Comparison of self-consistent field convergence acceleration techniques

The recently proposed ADIIS and LIST methods for accelerating self-consistent field (SCF) convergence are compared to the previously proposed energy-DIIS (EDIIS) + DIIS technique. We here show mathematically that the ADIIS functional is identical to EDIIS for Hartree-Fock wavefunctions. Convergence failures of EDIIS + DIIS reported in the literature are not reproduced with our codes. We also show that when correctly implemented, the EDIIS + DIIS method is generally better than the LIST methods, at least for the cases previously examined in the literature. We conclude that, among the family of DIIS methods, EDIIS + DIIS remains the method of choice for SCF convergence acceleration

Convergence of gradient-based algorithms for the Hartree-Fock equations

The numerical solution of the Hartree-Fock equations is a central problem in quantum chemistry for which numerous algorithms exist. Attempts to justify these algorithms mathematically have been made, notably in by Cances and Le Bris in 2000, but, to our knowledge, no complete convergence proof has been published. In this paper, we prove the convergence of a natural gradient algorithm, using a gradient inequality for analytic functionals due to Lojasiewicz. Then, expanding upon the analysis of Cances and Le Bris, we prove convergence results for the Roothaan and Level-Shifting algorithms. In each case, our method of proof provides estimates on the convergence rate. We compare these with numerical results for the algorithms studied

Optimal damping algorithm for unrestricted Hartree-Fock calculations

We have developed a couple of optimal damping algorithms (ODAs) for unrestricted Hartree-Fock (UHF) calculations of open-shell molecular systems. A series of equations were derived for both concurrent and alternate constructions of alpha- and beta-Fock matrices in the integral-direct self-consistent-field (SCF) procedure. Several test calculations were performed to check the convergence behaviors. It was shown that the concurrent algorithm provides better performance than does the alternate one.Comment: 4 color figure

Shanks and Anderson-type acceleration techniques for systems of nonlinear equations

This paper examines a number of extrapolation and acceleration methods, and introduces a few modifications of the standard Shanks transformation that deal with general sequences. One of the goals of the paper is to lay out a general framework that encompasses most of the known acceleration strategies. The paper also considers the Anderson Acceleration method under a new light and exploits a connection with quasi-Newton methods, in order to establish local linear convergence results of a stabilized version of Anderson Acceleration method. The methods are tested on a number of problems, including a few that arise from nonlinear Partial Differential Equations

Secant acceleration of sequential residual methods for solving large-scale nonlinear systems of equations

Sequential Residual Methods try to solve nonlinear systems of equations $F(x)=0$ by iteratively updating the current approximate solution along a residual-related direction. Therefore, memory requirements are minimal and, consequently, these methods are attractive for solving large-scale nonlinear systems. However, the convergence of these algorithms may be slow in critical cases; therefore, acceleration procedures are welcome. In this paper, we suggest to employ a variation of the Sequential Secant Method in order to accelerate Sequential Residual Methods. The performance of the resulting algorithm is illustrated by applying it to the solution of very large problems coming from the discretization of partial differential equations

Mathematical and Numerical Aspects of Quantum Chemistry Problems

This workshop was aimed at strengthtening the interactions between well established experts in quantum chemistry, mathematical analysis, numerical analysis and computational metodology. Most of the mathematicians present in the worskhop have already contributed to the theoretical and numerical study of models in quantum physics and chemistry. Some others, familiar with contiguous fiels, were new to chemistry. Several distinguished researchers in theoretical chemistry participated in the workshop, and presented the mathematical and computational challenges of the field

Development of Tools for the Study of Heavy-Element Containing Periodic Systems in the CRYSTAL Code and their Application

This thesis investigates the development of first-principles methods for the study of heavy-element containing periodic systems, as well as their application, in particular to crystalline lanthanide oxides. The Generalized Kohn-Sham Density Functional Theory (GKS-DFT, i.e. in which density functional approximations are built directly from KS orbitals, using so-called hybrid functionals) was shown to provide a particularly effective means to correct for self-interaction errors that plague more conventional local or semi-local formulations in a scalar-relativistic (SR) context. As such, the SR GKS-DFT scheme allowed for a detailed characterization of the electronic structure of the lanthanide sesquioxide series, and enabled (for the first time) to rationalize all known electronic and structural pressure-induced phase transitions in the prototypical strongly-correlated and mixed-valence material EuO. But the hybrid functional approach proved even more useful when developing instead fully relativistic theories and algorithms, which include not only SR effect, but also spin-dependent relativistic effects, such as spin-orbit coupling (SOC). Coincidentally, this thesis reports the first implementation for a self-consistent treatment of SOC in periodic systems with a fraction of exact non-local Fock exchange in a two- component spinor basis (2c-SCF). The numerous advantages of using such a formulation, as opposed to the more approximate treatments of previously existing implementations, are discussed. These advantages originate from the ability of the Fock exchange operator to locally rotate the magnetization of the system with respect to a starting guess configuration (local magnetic torque). In addition, the non-local Fock exchange operator permits to include in the two-electron potential the contribution of the spinors that are mapped to certain spin-blocks of the single-particle density matrix. This allows for a proper treatment of the orbital relaxation of current densities, and their coupling with the other density variables. As a result, it is shown that the lack of Fock exchange (or even its more approximate treatment in a one-component basis, as with previous implementations) from more conventional formulations of the KS-DFT means that the calculation would not allow to access the full range of time-reversal symmetry broken states. This is because, it is shown that in the absence of Fock exchange, the band structure is constrained by a sum rule, linking the one-electron energy levels at opposite points in the first Brillouin zone (kj and −kj)