Hamiltonian systematization

The basis of the value approach is a Hamiltonian systematization of the mathematical model of a conq)lex reaction with the extraction of the characteristics (functional) of a target reaction and with the kinetic comprehension of conjugate variables. Regardless of a reaction s conplexity, pivotal in this approach is a universal numerical determination of kinetic trajectories of value units (the kinetic significance) of individual steps and chemical species of a complex reaction, according to the selected target reaction characteristic. 

Now assuming as a basis the methodology of considering the chain reactions as reaction systems let us comment on the application capabhlities of the value analysis method, on the basis of the Hamiltonian systematization of kinetic models [9,12-25]. 

It is worthy of note that deriving explosion limits hy the method of the Hamiltonian systematization of dynamic systems is also proved hy the example of the reaction between hydrogen and oxygen involving 53 steps [23]. The results obtained are in good agreement with the calculation data of [43], where the extremal behavior of the sensitivity of the maximum concentration of hydrogen atoms relative to the reaction initial conditions is selected as a criterion for the explosion limit. 

Numerical calculation of the value contributions of individual steps of the kinetic model As mentioned above, parametric corollaries of the Hamiltonian systematization of reaction systems, namely, the value contributions of steps specify their kinetic significance. 

We believe that the objective quantitative interpretation of the kinetics, the time evolution of a chemical reaction, has to be in harmony with generally accepted approaches for the description of system dynamics. In accordance with this intention, we used the method of the Hamiltonian systematization of kinetic models of reaction systems. At the same time, the physical-chemical, kinetic comprehension of initial mathematical characteristics enabled to come to new systemic concepts in chemical kinetics, such as the value contributions of species and individual steps, specifying their kinetic significance in miltistep processes. 

Recent efforts of the authors resulted in development of a munerical method for the analysis of reaction mechanisms, based on their Hamiltonian systematization with marking out target characteristics of reactions and with the kinetic comprehension of conjugate variables. The core factor of this approach is the ability to calculate numerically the dynamics of value magnitudes, which characterize the systenuc kinetic significances of the chemical species of a reaction and its individual steps. Such information makes it possible to realize chemically the mechanism of a complex transformation, and particularly, to carry out the purposeful selection of efficient ways to control the reactions. 

Mead and Truhlar [10] broke new ground by showing how geometric phase effects can be systematically accommodated in scattering as well as bound state problems. The assumptions are that the adiabatic Hamiltonian is real and that there is a single isolated degeneracy hence the eigenstates n(q-, Q) of Eq. (83) may be taken in the form... 

Calculating the exact response of a semiconductor heterostructure to an ultrafast laser pulse poses a daunting challenge. Fortunately, several approximate methods have been developed that encompass most of the dominant physical effects. In this work a model Hamiltonian approach is adopted to make contact with previous advances in quantum control theory. This method can be systematically improved to obtain agreement with existing experimental results. One of the main goals of this research is to evaluate the validity of the model, and to discover the conditions under which it can be reliably applied. 

In Chapter 4 (Sections 4.7 and 4.8) several examples were presented to illustrate the effects of non-coincident g- and -matrices on the ESR of transition metal complexes. Analysis of such spectra requires the introduction of a set of Eulerian angles, a, jS, and y, relating the orientations of the two coordinate systems. Here is presented a detailed description of how the spin Hamiltonian is modified, to second-order in perturbation theory, to incorporate these new parameters in a systematic way. Most of the calculations in this chapter were first executed by Janice DeGray.1 Some of the details, in the notation used here, have also been published in ref. 8. 

To describe these experiments, introducing new degrees of freedom to manipulate systematically both spin and spatial tensor components, it proves convenient to recast the internal Hamiltonian in (5) in the form... 

Our approach is based on a systematic semiclassical study of the Dirac equation. After separating particles and anti-particles to arbitrary powers in h, a semiclassical expansion of the quantum dynamics in the Heisenberg picture is developed. To leading order this method produces classical spin-orbit dynamics for particles and anti-particles, respectively, that coincide with the findings of Rubinow and Keller Hamiltonian relativistic (anti-) particles drive a spin precession along their trajectories. A modification of that method leads to a semiclassical equivalent of the Foldy-Wouthuysen transformation resulting in relativistic quantum Hamiltonians with spin-orbit coupling. 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

The transition from (1) and (2) to (5) is reversible each implies the other if the variations 5l> admitted are completely arbitrary. More important from the point of view of approximation methods, Eq. (1) and (2) remain valid when the variations 6 in a trial function are constrained in some systematic way whereas the solution of (5) subject to model or numerical approximations is technically much more difficult to handle. By model approximation we shall mean an approximation to the form of as opposed to numerical approximations which are made at a lower level once a model approximation has been made. That is, we assume that H, the molecular Hamiltonian is fixed (non-relativistic, Born-Oppenheimer approximation which itself is a model in a wider sense) and we make models of the large scale electronic structure by choice of the form of and then compute the detailed charge distributions, energetics etc. within that model. 

