We introduce a novel energy functional for ground-state electronic-structure calculations. Its fundamental variables are the natural spin-orbitals of the implied singlet many-body wave function and their joint occupation probabilities. The functional derives from a sequence of controlled approximations to the two-particle density matrix. Algebraic scaling of computational cost with electron number is obtainable in general, and Hartree-Fock scaling in the seniority-zero version of the theory. Results obtained with the latter version for saturated small molecular systems are compared with those of highly-accurate quantum-chemical computations. The numerical results are variational, capturing most of the correlation energy from equilibrium to dissociation. Their accuracy is considerably greater than that obtainable with current density-functional theory approximations and with current functionals of the one-particle density matrix only.
We introduce a novel energy functional for ground-state electronic-structure calculations. Its fundamental variables are the natural spin-orbitals of the implied singlet many-body wave function and their joint occupation probabilities. The functional derives from a sequence of controlled approximations to the two-particle density matrix. Algebraic scaling of computational cost with electron number is obtainable in general, and Hartree-Fock scaling in the seniority-zero version of the theory. Results obtained with the latter version for saturated small molecular systems are compared with those of highly-accurate quantum-chemical computations. The numerical results are variational, capturing most of the correlation energy from equilibrium to dissociation. Their accuracy is considerably greater than that obtainable with current density-functional theory approximations and with current functionals of the one-particle density matrix only.
We introduce an energy functional for ground-state electronic structure calculations. Its variables are the natural spin-orbitals of singlet many-body wave functions and their joint occupation probabilities deriving from controlled approximations to the two-particle density matrix that yield algebraic scaling in general, and Hartree-Fock scaling in its seniority-zero version. Results from the latter version for small molecular systems are compared with those of highly accurate quantum-chemical computations. The energies lie above full configuration interaction calculations, close to doubly occupied configuration interaction calculations. Their accuracy is considerably greater than that obtained from current density-functional theory approximations and from current functionals of the one-particle density matrix.
Density-Functional Theory (DFT), although widely used and very successful in the calculation of several observables, fails to correctly describe strongly correlated materials. In the first part of this work we, therefore, introduce reduced-density-matrix-functional theory (RDMFT) which is one possible way to treat electron correlation beyond DFT. Within this theory the one-body reduced density matrix (1-RDM) is used as the basic variable. Our main interest is the calculation of the fundamental gap which proves very problematic within DFT. In order to calculate the fundamental gap we generalize RDMFT to fractional particle numbers M by describing the system as an ensemble of an N and an N+1 particle system (with N≤ M≤N+1). For each fixed particle number, M, the total energy is minimized with respect to the natural orbitals and their occupation numbers. This leads to the total energy as a function of M. The derivative of this function with respect to the particle number has a discontinuity at integer particle number which is identical to the gap. In addition, we investigate the necessary and sufficient conditions for the 1-RDM of a system with fractional particle number to be N-representable. Numerical results are presented for alkali atoms, small molecules, and periodic systems. Another problem within DFT is the description of non-relativistic many- electron systems in the presence of magnetic fields. It requires the paramagnetic current density and the spin magnetization to be used as basic variables besides the electron density. However, electron-gas-based functionals of current-spin-density-functional Theory (CSDFT) exhibit derivative discontinuities as a function of the magnetic field whenever a new Landau level is occupied, which makes them difficult to use in practice. Since the appearance of Landau levels is, intrinsically, an orbital effect it is appealing to use orbital-dependent functionals. We have developed a CSDFT version of the optimized effective potential (OEP) method which allows for the use of explicitly orbital-dependent functionals in the context of CSDFT. We present the derivation of corresponding equations and show results for a quantum dot in external magnetic fields in the second part of this thesis.
Currently, there is a growing interest in the development of a new hierarchy of methods based on the concept of seniority, which has been introduced quite recently in quantum chemistry. Despite the enormous potential of these methods, the accurate description of both dynamical and static correlation effects within a single and in-principle-exact approach remains a challenge. In this work, we propose an alternative formulation of reduced density-matrix functional theory (RDMFT) where the (one-electron reduced) density matrix is mapped onto an ab initio seniority-zero wave function. In this theory, the exact natural orbitals and their occupancies are determined self-consistently from an effective seniority-zero calculation. The latter involves a universal higher seniority density matrix functional for which an adiabatic connection (AC) formula is derived and implemented under specific constraints that are related to the density matrix. The pronounced curvature of the (constrained) AC integrand, which is numerically observed in prototypical hydrogen chains and the helium dimer, indicates that a description of higher seniority correlations within second-order perturbation theory is inadequate in this context. Applying multiple linear interpolations along the AC or connecting second-order perturbation theory to a full-seniority treatment via Pad\'e approximants are better strategies. Such information is expected to serve as a guide in the future design of higher seniority density-matrix functional approximations.
