# 12.12.1 Theory

XPol is an approximate, fragment-based molecular orbital method that was developed as a “next-generation” force field.279, 1074, 1076, 1075 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 10.2.1). 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,

 $|\Psi\rangle=\prod_{A}^{N_{\mathrm{frag}}}|\Psi_{\!A}\rangle,$ (12.39)

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 is1076, 422

 $E_{\rm XPol}=\sum_{A}\left[2\sum_{a}\mathbf{c}_{a}^{\dagger}\left(\mathbf{h}^{% A}+\mathbf{J}^{A}-\tfrac{1}{2}\mathbf{K}^{A}\right)\mathbf{c}_{a}+E_{\mathrm{% nuc}}^{A}\right]+E_{\mathrm{embed}}.$ (12.40)

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. (12.40),

 $E_{\mathrm{embed}}=\tfrac{1}{2}\sum_{A}\sum_{B\neq A}\sum_{J\in B}\left(-2\sum% _{a}\mathbf{c}_{a}^{\dagger}\mathbf{I}_{J}\mathbf{c}_{a}+\sum_{I\in A}L_{IJ}% \right)q_{J},$ (12.41)

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

 $\left(\mathbf{I}_{J}\right)_{\mu\nu}=\left\langle\mu\left|\frac{1}{\bigl{|}% \vec{r}-\vec{R}_{J}\bigr{|}}\right|\nu\right\rangle,$ (12.42)

and $L_{IJ}$ is given by

 $L_{IJ}=\frac{Z_{I}}{\bigl{|}\vec{R}_{I}-\vec{R}_{J}\bigr{|}}.$ (12.43)

According to Eqs. (12.40) and (12.41), 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. (12.41) 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,473 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.1074, 1076, 1075 uses Mulliken charges for the embedding charges $q_{J}$ in Eq. (12.41), though other charge schemes could be envisaged. In non-minimal basis sets, the use of Mulliken charges is beset by severe convergence problems,422 and Q-Chem’s implementation of XPol offers the alternative of using either Löwdin charges, Charge Model 5 (CM5) charges,634 or ChElPG charges,103 the latter being derived from the electrostatic potential as discussed in Section 10.2.1. The ChElPG charges are found to be stable and robust, albeit with a somewhat larger computational cost as compared to Mulliken or Löwdin charges.422, 375 An algorithm to compute ChElPG charges using atom-centered Lebedev grids rather than traditional Cartesian grids is available (see Section 10.2.1),394 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. A cost-effective and slightly more accurate alternative to the ChElPG charges are the CM5 charges.596 The CM5 charge derivatives are significantly cheaper to compute than those for ChElPG, and because XPol must iteratively update the charges the CM5 charges are considerably less expensive.

Researchers who use Q-Chem’s XPol code are asked to cite Refs. 422, 375.