Generalized Wannier Functions Research Articles

Wannier Function Localization Using Bloch Intrinsic Atomic Orbitals.

We extend the intrinsic atomic orbital (IAO) method for the localization of molecular orbitals to calculate well-localized generalized Wannier functions in crystals in the spirit of the Pipek-Mezey method. We furthermore present a one-shot diabatic Wannierization procedure that aligns the phases of the Bloch functions, providing immediate Wannier localization, which serves as an excellent initial guess for optimization. We test our Wannier localization implementation on a number of solid-state systems, highlighting the effectiveness of the diabatic preparation, especially for localizing core bands. Partial charges of Wannier functions generated using Bloch IAOs align well with chemical intuition, which we demonstrate through the example of the adsorption of CO on a MgO surface.

Existence and Computation of Generalized Wannier Functions for Non-Periodic Systems in Two Dimensions and Higher

Exponentially-localized Wannier functions (ELWFs) are an orthonormal basis of the Fermi projection of a material consisting of functions which decay exponentially fast away from their maxima. When the material is insulating and crystalline, conditions which guarantee existence of ELWFs in dimensions one, two, and three are well-known, and methods for constructing the ELWFs numerically are well-developed. We consider the case where the material is insulating but not necessarily crystalline, where much less is known. In one spatial dimension, Kivelson and Nenciu-Nenciu have proved ELWFs can be constructed as the eigenfunctions of a self-adjoint operator acting on the Fermi projection. In this work, we identify an assumption under which we can generalize the Kivelson-Nenciu-Nenciu result to two dimensions and higher. Under this assumption, we prove that ELWFs can be constructed as the eigenfunctions of a sequence of self-adjoint operators acting on the Fermi projection. We conjecture that the assumption we make is equivalent to vanishing of topological obstructions to the existence of ELWFs in the special case where the material is crystalline. We numerically verify that our construction yields ELWFs in various cases where our assumption holds and provide numerical evidence for our conjecture.

Massively parallel linear-scaling Hartree-Fock exchange and hybrid exchange-correlation functionals with plane wave basis set accuracy.

We extend our linear-scaling approach for the calculation of Hartree-Fock exchange energy using localized in situ optimized orbitals [Dziedzic et al., J. Chem. Phys. 139, 214103 (2013)] to leverage massive parallelism. Our approach has been implemented in the onetep (Order-N Electronic Total Energy Package) density functional theory framework, which employs a basis of non-orthogonal generalized Wannier functions (NGWFs) to achieve linear scaling with system size while retaining controllable near-complete-basis-set accuracy. For the calculation of Hartree-Fock exchange, we use a resolution-of-identity approach, where an auxiliary basis set of truncated spherical waves is used to fit products of NGWFs. The fact that the electrostatic potential of spherical waves (SWs) is known analytically, combined with the use of a distance-based cutoff for exchange interactions, leads to a calculation cost that scales linearly with the system size. Our new implementation, which we describe in detail, combines distributed memory parallelism (using the message passing interface) with shared memory parallelism (OpenMP threads) to efficiently utilize numbers of central processing unit cores comparable to, or exceeding, the number of atoms in the system. We show how the use of multiple time-memory trade-offs substantially increases performance, enabling our approach to achieve superlinear strong parallel scaling in many cases and excellent, although sublinear, parallel scaling otherwise. We demonstrate that in scenarios with low available memory, which preclude or limit the use of time-memory trade-offs, the performance degradation of our algorithm is graceful. We show that, crucially, linear scaling with system size is maintained in all cases. We demonstrate the practicability of our approach by performing a set of fully converged production calculations with a hybrid functional on large imogolite nanotubes up to over 1400 atoms. We finish with a brief study of how the employed approximations (exchange cutoff and the quality of the SW basis) affect the calculation walltime and the accuracy of the obtained results.

