EFP calculations

EFP energy terms

LibEFP can compute four inter-fragment energy terms:

  • electrostatic

  • polarization

  • dispersion

  • exchange-repulsion

Electrostatic term

Electrostatic energy is computed as a combination of charge-charge, charge-dipole, charge-quadrupole, charge-octupole, dipole-dipole, dipole-quadrupole, and quadrupole-quadrupole contributions.

The following parts of the EFP potential are used for electrostatic energy calculations:

Out of those, only COORDINATES section is mandatory as it determines fragment internal coordinate frame. All electrostatic terms (and corresponding parameters) are optional.

Charge-penetration screening

Two options exist for accounting for charge-penetration contribution to electrostatic energy, controlled by elec_damp keyword (see Short-range damping parameters):

  • exponential screening (“smearing”) of charges. This is achieved by invoking screening parameters defined in SCREEN2 section of the .efp potential (see SCREEN and SCREEN2). The charge-penetration energy is not printed separately but included in the electrostatic energy.

  • overlap-based screening. This is a separate energy term derived assuming that localized orbitals can be modeled as spherical gaussions (the same approximation is used in the exchange-repulsion term). This calculation will utilize exchange-repulsion parameters (FOCK MATRIX ELEMENTS, PROJECTION WAVEFUNCTION, PROJECTION BASIS SET, LMO CENTROIDS). This overlap-based charge-penetration energy is printed as a separate energy term.

Detailed description of damping functions and their benchmarks are published in Damping functions for electrostatic term and Short-range damping functions papers.

Polarization term

Polarization energy is computed using distributed anisotropic dipole polarizabilities. Induced dipoles, originating at the polarizability points, are converged until self-consistency. The default procedure is to solve for induced dipoles iteratively; the direct diagonalization of the induced dipole matrix is implemented but not parallelized, making its applicability limited to systems with a few thousands polarizability points (see Polarization solver). Detailed description of the EFP polarization term can be found in the first EFP paper (1996) and gradients of polarization energy paper.

The relevant sections of the EFP potential are:

POLARIZABLE POINTS groups provides coordinates and values of the polarizability tensors. Other sections specify positions and values of electrostatic multipoles that are used to compute static electric field on polarizability points.

Polarization energies are screened at short range with the Tang-Toennies (or gaussian-type) damping functions described in the short-range damping functions paper. A value of the damping parameter is controlled by an optional POLAB keyword; smaller values provide stronger screening of polarization energies which might be necessary for fragments with large multiple moments (charged or strongly polar species) or large polarizabilities (e.g., large conjugated/aromatic molecules).

Dispersion term

Dispersion energy term captures the London interaction between the molecules. Formally, it can be expanded in series of (1/R) operator as

\(E_{disp} = \frac{C_6}{R^6} + \frac{C_8}{R^8} + \frac{C_{10}}{R^{10}} + ....\)

In the case of distributed approach where dispersin contributions are computed as a sum of contributions due to individual parts of a molecules, the odd terms \(\frac{C_7}{R^7}, \frac{C_9}{R^9}\) etc are also non-zero.

LibEFP describes dispersion energy as the leading \(R^{-6}\) term using dipole dymamic polarizability tensors \(\alpha\) precomputed for each fragment:

\(E^{disp} = -\frac{4}{3} \sum_{p \in A, q \in B, B \notin A} \frac{C_6^{pq}}{R_{pq}^6}\)

\(C_6^{pq} = \int_0^{\infty} d\nu \bar{\alpha}^p(i\nu) \bar{\alpha}^q(i\nu)\)

The short-range behavior of the dispersion energies can be improved with damping functions of two types (see Short-range damping functions paper for detail), controlled by disp_damp keyword (described here Short-range damping parameters):

The relevant sections of the EFP potential are:

DYNAMIC POLARIZABLE POINTS group section provides coordinates and values of anisotropic dynamic polarizability tensors for computing dispersion energy.

Exchange-repulsion term

Exchange-repulsion term accounts for the contributions due to antisymmetry of the wave function of the fragments. It is modelled by on-the-fly computing inter-fragment kinetic and overlap integrals. The exchange-repulsion term is the most expansive part of typical EFP calculations. However, for larger systems, applying a shorter long-distance cutoff xr_cutoff can save some computational cost.

The relevant sections of the EFP potential are:

PROJECTION BASIS SET provides details of the basis set used for calculation of the exchange repulsion energy, MULTIPLICITY contains information on the multiplicity of the fragment (LibEFP works only on fragments with multiplicity 1), PROJECTION WAVEFUNCTION provides the localized wave function of the fragment, while FOCK MATRIX ELEMENTS and LMO CENTROIDS contain information regarding the elements of the Fock matrix of the fragment in the localized basis, and the coordinates of the localized molecular orbital, respectively.