact exchange) DTCAM-AEE as shown in Table 2.⁵⁴

Figure 7.

MAEs in eV are based on Thiel’s benchmark set.⁵⁶ The top figure presents the MAE results using the original XC functionals, while the bottom figure shows the results obtained with the “double tuning” approach. All calculations were performed using MRSF-TDDFT with the 6-31G(d) basis set. The values in parentheses represent c_HF for each functional.

Table 2. New Exchange-Correlation Functionals for MRSF-TDDFT^a.

Functional	Description
DTCAM-VAEE	Optimized for VEEs using both the “double tuning” and “valence attenuation” concepts.
DTCAM-VEE	Double Tuned for VEEs based on Thiel’s benchmark set.
DTCAM-AEE	Double Tuned for VEE, optimized based on Thiel’s set, CoIns in trans-butadiene and thymine, and nonadiabatic molecular dynamics (NAMD) simulations on thymine.
DTCAM-XI	Optimized for core ionization potentials (cIPs) and valence ionization potentials (vIPs) by “valence attenuation”.
DTCAM-XIV	Optimized for simultaneously predicting both VEEs and ionization potentials (IPs) based on the “double tuning“ and ”valence attenuation” concepts.

^aDescribed in refs (54) and (55).

Another concept, “valence attenuation,” was also introduced to accurately reproduce both VEE and ionization potentials.⁵⁵ The logic behind this approach is based on the observation that selectively reducing the amount of exact exchange in the frontier orbital regions, while keeping it large in the core, can simultaneously improve both properties, resulting in improved functionals such as DTCAM-VAEE, DTCAM-XI, and DTCAM-XIV.⁵⁵ (see Table 2)

The prediction accuracy of MRSF-TDDFT can be improved with the help of nonconventional XC functional optimizations. While the new functionals were optimized using only the 6-31G(d) basis set, similar MAEs were observed with cc-pVDZ basis sets, as shown in Table 3, highlighting their transferability. The BH&HLYP functional serves as a reliable general-purpose choice for a wide range of systems. For cases where both valence and core states need to be considered simultaneously, functionals like DTCAM-XI and DTCAM-XIV can be considered. Meanwhile, for processes involving valence excitations, DTCAM-AEE, DTCAM-VEE, and DTCAM-VAEE can be effective options. Further improvements can also be expected in the near future, which will be beneficial for theoretical studies in general.

Table 3. MAEs of Various XC Functionals against Three Benchmark Sets of cIP, VEE and vIP Using 6-31G(d) and cc-pVDZ in Parentheses.

Functional	cIP	VEE	vIP	Average
DTCAM-XI	0.51 (0.52)	0.97 (0.87)	0.30 (0.31)	0.59 (0.57)
DTCAM-XIV	0.64 (0.55)	0.67 (0.62)	0.61 (0.64)	0.64 (0.60)
DTCAM-VAEE	3.47 (3.54)	0.57 (0.51)	1.41 (1.27)	1.82 (1.77)

3.6. MRSF-TDDFT as a Method of Choice for Nonadiabatic Dynamics

As demonstrated in the previous sections, MRSF-TDDFT meets the three essential requirements for applicability to NAMD. It (i) provides a balanced treatment of electron correlations, including critical double excitations, ensuring an accurate and consistent description of various excited states; (ii) yields the correct topology of CoIns; and (iii) remains computationally feasible.

NAMD with MRSF-TDDFT has already been successfully applied to resolve mechanistic challenges posed by CASSCF in thymine dynamics.¹⁹ At short simulation times, t ≲ 50 fs (see Figure 8a), the pronounced synchronicity of bond length alternation (BLA) suggests that a concerted BLA leads to an ultrafast transfer of the S₂ population to S₁, a phenomenon recently confirmed experimentally.⁵⁷ This ultrafast transfer cannot be predicted by CASSCF due to its significant overestimation of the bright state (the dotted blue line), as shown in Figure 8b, which eliminates the conical intersection between S₂ to S₁. This issue aligns with the same observation found in Figure 6a.

Figure 8.

Panel (a) shows the time evolution of the adiabatic S₁ (red solid line) and S₂ (blue solid line) populations of thymine by MRSF-TDDFT. Panel (b) shows the potential energy surfaces of the FC → CI_21,BLA → S_1,min geometries, along the corresponding BLA coordinate. The latter two geometries correspond to the conical intersection between S₂ and S₁, and the minimum of S₁. It compares the MRSF-TDDFT paths (solid lines) with the CASSCF curves (dashed lines). Reproduced from ref (19). Copyright 2021 American Chemical Society.

In another NAMD study using MRSF-TDDFT, a plausible mechanism for uracil photohydration was proposed,²⁶ and new dynamic aspects of excited-state aromatization were uncovered,⁴ among other applications. As a result, MRSF-TDDFT is positioned to become the de facto standard quantum chemistry method for NAMD simulations, representing a major breakthrough in this field.

3.7. Perspectives

A relativistic version of MRSF-TDDFT for spin–orbit coupling,²⁴ EKT-MRSF-TDDFT (EKT: Extended Koopmans’ Theorem) for ionization potential and electron affinity,¹⁵ and a new protocol for core-hole relaxation in X-ray absorption⁵⁸ have been developed to expand the application scope of MRSF-TDDFT.

