Hamiltonian advantages

For this reason, there has been much work on empirical potentials suitable for use on a wide range of systems. These take a sensible functional form with parameters fitted to reproduce available data. Many different potentials, known as molecular mechanics (MM) potentials, have been developed for ground-state organic and biochemical systems [58-60], They have the advantages of simplicity, and are transferable between systems, but do suffer firom inaccuracies and rigidity—no reactions are possible. Schemes have been developed to correct for these deficiencies. The empirical valence bond (EVB) method of Warshel [61,62], and the molecular mechanics-valence bond (MMVB) of Bemardi et al. [63,64] try to extend MM to include excited-state effects and reactions. The MMVB Hamiltonian is parameterized against CASSCF calculations, and is thus particularly suited to photochemistry. 

U. Ascher and S. Reich. The midpoint scheme and variants for Hamiltonian systems advantages and pitfalls. SIAM J. Sci. Comput., to appear. 

Large stepsizes result in a strong reduction of the number of force field evaluations per unit time (see left hand side of Fig. 4). This represents the major advantage of the adaptive schemes in comparison to structure conserving methods. On the right hand side of Fig. 4 we see the number of FFTs (i.e., matrix-vector multiplication) per unit time. As expected, we observe that the Chebyshev iteration requires about double as much FFTs than the Krylov techniques. This is due to the fact that only about half of the eigenstates of the Hamiltonian are essentially occupied during the process. This effect occurs even more drastically in cases with less states occupied. 

The commonly used method for the determination of association constants is by conductivity measurements on symmetrical electrolytes at low salt concentrations. The evaluation may advantageously be based on the low-concentration chemical model (lcCM), which is a Hamiltonian model at the McMillan-Mayer level including short-range nonelectrostatic interactions of cations and anions [89]. It is a feature of the lcCM that the association constants do not depend on the physical... 

Semi-empirical methods, such as those implemented in the MOPAC [9] program, simplify the equations considerably by neglecting many terms, but then compensate for this by parameterising some of them so that the calculations reproduce experimental information on, for example, the heat of formation. Once the various approximations are made, the molecular properties to which the parameters are fitted, and the molecules used in the fitting, define a model Hamiltonian, of which the most commonly used are the AMI and the PM3 Hamiltonians found in MOPAC. A major advantage of semi-empirical methods is... 

The EPR spectrum is a reflection of the electronic structure of the paramagnet. The latter may be complicated (especially in low-symmetry biological systems), and the precise relation between the two may be very difficult to establish. As an intermediate level of interpretation, the concept of the spin Hamiltonian was developed, which will be dealt with later in Part 2 on theory. For the time being it suffices to know that in this approach the EPR spectrum is described by means of a small number of parameters, the spin-Hamiltonian parameters, such as g-values, A-values, and )-values. This approach has the advantage that spectral data can be easily tabulated, while a demanding interpretation of the parameters in terms of the electronic structure can be deferred to a later date, for example, by the time we have developed a sufficiently adequate theory to describe electronic structure. In the meantime we can use the spin-Hamiltonian parameters for less demanding, but not necessarily less relevant applications, for example, spin counting. We can also try to establish... 

These are the Hamiltonian or canonical form of the equations of motion. The advantage of the Hamiltonian equations over the Lagrangian is that they contain 6n partial differential equations of the first order rather than 3n of the second order. 

Note that the eigenvalue pfx) does not depend on time, which is a consequence of Eq. (17). In particular, for a quadratic Hamiltonian, the operator satisfying Eq. (17) can be obtained explicitly. This canonical method has an advantage that quantum statistical information can... 

We applied the Liouville-von Neumann (LvN) method, a canonical method, to nonequilibrium quantum phase transitions. The essential idea of the LvN method is first to solve the LvN equation and then to find exact wave functionals of time-dependent quantum systems. The LvN method has several advantages that it can easily incorporate thermal theory in terms of density operators and that it can also be extended to thermofield dynamics (TFD) by using the time-dependent creation and annihilation operators, invariant operators. Combined with the oscillator representation, the LvN method provides the Fock space of a Hartree-Fock type quadratic part of the Hamiltonian, and further allows to improve wave functionals systematically either by the Green function or perturbation technique. In this sense the LvN method goes beyond the Hartree-Fock approximation. 

In this section, we introduce the model Hamiltonian pertaining to the molecular systems under consideration. As is well known, a curve-crossing problem can be formulated in the adiabatic as well as in a diabatic electronic representation. Depending on the system under consideration and on the specific method used, both representations have been employed in mixed quantum-classical approaches. While the diabatic representation is advantageous to model potential-energy surfaces in the vicinity of an intersection and has been used in mean-field type approaches, other mixed quantum-classical approaches such as the surfacehopping method usually employ the adiabatic representation. 

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