[Paper Review] Wannier Functions Dually Localized in Space and Energy

Aaron Mahler, J. Williams+2 more|arXiv (Cornell University)|Jan 19, 2022

Advanced Chemical Physics Studies4 citations

TL;DR

This paper introduces dually localized Wannier functions (DLWFs) that are simultaneously localized in real space and energy by minimizing a weighted sum of spatial and energy variances. The method produces Wannier functions associated with specific energy levels, including fractionally occupied orbitals, enabling a unified description of both occupied and unoccupied frontier orbitals—critical for correcting delocalization error in density functional theory without separate treatment of occupied and unoccupied spaces.

ABSTRACT

The construction of Wannier functions from Bloch orbitals offers a unitary freedom that can be exploited to yield Wannier functions with advantageous properties. Minimizing the spatial variance is a well-known choice; another, previously proposed for Wannier functions constructed from the occupied Bloch manifold, minimizes a weighted sum of spatial and energy variance. Departing from all previous work, we extend dual localization to include both valence and conduction bands together. Near the Fermi energy, these dually localized Wannier functions yield frontier (bonding and antibonding) orbitals in bulk silicon and molecular ethylene, as well as $d$-orbital character in metallic copper. Because they are both localized and retain information about the orbital energy spectrum, dually localized Wannier functions are well suited to orbital-dependent methods that associate Wannier functions with specific energy ranges. They naturally induce fractional occupations, allowing for corrections to the DFA total energy.

Motivation & Objective

• To develop a unified framework for constructing Wannier functions that are localized in both space and energy, overcoming limitations of traditional maximally localized Wannier functions (MLWFs).
• To address the challenge of describing both occupied and unoccupied frontier orbitals in a single, consistent Wannier function framework without separate localization procedures.
• To enable accurate representation of chemical bonding and antibonding character in systems like silicon and ethylene using a single set of Wannier functions with defined energy associations.
• To support orbital-dependent density functional theory methods by providing Wannier functions with intrinsic fractional occupations tied to specific energy levels.
• To overcome the limitations of sequential disentanglement and localization in entangled band structures by integrating energy localization into the core optimization.

Proposed method

• Minimize a cost function F = (1−γ)Ω + γE that balances spatial variance (Ω) and energy variance (E), where γ controls the trade-off between spatial and energy localization.
• Use a generalized Wannier function construction that allows unitary mixing of bands at each k-point, enabling the inclusion of both occupied and unoccupied bands in the same Wannier function set.
• Apply the Loschmidt (LOSC) cost function (γ = 0.47714) to achieve optimal energy localization while preserving spatial localization, as validated in silicon and ethylene.
• Construct Wannier functions via Fourier transform of k-space Bloch orbitals, with phase and unitary freedom optimized to minimize F, ensuring smooth k-dependence and localization.
• Utilize isosurfaces and isovalue thresholds (e.g., 160, 60, 30) to visualize the spatial distribution of DLWFs and identify bonding/antibonding character.
• Leverage the energy localization to assign fractional occupations to Wannier functions, enabling direct use in methods correcting delocalization error in DFT.

Experimental results

Research questions

• RQ1 Can Wannier functions be simultaneously localized in both real space and energy to provide a unified description of occupied and unoccupied electronic states?
• RQ2 How does energy localization in Wannier functions affect their ability to represent chemically meaningful orbitals such as σ and π bonding/antibonding states?
• RQ3 To what extent can dually localized Wannier functions with fractional occupations improve the description of frontier orbitals in systems like silicon and ethylene?
• RQ4 Can the unified localization approach avoid the pitfalls of sequential disentanglement and localization in entangled band structures?
• RQ5 Why do Wannier functions with occupations far from 1 exhibit significant imaginary character, and how does this affect their physical interpretation?

Key findings

• DLWFs constructed with γ = 0.47714 successfully recover the σ and π bonding/antibonding orbitals in ethylene, with distinct isosurface patterns reflecting their chemical character.
• The method produces fractionally occupied Wannier functions that are naturally associated with specific energy levels, enabling a direct description of frontier orbitals without separate treatment of occupied and unoccupied spaces.
• DLWFs with occupation near 1 remain real and exhibit atomic-like or tight-binding character, preserving symmetry and localization, while those with non-integer occupations show significant imaginary components.
• Energy localization allows the inclusion of widely separated bands (e.g., occupied and unoccupied) in the same Wannier function set without unphysical mixing, improving the physical consistency of the representation.
• The approach provides a unified alternative to sequential disentanglement and localization, potentially avoiding suboptimal solutions in entangled band systems.
• The LOSC cost function (γ = 0.47714) yields chemically intuitive Wannier functions for both silicon (diamond lattice) and ethylene, including a π* antibonding orbital, demonstrating broad applicability.