Beyond Kohn–Sham Approximation: Hybrid Multistate Wave Function and Density Functional Theory

Abstract

A multistate density functional theory (MSDFT) is presented in which the energies and densities for the ground and excited states are treated on the same footing using multiconfigurational approaches. The method can be applied to systems with strong correlation and to correctly describe the dimensionality of the conical intersections between strongly coupled dissociative potential energy surfaces. A dynamic-then-static framework for treating electron correlation is developed to first incorporate dynamic correlation into contracted state functions through block-localized Kohn–Sham density functional theory (KSDFT), followed by diagonalization of the effective Hamiltonian to include static correlation. MSDFT can be regarded as a hybrid of wave function and density functional theory. The method is built on and makes use of the current approximate density functional developed in KSDFT, yet it retains its computational efficiency to treat strongly correlated systems that are problematic for KSDFT but too large for accurate WFT. The results presented in this work show that MSDFT can be applied to photochemical processes involving conical intersections.

Diabatic-At-Construction Method for Diabatic and Adiabatic Ground and Excited States Based on Multistate Density Functional Theory

“…18,50 In particular, the diagonal matrix elements of the effective Hamiltonian are simply the KSDFT energies for the corresponding states {|Ψ A ⟩; A = 1,⋯, N p }. 50 In the present case of LiH, N p is 7, as constructed from 12 BLKS determinants.…”

“…However, a transition density functional (TDF) 18 E TDF [ ρ AB ( r )] between states |Ψ A ⟩ and |Ψ B ⟩ does not exist within the KS-DFT framework. E TDF [ ρ AB ( r )] might be derived by multiconfigurational approaches 18 and by analogy to methods 79 used for electron scattering. Here though, we approximate it in MSDFT by two contributions $$H_{\mathit{\text{AB}}}\equiv E^{\text{TDF}}false[{\rho }_{\mathit{\text{AB}}}false(\mathrm{boldr}false)false]=false⟨{\mathrm{\Psi }}_{A}false|Hfalse|{\mathrm{\Psi }}_{B}false⟩+V_c^{\text{TDF}}false[{\rho }_{\mathit{\text{AB}}}false(\mathrm{boldr}false)false]$$ where H is the electronic Hamiltonian.…”

“…3,56 The second term in eq 4 is a transition density contribution. 3,18,50,51,56,67 Note that both exchange and static correlation are included in the first term, whereas only dynamic correlation is in the second term. 18 We then approximated the second term of eq 4 by an overlap-weighted average of the KS correlation energies for the two interacting states 50,51 $$V_c^{\text{TDF}}false[{\rho }_{\mathit{\text{AB}}}false(\mathrm{boldr}false)false]\approx \frac{\lambda }{2}{S}_{\mathit{\text{AB}}}false(E_c^{\text{KS}}false[{\rho }_A^{\text{ms}}false(\mathrm{boldr}false)false]+E_c^{\text{KS}}false[{\rho }_B^{\text{ms}}false(\mathrm{boldr}false)false]false)$$ where λ is unity in the present work (in later work this parameter or functional could be optimized to fit experimental results), where S AB is the overlap integral ⟨Ψ A |Ψ B ⟩, and where $E_c^{\text{KS}}$ specifies the correlation energy in the corresponding diagonal matrix element (as obtained from the exchange-correlation functional).…”

“…13,14 (This shortcoming can be overcome by modifying the theory, 15–17 but we shall not pursue that line of approach in the present article.) The present article considers a combination of DFT and WFT that goes beyond the current Kohn−Sham approximation 18 and has advantages compared to using either alone. This article has two objectives: (1) to describe a multistate density functional theory (MSDFT) for constructing quasidiabatic states and (2) to establish a relationship between the diabatic states obtained from MSDFT and those by the 3- fold-way diabatization approach.…”

“…3,56 The use of BLMO or block-localized Kohn−Sham (BLKS) orbitals reduces a large number of valence bond configurations to a manageable number; only those critical to a given system are retained in a way analogous to the spin-coupled valence bond theory. 64,65 This method has been extended to density functional theory, in which case it is called multistate density functional theory (MSDFT), 18,50,51,66,67 and this is the method that is used in the present work.…”