XPol is an approximate, fragment-based molecular orbital method that was
developed as a “next-generation” force
field.^{283, 1015, 1017, 1016} The basic idea of the method
is to treat a molecular liquid, solid, or cluster as a collection of fragments,
where each fragment is a molecule. Intra-molecular interactions are treated
with a self-consistent field method (Hartree-Fock or DFT), but each fragment
is embedded in a field of point charges that represent electrostatic
interactions with the other fragments. These charges are updated
self-consistently by collapsing each fragment’s electron density onto a set of
atom-centered point charges, using charge analysis procedures (Mulliken,
Löwdin, or ChElPG, for example; see Section 11.2.1). This
approach incorporates many-body polarization, at a cost that scales linearly
with the number of fragments, but neglects the anti-symmetry requirement of the
total electronic wave function. As a result, intermolecular exchange-repulsion
is neglected, as is dispersion since the latter is an electron correlation
effect. As such, the XPol treatment of polarization must be augmented with
empirical, Lennard–Jones-type intermolecular potentials in order to obtain
meaningful optimized geometries, vibrational frequencies or dynamics.

The XPol method is based upon an *ansatz* in which the super-system wave
function is written as a direct product of fragment wave functions,

$$|\mathrm{\Psi}\u27e9=\prod _{A}^{{N}_{\mathrm{frag}}}|{\mathrm{\Psi}}_{A}\u27e9,$$ | (13.29) |

where ${N}_{\mathrm{frag}}$ is the number of fragments. We assume
here that the fragments are molecules and that covalent bonds remain intact.
The fragment wave functions are anti-symmetric with respect to exchange of
electrons within a fragment, but not to exchange between fragments. For
closed-shell fragments described by Hartree-Fock theory, the XPol total energy
is^{1017, 414}

$${E}_{\mathrm{XPol}}=\sum _{A}\left[2\sum _{a}{\mathbf{c}}_{a}^{\u2020}\left({\mathbf{h}}^{A}+{\mathbf{J}}^{A}-\frac{1}{2}{\mathbf{K}}^{A}\right){\mathbf{c}}_{a}+{E}_{\mathrm{nuc}}^{A}\right]+{E}_{\mathrm{embed}}.$$ | (13.30) |

The term in square brackets is the ordinary Hartree-Fock energy expression for fragment $A$. Thus, ${\mathbf{c}}_{a}$ is a vector of occupied MO expansion coefficients (in the AO basis) for the occupied MO $a\in A$; ${\mathbf{h}}^{A}$ consists of the one-electron integrals; and ${\mathbf{J}}^{A}$ and ${\mathbf{K}}^{A}$ are the Coulomb and exchange matrices, respectively, constructed from the density matrix for fragment $A$. The additional terms in Eq. (13.30),

$${E}_{\mathrm{embed}}=\frac{1}{2}\sum _{A}\sum _{B\ne A}\sum _{J\in B}\left(-2\sum _{a}{\mathbf{c}}_{a}^{\u2020}{\mathbf{I}}_{J}{\mathbf{c}}_{a}+\sum _{I\in A}{L}_{IJ}\right){q}_{J},$$ | (13.31) |

arise from the electrostatic embedding. The matrix ${\mathbf{I}}_{J}$ is defined by its AO matrix elements,

$${\left({\mathbf{I}}_{J}\right)}_{\mu \nu}=\u27e8\mu \left|\frac{1}{\left|\overrightarrow{r}-{\overrightarrow{R}}_{J}\right|}\right|\nu \u27e9,$$ | (13.32) |

and ${L}_{IJ}$ is given by

$${L}_{IJ}=\frac{{Z}_{I}}{\left|{\overrightarrow{R}}_{I}-{\overrightarrow{R}}_{J}\right|}.$$ | (13.33) |

According to Eqs. (13.30) and (13.31), each fragment is embedded in the electrostatic potential arising from a set of point charges, $\{{q}_{J}\}$, on all of the other fragments; the factor of $1/2$ in Eq. (13.31) avoids double-counting. Exchange interactions between fragments are ignored, and the electrostatic interactions between fragments are approximated by interactions between the charge density of one fragment and point charges on the other fragments.

Crucially, the vectors ${\mathbf{c}}_{a}$ are constructed within the ALMO
*ansatz*,^{464} so that MOs for each fragment are represented
in terms of only those AOs that are centered on atoms in the same fragment.
This choice affords a method whose cost grows linearly with respect to
${N}_{\mathrm{frag}}$, and where basis set superposition error is
excluded by construction. In compact basis sets, the ALMO *ansatz*
excludes inter- fragment charge transfer as well.

The original XPol method of Xie *et al.*^{1015, 1017, 1016} uses
Mulliken charges for the embedding charges ${q}_{J}$ in Eq. (13.31),
though other charge schemes could be envisaged. In non-minimal basis sets,
the use of Mulliken charges is beset by severe convergence
problems,^{414} and Q-Chem’s implementation of XPol offers the
alternative of using either Löwdin charges or ChElPG
charges,^{118} the latter being derived from the electrostatic
potential as discussed in Section 11.2.1. The ChElPG charges
are found to be stable and robust, albeit with a somewhat larger computational
cost as compared to Mulliken or Löwdin
charges.^{414, 369} An algorithm to compute ChElPG
charges using atom-centered Lebedev grids rather than traditional Cartesian
grids is available (see Section 11.2.1),^{387}
which uses far fewer grid points and thus can significantly improve the
performance for the XPol/ChElPG method, where these charges must be iteratively updated.