Local Orbital Projection (PLO) Methods

• Local Orbital Projection (PLO) is a method that projects delocalized Bloch or Kohn–Sham states onto localized trial orbitals while preserving symmetry and chemical character.
• The approach uses techniques like Löwdin orthonormalization and symmetry-adapted trial functions to construct robust Wannier-like bases without iterative localization.
• PLO frameworks are applied in DFT, DFT+DMFT, and wavefunction methods to model correlated electrons and topological properties with improved computational efficiency.

Local orbital projection (PLO) refers to a set of methodologies in electronic structure theory that construct symmetry-adapted, localized bases—often Wannier-like or virtual orbital spaces—via a direct projection of delocalized solutions (e.g., Bloch or canonical orbitals) onto subspaces spanned by local trial functions. These schemes are increasingly favored in both density functional theory (DFT) and correlated-electron approaches due to their efficiency, symmetry control, and seamless incorporation into modern ab initio codes (Koepernik et al., 2021, 1804.02055, Mussard, 2018).

1. Formalism and Definitions

Local orbital projection defines a map from a delocalized basis (e.g., Kohn–Sham or Bloch bands) to a set of localized orbitals designed to capture specific physical or chemical properties. For a set of trial orbitals $\{\chi_L(\mathbf r)\}$, the projector onto the correlated subspace $C$ in DFT is

$P = \sum_L |\chi_L\rangle \langle \chi_L| \,,$

and the matrix elements for any eigenstate $|\psi_{\nu\mathbf k}\rangle$ are $$P_{L,\nu}(\mathbf k) = \langle \chi_L | \psi_{\nu\mathbf k}\rangle$$ (1804.02055).

In the context of Wannier function construction, the approach projects Bloch bands onto trial functions (themselves linear combinations of local orbitals),

$$\phi_{R c i}(r) = \sum_{R' s \nu} \Phi_{R' s \nu}(r) U_{R' s \nu, R c i} \,,$$

followed by Löwdin orthonormalization to yield a strictly orthonormal, symmetry-adapted basis (Koepernik et al., 2021).

2. Key Methodologies

2.1 Symmetry Conserving Maximally Projected Wannier Functions (SCMPWF)

Implemented in FPLO, SCMPWF projects DFT Bloch eigensolutions onto symmetry-adapted trial functions derived from the chemical basis (FPLO local orbitals or suitable combinations), producing Wannier-type functions that:

• Are strictly orthonormal
• Explicitly conserve space-group symmetry
• Inherit the chemical character and localization of the trial orbitals

Projection is combined with a single Löwdin orthonormalization:

$w^k = \tilde w^k (O^k)^{-1/2}$

where $O^k$ is the projected overlap matrix. This avoids the iterative optimization of the Marzari–Vanderbilt localization functional and provides unambiguous symmetry and gauge control (Koepernik et al., 2021).

2.2 Projected Localized Orbitals in PAW/DFT+DMFT

Within the projector augmented wave (PAW) framework, as in VASP, the PLOs are obtained by projecting KS (pseudo-)wavefunctions onto atomic-like channels inside augmentation spheres. The optimal projector for each angular channel is formed by maximizing overlap with targeted bands within a chosen energy window, followed by orthonormalization. This is crucial for embedding many-body self-energies (as in DFT+DMFT or DFT+U), ensuring stable, physically meaningful correlated subspaces even as the number of basis states or augmentation channels grows (1804.02055).

2.3 Projected Oscillator Orbitals (pOOs)

For correlated wavefunction methods, the pOO construction systematically generates localized virtual spaces by multiplying localized occupied (LMO) orbitals by harmonics and projecting out the occupied space. The general procedure for the $n$-th order pOO is

$$|i^{(n)}_{\alpha_1 \dots \alpha_n}\rangle = \hat P \prod_{p=1}^n ( \hat r_{\alpha_p} - D^i_{\alpha_p} ) |i\rangle \,,$$

where $C$0 is the orbital centroid and $C$1 projects out the occupied subspace. This enables compact representations with high correlation-energy recovery, especially for long-range effects like dispersion (Mussard, 2018).

3. Symmetry, Gauge, and Localization Considerations

PLO frameworks enforce symmetry by requiring the trial functions or projectors to form a complete set under the relevant space-group operations. In SCMPWF, for example, the set of projectors is symmetrized so that the resulting Wannier functions inherit both real-space localization and correct transformation under all symmetry operations (Koepernik et al., 2021).

