Interchangeable nuclei

It might be thought that inverting the molecule-fixed coordinates of all particles is equivalent to inverting the space-fixed coordinates of all particles, but this is not so. The direction of the z axis is defined as going from nucleus a to nucleus b, since the xyz axes are rigidly connected to the nuclear framework when we invert the coordinates of electrons and nuclei, we interchange nuclei a and b, and thereby reverse the direction of... 

These wave functions have important symmetry properties. For aa, pp, and (aP + Pa)/V, the two A nuclei may be interchanged to give the identical initial wave functions back. Such wave functions are said to be symmetric. In contrast, interchanging nuclei in(aP — Pa)/ /2 gives (Pa — aP)/V2, which is the negative of the original wave function. Such wave functions are said to be antisymmetric. There is a second selection rule when wave functions have these properties, namely, that transitions are allowed only between wave functions of the same symmetry. By contrast, transitions are forbidden between symmetric and antisymmetric wave functions. 

In the Bom-Oppenheimer approximation the electronic ground-state wave function of H2 has to be the eigenfunction of the nuclear inversion symmetry operator i interchanging nuclei a and h (cf. Appendix C). Since P = 1, the eigenvalues can be either —1 (called u symmetry) or -1-1 g symmetry). The ground-state is of g symmetry, therefore the projection operator will take care of that (it... 

As was shown in the preceding discussion (see also Sections Vin and IX), the rovibronic wave functions for a homonuclear diatomic molecule under the permutation of identical nuclei are symmetric for even J rotational quantum numbers in and E electronic states antisymmeUic for odd J values in and E elecbonic states symmetric for odd J values in E and E electronic states and antisymmeteic for even J values in Ej and E+ electeonic states. Note that the vibrational ground state is symmetric under pemrutation of the two nuclei. The most restrictive result arises therefore when the nuclear spin quantum number of the individual nuclei is 0. In this case, the nuclear spin function is always symmetric with respect to interchange of the identical nuclei, and hence only totally symmeUic rovibronic states are allowed since the total wave function must be symmetric for bosonic systems. For example, the nucleus has zero nuclear spin, and hence the rotational levels with odd values of J do not exist for the ground electronic state f EJ") of Cr. 

The system to be treated consists of two nuclei A and B and two electrons 1 and 2. In the unperturbed state two hydrogen atoms are assumed, so that the zeroth-order energy is 2WH-If the first electron is attached to nucleus A and the second to nucleus B, the zeroth-order eigenfunction is ip (1)

zeroth-order energy, so that the system... 

In this expression, the term hydrogen atoms A and B. The term interaction with the electrons interchanged. However, the term A2 represents both electrons 1 and 2 interacting with nucleus A. That means the structure described by the wave function is ionic, HA HB+. In an analogous way, the term B1 Bj2 represents both electrons interacting with nucleus B, which corresponds to the structure HA+ Hb . Therefore, what we have devised for a molecular wave function actually describes the hydrogen molecule as a "hybrid" (a valence bond term that is applied incorrectly) of... 

Nuclear fusion processes derive energy from the formation of low-mass nuclei, which have a different binding energy. Fusion of two nuclear particles produces a new nucleus that is lighter in mass than the masses of the two fusing particles. This mass defect is then interchangeable in energy via Einstein s equation E = me2. Specifically, the formation of an He nucleus from two protons and two neutrons would be expected to have mass ... 

The terms shell, orbital, and energy level are sometimes used interchangeably. In this text we use the term shells most often because it is descriptive and conveys the image of three-dimensional layers or structures of electrons smrounding the nucleus. The term... 

For indistinguishable particles, this has to by symmetrized appropriately. The 4He nucleus is a boson, and hence the total wave function does not change sign when interchanging the nuclei. One obtains for the properly symmetrized scattering amplitude... 

Interchange by Inversion Through a Center of Symmetry (i) A center of symmetry exists if a line drawn from each nucleus through the selected center encounters the same nucleus or group at the same distance on the other side of the center (Figure 3.43, structure d). 

Bile salts are substances derived from sterols, which make up a substantial part of the solid matter in bile and which play a central role in lipid absorption, by virtue of their surface-active properties. The structure and properties of these salts have been reviewed by Haslewood (305) and Heaton (316). Bile salts essentially have molecules of detergent type hydrocarbon, with a fat-dissolving part and a polar, water-attracting part. The fat-dissolving part consists of the bulk of the steroid nucleus. The hydroxyl groups are so distributed that hydration can readily take place the remainder of the molecule will dissolve the fatty phase. Emulsification of fat/water complexes can thus occur easily. The terms bile acid and bile salt are used somewhat interchangeably in the literature. 

Direction of Larmor precession in lower and upper magnetic states Case A for a nucleus of spin /= 1/2 in an external field B Case B for an electron of spin S = 1/2 in an external field B. The transition implies an interchange of Larmor precession direction from counterclockwise to clockwise. 

---