Block-Localized Excitation for Excimer Complex and Diabatic Coupling

Abstract

We describe a block-localized excitation (BLE) method to carry out constrained optimization of block-localized orbitals for constructing valence bond-like, diabatic excited configurations using multistate density functional theory (MSDFT). The method is an extension of the previous block-localized wave function method through a fragment-based ΔSCF approach to optimize excited determinants within a molecular complex. In BLE, both the number of electrons and the electronic spin of different fragments in a whole system can be constrained, whereas electrostatic, exchange, and polarization interactions among different blocks can be fully taken into account of. To avoid optimization collapse to unwanted states, a ΔSCF projection scheme and a maximum overlap of wave function approach have been presented. The method is illustrated by the excimer complex of two naphthalene molecules. With a minimum of eight spin-adapted configurational state functions, it was found that the inversion of L_a⁻ and L_b⁻ states near the optimal structure of the excimer complex is correctly produced, which is in quantitative agreement with DMRG-CASPT2 calculations and experiments. Trends in the computed transfer integrals associated with excited-state energy transfer both in the singlet and triplet states are discussed. The results suggest that MSDFT may be used as an efficient approach to treat intermolecular interactions in excited states with a minimal active space (MAS) for interpretation of the results and for dynamic simulations, although the selection of a small active space is often system dependent.

Graphical Abstract

1. INTRODUCTION

It is often more convenient to use a diabatic representation of localized initial and final states than the delocalized adiabatic ground and excited states to determine reaction rates and to model solvent reorganization in electron transfer (ET) and excited-state energy transfer (EET) processes.^1,2 Furthermore, intermolecular interactions of aromatic compounds can have drastic changes in the excited state, sometimes forming stronger excimer complexes than that in the ground state.³ The subsequent nonadiabatic dynamics of the excimer complexes can undergo radiative emission or conical intersection to return to the ground state.² In this case, calculations of nonadiabatic couplings require derivatives of the wave function, and the derivative coupling terms may become singular at the crossing points.⁴ In contrast, the potential energy surfaces of diabatic states are smooth.⁵ However, diabatic states are not unique,⁶ and many computational techniques have been described to approximate diabatic states.^4,7–14 To this end, we have proposed a generalized diabatic at construction (GDAC) approach on the basis of an effective valence bond (VB) classification,^15,16 which maximally constrains these diabatic states in bonding characters to the asymptotic dissociation limit for the processes of interest. GDAC diabatic states are physically interpretable configurations that form an active space to describe the adiabatic potential energy surfaces. Therefore, the validity and adequacy of GDAC diabatic states can be verified by comparison of experimental observables with properties determined from their resulting adiabatic states. In this article, we present a block-localized excitation (BLE) method to define locally excited configurations for treating excited energy transfer and for describing intermolecular interactions of excimer complexes.

VB theory naturally has orbital constraints, which can be generalized to represent diabatic states. The large number of configurations in ab initio VB theory prevents its practical use on large and condensed phase systems,¹⁷ though modern approaches are improving its applicability.^18–20 On the other hand, block-localized wave function (BLW) theory allows for the construction of an effective VB-like configuration using a single determinant.²¹ In BLW, a diabatic state is represented in terms of block-localized orbitals (BLO), an effective representation of the ab initio VB configurations within a molecular fragment.^22–25 The BLOs can be optimized in wave function theory (WFT)^21,22,24 and using Kohn–Sham density functional theory (KS-DFT).^26,27 In this regard, the block-localized Kohn–Sham orbitals are strictly localized on a given fragment by construction that is constrained in orbital space.^22,25 Related localization methods include that developed by Stoll and coworkers,²⁸ the self-consistent field for molecular interactions (SCF-MI),^29–31 locally projected molecular orbitals,^32,33 absolutely localized molecular orbitals,^34,35 and gradient methods with respect to orbital coefficients.^28,36–39 The BLW configurations can be used in nonorthogonal configuration interaction (NOCI) either in WFT,⁴⁰ called mixed molecular orbital and VB theory,^23,41 or in multistate density functional theory (MSDFT),^27,42–44 to yield the adiabatic ground and excited states. MSDFT has been used in a number of applications,^{15,16,27,43,45–47} including proton-coupled electron transfer (PCET)⁴⁸ and singlet fission⁴⁹ with approximate local-excitations.

A closely related approach is constrained density functional theory-configuration interaction (CDFT-CI) based on density confinement in a certain spatial domain.^50–52 CDFT-CI has been used to estimate electronic couplings for ET and EET reactions, but the results heavily depend on basis functions used (refs 53 and 54 and references there in). Recently, non-Aufbau occupation of the orbitals was employed to construct localized singlet excited state in singlet fission.⁵⁵ One question here is physically constraining electrons within a spatial region on a property that is in fact distributed in the full space. Consequently, the computation also depends on the particular population method used because it is not a physical observable,⁵⁴ and the constrained integration number of electrons actually is a composite from the wave functions from different target states.⁵⁶ Another related approach is the Frenkel–Davydov exciton approach using states derived from time-dependent density functional theory calculations for isolated monomers,^57–59 with which both dipole-coupling and Dexter exchange interactions are directly determined as in the study of singlet fission using MSDFT⁴⁹ and PCET.⁴⁸ In the latter case, local states are variationally optimized in the presence of the environment through a QM/MM interaction Hamiltonian.⁴⁹

In this work, we extended the block-localization method to treat excited states constrained within individual fragments. Similar to previous works for the ground state, the main focus is to obtain BLOs in the excited states. In principle, such calculations can be performed using CASSCF or time-dependent density functional theory (TDDFT) within a fixed molecular fragment.⁶⁰ However, a single-determinant representation rather than a multiconfigurational approach is desired to keep spirit of computational efficiency of KS-DFT for the ground state in MSDFT for the excited state. Here, we find a ΔSCF type of models suitable for our aim to construct localized excited states.⁶¹ Of course, the performance of ΔSCF methods is rather system-dependent, encountering optimization collapse to the ground state or to undesired configurations. To resolve these problems, a number of techniques have been developed such as the maximum overlap (MOM),⁶² improved MOM,⁶³ the MOM square methods,⁶⁴ and square gradient minimization (SGM).⁶⁵ These approaches can be directly adopted to construct diabatic states in the active space for MSDFT. In particular, the present BLE approach is a variation of ΔSCF methods, which is illustrated by the intermolecular interactions of a naphthalene dimer complex both in its first triplet and in two singlet excited states.^66–69 The latter features an interesting curve crossing of two low-lying singlet states as the two aromatic compounds are in close proximity. This was clearly resolved recently using a DMRG-CASPT2 approach including the full active space of 20 electrons in 20 orbitals, but such a curve crossing is only revealed by adding post-SCF perturbation corrections for dynamic correlation.⁶⁸

The remainder of this article is organized as follows. In Section 2, we present the theory and computational details of a BLE method to obtain single-determinant representation of excited configurations in the MSDFT active space. An SCF projection method and an MOM of wave function procedure through singular value decomposition are described to ensure ΔSCF convergence. Numerical results on the naphthalene excimer complex are then presented in Section 3, where we apply the present method to determine the transfer integrals associated with excited energy transfer both in the lowest singlet and in the triplet states, and binding interactions of excimer formation in these two states. Finally, the article is concluded with summary remarks in Section 4.

2. THEORY

2.1. BLW Method for the Ground State.

For completeness, we begin with a brief review of the BLW method for the ground state, which will be used later to construct diabatic states of locally excited configurations. In BLW, the determinant wave function is written as follows^21,25

$${\Psi }_{\text{u}}=N_{\text{u}}\widehat{\text{A}}\left\{{\Phi }^1{\Phi }^2\dots {\Phi }^K\right\}$$ (1)

where N_u is the normalization constant, $\widehat{\text{A}}$ is the antisymmetrization operator, K is the number of blocks, and Φ^A denotes a product of spin-orbitals of block A, ${\Phi }^A={\phi }_1^A\alpha {\phi }_1^A\beta \dots {\phi }_{nA\alpha }^A\alpha {\phi }_{nA\beta }^A\beta$, which are localized by construction

$${\mathit{\varphi }}^A=\chi T^A$$ (2)

where φ^A is a matrix of BLOs in block A and χ a row vector of basis orbitals, and T^A is a matrix of orbital coefficients. In general, BLOs from different blocks are nonorthogonal.

