Multistate Density Functional Theory Research Articles

Leveling the Mountain Range of Excited-State Benchmarking through Multistate Density Functional Theory.

The performance of multistate density functional theory (MSDFT) with nonorthogonal state interaction (NOSI) is assessed for 100 vertical excitation energies against the theoretical best estimates extracted to the full configuration interaction accuracy on the database developed by Loos et al. in 2018 (Loos2018). Two optimization techniques, namely, block-localized excitation and target state optimization, are examined along with two ways of estimating the transition density functional (TDF) for the correlation energy of the Hamiltonian matrix density functional. The results from the two optimization methods are similar. It was found that MSDFT-NOSI using the spin-multiplet degeneracy constraint for the TDF of spin-coupling interaction, along with the M06-2X functional, yields a root-mean-square error (RMSE) of 0.22 eV, which performs noticeably better than time-dependent density functional theory (DFT) at an RMSE of 0.43 eV using the same functional and basis set on the Loos2018 database. In comparison with wave function theory, NOSI has smaller errors than CIS(D∞), LR-CC2, and ADC(3) all of which have an RMSE of 0.28 eV, but somewhat greater than STEOM-CCSD (RMSE of 0.14 eV) and LR-CCSD (RMSE of 0.11 eV) wave function methods. In comparison with Kohn-Sham (KS) DFT calculations, the multistate DFT approach has little double counting of correlation. Importantly, there is no noticeable difference in the performance of MSDFT-NOSI on the valence, Rydberg, singlet, triplet, and double-excitation states. Although the use of another hybrid functional PBE0 leads to a greater RMSE of 0.36 eV, the deviation is systematic with a linear regression slope of 0.994 against the results with M06-2X. The present benchmark reveals that density functional approximations developed for KS-DFT for the ground state with a noninteracting reference may be adopted in MSDFT calculations in which the state interaction is key.

Generalized Nuclear-Electronic Orbital Multistate Density Functional Theory for Multiple Proton Transfer Processes.

Proton transfer and hydrogen tunneling play pivotal roles in many chemical and biological processes. The nuclear-electronic orbital multistate density functional theory (NEO-MSDFT) approach was developed to describe hydrogen tunneling systems within the multicomponent NEO framework, where the transferring proton is quantized and treated with molecular orbital techniques on the same level as the electrons. Herein, the NEO-MSDFT framework is generalized to an arbitrary number of quantum protons to allow applications to systems involving the transfer and tunneling of multiple protons. The generalized NEO-MSDFT approach is shown to produce delocalized, bilobal proton densities and accurate tunneling splittings for fixed geometries of the formic acid dimer and asymmetric substituted variants, as well as the porphycene molecule. Investigation of a protonated water chain highlights the applicability of this approach to proton relay systems. This work provides the foundation for nuclear-electronic quantum dynamics simulations of a wide range of multiple proton transfer processes.

Ab initio Valence Bond Theory with Density Functional

Abstract: The accurate description of strongly correlated systems, also known as multireference systems, requires a balanced treatment of static and dynamic correlations and is an important target for developing quantum chemical methods. An appealing treatment to economically describe strongly correlated systems is the multireference density function theory (MRDFT) approach, in which the static correlation is included in the multiconfigurational wave function, while the density functional includes the dynamic correlation. This mini-review focuses on the recent progress and applications of the density functional methods based on valence bond theory. A series of density functional valence bond (DFVB) methods are surveyed, including the dynamic correlation correction- based and Hamiltonian matrix correction-based DFVB methods, the hybrid one-parameter DFVB methods, the block-localized density functional theory and the multistate density functional theory. These methods have been applied to various chemical and physical property calculations of strongly correlated systems, including resonance energies, potential energy curves, spectroscopic constants, atomization energies, spin state energy gaps, excitation energies, and reaction barriers. Most of the test results show that the density functional methods based on VB theory give comparable accuracy but require lower computational cost than high-level quantum computational methods and thus provide a promising strategy for studying strongly correlated systems.

Excimer Energies.