Choice of phase ("gauge") in Bloch sums impacts observable properties, particularly Berry connections and curvatures. The "relative gauge" simplifies transformation laws and makes many Berry and position-operator formulae tractable, especially important for anomalous Hall and topological studies.

PLOs enable both atom-centered and bond-centered localization. The flexibility to construct projectors as arbitrary linear combinations (e.g., to form $C$2 hybrids or bond-centered functions) broadens the range of systems amenable to efficient localized representation.

4. Algorithmic Framework and Practical Guidelines

The standard workflow involves:

1. Band Selection: Specify energy windows to select the subset of bands for projection.
2. Trial Function Definition: Define (and, if possible, optimize) local projectors at relevant centers, ensuring closure under symmetry.
3. Projection: Compute overlaps between trial/projector functions and chosen Bloch or KS states.
4. Orthonormalization: Apply Löwdin or similar procedures to enforce orthonormality across the projected set.
5. Real-Space Construction: Fourier transform to yield localized Wannier (or virtual) functions.
6. Hamiltonian and Operator Transformation: Transform relevant operators (Hamiltonian, position, etc.) into the PLO/Wannier basis, symmetrizing as needed.
7. Many-Body Embedding (if needed): For DFT+DMFT or related schemes, "upfold" local self-energies to the extended basis using the computed projectors, and iterate to self-consistency (1804.02055, Koepernik et al., 2021).

In PAW-based codes such as VASP, the optimization step is crucial. It maximizes physical fidelity, suppresses spurious mixing with high-lying states, and ensures stability upon changing the number of states or projectors. In practice, optimized projectors recover correct atomic or molecular character and remove sensitivity to details such as the size of the PAW channel set (1804.02055).

5. Representative Applications and Performance Analysis

PLO-based techniques are widely deployed in:

• Tight-binding parameterization (downfolding in cuprates, pnictides, Heuslers)
• Transport calculations: Berry curvature, anomalous Hall/Nernst/spin Hall conductivities, Berry dipole responses in ferromagnets and Weyl semimetals
• Dynamical mean-field theory: Construction of correlated subspaces, e.g., for NiO and SrVO$C$3 monolayer systems, showing improved robustness and reduced sensitivity to double-counting parameters via optimized projectors (1804.02055).
• Wavefunction methods: Projected oscillator orbitals achieve rapid convergence of correlation energies, especially for dispersion interactions, with a drastically reduced number of virtual functions compared to canonical approaches (Mussard, 2018).
• Topological indices, surface-state spectroscopy, model Hamiltonian construction, and molecular cluster analysis (Koepernik et al., 2021).

Performance-wise, SCMPWF in FPLO introduces a negligible computational overhead (~10–30% over standard DFT), with robust exponential localization and exact symmetry preservation. Optimized PLOs in PAW/DFT+DMFT avoid discontinuities and loss of physical content in the correlated subspace, providing stable charge self-consistent solutions (1804.02055, Koepernik et al., 2021).

6. Comparative Perspective and Limitations

Compared to iterative maximization approaches (e.g., maximally localized Wannier functions via Wannier90), PLO methods:

• Bypass disentanglement and iterative minimization, yielding results via closed formulas and direct projection
• Offer explicit symmetry and gauge control, critical for high-symmetry and topological systems
• Depend heavily on physically meaningful trial function selection and projector optimization

A plausible implication is that, while PLO approaches offer exceptional robustness, localization, and efficiency, the quality of the resulting basis remains sensitive to the informed definition of trial/projector functions and the appropriate choice of projection windows. For strongly entangled or strongly-correlated systems, these parameters may require careful tuning.

7. Connections, Extensions, and Outlook

The systematic improvability of PLO bases (notably in pOOs, where higher-order harmonics increase flexibility) suggests potential for controlled accuracy within both mean-field and beyond-mean-field methods (Mussard, 2018). The generality of the projection formalism facilitates seamless interfacing of DFT with DMFT, GW, RPA, and other correlated methods, as well as high-throughput computational materials workflows.

Further development directions involve enhanced projector optimization schemes, automated selection of chemically relevant subspaces, and extensions to treat explicitly time-reversal/topological protection and multiorbital entanglement, driven by the growing demands of quantum materials modeling (Koepernik et al., 2021, 1804.02055).