The total coefficient matrix T is block-diagonal in the original BLW method when different blocks have different basis orbitals. In this work, that restriction has been lifted such that different blocks can share a group of common basis functions. The total density matrix D is given by²⁷

$$D=T{\left(T^{†}ST\right)}^{-1}{T}^{†}$$ (3)

where S = χ^†χ is the overlap matrix of the atomic orbitals. Then, the energy of such a block-constrained system is given by

$$E=\text{Tr}(Dh)+\frac{1}{2}\text{Tr}(DJD)+E_{\text{xc}}(\rho )$$ (4)

where h and J are the usual one-electron and Coulomb integrals, and E_xc(ρ) is the exchange–correlation energy in standard KS-DFT. The constrained optimization of the BLOs can be achieved using the Broyden–Fletcher–Goldfarb–Shanno updating scheme⁷⁰ with the energy gradient given as follows^28,36–39

$$\begin{gathered} \delta E=\text{Tr}(\delta D\cdot F) \\ =2\text{Tr}\left[\left(I-S{D}_{\alpha }\right)F_{\alpha }{T}_{\alpha }{\left(T_{\alpha }^{†}S{T}_{\alpha }\right)}^{-1}\delta T_{\alpha }^{†}+\left(I-S{D}_{\beta }\right)F_{\beta }{T}_{\beta }{\left(T_{\beta }^{†}S{T}_{\beta }\right)}^{-1}\delta T_{\beta }^{†}\right] \end{gathered}$$ (5)

where F is the Fock matrix.

The gradient optimization method can be applied to closed shell, restricted open shell, and unrestricted open shell cases. Both analytic orbital gradient and numerical orbital Hessian have been implemented in our program, though the latter is time-consuming. The gradient optimization method is used when common basis functions are used in different blocks.^37,38 Three different forms of SCF equations have been described in the literature.^{27,29–35,71} In Appendix A, we provide another direct derivation and highlight the properties of the non-orthogonal block SCF equations. These methods are derived on the basis of zero energy gradient, and their computational costs are essentially the same when the number of the blocks is not large. In this work, the approach described in ref 27 is used for the ground state calculations, and is extended to the localized orbitals for excited states (below).

To accelerate the convergence of the SCF, we employ the DIIS (direct inversion in the iterative subspace) method⁷² by using energy gradients (Appendix A) as the error vectors and updating the projected Fock matrix and the effective overlap matrix for each block. We have also implemented a DIIS by updating the Fock matrix and coefficients of the whole system. Efficiencies of the two methods are found to be similar for the naphthalene dimer system.

2.2. Block-Localized Excitation.

In the present study, we are interested in obtaining a computationally efficient representation of locally excited configurations within a given block. Traditional methods for excited states are usually of multiconfigurational nature, even for the simplest excited state approaches such as configuration interaction with single excitations (CIS) and TDDFT. Clearly, a multiconfigurational wave function technique within a fixed block would have not been desirable for a DFT-based method for the intended applications. To this end, we employ a ΔSCF-like approach to represent locally excited configurations.⁶¹ ΔSCF is a single determinant technique, with which the exchange–correlation functional developed for KS-DFT can be directly used, keeping in mind that the present Kohn–Sham determinant using BLO is constrained in the framework presented in Section 2.1. In a ΔSCF scheme, the localized orbitals can be optimized in a way that is very similar to the ground state Hartree–Fock–Roothaan equation. However, it is known that the original ΔSCF method often suffers from SCF convergence,⁶¹ and when it converges, an unwanted excited state or the ground state could be obtained.

A number of local excitation ΔSCF methods have been reported,^73–86 including the MOM method⁶² and MOM squared,⁶⁴ the SGM,⁶⁵ the local SCF method for core-excited states,⁷⁴ and a constricted variational DFT (SCF-CV-DFT) by one electron excitation to the virtual orbital space.⁷⁵ The single determinant approach can also be achieved by orthogonality constraints on the excited state. The “big shift” method was formulated by setting a large value (more than 10¹⁰ au) to the diagonal element of the Fock matrix corresponding to ground state orbitals.⁷⁶ Similarly, an asymptotic projection method was suggested by adding an infinite projector to the frozen orbital in the Hamiltonian operator.⁷⁷ Yet, another alternative is the guided SCF approach by transformation of the Fock matrix from the atom orbitals (AO) basis to the excited state orbitals basis.⁷⁸ Similar to the BLW method, but enforcing orbital orthogonality between the core and the valence is the projection configuration interaction method.⁷⁹

In this work, we adopt a ΔSCF approach to represent locally excited non-Aufbau configurations and employ two techniques to ensure SCF convergence. In the first method, we project the ΔSCF equations to the BLO space of the ground state in each iteration to retain the desired order of orbitals for the subsequent SCF iteration. Either the HF or KS-DFT method can be used. The approach in this study can be summarized below, making use of a general case of a single electron excitation from an occupied BLO j to a virtual BLO b. For clarity, only the α spin is shown.

1. We obtain BLOs of the ground state by solving the generalized secular equation F⁰T⁰ = ST⁰E⁰, where the superscript emphasizes ground state orbitals.

2. The initial guess determinant corresponding to the excitation ${\phi }_j^{\mathit{0}}(\alpha )\to {\phi }_b^{\mathit{0}}(\alpha )$ is constructed by switching the occupied and virtual BLOs: ${\Psi }_{b0}^{\text{BLE}}={|\left({\phi }_1^0{\phi }_2^0\dots {\phi }_b^0\dots {\phi }_n^0\right)\left({\phi }_{n+1}^0\dots {\phi }_j^0\dots {\phi }_m^0\right)\rangle }_{\alpha }$. Here, the two pairs of parentheses denote occupied and virtual spaces, and there is no change in the occupation of β electrons.

3. To solve the eigen equation f|φ〉 = |φ〉ε, we set |φ〉 = χT⁰T, and left multiply $\langle {\phi }^0|={\left(\chi T^0\right)}^{†}$. This has the effect of diagonalizing the Fock-like matrix in the BLO basis of the ground state, (T^0†FT⁰)T = (T^0†ST⁰)TE^BLE, to yield a new set of orbitals T for the block-localized excited determinant. After diagonalization, it is not needed to resort the eigenvectors according to the energy of each orbital as the order of ground state molecular orbitals is kept.

4. The effective Fock matrix for the BLE configuration can be obtained in the AO basis using the transformation matrix T_new = T⁰T^BLE for the next SCF iteration in step 3 until convergence, where T^BLE is T with the orbitals φ_j(α) → φ_b(α) swapped. Usually, good convergence can be obtained by projecting the ground state orbitals. We can also use the orbitals produced in a previous iteration to do the projection. Alternatively, we can use a permutated T⁰ to realize the ΔSCF excitation, then follow the process above without eigenvector permutation.

The above procedure is general in that any combinations of α and β excitations, including multiple excitations can be included. We note that the guided SCF method also employs a similar projection method.⁷⁸ The determinant that is optimized using the method presented above is not orthogonal to the ground state, but the overlap with the ground-state determinant is usually small. To obtain a localized excited state that is orthogonal to the localized ground state, the particle orbital can be obtained from the projection of unoccupied orbital space after obtaining the hole orbital from a full space projection. The derivation within localized constraint is presented in Appendix A. Furthermore, the adiabatic ground and excited states can be obtained using MSDFT through configuration interaction among these determinant configurations (see the next section) such that conical intersection between the ground and first excited states can be obtained.⁴³

The second method that we have implemented to improve the ΔSCF convergence is based on the maximum overlap of wave function (MOW) with the initial determinant for the excited configuration, denoted as $T_{b0}^{\text{BLE}}={\left({\phi }_1^0,\dots ,{\phi }_b^0,\dots ,{\phi }_n^0\right)}_{\alpha }$. All steps in the MOW procedure are identical to those outlined above, except step 4 to choose the correct n occupied orbitals of the BLE configuration. Here, at the Ith iteration, the goal is to construct a determinant for a specific BLE configuration that maximizes the overlap with the initial guess ${\Psi }_{b0}^{\text{BLE }}:\underset{\phi }{\text{max}}\left\{\langle {\Psi }_{0b}^{\text{BLE}}\mid {\Psi }_{ib}\rangle =\left|T_{0b}^{\text{BLE}†}S{T}_{\text{I}b}\right|\right\}$. It would be unfeasible to try out all $C_m^n$ possibilities by brute force. Instead, we choose the n orbitals that maximize its Frobenius norm, ${\Vert T_{0b}^{\text{BLE}†}S{T}_{\text{I}b}\Vert }_F=\sqrt{{\sum }_i^n{\sum }_j^n{O}_{i^{0I}j}^2}$, which can be easily accomplished by singular value decomposition. This is equivalent to an approach called MOM squared.^62,64

With the above ΔSCF projection and the MOW method, BLOs for an excited configuration within a given block A can be obtained using the approach described in Subsection 2.1. Because the excited determinants are fully localized, there is no charge transfer contribution between different blocks. We have adopted an unrestricted open-shell treatment both in the ΔSCF projection and in the MOW method, which yield identical results at convergence. All results in the examples illustrated below are obtained by using the block-localized ΔSCF projection scheme.

