Spin approximate Hamiltonians

In this work, relativistic effects are included in the no-pah or large component only approximation [13]. The total electronic Hamiltonian is H (r R) = H (r R) + H (r R), where H (r R) is the nom-elativistic Coulomb Hamiltonian and R) is a spin-orbit Hamiltonian. The relativistic (nomelativistic) eigenstates, are eigenfunctions of R)(H (r R)). Lower (upper)... 

In a second part we study the propagation of coherent states in general spin-orbit coupling problems with semiclassical means. This is done in two semiclassical scenarios h 0 with either spin quantum number s fixed (as above), or such that hs = S is fixed. In both cases, first approximate Hamiltonians are introduced that propagate coherent states exactly. The full Hamiltonians are then treated as perturbations of the approximate ones. The full quantum dynamics is seen to follow appropriate classical spin-orbit trajectories, with a semiclassical error of size yfh. As opposed to the first case,... 

The first case has already been considered section 2.0 the second case leads to a strong classical spin-orbit coupling, which is reflected in a Hamiltonian nature of the classical combined dynamics. In both situations the procedure is to find a suitable approximate Hamiltonian Hq( ) that propagates coherent states exactly along appropriate classical spin-orbit trajectories (x(l,),p(t),n(l,)). (For problems with only translational degrees of freedom this has been suggested in (Heller, 1975) and proven in (Combescure and Robert, 1997).) Then one treats the full Hamiltonian as a perturbation of the approximate one and calculates the full time evolution in quantum mechanical perturbation theory (via the Dyson series), i.e., one iterates the Duhamel formula... 

We have not mentioned open shells of electrons in our general considerations but then we have not specifically mentioned closed shells either. Certainly our examples are all closed shell but this choice simply reflects our main area of interest valence theory. The derivations and considerations of constraints in the opening sections are independent of the numbers of electrons involved in the system and, in particular, are independent of the magnetic properties of the molecules concerned simply because the spin variable does not occur in our approximate Hamiltonian. Nevertheless, it is traditional to treat open and closed shells of electrons by separate techniques and it is of some interest to investigate the consequences of this dichotomy. The independent-electron model (UHF - no symmetry constraints) is the simplest one to investigate we give below an abbreviated discussion. 

Here MA, MB, MA , and MB are the z-components of the spins of A and of B. Such collisions are usually treated7,34,142,177 in an adiabatic approximation using spin-free Hamiltonian and spin-free potential curves. Thus, MA and Mb are only approximate local quantum numbers during a collision and so may change. The total M quantum number is, however, conserved... 

This model, called the spin-boson Hamiltonian, is probably the only problem (except maybe for some very artificial ones) whose full solution can be obtained without any additional approximations. The equation of motion for the expectation value (crz) in the weak coupling limit has a... 

Since spin-orbit coupling is very important in heavy element compounds and the structure of the full microscopic Hamiltonians is rather complicated, several attempts have been made to develop approximate one-electron spin-orbit Hamiltonians. The application of an (effective) one-electron spin-orbit Hamiltonian has several computational advantages in spin-orbit Cl or perturbation calculations (1) all integrals may be kept in central memory, (2) there is no need for a summation over common indices in singly excited Slater determinants, and (3) matrix elements coupling doubly excited configurations do not occur. In many approximate schemes, even the tedious four-index transformation of two-electron integrals ceases to apply. The central question that comes up in this context deals with the accuracy of such an approximation, of course. 

Most common among the approximate spin-orbit Hamiltonians are those derived from relativistic effective core potentials (RECPs).35-38 Spin-orbit coupling operators for pseudo-potentials were developed in the 1970s.39 40 In the meantime, different schools have devised different procedures for tailoring such operators. All these procedures to parameterize the spin-orbit interaction for pseudo-potentials have one thing in common The predominant action of the spin-orbit operator has to be transferred from... 

Tike all effective one-electron approaches, the mean-field approximation considerably quickens the calculation of spin-orbit coupling matrix elements. Nevertheless, the fact that the construction of the molecular mean-field necessitates the evaluation of two-electron spin-orbit integrals in the complete AO basis represents a serious bottleneck in large applications. An enormous speedup can be achieved if a further approximation is introduced and the molecular mean field is replaced by a sum of atomic mean fields. In this case, only two-electron integrals for basis functions located at the same center have to be evaluated. This idea is based on two observations first, the spin-orbit Hamiltonian exhibits a 1/r3 radial dependence and falls off much faster... 

Approximate Spin-Orbit Hamiltonians in Light Conjugated Molecules The Fine-Structure Splitting of HC6H+, NC5H+, and NC4N+. 

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. 

It is a fundamental fact of quantum mechanics, that a spin-independent Hamiltonian will have pure spin eigenstates. For approximate wave functions that do not fulfill this criterion, e.g. those obtained with various unrestricted methods, the expectation value of the square of the total spin angular momentum operator, (5 ), has been used as a measure of the degree of spin contamination. is obviously a two-electron operator and the evaluation of its expectation value thus requires knowledge of the two-electron density matrix. 

For proton spins the Hamiltonian of the system can be approximated quite well by the dipolar interaction alone. Compared to the size of the homo-nuclear dipolar Hamiltonian, all other terms are small and can be neglected. The fluctuating part of the Hamiltonian of Equation (4.9) consists mainly of... 

In summary, the calculations on this special impurity system, with only one open-shell electron and simple manifolds, in which the assigmnents of the 5/<-> 6d absorption/emission bands were specially clear, even though not complete, and where absorption/emission bands built on a single origin and recorded with an extremely hi resolution exist, allow to be initially optimistic in the evaluation of the applicability of present day ab initio methods of the Quantum Chemistry in structural and spectroscopic studies of actinide ion impurities in ionic crystals. The quality of the approximate Hamiltonians and of the approximate procedures for decoupling the treatment of electron correlation and spin-orbit seems to be acceptable. The performance in other impurities with several open-shell electrons and large manifolds will be presented in the next two sections. 

Historically, the first derivations of approximate relativistic operators of value in molecular science have become known as the Pauli approximation. Still, the best-known operators to capture relativistic corrections originate from those developments which provided well-known operators such as the spin-orbit or the mass-velocity or the Darwin operators. Not all of these operators are variationally stable, and therefore they can only be employed within the framework of perturbation theory. Nowadays, these difficulties have been overcome by, for instance, the Douglas-Kroll-Hess hierarchy of approximate Hamiltonians and the regular approximations to be introduced in a later section, so that operators such as the mass-velocity and Darwin terms are no... 

We now consider reasons why the Born-Oppenheimer approximation can break down, i.e., why the right side of Eq. 3.12 can become appreciable. Since kk = ) for k k, large spin-orbital Hamiltonians figo can be a... 

The first approximation method for the Breit-Pauli spin-orbit Hamiltonian is to neglect the contribution from the two-electron terms. Justification... 

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