Wavefunctions subsystems

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. 

I l ) = I I a I b), and one says that is multiplicatively separable in the two subsystems, recognizing that in quantum mechanics ) is separable only up to an overall antisymmetrization (or a symmetrization, in the case of bosons) that renders all coordinates equivalent. The separation of the wavefunction in Eq. (12) is equivalent, in a necessary and sufficient sense, to the block structure of the Hamiltonian in Eq. (11) [32, 50-52]. 

Consider the RDMs obtained from the separable wavefunction in Eq. (12). Since a and b are strongly orthogonal, it follows from Eq. (8) that ( a b a flj I a b) = 0 unless 0, and (f)j are associated with the same subsystem. Thus the 1-RDM separates into subsystem 1-RDMs,... 

This form of Dp implies that Ap = 0 for each p > 1, a reflection of the fact that an independent-electron wavefunction consists of one-electron subsystems coupled only by exchange. 

Whenever an A -eleotron system is represented in terms of subsystems A, B,..., using a generalized product ansatz, the entire wavefunction can be optimized by repeated optimization of subsystem wavefunctions... 

Any subsystem wavefunction, refers only to Nr electrons with an effective field Hamiltonian, whose form depends on the forms of the 1-electron density matrices of all subsystems all such functions can be optimized, in an iterative manner, by standard methods and without the constraints implied by any a priori partitioning of the global basis... 

The resultant wavefunction provides an optimum separation of the whole system, many of whose properties may then be expressed as sums of subsystem contributions... 

Bardeen considers two separate subsystems first. The electronic states of the separated subsystems are obtained by solving the stationary Schrodinger equations. For many practical systems, those solutions are known. The rate of transferring an electron from one electrode to another is calculated using time-dependent perturbation theory. As a result, Bardeen showed that the amplitude of electron transfer, or the tunneling matrix element M, is determined by the overlap of the surface wavefunctions of the two subsystems at a separation surface (the choice of the separation surface does not affect the results appreciably). In other words, Bardeen showed that the tunneling matrix element M is determined by a surface integral on a separation surface between the two electrodes, z = zo. 

Fig. 1.20. The Bardeen approach to tunneling theory. Instead of solving the Schrddinger equation for the coupled system, a, Bardeen (1960) makes clever use of perturbation theory. Starting with two free subsystems, b and c, the tunneling current is calculated through the overlap of the wavefunctions of free systems using the Fermi golden rule.

If the initial state of the system is described by t /ift /os(io), where /oXi o) is the initial wavefunction for the slow variable subsystem at t = 0, then the k-hop contribution to the component of the system in fast variable state //at time t is given by... 

The applicability of the discussed two-step algorithms for calculation of wavefunctions of molecules with heavy atoms is a consequence of the fact that the valence and core electrons may be considered as two subsystems, interaction between which is described mainly by some integrated properties of these subsystems. The methods for consequent calculation of the valence and core parts of electronic structure of molecules give us a way to combine the relative simplicity and accessibility both of molecular RFCP calculations in gaussian basis set, and of relativistic finite-difference one-center calculations inside a sphere with the atomic core radius. 

When the two systems A and B are brought from infinity to their equilibrium postions, the wavefunctions FA and TB of the subsystems will be overlapping. The Pauli principle is obeyed by explicitly antisymmetrizing (operator A) and renormalizing (factor N) the product wavefunction ... 

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 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],... 

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. 

The wavefunctions of the quantum mechanical subsystem are subjected to the following requirements we have that 0 > at time t is... 

The main point of the preceding discussion is an assumption about the adiabaticity of intramolecular motion with reference to the intermolecular one, that is, division of the system into two subsystems the fast one involving electrons and intramolecular vibrations and the slow one incorporating the intermolecular vibrations. This division of the motions was called a double adiabatic approximation (DAA) and was applied earlier in the theory of proton transfer in the reorganizing medium (see, e.g., refs. 10, 14, 28, and 31-33). The wavefunctions in DAA are presented as the products of the wavefunctions of the fast and slow subsystems ... 

Going, in Eqn. (50), from summation to integration by E (in this case the wavefunctions of the final state of the intramolecular subsystem are normalized by the energy), we come to... 

In most practical applications different basis sets are used, for example, atomic orbitals associated or other wavefunctions localized in the different subsystems. 

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