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. They, along with Martin Karplus, won the 2013 Nobel Prize in Chemistry for "the development of multiscale models for complex chemical systems".
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(N2.7)). 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.
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.
In this equation 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:
The index labels the nuclei in the QM region whereas 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 or will be a QM atom with the others being MM atoms.
Reducing the computational cost of calculating QM-MM interactions
Evaluating the charge-charge term in the QM/MM interaction equation given previously can be very computationally expensive (consider the number of evaluations required a system with 106 grid points for the electron density of the QM system and 104 MM atoms). A method by which this issue can be mitigated is to construct three concentric spheres around the QM region and evaluate which one of these spheres the MM atoms lie within. If the MM atoms reside within the innermost sphere their interactions with the QM system are treated as per the equation for . The MM charges that lie within the second sphere (but not the first) interact with the QM region by giving the QM nuclei constructed charges. These charges are determined by the RESP approach in an attempt to mimic electron density. Using this approach the changing charges on the QM nuclei during the course of a simulation are accounted for.
In the third outermost region the classical charges interact with the multipole moments of the quantum charge distribution. By calculating charge-charge interactions by using successively more approximate methods it is possible to obtain a very significant reduction in computational cost whilst not suffering a significant loss in accuracy.