Double Excitations from Ensemble Density Functionals: Theory and Approximations

Double excitations, which are dominated by a Slater-determinant with both electrons in the highest occupied molecular orbital promoted to the lowest unoccupied orbital(s), pose significant challenges for low-cost electronic structure calculations based on density functional theory (DFT). Here, we demonstrate that recent advances in ensemble DFT [<i>Phys. Rev. Lett.</i> <b>125</b>, 233001 (2020)], which extend concepts of ground-state DFT to excited states via a rigorous physical framework based on the ensemble fluctuation-dissipation theorem, can be used to shed light on the double excitation problem. We find that the exchange physics of double excitations is reproducible by standard DFT approximations using a linear combination formula, but correlations are more complex. We then show, using selected test systems, that standard DFT approximations may be adapted to tackle double excitations based on theoretically motivated simple formulae that employ ensemble extensions of expressions that use the on-top pair density.<br><br>

Double Excitations from Ensemble Density Functionals: Theory and Approximations

Double excitations, which are dominated by a Slater-determinant with both electrons in the highest occupied molecular orbital promoted to the lowest unoccupied orbital(s), pose significant challenges for low-cost electronic structure calculations based on density functional theory (DFT). Here, we demonstrate that recent advances in ensemble DFT [<i>Phys. Rev. Lett.</i> <b>125</b>, 233001 (2020)], which extend concepts of ground-state DFT to excited states via a rigorous physical framework based on the ensemble fluctuation-dissipation theorem, can be used to shed light on the double excitation problem. We find that the exchange physics of double excitations is reproducible by standard DFT approximations using a linear combination formula, but correlations are more complex. We then show, using selected test systems, that standard DFT approximations may be adapted to tackle double excitations based on theoretically motivated simple formulae that employ ensemble extensions of expressions that use the on-top pair density.<br><br>

Reduction of the molecular hamiltonian matrix using quantum community detection

Quantum chemistry is interested in calculating ground and excited states of molecular systems by solving the electronic Schrödinger equation. The exact numerical solution of this equation, frequently represented as an eigenvalue problem, remains unfeasible for most molecules and requires approximate methods. In this paper we introduce the use of Quantum Community Detection performed using the D-Wave quantum annealer to reduce the molecular Hamiltonian matrix in Slater determinant basis without chemical knowledge. Given a molecule represented by a matrix of Slater determinants, the connectivity between Slater determinants (as off-diagonal elements) is viewed as a graph adjacency matrix for determining multiple communities based on modularity maximization. A gauge metric based on perturbation theory is used to determine the lowest energy cluster. This cluster or sub-matrix of Slater determinants is used to calculate approximate ground state and excited state energies within chemical accuracy. The details of this method are described along with demonstrating its performance across multiple molecules of interest and bond dissociation cases. These examples provide proof-of-principle results for approximate solution of the electronic structure problem using quantum computing. This approach is general and shows potential to reduce the computational complexity of post-Hartree–Fock methods as future advances in quantum hardware become available.

Many-electron Systems and the Pauli Principle

It is shown how the quantum Hamiltonian for a general molecule is set up, using the ‘quantum recipe’ of chapter 3. In the most restrictive Born Oppenheimer approximation, the nuclei are held fixed and the Schrodinger equation (SE) is set up for the electrons only. The wavefunction depends on the positions and spin projections of all electrons. The electron spin projection is introduced heuristically as another two-valued electron degree of freedom. The electronic SE cannot be solved exactly, and (spin-) orbitals are introduced to construct an approximate wavefunction. The Pauli principle demands that a many-electron wavefunction is antisymmetric upon the exchange of electron labels, which leads to the construction of the approximate orbital-model wavefunction as a Slater determinant rather than a simple Hartree product. The orbital model wavefunction does not describe the Coulomb electron correlation, but it incorporates the (Fermi) correlation leading to the Pauli exclusion.

Self-consistent Field Orbital Methods

This chapter discusses the concepts underlying the Hartree-Fock (HF) electronic structure method. First, it is shown how the energy expectation value is calculated for a Slater determinant (SD) wavefunction in the case of orthonormal orbitals. This leads to the definition of the electron repulsion integrals (ERIs). Next, the energy is minimized subject to the orthonormality constraints. This leads to the HF equation for the orbitals. The HF orbital energies are Langrange multipliers representing the constraints. An unknown set of orbitals can be determined from an initial guess via a self-consistent field (SCF) cycle. The HF scheme is discussed for closed-shell versus open shell systems, leading to the distinction between spin restricted and unrestricted HF (RHF, UHF). Kohn-Sham density functional theory (DFT) is introduced and its approximate version is placed in the context of ab-initio versus semi-empirical quantum chemistry methods.

The nucleon momentum distribution in light nuclei

The nucléon momentum distribution in light nuclei is studied by means of a single particle potential model which consists of an attractive harmonic oscillator potential Va=½mω²r² and also of a repulsive one of the form Vr=B/r², Β > 0. The latter simulates to some extend effects which would result if short range correlations were included (e.g. by a Jastrow factor) in a nuclear wave function, having as uncorrelated part a Slater determinant of harmonic oscillator orbitals. The main advantage of this model is that it leads to fairly simple analytic expressions for the momentum distribution of light nuclei and also for the density distributions and the elastic form factors. These expressions are quite useful in obtaining, for example, the asymptotic form of η(k) for large k from which it is seen that the steep decrease of the nucléon momentum distribution observed with the harmonic oscillator model in this region is improved. Numerical results using various least squares fittings are obtained and discussed for a number of nuclei of the 1s, 1p shell.