Orthant sampling

To form a microcanonical ensemble random values for the P( and Qt are chosen so that there is a uniform distribution in the classical phase space of ( , Q) [17]. Two ways are described here to accomplish this. For one method, called microcanonical normal-mode sampling [18], random values are chosen for the mode energies Et, which are then transformed to random values for Pt and Qt. In the second method, called orthant sampling [11], random values for P and Q are sampled directly from the phase space. 

Orthant sampling [10] works in the classical phase space of the molecular Hamiltonian H(P, Q). For a microcanonical ensemble, each phase-space point in the volume element dP dQ has equal probability [17]. The classical density of states is then proportional to the surface integral of the phase-space shell with H P, Q) = and is given by [17]... 

Both microcanonical normal-mode and orthant sampling give random sampling of the phase space and prepare microcanonical ensembles. For each, the average energy in a normal-mode is E/n. 

To form a microcanonical ensemble for the total Hamiltonian, H = HTib + Hrot, orthant sampling may be used for energy E = H. A (2n + 3)-dimen-sional random unit vector is chosen and projected onto the semiaxes for jx, jy, and jz [e.g., the semiaxis for jx is (2fx )1/2] as well as the semiaxes for Q and P. Since rotation has one squared-term in the total energy expression, whereas vibration has two, the average energy in a rotational degree of freedom will be one-half of that in a vibrational degree of freedom. 

Orthant sampling follows the same general procedure as described above for harmonic oscillators however, the sampling is done in Cartesian coordinates and momenta. The steps are as follows ... 

The remaining steps for orthant sampling are the same as steps 2 and 3 described above for microcanonical normal-mode sampling. 

Orth ant sampling Orthant sampling works in the classical phase space of the molecular Hamiltonian H(p,q). For a microcanonical ensemble. 

In orthant sampling an initial condition for a microcanical ensemble is chosen by projecting a random unit vector of dimension 2n, with components Xi, onto the E = ff(p,q) energy shell ... 

The classical trajectory calculations were performed at fixed total energies with both random and nonrandom excitation of HCC. Orthant sampling was used to choose the random initial conditions. 

Fig. 6. Dependence of lifetime distribution on surface type and equilibrium geometry. Dashed lines are random lifetime distributions. Histograms are for a total energy of 50 kcal/mol. Excitation of HCC is provided using orthant (random) sampling. All time axes have the identical scale with a maximum time of 1.5 x 10 " s.

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