2.3. MSDFT and Electronic Coupling of BLE Diabatic States.

The adiabatic ground and excited states of the delocalized system can be obtained using MSDFT through NOCI by solving the generalized secular equation,^{23,27,41,42,44,87} for which the BLE configurations form a minimal active space (MAS) for the adiabatic excited states of interest

$$HC=SCE$$ (6)

where E is a diagonal matrix of adiabatic energies, C is a matrix of the CI-coefficient, H is the CI-Hamiltonian, and S is the overlap matrix, which is determined using the separately optimized Kohn–Sham BLE-determinants $\left(S_{AB}=\langle {\Psi }_A^{\text{BLE}}\mid {\Psi }_B^{\text{BLE}}\rangle \right)$.²³ The adiabatic energies of the ground and excited states are given using^27,42

$$E_I^{\text{MS}}\left[{\rho }_I\right]=\sum _A{c}_{AI}^2{H}_A^{\text{BLE}}\left[{\rho }_A\right]+2\sum _{A>B}{c}_{AI}{c}_{BI}{H}_{AB}^{\text{TDF}}\left[{\rho }_{AB},{\mathbf{\text{Ψ}}}_A^{\text{BLE}},{\mathbf{\text{Ψ}}}_B^{\text{BLE}}\right]$$ (7)

In MSDFT, the diagonal matrix elements of the CI Hamiltonian $H_A^{\text{BLE}}\left[{\rho }_A\right]$ are the energies determined using the methods described in Section 2.1 for the ground state determinant and in Section 2.2 for block-localized excited configurations. Each off-diagonal element consists of two terms: a standard interaction term between two nonorthogonal determinants, ${\Psi }_A^{\text{BLE}}$ and ${\Psi }_B^{\text{BLE}}$, and a transition density functional (TDF) energy^{15,16,27,42,44}

$$H_{AB}^{\text{TDF}}=\langle {\Psi }_A^{\text{BLE}}\left|H\right|{\Psi }_B^{\text{BLE}}\rangle +E_{AB}^{\text{TDF}}\left[{\rho }_{AB},{\Psi }_A^{\text{BLE}},{\Psi }_B^{\text{BLE}}\right]$$ (8)

where $E_{AB}^{\text{TDF}}$ is the dynamic correlation contribution to the TDF, or simply, the TDF that depends on the transition density ρ_AB as well as the determinant functions ${\Psi }_A^{\text{BLE}}$ and ${\Psi }_B^{\text{BLE}}$. TDF is a novel class of correlation functional, which does not exist in KS-DFT, accounting for dynamic correlation in electronic coupling and playing a role in removing any double counting of electron correlation in a multiconfigurational representation of DFT.⁴³ As in KS-DFT, the exact functional form of $E_{AB}^{\text{TDF}}$ is not known, but under special conditions, such as the spin coupling of two unpaired electrons forming a pair of spin-adapted singlet and triplet states, the value of $E_{AB}^{\text{TDF}}$ can be obtained consistently with a particular KS-DFT functional approximation to the two spin-mixed determinant configurations.^43,88 This is achieved by enforcing the energy degeneracy of the high (M_s = 1) and low (M_s = 0) spin projections of the triplet states proposed by Ziegler et al.⁶¹ A related approach is the spin-restricted density functional theory for biradical calcualtions.^89–91 Here, the correlation energy of the high-spin projection state $E_{\text{c}}^{\text{KS}}\left[\rho \left(|1,1\rangle \right)\right]$, which can be adequately represented by a single determinant in KS-DFT, along with that of the mixed low-spin determinant $E_{\text{c}}^{\text{KS}}\left[\rho \left(|1,0\rangle \right)\right]$, is used to define the magnitude of the correlation contribution⁴³

$$E_{AB}^{\text{TDF}}\left[{\rho }_{AB},{\Psi }_A^{\text{BLE}},{\Psi }_B^{\text{BLE}}\right]=E_{\text{c}}^{\text{KS}}[\rho (\mid 1,1>)]-E_{\text{c}}^{\text{KS}}[\rho (\mid 1,0>)]$$ (9)

In general applications, one possible approximation for $E_{\text{c}}^{\text{TDF}}$ is a determinant-weighted average of correlation energies of the two interacting states⁴³

$$E_{AB}^{\text{TDF}}\approx \frac{\langle {\Psi }_A^{\text{BLE}}\left|H\right|{\Psi }_B^{\text{BLE}}\rangle }{E_A^{\text{HF}}\left[{\Psi }_A^{\text{BLE}}\right]+E_B^{\text{HF}}\left[{\Psi }_B^{\text{BLE}}\right]}\left[E_A^{\text{c}}\left({\rho }_A\right)+E_B^{\text{c}}\left({\rho }_B\right)\right]$$ (10)

where $E_A^{\text{HF}}\left[{\Psi }_A^{\text{BLE}}\right]$ and $E_B^{\text{HF}}\left[{\Psi }_B^{\text{BLE}}\right]$ are HF energies determined by using the KS-determinants for the BLE configurations, and $E_A^{\text{c}}\left({\rho }_A\right)$ and $E_B^{\text{c}}\left({\rho }_B\right)$ are the corresponding correlation energies. Alternatively, the TDF energy can be approximated by an overlap scaled average of the correlation energies of the two diabatic states.^27,42

$$E_{AB}^{\text{TDF}}\approx \frac{1}{2}{M}_{AB}^{\text{KS}}\left[E_A^{\text{c}}\left({\rho }_A\right)+E_B^{\text{c}}\left({\rho }_B\right)\right]$$ (11)

where $M_{AB}^{\text{KS}}$ is the overlap between two Kohn–Sham determinants. The numerical results from these approximate approaches are similar.

These block-localized configurations may be considered as charge- and energy-localized diabatic states, and their electronic coupling elements are related to the transfer integrals for electron and excited energy transfer applications. Within the MSDFT framework and CDFT-CI, the ET integral between any two diabatic states can be calculated.^27,54,56 Because the BLE configurational states are generally non-orthogonal, the effective coupling term $V_{AB}^{\perp }$ between states A and B can be calculated through Löwdin orthogonalization

$$V_{AB}^{\perp }=\frac{H_{AB}-M_{AB}\left(H_A+H_B\right)/2}{1-M_{AB}{}^2}$$ (12)

where M_AB is the overlap integral of states A and B.

3. COMPUTATIONAL DETAILS

The naphthalene monomer structure was optimized with the D_2h symmetry using Gaussian-16 at the B3LYP/6–31+G(d) level of theory, following the work of Shirai et al.⁶⁸ Then, the monomer structure is kept fixed and is used to construct a face-to-face dimer configuration at various separation distances, ranging from 2.9 to 20 Å. A structure at 50 Å was also built to verify asymptotic energies. It is known that the face-to-face structure of the excimer complex has the lowest energy in both the singlet and triplet states.^{66–69,92–94} Further, upon relaxation from the monomer geometry, the Franck–Condon approximation suggests that the diabatic couplings should not change significantly.^67,69 In BLE calculations, the dimer is separated into two blocks, each containing one naphthalene molecule. Single-point energy calculations are performed using MSDFT, for which a total of 16 block-localized configurations plus the adiabatic ground state have been obtained using block-localized KS-DFT. The PBE0 functional and the cc-pVDZ basis set are used in all calculations.⁹⁵ For the excitations on both monomers and with all three combinations of source and target orbitals, the separately optimized open-shell singlet determinants with αβ and βα spin combinations are converged to the same set of BLO, based on which the spin-coupled singlet and triplet states are obtained. This is equivalent to switch α and β electron spins of an open-shell configuration to yield a spin pair. The all-spin up triplet configuration (M_s = 1) at each geometry was obtained by separate open-shell calculations. The energy for the M_s = 1 state is used to determine the correlation energy term $E_{AB}^{\text{TDF}}$ for the TDF of spin-pair coupling (eq 6). This makes the orbitals not entirely compatible, but the energy difference is less than 0.1 eV, smaller than uncertainties from the method itself, compared with the results obtained using a common set of orbitals. Although this is not a conventional procedure for spin-adapted configurations,⁹⁶ it is not without precedents and we have used it in a number of applications.^15,16,42,43 This is also consistent with the use of a single determinant to treat open-shell systems using KS-DFT,^97,98 yielding lower energy than that adopting a restricted open-shell approach. All other computational details are the same as in other studies.

