Space isotropy

Owing to space isotropy of collision in the coordinate frame strongly coupled with a molecule s frame and hereafter referred to as the molecular system (MS), the kernel may be represented as... 

Owing to space isotropy and isotropy of collisions in free space, only... 

In general for a /-component fluid mixture one has a (z/ + 3 l 1)-component set p yd of dynamic variables, containing //-component of the number densities / k,a, the three components of the total current density Jk, and the total energy density E. However, as follows from the symmetric properties, the number ttk.a and energy densities are coupled only with the longitudinal component of Jk, directed along k. This is due to the space isotropy of the system. As a result, one may split the set of the hydrodynamic variables into two separate subsets ... 

Thus, it turns out that invariance of the equation of motion with respect to an arbitrary translation in time (time homogeneity) results in the energy conservation principle with respect to translation in space (space homogeneity) gives the total momentum conservation principle and with respect to rotation in space (space isotropy) implies the total angular momentum conservation principle. 

Let us stress that our eonclusion pertains to the toted wave funetion, whieh has to reflect the space isotropy leading to the zero dipole moment, because all orientations in space are equally probable. If one applied the transformation r —r only to some particles in the molecule (e.g.,... 

The Hamiltonian is also invariant with respect to some other symmetry operations, like changing the sign of the x coordinates of all particles, or similar operations that are products of inversion and rotation. If one changed the sign of all the x coordinates, it would correspond to a mirror reflection. Since rotational symmetry stems from space isotropy (which we will treat as trivial), the mirror reflection may be identified with parity P. 

In the case of polyatomics, the function /t (R) may be more complicated because some vibrations (e.g., a rotation of the CH3 group) may contribute to the total angular momentum, which has to be conserved (this is related to space isotropy cf., p. 69). 

The assumption that there exists an appropriate self-consistent field implies break-down of the space isotropy in some way. Such anisotropy is introduced either hy a statement that the initial point preserves its position r1[0) s 0 or by additional fixing of the second end of the chain at f(L) = h. In the first case, the field is spherically symmetrical about the origin of coordinate. In the second case, (f(0) = 0 and r[L) = h are fixed), the field has the symmetry Dooh with respect to these two points (focuses). 

In addition to break-down of the space isotropy, introduction of the self-consistent field implies the approximation of Markov s processes for an actually non-Markov one described by Equation 209. Hence, if VscF the self-consistent field, then Markov s approximation of Equation 205 is contained in... 

Suppose we wish to know the dipole moment of, say, the HCl molecule, the quantity that tells us important information about the charge distribution. We look up the output and we do not find anything about dipole moment. The reason is that all molecules have the same dipole moment in any of their stationary state y, and this dipole moment equals to zero, see, e.g., Piela (2007) p. 630. Indeed, the dipole moment is calculated as the mean value of the dipole moment operator i.e., ft = (T l/i l ) = ( F (2, q/r,) T), index i runs over all electrons and nuclei. This integral can be calculated very easily the integrand is antisymmetric with respect to inversion and therefore ft = 0. Let us stress that our conclusion pertains to the total wave function, which has to reflect the space isotropy leading to the zero dipole moment, because all orientations in space are equally probable. If one applied the transformation r -r only to some particles in the molecule (e.g., electrons), and not to the other ones (e.g., the nuclei), then the wave function will show no parity (it would be neither symmetric nor antisymmetric). We do this in the adiabatic or Born-Oppenheimer approximation, where the electronic wave function depends on the electronic coordinates only. This explains why the integral ft = ( F F) (the integration is over electronic coordinates only) does not equal zero for some molecules (which we call polar). Thus, to calculate the dipole moment we have to use the adiabatic or the Born-Oppenheimer approximation. 

The existence of an inertial reference system is a sequence of definite characteristics of space and time the uniformity and isotropy of space and uniformity of time. The uniformity of space and time means the equivalence of all positions of free bodies in space at all instants of time, and space isotropy means the equivalence of different directions. Therefore, it is possible to give another definition of an inertial reference system as a system relative to which space is homogeneous and isotropic and time is uniform. 

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