Calculation of Berry curvature using non-orthogonal atomic orbitals

We present a derivation of the full formula to calculate the Berry curvature on non-orthogonal numerical atomic orbital (NAO) bases. Because usually, the number of NAOs is larger than that of the Wannier bases, we use a orbital contraction method to reduce the basis sizes, which can greatly improve the calculation efficiency without significantly reducing the calculation accuracy. We benchmark the formula by calculating the Berry curvature of ferroelectric BaTiO3 and bcc Fe, as well as the anomalous Hall conductivity for Fe. The results are in excellent agreement with the finite-difference and previous results in the literature. We find that there are corrections terms to the Kubo formula of the Berry curvature. For the full NAO base, the differences between the two methods are negligibly small, but for the reduced bases sets, the correction terms become larger, which may not be neglected in some cases. The formula developed in this work can readily be applied to the non-orthogonal generalized Wannier functions.

Iterated projected position algorithm for constructing exponentially localized generalized Wannier functions for periodic and nonperiodic insulators in two dimensions and higher

Localized bases play an important role in understanding electronic structure. In periodic insulators, a natural choice of localized basis is given by the Wannier functions which depend on a choice of unitary transform known as a gauge transformation. Over the past few decades, there have been many works that have focused on optimizing the choice of the gauge so that the corresponding Wannier functions are maximally localized or reflect some symmetry of the underlying system. In this work, we consider fully nonperiodic materials where the usual Wannier functions are not well defined and gauge optimization is impractical. To tackle the problem of calculating exponentially localized generalized Wannier functions in both periodic and nonperiodic systems, we discuss the `iterated projected position (IPP)'' algorithm. The IPP algorithm is based on matrix diagonalization and therefore unlike optimization-based approaches, it does not require initialization and cannot get stuck at a local minimum. Furthermore, the IPP algorithm is guaranteed by a rigorous analysis to produce exponentially localized functions under certain mild assumptions. We numerically demonstrate that the IPP algorithm can be used to calculate exponentially localized bases for the Haldane model, the Kane-Mele model (in both ${\mathbb{Z}}_{2}$ invariant even and ${\mathbb{Z}}_{2}$ invariant odd phases), and the ${p}_{x}+i{p}_{y}$ model on a quasicrystal lattice.

The ONETEP linear-scaling density functional theory program.

We present an overview of the onetep program for linear-scaling density functional theory (DFT) calculations with large basis set (plane-wave) accuracy on parallel computers. The DFT energy is computed from the density matrix, which is constructed from spatially localized orbitals we call Non-orthogonal Generalized Wannier Functions (NGWFs), expressed in terms of periodic sinc (psinc) functions. During the calculation, both the density matrix and the NGWFs are optimized with localization constraints. By taking advantage of localization, onetep is able to perform calculations including thousands of atoms with computational effort, which scales linearly with the number or atoms. The code has a large and diverse range of capabilities, explored in this paper, including different boundary conditions, various exchange-correlation functionals (with and without exact exchange), finite electronic temperature methods for metallic systems, methods for strongly correlated systems, molecular dynamics, vibrational calculations, time-dependent DFT, electronic transport, core loss spectroscopy, implicit solvation, quantum mechanical (QM)/molecular mechanical and QM-in-QM embedding, density of states calculations, distributed multipole analysis, and methods for partitioning charges and interactions between fragments. Calculations with onetep provide unique insights into large and complex systems that require an accurate atomic-level description, ranging from biomolecular to chemical, to materials, and to physical problems, as we show with a small selection of illustrative examples. onetep has always aimed to be at the cutting edge of method and software developments, and it serves as a platform for developing new methods of electronic structure simulation. We therefore conclude by describing some of the challenges and directions for its future developments and applications.

