Nuclear Spin Contribution

TEMPO, and any of its (not too bulky) derivatives, is comparable in molecular mass with the spin trap DMPO, so the tumbling in water at ambient temperature should again average out all anisotropy. The spectrum is even simpler (namely, three identical lines see the high-temperature traces in Figure 10.4) than that of the hydrox-yl-DMPO adduct because only the 14N nuclear spin contributes to the spectrum ... 

It will be recalled that in section 3.10 we developed the nuclear Hamiltonian by calculating the magnetic vector potential Aea at nucleus a arising from the spin and orbital motion of the electrons. Clearly we should now also include the nuclear spin contribution to Aa, the complete magnetic vector potential being (3.249) plus the additional term from the other nucleus a, ... 

It should be appreciated that the Zeeman interactions are usually dominated by the first two terms in (7.232), the electron spin and orbital terms. The other terms are typically between two and four orders of magnitude smaller. For a molecule in a closed shell1 state, only the rotational Zeeman term, the nuclear spin contribution and the susceptibility term survive. 

Since atoms retain their nuclear spins unaltered in all processes except those involving ortho-para conversions, there is no change in the nuclear spin entropy. It is consequently the common practice to omit the nuclear spin contribution, leaving what is called the practical entropy or virtual entropy. ... 

It is of importance to note that, except for hydrogen and deuterium molecules, the entropy derived from heat capacity measurements, i.e., the thermal entropy, as it is frequently called, is equivalent to the practical entropy in other words, the nuclear spin contribution is not included in the former. The reason for this is that, down to the lowest temperatures at which measurements have been made, the nuclear spin does not affect the experimental values of the heat capacity used in the determination of entropy by the procedure based on the third law of thermodynamics ( 23b). Presumably if heat capacities could be measured right down to the absolute zero, a temperature would be reached at which the nuclear spin energy began to change and thus made a contribution to the heat capacity. The entropy derived from such data would presumably include the nuclear spin contribution of R In (2i + 1) for each atom. Special circumstances arise with molecular hydrogen and deuterium to which reference will be made below ( 24n). 

The thermal entropy of normal deuterium was found to be 33.90 E.u. mole . Normal deuterium consists of two parts of ortho- to one part of para-molecules at low temperatures the former occupy six and the latter nine closely spaced levels. The spin of each deuterium nucleus is 1 unit. Show that the practical standard entropy of deuterium gas at 25 C is 34.62 e.u. mole" (Add the entropy of mixing to the thermal entropy and subtract the nuclear spin contribution.) Compare the result with the value which would be obtained from statistical calculations, using moment of inertia, etc. in Chapter VI. 

Nuclear spin contribution to rotational heat capacity should be taken into account. The Cf orth0) and Cj para) are obtained from the corresponding partition functions... 

As mentioned earlier, the neglect of nuclear spin contributions is not possible in the presence of quadrupole nuclei such as, for instance, (nuclear spin / = 1, nuclear quadrupole moment =0.190 x 10 3 esu cm 2, 65) nuclear magnetic g-value g/ =0.4036). For a detailed discussion of the nuclear quadrupole interactions in the absence of exterior fields the reader is referred to the review article by W. Zeil in Vol. 30 (1972) of this series as well as other references... 

The nuclear spin partition function is given by (2i + 1) where i is the nuclear spin quantum number, since the energy of nuclear orientations is very small compared with kT Thus, for the hydrogen atom ( H), / = 1/2 and for the chlorine atom i = 3/2 giving nuclear spin contributions of 2 and 4, respectively, to the partition function. However, it is conventional to omit these factors from calculated entropies. 

Nuclear Spin Contribution. The nuclear spin partition function is the product of the nuclear spin multiplicity 2ij -H 1) for all the atoms in the molecule, where /V is the nuclear spin of the yth atom. Since, apart from processes involving molecular hydrogen and its isotopes (and the transmutation of the elements), nuclear spins are conserved, this contribution is conventionally omitted from the total entropy leaving the practical or virtual entropy (these adjectives are frequently omitted also). 

The interaction of the electron spin s magnetic dipole moment with the magnetic dipole moments of nearby nuclear spins provides another contribution to the state energies and the number of energy levels, between which transitions may occur. This gives rise to the hyperfme structure in the EPR spectrum. The so-called hyperfme interaction (HFI) is described by the Hamiltonian... 

While all contributions to the spin Hamiltonian so far involve the electron spin and cause first-order energy shifts or splittings in the FPR spectmm, there are also tenns that involve only nuclear spms. Aside from their importance for the calculation of FNDOR spectra, these tenns may influence the FPR spectnim significantly in situations where the high-field approximation breaks down and second-order effects become important. The first of these interactions is the coupling of the nuclear spin to the external magnetic field, called the... 

We now compare the results calculated for the fundamental frequency of the symmetric stretching mode with the only available experimental datum [78] of 326 cm . The theoretical result is seen to exceed experiment by only 8.3%. It should be recalled that the Li3 and Li3 tiimers have for lowest J the values 0 and respectively. Thus, the istopic species Li3 cannot contribute to the nuclear spin weight in Eq. (64), since the calculations for half-integer J should employ different nuclear spin weights. Note that atomic masses have been used... 

The values of S° represent the virtual or thermal entropy of the substance in the standard state at 298.15 K (25°C), omitting contributions from nuclear spins. Isotope mixing effects are also excluded except in the case of the H—system. 

The equivalent of the spin-other-orbit operator in eq. (8.30) splits into two contributions, one involving the interaction of the electron spin with the magnetic field generated by the movement of the nuclei, and one describing the interaction of the nuclear spin with the magnetic field generated by the movement of the electrons. Only the latter survives in the Born-Oppenheimer approximation, and is normally called the Paramagnetic Spin-Orbit (PSO) operator. The operator is the one-electron part of... 

There are in principle also energy levels associated with nuclear spins. In the absence of an external magnetic field, these are degenerate and consequently contribute a constant term to the partition function. As nuclear spins do not change during chemical reactions, we will ignore this contribution. 

The contributions of the second order terms in for the splitting in ESR is usually neglected since they are very small, and in feet they correspond to the NMR lines detected in some ESR experiments (5). However, the analysis of the second order expressions is important since it allows for the calculation of the indirect nuclear spin-spin couplings in NMR spectroscoi. These spin-spin couplings are usually calcdated via a closed shell polarization propagator (138-140), so that, the approach described here would allow for the same calculations to be performed within the electron Hopagator theory for open shell systems. 

In ESR, it is also customary to classify relaxation processes by their effects on electron and nuclear spins. A process that involves an electron spin flip necessarily involves energy transfer to or from the lattice and is therefore a contribution to Tx we call such a process nonsecular. A process that involves no spin flips, but which results in loss of phase coherence, is termed secular. Processes that involve nuclear spin flips but not electron spin flips are, from the point of view of the electron spins, nonsecular, but because the energy transferred is so small (compared with electron spin flips) these processes are termed pseudosecular. 

---