The potential energies curves for the naphthalene dimer have also been computed using MSDFT with the M06–2X density functional and the cc-pVDZ basis, yielding the same trends with the benefit that additional and empirical dispersion corrections are not needed. The results are given in the Supporting Information.

The method described in the previous section has been implemented in a locally modified version of the GAMESS-US program,⁹⁹ with which all calculations have been performed.

4. RESULTS AND DISCUSSION

An important application of the present BLE method is to understand dimer interactions of aromatic species in the excited states and to determine the electronic coupling in excitation energy transfer (EET) processes from one part of the molecular system to another.¹⁰⁰ We first apply the BLE method to determine the diabatic and adiabatic potential energy curves for the naphthalene dimer in the face-to-face configuration. This is followed by calculations of the EET couplings both in the singlet and triplet states. This system is chosen because it has been extensively studied and there is a challenging curve-crossing that is only obtained with a method balancing static and dynamic correlation.^{66–69,92–94,101} Here, we chose the face-to-face stacked parallel configuration in our study, corresponding to the minimum-energy configuration in the first singlet and triplet excited states of the dimer complex (excimer).^{67–69,92,102–104}

4.1. Excimer Interactions in the Singlet State.

Shirai et al.⁶⁸ reported a most comprehensive investigation of the naphthalene excimer complex in the singlet state using DMRG-CASPT2 with cc-pVDZ and cc-pVTZ basis functions, which revealed an inversion of the ¹L_a⁻ and ¹L_b⁻ states along the dimer separation path. In that study, the full π electrons in an active space of [20e, 20o] were used. Dynamic correlation is critical for the excimer calculations, without which the qualitative trends of the two low-lying states are wrong at the DMRG-CASSCF level.

MSDFT follows a dynamic-then-static (DTS) ansatz, in which dynamic correlation is incorporated in the optimization of BLOs through Kohn–Sham density functionals for each individual determinant configuration.^27,42 Moreover, each configuration in the active space for MSDFT is separately optimized, further introducing orbital relaxation. Consequently, it is expected that a minimal number of essential configurations would be sufficient to describe the properties of the relevant low-energy states. Following the insights provided by Shirai et al. on the naphthalene excimer complex, we tested the idea of a MAS for the two low-lying excited states of naphthalene excimer by MSDFT (Figure 1). This includes three local singly excited configurations for each monomer (eqs 13–20), plus a pair of charge transfer states between two monomers (eqs 19 and 20), which are depicted in Figure 1.

$${\mathbf{\text{Ω}}}_{A*B}{}^{s_{2\to 3}{s}_0}=\frac{1}{\sqrt{2}}\left\{{\Psi }_{A*B}\left(S_2{}^{\overline{3}}{S}_0\right)-{\Psi }_{A*B}\left(S_{\overline{2}}{}^3{S}_0\right)\right\}$$ (13)

$${\mathbf{\text{Ω}}}_{A*B}{}^{s_{2\to 4}{s}_0}=\frac{1}{\sqrt{2}}\left\{{\Psi }_{A*B}\left(S_2{}^{\overline{4}}{S}_0\right)-{\Psi }_{A*B}\left(S_{\overline{2}}{}^4{S}_0\right)\right\}$$ (14)

$${\mathbf{\text{Ω}}}_{A*B}{}^{s_{1\to 3}{s}_0}=\frac{1}{\sqrt{2}}\left\{{\Psi }_{A*B}\left(S_1{}^{\overline{3}}{S}_0\right)-{\Psi }_{A*B}\left(S_{\overline{1}}{}^3{S}_0\right)\right\}$$ (15)

$${\mathbf{\text{Ω}}}_{AB*}{}^{s_0{s}_{{}_{2\to 3}}}=\frac{1}{\sqrt{2}}\left\{{\Psi }_{AB*}\left(S_0{S}_2{}^{\overline{3}}\right)-{\Psi }_{AB*}\left(S_0{S}_{\overline{2}}{}^3\right)\right\}$$ (16)

$${\mathbf{\text{Ω}}}_{AB*}{}^{s_0{s}_{{}_{2\to 4}}}=\frac{1}{\sqrt{2}}\left\{{\Psi }_{AB*}\left(S_0{S}_2{}^{\overline{4}}\right)-{\Psi }_{AB*}\left(S_0{S}_{\overline{2}}{}^4\right)\right\}$$ (17)

$${\mathbf{\text{Ω}}}_{AB*}{}^{s_0{s}_{{}_{1\to 3}}}=\frac{1}{\sqrt{2}}\left\{{\Psi }_{AB*}\left(S_0{S}_1{}^{\overline{3}}\right)-{\Psi }_{AB*}\left(S_0{S}_{\overline{1}}{}^3\right)\right\}$$ (18)

$${\mathbf{\text{Ω}}}_{A^{-}{B}^{+}}=\frac{1}{\sqrt{2}}\left\{{\Psi }_{A^{-}{B}^{+}}\left(D^{\alpha }{D}_{\beta }\right)-{\Psi }_{A^{-}{B}^{+}}\left(D^{\beta }{D}_{\alpha }\right)\right\}$$ (19)

$${\mathbf{\text{Ω}}}_{A^{+}{B}^{-}}=\frac{1}{\sqrt{2}}\left\{{\Psi }_{A^{+}{B}^{-}}\left(D_{\beta }{D}^{\alpha }\right)-{\Psi }_{A^{+}{B}^{-}}\left(D_{\alpha }{D}^{\beta }\right)\right\}$$ (20)

Figure 1.

Schematic depiction of (a) block-locally excited configurations of monomer A* in the presence of monomer B in its ground state, (b) block-locally excited configurations of monomer B* in the presence of monomer A in its ground state, and (c) two charge transfer states between monomers A and B. Each yellow block shows one block-localized determinant configuration of the whole dimer, and the left and right halves within it show the four relevant fragment orbitals of the two monomers, respectively. The four BLOs used in the figure include the two highest occupied fragment orbitals, φ₁ and φ₂, and the two lowest unoccupied fragment orbitals, φ₃ and φ₄, of the excited monomer. The locally excited monomer is indicated by a star. Only singlet spin-adapted states are shown, whereas the symmetric (in-phase) combination of two determinants corresponds to a triplet state.

In eqs 13–20, the notation $S_i^{\overline{a}}$ or $S_{\overline{i}}^a$ denotes a single excitation within a given singlet block in which the β-spin of an electron in the corresponding orbital is represented by an over-hat, and D specifies a doublet block with α and β spin of the unpaired electrons in the occupied (subscript) and virtual (superscript) orbitals (with the reference ground-state configuration, S₀). In short, eqs 13–20 represent single excitations in a blocked 4-electron and 4-orbital space minus the largest energy transition, in which each individual determinant is separately optimized using BLE to yield spin-adapted configurational state functions for singlet (as written) and triplet states (symmetric combinations). Note that MAS is not necessarily a block-box selection in that a great deal of insights of the configuration space ought to be known in the first place, such as the naphthalene dimer in the present study. Furthermore, the configurational state functions in MAS are not equivalent to the orthogonal configurations in CASSCF. However, MAS, when verified by comparison of its resulting adiabatic energies with experiment, can be quite useful for interpretation from energy decomposition analysis²⁵ and for carrying out dynamic simulations of nonadiabatic processes.

