Hamiltonian reduced-dimensionality

A very perceptive treatment of chemical reaction dynamics, called the reaction path Hamiltonian analysis, states that the reactive trajectory is determined as the minimum energy path, and small displacements from that path, on the potential-energy surface [64-71]. The usual analysis keeps the full dimensionality of the reacting system, albeit with a focus on motion along and orthogonal to the minimum energy path. It is also possible to define a reaction path in a reduced dimensionality representation. 

The definition of a reduced dimensionality reaction path starts with the full Cartesian coordinate representation of the classical A-particle molecular Hamiltonian,... 

Using this transformation, it has been shown in Refs. [54,72] that the effective-mode Hamiltonian Heg by itself reproduces the short-time dynamics of the overall system exactly. This is reflected by an expansion of the propagator, for which it can be shown that the first few terms of the expansion - relating to the first three moments of the overall Hamiltonian - are exactly reproduced by the reduced-dimensional Hamiltonian Heg. 

The distance between two electrons at a given site is given as ri2. The electron wave function for one of the electrons is given as (p(ri) and the wave function for the second electron, with antiparallel spin, is Hubbard intra-atomic energy and it is not accounted for in conventional band theory, in which the independent electron approximation is invoked. Finally, it should also be noted that the Coulomb repulsion interaction had been introduced earlier in the Anderson model describing a magnetic impurity coupled to a conduction band (Anderson, 1961). In fact, it has been shown that the Hubbard Hamiltonian reduces to the Anderson model in the limit of infinite-dimensional (Hilbert) space (Izyumov, 1995). Hence, Eq. 7.3 is sometimes referred to as the Anderson-Hubbard repulsion term. 

REDUCED DIMENSIONAL HAMILTONIAN FOR THE X + YCZ3 SYSTEM AND BASIS SET EXPANSION... 

Hamiltonians (for finite- or infinite-dimensional M ). However, in the case of several incommensurate frequencies (quasi-periodic Hamiltonian), reducibility is not always satisfied, even for finite-dimensional Jtf [19,33-36], We remark that for finite-dimensional ffl, reducibility is equivalent to the property of K having no continuous spectrum [19],... 

Distorted wave Born approximation resonance energies and widths were calculated numerically using Equation 17. for the reduced-dimensionality hamiltonian given by Equation 35. and employing Equation 36. for V (r,t) up to second order. The coefficients a(t) and b(t) were determined numerically from the ab imitio potential surface. Zero-order wavefimctions X,(t) Xj(t) were deter-... 

The reduced dimensionality Hamiltonian for the X + YCZ3 system can be written as [40]... 

The above discussion of the intermolecular potentials and of the stability of collision systems in reduced dimensionality provides the microscopic justification for studying an ensemble of polar molecules in two-dimensions interacting via (modified) dipole-dipole potentials. At low temperatures T < ficoj., the general many-body Hamiltonian has the form of Equation 12.5. As an example of the possibilities offered by potential engineering to realize novel many-body quantum phases, we here focus on bosonic particles interacting via the effective potential =D/p of Equation 12.6,... 

The calculation of the molecular eigenstates with the MVCM model, necessary in traditional time-independent methods, can prove to be very cumbersome or even unfeasible. However, time-independent effective solutions, practicable for reduced-dimensionality models (in practice when the number of relevant normal coordinates is less than 10), may be obtained by taking advantage of the Lanczos iterative tridiagonalization of the Hamiltonian matrix [130]. The Lanczos algorithm proves to be very suitable for the computation of low-resolution spectra however, its effectiveness is better highlighted in a time-dependent framework. In fact, it can be easily realized that Lanczos states are only sequentially coupled, and it is therefore clear that only a limited number of states is necessary to describe short-time dynamics since the latter is the only relevant information for low-resolution spectra (see Chapter 10). 

Since the only angle dependence conies from 0 , and the actions /, L are constant. From this point onwards we concentrate on motion under the reduced Hamiltonian which depends, apart from the scaling parameter y, only on the values of scaled coupling parameter p and the scaled detuning term p. In other words, we investigate the monodromy only in a fixed J (or polyad number N = 2J) section of the three-dimensional quanmm number space. 

The point q = p = 0 (or P = Q = 0) is a fixed point of the dynamics in the reactive mode. In the full-dimensional dynamics, it corresponds to all trajectories in which only the motion in the bath modes is excited. These trajectories are characterized by the property that they remain confined to the neighborhood of the saddle point for all time. They correspond to a bound state in the continuum, and thus to the transition state in the sense of Ref. 20. Because it is described by the two independent conditions q = 0 and p = 0, the set of all initial conditions that give rise to trajectories in the transition state forms a manifold of dimension 2/V — 2 in the full 2/V-dimensional phase space. It is called the central manifold of the saddle point. The central manifold is subdivided into level sets of the Hamiltonian in Eq. (5), each of which has dimension 2N — 1. These energy shells are normally hyperbolic invariant manifolds (NHIM) of the dynamical system [88]. Following Ref. 34, we use the term NHIM to refer to these objects. In the special case of the two-dimensional system, every NHIM has dimension one. It reduces to a periodic orbit and reproduces the well-known PODS [20-22]. 

