Electrostatic interactions

Second order

Isotropic electrostatics

The isotropic electrostatic in a shell-resolved formulation is given by the parametrized Coulomb interaction between shellwise partial charges

\[E_\text{IES} = \frac12 \sum_{\text{A},\text{B}} \sum_{l,l'}^{s,p,d} q^{l}_\text{A} \gamma^{ll'}_\text{AB} q^{l'}_\text{B}\]

The interaction potential is parametrized by a Klopman–Ohno type potential in the xTB Hamiltonian or the γ-functional as used in the DFTB Hamiltonian.

Klopman–Ohno kernel

The interaction kernel for the Klopman–Ohno electrostatic is given by

\[\gamma^{ll'}_\text{AB} = \left( R_\text{AB}^g + f_\text{av}(\eta_A^l, \eta_B^{l'})^{-g} \right)^{-\frac1g}\]

where η:sub:A/B are the chemical hardness parameters of the respective shells and g is the exponent to manipulate the potential shape.

For three-dimensional periodic systems with \(g=2\), the kernel is evaluated using a generalized Ewald partition.[1] For a lattice translation \(\mathbf T\), the Klopman–Ohno kernel has the binomial expansion

\[\left(\lvert\mathbf R_{AB}+\mathbf T\rvert^2 +\eta_{Al,Bl'}^{-2}\right)^{-1/2} = \sum_{j=0}^{\infty}\binom{-1/2}{j} \frac{\eta_{Al,Bl'}^{-2j}} {\lvert\mathbf R_{AB}+\mathbf T\rvert^{2j+1}}.\]

Retaining the two long-range terms gives the exact partition

\[\begin{split}\begin{split} \gamma_{Al,Bl'}^{\text{PBC}} ={}& \sum_{\mathbf T}^{\prime} \left[ \left(r_{AB,\mathbf T}^2+\eta_{Al,Bl'}^{-2}\right)^{-1/2} -r_{AB,\mathbf T}^{-1} +\frac12\eta_{Al,Bl'}^{-2}r_{AB,\mathbf T}^{-3} \right] \\ &+S_1(\mathbf R_{AB}) -\frac12\eta_{Al,Bl'}^{-2}S_3(\mathbf R_{AB}), \end{split}\end{split}\]

where \(r_{AB,\mathbf T}=\lvert\mathbf R_{AB}+\mathbf T\rvert\) and the prime excludes \(r_{AB,\mathbf T}=0\). The residual in square brackets decays as \(r^{-5}\) and is summed in real space. With Ewald parameter \(\alpha=\sqrt{\pi}K\), the Coulomb lattice sum is

\[\begin{split}\begin{split} S_1(\mathbf R) ={}& \sum_{\mathbf T}^{\prime} \frac{\operatorname{erfc}(\alpha r_{\mathbf T})}{r_{\mathbf T}} +\frac{4\pi}{V}\sum_{\mathbf G\ne0} \frac{\exp[-G^2/(4\alpha^2)]}{G^2} \cos(\mathbf G\cdot\mathbf R) \\ &-\delta_{\mathbf R,0}\frac{2\alpha}{\sqrt\pi}, \end{split}\end{split}\]

and the cubic lattice sum is

\[\begin{split}\begin{split} S_3(\mathbf R) ={}& \sum_{\mathbf T}^{\prime}\left[ \frac{\operatorname{erfc}(\alpha r_{\mathbf T})}{r_{\mathbf T}^3} +\frac{2\alpha\exp(-\alpha^2r_{\mathbf T}^2)} {\sqrt\pi r_{\mathbf T}^2}\right] \\ &+\frac{2\pi}{V}\sum_{\mathbf G\ne0} E_1\!\left(\frac{G^2}{4\alpha^2}\right) \cos(\mathbf G\cdot\mathbf R) \\ &+\frac{4\pi}{V}\left[ \ln\!\left(\frac{\alpha}{\sqrt\pi}\right) +\frac12\left(\ln\pi-\psi\!\left(\frac32\right)\right) \right] -\delta_{\mathbf R,0}\frac{4\pi}{3} \left(\frac{\alpha}{\sqrt\pi}\right)^3. \end{split}\end{split}\]

Here \(V\) is the unit-cell volume, \(\mathbf G\) is a reciprocal lattice vector, \(E_1(x)=\Gamma(0,x)\) is the exponential integral, and \(\psi\) is the digamma function. The third line of \(S_3\) contains the \(\mathbf G=0\) contribution; it is required to make the result independent of \(\alpha\). The final terms in \(S_1\) and \(S_3\) remove the Gaussian self-interaction.

γ-functional kernel

The interaction kernel for the DFTB γ-functional is derived from the integral of two exponential densities

\[\begin{split}\begin{split} \gamma^{ll'}_\text{AB} = \frac1{R_\text{AB}} - \exp[-\tau_\text{A}R] \left( \frac{\tau_\text{B}^4\tau_\text{A}}{2(\tau_\text{A}^2-\tau_\text{B}^2)^2} - \frac{\tau_\text{B}^6\tau_\text{A} - 3\tau_\text{B}^4\tau_\text{A}^2} {(\tau_\text{A}^2-\tau_\text{B}^2)^3 R_\text{AB}} \right) \\ - \exp[-\tau_\text{B}R] \left( \frac{\tau_\text{A}^4\tau_\text{B}}{2(\tau_\text{B}^2-\tau_\text{A}^2)^2} - \frac{\tau_\text{A}^6\tau_\text{B} - 3\tau_\text{A}^4\tau_\text{B}^2} {(\tau_\text{B}^2-\tau_\text{A}^2)^3 R_\text{AB}} \right) \end{split}\end{split}\]

where τ:sub:A/B are scaled Hubbard parameters of the respective shells and R is the distance between the atomic sides.

Anisotropic electrostatics

The anisotropic electrostatic in an atom-resolved formulation is given by the multipole interactions between the different moments:

\[E_\text{AES} = \sum_{\text{A},\text{B}} \sum_{k}^{x,y,z} q_\text{A} \gamma^{k}_\text{AB} \mu^{k}_\text{B} + \frac12 \sum_{\text{A},\text{B}} \sum_{k,k'}^{x,y,z} \mu^{k}_\text{A} \gamma^{kk'}_\text{AB} \mu^{k'}_\text{B} + \sum_{\text{A},\text{B}} \sum_{k,k'}^{x,y,z} q_\text{A} \gamma^{kk'}_\text{AB} \theta^{kk'}_\text{B}\]

Third order

The isotropic third-order contributions are included as the trace of the on-site shell-resolved Hubbard derivatives.

\[E_\text{IXC} = \frac13 \sum_\text{A} \sum_{l} \Gamma^l_\text{A} (q^l_\text{A})^3\]

Literature

[1]

Alexander Buccheri, Rui Li, J. Emiliano Deustua, S. Mohamad Moosavi, Peter J. Bygrave, and Frederick R. Manby. Periodic GFN1-xTB tight binding: a generalized Ewald partitioning scheme for the Klopman–Ohno function. J. Chem. Theory Comput., 21(4):1615–1625, 2025. doi:10.1021/acs.jctc.4c01234.