We begin with the low-lying excited states of a naphthalene monomer, whose active space in MSDFT can be represented using eqs 13–15 (omitting the orbitals on fragment B). There are two singlet excited states of similar energies, corresponding to ${}^1{B}_{3u}\left({}^1{L}_{\text{b}}\right)$ and ${}^1{B}_{2u}\left({}^1{L}_{\text{b}}\right)$ symmetry.¹⁰⁵ The latter is of HOMO → LUMO character given by eq 13 (Figure 1a), and the complementary symmetric combination of the two determinants, $\left\{{\Psi }_{A*B}\left(S_2{}^{\overline{3}}{S}_0\right)+{\Psi }_{A*B}\left(S_{\overline{2}}{}^3{S}_0\right)\right\}/\sqrt{2}$, corresponds to the M_S = 0 multiplet of the first triplet excited state |1,0〉. The M_S = 1 state of the triplet (|1,1〉) can be adequately treated by KS-DFT, and a vertical excitation energy of 3.13 eV above the ground state is obtained with PBE0/cc-pVDZ. The value is in accord with experiment (3.0 eV), and is used to define⁶¹ the TDF for spin-coupling interaction (eq 9).⁴³ We obtained a vertical excitation energy of 4.52 eV ¹L_a singlet state (eq 10) from MSDFT@PBE0, falling in the range of experimental observations (4.45^106,107 and 4.7 eV).¹⁰⁸

For the diabatic coupling between the symmetry-allowed spin-adapted states $E_{14-15}^{\text{TDF}}$ (eqs 14 and 15), we used the average of the two spin-coupling values determined with eq 9^106,107 because the transition energies are nearly degenerate for HOMO – 1 → LUMO (4.48 eV) and HOMO → LUMO + 1 (4.51 eV) single excitations.⁶⁸ This yields an energy for ¹L_b of 4.14 eV in comparison with the experimental value of 3.97 EV.^106,107 The $E_{14-15}^{\text{TDF}}$ value for naphthalene is used for the local (monomer) ¹L_b excitation in the dimer complex at all geometries because it is intrinsically a local (monomer) property. For comparison, CASPT2/cc-pVTZ calculations yielded vertical excitation energies of 3.83 and 4.34 eV for the ¹L_b and ¹L_a states,⁶⁸ whereas 4.03 and 4.56 eV were obtained by Roos and coworkers using natural atomic orbital basis functions.¹⁰⁹ Overall, we found that with an MAS of three spin-adapted diabatic states in MSDFT, the computed energies of the three low-lying excited states (two singlet and one triplet) are within 0.2 eV of experimental values.^106,107

Second, Figure 2 shows the potential energy curves for the ground state and three low-lying excited states determined from MSDFT@PBE0 calculations (solid curves), along with the corresponding curves for the singlet states by DMRG-CASPT2 (dashed curves) and DMRG-CASSCF (dotted curves) from ref 68. Because the energies for the excited states are overestimated by 0.3–0.5 eV from DMRG-CASPT2 calculations of the naphthalene monomer, this trend is transferred to the potential energy curves of the dimer complex. At large interfragment separations (>3.5 Å), the two low-lying singlet states of naphthalene in the excimer complex follow the same order in energy as that of the monomer: ${}^1{L}_{\text{a}}^{-}\left({}^1{B}_{3g}\right){>}^1{L}_{\text{b}}^{-}\left({}^1{B}_{2g}\right)$. However, the charge-transfer (CT) state (Figure 1c and eqs 19 and 20) is strongly mixed in ¹L_a⁺ at short distances (Figure 3), stabilizing intermolecular interactions between two naphthalene molecules in the excimer complex. This leads to an inversion of the ¹L_a⁻ and ¹L_b⁻ energy levels to bring ¹L_a⁻ as the first singlet excited state of the excimer complex. MSDFT calculations show that the singlet energies are inverted at a distance of about 3.9 Å (Figure 2), somewhat longer than the DMRG-CASPT2 prediction (distances of inversion at 3.26–3.62 Å using, respectively, cc-pVDZ and cc-pVTZ basis sets).⁶⁸ The qualitative trend of these two low-lying states are critically dependent on the active space used in CASPT2 calculations; the correct order is only obtained when the active space is greater than [12e, 12o]. The order of energy levels is not correctly obtained using the single-reference EOM-CCSD/cc-pVTZ method.⁶⁸ Furthermore, the work of Shirai et al. demonstrated that dynamic correlation is absolutely necessary for the excimer complex, without which ¹L_b⁻ remains the lowest state incorrectly at all separations, below ¹L_a⁻ by as much as 1.6 eV using DMRG-CASSCF [20e, 20o].

Figure 2.

Computed potential energy curves of the adiabatic ground and the first triplet (T1) and first two singlet (S1) excited states from MSDFT along with those determined by using DMRG-CASPT2 (DMRG) and DMRG-CASSCF (CAS) with 20 electrons in 20 orbitals for the singlet states (ref 68). The PBE0 functional is used in MSDFT calculations.

Figure 3.

Computed adiabatic singlet excited states, along with BLE diabatic states, excitonic resonance (ER) diabatic states, and CT resonance diabatic states. Energies are given relative to the separated naphthalene molecules in the ground state. All computations have been performed using MSDFT with the PBE0 density functional (MSDFT@PBE0) and the cc-pVDZ basis set.

An anonymous referee noted that the energy of the ¹L_b⁻ state is more repulsive at short distances than that determined by DMRG-PT2, which may reflect CT effects not included in the small active space used in this study. While CT effects are captured in the ¹L_a⁻ state, high-lying CT states at longer separations may indeed become competitive to make contributions at a short interfragment distance, and this has been observed in the study of the photochemical dissociation of LiH.¹⁵

Third, using the active states defined by eqs 13–20, we obtained vertical emission (fluorescent) energies of 3.0 and 4.0 eV from ¹L_a⁻ and ¹L_b⁻, respectively, which are similar to DMRG-CASPT2 values of 3.25 and 3.65 eV (Supporting Information, see ref 68), and in reasonable accordance with an experimental value of 3.13 eV for the first state.¹¹⁰ These values are dependent on the structures of the excimer complex, which were not optimized in the present study. We obtained a binding energy of 1.44 eV for the ¹L_a⁻ state of the excimer (including Grimme’s D3 dispersion correction applied to all states), stronger than that predicted using DMRG-CASPT2 (1.23 and 1.33 eV using cc-pVDZ and cc-pVTZ basis sets) and a value of 0.99 eV using SOS-CIS(D₀)/aug-cc-pVDZ.⁶⁹ Basis set superposition errors (BSSE), which decrease the binding energy by 0.33 and 0.17 eV for the two basis sets, have been noted.⁶⁸ In principle, the present BLE configurations in MSDFT do not have BSSE by construction, which was pioneered by Gianinetti et al.^29–31 The binding energy is much weaker for the ¹L_a⁻ state at about 0.36 eV from MSDFT, comparable to the DMRG-CASPT2 value (ca. 0.5 eV).⁶⁸

Finally, it is of interest to note that only eight spin-adapted configurational state functions (16 determinants) plus the ground state are used in the present MSDFT calculations, whereas the DMRG study included all π-electron in a [20e, 20o] active space, consisting of more than 3 × 10¹⁰ configurations.⁶⁸ Both qualitative trends as well as quantitative results from MSDFT with an MAS are at least of the quality of DMRG-CASPT2. It is sobering to note that without perturbation correction for dynamic correlation, ¹L_a⁻ from DMRG-CASSCF [20e, 20o] is more than 2 eV too high than the experimental value at large dimer separation, while the result for ¹L_b⁻ is nearly perfect because there is little dynamic correlation (Figure 2). The agreement highlights the usefulness of a DTS approach for balancing dynamic and static correlation in excited state calculations at a cost comparable to KS-DFT for the ground state.

The nature of resonance stabilization of ¹L_a⁻ relative to that of ¹L_b⁻ is revealed by comparing diabatic states in the two symmetry groups in Figure 3. For ${}^1{L}_{\text{b}}^{-}\left(1^1{B}_{2g}\right)$, the BLEs of each naphthalene molecule in the presence of the second monomer are constructed as follows

$$|A*B\left(L_{\text{b}}\right)\rangle =a_{2\to 4}^A{\mathbf{\text{Ω}}}_{A*B}{{}^{S_{2\to 4}}}^{S_0}-a_{1\to 3}^A{\mathbf{\text{Ω}}}_{A*B}{}^{S_{1\to 3}{S}_0}$$ (21)

$$|AB*\left(L_{\text{b}}\right)\rangle =a_{2\to 4}^B{\mathbf{\text{Ω}}}_{AB*}{}^{S_0{S}_{2\to 4}}-a_{1\to 3}^B{\mathbf{\text{Ω}}}_{AB*}{}^{S_0{S}_{1\to 3}}$$ (22)

where the coefficients $a_{ia}^{A/B}$ are determined by using MSDFT (eqs 14 and 15 for |A*B〉 and eqs 17 and 18 for |AB*〉). These two diabatic states are completely degenerate shown as a single curve in Figure 3 (red). Interestingly, the presence of the second monomer lowers the BLE state |A*B(L_b)〉 by 0.30 eV relative to an isolated naphthalene, largely because of the empirical D3 dispersion term. The resonance delocalization of these two BLEs is weak, resulting in a pair of adiabatic states ¹L_b⁻ and ¹L_b⁺ (blue dashed curves), whose energy splitting is only 0.17 eV at the minimum of the complex (3.4 Å), gaining only 0.06 eV in binding energy for the lower state ¹L_b⁻. This is consistent with the results from DMRG calculations with and without the PT2 correction (Figure 2).

