Classical subsystem

To begin with, we compare the stepsizes used in the simulations (Fig. 3). As pointed out before, it seems to be unreasonable to equip the Pickaback scheme with a stepsize control, because, as we indeed observe in Fig. 3, the stepsize never increases above a given level. This level depends solely on the eigenvalues of the quantum Hamiltonian. When analyzing the other integrators, we observe that the stepsize control just adapts to the dynamical behavior of the classical subsystem. The internal (quantal) dynamics of the Hydrogen-Chlorine subsystem does not lead to stepsize reductions. 

Smaller values of the activation free energy due to (i) the distortion of the shape of the free energy surfaces and (ii) the increase of the resonance splitting, AJF, of the potential free energies for the classical subsystem due to the increased overlapping of the wave functions of the quantum particles. 

Let us assume that all the nuclear subsystems may be separated into several subsystems (R, q9 Q, s,...) characterized by different times of motion, for example, low-frequency vibrations of the polarization or the density of the medium (q), intramolecular vibrations, etc. Let (r) be the fastest classical subsystem, for which the concept of the transition probability per unit time Wlf(q, Q,s) at fixed values of the coordinates of slower subsystems q, Q, s) may be introduced. 

Changes in the degrees of freedom in a reaction can be classified in two ways (1) classical over the barrier for frequencies o) such that hot) < kBT and (2) quantum mechanical through the barrier for two > kBT. In ETR, only the electron may move by (1) all the rest move by (2). Thus, the activated complex is generated by thermal fluctuations of all subsystems (solvent plus reactants) for which two < kBT. Within the activated complex, the electron may penetrate the barrier with a transmission coefficient determined entirely by the overlap of the wavefunctions of the quantum subsystems, while the activation energy is determined entirely by the motion in the classical subsystem. 

In the system of quantum dipoles, dipole and momentum variables have to be replaced by the quantum operators, and quantum statistical mechanics has to be applied. Now, the kinetic energy given by Eq. 9 does affect the thermal average of quantity that depends on dipole variables, due to non-commutivity of dipole and momentiun operators. According to the Pl-QMC method, a quantum system of N dipoles can be approximated by P coupled classical subsystems of N dipoles, where P is the Trotter munber and this approximation becomes exact in the limit P oo. Each quantiun dipole vector is replaced by a cychc chain of P classical dipole vectors, or beads , i.e., - fii -I-. .., iii p, = Hi,I. This classical system of N coupled chains... 

The next and necessary step is to account for the interactions between the quantum subsystem and the classical subsystem. This is achieved by the utilization of a classical expression of the interactions between charges and/or induced charges and a van der Waals term [45-61] and we are able to represent the coupling to the quantum mechanical Hamiltonian by interaction operators. These interaction operators enable us to include effectively these operators in the quantum mechanical equations for calculating the MCSCF electronic wavefunction along with the response of the MCSCF wavefunction to externally applied time-dependent electromagnetic fields when the molecule is exposed to a structured environment [14,45-56,58-60,62,67,69-74],... 

The operator, HqM/cm, for the interactions between the quantum mechanical and the classical subsystems is composed of three parts ... 

We represent the electrostatic interactions between the electrons and nuclei in the quantum subsystem and the charges in the classical subsystem as... 

The van der Waals interactions between the quantum and classical subsystems are given by the term Hviw and it is defined as... 

We have the following for the operator representing the polarization interactions between the charges in the quantum mechanical subsystem and the induced dipoles in the classical subsystem... 

Following, we determine the effects of the interactions between the quantum and classical subsystems on the optimization procedures of the MCSCF electronic wavefunction by evaluating the contributions of the quantum-classical interactions to the gradient and Hessian terms in the above equation. 

In the following we give a short summary of the applications of multiconfigurational self-consistent field classical mechanics response method based on the work by Poulsen and co-workers [15-17], The investigated QM/CM system is represented as a sample of 128 H20 molecules, one of which is selected as the quantum mechanical subsystem while the remaining 127 H20 molecules represent the classical subsystem. 

The MCSCF/CM response method provide procedures for obtaining frequency-dependent molecular properties when investigating a molecule coupled to a structured environment and the basis is achieved by treating the quantum mechanical subsystem on a quantum mechanical level and the structured environment as a classical subsystem described by a molecular mechanics force field. The important interactions between the two subsystems are included directly in the optimized wave function. 

The above comprises the derivation of the expression for the PES of the complex system which is not only free from the necessity to recalculate the wave function of the classical subsystem in each point, but formally not requiring any wave function of the M-system at all, since the result is expressed in terms of the generalized observables - one-electron Green s functions and the polarization propagator of the free M-system. Reality is of course more harsh as the necessary quantities must be known for a system we know too little about, except the initial assumption that its orbitals do exist. Section 3.5 will be devoted to reducing this uncertainty. 

The interactions between the quantum mechanical and the classical subsystems are given by the term Eqm/mm, in Eq. (13-1). The last term in Eq. (13-1), Emm, provides the energy of the classically treated part of the total system and is represented by molecular mechanics. 

The electrostatic interactions between the partial charges in the classical subsystem and the electrons and the nuclei in the QM system are described by this term. We let the term Efff° denote the interactions between the MM partial charges and the QM nuclei and the index s runs over all the sites in the MM system. Typically, the sites in the MM system are located at the MM atoms having the partial charge qs and positioned according to the position vector Rs. 

This section contains the background for the combination of density functional theory and molecular mechanics. Following the basic philosophy of quantum mechan-ics/molecular mechanics approaches we partition the total system into at least two parts which can be treated simultaneously. The quantum mechanical subsystem is described using DFT and the classical subsystem is given by molecular mechanics. Based on the QM/MM approach we have that the total energy of the system is... 

For the optimization of the coupled cluster wave function in the presence of the classical subsystem we write the CC/MM Lagrangian as [24]... 

The next developments of the FC approach were in papers by (R. A.) Marcus,41 49 and a later series from the Soviet Union. About the same time Hush50 introduced other concepts, to be discussed below. The early work of Marcus41 considered the Inner Sphere to be invariant with frozen bonds and vibrational coordinates up to the time of electron transfer. The classical subsystem for ion activation has its ground state floating on a continuum of classical levels, i.e., vibrational-librational-hindered translational motions of solvent molecules in thermal equilibrium with the ground state of the frozen solvated ion. 

The interaction operator, Wpoi, that describes the interactions between the quantum mechanical and classical subsystems is given by... 

Concluding this section, we note that the interaction operator describing the interactions between the molecular compound (the quantum mechanical subsystem) and the structured environment (the classical subsystem) is effectively written as a product of two one-electron operators. 

We make use of the following notation for the quantum and classical subsystems (i) i and r/, are the index and coordinates for the electrons, respectively, (ii) m and Rm, denote, respectively, the index and coordinates for the nuclei in the quan-... 

---