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arXiv:1008.1478 · Topics in current chemistry

Many-Body Perturbation Theory (MBPT) and Time-Dependent Density-Functional Theory (TD-DFT): MBPT Insights About What Is Missing In, and Corrections To, the TD-DFT Adiabatic Approximation

Mark E. Casida, Miquel Huix‐Rotllant

Published Jan 1, 2015• 32 citations
arXivHTMLPDF
DOI
10.1007/128_2015_632

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In many cases, density-functional theory (DFT) with current standard approximate functionals offers a relatively accurate and computationally cheap description of the short-range dynamic electron correlation effects. However, in general, standard DFT does not treat the dispersion interaction effects adequately which, on the other hand, can be described by many-body perturbation theory (MBPT). It is therefore of interest to develop a hybrid model which combines the best of both the MBPT and DFT approaches. This can be achieved by splitting the two-electron interaction into long-range and short-range parts; the long-range part is then treated by MBPT and the short-range part by DFT. This work deals with the formulation of a general MBPT-DFT model (i.e., valid for any type of zeroth-order Hamiltonian) based on such a range separation. Applying the Rayleigh-Schr\"odinger formalism in this context, one finds that the generalized Bloch equation becomes self-consistent at each order of perturbation. This complication arises because the short-range part of the energy is a functional of the exact electron density, which is expanded in a perturbation series. In order to address this ``self-consistency problem'' and provide computable orbital-based expressions for any order of perturbation, a general one-electron reduced-density-matrix-based formalism is proposed. Two applications of our general formalism are presented: The derivation of a hybrid second-order M\o{}ller-Plesset-DFT model and the formulation of a range-separated optimized effective potential method based on a hybrid G\"orling-Levy-type MBPT-DFT model.
For the calculation of neutral excitations, time-dependent density functional theory (TDDFT) is an exact reformulation of the many-body time-dependent Schrödinger equation, based on knowledge of the density instead of the many-body wavefunction. The density can be determined in an efficient scheme by solving one-particle non-interacting Schrödinger equations—the Kohn–Sham equations. The complication of the problem is hidden in the—unknown—time-dependent exchange and correlation potential that appears in the Kohn–Sham equations and for which it is essential to find good approximations. Many approximations have been suggested and tested for finite systems, where even the very simple adiabatic local-density approximation (ALDA) has often proved to be successful. In the case of solids, ALDA fails to reproduce optical absorption spectra, which are instead well described by solving the Bethe–Salpeter equation of many-body perturbation theory (MBPT). On the other hand, ALDA can lead to excellent results for loss functions (at vanishing and finite momentum transfer). In view of this and thanks to recent successful developments of improved linear-response kernels derived from MBPT, TDDFT is today considered a promising alternative to MBPT for the calculation of electronic spectra, even for solids. After reviewing the fundamentals of TDDFT within linear response, we discuss different approaches and a variety of applications to extended systems.
We present a variational formulation of the random phase approximation (RPA) that places density functional theory (DFT), linear-response time-dependent density functional theory (LR-TDDFT), one-body reduced density matrix functional theory (1RDMFT), and Green's function many-body perturbation theory (MBPT) into a common source-variable hierarchy. The central claim is that a broad class of RPA constructions can be organized, independently of any one problem-specific formula, diagrammatic resummation, or small-amplitude equation of motion, as closure approximations to the exact Hessian of an effective functional. In this language, exact linear response is governed by the Hessian of the corresponding effective functional, while RPA is obtained by retaining a reference contribution together with an explicit interaction kernel and discarding the irreducible remainder. The hierarchy has two independent enrichments of the density-level description. One may enlarge the static local density to a time-dependent density, giving the dynamical density channel of LR-TDDFT, or enlarge it to an equal-time bilocal one-body reduced density matrix, giving the static bilocal channel of 1RDMFT. The Green's function level combines both enrichments, since the one-particle Green's function is bilocal in both space and time. This picture clarifies the relation between DFT, LR-TDDFT, 1RDMFT, and MBPT through exact forward reductions and source restrictions, while emphasizing that the corresponding RPA closures need not commute under projection. The hierarchy also distinguishes the local branch-wise Legendre geometry common to all four levels from the stronger global convex duality that may emerge only in source sectors possessing additional positivity and global regularity.