QM subsystem

The QM/MM interactions (Eqm/mm) are taken to include bonded and non-bonded interactions. For the non-bonded interactions, the subsystems interact with each other through Lennard-Jones and point charge interaction potentials. When the electronic structure is determined for the QM subsystem, the charges in the MM subsystem are included as a collection of fixed point charges in an effective Hamiltonian, which describes the QM subsystem. That is, in the calculation of the QM subsystem we determine the contributions from the QM subsystem (Eqm) and the electrostatic contributions from the interaction between the QM and MM subsystems as explained by Zhang et al. [13],... 

Geometry optimizations are carried out by an iterative minimization procedure as described by Zhang et al. [13] In this procedure one iteration consists of a complete optimization of the QM subsystem, followed by a complete optimization of the MM subsystem. At each point the subsystem not being optimized is held fixed at the geometry obtained from the previous iteration QM/MM interactions are also included at each iteration. The iterations are continued until the geometries of both systems no longer change. 

When the MM subsystem is being optimized, or a molecular dynamics simulation is being carried out on the MM subsystem, the QM/MM electrostatic interactions are approximated with fixed point charges on the QM atoms which are fitted to reproduce the electrostatic potential (ESP) of the QM subsystem [37],... 

The path optimizations are carried out by an iterative optimization procedure [25]. In the case of enzyme systems, because of the large number of degrees of freedom, we partition them into a core set and an environmental set. The core set is small and contains all the degrees of freedom that are involved with the chemical steps of the reaction, while all the remaining degrees of freedom are included in the environmental set. In all the QM/MM calculations presented below, the core set is defined by the QM subsystem and the environmental set by the MM subsystem. 

Relative free energy changes along the reaction coordinate may be calculated via MD simulations. Subsequently, they may be combined with the QM total energies of the reacting QM subsystem to obtain the overall free energy profiles associated with the reaction steps in the following way... 

It is important to point out that Eq. (3-6) has the same form as Jorgensen s approach [14, 15]. However, this approach has two major differences. First, Xq is defined as the MEP obtained from the modified NEB procedure (Section 3.2.3) or from the CD method (Section 3.2.1), where the QM subsystem interacts with the enzyme environment. In the case of the QM-FE approach, the QM subsystem is isolated. 

Second, AEQM xgj ) is calculated as the difference between the QM subsystem energy computed form the ab initio calculation (Eqm(QM)), and the QM/MM electrostatic energy computed classically (Eeiectrostatics(QM/MM)) [13]. 

The calculation of the free energies of enzymes by this procedure provides several advantages. In this approach, the MEP is determined in the enzyme environment with a smooth connection between the QM and MM subsystems by means of the pseudobond QM/MM method [39], Also, the polarization effects of the enzyme environment on the QM subsystem have been included [13]. 

It is important to note that in this method, the dynamic fluctuations associated with the QM subsystem are assumed to be independent of the fluctuations from the MM subsystem. Also, in this method we assume that the contributions of the fluctuations of the QM subsystem to the total free energy are the same along the reaction coordinate. We have recently addressed these approximations by developing a novel reaction path potential method where the dynamics of the system are sampled by employing an analytical expression of the combined QM/MM PES along the MEP [40],... 

Table 3-1. Calculated potential and free energy differences for path B (in kcal/mol) between the determined structure and the reactant (ES complex), where Ait is the total HF potential energy difference, A Eqm refers to the QM energy difference between two QM subsystems. A 1 qm/MM is the free energy change in the QM/MM interaction, and A F = AEqm + A I qm/MM- Numbers without parentheses correspond to the present work and numbers in parentheses correspond to our previous determinations (path D) [33]...

In order to determine where the energy differences stem from, the effects of the individual residues on the stabilization(destabilization) of the QM subsystem were analyzed as explained in Ref. [33], These results are shown in Tables 3-3 and 3-4,... 

At each simulation step, the property or properties of interest are included in the ensemble average. For Table 13.2, the property is the evaluation of the dipole moment operator as an electronic expectation value over the QM subsystem. Thus, the QM/MM result for diis case is an MC ensemble expectation value of a quantum mechanical operator expectation value. 

Further elaboration of the hybrid models stipulated by the necessity to model chemical processes in polar solvents or in the protein environment of enzymes, or in oxide-based matrices of zeolites, requires the polarization of the QM subsystem by the charges residing on the MM atoms of the classically treated solvent, or protein, or oxide matrix. This polarization is described by renormalizing the one-electron part of the effective Hamiltonian for the QM subsystem ... 

The MM subsystem in its turn also affects the parameters of the QM subsystem as any geometry variation in the MM subsystem induces changes in pseudo- and quasirotation angles defining hybridization of the frontier atom. The corrections to the QM one-center Hamiltonian parameters (in the linear approximation) are ... 

The resonance integrals in the QM subsystem are also modified. In the respective DCFs the corrections can be expressed as ... 

In the QM part of the system, the variation of the bond orders can also take place. In variance with the pure SLG picture [11,12] used here as the QM method underlying the MM part of the system, the atoms in the QM part of the combined system may have off-diagonal elements of the one-electron density matrix between orbitals ascribed to the QM subsystem. The latter are obviously the (Coulson) bond orders for the QM part of the system. The corresponding contribution to the energy reads ... 

If these variations are taken into account in the calculations on the QM part of the complex system, the effect of the MM system on the parameters of the effective Hamiltonian for the QM part turns out to be taken into account in the first order. It should be stressed that changes in the hybridization of the frontier atom due to participation of one orbital in the QM subsystem are not taken into account in any of the existing QM/MM schemes. This effect is not very large, so the first-order correction for taking it into account seems to be adequate. 

Accordingly, the modifications to the KS operator are twofold (i) a static contribution through the static multipole moments (here charges) of the solvent molecules and (ii) a dynamical contribution which depends linearly on the electronic polarizability of the environment and also depends on the electronic density of the QM region. Due to the latter fact we need within each SCF iteration to update the DFT/MM part of the KS operator with the set of induced dipole moments determined from Eq. (13-29). We emphasize that it is the dynamical contribution that gives rise to polarization of the MM subsystem by the QM subsystem. 

---