Parseval Frames of Exponentially Localized Magnetic Wannier Functions

Motivated by the analysis of gapped periodic quantum systems in presence of a uniform magnetic field in dimension $d \le 3$, we study the possibility to construct spanning sets of exponentially localized (generalized) Wannier functions for the space of occupied states. When the magnetic flux per unit cell satisfies a certain rationality condition, by going to the momentum-space description one can model $m$ occupied energy bands by a real-analytic and $\mathbb Z^{d}$-periodic family $\{P({\bf k})\}_{{\bf k} \in \mathbb R^{d}}$ of orthogonal projections of rank $m$. A moving orthonormal basis of $\mathrm{Ran} P({\bf k})$ consisting of real-analytic and $\mathbb Z^d$-periodic Bloch vectors can be constructed if and only if the first Chern number(s) of $P$ vanish(es). Here we are mainly interested in the topologically obstructed case. First, by dropping the generating condition, we show how to algorithmically construct a collection of $m-1$ orthonormal, real-analytic, and periodic Bloch vectors. Second, by dropping the linear independence condition, we construct a Parseval frame of $m+1$ real-analytic and periodic Bloch vectors which generate $\mathrm{Ran} P({\bf k})$. Both algorithms are based on a two-step logarithm method which produces a moving orthonormal basis in the topologically trivial case. A moving Parseval frame of analytic, periodic Bloch vectors corresponds to a Parseval frame of exponentially localized composite Wannier functions. We extend this construction to the case of magnetic Hamiltonians with an irrational magnetic flux per unit cell and show how to produce Parseval frames of exponentially localized generalized Wannier functions also in this setting. Our results are illustrated in crystalline insulators modelled by $2d$ discrete Hofstadter-like Hamiltonians, but apply to certain continuous models of magnetic Schr\"{o}dinger operators as well.

Exact exchange with non-orthogonal generalized Wannier functions.

The evaluation of exact exchange (EXX) is an important component of quantum chemical calculations performed with ab initio and hybrid density functional methods. While evaluating exact exchange is routine in molecular quantum chemical calculations performed with localized basis sets, the non-local nature of the exchange operator presents a major impediment to the efficient use of exact exchange in calculations that employ planewave basis sets. Non-orthogonal generalized Wannier functions (NGWFs) corresponding to planewave expansions of localized basis functions are an alternative form of basis set that can be used in quantum chemical calculations. The periodic nature of these functions renders them suitable for calculations of periodic systems, while the contraction of sets of planewaves into individual basis functions reduces the number of variational parameters, permitting the construction and direct diagonalization of the Fock matrix. The present study examines how NGWFs corresponding to Fourier series representations of conventional atom-centered basis sets can be used to evaluate exact exchange in periodic systems. Specifically, an approach for constructing the exchange operator with NGWFs is presented and used to perform Hartree-Fock calculations with a series of molecules in periodically repeated simulation cells. The results demonstrate that the NGWF approach is significantly faster than the EXX method, which is a standard approach for evaluating exact exchange in periodic systems.

Bosons in a two-dimensional bichromatic quasiperiodic potential: Analysis of the disorder in the Bose-Hubbard parameters and phase diagrams

Experimental realizations of disorder in optical lattices generate a distribution of the Bose-Hubbard (BH) parameters, like on-site potentials, hopping strengths, and interaction energies. We analyze this distribution for bosons in a two-dimensional bichromatic quasiperiodic potential by determining the generalized Wannier functions and calculating the corresponding BH parameters. Using a local mean-field cluster analysis, we study the effect of the corresponding disorder on the phase diagrams. We find a substantial amount of disorder in the hopping strengths, which produces strong deviations from the phase diagram of the disordered BH model with purely random on-site potentials.

