First-order nuclear frequencies

Under the assumption the first-order nuclear frequencies... 

A further contribution to the first order ENDOR frequencies arises from the nuclear dipole-dipole interaction = IDK between the two nuclei I and K. The shifts of the ENDOR lines of nucleus I due to %CD are described by mKD33(ms), with D33(m ) = R3CI(ms)DCK(ms)R3/cI(ms)cK(ms). In transition metal complexes this interaction is... 

First order ENDOR frequencies of nonequivalent nuclei or of pairs of magnetically equivalent nuclei are given by Eq. (3.3) which is derived from the direct product spin base. To obtain correct second order shifts and splittings, however, adequate base functions have to be used. We start the discussion of second order contributions with the most simple case of a single nucleus and will then proceed to more complex nuclear spin systems. 

The situation with a pair of spins is illustrated in Fig. 1, where the static field B0 is along the z-axis. To first order and neglecting terms in the expansion of Eq. (2) which lead to a change in nuclear quantum numbers of 1 or 2 (i.e. Am — 1 and + 2 transitions), the effect of is to split the Zeeman levels into many closely spaced energy levels, thereby causing a distribution of resonant frequencies and consequently a broad line. Eq. (2) has been simplified by the van Vleck formula 2... 

Fourthly, two unpaired electrons interact because of the overlap of their electronic orbitals. This gives rise to the so-called exchange energy, which again changes the resonance frequency of the individual electrons compared to that of the free electron. In Table 3 the four interactions are tabulated, together with their mathematical expressions. We have neglected the nuclear Zeeman interaction as this is more than six hundred times smaller than the electronic Zeeman interaction and, to first order, does not influence the EPR resonance. 

Application of B at the resonance frequency results in both energy absorption (+ nuclei become -and emission (— nuclei become + ). Because initially there are more + than -1 nuclei, the net effect is absorption. As B irradiation continues, however, the excess of +5 nuclei disappears, so that the rates of absorption and emission eventually become equal. Under these conditions, the sample is said to be approaching saturation. The situation is ameliorated, however, by natural mechanisms whereby nuclear spins move toward equilibrium from saturation. Any process that returns the z magnetization to its equilibrium condition with the excess of spins is called spin-lattice, or longitudinal, relaxation and is usually a first-order process with time constant T. For a return to equilibrium, relaxation also is necessary to destroy magnetization created in the xy plane. Any process that returns the X and y magnetizations to their equilibrium condition of zero is called spin-spin, or transverse, relaxation and is usually a first-order process with time constant T2. 

Notice that the next set of energy derivatives are also of considerable chemical interest the second derivatives of the energy with respect to nuclear positions are the (harmonic) force constants which determine the vibration frequencies of the molecule. To obtain these numbers it is necessary to obtain the (first-order) derivatives of the variational parameters with respect to the nuclear coordinates (with n = 1, 2n -f-1 = 3). 

The most precise measurements of the fine-structure parameters D and E have in fact been carried out using zero-field resonance. Figure 7.6 shows the three zero-field transitions in the Ti state of naphthalene molecules in a biphenyl crystal at T = 83 K. In these experiments, the absorption of the microwaves was detected as a function of their frequency [5]. The lines are inhomogeneously broadened and nevertheless only about 1 MHz wide. Owing to the small hnewidth of the zero-field resonances, the fine-structure constants can be determined with a high precision. This small inhomogeneous broadening is due to the hyperfine interaction with the nuclear spins of the protons (see e.g. [M2] and [M5]). For triplet states in zero field, the hyperfine structure vanishes to first order in perturbation theory, since the expectation value of the electronic spins vanishes in all three zero-field components (cf Sect. 7.2). The hyperfine structure of the zero-field resonances is therefore a second-order effect [5]. 

The quantity Gm contains contributions from the hyperfine tensor A and the nuclear Zeeman term i n- A quadrupole energy term Pm contributes when I > Vi. By applying 2nd order corrections frequencies can be obtained with better accuracy when the condition Gm Pm does not strictly apply. Equation (3.33) for the ENDOR transition probability first given by Toriyama et al. [45b] applies for species with small g anisotopy. 

The nuclear quadrupole frequency parameter P contains second-order corrections to the first-order part. 

---