Unified Hamiltonian

In this chapter, we wiU review electrochemical electron transfer theory on metal electrodes, starting from the theories of Marcus [1956] and Hush [1958] and ending with the catalysis of bond-breaking reactions. On this route, we will explore the relation to ion transfer reactions, and also cover the earlier models for noncatalytic bond breaking. Obviously, this will be a tour de force, and many interesting side-issues win be left unexplored. However, we hope that the unifying view that we present, based on a framework of model Hamiltonians, will clarify the various aspects of this most important class of electrochemical reactions. 

In the early 1990s a few classical semimoleculai and molecular models of electron transfer reactions involving bond breaking appeared in the literature. A quantum mechanical treatment of a unified mr el of electrochemical electron and ion transfer reactions involving bond breaking was put forward by Schmickler using Anderson-Newns Hamiltonian formalism (see Section V.2). 

From the conceptual point of view, there are two general approaches to the molecular structure problem the molecular orbital (MO) and the valence bond (VB) theories. Technical difficulties in the computational implementation of the VB approach have favoured the development and the popularization of MO theory in opposition to VB. In a recent review [3], some related issues are raised and clarified. However, there still persist some conceptual pitfalls and misinterpretations in specialized literature of MO and VB theories. In this paper, we attempt to contribute to a more profound understanding of the VB and MO methods and concepts. We briefly present the physico-chemical basis of MO and VB approaches and their intimate relationship. The VB concept of resonance is reformulated in a physically meaningful way and its point group symmetry foundations are laid. Finally it is shown that the Generalized Multistructural (GMS) wave function encompasses all variational wave functions, VB or MO based, in the same framework, providing an unified view for the theoretical quantum molecular structure problem. Throughout this paper, unless otherwise stated, we utilize the non-relativistic (spin independent) hamiltonian under the Bom-Oppenheimer adiabatic approximation. We will see that even when some of these restrictions are removed, the GMS wave function is still applicable. 

In Section 3, the Reporter has attempted to cast VB theory into as compact and unified a form as possible by making considerable use of group theoretical techniques. This is followed by a discussion of the various improvements and extensions that have been made over the past few years. The basic difficulty in VB theory is the calculation of the matrix elements of the hamiltonian when there is no orthogonality between the orbitals involved. This problem is also discussed at some length in this section, together with a survey of the various approaches that have been tried or proposed for its solution. 

As an example of application of the method we have considered the case of the acrolein molecule in aqueous solution. We have shown how ASEP/MD permits a unified treatment of the absorption, fluorescence, phosphorescence, internal conversion and intersystem crossing processes. Although, in principle, electrostatic, polarization, dispersion and exchange components of the solute-solvent interaction energy are taken into account, only the firsts two terms are included into the molecular Hamiltonian and, hence, affect the solute wavefunction. Dispersion and exchange components are represented through a Lennard-Jones potential that depends only on the nuclear coordinates. The inclusion of the effect of these components on the solute wavefunction is important in order to understand the solvent effect on the red shift of the bands of absorption spectra of non-polar molecules or the disappearance of... 

The efficient combination of RF pulses with MAS requires theoretical tools. While the initial multiple-pulse sequences were based on the average Hamiltonian theory (AHT) [14,25,26,32,33], some of the later schemes were more effectively formulated on the basis of Floquet theory [52], combinations of both AHT and Floquet theory [51], synchronisation arguments [42,43,46,55], and numerical methods [50]. A combination of RF pulses with MAS was also visualised in terms of certain symmetry conditions based on AHT [77]. An attempt to present all these in a unified picture is perhaps not out of place for a better understanding of the underlying phenomena. 

General Description of the Cluster Variation Method. In the previous paragraphs we have described several approximation schemes for treating the statistical mechanics of lattice-gas Hamiltonians like those introduced in the previous section. These approximations and systematic improvements to them are afforded a unifying description when viewed from the perspective of the cluster variation method. The evaluation of the entropy associated with the alloy can be carried out approximately but systematically by recourse to this method. The idea of the cluster variation method is to introduce an increasingly refined description of the correlations that are present in the system, and with it, to produce a series of increasingly improved estimates for the entropy. 

The beautiful point of this formulation is that the Marcus-Levich-Dogonadze result Equation (5) is the solution of the Hamiltonian Equation (7) in the deep tunneling limit. In addition, the solution of the Hamiltonian Equation (7) in the classical limit reproduces the TST result, corrected for recrossings of the barrier and for memory effects.12 These results mean that the Zwanzig Hamiltonian provides a unified description of proton transfer reactions in all the three parameter regions defined earlier in this section. 

An even coarser description is attempted in Ginzburg-Landau-type models. These continuum models describe the system configuration in terms of one or several, continuous order parameter fields. These fields are thought to describe the spatial variation of the composition. Similar to spin models, the amphiphilic properties are incorporated into the Hamiltonian by construction. The Hamiltonians are motivated by fundamental symmetry and stability criteria and offer a unified view on the general features of self-assembly. The universal, generic behaviour—the possible morphologies and effects of fluctuations, for instance—rather than the description of a specific material is the subject of these models. 

Within the electroweak model, which unifies the electromagnetic and the weak interaction, the electroweak Hamiltonian Hew does not commute with the parity operator. Hew can be split into a parity conserving term Hpc behaving even under parity and a parity violating term Hpv that behaves... 

When dealing with homonuclear multiple-spin systems, polarization transfer in the zero-quantum subspace of the effective dipolar Hamiltonian is advantageous as the spin dynamics of the process may be described by a kinetic matrix. The USEME (unified spin echo and magic echo) and RIL (rotating/laboratory frame) sequences follow this zero-quantum-subspace philosophy and aim at broadband properties of the pulse sequence. 

We have presented a unified formulation of dissipative dynamics based on the quantum theory of resonances. The reversible and dissipative confri-bufions fo fhe dynamics are gafhered in small-dimensional non-Hermifian effective Hamiltonians and effecfive Liouvillians wifh well-defined fheoref-ical sfafus. The formulation fends fo fill the gap between the dynamics and the thermodynamics. It has many advantages ... 

In order to unify, in fhe spirit of quantum defect theory, the treatment of discrefe and confinuous spectra in the presence of discrete Rydberg and valence states and of resonances, Komninos and Nicolaides [82, 83] developed K-mafrix-based Cl formalism that includes the bound states and the Rydberg series, and where the state-specific correlated wavefunc-tions (of the multi-state o) can be obtained by the methods of the SSA. The validity and practicality of fhis unified Cl approach was first demonstrated with the He P° Rydberg series of resonances very close to the n = 2 threshold [76], and subsequently in advanced and detailed computations in the fine-structure spectrum of A1 using fhe Breit-Pauli Hamiltonian [84, 85], which were later verified by experiment (See the references in Ref. [85]). 

Kaufman, A. N. (1984). Dissipative Hamiltonian systems a unifying principle. Phys. Letters, lOOA, 419-22. 

---