Density functional theory is currently the most widely applied method in electronic structure theory. The Kohn-Sham method, based on a fictitious system of non-interacting particles, is the work horse of the theory. The particular form of the Kohn-Sham wavefunction admits only idem-potent one electron density matrices whereas wavefunctions of correlated electrons in post-Hartree-Fock methods invariably have fractional occupation numbers. Here we show that by generalizing the orbital concept, and introducing a suitable dot-product as well as a probability density a non-interacting system can be chosen that can represent the one-electron density matrix of any system, even one with fractional occupation numbers. This fictitious system ensures that the exact electron density is accessible within density functional theory. It can also serve as the basis for reduced density matrix functional theory. Moreover, to aid the analysis of the results the orbitals may be assigned energies from a mean-field Hamiltonian. This produces energy levels that are akin to Hartree-Fock orbital energies such that conventional analyses based on Koopmans theorem are available. Finally, this system is convenient in formalisms that depend on creation and annihilation operators as they are trivially applied to single determinant wavefunctions.
Strong electron correlation remains difficult to describe within conventional density functional approximations, particularly when natural-orbital occupations become strongly fractional. We introduce a systematic particle-hole expansion that generalizes density functional theory (DFT) through an explicit functional of the one-electron reduced density matrix. The previously derived cumulant-based correction emerges as the leading term and the thermal (entropic) correction as an infinite-order member whose coefficients are fixed by a single parameter. The expansion can be formulated as a moment problem solved by semidefinite programming with computational scaling comparable to DFT. We show that both the first- and second-order corrections yield closed-form natural-orbital occupations. Applications to periacenes and Fe-porphyrin demonstrate that the second-order correction substantially reduces excessive orbital fractionalization, yielding natural-orbital occupations and fundamental gaps in improved agreement with correlated calculations.
Previously unknown properties of the natural orbitals (NOs) pertaining to singlet states (with natural parity, if present) of electronic systems with even numbers of electrons are revealed upon the demonstration that, at the limit of n → ∞, the NO ψ n ( r⃗ ) with the n th largest occupation number ν n approaches the solution ψ̃ n ( r⃗ ) of the zero-energy Schrödinger equation that reads T̂ ([ ρ 2 ( r⃗, r⃗ )] −1/8 ψ̃ n (r⃗)) − (π 2 / ṽ n ) 1/4 [ρ 2 ( r⃗, r⃗ )] 1/4 ([ρ 2 ( r⃗, r⃗ )] −1/8 ψ̃ n (r⃗)) = 0 (where T̂ is the kinetic energy operator), whereas ν n approaches ν̃ n . The resulting formalism, in which the “on-top” two-electron density ρ 2 ( r⃗, r⃗ ) solely controls the asymptotic behavior of both ψ n ( r⃗ ) and ν n at the limit of the latter becoming infinitesimally small, produces surprisingly accurate values of both quantities even for small n . It opens entirely new vistas in the elucidation of their properties, including single-line derivations of the power laws governing the asymptotic decays of ν n and ⟨ψ n ( r⃗ )| T̂ |ψ n ( r⃗ )⟩ with n, some of which have been obtained previously with tedious algebra and arcane mathematical arguments. These laws imply a very unfavorable asymptotics of the truncation error in the total energy computed with finite numbers of natural orbitals that severely affects the accuracy of certain quantum-chemical approaches such as the density matrix functional theory. The new formalism is also shown to provide a complete and accurate elucidation of both the observed order (according to decreasing magnitudes of the respective occupation numbers) and the shapes of the natural orbitals pertaining to the 1 Σ g + ground state of the H 2 molecule. In light of these examples of its versatility, the above Schrödinger equation is expected to have its predictive and interpretive powers harnessed in many facets of the electronic structure theory.