A multistate energy decomposition analysis (MS-EDA) method is introduced for excimers using density functional theory. Although EDA has been widely applied to intermolecular interactions in the ground state, few methods are currently available for excited-state complexes. Here, the total energy of an excimer state is separated into exciton excitation energy ΔEEx(|ΨX·ΨY⟩*), resulting from the state interaction between locally excited monomer states |ΨX*·ΨY⟩ and |ΨX·ΨY*⟩ , a superexchange stabilization energy ΔESE, originating from the mutual charge transfer between two monomers |ΨX+·ΨY⟩ and |ΨX-·ΨY+⟩ , and an orbital-and-configuration delocalization term ΔEOCD due to the expansion of configuration space and block-localized orbitals to the fully delocalized dimer system. Although there is no net charge transfer in symmetric excimer cases, the resonance of charge-transfer states is critical to stabilizing the excimer. The monomer localized excited and charge-transfer states are variationally optimized, forming a minimal active space for nonorthogonal state interaction (NOSI) calculations in multistate density functional theory to yield the intermediate states for energy analysis. The present MS-EDA method focuses on properties unique to excited states, providing insights into exciton coupling, superexchange and delocalization energies. MS-EDA is illustrated on the acetone and pentacene excimer systems; three configurations of the latter case are examined, including the optimized excimer, a stacked configuration of two pentacene molecules and the fishbone orientation. It is found that excited-state energy splitting is strongly dependent on the relative energies of the monomer excited states and the phase-matching of the monomer wave functions.

Target State Optimized Density Functional Theory for Electronic Excited and Diabatic States.

A flexible self-consistent field method, called target state optimization (TSO), is presented for exploring electronic excited configurations and localized diabatic states. The key idea is to partition molecular orbitals into different subspaces according to the excitation or localization pattern for a target state. Because of the orbital-subspace constraint, orbitals belonging to different subspaces do not mix. Furthermore, the determinant wave function for such excited or diabatic configurations can be variationally optimized as a ground state procedure, unlike conventional ΔSCF methods, without the possibility of collapsing back to the ground state or other lower-energy configurations. The TSO method can be applied both in Hartree-Fock theory and in Kohn-Sham density functional theory (DFT). The density projection procedure and the working equations for implementing the TSO method are described along with several illustrative applications. For valence excited states of organic compounds, it was found that the computed excitation energies from TSO-DFT and time-dependent density functional theory (TD-DFT) are of similar quality with average errors of 0.5 and 0.4 eV, respectively. For core excitation, doubly excited states and charge-transfer states, the performance of TSO-DFT is clearly superior to that from conventional TD-DFT calculations. It is shown that variationally optimized charge-localized diabatic states can be defined using TSO-DFT in energy decomposition analysis to gain both qualitative and quantitative insights on intermolecular interactions. Alternatively, the variational diabatic states may be used in molecular dynamics simulation of charge transfer processes. The TSO method can also be used to define basis states in multistate density functional theory for excited states through nonorthogonal state interaction calculations. The software implementing TSO-DFT can be accessed from the authors.

Nonadiabatic Dynamics of Hydrogen Tunneling with Nuclear-Electronic Orbital Multistate Density Functional Theory.

Proton transfer reactions play a critical role in many chemical and biological processes. The development of computationally efficient approaches to describe the quantum dynamics of proton transfer, which often involves hydrogen tunneling, is challenging. Herein, the nuclear-electronic orbital multistate density functional theory (NEO-MSDFT) method is combined with both Ehrenfest and surface hopping nonadiabatic dynamics methods to describe hydrogen tunneling. The NEO-MSDFT method treats the transferring hydrogen nucleus quantum mechanically on the same level as the electrons and incorporates both static and dynamical correlation by mixing localized NEO-DFT solutions with a nonorthogonal configuration interaction scheme. The other nuclei are propagated on the NEO-MSDFT vibronic surfaces during the Ehrenfest or surface hopping dynamics. These methods are applied to proton transfer in malonaldehyde as a prototypical hydrogen tunneling system. The inclusion of vibronically nonadiabatic effects is found to significantly impact the proton transfer time and tunneling dynamics. This approach is applicable to a wide range of other proton transfer reactions.

Minimal Active Space: NOSCF and NOSI in Multistate Density Functional Theory.

In this Perspective, we introduce a minimal active space (MAS) for the lowest N eigenstates of a molecular system in the framework of multistate density functional theory (MSDFT), consisting of no more than N2 nonorthgonal Slater determinants. In comparison with some methods in wave function theory in which one seeks to expand the ever increasing size of an active space to approximate the wave functions, it is possible to have an upper bound in MSDFT because the auxiliary states in a MAS are used to represent the exact N-dimensional matrix density function D(r). Here, we partition the total Hamiltonian matrix functional into an orbital-dependent part, including multistate kinetic energy Tms and Coulomb-exchange energy EHx plus an external potential energy ∫dr v(r)D(r), and a correlation matrix density functional . The latter accounts for the part of correlation energy not explicitly included in the minimal active space. A major difference from Kohn-Sham DFT is that state interactions are necessary to represent the N-matrix density D(r) in MSDFT, rather than a noninteracting reference state for the scalar ground-state density ρo(r). Two computational approaches are highlighted. We first derive a set of nonorthogonal multistate self-consistent-field (NOSCF) equations for the variational optimization of . We introduce the multistate correlation potential, as the functional derivative of , which includes both correlation effects within the MAS and that from the correlation matrix functional. Alternatively, we describe a nonorthogonal state interaction (NOSI) procedure, in which the determinant functions are optimized separately. Both computational methods are useful for determining the exact eigenstate energies and for constructing variational diabatic states, provided that the universal correlation matrix functional is known. It is hoped that this discussion would stimulate developments of approximate multistate density functionals both for the ground and excited states.

