Symmetric method

Long term simulations require structurally stable integrators. Symplec-tic and symmetric methods nearly perfectly reproduce structural properties of the QCMD equations, as, for example, the conservation of the total energy. We introduced an explicit symplectic method for the QCMD model — the Pickaback scheme— and a symmetric method based on multiple time stepping. 

Note that the same results have not been shown for symmetric (time-reversible) integration methods, although symmetric methods seem to perform quite well in practice. For a discussion of symmetric methods in the context of the QCMD model see [16, 17, 13]. 

We have derived time-reversible, symplectic, and second-order multiple-time-stepping methods for the finite-dimensional QCMD model. Theoretical results for general symplectic methods imply that the methods conserve energy over exponentially long periods of time up to small fluctuations. Furthermore, in the limit m —> 0, the adiabatic invariants corresponding to the underlying Born-Oppenheimer approximation will be preserved as well. Finally, the phase shift observed for symmetric methods with a single update of the classical momenta p per macro-time-step At should be avoided by... 

T. E. Simos, A family of trigonometrically-fitted symmetric methods for the efficient solution of the Schrodinger equation and related problems, J. Math. Chem., 2003, 34(1-2), 39-58. 

Even though the data of Problems 7 and 8 fit a linear Equation (except for the deviant points) they can be forced to fit the Equation for the quasi-symmetric method (Equations 11 or 12) to yield a value for Ap of 0.1 units (with an uncertainty of 0.1). Indicate how this value affects the conclusions that can be drawn regarding the concertedness or otherwise of the displacement reaction. Compare the Ap with that for the pyridinolysis of N-triazinyl-pyridinium ions." ... 

Another problematic point appears in the treatment of electron loss due to heavy (neutral) targets. In this case, unrealistic capture processes come into play where the projectile electron is transferred into populated bound target states. In principle, this problem may be circumvented by using the multielectron anti-symmetrization method, where the Pauli exclusion principle is enforced for the transitions amplitudes. Thus, an explicit and time-consuming treatment of these occupied bound states would then be necessary. 

Important savings are possible by using symmetry efficiently. A systematic procedure was given by King and coworkers (Dupuis and King, 1978 Takada et al, 1981,1983), who generalized the symmetrization method of Dacre (1970) to derivative calculations. The new problem is that the derivatives, in contrast to the energy, are not totally symmetrical. 

Clearly, standard Rayleigh-Schrodinger perturbation theory is not applicable and other perturbation methods have to be devised. Excellent surveys of the large and confusing variety of methods, usually called exchange perturbation theories , that have been developed are available [28, 65]. Here it is sufficient to note that the methods can be classified as either symmetric or symmetry-adapted . Symmetric methods start with antisymmetrized product functions in zeroth order and deal with the non-orthogonality problem in various ways. Symmetry-adapted methods start with non-antisymmetrized product functions and deal with the antisymmetry problem in some other way, such as antisymmetrization at each order of perturbation theory. 

Unlike the molecular case, this time the matrix to diagonalize is Hermitian, and is not necessarily symmetric. Methods of diagonalization exist for such matrices, and there is a guarantee that their eigenvalues are real. 

Of particular interest for sampling the canonical distribution are symmetric Langevin methods. We believe these are likely to be the most useful class of methods for practitioners, as by symmetrizing the expansion the odd order terms in (7.16) vanish identically using the Jacobi identity in the BCH expansions. This implies that a symmetric scheme gives a second order error in computed averages. Many symmetric methods can be constructed that require only one evaluation of the force per iteration (effectively making them as inexpensive as a first order method). 

G. A. Panopoulos, Z. A. Anastassi and T. E. Simos, Two New Optimized Eight-Step Symmetric Methods for the Efficient Solution of the Schrodinger Equation and Related Problems, MATCH Common. Math. Comput. Chem., 2008, 60, 3. 

The only way to produce efficient dissipative (i.e. non-symmetric) methods for the numerical solution of problems described in Section 1 is the exponentially fitted and the trigonometrically fitted version of these methods. The above methods are based on the well known exponential fitting procedure, first introduced by Lyche. ... 

This method, although requiring longer reaction times than the symmetrical method, results in products virtually free of impurities arising horn enantio-merization and dipeptide formation (6). It involves formation of a mixed anhydride between an Fmoc-amino acid derivative and 2,6-dichlorobenzoyl chloride (DCB) in DMF/pyridine (Figure 3). This anhydride effects esterification of resin-bound hydroxyl groups in yields of typically > 70% in 18 h. Levels of D-isomer and dipeptide formation for most amino acids are typically less than 1%, with 5-acetamidomethyl cysteine giving the most enantio-merization (6.5% D-isomer) of the amino acids tested (8). 

B. New trigonometrically fitted six-step symmetric methods for the efficient solution of the Schrodinger equation... 

We consider the multistep symmetric method of Jenkins, with six steps and sixth algebraic order ... 

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