Hamiltonian electronic subsystem

To illustrate this point, consider a composite system composed of two noninteracting subsystems, one with p electrons (subsystem A) and the other with q = N — p electrons (subsystem B). This would be the case, for example, in the limit that a diatomic molecule A—B is stretched to infinite bond distance. Because subsystems A and B are noninteracting, there must exist disjoint sets Ba and Bb of orthonormal spin orbitals, one set associated with each subsystem, such that the composite system s Hamiltonian matrix can be written as a direct sum. 

The LVC model further allows one to introduce coordinate transformations by which a set of relevant effective, or collective modes are extracted that act as generalized reaction coordinates for the dynamics. As shown in Refs. [54, 55,72], neg = nei(nei + l)/2 such coordinates can be defined for an electronic nei-state system, in such a way that the short time dynamics is completely described in terms of these effective coordinates. Thus, three effective modes are introduced for an electronic two-level system, six effective modes for a three-level system etc., for an arbitrary number of phonon modes that couple to the electronic subsystem according to the LVC Hamiltonian Eq. (7). In order to capture the dynamics on longer time scales, chains of such effective modes can be introduced [50,51,73]. These transformations, which are briefly summarized below, will be shown to yield a unique perspective on the excited-state dynamics of the extended systems under study. 

Following the analysis of Refs. [54,55,72], we now make use of the fact that the nuclear modes of the Hamiltonian Eq. (8) produce cumulative effects by their coupling to the electronic subsystem. From Eq. (9), the electron-phonon interaction can be absorbed into the following collective modes,... 

The electronic Hamiltonian for the whole system is now a sum of subsystem Hamiltonians and of their interaction which is taken to comprise the terms of two types - the Coulomb Wc and the resonance (electron transfer) Wr interactions ... 

Averaging the interaction operators in eq. (1.246) - they are both two-electronic ones - over the ground states of each subsystem does not touch the fermi-operators of the other subsystem. The averaging of the two-electron operators PWCP and PwrrP yields the one-electron corrections to the bare subsystem Hamiltonians. The wave functions and d, )7 are calculated in the presence of each other. The effective operator iTff describes the electronic structure of the R-system in the presence of the medium, whereas HIf describes the medium in the presence of the R-system. 

Let the evolution of the nuclear subsystem be given by a trajectory Q = Q(t). Consequently, the electronic Hamiltonian f/el becomes time dependent [through g(t)], and the state (j>(Q(t), q) of the electronic subsystem is, in general, nonstationary (j)(Q(t), q) obeys the time-dependent Schrodinger equation (20)... 

Here /Tel is the Hamiltonian of the electronic subsystem, —that of the nuclear subsystem (each a sum of kinetic energy and potential energy operators) and kei-N(r, R) is the electrons-nuclei (electrostatic) interaction that depends on the electronic coordinates r and the nuclear coordinates R. The BO approximation relies on the large mass difference between electron and nuclei that in turn implies that electrons move on a much faster timescale than nuclei. Exploring this viewpoint leads one to look for solutions for eigenstates of H of the form iA ,v(r,R) = < (r,R)/ R), or a linear combination of such products. Here (r, R) are solutions of the electronic Schrodinger equation in which the nuclear configuration Ris taken constant... 

A straightforward relativistic ab initio AE calculation might appear to be the most rigorous approach to a problem in electronic stmcture theory, however, one has to keep in mind that the methods to solve the Schrodinger equation used in connection with both the AE Hamiltonian and the ECP VO model Hamiltonian usually also rely on approximations, e.g., the choice of the one- and many-particle basis sets, and therefore lead to more or less significant errors in the results. In some cases the introduction of ECPs even helps to avoid or reduce errors, e.g., the basis set superposition error (BSSE), or allows a higher quality treatment of the chemically relevant valence electron subsystem compared to the AE case. 

An effective Hamiltonian of file electron subsystem can be constructed with the displaced phonon operator method (Elliott et al. 1972, Young 1975) or the method of canonical transformation (Mutscheller and Wagner 1986) analoguous results are given by a perturbation method in the second order in the electron-qrhonon interaction (13) (Baker 1971). 

