Potential rotating nuclei

The fact that H commutes with Ez, Ex, and Ey and hence E2 is a result of the fact that the total coulombic potential energies among all the electrons and the nucleus is invariant to rotations of all electrons about the z, x, or y axes (H does not commute with L ) since if... 

So far, we have not specified the external vector field but summarized all individual contributions in A = ( , A ). Each (moving) nucleus also gives rise to a vector potential / (through translational and rotational motion as well as spin) in addition to a truly external vector field A0 applied in the laboratory setup,... 

As we have seen, the nucleons reside in well-defined orbitals in the nucleus that can be understood in a relatively simple quantum mechanical model, the shell model. In this model, the properties of the nucleus are dominated by the wave functions of the one or two unpaired nucleons. Notice that the bulk of the nucleons, which may even number in the hundreds, only contribute to the overall central potential. These core nucleons cannot be ignored in reality and they give rise to large-scale, macroscopic behavior of the nucleus that is very different from the behavior of single particles. There are two important collective motions of the nucleus that we have already mentioned that we should address collective or overall rotation of deformed nuclei and vibrations of the nuclear shape about a spherical ground-state shape. 

Thus when (cua + (Ux)Tr is not much less than 1, where is the molecular rotational correlation time, the potential maximum enhancement is not attained and it is not possible directly to use the size of the observed NOE to deduce the relative contribution of the dipole-dipole mechanism to the overall longitudinal relaxation of the observed nucleus. (228) The extreme narrowing condition is more stringent at higher measuring fields, but even for protons at say 300 MHz it is easily fulfilled (229) by small molecules for which Tr 10 -10 s. However, with large molecules such as globular... 

Since atoms are strongly affected by the central potential of the nucleus, an important part in electron—atom collision theory is played by states that are invariant under rotations. From the general dynamical principle that invariance under change of a dynamical variable implies a conservation law for the canonically-conjugate variable we expect rotational invariance to imply conservation of angular momentum. Hence angular momentum... 

The polarographic behavior of the 1-oxides and 1,4-dioxides of pyrazine, 2,5-dimethylpyrazine, and tetramethylpyrazine at various pH values has been investigated. It was assumed that at lower pH values, the A -oxide group was reduced in its protonated form. In acid media the 1-oxides exhibited double waves, the first of which is attributable to the reduction of jV-oxide groups and the second to that of the pyrazine nucleus (production of 1,4-dihydro compounds). Reduction of both A -oxide groups of pyrazine-1,4-dioxide proceeded simultaneously (588). Half-wave potentials of the voltammetric oxidation and reduction of pyrazine mono- and di-A -oxides have been measured in dimethylformamide, and in acetonitrile by the technique of a rotating platinum electrode (750). 

The hydrogen molecnlar ion is sketched in Fignre 6.1. The two nuclei, for convenience labeled A and B, are separated by the distance Rab along the internnclear axis, chosen by convention to be the x-axis. The electron is located at distance Ta from nucleus A and at distance tb from nucleus B. The angle describes rotation about the internuclear axis. For a fixed value of Rab> the position of the electron is more conveniently specified by the values of (ta, tb, 4>) than by (x, y, z) because the former set reflects the natural symmetry of the system. The internal potential energy is given by... 

Since detailed investigation of properties of molecules began, the motions of the atomic nuclei within a molecule have been of great interest because they reflect the dynamics of the molecule. Usually the internal motions are adequately described as vibrations, but if there is not a one-to-one correspondence of potential well and nucleus, we speak of inversion or hindered internal rotation. In the latter case, which is the subject of this paper, the change of sites of the nuclei is performed by a rotation of a part of the molecule (top) against the rest of the molecule (frame). Internal rotation and torsion are used synonymously in this paper. 

In some cases, the nucleus may not be at the centre of the confining potential. If there is rotational symmetry about the line of nuclear displacement, the elliptic coordinates may lead to separable solutions. For example, in the case of a hydrogen atom with the nucleus at the focus (0,0, —R/2) of a confining ellipsoid, one can take the elliptic coordinates of the electron as... 

To construct the potential due to the off centre nuclei, for each of them we choose a coordinate system with its origin at the expansion centre and the z axis passing through the nucleus for which we expand the potential. In doing so we obtain an expression for each of the off centre nuclei. The next step is to rotate all these contributions to a common coordinate system which requires the Wigner rotation matrices. The explicit results for the many symmetries applied are published elsewhere [45] and we write the potential created by the protons located at a distance R from the centre of expansion in the generic form ... 

---