In contrast, it is no longer physically meaningful to define a BLE (HOMO → LUMO) diabatic state for ${}^1{L}_{\text{a}}^{-}\left(1^1{B}_{3g}\right)$ because CT configurations dominate coupling interactions and it is not a local (monomer) property. Of course, one can nevertheless construct a BLE for the “covalent” diabatic configuration (not shown) similar to eqs 18 and 19, and its energy monotonically decreases as the dimer distance shortens. In this case, we construct two pairs of resonance states from the local HOMO → LUMO excited configurations, ER, and from the CT configurations across monomer boundaries, charge-transfer resonance (CR)

$${\left(L_{\text{a}}{}^{\pm }\right)}_{\text{ER}}=\frac{1}{\sqrt{2}}\left\{{\mathbf{\text{Ω}}}_{A*B}{}^{S_{2\to 3}{S}_0}\pm {\mathbf{\text{Ω}}}_{AB*}{}^{S_0{S}_{2\to 3}}\right\}$$ (23)

$${(\text{CT})}_{\text{CR}}{}^{\pm }=\frac{1}{\sqrt{2}}\left\{{\mathbf{\text{Ω}}}_{A^{-}{B}^{+}}\pm {\mathbf{\text{Ω}}}_{A^{+}{B}^{-}}\right\}$$ (24)

Figure 3 shows that the resonance splitting (0.88 eV) in (L_a^±)_ER is significantly greater than that in the ¹L_b⁻ state, gaining a resonance stabilization energy of 0.65 eV, and the resonance propagates to long dimer separations without vanishing at 10 Å (black and gray curves). The interaction between the CT configurations is less pronounced (green and orange curves). Importantly, electronic coupling of the BLE resonance diabatic state (L_a⁻)_ER and that of the CT state (CT)_CR⁻ lowers the complexation energy of the BLE resonance state by 0.79 eV and shifts the minimum of the ER state from 3.3 to 3.1 Å (Figure 3). Therefore, both the resonance delocalization of BLE and CT stabilization contributes significantly, with the latter playing a slightly greater role (55 vs 45%) in the total excimer binding energy.

4.2. Excimer in the Triplet State.

The computed first triplet excited state is shown in Figure 2 and its diabatic state composition is separated in Figure 4. Although a number of higher energy states, both singlet and triplet, can be obtained, other diabatic configurations may or may not be needed to fully characterize these adiabatic states. Because the main goal is to illustrate the BLE optimization method to define configurational states for MSDFT calculations, these states are not further analyzed and shown. Although there has been controversy on the existence of a triplet excimer of naphthalene,^{67,69,111–114} the work of Pabst et al. convincingly demonstrated that the triplet state excimer is as important and as stable as that for the singlet configurations.⁶⁷ The reason is simple from the present diabatic perspective because the excimer complex in the triplet state ³L_a⁻ has an equivalent manifold of interactions as that of the excimer complex in the singlet state ¹L_a⁻, except that electronic couplings in the triplet states are somewhat weaker. Consequently, a binding energy of 0.85 eV is obtained for the excimer complex at a separation of 3.2 Å from MSDFT calculations. Thus, the triplet excimer also forms a strong complex. This may be compared with a value of 0.56 eV predicted by Pabst et al. using CC2/cc-pVTZ after the BSSE correction,⁶⁷ and values of 0.79 and 0.60 eV with frozen and relaxed monomer structures by Kim.⁶⁹ The interfragment distance for the triplet excimer was found to be 3.08 Å in the work of Pablst et al. Another key spectroscopic quantity of interest is the adiabatic phosphorescence energy, which is estimated to be 2.27 eV from MSDFT, in reasonable agreement with the result of Pabst et al. (2.54 eV).⁶⁷ Phosphorescence in the present face-to-face configuration is symmetry forbidden, but deviations from the minimum structure could produce emissions and experimental observations of naphthalene triplet excimers show an energy of 2.3–2.4 eV in solution.^113,114

Figure 4.

Computed adiabatic triplet excited state (T1) along with BLE diabatic states, ER diabatic states, and CT resonance diabatic states. Energies are given relative to the separated monomers in the ground state. All computations have been performed using MSDFT with the PBE0 density functional (MSDFT@PBE0) and the cc-pVDZ basis set.

The present study also yield a singlet–triplet (S–T) energy gap, which is 1.0 eV between ³L_a and ¹L_b for naphthalene, and 0.79 eV between ³L_a⁻ and ¹L_a⁻ in the excimer complex. Thus, there is a small red-shift in the energy gap of 0.21 eV between the monomer and excimer complexes. S–T energy gaps of 1.16 and 1.38 were obtained by Kim for naphthalene in the ground monomer geometry and relaxed structure, respectively, which are correspondingly red-shifted by 0.2 in the monomer and 0.3 eV in the excimer complex.⁶⁹

4.3. Excitonic Coupling between Singlet Excited States of Naphthalene Molecules.

EET plays an important role in many artificial and natural molecular devices such as organic light-emitting diodes, photovoltaics, and the photosynthetic system.^115,116 In the weak coupling limit, the EET rate constant can be calculated using the Fermi’s golden rule⁴

$$k_{\text{EET}}=\frac{2\pi }{ħ}|V{|}^2\delta \left(E_i-E_{\text{f}}\right)$$ (25)

where V is the electronic coupling between the initial state E_i and final state E_f. Therefore, the electronic coupling parameter is a key quantity to determine the EET rate constants.

The excitonic coupling between two locally excited states (covalent configuration) in the ¹L_a⁻ state has been determined in the range of 2.5–20 Å between dimer separation (Figure 5), which may be considered to associate with singlet excitation energy transfer (SEET) from the initial (eq 13) to the final (eq 16) states.¹⁰¹ At distances greater than 6 Å, the excitonic coupling is reciprocally proportional to the cube of the separation distance, indicating that the energy resonance in the excited state is dominated by long-range electrostatic interactions, consistent with a Förster dipole–dipole interaction model.⁴ At close proximity, Dexter exchange coupling becomes important and deviation from the linearity of long-range behavior occurs. Figure 5 shows that the effect of the basis set size on the computed BLE coupling is small at all distances, suggesting that the present BLE approach is a robust procedure for modeling exciton coupling in the singlet state.

Figure 5.

Computed transfer integral (V) for the excited energy transfer in the first singlet excited state between two naphthalene molecules as a function of the intermolecular separation. The PBE0 density functional is used in all calculations with different basis sets. Spin-flip using orbitals optimized in the triplet-state (T-SF) with the 6–31G(d) basis set was used; ionic configurations determined by perturbation theory (IC-PT); and IC by solving an eigenvalue equation (IC-EIG). ICs were calculated using the aug-cc-pVDZ basis set. The slopes were fitted using data in the range from 8 to 20 Å.

The SEET coupling parameters have also been computed on the basis of locally excited diabatic states constructed using BLOs optimized in the triplet state (M_s = 1) followed by an α → β spin flip (T-SF). In these calculations, the 6–31G(d) basis set was used. The computed electronic coupling values are not sensitive with respect to orbital optimization in the singlet or triplet states (Figure 5). We further examined the effect of ionic configurations (ICs) given in eqs 19 and 20, which could be important for excited energy transfer through a super-exchange mechanism.⁴⁹ Two approaches were used to determine IC contributions. Using a perturbation theory (IC-PT) with the aug-cc-pVDZ basis set,¹¹⁷ we noticed significant deviations from results obtained using other models at short interchromophore distances. Not surprisingly, perturbation theory is expected to be valid when contributions from ICs are small at long distances. The IC contribution can be determined in the initial (A excitation) and final (B excitation) states using MSDFT (IC-EIG). Figure 5 shows that IC contributions cause a modest enhancement to the computed transfer integrals obtained directly using locally excited configurations at short distances, whereas at long ranges this effect diminishes beyond 5 Å.