Multistate Density Functional Theory of Excited States.

We report a rigorous formulation of density functional theory for excited states, providing a theoretical foundation for a multistate density functional theory. We prove the existence of a Hamiltonian matrix functional of the multistate matrix density D(r) in the subspace spanned by the lowest N eigenstates. Here, D(r) is an N-dimensional matrix of state densities and transition densities. Then, a variational principle of the multistate subspace energy is established, whose minimization yields both the energies and densities of the individual N eigenstates. Furthermore, we prove that the N-dimensional matrix density D(r) can be sufficiently represented by N2 nonorthogonal Slater determinants, based on which an interacting active space is introduced for practical calculations. This work establishes that the ground and excited states can be treated on an equal footing in density functional theory.

Minimal Active Space for Diradicals Using Multistate Density Functional Theory.

This work explores the electronic structure as well as the reactivity of singlet diradicals, making use of multistate density functional theory (MSDFT). In particular, we show that a minimal active space of two electrons in two orbitals is adequate to treat the relative energies of the singlet and triplet adiabatic ground state as well as the first singlet excited state in many cases. This is plausible because dynamic correlation is included in the first place in the optimization of orbitals in each determinant state via block-localized Kohn–Sham density functional theory. In addition, molecular fragment, i.e., block-localized Kohn–Sham orbitals, are optimized separately for each determinant, providing a variational diabatic representation of valence bond-like states, which are subsequently used in nonorthogonal state interactions (NOSIs). The computational procedure and its performance are illustrated on some prototypical diradical species. It is shown that NOSI calculations in MSDFT can be used to model bond dissociation and hydrogen-atom transfer reactions, employing a minimal number of configuration state functions as the basis states. For p- and s-types of diradicals, the closed-shell diradicals are found to be more reactive than the open-shell ones due to a larger diabatic coupling with the final product state. Such a diabatic representation may be useful to define reaction coordinates for electron transfer, proton transfer and coupled electron and proton transfer reactions in condensed-phase simulations.

Analytical gradients for nuclear–electronic orbital multistate density functional theory: Geometry optimizations and reaction paths

Hydrogen tunneling plays a critical role in many biologically and chemically important processes. The nuclear-electronic orbital multistate density functional theory (NEO-MSDFT) method was developed to describe hydrogen transfer systems. In this approach, the transferring proton is treated quantum mechanically on the same level as the electrons within multicomponent DFT, and a nonorthogonal configuration interaction scheme is used to produce delocalized vibronic states from localized vibronic states. The NEO-MSDFT method has been shown to provide accurate hydrogen tunneling splittings for fixed molecular systems. Herein, the NEO-MSDFT analytical gradients for both ground and excited vibronic states are derived and implemented. The analytical gradients and semi-numerical Hessians are used to optimize and characterize equilibrium and transition state geometries and to generate minimum energy paths (MEPs), for proton transfer in the deprotonated acetylene dimer and malonaldehyde. The barriers along the resulting MEPs are lower when the transferring proton is quantized because the NEO-MSDFT method inherently includes the zero-point energy of the transferring proton. Analysis of the proton densities along the MEPs illustrates that the proton density can exhibit symmetric or asymmetric bilobal character associated with symmetric or slightly asymmetric double-well potential energy surfaces and hydrogen tunneling. Analysis of the contributions to the intrinsic reaction coordinate reveals that changes in the C-O bond lengths drive proton transfer in malonaldehyde. This work provides the foundation for future reaction path studies and direct nonadiabatic dynamics simulations of a wide range of hydrogen transfer reactions.

Generation of singlet oxygen catalyzed by the room-temperature-stable anthraquinone anion radical.

