The LocalOrbitals procedure generates a set of spatially localized orbitals from a set of canonical molecular orbitals (MOs).

Local orbitals offer an alternative, spatially localized rotation of the canonical MOs, which can be particularly useful for visualizing chemical bonding, improving the convergence and cost of correlated methods, and selecting active spaces in active-space methods like CI-CASSCF (ActiveSpaceCI or ActiveSpaceSCF) and V2RDM-CASSCF (Variational2RDM).   

Two arguments, molecule and mos, are required where  molecule defines the atom or molecule while  mos is a Matrix of canonical molecular orbitals coefficients returned as the entry of the output table with the key mo_coeff from one of the electronic structure methods such as 'HartreeFock', 'DensityFunctional', 'RDMFunctional', 'ActiveSpaceCI', 'ActiveSpaceSCF', 'CoupledCluster', 'FullCI', 'MP2', 'Parametric2RDM', 'ContractedSchrodinger', or 'Variational2RDM'.

The localization string (optional) -- selects the localization technique: "Boys", "PM", and "ER".  Boys Localization ("Boys") creates orbitals by keeping electron density as close together as possible in space; Pipek-Mezey Localization ("PM") generates orbitals that resemble atomic orbitals as much as possible, and Edmiston-Ruedenberg Localization ("ER") produces orbitals that minimize electron repulsion.

The mo_indices list of integers (optional) -- denotes the MOs to be localized. If not provided, the command localizes all MOs.

The command also supports options to describe the molecular calculation that produced the mos such as basis, charge, spin, symmetry, ghost, and unit.  Note that it is critical to provide these keywords if you are not using their default values; for example, if mos was not produced with the default "sto-3g" basis set, you must include the basis keyword with the correct basis set.

On the Windows operating system the LocalOrbitals command requires the installation of Microsoft's Windows Subsystem for Linux (WSL).  For Windows 10 (version 2004 and higher) and Windows 11 you can install the WSL by opening the Command Prompt in administrator mode and entering the command: wsl --install -d Ubuntu  For additional details, please refer to: https://learn.microsoft.com/en-us/windows/wsl/install

S. F. Boys, Reviews of Modern Physics 32 (2): 296–299 (1960). "Construction of molecular orbitals to be minimally variant for changes from one molecule to another"

C. Edmiston and K. Ruedenberg, Reviews of Modern Physics. 35 (3): 457–465 (1963). "Localized Atomic and Molecular Orbitals"

J. Pipek and P. G. Mezey, The Journal of Chemical Physics. 90 (9): 4916 (1989). "A fast intrinsic localization procedure applicable for ab initio and semiempirical linear combination of atomic orbital wave functions"

$\mathrm{with}\left(\mathrm{QuantumChemistry}\right)\:$

$\mathrm{mol} ≔ \mathrm{MolecularGeometry}\left(''hydroxyurea''\right)\;$

$\mathrm{mol}≔\left[\left[''O''\,1.88790000\,\mathrm{-0.00950000}\,\mathrm{-0.00270000}\right]\,\left[''O''\,\mathrm{-0.48550000}\,\mathrm{-1.30440000}\,\mathrm{-0.00030000}\right]\,\left[''N''\,0.67610000\,0.69460000\,0.00260000\right]\,\left[''N''\,\mathrm{-1.61350000}\,0.69390000\,\mathrm{-0.00290000}\right]\,\left[''C''\,\mathrm{-0.46490000}\,\mathrm{-0.07450000}\,0.00330000\right]\,\left[''H''\,0.71730000\,1.68970000\,0.18780000\right]\,\left[''H''\,\mathrm{-1.57330000}\,1.70750000\,\mathrm{-0.00260000}\right]\,\left[''H''\,\mathrm{-2.53340000}\,0.26600000\,\mathrm{-0.00670000}\right]\,\left[''H''\,2.08350000\,\mathrm{-0.00760000}\,\mathrm{-0.95500000}\right]\right]$

$\mathrm{data\_dft} ≔ \mathrm{DensityFunctional}\left(\mathrm{mol}\, \mathrm{basis}\=''pc0''\right)\;$

$\mathrm{local\_mo\_coeff} ≔ \mathrm{LocalOrbitals}\left(\mathrm{mol}\, \mathrm{data\_dft}\left[\mathrm{mo\_coeff}\right]\, \mathrm{basis}\=''pc0''\,\mathrm{localization}\=''ER''\right)\;$

$\mathrm{data\_dft}\left[\mathrm{mo\_coeff}\right] ≔ \mathrm{local\_mo\_coeff}\;$

$\mathrm{DensityPlot3D}\left(\mathrm{mol}\,\mathrm{data\_dft}\,\mathrm{basis}\=''pc0''\,\mathrm{orbitalindex}\=20\right)\;$