Natural bond orbital analysis in the ONETEP code: Applications to large protein systems

First principles electronic structure calculations are typically performed in terms of molecular orbitals (or bands), providing a straightforward theoretical avenue for approximations of increasing sophistication, but do not usually provide any qualitative chemical information about the system. We can derive such information via post-processing using natural bond orbital (NBO) analysis, which produces a chemical picture of bonding in terms of localized Lewis-type bond and lone pair orbitals that we can use to understand molecular structure and interactions. We present NBO analysis of large-scale calculations with the ONETEP linear-scaling density functional theory package, which we have interfaced with the NBO 5 analysis program. In ONETEP calculations involving thousands of atoms, one is typically interested in particular regions of a nanosystem whilst accounting for long-range electronic effects from the entire system. We show that by transforming the Non-orthogonal Generalized Wannier Functions of ONETEP to natural atomic orbitals, NBO analysis can be performed within a localized region in such a way that ensures the results are identical to an analysis on the full system. We demonstrate the capabilities of this approach by performing illustrative studies of large proteins--namely, investigating changes in charge transfer between the heme group of myoglobin and its ligands with increasing system size and between a protein and its explicit solvent, estimating the contribution of electronic delocalization to the stabilization of hydrogen bonds in the binding pocket of a drug-receptor complex, and observing, in situ, the n → π* hyperconjugative interactions between carbonyl groups that stabilize protein backbones.

Analytical optimization of spread and change of exponential decay of generalized Wannier functions in one dimension

Generalized Wannier functions of a couple of bands in a one-dimensional crystal are investigated. A lower bound for the global minimum of the total spread is obtained. Assumption of such a value being the minimum leads to a first-order differential equation for the transformation matrix. Simple analytical solutions leading to real generalized Wannier functions are presented for consecutive bands in a crystal with inversion symmetry. Results are displayed for a particle in a diatomic Kronig-Penney potential. For the lowest couple of bands, calculated single-band Wannier functions resemble orbitals of a diatomic molecule, whereas generalized Wannier functions seem like orthogonalized atomic orbitals. The latter functions are neither symmetric nor antisymmetric and display increased exponential decay. For the next two pairs of bands, Wannier functions retain their centers and symmetries, and exponential decay does not increase. Results are also shown to be in agreement with solutions of the eigenvalue problem of the band-projected position operator.

Generalized Wannier functions: A comparison of molecular electric dipole polarizabilities

Localized Wannier functions provide an efficient and intuitive means by which to compute dielectric properties from first principles. They are most commonly constructed in a post-processing step, following total-energy minimization. Nonorthogonal generalized Wannier functions (NGWFs) [Skylaris et al., Phys. Rev. B 66, 035119 11 (2002); Skylaris et al., J. Chem. Phys. 122, 084119 (2005)] may also be optimized in situ, in the process of solving for the ground-state density. We explore the relationship between NGWFs and orthonormal, maximally localized Wannier functions (MLWFs) [Marzari and Vanderbilt, Phys. Rev. B 56, 12847 (1997); Souza, Marzari, and Vanderbilt, ibid. 65, 035109 (2001)], demonstrating that NGWFs may be used to compute electric dipole polarizabilities efficiently, with no necessity for post-processing optimization, and with an accuracy comparable to MLWFs.

Maximally localized Wannier functions for ultracold atoms in one-dimensional double-well periodic potentials

We discuss a method for constructing generalized Wannier functions that are maximally localized at the minima of a one-dimensional periodic potential with a double well per unit cell. By following the approach of Marzari and Vanderbilt (1997 Phys. Rev. B 56 12847), we consider a set of band-mixing Wannier functions with minimal spread and design a specific two-step gauge transformation of the Bloch functions for a composite two-band system. This method is suited for efficiently computing the tight-binding coefficients needed for mapping the continuous system to a discrete lattice model. The behaviour of the tight-binding coefficients is analyzed here as a function of the symmetry properties of the double well (including the possibility of parity-breaking), in a range of feasible experimental parameters.

Projector self-consistentDFT+Uusing nonorthogonal generalized Wannier functions

We present a formulation of the density-functional theory + Hubbard model (DFT+U) method that is self-consistent over the choice of Hubbard projectors used to define the correlated subspaces. In order to overcome the arbitrariness in this choice, we propose the use of non-orthogonal generalized Wannier functions (NGWFs) as projectors for the DFT+U correction. We iteratively refine these NGWF projectors and, hence, the DFT+U functional, such that the correlated subspaces are fully self-consistent with the DFT+U ground-state. We discuss the convergence characteristics of this algorithm and compare ground-state properties thus computed with those calculated using hydrogenic projectors. Our approach is implemented within, but not restricted to, a linear-scaling DFT framework, opening the path to DFT+U calculations on systems of unprecedented size.

