Nonorthogonal Localized Orbitals Research Articles

Direct Unconstrained Variable-Metric Localization of One-Electron Orbitals.

Spatially localized one-electron orbitals, orthogonal and non-orthogonal, are widely used in electronic structure theory to describe chemical bonding and speed up calculations. In order to avoid linear dependencies of localized orbitals, the existing localization methods either constrain orbital transformations to be unitary, that is, metric preserving, or, in the case of variable-metric methods, fix the centers of non-orthogonal localized orbitals. Here, we developed a different approach to orbital localization, in which these constraints are replaced with a single restriction that specifies the maximum allowed deviation from the orthogonality for the final set of localized orbitals. This reformulation, which can be viewed as a generalization of existing localization methods, enables one to choose the desired balance between the orthogonality and locality of the orbitals. Furthermore, the approach is conceptually and practically simple as it obviates the necessity in unitary transformations and allows one to determine optimal positions of the centers of non-orthogonal orbitals in an unconstrained and straightforward minimization procedure. It is demonstrated to produce well-localized orthogonal and non-orthogonal orbitals with the Berghold and Pipek--Mezey localization functions for a variety of molecules and periodic materials including large systems with nontrivial bonding.

First-principles calculation of optical responses based on nonorthogonal localized orbitals

Based on ab initio software packages using nonorthogonal localized orbitals, we develop a general scheme of calculating response functions. We test the performance of this method by calculating nonlinear optical responses of materials, like the shift current conductivity of monolayer WS2, and achieve good agreement with previous calculations. This method bears many similarities to Wannier interpolation, which requires a challenging optimization of Wannier functions due to the conflicting requirements of orthogonality and localization. Although computationally heavier compared to Wannier interpolation, our procedure avoids the construction of Wannier functions and thus enables automated high throughput calculations of linear and nonlinear responses related to electrical, magnetic and optical material properties.

Calculation of wave‐functions with frozen orbitals in mixed quantum mechanics/molecular mechanics methods. II. Application of the local basis equation

The application of the local basis equation (Ferenczy and Adams, J. Chem. Phys. 2009, 130, 134108) in mixed quantum mechanics/molecular mechanics (QM/MM) and quantum mechanics/quantum mechanics (QM/QM) methods is investigated. This equation is suitable to derive local basis nonorthogonal orbitals that minimize the energy of the system and it exhibits good convergence properties in a self-consistent field solution. These features make the equation appropriate to be used in mixed QM/MM and QM/QM methods to optimize orbitals in the field of frozen localized orbitals connecting the subsystems. Calculations performed for several properties in divers systems show that the method is robust with various choices of the frozen orbitals and frontier atom properties. With appropriate basis set assignment, it gives results equivalent with those of a related approach [G. G. Ferenczy previous paper in this issue] using the Huzinaga equation. Thus, the local basis equation can be used in mixed QM/MM methods with small size quantum subsystems to calculate properties in good agreement with reference Hartree-Fock-Roothaan results. It is shown that bond charges are not necessary when the local basis equation is applied, although they are required for the self-consistent field solution of the Huzinaga equation based method. Conversely, the deformation of the wave-function near to the boundary is observed without bond charges and this has a significant effect on deprotonation energies but a less pronounced effect when the total charge of the system is conserved. The local basis equation can also be used to define a two layer quantum system with nonorthogonal localized orbitals surrounding the central delocalized quantum subsystem.

Linear-scaling quantum calculations using non-orthogonal localized molecular orbitals

An absolute energy minimum variational principle is used for carrying out linear-scalingcalculations with non-orthogonal localized orbitals. Comparing with results based onorthogonal localized molecular orbitals, the method is shown to give significantly moreaccurate results when the localized molecular orbitals are allowed to be non-orthogonal.This is made possible by introducing a second minimization for approximatingthe inverse overlap matrix. We also show how an exact line search may be usedefficiently with the conjugate gradient method for minimizing the energy functional.

Nonlinear multigrid eigenvalue solver utilizing nonorthogonal localized orbitals

AbstractA nonlinear multigrid eigenvalue solver utilizing nonorthogonal localized orbitals is formulated and implemented on a real‐space grid. The localization of orbitals is necessary to achieve linear scaling in the computational effort. Numerical tests are performed on the benzene molecule, C20, and C60. The localization centers for the orbitals are allowed to move so as to lower the total energy. The convergence rate depends on the radius of the confined regions. Also, the convergence rate slows when the number of atoms in the system increases, and/or when unoccupied orbitals are included. The slowed convergence is due to ill‐conditioning, which is related to the kinetic contribution to the total energy. Work is in progress to alleviate the ill‐conditioning. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Optimized nonorthogonal localized orbitals for linear scaling quantum Monte Carlo calculations

