The hybrid QM/MM (quantum mechanics/molecular mechanics) approach is a molecular simulation method that combines the strengths of ab initio QM calculations (accuracy) and MM (speed) approaches, thus allowing for the study of chemical processes in solution and in proteins. The QM/MM approach was introduced in the 1976 paper of Warshel and Levitt.^[1] They, along with Martin Karplus, won the 2013 Nobel Prize in Chemistry for "the development of multiscale models for complex chemical systems".^[2][3]

Efficiency

An important advantage of QM/MM methods is their efficiency. The cost of doing classical molecular mechanics (MM) simulations in the most straightforward case scales as O(N²), where N is the number of atoms in the system. This is mainly due to electrostatic interactions term (every particle interacts with everything else). However, use of cutoff radius, periodic pair-list updates and more recently the variations of the particle mesh Ewald (PME) method has reduced this to between O(N) to O(N²). In other words, if a system with twice as many atoms is simulated then it would take between twice to four times as much computing power. On the other hand, the simplest ab initio calculations formally scale as O(N³) or worse (restricted Hartree–Fock calculations have been suggested to scale ~O(N^2.7)). Here in the ab initio calculations, N stands for the number of basis functions rather than the number of atoms. Each atom has at least as many basis functions as is the number of electrons. To overcome the limitation, a small part of the system that is of major interest is treated quantum-mechanically (for instance, the active site of an enzyme) and the remaining system is treated classically.^[4][5]

Calculating the energy of the combined system

The energy of the combined system may be calculated in two different ways. The simplest is referred to as the 'subtractive scheme' which was proposed by Maseras and Morokuma in 1995. In the subtractive scheme the energy of the entire system is calculated using a molecular mechanics force field, then the energy of the QM system is added (calculated using a QM method), finally the MM energy of the QM system is subtracted.

${\displaystyle E=E^{QM}(QM)+E^{MM}(QM+MM)-E^{MM}(QM)}$

In this equation ${\displaystyle E^{MM}(QM)}$ would refer to the energy of the QM region as calculated using molecular mechanics. In this scheme, the interaction between the two regions will only be considered at a MM level of theory.

In practice, a more widely used approach is a more accurate, additive method. The equation for this consists of three terms: ${\displaystyle {\begin{aligned}E(QM/MM)&=\sum _{I'=1}^{MM}\left[\int \operatorname {d} \mathbf {r} {q_{I'}\rho (\mathbf {r} ) \over \left\vert \mathbf {R} _{I'}-\mathbf {r} \right\vert }+\sum _{I=1}^{QM}{\frac {q_{I'}q_{I}}{\left\vert \mathbf {R} _{I'}-\mathbf {R} _{I}\right\vert }}\right]\\&+\sum _{\text{non-bonded pairs}}\left({\frac {A_{II'}}{R_{II'}^{12}}}-{\frac {B_{II'}}{R_{II'}^{6}}}\right)+\sum _{\text{bonds}}k_{r}(R_{II'}-r_{0})^{2}\\&+\sum _{\text{angles}}k_{\theta }(\theta -\theta _{0})^{2}+\sum _{\text{torsions}}\sum _{n}k_{\phi ,n}[\cos(n\phi +\delta _{n})+1]\end{aligned}}}$

The index ${\displaystyle I}$ labels the nuclei in the QM region whereas ${\displaystyle I'}$ labels the MM nuclei. The first two terms represent the interaction between the total charge density (due to electrons and cores) in the QM region and classical charges of the MM region. The third term accounts for dispersion interactions across the QM/MM boundary. Any covalent bond-stretching potentials that cross the boundary are accounted for by the fourth term. The final two terms account for the energy across the boundary that arises from bending covalent bonds and torsional potentials. At least one of the atoms in the angles ${\displaystyle \theta }$ or ${\displaystyle \phi }$ will be a QM atom with the others being MM atoms.^{[6]: 422–3}

The electrostatic QM-MM interaction

Electrostatic interactions between the QM and MM region may be considered at different levels of sophistication. These methods can be classified as either mechanical embedding, electrostatic embedding or polarized embedding.

Problems involved with QM/MM

Even though QM/MM methods are often very efficient, they are still rather tricky to handle. A researcher has to limit the regions (atomic sites) which are simulated by QM, however methods have been developed that allow particles to move between the QM and MM region.^[8] Moving the limitation borders can both affect the results and the time computing the results. The way the QM and MM systems are coupled can differ substantially depending on the arrangement of particles in the system and their deviations from equilibrium positions in time. Usually, limits are set at carbon-carbon bonds and avoided in regions that are associated with charged groups, since such an electronically variant limit can influence the quality of the model.^[9]

Covalent bonds across the QM-MM boundary

Directly connected atoms, where one is described by QM and the other by MM are referred to as Junction atoms. Having the boundary between the QM region and MM region pass through a covalent bond may prove problematic however this is sometimes unavoidable. When it does occur it is important that the bond of the QM atom be capped in order to prevent the appearance of bond cleavage in the QM system.^[9]

See also

• ONIOM: "our own n-layered integrated molecular orbital and molecular mechanics"
• List of quantum chemistry and solid state physics software
• List of software for molecular mechanics modeling

References