Nuclear spin-vibration interaction

We note that the nuclear spin-vibration interaction, anticipated in equation (4.15), is actually identically zero. This is because the only vibrational mode for a diatomic molecule is that associated with bond stretching. Such motion does not generate any angular momentum and so does not produce a magnetic field with which the nuclear spin can interact. 

The first of these two terms describes the interaction of the magnetic moments due to electron spin and nuclear rotation and it is therefore called the spin-rotation interaction the second term is the corresponding spin-vibration interaction. Similarly we have,... 

These represent the nuclear spin Zeeman interaction, the rotational Zeeman interaction, the nuclear spin-rotation interaction, the nuclear spin-nuclear spin dipolar interaction, and the diamagnetic interactions. Using irreducible tensor methods we examine the matrix elements of each of these five terms in turn, working first in the decoupled basis set rj J, Mj /, Mi), where rj specifies all other electronic and vibrational quantum numbers this is the basis which is most appropriate for high magnetic field studies. In due course we will also calculate the matrix elements and energy levels in a ry, J, I, F, Mf) coupled basis which is appropriate for low field investigations. Most of the experimental studies involved ortho-H2 in its lowest rotational level, J = 1. If the proton nuclear spins are denoted I and /2, each with value 1 /2, ortho-H2 has total nuclear spin / equal to 1. Para-H2 has a total nuclear spin / equal to 0. 

The interaction of a molecular species with electromagnetic fields can cause transitions to occur among the available molecular energy levels (electronic, vibrational, rotational, and nuclear spin). Collisions among molecular species likewise can cause transitions to occur. Time-dependent perturbation theory and the methods of molecular dynamics can be employed to treat such transitions. 

The quantum alternative for the description of the vibrational degrees of freedom has been commented by Westlund et al. (85). The comments indicate that, to get a reasonable description of the field-dependent electron spin relaxation caused by the quantum vibrations, one needs to consider the first as well as the second order coupling between the spin and the vibrational modes in the ZFS interaction, and to take into account the lifetime of a vibrational state, Tw, as well as the time constant,T2V, associated with a width of vibrational transitions. A model of nuclear spin relaxation, including the electron spin subsystem coupled to a quantum vibrational bath, has been proposed (7d5). The contributions of the T2V and Tw vibrational relaxation (associated with the linear and the quadratic term in the Taylor expansion of the ZFS tensor, respectively) to the electron spin relaxation was considered. The description of the electron spin dynamics was included in the calculations of the PRE by the SBM approach, as well as in the framework of the general slow-motion theory, with appropriate modifications. The theoretical predictions were compared once again with the experimental PRE values for the Ni(H20)g complex in aqueous solution. This work can be treated as a quantum-mechanical counterpart of the classical approach presented in the paper by Kruk and Kowalewski (161). 

We come now to the second study, described ten years later [23]. The main development was the employment of a tunable dye laser to pump the A <— X + transition. Rotational levels in the ground state with J = 1 to 29, in the v = 0 vibrational level, were pumped by the laser and radioffequency hyperfine transitions studied. The range of J levels studied meant that the effective Hamiltonian required the addition of terms describing the dipolar and scalar interactions between the 23Na nuclear spins. These terms were given earlier in our discussion of the D2 molecule, and the complete effective Hamiltonian is ... 

Note, for completeness, that the vibrational and electronic spin parts of the total wave function are both unaffected by I i, that is, they are symmetric. We now have all the information required to derive expressions for the zero-field spin spin and spin rotation energies of the N, J levels for both para- and ortho-H2, excluding nuclear magnetic hyperfine interaction for ortho-H2 which we will come to in due course. These are given in table 8.6. 

The agreement between experiment and theory is now much better than before, the discrepancy having been reduced from 5.444 to 0.182 MHz, but it is still poor compared with the experimental accuracy which is quoted as 0.01 MHz. However, our theory is still approximate because the electron spin spin interaction mixes N = 2 with N = 4, which introduces more hyperfine matrix elements off-diagonal in both N and J. The nuclear spin-rotation term, equation (8.271), does not contribute to the first-order energy of the N = 0 level, and makes a negligible second-order contribution. We will not pursue this analysis any further, our aim having been to illustrate the complexity of the fitting process moreover this was achieved for 13 different vibrational levels. 

There are numerous interactions which are ignored by invoking the Born-Oppenheimer approximation, and these interactions can lead to terms that couple different adiabatic electronic states. The full Hamiltonian, H, for the molecule is the sum of the electronic Hamiltonian, the nuclear kinetic energy operator, Tf, the spin-orbit interaction, H, and all the remaining relativistic and hyperfine correction terms. The adiabatic Born-Oppenheimer approximation assumes that the wavefunctions of the system can be written in terms of a product of an electronic wavefunction, (r, R), a vibrational wavefunction, Xni( )> rotational wavefunction, and a spin wavefunction, Xspin- However, such a product wave-function is not an exact eigenfunction of the full Hamiltonian for the... 

Nonconformity with the third law also arises when a substance can exist in more than one state of such low energy that the distribution among these states is not influenced by the falling temperature down to the lowest attainable T. These states are frequently due to interactions of electronic or nuclear magnetic dipoles. An extrapolation to zero using the Debye law would reduce the system to a state of zero vibrational entropy, but the rotational entropy due to nuclear spin or the entropy associated with random orientation of magnetic dipoles in... 

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