Hybrid QM/MM procedures

A. L. Tchougreeff, A. M. Tokmachev. Physical principles of constructing hybrid QM/MM procedures. In J. Maruani, R. Lefebvre, E. Brandas, editors. Advanced Topics in Theoretical Chemical Physics, volume 12 of Progress in Theoretical Chemistry and Physics, pages 207-246. Kluwer, Dordrecht, 2003. 

The approach with the partitioning of the system into a QM and a classical molecular mechanical (MM) part, thus usually termed hybrid QM/MM procedure, provides a reasonable reduction of the computational effort by restricting the time-consuming QM calculation of forces to the most relevant part of the liquid system. The main error sources in this approach are a too small choice of the QM region, an inadequate level of theory for the QM calculation, the choice of suitable potentials for the MM part of the system, and smooth transitions of particles between QM and MM region. In conventional QM/MM procedures, the whole system is first evaluated at MM level and then corrected by the QM data. This means that classical potential functions (with all their problems and difficulty of construction) are needed for all components of the system. A recently developed methodology can reduce the need for such potentials to the solvent only, as will be outlined below. 

In the frame of the target hybrid QM/MM procedure, only the electronic structure of the R-system is calculated explicitly. For this reason, we consider its effective Hamiltonian eq. (1.235) in more detail. It contains the operator terms coming from (1) the Coulomb interaction of the effective charges in the M-system with electrons in the R-system 5VM and (2) from the resonance interaction of the R- and M-systems. 

To overcome these limitations, the hybrid QM-MM potential can employ ad initio or density function methods in the quantum region. Both of these methods can ensure a higher quantitative accuracy, and the density function methods offer a computaitonally less expensive procedure for including electron correlation [5]. Several groups have reported the development of QM-MM programs that employ ab initio [8,10,13,16] or density functional methods [10,41-43]. 

The GEPOL procedure we have mentioned becomes very expensive and leads to a too large number of tesserae when applied to solutes containing a large number of atoms. We started a couple of years ago the development of an alternative model, mostly addressed to large molecules (i.e. from 500 atoms upwards) but also applicable to smaller solutes [60]. The main field of application is not addressed to full QM ab initio calculations, but rather to molecular mechanic (MM) or hybrid QM/MM description of the solute [61] which we are now developing. 

It should be noted that the hybrid quantum/classical schemes apply not only for determination of geometries, energies, and reaction mechanisms. The Monte Carlo [67, 68] and molecular dynamics (MD) [69-72] simulations are quite popular as frameworks for which various QM/MM procedures serve as subroutines . Before employing hybrid schemes the large-scale MD simulations were performed only with low-level approximations for force fields. The use of hybrid schemes extends significantly the scope of their application, improve precision of the results that allows to improve the understanding of statistical properties and dynamical processes in liquids and biopolymers. 

It is interesting to compare the possibilities and errors of different hybrid QM/MM schemes. The careful examination and comparison of link atom and LSCF techniques was performed in Ref. [128] using the CHARMM force field [114] and the AMI method [143] as a quantum chemical procedure. In the case of the link atom procedure two options were used QQ - the link atom does not interact with the MM subsystem and HQ - link atom interacts with all MM atoms. The main conclusion of this consideration is that the LSCF and the link atom schemes are of similar quality. The error in the proton affinity determination induced by these schemes is several kcal/mol. It is noteworthy that all the schemes work rather badly in description of conformational properties of n-butane. The large charge on the MM atoms in the proximity of the QM subsystem (especially on the boundary atom) cause significant errors in the proton affinity estimates for all methods (especially, in the case of the LSCF approach where the error can be of tens of kcal/mol). This is not surprising since the stability and transferability of intrabond one- and two-electron density matrix elements Eq. (19) is broken here. It proves that the simple electrostatic model is not well appropriate for these schemes and that a detailed analysis of the... 

An early application of the hybrid QM/MM method was the investigation of ground and excited state potential surfaces of conjugated molecules. The procedure is based on a formal separation of the a and n electrons, with the a framework of the... 

Now that the MM induced dipoles [ps] are influenced by the QM electric field, while the wavefiinction vP in the QM region depends on j, an iterative procedure must be used in the determination of the MM induced dipoles and QM wavefunction to ensure the convergence of the total energy of the system. It may be noted that the use of equation (14) implies a mean field approximation. This procedure leads to a significant increase in the computational time. As a result, there has been only a limited number of applications reported in the literature, which include explicit treatment of the MM polarization in hybrid QM/MM calculations. ... 

Hartree-Fock, DFT or CCSD levels. Because they can reproduce such quantities, APMM procedures should account for an accurate description of the interactions including polarization cooperative effects and charge transfer. They should also enable the reproduction of local electrostatic properties such as dipole moments an also facilitate hybrid Quantum Mechanical/Molecular Mechanical (QM/MM) embeddings. 

The description of the electronic structure of the complex molecular system given by the system eq. (1.246) is perfectly sufficient when it goes about the hybrid QM/QM methods, when both the parts of the complex system are described by some QM methods. In the case of the hybrid methods in a narrow sense i.e. of the QM/MM methods, further refinements are necessary. The problem is that the description provided by eq. (1.246) suffers from the need to calculate the expectation values in these expressions over the wave function i.e. over the solution of the self-consistency equations eq. (1.246) in the presence of the R-system. This result does not seem to be particularly attractive since the functions < > Y are not known and are not supposed to be calculated in the frame of the MM procedure. Thus the theory must be reformulated in a spirit of the theory of intermolecular interactions [67] and to express necessary quantities in terms of the observable characteristics of free parts of the complex system. 

Biological systems can be treated at various different levels within a Car-Parrinello approach. One possibility is to use intelligently designed cluster models of the active site. Currently, systems of typically a few himdred atoms can be treated at the full quantum level. In addition, a static external field that captures the electrostatic field of the surrounding protein can be introduced. A common procedure is to parameterize the external electrostatic field in terms of point charges from empirical protein force fields. The most comprehensive approach for the treatment of biological systems within a Car-Parrinello framework are mixed quantum/classical QM/MM simulations. In these hybrid simulations, the reactive part of the system is treated within a standard Car-Parrinello scheme, whereas the surroimding protein is described with an empirically derived force field. In this way, electrostatic as well as steric and mechanic effects of the environment can be taken explicitly into account. 

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