The Hamiltonian for the M-subsystem is a sum of the free M-subsystem Hamiltonian H q) and of the attraction of electrons in the M-subsystem to the cores of the / -subsystem V q). Analogous subdividing is true for the i -subsystem. The exact wave function of the system can be represented by generalized group function with number of electrons in subsystems not fixed ... 

The spin-boson model can be motivated in terms of the physical problan of a quantum particle (e.g., an electron, muon, or proton) tunneling in an asymmetric double well (Figure 11.4). The relevant energy scales are the asymmetry he, tunneling HAq, and thermal k T, which can be all comparable but are much smaller than the barrier height Eq. Under this circumstance, the dynamics of the quantum particle moving in the double well can be described by what we shall refer to as the subsystem Hamiltonian... 

What about the environment Think of the particle as a proton or a positive muon (. i+) which, being positively charged, would carry with it the electron cloud of, for example, a metallic system like niobium (Nb) [4], While the subsystem Hamiltonian entails coherent tunneling with a (resultant) frequency A... 

In the new coordinates, the bath Hamiltonian takes a hierarchical form The effective modes Xg couple directly to the electronic subsystem, while the remaining (residual) Nb —Ngs bath modes couple in turn to the effective modes. The new bath Hamiltonian Hb of Eq. 15.8 can thus be split as follows ... 

As can be seen from Eq. 16.15 above, only three modes contribute to Heg. The remaining Nb — i modes (Nb is the number of bath modes) of the environment appear in VBath- In the numerical calculations we will (completely) neglect VBath, because this part of the Hamiltonian does not couple directly to the electronic subsystem. 

Let us examine a special but more practical case where the total molecular Hamiltonian, H, can be separated to an electronic part, W,.(r,s Ro), as is the case in the usual BO approximation. Consequendy, the total molecular wave function fl(R, i,r,s) is given by the product of a nuclear wave function X uc(R, i) and an electronic wave function v / (r, s Ro). We may then talk separately about the permutational properties of the subsystem consisting of electrons, and the subsystemfs) formed of identical nuclei. Thus, the following commutative laws Pe,Hg =0 and =0 must be satisfied X =... 

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],... 

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 theory of solvent effects on the electronic structure of a given solnte leads to a representation of the subsystem embedded in a larger one with the help of effective Hamiltonians, wave functions, and eigenvalues. Since the whole electronic system is quantum mechanical in nature, and in principle nonseparable, the theory for the ground electronic state permits defining under which conditions the solnte and solvent separability is an acceptable hypothesis. It is possible to distinguish passive from... 

Here, He(j) is Hamiltonian of a free electron, V,-(r) is Coulomb s interaction of the electron with the donor ion residue, Hlv( q ) is Hamiltonian of the vibration subsystem depending on the set of the vibration coordinates qj that corresponds to the movement of nuclei without taking into account the interaction of the electron with the vibrations. The short-range (on r) potential Ui(r, q ) describes the electron interaction with the donor ion residue and with the nuclear oscillations. The wave function of the system donor + electron may be represented in MREL in the adiabatic approach (see Section 2 of Chapter 2) ... 

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],... 

Hamiltonians thus defined contain large one-electron terms describing the attraction of electrons to the unscreened atomic cores in an alien subsystem ... 

One-electron transfers between the subsystems finally contribute to the effective Hamiltonian the following energy dependent term ... 

The idea of chemical nonactivity of the M-system assumes among other features that the energies of the states with electrons transferred between the subsystems (the poles of the resolvent eq. (1.242)) are much larger than the energies of the complex system which are of interest to us. For that reason in order to estimate the effective Hamiltonian eq. (1.232) one may set E = 0. By this we immediately arrive at the... 

Here the 7r-system is treated with a very simple, but still quantum mechanical method e.g. by the Hiickel Hamiltonian and MO LCAO approximation (which in the particular case of the Hiickel Hamiltonian gives the exact answer). No explicit interaction, i.e. junction, between the subsystems was assumed at that time however, the effects of the geometry of the classically moving nuclei were very naturally reproduced by a linear dependence of the one-electron hopping matrix elements of the bond length ... 

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