Hamiltonian, simple model

In a seminal paper [4], Su, Schrieffer, and Heeger discussed the energetics of solitons using a simple model of free electrons interacting with the lattice described by the Hamiltonian ... 

The usual EVB procedure involves diagonalizing this 3x3 Hamiltonian. However, here we wish to use a very simple model for our reaction and represent the potential surface and wavefunction of the reacting system using only two electronic states. Using a two-state system will preserve most of the important features of the potential energy surface while at the same time provide a simple model that will be more amenable to discussion than the three-state system. For the two-state system we define the following states as the reactant and product wavefunctions ... 

As a simple model, we confine our attention just to a single mode Ha(t) of the Hamiltonian (23). Note that neither any instantaneous eigenstate of Ha(t) is an exact quantum state nor e-/3ii W is a density operator. To calculate the thermal expectation value of an operator A, one needs either the Heisenberg operator Ah or the density operator pa(t) = UapaUa Now we use the time-dependent creation and annihilation operators (24), invariant operators, to construct the Fock space. 

To account for photochemical processes, we adopt a simple model that was proposed by Seidner and Domcke for the description of cis-trans isomerization processes [164]. In addition to the normal-mode expansion above, they introduced a Hamiltonian exhibiting torsional motion. The diabatic matrix elements of the Hamiltonian are given as... 

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 order to compare our approach with other approaches dealing with adiabatic corrections we perform simple model calculations for adiabatic corrections to ground state energy. We start with adiabatic Hamiltonian (32). We now perform the following approximation. We limit ourselves to finite orders of Taylor expansion of the operators H and H g We shall use similar approximation as in [25]. The diagrammatic representation of our approximate Hamiltonian will be... 

We consider the same reaction model used in previous studies as a simple model for a proton transfer reaction. [31,57,79] The subsystem consists of a two-level quantum system bilinearly coupled to a quartic oscillator and the bath consists of v — 1 = 300 harmonic oscillators bilinearly coupled to the non-linear oscillator but not directly to the two-level quantum system. In the subsystem representation, the partially Wigner transformed Hamiltonian for this system is,... 

In the present simulation we apply a simple model consisting of two vibrational modes in the relevant system. The first one will contribute to the Stokes shift as well as to the Herzberg-Teller correction, while the second one only to the Stokes shift. Next, we will assume a mutual coupling of the modes in the bilinear form Q Qi for the excited state. This type of coupling is usually omitted in literature. The Hamiltonian 77s is written as... 

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... 

One of the simple models of high-Tc superconducting copper oxides is a two-band Hubbard Hamiltonian (so called Emery model) [46]... 

The energies of the unperturbed (sub)states have been defined above. Ha is the configuration interaction Hamiltonian, which describes essentially the electron-electron interaction [128]. The off-diagonal terms are taken as the perturbation. The aim of this discussion is, at least for this simple model, to present the structure of the corresponding perturbational formulas. For example, it will be shown that different energy denominators occur, which are connected to the different states involved. In this model, we neglect any diagonal contributions to the model Hamilton operator and treat the matrix elements as real for simplicity. 

Here we employ the quantum kicked rotor as a simple model of quanmm chaos systems. The Hamiltonian of a kicked rotor is written as... 

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