Local Hamiltonians

Numerical solution of Eq. (51) was carried out for a nonlocal effective Hamiltonian as well as for the approximated local Hamiltonian obtained by applying a gradient expansion. It was demonstrated that the nonlocal effective Hamiltonian represents quite well the lateral variation of the film density distribution. The results obtained showed also that the film behavior on the inhomogeneous substrate depends crucially on the temperature regime. Note that the film exhibits different wetting temperatures on both parts of the surface. For chemical potential below the bulk coexistence value the film thickness on both parts of the surface tends to appropriate assymptotic values at x cx) and obeys the power law x. Such a behavior of the film thickness is a consequence of van der Waals tails. The above result is valid when both parts of the surface exhibit either continuous (critical) or first-order wetting. 

Here, H0 is a Hartree local Hamiltonian that includes the Coulomb effects of both nuclei and average electronic charge distributions,... 

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... 

Etor state Torsional potential energy local Hamiltonian Magnetic field strength at nucleus of a... 

In order to find the Hamiltonian for which the wave functions (83) and (84) are the exact ground state wave functions, we represent the Hamiltonian as a sum of local Hamiltonians hi defined on three neighboring sites (periodic boundary conditions are supposed) ... 

The basis of three-site local Hamiltonians hi consists of 64 states, while only eight of them are present in 4>i and 4>2. These 8 states are... 

The local Hamiltonian hi for which all the functions (86) are the exact ground state wave functions can be written as the sum of the projectors onto other 56 states Xk)... 

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

In order to find the Hamiltonian for which the wave function (93) is the exact ground state wave function, it is necessary to consider what states are present on the two nearest rungs in It turns out that there are only 16 states from the total 256 ones in the product i Xfi(i)g i up(i + 1). The local Hamiltonian hi acting on two nearest rungs i and i + 1 can be written in the form of (87) with the projectors onto the 240 missing states. The total Hamiltonian is the sum of local ones (85). The explicit form of this Hamiltonian is very cumbersome and, therefore, it is not given here. 

This result is equivalent to the renormalized perturbation theory of ref. 53, which these authors found to be at least qualitatively valid for ID and 2D lattices. Thus, at the scale of our n-domains, we may forget about interdomain interactions Bn and take a localized hamiltonian ... 

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). 

A local Hamiltonian may be presented for the pyrocatechin case. It just a matter of a mathematic model, since the interaction between the two hydroxyl groups ought to be relatively strong, due to an intramolecular hydrogen bond formation. 

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 the present work, simple analytical forms were deduced for the potential energy functions of single rotation in phenol (22), of double rotations in benzaldehyde (25) pyrocatechin (30), and acetone (47), of double rotation and inversion in non-planar pyrocatechin (41) and pyramidal acetone(52). Approximate potential energy functions were proposed for some of these non-rigid systems in the local Hamiltonian approach (75), (80), (84), (86), (90) and (96). 

G.A. Natanson, On the Invariance of the Localized Hamiltonians under Feasible Elements of the Nuclear Permutation-Inversion Group in Advances in Chemical Physics, Vol.58, Prigogine Ed., John Wiley,New York, 1985. 

They are equivalent to the Wannier functions used by Anderson for periodic lattices (see below) [54], Under these conditions, OMO s derived by Eqs. 42 and 43 are not strictly equivalent, simply due to the fact that the one-electron Hamiltonian of A-B is not the sum of the local Hamiltonians for A and B, considered separately. However, both types of OMO s show the same defect of locaUzation. hi addition, from a practical point of view, the OMO approach leads to much simpler calculations, as shown by Anderson [54], whereas the NMO approach is closer to the real mechanism involved in the nature of interaction and will favour the use of more reahstic molecular integrals. From now and for clarity, magnetic orbitals will be written without the prime (0 notation. 

If it is decided to treat the system by specifying a set of three rotation angles and 3Nt — 6 internal coordinates, then problems arise with the vector bundle idea. This is because such a separation requires that the vector space 3 js decomposed into the manifold S3 / R, V7 6. But this manifold cannot be coordinatized globally, of course so any account of electronic structure given in this way can at best be only local. The local Hamiltonian can describe only a subset of the states accessible in the full problem. 

Because we are dealing with double proton transfer, we need a more general Hamiltonian than Eq. (29.20). Following the approach introduced in the preceding section, we can construct such a Hamiltonian as the sum of two local Hamiltonians describing the transfer of each of the protons together with a proton-proton coupling ... 

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