We derive an automatic procedure for generating a set of highly localized, nonorthogonal orbitals for linear scaling quantum Monte Carlo (QMC) calculations. We demonstrate the advantage of these orbitals for calculating the total energy of both semiconducting and metallic systems by studying bulk silicon and the homogeneous electron gas. For silicon, the improved localization of these orbitals reduces the computational time by a factor of 5 and the memory by a factor of 6 compared to localized, orthogonal orbitals. For jellium at typical metallic densities, we demonstrate that the total energy is converged to 3 meV per electron for orbitals truncated within spheres with radii $7{r}_{s}$, opening the possibility of linear scaling QMC calculations for realistic metallic systems.

Electron density of periodic systems derived from non-orthogonal localized orbitals

Development of a quantum cluster embedding scheme based on non-orthogonal localizedorbitals requires an efficient numerical method for calculating the electron density of thereference periodic crystal. We demonstrate that an existing method due to Löwdin basedon the inverse overlap matrix expansion can only be used when the orbitals arehighly localized (e.g. ionic systems). In other cases including covalent crystalsor those with an intermediate type of chemical bonding this method may beeither numerically inefficient or fail altogether. Instead, we suggest an exact andnumerically efficient method which can be used for orbitals of practically arbitrarylocalization. Theory is illustrated by numerical calculations on a model system.

Ab initio incremental correlation treatment with non-orthogonal localized orbitals

The local incremental expansion of the correlation energy of extended systems is applied to non-orthogonal localized orbitals and compared to the standard approach, which uses orthogonal Foster–Boys orbitals. Several methods of how to generate suitable non-orthogonal orbitals for the investigated covalent systems, bulk silicon and beryllium rings, are discussed. For the non-orthogonal orbitals the correlation energy contributions from increments involving more than one correlated orbital decay faster with the distance between these orbitals than for standard Foster–Boys orbitals. Also, the transferability of the individual energy increments from one cluster to another cluster is better in case of the non-orthogonal orbitals. Yet, the convergence of the incremental series with the order of the increments, i.e. the number of correlated bonds involved, is somewhat slower compared to the orthogonal Foster–Boys orbitals.

Dynamics of breathers and kink-antikink collisions in trans-polyacetylene studied by an all-valence electron tight-binding Hamiltonian

We present a study of the dynamics of trans-polyacetylene chains using an O( N)-scaling all-valence electron tight-binding Hamiltonian. The O( N)-scheme based on a set of non-orthogonal localized orbitals turns out to be capable of efficiently describing fine details in the dynamics of such large molecular systems. In this work we focus on nonlinear vibrational excitations as well as on their interactions.

Absolute-energy-minimum principles for linear-scaling electronic-structure calculations

Two absolute energy minimum principles are developed for first-principle linear-scaling electronic structure calculations. One is with a normalization constraint and the other without any constraint. The density matrix is represented by a set of nonorthogonal localized orbitals and an auxiliary matrix which at the minimum becomes a generalized inverse of the overlap matrix of the localized orbitals. The number of localized orbitals is allowed to exceed the number of occupied orbitals. Comparison with other variational principles is made and numerical tests presented. @S0163-1829~97!01939-5# Most electronic structure calculations are based on a oneelectron Hamiltonian, either within semiempirical approaches such as tight-binding and semiempirical quantum chemical models, or within first-principle approaches such as the density functional and Hartree-Fock-based method. The one-electron Hamiltonian provides building blocks for the construction of the electronic wave function or the electron density. Conventional direct and iterative diagonalization algorithms for obtaining eigenvalues require an overall computational effort which grows as the cubic of the number of atoms. This has been a major limiting factor for the application of the electronic structure methods to large systems. The possibility of obtaining the sum of eigenvalues in electronic structure calculations with effort scaling linearly with the size of the system was first demonstrated in the

Non-orthogonal localized orbitals and orthogonal atomic hybrids derived from Mulliken's population analysis

The construction of non-orthogonal localized molecular orbitals (Wannier functions of extended systems) with maximal degree of localization according to the Magnasco-Perico localization criterion derived from Mulliken's population analysis is discussed and several forms of the equations giving these orbitals are presented. A partial orthogonality property of these orbitals permits an easy determination of the effective orthogonal atomic orbitals characterizing the actual state of the atom in the molecule; this helps to find connections between large-scale ab initio SCF calculations and traditional atomic orbital concepts. Calculations indicate full agreement with the traditional AO picture: for all ‘ordinary’ compounds one gets as many molecular orbitals appreciably localized on the given atom as are contained in the classical ‘minimal basis’ for that atom.