The chemical nature and the catalytic selectivity of the complex of anthraquinone and potassium tert-butoxide, AQ-KOtBu, in generating singlet oxygen (1O2) have been studied using a high-level ab initio method and density functional theory (DFT). The results suggest that the stable catalytic center of the AQ anion radical (semiquinone, [AQ˙]-) can be produced at room temperature, which is due to the strong delocalization characteristics of electrons in potassium atoms. Two experimentally observed complexes, the ground state AQ-KOtBu, i.e., C(1), and the photoexcited AQ-KOtBu, i.e., C(2), can be distinguished via the two different electronic states (π-type and σ-type) of the tert-butoxide group. More interestingly, the catalytic selectivity of AQ-KOtBu to generate 1O2 was investigated using multistate density functional theory (MSDFT), and the results suggest that only open-shell 1O2 rather than the closed-shell component can be generated. This work explores the electronic structure and the catalytic nature of AQ-KOtBu, which is of great importance for the application of AQ and its derivatives.

Minimal-active-space multistate density functional theory for excitation energy involving local and charge transfer states

Multistate density functional theory (MSDFT) employing a minimum active space (MAS) is presented to determine charge transfer (CT) and local excited states of bimolecular complexes. MSDFT is a hybrid wave function theory (WFT) and density functional theory, in which dynamic correlation is first incorporated in individual determinant configurations using a Kohn–Sham exchange-correlation functional. Then, nonorthogonal configuration-state interaction is performed to treat static correlation. Because molecular orbitals are optimized separately for each determinant by including Kohn–Sham dynamic correlation, a minimal number of configurations in the active space, essential to representing low-lying excited and CT states of interest, is sufficient to yield the adiabatic states. We found that the present MAS-MSDFT method provides a good description of covalent and CT excited states in comparison with experiments and high-level computational results. Because of the simplicity and interpretive capability through diabatic configuration weights, the method may be useful in dynamic simulations of CT and nonadiabatic processes.

Dynamic-then-Static Approach for Core Excitations of Open-Shell Molecules

Delta self-consistent-field methods are widely used in studies of electronically excited states. However, the nonaufbau determinants are generally spin-contaminated. Here, we describe a general approach for spin-coupling interactions of open-shell molecules, making use of multistate density functional theory (MSDFT). In particular, the effective exchange integrals that determine spin coupling are obtained by enforcing the multiplet degeneracy of the S+1 state in the MS = S manifold. Consequently, they are consistent with the energy of the high-spin state that is adequately treated by Kohn-Sham density functional theory (DFT) and, thereby, free of double counting of correlation. The method was applied to core excitations of open-shell molecules and compared with those by spin-adapted time-dependent DFT. An excellent agreement with experiment was found employing the BLYP functional and aug-cc-pCVQZ basis set. Overall, MSDFT provides an effective combination of the strengths of DFT and wave function theory to achieve efficiency and accuracy.

Generalization of Block-Localized Wave Function for Constrained Optimization of Excited Determinants.

The block-localized wave function method is useful to provide insights on chemical bonding and intermolecular interactions through energy decomposition analysis. The method relies on block localization of molecular orbitals (MOs) by constraining the orbitals to basis functions within given blocks. Here, a generalized block-localized orbital (GBLO) method is described to allow both physically localized and delocalized MOs to be constrained in orbital-block definitions. Consequently, GBLO optimization can be conveniently tailored by imposing specific constraints. The GBLO method is illustrated by three examples: (1) constrained polarization response orbitals through dipole and quadrupole perturbation in a water dimer complex, (2) the ground and first excited-state potential energy curves of ethene about its C-C bond rotation, and (3) excitation energies of double electron excited states. Multistate density functional theory is used to determine the energies of the adiabatic ground and excited states using a minimal active space (MAS) comprising specifically charge-constrained and excited determinant configurations that are variationally optimized by the GBLO method. We find that the GBLO expansion that includes delocalized MOs in configurational blocks significantly reduces computational errors in comparison with physical block localization, and the computed ground- and excited-state energies are in good accordance with experiments and results obtained from multireference configuration interaction calculations.

Nuclear-Electronic Orbital Multistate Density Functional Theory.

Hydrogen tunneling is essential for a wide range of chemical and biological processes. The description of hydrogen tunneling with multicomponent quantum chemistry approaches, where the transferring hydrogen nucleus is treated on the same level as the electrons, is challenging due to the importance of both static and dynamical electron-proton correlation. Herein the nuclear-electronic orbital multistate density functional theory (NEO-MSDFT) method is presented as a strategy to include both types of correlation. In this approach, two localized nuclear-electronic wave functions obtained with the NEO-DFT method are combined with a nonorthogonal configurational interaction approach to produce bilobal, delocalized ground and excited vibronic states. By including a correction function, the NEO-MSDFT approach can produce quantitatively accurate hydrogen tunneling splittings for fixed geometries of systems such as malonaldehyde and acetoacetaldehyde. This approach is computationally efficient and can be combined with methods such as vibronic coupling theory to describe tunneling dynamics and to compute vibronic couplings in many types of systems.

