Molecular system subsystem

This argument can be generalized to any number of subsystems and energy levels. For the case of a molecular system in a given electronic state, the factorization into translational, vibrational, and rotational contributions gives... 

One of the necessary conditions for a many-body description is the validity of the decomposition of the system under consideration on separate subsystems. In the case of very large collective effects we cannot separate the individual parts of the system and only the total energy of the system can be defined. However, in atomic systems the inner-shell electrons are to a great extent localized. Therefore, even in metals with strong collective valence-electron interactions, atoms (or ions) can be identified as individuals and we can define many-body interactions. The important role in this separation plays the validity for atom- molecular systems the adiabatic or the Born-Oppenheimer approximations which allow to describe the potential energy of an N-atom systeni as a functional of the positions of atomic nuclei. 

We do not consider here the case in which a nuclear magnetic subsystem or any other subsystem that has a limited number of quantum states may be considered as a thermodynamic system separate from the other parts of the total molecular system. 

Studies of the electronic structure of the P4 molecular system were performed in the Td symmetry in all charge states (for P both in the singlet state with total spin value 5 = 0 and in the triplet one with 5 = 1). To study any possible instability of the electronic system to the lowering of symmetry in the fixed Td configuration of the nuclear subsystem, the calculations were performed using the following three variants ... 

At the same time, as it is known, the molecular systems in the degenerated states are dynamically unstable with respect to symmetry distortions due to the Jahn-Teller effect [9]. Such kinds of distortions are expected both in the electronic and nuclear subsystems (vibronic instability). The Jahn-Teller effect destabilizes the tetrahedral geometry of the nuclear system, which results in the bond order s changes too. In particular, according to the theory of the Jahn-Teller effect [9] the tetrahedral molecular systems in the 2Ti(P ),lT2(Pi ) or 37 (P4 ) states are dynamically unstable to the e- and t2-type nuclear distortions, one of which is presented in Fig. 7.1 Evidently such a kind of distortion to the butterfly geometry... 

The Hamiltonians and the energy functionals for molecules interacting with a structured environment method are obtained by dividing a large system into two subsystems. One of these subsystems is the molecular system of interest and that part of the system is described by quantum mechanics. The other subsystem is not of principal interest and it is therefore treated by a much coarser method. Approaches along these lines have been presented within quantum chemistry [13,14,45,46-77] and molecular reaction dynamics [62,78-81],... 

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. 

Now we pass to the formal derivations of a hybrid method. We assume that the orbitals forming the basis for the entire molecular system may be ascribed either to the chemically active part of the molecular system (reactive or R-states) or to the chemically inactive rest of the system (medium or M-states). In the present context, the orbitals are not necessarily the basis AO, but any set of their orthonormal linear combinations thought to be distributed between the subsystems. The numbers of electrons in the R-system (chemically active subsystem) Nr and in the M-system (chemically inactive subsystem) NM = Ne — Nr, respectively, are good quantum numbers at least in the low energy range. We also assume that the orbital basis in both the systems is formed by the strictly local orbitals proposed in [59]. The strictly local orbitals are orthonormalized linear combinations of the AOs centered on a single atom. In that sense they are the classical hybrid orbitals (HO) ... 

In Chapter 3 we introduce the formal construction and testing of an intermediate procedure bridging QM and MM procedures. This will be a mechanistic treatment, derived from the quantum description of the molecular system. Then this technique will be used to define the one-electron states of the frontier atoms - the key elements of the intersubsystem border/junction the shapes of the one-electron states at the frontier atoms, their electronic densities and the response of either subsystem to the variables characterizing each subsystem. 

A molecular system consists of electrons and nuclei. Their position vectors are denoted hereafter as rel and qa, respectively. The potential energy function of the whole system is V(rel, qa). For simplicity, we skip the dependence of the interactions on the spins of the particles. The nuclei, due to their larger mass, are usually treated as classical point-like objects. This is the basis for the so called Bom-Oppenheimer approximation to the Schroedinger equation. From the mathematical point of view, the qnuc variables of the Schroedinger equation for the electrons become the parameters. The quantum subsystem is described by the many-dimensional electron wave function rel q ). 

In order to make the fragment decoupling continuous in this generalized description, the input probabilities p i), p°(i ) have to be replaced by the separate distributions reflecting the actual participation of zth AO in the chemical bonds (communications) of the molecule. Therefore, they both have to be related explicitly to the zth row in the conditional probability matrix P(b a) = P(j i), which reflects all communications (bonds) between this orbital input and all orbital outputs / (columns in P(b a)). This link must generate the separate subsystem probabilities p°, when the fragment becomes decoupled from the rest of the molecular system, a - a0, when P(b aa) - P(ba aa)8a/p], where P(ba aa) = [P(a a). Indeed, for the decoupled subsystem a0 = (a, a, ...) only the internal communications of the corresponding block of the molecular conditional probabilities P(ba aa) = P(a a) are allowed. They also characterize the internal conditional probabilities in a0 since... 

The properties of the topologically defined atoms and their temporal changes are identified within a general formulation of subspace quantum mechanics. It is shown that the quantum mechanical partitioning of a system into subsystems coincides with the topological partitioning both are defined by the set of zero flux surfaces in Vp(r). Consequently the total energy and any other property of a molecular system are partitioned into additive atomic contributions. 

Coupling of quantum mechanical molecular subsystems with larger classically treated subsystems has traditionally involved electronic structure models describing molecules embedded in a dielectric medium and this is a research area that has expanded tremendously over the last three decades [2-36]. Most of this work has involved electronic structure methods that have been based on uncorrelated electronic structure methods [2-12,15-19]. Accurate description of the electronic structure of molecular systems requires that the correlated electronic motion in the molecule is incorporated and therefore a number of correlated electronic structure methods have been developed such as the second order Moller-Plesset (MP2) [28,30,90,91], the multiconfigurational self-consistent reaction field (MCSCRF) [13,20] and the coupled-cluster self-consistent reaction field (CCSCRF) method [36]. 

We consider correlated electronic structure methods for molecular systems interacting with a structured environment and the utilization of response theory makes it possible to calculate molecular properties of the molecular subsystem coupled to an aerosol particle and the molecular properties could be... 

For both methods, we describe the interactions between the quantum subsystem and the classical subsystem as interactions between charges and/or induced charges/dipoles and a van der Waals term [2-18]. The coupling between the quantum subsystem and the classical subsystem is introduced into the quantum mechanical Hamiltonian by finding effective interaction operators for the interactions between the two subsystems. This provides an effective Schrodinger equation for determining the MCSCF electronic wave function of the molecular system exposed to a classical environment, a structured environment, such as an aerosol particle. 

We will in this section introduce the representation of the electronic wave function of the molecular system and describe how we determine the wave function of the quantum mechanical subsystem including interactions with the structured environment. We consider the situation where the MCSCF electronic wave function of the quantum mechanical subsystem is optimized while interacting with a classical system represented by charges, polarization sites and van der Waals sites. We start out by expressing the total electronic free energy for the QM/MM-system as... 

It is important to remark that this formulation for Ve/ makes it possible to treat solvent-separated subsystems (for example, the reactants A and B) at the same level as the composite molecular system (i.e. A-B in our... 

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