Total-energy global optimizations using nonorthogonal localized orbitals.

An energy functional for orbital-based O(N) calculations is proposed, which depends on a number of nonorthogonal, localized orbitals larger than the number of occupied states in the system, and on a parameter, the electronic chemical potential, determining the number of electrons. We show that the minimization of the functional with respect to overlapping localized orbitals can be performed so as to attain directly the ground-state energy, without being trapped at local minima. The present approach overcomes the multiple-minima problem present within the original formulation of orbital-based O(N) methods; it therefore makes it possible to perform O(N) calculations for an arbitrary system, without including any information about the system bonding properties in the construction of the input wave functions. Furthermore, while retaining the same computational cost as the original approach, our formulation allows one to improve the variational estimate of the ground-state energy, and the energy conservation during a molecular dynamics run. Several numerical examples for surfaces, bulk systems, and clusters are presented and discussed.

Non-orthogonal localized orbitals to study delocalization effects

A theorem is proven showing that one can always fin d a special non-orthogonal set of localized orbitals each of which contains only one of a starting set of approximate occupied orbitals (e.g., bonding orbitals of some subunits). A method of a posterior determination and a specific perturbation theory for such orbitals are described.

An effective (exchange) Hamiltonian for many-electron systems with degenerate and nonorthogonal localized orbitals

Using biorthonormal basis sets, the orthonormal formulation of magnetic interactions between ions in an insulator, previously given by Fuchikami and Tanabe, is extended to nonorthogonal function spaces. The nonorthogonality is entirely absorbed in the coefficients of each operator appearing in the effective Hamiltonian. To make a numerical comparison, both methods are applied to a few simple and exactly solvable problems. We discuss the effect of nondiagonal matrix elements between spatially equivalent excited configurations in connection with the perturbational expansion applied.

Expansion methods for Adams–Gilbert equations. I. Modified Adams–Gilbert equation and common and fluctuating basis sets

Adams–Gilbert (AG) equation for nonorthogonal localized orbitals of a single-determinant wavefunction has been modified so as to enable one to compute wavefunctions of large polyatomic systems by the expansion method. This equation is named as modified Adams–Gilbert (MAG) equation. One solves the AG or the MAG equation by each subsystem and, collecting all the orbitals obtained, one constructs wavefunction of the system. It is shown that when one employs the expansion method, one must actually use basis functions common to all the subsystems (common basis set) to solve the AG equation, while one can employ, by each subsystem, different basis functions appropriate to the subsystem (fluctuating basis set) to solve the MAG equation. An expansion method suitable for solving the AG and the MAG equations has been presented. Application of the method to HF, H2O, and CH4 has revealed that (1) the method proposed is workable, (2) actually so many basis functions are not needed for describing some subsystems, especially for core electrons, and (3) it is necessary to orthogonalize approximately, not necessarily rigorously, the orbitals in the system.

Localized orbitals and short-range molecular interactions. III. Rotational barriers in C2H6 and H2O2

Ab-initio calculations for eclipsed and staggered C2H6 and for several conformers of H2O2 are performed using a single determinant of doubly occupied nonorthogonal localized bond orbitals constructed from a minimal set of Slater AO's. The energies calculated with optimized BO's closely follow the pattern of the full SCF calculations. A component analysis of the rotation barriers is given along the lines suggested in Paper I [V. Magnasco and G. F. Musso, J. Chem. Phys. 60, 3744 (1974)]. The barrier in ethane (2.67 kcal) is primarily due to the penetration of nonorthogonal electron pairs belonging to vicinal C–H bonds into the core provided by nuclei and unperturbed electrons of all other groups. Although the minimal set is not suited to account for the small trans barrier in H2O2 both in BO and full SCF calculations, the calculated cis barrier in H2O2 (13.79 kcal) is seen to have a nature similar to that found in ethane, except for an enhanced importance of the Coulombic contribution. Delocalization effects are found to be of little importance in determining the conformational behavior of both molecules.

Localized orbitals and short-range molecular interactions. II. Nonadditivity in He3 and He4

Short-range interatomic interactions in singlet He3 and He4 are discussed in terms of a single determinant of doubly occupied nonorthogonal localized orbitals through the formalism described in Paper I [V. Magnasco and G. F. Musso, J. Chem. Phys. 60, 3744 (1974)]. Ab-initio calculations using a minimal set of Slater 1s orbitals are reported for several nuclear configurations. Nonadditive terms can be very well represented as resulting from penetration of pairs of nonorthogonal localized electron groups into the core provided by nuclei and ``unperturbed'' electrons of all other groups. Even in the small overlap region it is essential to include all the penetration effects, and this is easily done by the present formalism.