Variational Multistate Density Functional Theory for a Balanced Treatment of Static and Dynamic Correlations.

Here, an approach to variational multistate density functional theory (vMSDFT) is explored. In this approach, the Kohn-Sham orbitals as well as configuration coefficients were simultaneously optimized, thus yielding a full variational minimum. Furthermore, this work also proposes two important improvements on the MSDFT framework. First, a "point-to-point correction" is used to correct the static correlation present in the DFT framework. Therefore, double counting of static correlation in vMSDFT is mitigated. Second, a general form to construct the transition density functional in the vMSDFT framework is proposed, which allows for the properties of vMSDFT wave functions to be standardized to the complete active space self-consistent field properties. The utility of vMSDFT is illustrated on molecular systems of interest including bond breaking, diradicals, excited states, and conical intersections. The numerical results suggest that the accuracy of vMSDFT is in close agreement with the high-level multireference methods.

Multistate density functional theory applied with 3 unpaired electrons in 3 orbitals: The singdoublet and tripdoublet states of the ethylene cation

Multistate density functional theory (MSDFT) is extended to facilitate treatment of situations involving more than two open-shell electrons. The method is applied to determine energies for the two doublet state (tripdoublet and singdoublet) and the quartet-state components that arise when two electrons of one spin type and one electron of the other singly occupy three orbitals. A test system, the (π,π*) excitation of the ethylene cation, is utilized, with MSDFT delivering energies that are numerically superior to those from time-dependent density-functional theory (TD-DFT) and states free from spin contamination.

Diabatic States at Construction (DAC) through Generalized Singular Value Decomposition.

A procedure, called generalized diabatic-at-construction (GDAC), is presented to transform adiabatic potential energy surfaces into a diabatic representation by generalized singular value decomposition. First, we use a set of localized, valence bond-like configuration state functions, called DAC, as the basis states. Then, the adiabatic ground and relevant excited states are determined using multistate density functional theory (MSDFT). GDAC differs in the opposite direction from traditional approaches based on adiabatic-to-diabatic transformation with certain property restraints. The method is illustrated with applications to a model first-order bond dissociation reaction of CH3OCH2Cl polarized by a solvent molecule, the ground- and first-excited-state potential energy surfaces near the minimum conical intersection for the ammonia dimer photodissociation, and the multiple avoided curve crossings in the dissociation of lithium hydride. The GDAC diabatization method may be useful for defining charge-localized states in studies of electron transfer and proton-coupled electron transfer reactions in proteins.

Combined Multistate and Kohn-Sham Density Functional Theory Studies of the Elusive Mechanism of N-Dealkylation of N,N-Dimethylanilines Mediated by the Biomimetic Nonheme Oxidant FeIV(O)(N4Py)(ClO4)2.

The oxidative C-H bond activation mediated by heme and nonheme enzymes and related biomimetics is one of the most interesting processes in bioinorganic and oxidative chemistry. However, the mechanisms of these reactions are still elusive and controversy due to the involvement of highly reactive metal-oxo intermediates with multiple spin states, despite extensive experimental efforts, especially for the N-dealkylation of N,N-dialkyalinines. In this work, we employed multistate density functional theory (MSDFT) and the Kohn-Sham DFT to investigate the mechanism of N-demethylation of N,N-dimethyalinines oxidized by the reaction intermediate FeIV(O)(N4Py)(ClO4)2. The Kohn-Sham DFT study demonstrated that the reaction proceeds via a rate-limiting hydrogen atom transfer (HAT) step and a subsequent barrier-free oxygen rebound step to form the carbinol product. The MSDFT investigation on the first C-H activation further showed that this step is an initial hydrogen atom abstraction that is highly correlated between CEPT and HAT, i.e., both CEPT and HAT processes make significant contributions to the mechanism before reaching the diabatic crossing point, then the valence bond character of the adiabatic ground state is switched to the CEPT product configuration. The findings from this work may be applicable to other hydrogen abstraction process.

The charger transfer electronic coupling in diabatic perspective: A multi-state density functional theory study

The multi-state density functional theory (MSDFT) provides a convenient way to estimate electronic coupling of charge transfer processes based on a diabatic representation. Its performance has been benchmarked against the HAB11 database with a mean unsigned error (MUE) of 17 meV between MSDFT and ab initio methods. The small difference may be attributed to different representations, diabatic from MSDFT and adiabatic from ab initio calculations. In this discussion, we conclude that MSDFT provides a general and efficient way to estimate the electronic coupling for charge-transfer rate calculations based on the Marcus-Hush model.