# Mixed Quantum-Classical simulation techniques¶

In a molecular mechanics (MM) force field, the influence of electrons is expressed by empirical parameters that are assigned on the basis of experimental data, or on the basis of results from high-level quantum chemistry calculations. These are valid for the ground state of a given covalent structure, and the MM approximation is usually sufficiently accurate for ground-state processes in which the overall connectivity between the atoms in the system remains unchanged. However, for processes in which the connectivity does change, such as chemical reactions, or processes that involve multiple electronic states, such as photochemical conversions, electrons can no longer be ignored, and a quantum mechanical description is required for at least those parts of the system in which the reaction takes place.

One approach to the simulation of chemical reactions in solution, or in enzymes, is to use a combination of quantum mechanics (QM) and molecular mechanics (MM). The reacting parts of the system are treated quantum mechanically, with the remainder being modeled using the force field. The current version of GROMACS provides interfaces to several popular Quantum Chemistry packages (MOPAC 150, GAMESS-UK 151, Gaussian 152 and CPMD 153).

GROMACS interactions between the two subsystems are either handled as described by Field et al. 154 or within the ONIOM approach by Morokuma and coworkers 155, 156.

## Overview¶

Two approaches for describing the interactions between the QM and MM subsystems are supported in this version:

1. Electronic Embedding The electrostatic interactions between the electrons of the QM region and the MM atoms and between the QM nuclei and the MM atoms are included in the Hamiltonian for the QM subsystem:

$H^{QM/MM} = H^{QM}_e-\sum_i^n\sum_J^M\frac{e^2Q_J}{4\pi\epsilon_0r_{iJ}}+\sum_A^N\sum_J^M\frac{e^2Z_AQ_J}{e\pi\epsilon_0R_{AJ}},$

where $$n$$ and $$N$$ are the number of electrons and nuclei in the QM region, respectively, and $$M$$ is the number of charged MM atoms. The first term on the right hand side is the original electronic Hamiltonian of an isolated QM system. The first of the double sums is the total electrostatic interaction between the QM electrons and the MM atoms. The total electrostatic interaction of the QM nuclei with the MM atoms is given by the second double sum. Bonded interactions between QM and MM atoms are described at the MM level by the appropriate force-field terms. Chemical bonds that connect the two subsystems are capped by a hydrogen atom to complete the valence of the QM region. The force on this atom, which is present in the QM region only, is distributed over the two atoms of the bond. The cap atom is usually referred to as a link atom.

2. ONIOM In the ONIOM approach, the energy and gradients are first evaluated for the isolated QM subsystem at the desired level of ab initio theory. Subsequently, the energy and gradients of the total system, including the QM region, are computed using the molecular mechanics force field and added to the energy and gradients calculated for the isolated QM subsystem. Finally, in order to correct for counting the interactions inside the QM region twice, a molecular mechanics calculation is performed on the isolated QM subsystem and the energy and gradients are subtracted. This leads to the following expression for the total QM/MM energy (and gradients likewise):

$E_{tot} = E_{I}^{QM} +E_{I+II}^{MM}-E_{I}^{MM},$

where the subscripts I and II refer to the QM and MM subsystems, respectively. The superscripts indicate at what level of theory the energies are computed. The ONIOM scheme has the advantage that it is not restricted to a two-layer QM/MM description, but can easily handle more than two layers, with each layer described at a different level of theory.

## Usage¶

QMMM is currently not supported in GROMACS.