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10.3 Orbital Analysis

10.3.2 Orbital Localization

(April 13, 2024)

10.3.2.1 Overview

The concept of localized orbitals has already been visited in this manual in the context of perfect-pairing and methods. As the SCF energy is independent of the partitioning of the electron density into orbitals, there is considerable flexibility as to how this may be done. The canonical picture, where the orbitals are eigenfunctions of the Fock operator is useful in determining reactivity, for, through Koopmans’ theorem, the orbital energy eigenvalues give information about the corresponding ionization energies and electron affinities. As a consequence, the HOMO and LUMO are very informative as to the reactive sites of a molecule. In addition, in small molecules, the canonical orbitals lead us to the chemical description of σ and π bonds.

In large molecules, however, the canonical orbitals are often very delocalized, and so information about chemical bonding is not readily available from them. Here, orbital localization techniques can be of great value in visualizing the bonding, as localized orbitals often correspond to the chemically intuitive orbitals which might be expected.

Q-Chem has three post-SCF localization methods available. These can be performed separately over both occupied and virtual spaces. The localization scheme attributed to Boys 135 Boys S. F.
Rev. Mod. Phys.
(1960), 32, pp. 296.
Link
minimizes the radial extent of the localized orbitals, i.e., the second moment iii||𝐫1-𝐫2|2|ii, and although is relatively fast, does not separate σ and π orbitals, leading to two “banana-orbitals” in the case of a double bond. 981 Pipek J., Mezey P. G.
J. Chem. Phys.
(1989), 90, pp. 4916.
Link
Pipek-Mezey localized orbitals 981 Pipek J., Mezey P. G.
J. Chem. Phys.
(1989), 90, pp. 4916.
Link
maximize the locality of Mulliken populations, and are of a similar cost to Boys localized orbitals, but maintain σ-π separation. Edmiston-Ruedenberg localized orbitals 323 Edmiston C., Ruedenberg K.
Rev. Mod. Phys.
(1963), 35, pp. 457.
Link
maximize the self-repulsion of the orbitals, iii|1r|ii. This is more computationally expensive to calculate as it requires a two-electron property to be evaluated, but can be reduced to cubic-scaling cost (with respect to the number of occupied orbitals), via the resolution of identity approximation. 1189 Subotnik J. E. et al.
J. Chem. Phys.
(2004), 121, pp. 9220.
Link

BOYSCALC

BOYSCALC
       Specifies how Boys localized orbitals are to be calculated
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       0 Do not perform any Boys localization. 1 Localize core and valence together. 2 Do separate localizations on core and valence.
RECOMMENDATION:
       None

ERCALC

ERCALC
       Specifies how Edmiston-Ruedenberg localized orbitals are to be calculated
TYPE:
       INTEGER
DEFAULT:
       06000
OPTIONS:
       aabcd aa specifies the convergence threshold. If aa>3, the threshold is set to 10-aa. The default is 6. If aa=1, the calculation is aborted after the guess, allowing Pipek-Mezey orbitals to be extracted. b specifies the guess: 0 Boys localized orbitals. This is the default 1 Pipek-Mezey localized orbitals. c specifies restart options (if restarting from an ER calculation): 0 No restart. This is the default 1 Read in MOs from last ER calculation. 2 Read in MOs and RI integrals from last ER calculation. d specifies how to treat core orbitals 0 Do not perform ER localization. This is the default. 1 Localize core and valence together. 2 Do separate localizations on core and valence. 3 Localize only the valence electrons. 4 Use the $localize section.
RECOMMENDATION:
       ERCALC 1 will usually suffice, which uses threshold 10-6.

The $localize section may be used to specify orbitals subject to ER localization if require. It contains a list of the orbitals to include in the localization. These may span multiple lines. If the user wishes to specify separate beta orbitals to localize, include a zero before listing the beta orbitals, which acts as a separator, e.g.,

$localize
   2 3 4 0
   2 3 4 5 6
$end

10.3.2.2 Virtual Orbital Localization

Virtual orbitals can be advantageous to be localized in many scenarios. One scenario where this is useful is generalized valence bond (GVB) methods, where each bonding orbital is paired with its antibonding orbital through Sano procedure. Currently this is done in GVBMAN when PP or CCVB is run. An improved guess has been proposed that has been shown to converge faster. 37 Aldossary A., Head-Gordon M.
J. Chem. Phys.
(2022), 157, pp. 094102.
Link
The new subroutine is a stand-alone version that can generate these antibonding orbitals and exit without initiating a GVB calculation. It can do Boys, Pipek-Mezey, or Edmiston-Rudenberg localization for the occupied space depending on GVB_LOCAL = 1, 2, or 3, respectively, while 0 performs it on the canonical orbitals. The subroutine also prints out each occupied orbital’s Mulliken charge, delocalization measure, and variance, in which it automatically detects the bonding orbitals and generates an antibonding guess for each. A population analysis based on this effective minimal basis can also be done using EDA_POP_ANAL = 1. The number of bonds can be enforced by taking the highest GVB_N_PAIRS specified, with no guarantee of them being bonding, i.e. they can be core or lone pairs. This is currently implemented for restricted and restricted Open-shell spin symmetries; work on the unrestricted case is underway.

ANTIBOND

ANTIBOND
       Triggers Antibond subroutine to generate antibonding orbitals after a converged SCF
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       0 Does not localize the virtual space. 1 Localizes the virtual space, one antibonding for every bond. 2,3 Fill the virtual space with antibonding orbitals-like guesses. 4 Does Frozen Natural Orbitals and leaves them on scratch for future jobs or visualization.
RECOMMENDATION:
       None

DOMODSANO

DOMODSANO
       Specifies whether to do modified Sano or the original one
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       0 Does original Sano procedure (similar to GVBMAN). 1 Does an improved Sano procedure that’s more localized. 2 Does another variation of Sano.
RECOMMENDATION:
       1 is always better