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

Abstract

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.

Multistate Energy Decomposition Analysis of Molecular Excited States

“…Ψ and Φ are typically multistate wave functions, which are also used to represent the matrix density. The Greek letter Ξ is used to specify a single Slater determinant wave function for a given electronic configuration, typically monomer-block localized, to represent the corresponding electron density in block-localized excitation (BLE) and target state optimized density functional theory (TSO-DFT) calculations. , Occupied Kohn–Sham molecular spin orbitals, typically block localized, are denoted by χ i , χ j , ..., and unoccupied (virtual) orbitals are given as χ a , χ b , .... In the present MS-EDA method, we use the monomer block-localized Kohn–Sham (BLKS) determinant, Ξ ( X • Y ) KS , as a reference state, which is used to generate the initial guess for singly excited configurations Ξ i a , all of which are variationally optimized individually using BLE and TSO–DFT for block-localized excitation. , Together, these variationally optimized determinants states form a minimum active space denoted by a single subscript {Ξ A ; A = 1, ...}.…”

“…The adiabatic ground and intermediate states in the present MS-EDA analysis are written as linear combinations of determinant basis states {Ξ A } that form a minimal active space (MAS), consisting of the Kohn–Sham reference and singly excited configurations, where Ξ ( X • Y ) KS is the reference BLKS determinant for the block-localized complex (eq ), Ξ i a is a singly excited configuration, and { c } are configuration coefficients determined by NOSI by diagonalizing the Hamiltonian matrix functional . Although orbitals and { c } can be optimized simultaneously, in NOSI, the determinant states Ξ ( X • Y ) KS and in eq are variationally optimized first using the block-localized excitation (BLE) method. ,, In particular, each constrained BLKS-determinant wave function is written where χ j U ( j =1, ⋯, N U ) is the j th block-localized spinorbital that is expanded over the basis functions located on atoms of monomer U ( U = X , Y ), N U is the number of electrons, and χ a X indicates that the i th occupied spinorbital of monomer X in the BLKS reference Ξ ( X • Y ) KS is replaced by the a th virtual orbital. In MSDFT, we include all spin-complement configurations in the MAS, all of which are together labeled by a single index {Ξ A }.…”

“…Although orbitals and {c} can be optimized simultaneously, in NOSI, 63 the determinant states Ξ (X•Y) KS and in eq 21 are variationally optimized first using the block-localized excitation (BLE) method. 91,92,98 In particular, each constrained BLKSdeterminant wave function is written…”

“…The adiabatic ground and intermediate states in the present MS-EDA analysis are written as linear combinations of determinant basis states {Ξ A } that form a minimal active space (MAS), consisting of the Kohn–Sham reference and singly excited configurations, Ψ false( X • Y false) L = c 0 Ξ false( X • Y false) KS + ∑ i , a c i a Ξ i a where Ξ ( X • Y ) KS is the reference BLKS determinant for the block-localized complex (eq ), Ξ i a is a singly excited configuration, and { c } are configuration coefficients determined by NOSI by diagonalizing the Hamiltonian matrix functional H false[ bold-italicD false] . Although orbitals and { c } can be optimized simultaneously, in NOSI, the determinant states Ξ ( X • Y ) KS and in eq are variationally optimized first using the block-localized excitation (BLE) method. ,, In particular, each constrained BLKS-determinant wave function is written Ξ i a = Â false{ ( χ 1 X ··· χ i − 1 X χ a X χ i + 1 X ··· χ …”

Calculations of Excited Electronic States by Converging on Saddle Points Using Generalized Mode Following

“…Variational density functional calculations of excited states ,− are emerging as an attractive alternative to TDDFT. They typically involve similar computational cost as ground state calculations and can better describe long-range charge transfer, ,,, Rydberg, , core-level, ,, and other excitations ,, where a significant change in the electron density occurs. As the calculations are variational, they provide atomic forces that can be used in excited state geometry optimizations and classical dynamics simulations. − In a variational calculation within a mean-field approximation, an excited state is found as an optimal single Slater determinant with nonaufbau orbital occupation.…”