Models and corrections: Range separation for electronic interaction—Lessons from density functional theory

Abstract

a) Model Hamiltonians with long-range interaction yield energies that are corrected taking into account the universal behavior of the electron-electron interaction at short range. Although the intention of the paper is to explore the foundations of using density functionals combined with range separation, the approximations presented can be used without them, as illustrated by a calculation on Harmonium. In the regime when the model system approaches the Coulomb system, they allow the calculation of ground states, excited states, and properties, without making use of the Hohenberg-Kohn theorem. Asymptotically, the technique is improvable, allows for error estimates that can validate the results. Some considerations for correcting the errors of finite basis sets in this spirit are also presented. Being related to the present understanding of density functional approximations, the results are comparable to those obtained with the latter, as long as these are accurate.

Range‐separated multiconfigurational density functional theory methods

“…Steps in these directions have recently been taken 87 . In a broader perspective, the known small‐ and large‐ μ asymptotics of the exact eigenvalues and eigenfunctions of the range‐separated Hamiltonian may lead to the emergence of systematically improvable RS approximations which abandon density functionals 182 …”

Correcting Models with Long-Range Electron Interaction Using Generalized Cusp Conditions

“…One can replace the expansion in powers of r , eq , by a simple form that does not diverge as r → ∞. For example, one can ignore all terms arising from the external potential and the energy in the Schrödinger equation, in the limit μ → ∞, and we get, to order μ –4 (see ref 11 and eq 21 in ref ), ∫ 0 1 nobreak0em0.25em⁣ normald λ scriptI ( λ , μ ) scriptI ( λ = 0 , μ ) = μ 2 + 1.06385 μ + 0.31982 μ 2 + 1.37544 μ + 0.487806 A comparison of Figure a, f, or g to h shows that this approximation performs very well compared to those obtained from the GCC: the “walls” are at comparable values of μ. This is encouraging because the GCC require, in general, taking into account the external potential and the energy.…”

“…One way to achieve our aim is to introduce more parameters into the Hamiltonian, e.g., H ( R ; λ , μ ) = H ( R ; μ ) + λ [ H false( boldR false) − H false( boldR ; μ false) ] The eigenvalue and the corresponding eigenfunction of H ( R ; λ, μ) are, respectively, E (λ, μ) and Ψ( R , σ ;λ, μ); notice that E (μ) = E (λ, μ) | λ=0 . In the Hamiltonian (eq ), the interaction potential is v int ( r ; λ , μ ) = w ( r ; μ ) + λ w̅ ( r ; μ ) where w̅ ( r ; μ ) = 1 r − w ( r ; μ ) = 1 − erf ( μ r ) r = erfc ( μ r ) r Therefore, v int ( r ; λ , μ ) = ( 1 − λ …”

Projected Hybrid Density Functionals: Method and Application to Core Electron Ionization

“…The generalized range-separated adiabatic connection provides one way to optimize hybrid DFT’s underlying trade-offs. ,− This approach separates the electron–electron interaction operator V̂ e e into short-range and long-range pieces, for example, .25ex2ex V̂ e e = ∑ i > j 1 r i j = ∑ i > j erf ( μ r i j ) r i j + erfc false( μ r i j false) r i j infix= V̂ e e L R + V̂ e e S R Part of the interaction, typically the long-range part, is reintroduced into the noninteracting reference system. The Hohenberg–Kohn theorems ensure that the real system’s ground-state energy and density can be obtained from an exact wave function calculation on the long-range-interacting reference system, corrected by a density functional for the remaining short-range Hartree-exchange-correlation energy .…”