Nonzero moment

To see the type of differences that arises between an iterative solution and a simultaneous solution of the coefficient equations, we may proceed as follows. Bor the thirteen moment approximation, we shall allow the distribution function to have only thirteen nonzero moments, namely n, v, T, p, q [p has only five independent moments, since it is symmetric, and obeys Eq. (1-56)]. For the coefficients, we therefore keep o, a, a 1, k2), o 11 the first five of these... 

Table 13. i. Apparent partial electronic charge on H atoms based upon lowest nonzero moment and the corresponding calculated bond lengths, ST03G basis. 

The patterns of intensities can be understood on the basis of how much a given vibrational motion changes the molecular dipole moment. The donor bend, for example, acquires its high intensity because it turns the HF molecule with its large molecular moment in such a way as to give the complex a nonzero moment perpendicular to the H-bond axis. In contrast, the stretch of the two subunits away from each other in the v mode does little to... 

These features of molecules in degenerate states should be manifest in the experiments that depend critically on the first nonzero moment of the charge distribution. 

The initial conditions for the nonzero moments in the Riemann shock problem are shown in Figure 8.9. On the left half of the domain the normalized number density is Moo = 1, and on the right half it is 0.1. Initially the RMS velocity is 1 on both sides of the domain and the mean velocity is null. The moments are initialized as Maxwellian, so that the initial conditions are at local equilibrium. As time increases, a shock wave starting at x = 0 moves to the right and a deflation wave moves to the left. In the limit of t = 0, the solution is the same as for the Euler equation of gas dynamics. Sample results for the moments with T = 100 at time t = 0.5 are shown in Figure 8.10. For this case, the collisions are very weak, and thus there is little transfer of kinetic energy from the M-component to the... 

Figure 8.10. Nonzero moments for Riemann shock with t = 100 aU = 0.5.

Figure 8.17. Nonzero moments for crossing jets with t = 100 at t = 1.

Molecules may have two or three equal moments of inertia (and hence be symmetric or spherical tops) either by symmetry or accidentally. Che following elements of symmetry are sufficient conditions to ensure that the rotors with three nonzero moments are symmetrical or. spherical ... 

The simplest interpretion or explanation for a nonzero moment in Cm02 would be nonstoichiometry due to the preparation difficulties or radiation damage and lower-... 

It is not always well understood that the presence of a charge or the existence of a dipole in a molecule does not exclude the existence of a quadrupole. Any charge distribution can be expanded as a series of l-poles (monopole, dipole, quadrupole, octupole, hexadecapole, and so on), and it is a serious error to limit the series to the first nonzero moment. 

The value of the 1-pole moment of a charge distribution is independent of the choice of the origin only if this 1-pole moment is the first nonzero moment. In other words, the dipole moment of HCl and the quadrupole moment of II2 are independent of the choice of the origin, but the quadrupole moment of HCl and the hexadecapole moment of hydrogen depend on the choice of the origin. 

In a recent work (26,27) we have clearly shown the importance of higher moments than the first nonzero moments in molecular chemical physics. The situation is similar in nuclear physics. The presence of a charge (monopole) does not preclude the presence of higher multipole moments (28). 

The general nature of the MMCC theory, on which all renormalized and completely renormalized CC methods described here are based, allows us to proposed many other potentially useful approximations. We can, for example, introduce the MMCC(2,6) method, in which the CCSD results are corrected by considering all nonzero moments of the CCSD equations, including those corresponding to projections on pentuply and hextuply excited configurations. We can also introduce the active-space variants of the renormalized and completely renormalized CC approaches, in which we consider small subsets of the generalized moments of CC equations defined... 

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