Approximate Hamiltonians

The eigenfunctions of the zeroth order Hamiltonian define the projection of the DCB equation onto the subspace of electronic solutions. This is a first and necessary step to apply QED theory in quantum chemistry. The resulting second quantized formalism is compatible with the non-relativistic spin-orbital formalism if the connection (unbarred spinors - alpha-spinorbitals) and (barred spinors beta spinorbitals) is made. This correspondence allows transfer to the relativistic domain of non-relativistic algorithms after the differences between the two formalism are accounted for. 

The major difference between the relativistic and the non-relativistic theory concerns the algebra of the matrix elements. Because the non-relativistic Hamiltonian does not contain imaginary parts its eigenfunctions and associated matrix elements are usually chosen real. In the relativistic case the eigenspinors are, however, inherently complex functions that cannot be made real. Another difference is the fact that odd-barred matrix elements of h (see Eq.(36)) and g 

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Under the single-electron approximation, Hamiltonian (9-6) becomes... 

With the knowledge of g, we can estimate the inverse mean free path of a phonon with frequency co. As done originally within the TLS model, the quantum dynamics of the two lowest energies of each tunneling center are described by the Hamiltonian //tls = gcTz/2 + Aa /2. This expression, together with Eqs. (15) and (17), is a complete (approximate) Hamiltonian of... 

As before, we make the fundamental assumption of TST that the reaction is determined by the dynamics in a small neighborhood of the saddle, and we accordingly expand the Hamiltonian around the saddle point to lowest order. For the system Hamiltonian, we obtain the second-order Hamiltonian of Eq. (2), which takes the form of Eq. (7) in the complexified normal-mode coordinates, Eq. (6). In the external Hamiltonian, we can disregard terms that are independent of p and q because they have no influence on the dynamics. The leading time-dependent terms will then be of the first order. Using complexified coordinates, we obtain the approximate Hamiltonian... 

Prior to an effective Hamiltonian analysis it is, in order to get this converging to the lowest orders, typical to remove the dominant rf irradiation from the description by transforming the internal Hamiltonian into the interaction frame of the rf irradiation. This procedure is well established and also used in the most simple description of NMR experiments by transforming the Hamiltonian into the rotating frame of the Zeeman interaction (the so-called Zeeman interaction frame). In the Zeeman interaction frame the time-modulations of the rf terms are removed and the internal Hamiltonian is truncated to form the secular high-field approximated Hamiltonian - all facilitating solution of the Liouville-von-Neumann equation in (1) and (2). The transformation into the rf interaction frame is given by... 

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. 

Further Analysis of Solutions to the Time-Independent Wave Packet Equations of Quantum Dynamics II. Scattering as a Continuous Function of Energy Using Finite, Discrete Approximate Hamiltonians. 

Generally, it is not required to retain all the terms in the resulting approximate Hamiltonian, except those operators which describe the actual physical processes involved in the problem. For example, in the absence of an external electromagnetic field, the non-relativistic energy calculations only requires... 

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... 

The generalization of the pseudopotential method to molecules was done by Boni-facic and Huzinaga[3] and by Goddard, Melius and Kahn[4] some ten years after Phillips and Kleinman s original proposal. In the molecular pseudopotential or Effective Core Potential (ECP) method all core-valence interactions are approximated with l dependent projection operators, and a totally symmetric screening type potential. The new operators, which are parametrized such that the ECP operator should reproduce atomic all electron results, are added to the Hamiltonian and the one electron ECP equations axe obtained variationally in the same way as the usual Hartree Fock equations. Since the total energy is calculated with respect to this approximative Hamiltonian the separability problem becomes obsolete. 

Second, one solves the approximate Hamiltonian H° for the Z-electron problem... 

Using the basis functions which follow from the approximate Hamiltonian H° of equ. (1.3), it is the residual interaction H — H° which causes the Auger transitions. This operator, however, reduces to the Coulomb interaction if more than one electron changes its orbital.) Within the LS-coupling scheme this transition operator requires the following selection rules... 

The term does not imply that such calculations are exact. This is clear from the fact that most ab initio calculations use an approximate Hamiltonian, and all use a finite basis set. 

With these approximations, the coupled singles and doubles Eqs. (2-40) and (2-41) can be recast to a CIS-like form which can be subject to a comparison to CIS(D) or other correlation corrections to CIS. Using (1) to distinguish approximate Hamiltonian matrices using f2(1), Eq. (2 11) can be solved formally for xD ... 

The complete set of the molecular conversion operations which commute with this approximate Hamiltonian operator (19) will define another group, we call the local full NRG. This new group may be larger than the exact full NRG group. 

Hamiltonian operators. We shall see that with such approximate operators, we shall be able to deduce new and larger groups, which shall permit to introduce additional simplifications into the Hamiltonian matrix solutions. These new groups will be called Local Restricted Non-Rigid Groups [21,22]. The idea of using approximate Hamiltonian operators was already for warded by Bunker [8. ... 

In appropriate units, an approximate Hamiltonian of two ions in a Paul trap is given by... 

Each of these expressions is constructed by simply assigning the appropriate perturbational orders to each operator in Eq. [122] and retaining only the terms that correspond to the desired order, n. Using H as an approximate Hamiltonian, one may construct th-order Schrodinger equations of the form... 

Huang. Y.. Iyengar. S.S.. Kouri, D.J. and Hoffman. D.K. (1996) Further analysis of solutions to the time-independent wnve packet equations of quantum dynamics. 2. Scattering as a continuous function of energt- using finite, discrete approximate Hamiltonians, J. Chem. Phys. 105, 927-939. 

The meaning of the symbols is explained in Section III.C [Eq. (83)] H is the exact crystal Hamiltonian [Eq. (23)]. This time, however, we choose as the approximate Hamiltonian H0 a sum of single-particle Hamiltonians ... 

In this section, we outline a procedure for obtaining a Hamiltonian for the treatment of low-frequency vibrations in molecules. We do this, in particular, to point out the justification for some of the Hamiltonians used in the past and to make clear the nature of the approximations involved in arriving at a specific Hamiltonian. Since there is danger of overinterpreting the results obtained from approximate Hamiltonians, we indicate some of the pitfalls in doing so. 

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