4.4. Excitonic Coupling in the Triplet Excited States.

In contrast to the singlet state, excitonic coupling in the triplet excitation energy transfer (TEET) process is dominated by short-range coupling. Using MSDFT, both spin multiplets of the locally excited triplet state can be defined. In particular, the M_s = 0 state is defined by two BLE determinants through symmetric combination complementary of the singlet state in eqs 13 and 16.

$${\mathbf{\text{Ω}}}_{A*B}{}^{T_{2\to 3}{S}_0}=\frac{1}{\sqrt{2}}\left\{{\Psi }_{A*B}\left(S_2{}^{\overline{3}}{S}_0\right)+{\Psi }_{A*B}\left(S_{\overline{2}}{}^3{S}_0\right)\right\}$$ (26)

$${\mathbf{\text{Ω}}}_{AB*}{}^{S_0{T}_{2\to 3}}=\frac{1}{\sqrt{2}}\left\{{\Psi }_{AB*}\left(S_0{S}_2{}^{\overline{3}}\right)+{\Psi }_{AB*}\left(S_0{S}_{\overline{2}}{}^3\right)\right\}$$ (27)

Of course, the M_S = 1 state can be treated by a single determinant. We note that the condition for energy degeneracy of these spin multiplets defines the spin-coupling correlation energy of the TDF along with the PBE0 correlation energy.⁴³

The calculated TEET excitonic couplings are shown in Figure 6 for both the M_s = 0 and M_s = 1 triplet states. We first compare the “1 + 1”, “BLW”, and “relaxed” methods for M_s = 1 states and the BLE method for M_s = 0 states with the 6–31G(d) basis set. “1 + 1” means that the localized wave function is composed directly of the wave functions of two blocks. “Relaxed” means that the wave functions are relaxed without any restriction starting from BLW. The couplings of the “relaxed” method are the same starting either from BLW or “1 + 1”. The couplings of “1 + 1”, “BLW”, and BLE methods have small differences because the polarization interaction between two blocks is small for the eclipsed naphthalene molecules at certain distances. The “relaxed” method did not give a smooth curve because the obtained wave function may deviate from the optimized state when the distance is less than 4.5 Å. Therefore, the “relax” method may not be stable when the interaction between two blocks is strong.

Figure 6.

Computed electronic coupling vs monomer separation distance between two stacked locally excited naphthalene molecules in the first triplet state, both in the M_S = 0 and M_S = 1 spin-projection states. In all calculations, the hybrid PBE0 functional is used with basis functions indicated in the figure or in the text. T-SF with the 6–31G(d) basis set was used; IC-PT; IC-EIG; 1 + 1: localized state directly composed of fragment orbitals for the monomers in the triplet state; and relax: variationally optimized locally excited states using the 6–31G(d) basis set. ICs were calculated using the aug-cc-pVDZ basis set. The exponential parameters in the figures are 2.60 Å obtained using IC-EIG at distances of 3.5–7 Å and 2.48 Å⁻¹ without ICs at distances of 2.5–7 Å.

In contrast to SEET, where the computed transfer integrals do not show dependence on basis functions, Figure 6 reveals that inclusion of diffuse functions in computing the EET coupling is critical. Charge transfer contributions are also important, enhancing the coupling term by a factor of 2 over that without including ICs. Both the perturbation approach and eigenvalue solution yielded similar results except at distances shorter than 3 Å. The coupling values obtained with diffuse functions decay exponentially, V_TEET ∝ e^−βr, with β values of 2.48 Å⁻¹ without ICs, and 2.60 Å⁻¹ with the inclusion of ICs. These observations are consistent with the results reported in refs 101, 117–120.

The exponential decay of the TEET coupling is similar to ET coupling because the TEET can be viewed as two electron exchanges with different spins.⁴ We further calculated the couplings of ET and hole transfer (HT) at the aug-cc-pVDZ level from 3.5 to 7 Å using MSDDFT for two naphthalene molecules. The β values of ET and HT are 1.00 and 1.45 Å⁻¹, respectively. Thus, the β value of TEET is almost the sum of the β values of ET and HT.

Equation 8 shows that the computed electronic coupling is proportional to the overlap of the two diabatic states when the overlap is small and orbital variations can be ignored.

$$V_{\text{AB}}=\frac{M_{\text{AB}}}{1-M_{\text{AB}}{}^2}\left(F_{\text{AB}}-H_{\text{AA}}/2-H_{\text{BB}}/2\right)\approx M_{\text{AB}}{C}_{\text{AB}}$$ (28)

The approximation V_AB = CM_AB, where M_AB is the overlap integral between determinants A and B, has been applied to ET.¹²¹ This trend is indeed reproduced using MSDFT along with BLE states in Figure 7, in which all methods exhibit nearly perfect linear regression (R² > 0.99 from exponential root-mean-square logarithmic error analysis).^121,122

Figure 7.

Correlations between the coupling constants and the wavefunction overlaps for TEET. The labels for all the curves are the same as those in Figure 6.

5. CONCLUSIONS

In this work, we describe a BLE method on the basis of fragment-block localization and ΔSCF optimization such that a single-determinant representation for excited diabatic states can be approximated using density functional theory. We examined two optimization approaches to ensure the convergence of the block-localized molecular orbitals for a BLE, including a ΔSCF projection and an MOW technique. Within the framework of MSDFT, the use of the present BLE method in the construction of an effective, MAS is illustrated by the excimer complex formation of two naphthalene molecules both in the singlet and triplet states. Furthermore, excited energy transfer integrals in the singlet (SEET) and triplet (TEET) excited states of the naphthalene dimer are determined using block-localized excited diabatic states.

Numerical results show that the BLE method can provide a reasonable description of the transfer integrals associated with excited energy transfer in the singlet and triplet states. Using an MAS of eight spin-adapted configurations, the present MSDFT method correctly produced qualitatively and quantitatively the trends of an energy-level inversion involving the two lowest singlet excited states ¹L_a ⁻ and ¹L_b⁻, in good accordance with the DMRG-CASPT2 results. The latter approach used a large, complete active space of all 20 π electrons in 20 orbitals, consisting of more than 3 × 10¹⁰ configurations. Yet, without perturbation correction, it was still not possible to produce the correct order, with the ¹L_b⁻ state below that of ¹L_a ⁻ by as much as 1.6 eV for the excimer complex. Dynamic correlation proved to be absolutely critical to bring ¹L_a⁻ below ¹L_b⁻ in DMRG-CASPT2. The good performance of MAS-MSDFT in comparison with DMRG-CASPT2 illustrates the effectiveness of a DTS ansatz to reduce the size of the active space in excited state calculations. It should be reminded that the choice of an MAS is not necessarily a black-box approach and an understanding of the nature of the relevant excited states is required. The benefits of this representation include physical insights gained from a diabatic perspective and its connection to classical theories such as the Marcus theory for ET. Although the calculations were illustrated on a model system of two identical monomers, the method can certainly be applied to heterodimers and complex molecular blocks in photoreceptor proteins.^46,47

APPENDIX A

Calculation of the Block-Localized Ground and Excited State Energy Gradients

The operator form of the ground state energy gradient for RHF is given using⁴⁴

$$\frac{\partial E}{\partial \langle {\phi }_A|}=4(1-\rho )f|{\tilde{\phi }}_A\rangle$$ (A1)

where the reciprocal orbitals $|\tilde{\phi }\rangle$ are

$$\begin{gathered} |\tilde{\phi }\rangle =|{\phi }_A{\phi }_{\notin A}\rangle {\langle {\phi }_A{\phi }_{\notin A}\mid {\phi }_A{\phi }_{\notin A}\rangle }^{-1} \\ =|{\phi }_A{\phi }_{\notin A}\rangle {\left(\begin{array}{ll}\langle {\phi }_A\mid {\phi }_A\rangle \hfill & \langle {\phi }_A\mid {\phi }_{\notin A}\rangle \hfill \\ \langle {\phi }_{\notin A}\mid {\phi }_A\rangle \hfill & \langle {\phi }_{\notin A}\mid {\phi }_{\notin A}\rangle \hfill \end{array}\right)}^{-1} \end{gathered}$$ (A2)

here eq 〈φ|φ〉 is the overlap matrix of the molecular orbitals, not an integral value. By using the method of blockwise matrix inversion¹²³

$${\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)}^{-1}=\left(\begin{array}{ll}{\left(A-B{D}^{-1}C\right)}^{-1}\hfill & -{\left(A-B{D}^{-1}C\right)}^{-1}B{D}^{-1}\hfill \\ -D^{-1}C{\left(A-B{D}^{-1}C\right)}^{-1}\hfill & D^{-1}C{\left(A-B{D}^{-1}C\right)}^{-1}B{D}^{-1}+D^{-1}\hfill \end{array}\right)$$ (A3)

we set

$$\begin{gathered} {\left(\langle {\phi }_A\mid {\phi }_A\rangle -\langle {\phi }_A\mid {\phi }_{\notin A}\rangle {\langle {\phi }_{\notin A}\mid {\phi }_{\notin A}\rangle }^{-1}\langle {\phi }_{\notin A}\mid {\phi }_{\notin A}\rangle \right)}^{-1} \\ ={\langle {\phi }_A\left|\left(1-{\rho }_{\notin A}\right)\right|{\phi }_A\rangle }^{-1} \\ =\alpha \end{gathered}$$ (A4)

