Nuclear spatial coordinates

So far, we have not worried about nuclear spin—the wave function (4.28) for nuclear motion contains only the nuclear spatial coordinates. However, nuclei possess an intrinsic or spin angular momentum of magnitude... 

In the last section, we showed rot to have the eigenvalue (—1/ for inversion of the nuclei. Since inversion amounts to interchanging the nuclear spatial coordinates, and since the nuclear spin coordinates do not occur in proV this is also the eigenvalue of rot for the operator which interchanges the space and spin coordinates of the two nuclei. 

Spectroscopies such as X-ray, 2D nuclear magnetic resonance, neutron diffraction, and inelastic neutron scattering provide a representation of molecular structure in terms of nuclear spatial coordinates and a thermal noise. We shall indicate this set of coordinates with r,, for i = 1, 2,. . . , nuclei, and take its origin at the center of mass. 

We now consider planar molecules. The electronic wave function is expressed with respect to molecule-fixed axes, which we can take to be the abc principal axes of ineitia, namely, by taking the coordinates (x,y,z) in Figure 1 coincided with the principal axes a,b,c). In order to detemiine the parity of the molecule through inversions in SF, we first rotate all the electrons and nuclei by 180° about the c axis (which is peipendicular to the molecular plane) and then reflect all the electrons in the molecular ab plane. The net effect is the inversion of all particles in SF. The first step has no effect on both the electronic and nuclear molecule-fixed coordinates, and has no effect on the electronic wave functions. The second step is a reflection of electronic spatial coordinates in the molecular plane. Note that such a plane is a symmetry plane and the eigenvalues of the corresponding operator then detemiine the parity of the electronic wave function. 

Equation (4.15) would be extremely onerous to evaluate by explicit treatment of the nucleons as a many-particle system. However, in Mossbauer spectroscopy, we are dealing with eigenstates of the nucleus that are characterized by the total angular momentum with quantum number 7. Fortunately, the electric quadrupole interaction can be readily expressed in terms of this momentum 7, which is called the nuclear spin other properties of the nucleus need not to be considered. This is possible because the transformational properties of the quadrupole moment, which is an irreducible 2nd rank tensor, make it possible to use Clebsch-Gordon coefficients and the Wigner-Eckart theorem to replace the awkward operators 3x,xy—(5,yr (in spatial coordinates) by angular momentum operators of the total... 

The total number of spatial coordinates for a molecule with Q nuclei and N electrons is 3(Q + N), because each particle requires three cartesian coordinates to specify its location. However, if the motion of each particle is referred to the center of mass of the molecule rather than to the external spaced-fixed coordinate axes, then the three translational coordinates that specify the location of the center of mass relative to the external axes may be separated out and eliminated from consideration. For a diatomic molecule (Q = 2) we are left with only three relative nuclear coordinates and with 3N relative electronic coordinates. For mathematical convenience, we select the center of mass of the nuclei as the reference point rather than the center of mass of the nuclei and electrons together. The difference is negligibly small. We designate the two nuclei as A and B, and introduce a new set of nuclear coordinates defined by... 

Interchanging the nuclear coordinates does not affect R, but it does affect the electronic spatial coordinates since they are defined with respect to the molecule-fixed xyz axes, which are rigidly attached to the nuclei. To find the effect on el of interchanging the nuclear coordinates, we will first invert the space-fixed coordinates of the nuclei and the electrons, and then carry out a second inversion of the space-fixed electronic coordinates only the net effect will be the interchange of the space-fixed coordinates of the two nuclei. We found in the last section that inversion of the space-fixed coordinates of all particles left //e, unchanged for 2+,n+,... electronic states, but multiplied it by —1 for 2, II ,... states. Consider now the effect of the second step, reinversion of the electronic space-fixed coordinates. Since the nuclei are unaffected by this step, the molecule-fixed axes remain fixed for this inversion, so that inversion of the space-fixed coordinates of the electrons also inverts their molecule-fixed coordinates. But we noted in Section 1.19 that the electronic wave functions of homonuclear diatomics could be classified as g or m, according to whether inversion of molecule-fixed electronic coordinates multiplies ptl by + 1 or -1. We conclude that for 2+,2,7,11, IV,... electronic states, i//el is symmetric with respect to interchange of nuclear coordinates, whereas for... 

To derive the Hamiltonian operator corresponding to (8.104), we replace pe and pN by the corresponding operators. The resulting Hamiltonian contains the electron s spatial coordinates r, as well as electronic and nuclear spin coordinates. In this chapter, we are dealing only with spin... 

There are point-group selection rules in the presence of spin interactions.73,115117 172 We recall that a spin-free Hamiltonian //SF(Qeq) for a rigid nuclear framework Qeq has a point group SF which acts on electronic spatial coordinates, and that... 

In these expressions written with use of so-called atomic units (elementary charge, electron mass and Planck constant are all equal to unity) RQs stand as previously for the spatial coordinates of the nuclei of atoms composing the system r) s for the spatial coordinates of electrons Mas are the nuclear masses Zas are the nuclear charges in the units of elementary charge. The meaning of the different contributions is as follows Te and Tn are respectively the electronic and nuclear kinetic energy operators, Vne is the operator of the Coulomb potential energy of attraction of electrons to nuclei, Vee is that of repulsion between electrons, and Vnn that of repulsion between the nuclei. Summations over a and ft extend to all nuclei in the (model) system and those over i and j to all electrons in it. 

The effective operator is the product of one operator on three-dimensional spatial coordinates and another which acts on nuclear spin space. This distinction can be brought out even more clearly by making use of the operator replacement theorem in section 5.5.3 to give... 

The 3E and p-complex structures resemble each other because both consist of one unit of spin or electronic angular momentum (S or L) coupled to the nuclear rotation (R). However, since fj, operates exclusively on electron spatial coordinates, any resemblance between the rotational-branch intensity patterns for 3S —1E+ and p-complex —1E+ transitions would seem to be coincidental. A 3E —1E+ transition will look exactly like a p-complex —1E+ transition if, in addition to satisfying Eqs. (6.3.47), the cr-orbital of the 1E+ state is predominantly of scr united atom character. Then the transition moment ratio will be... 

The numbers in parentheses on the left-hand side of Eq. (5-1) symbolize the spatial coordinates of each of the n electrons. Thus, 1 stands for xi,> i,zi, or n, 6i, (j>i, etc. We shall use this notation frequently throughout this book. Since we are not here concerned with the quantum-mechanical description of the translational motion of the atom, there is no kinetic energy operator for the nucleus in Eq. (5-1). The index i refers to the electrons, so we see that Eq. (5-1) provides us with the desired kinetic energy operator for each electron, a nuclear electronic attraction term for each electron, and an interelectronic repulsion term for each distinct electron pair. (The summation indices guarantee that l/ri2 and l/r2i will not both appear in H. This prevents counting the same physical interaction twice. The indices also prevent nonphysical self-repulsion terms, such as l/r22, from occurring.) Frequently used alternative notations for the double summation in Eq. (5-1) are j 1/ 7. which counts each interaction twice... 

As noted above, Cohen s NFF descriptor was introduced in the spin free or conventional conceptual DFT formalism in order to measure (indirectly) the effect of changes in the external potential. The NFF becomes then defined in terms of the derivative of the force on the nucleus k, F, with respect to the number of electrons N, or equivalently, as the change of chemical potential upon variations in the spatial coordinates of nucleus k. Within the representation of spin polarized DFT, two nuclear... 

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