Chapter 2. Theoretical background
2.8 QM/MM methods
The idea behind quantum mechanics / molecular mechanics methods (QM/MM) is to retain the atomic degrees of freedom of the real molecular system one wants to model, by replacing most of the full quantum mechanical description with molecular mechanics (an approximation to quantum mechanics). The quantum mechanical part (usually in the center of the system) thus describes the main region of interest while molecular mechanics acts as an approximation to the environment surrounding the quantum mechanical part.
Almost all QM/MM methods can be described by the additive equation:80
EQM/MM = EQM(I) + EMM(II) + EQM-‐MM(I , II)
(64) ∆Gelstat=�ψ|H0− e 2φ rxn|ψ�+ e 2 � k Zkφrxnk − �ψ0|H0|ψ0�
Where EQM/MM is the total energy of the system, EQM(I) is the QM energy of the QM
region (I), EMM is the MM energy of the MM region (II) and EQM-‐MM(I , II) is the QM-‐MM
interaction term between the two regions.
The EQM(I) term is simply a normal quantum mechanics calculated energy using any
suitable method. The only prerequisite is that the QM method must be able to perform energy and gradient calculations in the presence of external point charges when using the electrostatic embedding scheme (see later).
The EMM energy expression consists of both bonded (bonds, angles and dihedrals) and
nonbonded (Lennard-‐Jones potential and the Coulomb potential of point charges) terms:
(65) where the constants kX, d0, θ0, δ, ε, σ, qA, qB are fitted so that the force field reproduces
conformational energies, interaction energies etc. of molecular systems for which the force field is intended. Several parameterised force fields are available to describe proteins, for example the CHARMM force field81,82 which is used in this work.
It is the EQM-‐MM(I , II) term that essentially distinguishes different QM/MM methods
from one another (apart from the use of different force fields) as it takes care of the coupling terms (bonded, van der Waals (vdW) and electrostatic interactions) between the different regions:
EQM-‐MM(I , II) = EbQM-‐MM + EvdWQM-‐MM + EelQM-‐MM
(66) The bonded and vdW terms of EQM-‐MM(I,II) are described at the MM level. The vdW
interaction is usually described by a Lennard-‐Jones potential, necessitating Lennard-‐ Jones parameters for all QM atoms. In theory, every atom of the QM region interacts with every atom of the MM region, but in practice only the atoms closest to the boundary contribute significantly.
The electrostatic QM-‐MM interaction, which is the most important term, can be described by several different methods, the models can be classified by their increasing complexity, as mechanical embedding, electrostatic embedding and polarised embedding. In the mechanical embedding scheme the electrostatic
interations between QM atoms and MM atoms is handled at the MM level where the MM charge model (usually rigid atomic point charges) is used to calculate the interaction between QM and MM atoms where point charges for all QM atoms must be defined. Mechanical embedding is most often used in a subtractive QM/MM approach where the EMM term is calculated for the whole system (I+II) right from the
beginning. The main problem with mechanical embedding is that the QM-‐based electron density is simplified into a simple model of point charges which is additionally not influenced directly by the charges of the MM region (usually).
The electrostatic embedding scheme is a natural extension to mechanical embedding. The electrostatic interaction is simply calculated at the same time as the EQM(I) term
by incorporating the point charges of the MM region as an electron point-‐charge term
EM M = � bonds kd(d−d0)2+ � angles kθ(θ−θ0)2+ � dihedrals kφ[1 +cos(nφ+δ)] + � nonbonded �AB[( σAB rAB) 12 −(σAB rAB) 6)] + 1 4π�0 qAqB rAB
and a nucleus point-‐charge into the Hamiltonian of the QM calculation of the QM region. The nucleus point-‐charge term is straightforward and the electron point-‐ charge term is simply calculated in the same way as the electron-‐nucleus term is calculated. The quantum mechanical electron density is thus influenced by the surroundings directly and can adapt to any changes occurring in the MM region.
A polarised embedding scheme takes electrostatic embedding one step further by using a MM charge model that can be polarised as well (by the QM electron density), sometimes involving completely mutual polarisation in which case a self-‐consistent cycle is required at each QM/MM energy evaluation. Several different polarised embedding approaches have been suggested: polarised point dipoles scheme where assigned atomic polarisabilities induce point dipoles, Drude oscillator/shell/charge-‐ on-‐spring models where an additional opposite-‐sign point charge is attached to the MM atom connected by a harmonic spring creating a dipole, fluctuating charges and more.80
Electrostatic embedding and polarised embedding QM/MM can also be looked at in the general context of effective embedding potentials. The QM system is embedded by an embedding potential, just like the effective potentials in the single electron
HF/Kohn-‐Sham equations. In electrostatic/polarised embedding QM/MM, the embedding potential is (usually) a simple Coulomb point charge potential (the point charge approximating the multi-‐particle atom), an approximation to the real smeared out Coulomb multi-‐particle term of the surroundings. Continuum solvation models can also be seen as embedding potentials. A higher-‐order multipole potential goes a step further towards the real potential but still neglects exchange-‐repulsion effects that can in principle be included as well in an embedding potential. The most complicated embedding potentials are those involving whole electron densities, usually frozen, as in the frozen density embedding methods.83-‐86
A specific problem remains that concerns the division of the system into QM and MM regions. Covalent bonds often end up being cut by the QM-‐MM boundary, especially for systems such as proteins. As the dangling bond of the QM region needs to be saturated, one of the simplest solutions is to include link atoms (usually H) as part of the QM region in the QM energy calculation. Since the added link atoms constitute an additional energy term, it should be corrected for, in practice though, this is rarely done. However, since the link atom will be very close to the first MM atom at the boundary, a strong artificial electrostatic interaction will be present. One approach to solve this problem is called charge-‐shifting which removes the charge from the MM atom and divides it among the nearest bonded MM atoms.87,88
Other methods to deal with the QM-‐MM boundary are boundary-‐atom schemes where the MM atom involved in the cut is transformed into a special boundary atom which mimics both the QM interaction and the MM interaction, and localised-‐orbital schemes where hybrid orbitals are placed at the boundary (saturating the dangling bond) and frozen so that they are not part of the SCF iterations.80
QM/MM calculations can nowadays be performed using many different programs, often these are quantum chemistry programs that have incorporated some molecular mechanics functionality or the other way around.
Chemshell, on the other hand, is a computational chemistry program which is neither.88,89 It uses a modular approach that links together different external
programs through a programmable interface based on the Tool Command Language (Tcl). The main speciality of the program is in hybrid QM/MM simulations where in principle any quantum chemistry program can be linked to any molecular mechanics program to perform QM/MM calculations where the Chemshell program is
responsible for the geometry optimisations/molecular dynamics simulations, the QM/MM coupling and the data management. The program allows one to do geometry optimisation and molecular dynamics simulations on small to large chemical systems using a variety of different quantum chemistry programs and molecular mechanics programs through pre-‐programmed interfaces.