Electron-nucleus coupling

No explicit temperature dependence is included in the equations for R m and Rim, except for cases where Curie spin relaxation is the dominant term (Section 3.6). In the latter case, Curie paramagnetism has a T x dependence and therefore relaxation depends on T 2. The effect of temperature on linewidths determined by Curie relaxation is dramatic also because of the xr dependence on temperature, as shown in Eq. (3.8). All the correlation times modulating the electron-nucleus coupling, either contact or dipolar, are generally temperature dependent, although in different ways, and their variation will therefore be reflected in the values of Rim and Rim-... 

PROBLEM 3.1.2. Bohr s 1913 derivation of the energy of the hydrogen atom (nuclear charge = e, electron charge = e, reduced mass of the electron-nucleus couple = m, electron-nucleus distance = r, linear momentum = p) is based on the classical energy... 

PROBLEM 3.1.3. The energy of a one-electron atom (nuclear charge Z e, electron charge — e, reduced mass of the electron-nucleus couple p) is obtained by solving the Schrodinger equation for the one-electron atom ... 

Similar conclusions can be drawn from the study of NMR spectra given by unsaturated closed-shell molecules (see 18 21>). However, the theoretical analysis of the hyperfine structure is more involved for NMR spectra than for ESR spectra, because the nuclear spin-spin coupling constants are second-order phenomena as compared with the electron-nucleus coupling constants. 

Whenever the electrons are constrained to move on a chain by the one-dimensional character of the structure, the electron—nucleus coupling becomes a very efficient relaxation... 

The interaction of water with the paramagnetic copper(II) ion of oxidized SOD has been monitored through nuclear magnetic reljix-ation dispersion (NMRD) 114,205-208). The interaction of water protons with the unpaired electron of copperdi) causes an enhancement on the proton NMR relaxation. Usually the water proton Tr measurements are performed at proton Larmor frequencies between 0.01 and 50 MHz. The profiles are of the type shown in Fig. 26. In a simplified model of the electron-nucleus coupling in macromolecules the enhancements are inversely proportional to the sixth power of... 

Electron-nucleus coupling The interaction between the magnetic moments of an unpaired electron and a magnetic nucleus gives rise to an anisotropic hyperiine coupling (Adip). The coupling Adip calculated with the point-dipolar approximation is (Appendix A2.1) ... 

Coefficients that should multiply the S(S + 1) term in the. unpaired electron-nucleus coupling contribution to nuclear relaxation in a magnetically coupled... 

We first divide the dynamical variables into two groups those of heavy particle , which are treated in the coordinate representation R, and those of internal degrees of freedom, which are represented by state labels a, /3,. The former are to be treated in the classical approximation. In his original work, Pechukas considered atomic collision problems. Here we consider a general electron-nucleus coupled dynamics and set the former as the nuclear degrees of freedom and the latter as the electronic degrees of freedom. 

We now consider an electron-nucleus coupled system. We denote the two-time extension of the total state as The corresponding Schrodinger equation, in the nuclear coordinate representation, becomes... 

The long-range nature of the electron-nucleus coupling in systems with anisotropic susceptibility has produced a growing interest in the field of biomolecular NMR. The protein backbone and sidechains offer a large pool of nuclei far from the... 

It is simpler to consider the electric field produced by the distribution of an electron in an orbital or, in effect, a time-averaged potential. Adophng this point of view, each electron moves as if in a static potential produced by the nucleus and the hme-averaged distibutions of the other electrons. As each electron moves in the field of the other electrons, it creates by its own distribution a held affecting the mohons of each of the others. Thus the motions of all the electrons are coupled delicately the term self-consistent held indicates the requirement that all the individual electron distribuhons must produce the proper potenhals for one another s motion. 

This term is called the Fermi contact term. That part of the electron-nucleus magnetic-dipole interaction represented by (8.104) depends on the angular coordinates of the electron and is therefore anisotropic in contrast, the Fermi contact energy (8.108) is isotropic. The contact term plays an important role in the electron-coupled nuclear spin-spin interactions seen in the NMR spectra of liquids. 

