10.4 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^{98, 99} minimizes the radial extent of the localized orbitals, i.e., ${\sum }_i⟨ii|{|{𝐫}_1-{𝐫}_2|}^2|ii⟩$, and although is relatively fast, does not separate $\sigma$ and $\pi$ orbitals, leading to two ‘banana-orbitals’ in the case of a double bond.⁷⁵⁸ Pipek-Mezey localized orbitals⁷⁵⁸ maximize the locality of Mulliken populations, and are of a similar cost to Boys localized orbitals, but maintain $\sigma -\pi$ separation. Edmiston-Ruedenberg localized orbitals²⁴⁰ 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 due to the work of Dr. Joe Subotnik,⁹²⁹ and later Prof. Young-Min Rhee and Westin Kurlancheek with Prof. Martin Head-Gordon at Berkeley, this has been reduced to asymptotic cubic-scaling cost (with respect to the number of occupied orbitals), via the resolution of identity approximation.

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