Local Hamiltonian operator

LOCAL NON-RIGID GROUPS. THE LOCAL HAMILTONIAN OPERATOR... 

If we compare this local Hamiltonian operator with the exact one (25), we can verify that the kinetic interaction terms, as well as the potential ones in sine X sine are dropped. The second order sine terms are known to introduce the cogwheel effect in the potential energy function, because the sine function depends on the rotation sense (34,35). 

The restricted local Hamiltonian operator should be written as [21] ... 

As a result the local Hamiltonian operator does not commute solely with the triple switch operator (35), but also with the double switch, V, and simple switch, U, independently. So, the local group may be easily deduced replacing in (39) the triple switch subgroup (VUY = [E + VU] by the subgroup product... 

When there are no cog-wheel effect betweai the rotating parts, the sine x sine products may be dropped in (41), (84) or (86). So, the local Hamiltonian operator may be expressed as ... 

The restricted local Hamiltonian operator for the double internal rotation in acetone neglecting the cog-wheel effect between the rotors is similar to that of pyrocatechine (80), except for the periodicity of the rotor. The restricted local NRG is deduced directly from that of pyrocatechin (81), taking into account (42) [21-22] ... 

The pyramidal acetone-like molecules, such as acetone in an electronic excited state, dimethylamine, yield good examples of applications of the local groups to Czv rotor molecules. As in non-planar pyrocatechin, three different local Hamiltonian operators may be considered. 

The local Hamiltonian operator and the local rNRG are given essentially by expressions (90) and (92), respectively, in which the threefold periodicity of the rotors has to be introduced. So, the local rNRG has the simple form ... 

The restricted Hamiltonian operator for a non-rigid molecular system may be regarded as a special case of a local Hamiltonian operator in which the external rotation term has been dropped. This case holds only when the external and internal motions are separable. Let us consider some typical examples. 

In a diabatic representation, the electronic wave functions are no longer eigenfunctions of the electronic Hamiltonian. The aim is instead that the functions are so chosen that the (nonlocal) non-adiabatic coupling operator matrix, A in Eq. (52), vanishes, and the couplings are represented by (local) potential operators. The nuclear Schrddinger equation is then written... 

We have stated several times that whenever the Hamiltonian can be written in terms of invariant (Casimir) operators of a chain, its eigenvalue problem can be solved analytically. This method can be applied to the construction of both local and normal Hamiltonians. For local Hamiltonians, one writes H in terms of Casimir invariants of Eq. (4.43). 

In tetratomic molecules there is only one such operator, Ci23. The local Hamiltonian (5.16) is diagonal in the basis (5.4) with eigenvalues... 

A special focus will be on phenomenological Hamiltonians involving electronic spin interactions. For this it is necessary to define atomic surrogate spin operators—so-called local spin operators—that may be directly related to the effective spins in... 

In molecules, the interaction of surrogate spins localized at the atomic centers is calculated describing a picture of spin-spin interaction of atoms. This picture became prominent for the description of the magnetic behavior of transition-metal clusters, where the coupling type (parallel or antiparallel) of surrogate spins localized at the metal centers is of interest. Once such a description is available it is possible to analyze any wave function with respect to the coupling type between the metal centers. Then, local spin operators can be employed in the Heisenberg Spin Hamiltonian. An overview over wave-function analyses for open-shell molecules with respect to local spins can be found in Ref. (118). 

The quantum mechanical mechanisms that underlie exchange coupling are complex, but can be modeled by a phenomenological Hamiltonian that involves the coupling of local spin operators Sa and SB, the so-called Heisenberg-Dirac-Van... 

Each local Hamiltonian hi is a non-negatively defined operator at x < 1. The following statements related to the Hamiltonian (88) are valid ... 

In such a case, the over2dl rotation variables, which may be expressed as rotational angles around some orthogonal axes, and the internal motion variables, which may be written as internal coordinates, are completely separable. Because of this separability, this approximate full Hamiltonian operator may be regarded as local... 

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. 

Up to now we have rigorously considered the restricted Hamiltonian operator for some molecular motions, in which all the intramolecular interactions were taken into account. In the following, we shall neglect some of these interactions we suppose small enough, and we shall deal with approximate or local... 

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 the previous section, we considered the relatively simple non-rigid systems and we have deduce their local rNRG s for different st es of simplification of the Hamiltonian operator. Next, we shall consider molecular systems bearing a Cz rotor. The study of these more complex molecules better illustrated the advantage of the using the local NRG. 

Acetone in an electronic excited state, in which the oxygen atom is wagging out-of-plane, furnishes an example of such a case. The corresponding Hamiltonian operator is given by expression (84) in which the threefold periodicity of the rotors are introduced. Similarly, the local rNRG may deduced from the expression (85) ... 

The corresponding Hamiltonian operator is given essentially by expression (86), and its local rNRG by expression (88), where the threefold periodicity of the rotors is introduce so, we have for the local rNRG ... 

Throughout this paper, we give potential energy functions, symmetry eigenvectors, as examples, for systems of one, two and three internal degrees of freedom, in the formalism of the restricted Hamiltonian operator, as well as in the local one. A generalization of these ideas can be found in the scientific literature [21,22] and [30-37]. 

The application of Walter Heftier and Fritz London s valence bond theory was the first description of the binding forces in the H2 molecule, the simplest neutral molecule. Linus Pauling and John Slater later extended the principles to larger molecules. The key element in their proposal was the synthesis of a bonding wavefunction resulting from a combination of atomic orbitals that link the two atoms in a bond. It was hugely important that this localized approach concurred with the Lewis dot model. For the simplest neutral molecule, H2, the Hamiltonian operator may be written... 

A special word on the calculation of the energy is in order. In the last equality in (If) the hamiltonian operator in the numerator can be pushed either to the left or to the right in such a way to operate directly on the trial state. Remembering the definition of the local energy, (4), we can write... 

Let the potential W(r) be a non-positive function, localized mainly in a compact region 2 and disappearing at infinity. For simplicity, one may suppose IT to be a function with a compact support 2, that is, W/r) = 0 for r 2. When the volume of a well is large enough, there are some bounded states for the Hamiltonian operator hw of the form hw = -jA + W (r). 

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