10.3.2 Orbital Localization

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 $\sigma$ and $\pi$ 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 ${\sum }_i⟨ii|{|{𝐫}_1-{𝐫}_2|}^2|ii⟩$, and although is relatively fast, does not separate $\sigma$ and $\pi$ 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 $\sigma -\pi$ 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, ${\sum }_i⟨ii|\frac{1}{r}|ii⟩$. 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

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

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.