Relaxation measurements provide a wealth of information both on the extent of the interaction between the resonating nuclei and the unpaired electrons, and on the time dependence of the parameters associated with the interaction. Whereas the dipolar coupling depends on the electron-nucleus distance, and therefore contains structural information, the contact contribution is related to the unpaired spin density on the various resonating nuclei and therefore to the topology (through chemical bonds) and the overall electronic structure of the molecule. The time-dependent phenomena associated with electron-nucleus interactions are related to the molecular system, and to the lifetimes of different chemical situations, for the resonating nucleus. Obtaining either structural or dynamic information, however, is only possible if an in-depth analysis of a series of experimental results provides sufficient data to characterize the system within the theoretical framework discussed in this chapter. 

Another mechanism to provide splitting of the S manifold is the hyperfine coupling between the unpaired electron and the metal nucleus. For example, at zero magnetic field an S = Vi / = V2 system gives two sets of levels of degeneracy 3 and 1, separated by A (see Appendix III) where A is the metal-nucleus-unpaired-electron hyperfine coupling. The effect of this splitting is... 

In Section 1.4, we discussed the history and foundations of MO theory by comparison with VB theory. One of the important principles mentioned was the orthogonality of molecular wave functions. For a given system, we can write down the Hamiltonian H as the sum of several terms, one for each of the interactions which will determine the energy E of the system the kinetic energies of the electrons, the electron-nucleus attraction, the electron-electron and nucleus-nucleus repulsion, plus sundry terms like spin-orbit coupling and, where appropriate, other perturbations such as an applied external magnetic or electric field. We now seek a set of wave functions P, W2,... which satisfy the Schrodinger equation ... 

In addition to the main hyperfine components discussed above, weak satellite lines are frequently detected, which correspond to transitions normally considered to be forbidden. Under certain circumstances the selection rule, that there is no change in orientation of the nuclear spin as the electron spin changes, breaks down, and simultaneous electron-nuclear transitions can occur. These are especially strong if the electron-nuclear coupling is very anisotropic and is comparable in magnitude to the direct interaction between the external field and the nucleus, which will, in general, have a different direction from that of the anisotropic interaction. Under these conditions the nucleus is affected by both fields, and quantum restrictions break down. 

This is a very important result. The first term in the last line of (4.13) represents the so-called Fermi contact interaction between the electron and nuclear spin magnetic moments, and the second term is the electron-nuclear dipolar coupling, analogous to the electron-electron dipolar coupling derived previously in (3.151). The Fermi contact interaction occurs only when the electron and nucleus occupy the same position in Euclidean space, as required by the Dirac delta function S(-i Rai). This seemingly... 

The magnitudes of the hyperfine splitting parameters also yield information about the electron distribution in the molecule. The theory of the electron-nuclear coupling interaction was first worked out by Fermi, who showed that the constant a depended on the electron density at the nucleus. For a free hydrogen atom, a is given by the Fermi contact interaction in the forrn ... 

The formalism developed so far is adequate whenever the motion of the atomic nuclei can be neglected. Then the electron-nucleus interaction only enters as a static contribution to the potential r(r, t) in Eq. (41). This is a good approximation for atoms in strong laser fields above the infrared frequency regime. When the nuclei are allowed to move, the nuclear motion couples dynamically to the electronic motion and the situation becomes more complicated. 

Clearly, a complete numerical solution of the coupled KS equations (67,68) for electrons and nuclei will be rather involved. Usually only the valence electrons need to be treated dynamically. The core electrons can be taken into account approximately by replacing the electron-nucleus interaction (65) by suitable pseudopotentials and by replacing the nuclear Coulomb potential in Eq. (64) by the appropriate ionic Coulomb potential [38]. This procedure reduces the number of electronic KS equations and hence the numerical effort considerably. 

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