Maximally localized Wannier functions for photonic lattices

A method is described for the computation of the generalized Wannier functions corresponding to a composite group of electromagnetic bands in a photonic crystal. The Wannier functions are optimally localized, in that they have minimum real-space spread, defined as the average over the set of the second moment of the functions. The mathematical approach follows that developed by N. Marzari and D. Vanderbilt [Phys. Rev. B 56, 12 847 (1997)] to obtain similarly well-localized Wannier functions for electrons in crystalline solids. Results are presented for the lowest few TE and TM bands in two-dimensional photonic crystals consisting of square and triangular lattices of holes and rods.

Nonorthogonal generalized Wannier function pseudopotential plane-wave method

We present a reformulation of the plane-wave pseudopotential method for insulators. This new approach allows us to perform density-functional calculations by solving directly for “nonorthogonal generalized Wannier functions” rather than extended Bloch states. We outline the theory on which our method is based and present test calculations on a variety of systems. Comparison of our results with a standard plane-wave code shows that they are equivalent. Apart from the usual advantages of the plane-wave approach such as the applicability to any lattice symmetry and the high accuracy, our method also benefits from the localization properties of our functions in real space. The localization of all our functions greatly facilitates the future extension of our method to linear-scaling schemes or calculations of the electric polarization of crystalline insulators.

Tight-binding method modelling of photonic crystal waveguides

We present here a tight-binding-like modelling of two-dimensional (2D) photonic crystals (PCs). Adopted from solid-state physics, the concept of generalized Wannier functions is used to construct a localized state basis that allows a parameter-free ab initio study of defects in PCs. We demonstrate here for a 2D triangular lattice of dielectric rods in air, the existence of this localized basis and the possibility to study large scale complex dielectric structures deviating from periodicity. Specific numerical simulations on a split waveguide embedded in this triangular lattice are performed, and they demonstrate the superiority of this method over plane wave based techniques.

Photonic crystal modelling using a tight-binding Wannier function method

Using the concept of generalized Wannier functions, adapted from the electronic theory of solids, we demonstrate for two-dimensional photonic crystals the existence of a localized state basis and we establish an efficient computational method allowing a tight-binding-like parameter free modelization of any dielectric structure deviating from periodicity. Examples of numerical simulations using this formalism, including modal analysis of microcavities and waveguides are presented to prove the ability of this approach to deal accurately with large scale systems and complex structures. A tight-binding version of the transfer matrix method is proposed to describe the transmission and reflection properties of finite samples of photonic crystals.

Generalized Wannier function method for photonic crystals

The concept of generalized Wannier functions, adopted from the electronic theory of solids, is used to build a localized representation of electromagnetic waves in dielectric materials. For two-dimensional photonic crystals, we demonstrate the existence of such a localized state basis, and we establish an efficient computational method, allowing a tight-binding-like parameter free modelization of any dielectric structure deviating from periodicity. Numerical simulations of a T-shaped photonic crystal waveguide prove its ability to deal with large-scale systems and complex structures.

The structure of electronic states in amorphous silicon

We illustrate the structure and dynamics of electron states in amorphous Si. The nature of the states near the gap at zero temperature is discussed and especially the way the structure of the states changes for energies ranging from midgap into either band tail (Anderson transition). We then study the effect of lattice vibrations on the eigenstates, and find that electronic states near the optical gap can be strongly influenced by thermal modulation of the atomic positions. Finally, we show the structure of generalized Wannier functions for amorphous Si, which are of particular interest for efficient ab initio calculation of electronic properties and forces for first principles dynamic simulation.