The current implementation of MRSF-TDDFT is based on the restricted open-shell Kohn–Sham (ROKS) formalism, which is chosen for its simplicity and for avoiding spin contamination in different α and β orbitals. However, ROKS is not particularly efficient when it comes to orbital optimizations through SCF. Recent developments based on the unrestricted Kohn–Sham (UKS) formalism are expected to address these limitations,⁵⁹ thereby enhancing the practicality of MRSF-TDDFT.

As the current implementation of MRSF utilizes the collinear formalism, it requires a larger fraction of exact exchange, such as in the BH&HLYP functional. This issue has been partially addressed by the development of customized XC functionals for MRSF-TDDFT. Further advancements are highly desirable for improving prediction accuracy.

Four out of six type C → V configurations (those shown by the gray arrows in Figure 9) remain unaccounted for, which is a residual source of spin contamination, primarily arising from quintet states that influence triplet states. Typically, these configurations represent high-lying excited states and make insignificant contributions to the low-lying states of organic molecules.¹ Thus, the effect of the missing configurations on spin contamination is effectively zero (<0.00001).³ However, more challenging scenarios, such as strong charge-transfer states in dimers, may require the missing configurations. One simple way to recover them is by adding the missing configurations through configuration interaction (CI) expansion, which is currently under investigation.

Figure 9.

C → V types of electronic configurations that can be generated by spin-flip linear responses are given by black and red arrows, where C and V are the closed and virtual orbitals. Likewise, O1 and O2 are the two singly occupied orbitals of ROHF reference. The black and red ones are generated from M_s = +1 and −1 references, respectively. Configurations that cannot be obtained in the MRSF-TDDFT are denoted by gray arrows.

As discussed in the introductions, the major aspects discussed in this account have been implemented in the recent quantum chemistry software, OpenQP.^6,7OpenQP (Open Quantum Platform) is a new open-source quantum chemistry library designed to address sustainability and interoperability challenges in computational chemistry. OpenQP offers various popular quantum chemical theories as autonomous modules, such as energy and gradient calculations for HF, DFT, TDDFT, SF-TDDFT, and MRSF-TDDFT, enabling seamless integration with third-party software. Community-wide collaborations are highly welcome through the platform.

4. Conclusions

Although conventional DFT and TDDFT may still be applicable in regions where electron correlation is relatively weak, such as the ground-state minimum (Franck–Condon region) of closed-shell systems, they tend to fail in other regions. In contrast, MRSF-TDDFT consistently performs well in both weak and strong electron correlation situations. In this work, we have demonstrated the effectiveness of Mixed-Reference Spin-Flip Time-Dependent Density Functional Theory (MRSF-TDDFT) in overcoming the limitations of traditional quantum chemical methods. Specifically, MRSF-TDDFT excels in handling complex quantum systems, such as diradicals, bond-breaking processes, and excited states, where it can reproduce results comparable to more computationally expensive methods, but with significantly reduced computational cost.

Additionally, the implementation of customized exchange-correlation functionals tailored for MRSF-TDDFT further improves its accuracy. Future developments, including the integration of the unrestricted Kohn–Sham formalism and the expansion of the configurational space, are expected to enhance the efficiency of MRSF-TDDFT and extend its applicability to a broader range of quantum chemical problems of increasing complexity.

Biographies

Seunghoon Lee was born in Busan, South Korea (1989). He received his B.Sc. in Chemistry (2014) and Ph.D. in Physical Chemistry (2019) from Seoul National University in South Korea, supervised by Prof. Sangyoub Lee and Cheol Ho Choi. He did his postdoctoral research in Prof. Garnet K. Chan’s group at Caltech (2019–2023). He started his independent career at Seoul National University in South Korea in 2023, and he is currently an assistant professor. His research focuses on developing electronic structure methods for strongly correlated systems, from diradicals to transition-metal complexes.

Woojin Park was born in Daegu, South Korea (1995) and received his B.Sc. and M.Sc. degrees in Chemistry from Kyungpook National University, where he worked in the group of Prof. Cheol Ho Choi. He is currently a Ph.D. candidate in the same group. His research focuses on light-induced chemical reactions, specifically exploring the subsequent processes following photon absorption, including vibrational relaxation, internal conversion, and intersystem crossing. He is also involved in the development of accurate yet computationally efficient quantum theoretical methods.

Cheol Ho Choi was born in Daegu, South Korea, in 1967, and grew up in Daegu and Seoul. He received his B.Sc. and M.Sc. degrees in Chemistry from Seoul National University, South Korea. In 1998, he obtained his Ph.D. (Hons) in Chemistry from Georgetown University. After completing postdoctoral research with Prof. Mark Gordon at Iowa State University, Choi began his independent academic career at Kyungpook National University, Daegu, South Korea, in 2001. He is currently a Full Professor of Chemistry and the Director of the BK21 Research Center in the Department of Chemistry. Choi is a member of the American Chemical Society and the Korean Chemical Society. His research focuses on the development and application of efficient electronic structure methods and their practical applications.

The authors declare no competing financial interest.