There is another case when the orientational Hamiltonian for nonpolar molecules (2.3.1) on a symmetric two-dimensional lattice is reducible to a quasidipole form, viz. the case of planar orientations of long molecular axes ( s = 90° in expression (2.3.2)) when one can invoke the transformation for doubled orientation angles qy of the unit vectors ej and r ... 

Consider an arbitrary two-dimensional Bravais lattice, with its sites R occupied by adsorbed molecules and molecular vibrations representing two modes, of a high and low frequency. Frequencies (ohh reduced masses mh/y vibrational coordinates w/,/(R), and momenta pA /(R) are accordingly labeled by subscripts h and / referring to the high-frequency and the low-frequency vibration. The most general form of the Hamiltonian appears as140... 

Of course, the Coulomb interaction appears in the Hamiltonian operator, H, and is often invoked for interpreting the chemical bond. However, the wave function, l7, must be antisymmetric, i.e., must satisfy the Pauli exclusion principle, and it is the only fact which explains the Lewis model of an electron pair. It is known that all the information is contained in the square of the wave function, 1I7 2, but it is in general much complicated to be analyzed as such because it depends on too many variables. However, there have been some attempts [3]. Lennard-Jones [4] proposed to look at a quantity which should keep the chemical significance and nevertheless reduce the dimensionality. This simpler quantity is the reduced second-order density matrix... 

So far we have assumed that the electronic structure of the crystal consists of one band derived, in our approximation, from a single atomic state. In general, this will not be a realistic picture. The metals, for example, have a complicated system of overlapping bands derived, in our approximation, from several atomic states. This means that more than one atomic orbital has to be associated with each crystal atom. When this is done, it turns out that even the equations for the one-dimensional crystal cannot be solved directly. However, the mathematical technique developed by Baldock (2) and Koster and Slater (S) can be applied (8) and a formal solution obtained. Even so, the question of the existence of otherwise of surface states in real crystals is diflBcult to answer from theoretical considerations. For the simplest metals, i.e., the alkali metals, for which a one-band model is a fair approximation, the problem is still difficult. The nature of the difficulty can be seen within the framework of our simple model. In the first place, the effective one-electron Hamiltonian operator is really different for each electron. If we overlook this complication and use some sort of mean value for this operator, the operator still contains terms representing the interaction of the considered electron with all other electrons in the crystal. The Coulomb part of this interaction acts in such a way as to reduce the effect of the perturbation introduced by the existence of a free surface. A self-consistent calculation is therefore essential, and the various parameters in our theory would have to be chosen in conformity with the results of such a calculation. 

In quantum wires formed in a two-dimensional electron gas (2DEG) by lateral confinement the Rashba spin-orbit interaction is not reduced to a pure ID Hamiltonian H[s = asopxaz. As was shown in Ref. [4] the presence of an inplane confinement potential qualitatively modifies the energy spectrum of the ID electrons so that a dispersion asymmetry appears. As a result the chiral symmetry is broken in quantum wires with Rashba coupling. Although the effect was shown [4] not to be numerically large, the breakdown of symmetry leads to qualitatively novel predictions. 

Equation (5.2) is a combination of the two two-dimensional Hamiltonians (2.39) and (3.15) which describe the vibrational and rotational excitations of BC separately. The Jacobi coordinates R, r, and 7 are defined in Figures 2.1 and 3.1 and P and p denote the linear momenta corresponding to R and r, respectively, j is the classical angular momentum vector of BC and 1 stands for the classical orbital angular momentum vector describing the rotation of A with respect to BC. For zero total angular momentum J=j+l = 0we have 1 = — j and the Hamilton function reduces to... 

As an example, consider the product of two arbitrary first-rank tensor operators 0 and It is nine-dimensional and can be reduced to a sum of compound irreducible tensor operators of ranks 2, 1, and 0, respectively. Operators of this type play a role in spin-spin coupling Hamiltonians. In terms of spherical and Cartesian components of 0 and J2, the resulting irreducible tensors are given in Tables 8 and 9, respectively.70... 

The eigenvalue problem of the Hamiltonian operator (1) is defined in an infinite-dimensional Hilbert space Q and may be solved directly only for very few simple models. In order to find its bound-state solutions with energies not too distant from the ground-state it is reduced to the corresponding eigenvalue problem of a matrix representing H in a properly constructed finite-dimensional model space, a subspace of Q. Usually the model space is chosen to be spanned by TV-electron antisymmetrized and spin-adapted products of orthonormal spinorbitals. In such a case it is known as the full configuration interaction (FCI) space [8, 15]. The model space Hk N, K, S, M) may be defined as the antisymmetric part of the TV-fold tensorial product of a one-electron space... 

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