After transformations, the following expressions are obtained

$$|{\tilde{\phi }}_A\rangle =|\left(1-{\rho }_{\notin A}\right)|{\phi }_A\rangle \alpha ,\left(1-\left(1-{\rho }_{\notin A}\right)|{\phi }_A\rangle \alpha \langle {\phi }_A|\right){\langle {\phi }_{\notin A}|{\phi }_{\notin A}\rangle }^{-1}\rangle$$ (A5)

$$\rho =|\tilde{\phi }\rangle \langle {\phi }_A{\phi }_{\notin A}|=\left(1-{\rho }_{\notin A}\right)|{\phi }_A\rangle \alpha \langle {\phi }_A|\left(1-{\rho }_{\notin A}\right)+{\rho }_{\notin A}$$ (A6)

$$\begin{gathered} \frac{\partial E}{\partial \langle {\phi }_A|}=4(1-\rho )f|{\tilde{\phi }}_A\rangle \\ =\left(1-\left(1-{\rho }_{\notin A}\right)|{\phi }_A\rangle \alpha \langle {\phi }_A|\right)\left(1-{\rho }_{\notin A}\right) \\ f\left(1-{\rho }_{\notin A}\right)|{\phi }_A\rangle \alpha \end{gathered}$$ (A7)

We can further set α as a unit matrix to make $\left(1-{\rho }_{\notin A}\right)|{\varphi }_A\rangle$ orthonormalized

$$\alpha =\langle {\phi }_A|\left(1-{\rho }_{\notin A}\right)\left(1-{\rho }_{\notin A}\right){|{\phi }_A\rangle }^{-1}={\langle {\Phi }_A\mid {\Phi }_A\rangle }^{-1}=I$$ (A8)

and obtain the energy gradient for block A

$$\begin{gathered} \frac{\partial E}{\partial \langle {\phi }_A|}=4\left(1-\left(1-{\rho }_{\notin A}\right)|{\phi }_A\rangle \langle {\phi }_A|\right)\left(1-{\rho }_{\notin A}\right) \\ f\left(1-{\rho }_{\notin A}\right)|{\phi }_A\rangle \end{gathered}$$ (A9)

At the point of the lowest energy, the energy gradient is zero. We obtain the SCF equation with

$$\langle {\phi }_i^A|\left(1-{\rho }_{\notin A}\right)f\left(1-{\rho }_{\notin A}\right)|{\phi }_i^A\rangle ={\epsilon }_i^A$$ (A10)

The SCF equation (in refs 30 and 34) can then be obtained by putting eq A11 into the above SCF equation. Because α is a unit matrix, eq A6 now becomes

$$\rho =\left(1-{\rho }_{\notin A}\right){\rho }_A\left(1-{\rho }_{\notin A}\right)+{\rho }_{\notin A}={\rho }_A^x+{\rho }_{\notin A}$$ (A11)

Equation S5 can then be obtained by inserting eqs A11 into S1.

We have the property

$$\left(1-{\rho }_{\notin A}\right)|{\phi }_{\notin A}\rangle =|{\phi }_{\notin A}\rangle -{\rho }_{\notin A}|{\phi }_{\notin A}\rangle =0$$ (A12)

If we set $|{\Phi }_i^A\rangle =\left(1-{\rho }_{\notin A}\right)|{\phi }_i^A\rangle$, then $\langle {\phi }_{\notin A}\mid {\Phi }_i^A\rangle =0$. By multiplying $\langle {\phi }_i^A|$ on the left of eq 8, using the idempotency relation $\left(1-{\rho }_{\notin A}\right)\left(1-{\rho }_{\notin A}\right)=\left(1-{\rho }_{\notin A}\right)$, and keeping Φ_A orthogonal, we obtain

$$\langle {\mathbf{\text{Φ}}}_i^A\left|f\right|{\mathbf{\text{Φ}}}_i^A\rangle =\langle {\mathbf{\text{Φ}}}_i^A\mid {\mathbf{\text{Φ}}}_i^A\rangle {\epsilon }_i^A={\epsilon }_i^A$$ (A13)

$$f|{\mathbf{\text{Φ}}}_i^A\rangle ={\epsilon }_i^A|{\mathbf{\text{Φ}}}_{\notin A}\rangle$$ (A14)

The above derivation from eqs A12–A14 is a reversed process of the generalized Phillips–Kleiman pseudopotential derivation in ref 75. This means that the projected wavefuction of block A orthogonalized to all other blocks is the eigenfuction of the whole system.

To constrain the excited state particle orbitals |φ_A〉 orthogonal to the ground state occupied orbitals $|{\phi }_A^0\rangle$, we need $\langle {\phi }_A\mid \left(1-{\rho }_{\notin A}^0\right){\phi }_A^0\rangle =0$. The following equation is solved with a Lagrange multiplier λ

$$\begin{gathered} \frac{\partial L}{\partial \langle {\phi }_A|}=\left(1-\left(1-{\rho }_{\notin A}\right)|{\phi }_A\rangle \langle {\phi }_A|\right)\left(1-{\rho }_{\notin A}\right) \\ f\left(1-{\rho }_{\notin A}\right)|{\phi }_A\rangle -\lambda \left(1-{\rho }_{\notin A}^0\right)|{\phi }_A^0\rangle \\ =0 \end{gathered}$$ (A15)

By multiplying $\langle {\phi }_A^0|$ on the left, the Lagrange multiplier λ can be obtained

$$\lambda =\langle {\phi }_A^0|\left(1-\left(1-{\rho }_{\notin A}\right)|{\phi }_A\rangle \langle {\phi }_A|\right)\left(1-{\rho }_{\notin A}\right)f\left(1-{\rho }_{\notin A}\right)|{\phi }_A\rangle$$ (A16)

The equation then becomes

$$\begin{gathered} \left(1-\left(1-{\rho }_{\notin A}^0\right)|{\phi }_A^0\rangle \langle {\phi }_A^0|\right)\left(1-\left(1-{\rho }_{\notin A}\right)|{\phi }_A\rangle \langle {\phi }_A|\right) \\ \left(1-{\rho }_{\notin A}\right)f\left(1-{\rho }_{\notin A}\right)|{\phi }_A\rangle \\ =0 \end{gathered}$$ (A17)

We can further rewrite the above equation into a symmetric form

$${\rho }^{\prime }\left(1-{\rho }_{\notin A}\right)f\left(1-{\rho }_{\notin A}\right){\rho }^{\prime }|{\phi }_i^A\rangle ={\rho }^{\prime }\left(1-{\rho }_{\notin A}\right)|{\phi }_i^A\rangle {\epsilon }_i^A$$ (A18)

where ${\rho }^{\prime }=1-\left(1-{\rho }_{\notin A}^0\right)|{\phi }_A^0\rangle \langle {\phi }_A^0|=\left(1-{\rho }_{\notin A}^0\right)|{\phi }_{Av}^0\rangle \langle {\phi }_{Av}^0|$ by using normalization property of complete nonorthogonal basis, $|{\phi }_{Av}^0\rangle$ is ground state unoccupied orbitals. To solve this eigenequation, we multiply $\langle {\phi }_{Av}^0|={\left(\chi T_{Av}^0\right)}^{†}$ on the left and set $|{\phi }_A\rangle =\chi T_A=\chi T_{Av}^0{T}_A$ to project the equation to ground state unoccupied orbital space. The following equation is then obtained

$$\left(T_{Av}^{0†}{F}_A^{\prime }{T}_{Av}^0\right)T_A^{\prime }=\left(T_{Av}^{0†}{S}_A^{\prime }{T}_{Av}^0\right)T_A^{\prime }{E}_A$$ (A19)

After the diagonalization, the particle orbitals will be selected according to the order of ground state unoccupied orbitals.