Exact Factorization-Based Density Functional Theory of Electrons and Nuclei

Abstract

The ground state energy of a system of electrons and nuclei is proven to be a variational functional of the conditional electronic density nR(r), the nuclear wavefunction χ(R) and an induced vector potential Aµ(R) and quantum geometric tensor Tµν (R) derived from the conditional electronic wavefunction ΦR(r) over nuclear configuration space, where r = r1, r2, . . . are electronic coordinates and R = R1, R2, . . . are nuclear coordinates. The ground state (nR, χ, Aµ, Tµν ) can be calculated by solving self-consistently (i) conditional Kohn-Sham equations containing an effective potential vs(r) that depends parametrically on R, (ii) the Schrödinger equation for χ(R) and (iii) Euler-Lagrange equations that determine Tµν . The theory is applied to the E ⊗ e Jahn-Teller model. The foundations of density functional theory (DFT) [1,2] are inextricably tied to the Born-Oppenheimer approximation. In DFT applications, e.g. electronic band structure calculations, it often suffices to treat the nuclei classically or to fix them to their equilibrium positions. Quantum nuclear effects such as tunneling, delocalization and zero-point energy are, however, relevant for several interesting problems, e.g. the phases of ice [3][4][5][6] and the local structure of water [7][8][9][10], and were recently reported to enable thermally-activated tunneling of protons through a graphene layer [11,12]. Some quantum nuclear effects can be included in DFT-based calculations by quantizing nuclear vibrations on the adiabatic ground state potential energy surface, but because such an approach relies on the Born-Oppenheimer approximation, it is not formally exact. When the nuclear variables and electron-nuclear coupling are treated exactly and fully quantum mechanically, the electrons feel, instead of the external potential v(r) of DFT, a "weighted" potential − i |χ(R)| 2 Z i e 2 /|r−R i |dR, modified by the delocalization of the nuclear probability density |χ(R)| 2 , but also additional interactions induced by nonadiabatic electron-nuclear correlations [13,14] not included in standard DFT functionals.Particularly in time-dependent processes such as photoinduced chemical bond dynamics [15], proton transfer in hydrogen-bonded systems [16], dissociative adsorption of H 2 on Pd(100) [17], and molecular processes involving conical intersections of Born-Oppenheimer (BO) potential energy surfaces [18], nonadiabatic and quantum nuclear effects may be significant. Mixed quantum-classical approaches, which couple quantum mechanical electrons to classical nuclear motion, usually adopt an effective single-particle description of the electrons, and DFT is often the only method capable of treating large systems of interest with sufficient accuracy.For the further development of theories capable of describing quantum nuclear effects in large systems, it would be useful to know whether it is in principle possible to include full quantum nuclear motion and electronicvibrational coupling while retaining a density-functional formulation of the electronic part of t...

Asymptotic analysis of the Berry curvature in theE⊗eJahn-Teller model

“…Although we have found numerically that the Berry curvature approaches a universal function in the limit M → 0, we were not able to find its analytical form in terms of special functions. Nevertheless, we have proposed a compact formula that we hope will prove helpful in designing functional approximations in a nonadiabatic generalization of density functional theory, in which the exchange-correlation energy depends on the Berry curvature [53].…”

“…The motivation for a detailed asymptotic analysis of the Berry curvature comes from a recent nonadiabatic generalization of density functional theory [53], where the exchange-correlation energy is a functional of the Berry curvature in addition to the density. Unlike standard density functional theory [54,55], which depends on the BO approximation, nonadiabatic density functional theory is an exact theory of electrons and nuclei.…”

Model Hamiltonian for strongly correlated systems: Systematic, self-consistent, and unique construction

“…Adapting the present theory to finite systems in which the strongly correlated subspace is taken to be spanned by just a few natural orbitals [180], such as localized orbitals in Kondo systems or hybridized transition metal orbitals in molecules, would constitute a novel embedding theory that might allow one to obtain more accurate ab initio results for systems with strong static correlation and partially circumvent the problem of memory dependence in TDDFT. This would be especially helpful in the modeling of coupled electron-ion dynamics within exact factorization density functional theory [181,182]. The role of the on-site d-orbital density matrix and orbital-dependent potentials in the DFT+U method [16,68,[183][184][185][186][187], as well as the uncertainties arising from the strongly correlated narrow Fe bands in semilocal DFT calculations of the pressure-induced spin state crossover in Fe x Mg 1−x SiO 3 perovskite [31], motivated the investigation of an effective single-particle Hamiltonian for a strongly correlated Hubbard model in a reduced density matrix approach [188].…”

Exact factorization-based density functional theory of electron-phonon systems

“…(13), modifies the electronic and nuclear kinetic energy operators, 109 the exact factorization scheme can still be straightforwardly applied to the resulting Schrödinger equation (see the supplemental material of Ref. 96). To keep our focus on the essential differences between the present theory and standard DFT calculations of electron-phonon systems, we neglect these modifications and, moreover, we restrict our attention to nonpolar solids.…”