Localized orbitals

Modules: lo

Introduction

A molecular orbital is usually delocalized, i.e. it has non-negligible amplitude over the whole system rather than only around some atom(s) or bond(s). However, one can choose a unitary rotation \(U\)

\[\phi = \psi U\]

such that the resulting orbitals \(\phi\) are as spatially localized as possible. This is typically achieved by one of two classes of methods. The first is to project the orbitals onto a predefined local set of orbitals, which can be e.g. atomic orbitals or pseudo-atomic orbitals. The second is to optimize a cost function \(f\), which measures the locality of the molecular orbitals. Because there is no unambiguous choice for the localization criterion, several criteria have been suggested. Boys localization minimizes the spread of the orbital

\[f(U) = \sum_{i} \langle\psi_i|r^2|\psi_i\rangle - \langle\psi_i|r|\psi_i\rangle^2\]

Boys localized orbitals [1] in periodic systems are typically termed maximally localized Wannier orbitals (MLWF) [2].

Pipek-Mezey (PM) localization [3] maximizes the population charges on the atoms

\[f(U) = \sum^{\mathrm{atoms}}_{I} \sum_{i} \left|q^{I}_{i} \right|^2\]

Note that PM localization depends on the choice of atomic orbitals used for the population analysis. Several choices of populations are available, e.g. Mulliken or based on (meta-) L"owdin orbitals. Intrinsic bond orbitals (IBOs) can be viewed as a special case of PM localization using intrinsic atomic orbitals (IAOs) as population method. See Ref. [4] for a summary of choices of orbitals. Note that PM localization preserves the separation between \(\sigma\) and \(\pi\) orbitals.

Edmiston-Ruedenberg (ER) localization [5] maximizes the orbital Coulomb self-repulsion,

\[f(U) = \sum_{i} (ii|ii)\]

ER localization, however, is computationally more expensive than the Boys or PM approaches.

Localized orbitals can be calculated via the pivoted Cholesky factorization of a density-like matrix \(\mathbf{D} = \mathbf{C} \mathbf{C}^\dagger\). [6] Since \(\mathbf{C}\) is generally a rectangular matrix containing only the subset of \(N\) orbitals intended for localization, the matrix \(\mathbf{D}\) is positive-semidefinite. It can be factored using a Cholesky decomposition with full column pivoting,

\[\mathbf{P}^\dagger \mathbf{D} \mathbf{P} = \mathbf{L} \mathbf{L}^\dagger ,\]

where \(\mathbf{L}\) is a lower triangular matrix and \(\mathbf{P}\) is a permutation matrix. In the end, the \(N\) leftmost columns of \(\mathbf{P L}\) are taken as the localized orbitals. While Cholesky orbitals are usually not as localized as, for example, PM or Boys orbitals, the procedure is non-iterative and produces unique result, except possibly for the impact of degeneracies. Cholesky orbitals can serve as an excellent guess for iterative localization procedures.

A summary of the functionality of the lo module is given below:

Method

optimization

cost function

PBC

ref

(meta-) L"owdin

No

yes

[7][8]

Natural atomic orbitals

No

gamma

[9]

Intrinsic atomic orbitals

No

yes

[10]

Cholesky orbitals

No

no

[6]

Boys

yes

dipole

no

[1]

Pipek-Mezey

yes

local charges

gamma

[3]

Intrinsic bond orbitals

yes

IAO charges

gamma

[10]

Edmiston-Ruedenberg

yes

coulomb integral

gamma

[5]

For example, to obtain the natural atomic orbital coefficients (in terms of the original atomic orbitals):

import numpy
from pyscf import gto, scf, lo

x = .63
mol = gto.M(atom=[['C', (0, 0, 0)],
                  ['H', (x ,  x,  x)],
                  ['H', (-x, -x,  x)],
                  ['H', (-x,  x, -x)],
                  ['H', ( x, -x, -x)]],
            basis='ccpvtz')
mf = scf.RHF(mol).run()

# C matrix stores the AO to localized orbital coefficients
C = lo.orth_ao(mf, 'nao')

References

1(1,2)

J. M. Foster and S. F. Boys. Canonical Configurational Interaction Procedure. Rev. Mod. Phys., 32(2):300–302, 1960.

2

N. Marzari and D. Vanderbilt. Maximally Localized Generalized Wannier Functions for Composite Energy Bands. Phys. Rev. B, 56(20):12847–12865, 1997.

3(1,2)

János Pipek and Paul G. Mezey. A fast intrinsic localization procedure applicable for ab-initio and semiempirical linear combination of atomic orbital wave functions. J. Chem. Phys., 90(9):4916, 1998.

4

Susi Lehtola and Hannes Jónsson. Pipek–mezey orbital localization using various partial charge estimates. Journal of chemical theory and computation, 10(2):642–649, 2014.

5(1,2)

Clyde Edmiston and Klaus Ruedenberg. Localized Atomic and Molecular Orbitals. Rev. Mod. Phys., 35(3):457–464, 1963.

6(1,2)

Francesco Aquilante, Thomas Bondo Pedersen, Alfredo Sánchez de Merás, and Henrik Koch. Fast noniterative orbital localization for large molecules. J. Chem. Phys., 125(17):174101, 2006. doi:10.1063/1.2360264.

7

Per-Olov Löwdin. On the Non-Orthogonality Problem Connected with the Use of Atomic Wave Functions in the Theory of Molecules and Crystals. J. Chem. Phys., 18(3):365, 1950.

8

Qiming Sun and Garnet Kin-Lic Chan. Exact and optimal quantum mechanics/molecular mechanics boundaries. J. Chem. Theory Comput., 10:3784, 2014.

9

Alan E. Reed, Robert B. Weinstock, and Frank Weinhold. Natural population analysis. J. Chem. Phys., 83(2):735–746, 1985.

10(1,2)

Gerald Knizia. Intrinsic Atomic Orbitals: An Unbiased Bridge between Quantum Theory and Chemical Concepts. J. Chem. Theory Comput., 9(11):4834–4843, 2013.