Hamiltonian splitting

Observe that, in principle, it is possible to introduce quaternions in the solution of the free rotational part of a Hamiltonian splitting, although there is no compelling reason to do so, since the rotation matrix is usually a more natural coordinatization in which to describe interbody force laws. 

Backward Error Analysis for Hamiltonian Splitting Methods 103... 

Gatti F, lung C, Leforestier C and Chapuisat X 1999 Fully coupled 6D calculations of the ammonia vibration-inversion-tunneling states with a split Hamiltonian pseudospectral approach J. Chem. Phys. Ill 7236 3... 

While all contributions to the spin Hamiltonian so far involve the electron spin and cause first-order energy shifts or splittings in the FPR spectmm, there are also tenns that involve only nuclear spms. Aside from their importance for the calculation of FNDOR spectra, these tenns may influence the FPR spectnim significantly in situations where the high-field approximation breaks down and second-order effects become important. The first of these interactions is the coupling of the nuclear spin to the external magnetic field, called the... 

Until now we have implicitly assumed that our problem is formulated in a space-fixed coordinate system. However, electronic wave functions are naturally expressed in the system bound to the molecule otherwise they generally also depend on the rotational coordinate 4>. (This is not the case for E electronic states, for which the wave functions are invariant with respect to (j> ) The eigenfunctions of the electronic Hamiltonian, v / and v , computed in the framework of the BO approximation ( adiabatic electronic wave functions) for two electronic states into which a spatially degenerate state of linear molecule splits upon bending. 

In Table I, 3D stands for three dimensional. The symbol symbol in connection with the bending potentials means that the bending potentials are considered in the lowest order approximation as already realized by Renner [7], the splitting of the adiabatic potentials has a p dependence at small distortions of linearity. With exact fomi of the spin-orbit part of the Hamiltonian we mean the microscopic (i.e., nonphenomenological) many-elecbon counterpart of, for example, The Breit-Pauli two-electron operator [22] (see also [23]). 

The construction of an efficient algorithm rests on the ability to separate the Hamiltonian into parts which are themselves integrable and also efficiently computable. Suppose that the MD Hamiltonian H defined by (6) is split into two parts as... 

For the model Hamiltonian used in this study it was assumed that bond stretching satisfactorily describes all internal vibrational motions for a system of linear molecules and the split parts of the Hamiltonian were of the form... 

A key feature required of a Hamiltonian system that leads to an efficient method based on splitting is the ability to separate the Hamiltonian into p-dependent and g-dependent terms. 

A convenient and constructive approach to attain symplectic maps is given by the composition of symplectic maps, which yields again a symplectic map. For appropriate Hk, the splittings (6) and (7) are exactly of this form If the Hk are Hamiltonians with respect to the whole system, then the exp rLnk) define the phase flow generated by these Hk- Thus, the exp TL-Hk) are symplectic maps on the whole phase space and the compositions in (6) and (7) are symplectic maps, too. Moreover, in order to allow for a direct numerical realization, we have to find some Hk for which either exp(rL-Kfc) has an analytic solution or a given symplectic integrator. 

The splitting technique, introduced above for the construction of symplec-tic schemes, is also adequate for symmetric ones. Now, the only condition is that we have to split symmetrically. To this end, let us consider the Liouville generator for the Hamiltonian li from above ... 

Here we suggest a different approach that propagates the system using multiple step-sizes, i.e., few steps with step-size At are taken in the slow classical part whereas many smaller steps with step-size 5t are taken in the highly oscillatory quantum subsystem (see, for example, [19, 4] for symplectic multiple-time-stepping methods in the context of classical molecular dynamics). Therefore, we consider a splitting of the Hamiltonian H = Hi +H2 in the following way ... 

In order to better understand the origin of the first term in (5.59) we separate from the Hamiltonian the part proportional to and average it over the equilibrium oscillators. This gives rise to an effective tunneling splitting A n,... 

A unique example of observation of tunneling splitting is given by Oppenlander et al. [1989]. Upon replacing the host benzoic acid dimer by a thioindigo molecule of nearly the same size, the resulting bias accidentally turns out to be small, of order of A. The 4x4 Hamiltonian of the complex of two dimers and the guest molecule is... 

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