Direct Unconstrained Variable-Metric Localization of One-Electron Orbitals

Direct Unconstrained Optimization of Molecular Orbital Coefficients in Density Functional Theory

One-electron orbitals in Kohn-Sham density functional theory (DFT) are typically constrained to be orthogonal during their variational optimization, leading to elaborate parameterization of the orbitals and complicated optimization algorithms. This work shows that orbital optimization can be performed with nonorthogonal orbitals if the DFT energy functional is augmented with a term that penalizes linearly dependent states. This approach, called variable-metric self-consistent field (VM SCF) optimization, allows us to use molecular orbital coefficients, natural descriptors of one-electron orbitals, as independent variables in a direct, unconstrained minimization, leading to very simple closed-form expressions for the electronic gradient and Hessian. It is demonstrated that efficient convergence of the VM SCF procedure can be achieved with a basic preconditioned conjugate gradient algorithm for a variety of systems, including challenging narrow-gap systems and spin-pure two-determinant states of singlet diradicals. This simple reformulation of the variational procedure can be readily extended to electron correlation methods with multiconfiguration states and to the optimization of excited-state orbitals.

Variable-Metric Localization of Occupied and Virtual Orbitals

The key idea of the variable-metric approach to orbital localization is to allow nonorthogonality between orbitals while, at the same time, preventing them from becoming linearly dependent. The variable-metric localization has been shown to improve the locality of occupied nonorthogonal orbitals relative to their orthogonal counterparts. In this work, numerous localization algorithms are designed and tested to exploit the conceptual simplicity of the variable-metric approach with the goal of creating a straightforward and reliable localization procedure for virtual orbitals. The implemented algorithms include the steepest descent, conjugate gradient (CG), limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS), and hybrid procedures as well as trust-region (TR) methods based on the CG and Cauchy-point subproblem solvers. Comparative analysis shows that the CG-based TR algorithm is the best overall method to obtain nonorthogonal localized molecular orbitals (NLMOs), occupied or virtual. The L-BFGS and CG algorithms can also be used to obtain NLMOs reliably but often at higher computational cost. Extensive tests demonstrate that the implemented methods allow us to obtain well-localized Boys-Foster (i.e., maximally localized Wannier functions) and Pipek-Mezey, orthogonal and nonorthogonal, and occupied and virtual orbitals for a variety of gas-phase molecules and periodic materials. The tests also show that virtual NLMOs, which have not been described before, are, on average, 13% (Boys-Foster) and 18% (Pipek-Mezey) more localized than their orthogonal counterparts.