Constant energy hypersurface

Consider, at t = 0, some non-equilibrium ensemble density P g(P. q°) on the constant energy hypersurface S, such that it is nonnalized to one. By Liouville s theorem, at a later time t the ensemble density becomes ((t) t(p. q)), where q) is die function that takes die current phase coordinates (p, q) to their initial values time (0 ago the fimctioii ( ) is uniquely detemiined by the equations of motion. The expectation value of any dynamical variable ilat time t is therefore... 

Also we must bear in mind that the advancement of the coordinates fidfds two fiinctions (i) accurate calculation of dynamical properties, especially over times as long as typical correlation times x (ii) accurately staying on the constant-energy hypersurface, for much longer times Exact time reversibility is highly desirable (since the original equations... 

Consider a system of N particles evolving in time under the constraint of constant energy. The state vectors form a (6N — l)-dimensional constant energy hypersurface of size... 

Figure 3.7 Constant energy hypersurface E representing microscopic states of NVE system.

For the points denoted by numbers in Figure 5 we performed additionally a calculation of vibration frequencies to determine the kind of extremal values. For 120, which corresponds to a local maximum of the energy curve or, more precisely, to a saddle point of the energy hypersurface, all but one force constants (as second derivatives of the total energy) are positive, which is the mathematical criterion for a saddle point181. 

Tf the origin of the potential energy is set for the equilibrium geometry, then Vo = 0. At the equilibrium position the molecule should be at the minimum of the potential energy hypersurface therefore all first order derivatives are zero. The following term containing the second derivatives is called the harmonic term. These derivatives are noted and are referred to as force constants ... 

P, Hennig, W. P, Kraemer and 6. H. F. Oierckseni A Compilation of Theoretical Spectroscopic Constants and Rotational-Vibrational Transition Frequencies for the Isoelectronic Series of Linear Triatomic Molecules HCN, HNC, HCO , HOC", HNN Obtained from Ab Initio Calculated Energy Hypersurface, Max-Plenck-Insti-tut fur Physik und Astrophysik, Munich 1977. 

All force constants obtained from the H. F. potential energy hypersurface F(Xoe constants for the OH stretching and HOH bending taken from isolated watex ranaining force constants taken from H.F. potential hypersurface... 

All force constants obtained from the H. F. potential energy hypersurface... 

It is generally accepted that the transition-state theory provides the most convenient framework for calculation of the rate constants. Its rigorous application, however, needs a knowledge of the energy hypersurface of a... 

Finally, potential energy hypersurfaces may be used for calculating the rate constant of elementary processes either in the framework of the transition-state theory (Glasstone et al., 1941) or by performing classical trajectories calculations. 

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