A Hybrid Quantum Mechanical/Molecular Mechanical Approach

Hybrids are popular from the ancient time:

So, it was a logical step to combine Quantum Chemical and Molecular Mechanical methods.

Why Do We Need a Hybrid QM/MM Approach?

Quantum chemical methods are generally applicable and allow the calculation of ground and excited state properties (molecular energies and structures, energies and structures of transition states, atomic charges, reaction pathways etc.)

Molecular Mechanical methods are restricted to the classes of molecule it have been designed for and their success strongly depends on the careful calibration of a large number of parameters.

The main bottleneck of quantum chemical methods is that they are CPU and memory hungry.

For example, for a small peptide of 126 atoms one energy evaluation requires:

*Semi-empirical PM3 method

In general, CPU and memory requirements are:

where N is a number of atoms.

The development of the hybrid QM/MM approaches is guided by the general idea that large chemical systems may be partitioned into an electronically important region which requires a quantum chemical treatment and a remainder which only acts in a perturbative fashion and thus admits a classical description:

The Simplest Hybrid QM/MM Model

Hamiltonian for the molecular system in the Born-Oppenheimer approximation:

In the presence of the external charges we have two additional terms in the Hamiltonian:

Using this simple model we shall be able to calculate effect of external charges on our quantum chemical system:

Computed Molecular Dipole Moments (Debye Units) in the gas phase and in water (H2O molecules are treated by MM Method) using AM1 method (J. Gao, J.Comp.Chem. 18(1997), 1061)

Application: MODELING OF ADSORPTION PROPERTIES OF ZEOLITES

The main drawbacks of this simple QM/MM model are:

it is impossible to optimize the position of the QM part relative to the external charges because QM nuclei will collapse on the negatively charged external charges.

some MM atoms possess no charge and so would be invisible to the QM atoms

the van der Waals terms on the MM atoms often provide the only difference in the interactions of one atom type versus another, i.e. chloride and bromide ions both have unit negative charge and only differ in their van der Waals terms.

So, it is quite reasonable to attribute the van der Waals parameters (as it is in the MM method) to every QM atom and the Hamiltonian describing the interaction between the QM and MM atoms can have a form:

The van der Waals term models also electronic repulsion and dispersion interactions, which do not exist between QM and MM atoms because MM atoms possess no explicit electrons. Such form of the Hamiltonian was suggested for the first time by A. Warshel and M. Levitt (A. Warshel, M. Levitt // Theoretical Studies of Enzymic Reactions: Dielectric, Electrostatic and steric stabilization of the carbonium ion in the reaction of lysozyme. // J.Mol.Biol. 103(1976), 227-49)

Now we can construct a "real" hybrid QM/MM Hamiltonian:

A "standard" MM force field can be used to determine the MM energy. For example, AMBER-like force field has a form:

Choice of QM method

... is a compromise between computational efficiency and the desired chemical accuracy.

The main advantage of semiempirical QM methods is that their computational efficiency is orders of magnitude greater than either the density functional or ab initio methods.

The use of ab initio methods limits the types of QM/MM studies that may be conducted at a single (X-ray structure) geometry.
         J. Bajorath et al., Proc.Natl.Acad.Sci. USA, 88(1991), 6423.
         J. Bajorath et al., Proteins: Struct.Funct.Genet., 9(1991), 217.
         J. Bajorath et al., Proteins: Struct.Funct.Genet., 11(1991), 263.

In contrast, a number of QM/MM calculations have now been performed using semiempirical QM methods to compute enzymic reaction mechanisms, potentials of mean force, and solvation energies using minimization and Monte Carlo techniques.
        J. Gao and X. Xia, Science, 258(1992), 631
        V.V. Vasilyev, J.Mol.Struct. (Theochem), 304(1994), 129.
        M.A. Thomson, J.Am.Chem.Soc., 117(1995), 11341.

Calibration of the QM/MM potential

Crucial aspect is how the interaction between QM and MM parts is determined.

In choosing the appropriate form, it is required that the balance between attractive and repulsive forces must be preserved and the QM/MM interactions must be of the correct magnitude with respect to the separate QM and MM contributions

Parameterization

1) Modification of the one-electron terms arising from interaction of the electron cloud of the QM fragment with the point charge of an MM atom.
2) By varying the radii in the van der Waals terms.
3) By varying 1) + 2)

Calibration

1) By hand, to find the optimum values of the parameters by calculating interaction curves for charge/ion systems and comparing them with the MP2/6-311++G** ab initio results.
        M.J. Field, P.A. Bash, M. Karplus, J.Comp.Chem., 11(1990), 700-733.

2) Fitting calculated H-bond energies to experimental data on ion-molecular complexes in the gas phase.
            V.V. Vasilyev, A.A. Bliznyuk, A.A. Voityuk, Int.J.Quant.Chem. 44(1992), 897-930.

3) Optimizing van der Waals parameters on QM atoms to reproduce the 6-31G(d) interaction energies for H-bonded complexes in the gas phase.
            P.A. Bash, L. Lawrence, A.D. MacKerell, Jr., D. Levine, P. Hallstrom, PNAS USA, 93(1996), 3698-703.

4) Optimizing van der Waals parameters on QM atoms to reproduce the MP2/6-31G(dp) interaction energies for H-bonded complexes in the gas phase.
            J. Gao // Toward a molecular Orbital Derived Empirical Potential for Liquid Simulations // J.Phys.Chem. B 101(1997), 657-63

5) By varying the radii in the van der Waals terms to reproduce experimental free energies of solvation using MD simulations.
           P.L. Cummins, J.E. Gready, J.Comp.Chem., 18(1997), 1496-512.

Special Case: Dividing Covalent Bonds across the QM and MM Regions

1) Using a hybrid orbital on the frontier MM atom

A. Warshel, M. Levitt // Theoretical Studies of Enzymic Reactions: Dielectric, Electrostatic and steric stabilization of the carbonium ion in the reaction of lysozyme. // J.Mol.Biol. 103(1976), 227-249
V. Thery, D. Rinaldi, J.-L. Rivail, B. Maigret, G.G. Ferenczy, J J.Comp.Chem. 15(1995), 269

1) Using "link" atoms (hydrogens, halogens)

"Link" atoms are used to gracefully cap the electron density. (1) For QM region, this is hydrogen atom. Its sigma and epsilon are set to zero, i.e. it interacts with MM region only electrostatically. (2) Charge for "link" MM atom is set to zero to avoid double counting the electrostatic interactions. (3) WdV interaction between QM and MM atoms which form 1-2 and 1-3 "bonded" pairs is not calculated. (4) Bond stretching, angle bending, and torsion interactions between QM and MM regions are calculated as those in MM if 1-2, 1-2-3, or 1-2-3-4 terms contain at least one MM atom AND one QM atom

References

Reviews

J. Gao, Reviews in Comp.Chem., 7(1996), 119-185 - Review on QM/MM