Quantum representations effective Hamiltonians

Passing to the Boson operators by aid of Table II, and after neglecting the zero-point-energy of the fast mode, we obtain a quantum representation we shall name I, in which the effective Hamiltonians of the slow mode corresponding respectively to the ground and first excited states of the fast mode are... 

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... 

The theoretical basis for a conserved quantity is the commutation of an effective Hamiltonian with the elements of some symmetry group. If this condition exists, then the irreducible representations of the group are good quantum numbers, i.e., are conserved. Conversely, the existence of good quantum numbers implies a Hamiltonian which commutes with an appropriate group. The most general molecular A-electron Hamiltonians... 

The Adiabatic Hamiltonian and the Effective Hamiltonians Quantum representations II and III ... 

For this purpose, consider selective canonical transformations leading to a new quantum representation that we name /// in order to diagonalize the effective Hamiltonian corresponding to k = 1, without affecting that corresponding to k = 0. This may be performed on the different effective operators B dealing with k = 0,1 with the aid of... 

Now, pass to quantum representation ///. Within it, the effective Hamiltonians corresponding to the ground-state fast mode is unchanged, whereas that corresponding to the first excited state is modified. Owing to Table VI, the following equations hold ... 

Effective Hamiltonian of the two H-bond bridge quantum harmonic oscillator in representation II when the two moieties fast modes are in their eigenstates... 

QMSTAT is an effective quantum chemical solvent model with an explicit solvent representation. Effective here means that the quantum chemical electronic Hamiltonian only pertains to a small subset of the total system (typically the solute), with the solvent entering as a perturbation operator to the Hamiltonian explicit solvent means that the solvent is described with a set of spatial coordinates and parametrized physical features significantly simplified compared to a full quantum chemical description. The explicit solvent representation implies that it is possible to go beyond the mean-field approximation inherent in the often used continuum... 

The condition of Eq. (179) is fulfilled if the matrix elements of both operators A and B are either purely real or purely imaginary. For example, the effective Hamiltonians for characteristic zero-quantum coupling topologies with coupling tensor types /, P, L, or O (see Section V.B) are symmetric in the usual product basis. Because the matrix representations of the operators /, and /y are either real-valued (a =x or z) or purely imaginary (a = y) in this basis, the condition of Eq. (179) if fulfilled. In a system consisting of N spins 1/2, the operators and / have the same norm Tr /, = Tr /y = 2 and... 

The adiabatic picture is the standard one in quantum chemistry for the reason that, not only is it mathematically well defined, but it is also that used in ab initio calculations, which solve the electronic Hamiltonian at a particular nuclear geometry. To see the effects of vibronic coupling on the potential energy surfaces one must move to what is called a diabatic representation [1,65,180, 181]. 

For quantum chemistry the expansion of e in a Gaussian basis is, of course, much more important than that of 1/r. The formalism is a little more lengthy than for 1/r, but the essential steps of the derivation are the same. For an even-tempered basis one has a cut-off error exp(—n/i) and a discretization error exp(-7//i), such that results of the type (2.15) and (2.16) result. Of course, e is not well represented for r very small and r very large. This is even more so for 1/r, but this wrong behaviour has practically no effect on the rate of convergence of a matrix representation of the Hamiltonian. This is very different for basis set of type (1.1). Details will be published elsewhere. 

A more general description of the effects of vibronic coupling can be made using the model Hamiltonian developed by Koppel, Domcke and Cederbaum [65], The basic idea is the same as that used in Section III.C, that is to assume a quasidiabatic representation, and to develop a Hamiltonian in this picture. It is a useful model, providing a simple yet accurate analytical expression for the coupled PES manifold, and identifying the modes essential for the non-adiabatic effects. As a result it can be used for comparing how well different dynamics methods perform for non-adiabatic systems. It has, for example, been used to perform benchmark full-dimensional (24-mode) quantum dynamics calculations... 

The two avenues above recalled, namely ab-initio computations on clusters and Molecular Dynamics on one hand and continuum model on the other, are somewhat bridged by those techniques where the solvent is included in the hamiltonian at the electrostatic level with a discrete representation [13,17], It is important to stress that quantum-mechanical computations imply a temperature of zero K, whereas Molecular Dynamics computations do include temperature. As it is well known, this inclusion is of paramount importance and allows also the consideration of entropic effects and thus free-energy, essential parameters in any reaction. 

In the quantum mechanical continuum model, the solute is embedded in a cavity while the solvent, treated as a continuous medium having the same dielectric constant as the bulk liquid, is incorporated in the solute Hamiltonian as a perturbation. In this reaction field approach, which has its origin in Onsager s work, the bulk medium is polarized by the solute molecules and subsequently back-polarizes the solute, etc. The continuum approach has been criticized for its neglect of the molecular structure of the solvent. Also, the higher-order moments of the charge distribution, which in general are not included in the calculations, may have important effects on the results. Another important limitation of the early implementations of this method was the lack of a realistic representation of the cavity form and size in relation to the shape of the solute. 

There has been considerable controversy in the literature as to whether the nonradiative decay rates kn T are to be evaluated using the adiabatic Bom-Oppen-heimer (ABO) or the crude Born-Oppenheimer (CBO) approximation mo,31,39-43) In Sect. 5 it was noted that the complimentary principle of quantum mechanics requires that the rates exactly calculated within these two schemes be the same provided that 4>s in both schemes is, as expected, a reasonable approximation to the true physical state. As noted also, in both the ABO and the CBO approximations. we have the same mechanistic schemes of (f>s coupled to the effective quasicontinuum 0,. Thus, both cases represent differing, yet reasonable representations of s and 10,). In the present discussion it is necessary to consider the remainder of the vibronic states of the moleculae, 0C in addition to f coupling matrix elements are no longer the effective values, but are actual matrix elements of the perturbing Hamiltonian. 

An overview of the salient features of the relativistic many-body perturbation theory is given here concentrating on those features which differ from the familiar non-relativistic formulation and to its relation with quantum electrodynamics. Three aspects of the relativistic many-body perturbation theory are considered in more detail below the representation of the Dirac spectrum in the algebraic approximation is discussed the non-additivity of relativistic and electron correlation effects is considered and the use of the Dirac-Hartree-Fock-Coulomb-Breit reference Hamiltonian demonstrated effects which go beyond the no virtual pair approximation and the contribution made by the negative energy states are discussed. 

---