Very recently, a modified zeroth-order Hamiltonian has been suggested by Ghigo and coworkers20 to accomplish this, which removes the systematic error and considerably improves both dissociation and excitation energies. Equation [5] can be written approximately as an interpolation between the two extreme cases... 

The point made in Eq. (3.31), namely, that the coupled, old, n action variables can be transformed to new, uncoupled, n - 1 conserved action variables is one to which we shall repeatedly return, in the quantum-algebraic context, in Chapters 4—6. Of course, we shall first discuss H0, which has n good quantum numbers, and which we shall call a Hamiltonian with a dynamical symmetry. At the next order of refinement we shall introduce coupling terms that will break the full symmetry but that will still retain some symmetry so that new, good, but fewer quantum numbers can still be exactly defined. In particular, we shall see that this can be done in a very systematic and sequential fashion, thereby establishing a hierarchy of sets of good quantum numbers, each successive set having fewer members. 

In this book we present an alternative approach. Our discussion in this introductory volume will put particular emphasis on the traditional concerns, namely, determing the levels and intensities of the corresponding transitions. The approach we present retains, at least in part, the simplicity of a Dunham-like approach in that, at least approximately, it provides the energy as an analytic function of the quantum numbers as in Equation (0.1). If this approximation is not sufficient, the method provides corrections derived in a systematic fashion. On the other hand, the method starts with a Hamiltonian so that one obtains not only eigenvalues but also eigenfunctions. It is for this reason that it can provide intensities and other matrix elements. 

We extend the method over all three rows of TMs. No systematic study is available for the heavier atoms, where relativistic effects are more prominent. Here, we use the Douglas-Kroll-Hess (DKH) Hamiltonian [14,15] to account for scalar relativistic effects. No systematic study of spin-orbit coupling has been performed but we show in a few examples how it will affect the results. A new basis set is used in these studies, which has been devised to be used with the DKH Hamiltonian. 

A systematic development of relativistic molecular Hamiltonians and various non-relativistic approximations are presented. Our starting point is the Dirac one-fermion Hamiltonian in the presence of an external electromagnetic field. The problems associated with generalizing Dirac s one-fermion theory smoothly to more than one fermion are discussed. The description of many-fermion systems within the framework of quantum electrodynamics (QED) will lead to Hamiltonians which do not suffer from the problems associated with the direct extension of Dirac s one-fermion theory to many-fermion system. An exhaustive discussion of the recent QED developments in the relevant area is not presented, except for cursory remarks for completeness. The non-relativistic form (NRF) of the many-electron relativistic Hamiltonian is developed as the working Hamiltonian. It is used to extract operators for the observables, which represent the response of a molecule to an external electromagnetic radiation field. In this study, our focus is mainly on the operators which eventually were used to calculate the nuclear magnetic resonance (NMR) chemical shifts and indirect nuclear spin-spin coupling constants. 

Based on a perturbation expansion using the KS Hamiltonian [26,27], recently a new systematic scheme for the derivation of orbital-dependent Ec has been proposed [12]. While this representation is exact in principle, an explicit evaluation requires the solution of a highly nonlinear equation, coupling Exc and the corresponding x>xc [19]. For a rigorous treatment of this Exc one thus has to resort to an expansion in powers of e, which allows to establish a recursive procedure for the evaluation of Exc and the accompanying Vxc-... 

After the energy is expressed as a functional of the 2-RDM, a systematic hierarchy of V-representabihty constraints, known as p-positivity conditions, is derived [17]. We develop the details of the 2-positivity, 3-positivity, and partial 3-positivity conditions [21, 27, 34, 33]. In Section II.E the formal solution of V-representability for the 2-RDM is presented through a convex set of two-particle reduced Hamiltonian matrices [7, 21]. It is shown that the positivity conditions correspond to certain classes of reduced Hamiltonian matrices, and consequently, they are exact for certain classes of Hamiltonian operators at any interaction strength. In Section II.F the size of the 2-RDM is reduced through the use of spin and spatial symmetries [32, 34], and in Section II.G the variational 2-RDM method is extended to open-shell molecules [35]. 

Rosina s theorem states that for an unspecified Hamiltonian with no more than two-particle interactions the ground-state 2-RDM alone has sufficient information to build the higher ROMs and the exact wavefunction [20, 51]. Cumulants allow us to divide the reconstruction functional into two parts (i) an unconnected part that may be written as antisymmetrized products of the lower RDMs, and (ii) a connected part that cannot be expressed as products of the lower RDMs. As shown in the previous section, cumulant theory alone generates all of the unconnected terms in RDM reconstruction, but cumulants do not directly indicate how to compute the connected portions of the 3- and 4-RDMs from the 2-RDM. In this section we discuss a systematic approximation of the connected (or cumulant) 3-RDM [24, 26]. 

Here, AE is the energy difference to transition from state A to B and jl the reciprocal thermal energy. Metropolis et al. [11] showed that such a scheme samples the Boltzmann distribution associated with the given Hamiltonian at the temperature specified by j>. For larger systems, such importance sampling is vastly superior to any systematic or random enumeration schemes, which scale extremely poorly with the number of degrees of freedom in the system [8]. 

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