Quantum subsystems

As ab initio MD for all valence electrons [27] is not feasible for very large systems, QM calculations of an embedded quantum subsystem axe required. Since reviews of the various approaches that rely on the Born-Oppenheimer approximation and that are now in use or in development, are available (see Field [87], Merz ]88], Aqvist and Warshel [89], and Bakowies and Thiel [90] and references therein), only some summarizing opinions will be given here. 

In the sequel, we assume that the quantum subsystem has been truncated to a finite-dimensional system by an appropriate spatial discretization and a corresponding representation of the wave function by a complex-valued vector Ip C. The discretized quantum operators T, V and H are denoted by T e V(q) E and H q) e respectively. In the following... 

Here we suggest a different approach that propagates the system using multiple step-sizes, i.e., few steps with step-size At are taken in the slow classical part whereas many smaller steps with step-size 5t are taken in the highly oscillatory quantum subsystem (see, for example, [19, 4] for symplectic multiple-time-stepping methods in the context of classical molecular dynamics). Therefore, we consider a splitting of the Hamiltonian H = Hi +H2 in the following way ... 

Unlike the simplest outer-sphere electron transfer reactions where the electrons are the only quantum subsystem and only two types of transitions are possible (adiabatic and nonadiabatic ones), the situation for proton transfer reactions is more complicated. Three types of transitions may be considered here5 ... 

The physical mechanism of entirely nonadiabatic and partially adiabatic transitions is as follows. Due to the fluctuation of the medium polarization, the matching of the zeroth-order energies of the quantum subsystem (electrons and proton) of the initial and final states occurs. In this transitional configuration, q, the subbarrier transition of the proton from the initial potential well to the final one takes place followed by the relaxation of the polarization to the final equilibrium configuration. 

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. 

The technical problem was to develop an adequate form of the intersubsystem junction for the case when the quantum subsystem reduces to the d-shell. The problem here is to keep clear advantage of the TFT taking into account the ligands electronic structure over the CFT reproducing it economically in the otherwise MM calculation. 

This will lead to two separate energy scales at the expense of losing the physical time information of the quantum subsystem dynamics. The according Lagrangian (Eq. 1) reads ... 

Figure 13.8 A 25-atom quantum subsystem embedded in an 8863-atom classical system to model the catalytic step in the conversion of D-2-phosphoglycerate to phosphoenolpyruvate by enolase. What factors influence the choice of where to set the boundary between the QM and MM regions Alhambra and co-workers found, using variational transition-state theory with a frozen MM region that was selected from a classical trajectory so as to make the reaction barrier and thermochemistry reasonable, that the breaking and making bond lengths were 1.75 and 1.12 A, respectively, for H, but 1.57 and 1.26 A, respectively, for D...

This equation gives the dynamics of the quantum-classical system in terms of phase space variables (R, P) for the bath and the Wigner transform variables (r,p) for the quantum subsystem. This equation cannot be simulated easily but can be used when a representation in a discrete basis is not appropriate. It is easy to recover a classical description of the entire system by expanding the potential energy terms in a Taylor series to linear order in r. Such classical approximations, in conjunction with quantum equilibrium sampling, are often used to estimate quantum correlation functions and expectation values. Classical evolution in this full Wigner representation is exact for harmonic systems since the Taylor expansion truncates. 

In many cases, in order to compute the dynamics of condensed phase systems, one invokes a basis representation for the quantum degrees of freedom in the system. Typically, one computes the dynamics of these systems in order to obtain quantities of interest, such as an average value, A(t) = Tr [Ap(t)], or a correlation function, as will be discussed below. Since such averages are basis independent one may project Eq. (8) onto any convenient basis. This is in principle a nice feature, and one that is often exploited to aid in calculations. However, it is important to note that the basis onto which one chooses to project the QCLE has important implications on how one goes about solving the resulting equations of motion. Ultimately the time-dependent average value of an observable is expressed as a trace over quantum subsystem... 

The surface-hopping trajectories obtained in the adiabatic representation of the QCLE contain nonadiabatic transitions between potential surfaces including both single adiabatic potential surfaces and the mean of two adiabatic surfaces. This picture is qualitatively different from surface-hopping schemes [2,56] which make the ansatz that classical coordinates follow some trajectory, R(t), while the quantum subsystem wave function, expanded in the adiabatic basis, is evolved according to the time dependent Schrodinger equation. The potential surfaces that the classical trajectories evolve along correspond to one of the adiabatic surfaces used in the expansion of the subsystem wavefunction, while the subsystem evolution is carried out coherently and may develop into linear combinations of these states. In such schemes, the environment does not experience the force associated with the true quantum state of the subsystem and decoherence by the environment is not automatically taken into account. Nonetheless, these methods have provided com-... 

The difficulty in simulating the full quantum dynamics of large many-body systems has stimulated the development of mixed quantum-classical dynamical schemes. In such approaches, the quantum system of interest is partitioned into two subsystems, which we term the quantum subsystem, and quantum bath. Approximations to the full quantum dynamics are then made such that... 

One noteworthy feature of Eqs. (2) and (4) is that they provide an exact quantum description for an arbitrary quantum subsystem bilinearly coupled to a quantum harmonic bath. Other aspects of this equation have been discussed previously in the literature [2,5]. 

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],... 

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

The two first terms form the time-independent Hamiltonian of the quantum subsystem where H0 is the Hamiltonian of molecular subsystem in vacuum and the operator WqM/cm represents the coupling between the molecular system and the structured environment. 

The operator V t) describes how the externally applied time-dependent electromagnetic field interacts with the quantum subsystem and it is represented by... 

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