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Orbital Localization

The canonical molecular orbitals from an SCF calculation are typically delocalized over the whole molecule. TeraChem can instead rotate the orbitals within a chosen energy block into spatially localized orbitals — compact core, lone-pair, and bond orbitals that map onto familiar chemical concepts — without changing the total wavefunction or energy. Three localization criteria are available: Pipek–Mezey, Foster–Boys, and Knizia's intrinsic bond orbitals (IBO). (This is the same localization machinery used by F-SAPT; the related IAO partial charges are documented on the Charges page.)

Quick start

Add localization yes and pick a loctype in an ordinary RHF calculation:

methanol_ibo.in
basis         cc-pvdz
method        hf
coordinates   methanol.xyz
charge        0
spinmult      1
run           energy
gpus          1

localization  yes
loctype       ibo      # pm | fb | ibo  (and iao for charges only)
locblock      occ
end

This performs the localization (reporting convergence and, for IBO, the per-orbital atomic-charge matrix) and writes the localized orbitals to a molden file in the scratch directory.

Localization schemes (loctype)

Pipek–Mezey (pm, default)

Pipek–Mezey localization1 rotates the orbitals to maximize the sum of squared atomic populations,

\[ \mathcal{L}_{\mathrm{PM}} = \sum_{\bar a}\sum_{A} \big(Q_{A\bar a}\big)^2 , \]

where \(Q_{A\bar a}\) is the population (Mulliken- or Löwdin-type) of localized orbital \(\bar a\) on atom \(A\). Maximizing this concentrates each orbital on as few atoms as possible. Its distinguishing feature is that it preserves the \(\sigma/\pi\) separation: a double bond comes out as a separate \(\sigma\) and \(\pi\) orbital, and the scheme cleanly produces lone pairs and classical two-center bonds — which is why it is the default.

Foster–Boys (fb)

Foster–Boys localization2 instead minimizes the total spatial spread of the orbitals — equivalently, it maximizes the sum of squared distances between their charge centroids,

\[ \mathcal{L}_{\mathrm{FB}} = \sum_{\bar a < \bar b} \big|\langle \bar a|\mathbf{r}|\bar a\rangle - \langle \bar b|\mathbf{r}|\bar b\rangle\big|^2 . \]

This yields maximally compact, well-separated orbitals, but (unlike Pipek–Mezey) it mixes \(\sigma\) and \(\pi\): multiple bonds emerge as equivalent, bent "banana" bonds. Foster–Boys is inexpensive and basis-set robust and is a good choice when equivalent bonds are desired.

Intrinsic bond orbitals (ibo)

Intrinsic bond orbitals3 are a Pipek–Mezey-type localization in which the atomic populations are taken from the intrinsic atomic orbital (IAO) charges (see Charges) rather than from Mulliken/Löwdin populations, making the result essentially basis-set independent. The occupied orbitals are rotated to maximize

\[ \mathcal{L}_{\mathrm{IBO}} = \sum_{\bar a}\sum_{X}\big(Q_{X\bar a}\big)^{p}, \qquad p = 4, \]

where \(Q_{X\bar a}\) is the IAO charge of localized orbital \(\bar a\) on atom \(X\). The optimization uses \(2\times2\) Jacobi rotations, and core and valence orbitals are localized separately. The per-orbital atomic charges \(Q_{X\bar a}\) (which sum to 1 over atoms for each orbital) are printed, and the localized orbitals are written to molden format.

Projection (MinAO) basis

The IAO construction underlying IBO projects onto a minimal cc-pvdz-minao basis, shipped with TeraChem at $TeraChem/basis/cc-pvdz-minao (independent of the main basis). It is a normal basis file and may also be used as a basis set in its own right.

Summary

loctype Method Notes
pm (default) Pipek–Mezey maximizes atomic populations; preserves \(\sigma/\pi\)
fb Foster–Boys minimizes orbital spread; mixes \(\sigma/\pi\) (banana bonds)
ibo Knizia intrinsic bond orbitals IAO-based Pipek–Mezey; basis-set robust
iao Knizia IAO charges only no orbital rotation — see Charges

Summary of relevant keywords

Keyword Type Default Description
localization bool no Perform orbital localization / IAO analysis
loctype string pm Localization method: pm, fb, ibo, or iao
locblock string occ Orbital block to localize: occ, virt, closed, active, ca, ov, all
locmaxiter int 1000 Maximum localization iterations
iao_dump bool no Dump the IAO matrix \(A\) and IBO coefficients to the scratch directory

  1. J. Pipek and P. G. Mezey, A fast intrinsic localization procedure applicable for ab initio and semiempirical linear combination of atomic orbital wave functions, J. Chem. Phys. 90, 4916 (1989). ↩

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

  3. G. Knizia, Intrinsic Atomic Orbitals: An Unbiased Bridge between Quantum Theory and Chemical Concepts, J. Chem. Theory Comput. 9, 4834 (2013). ↩