Electronic interactions in the nuclear Hamiltonian

In order to complete our derivation of the molecular Hamiltonian we must consider the nuclear Hamiltonian in more detail. A thorough relativistic treatment analogous to that for the electron is not possible within the limitations of quantum mechanics, since nuclei are not Dirac particles and they can have large anomalous magnetic moments. However, the use of quantum electrodynamics [18] shows that we can derive the correct Hamiltonian to order 1 /c2 by taking the non-relativistic Hamiltonian  

Substitution of (3.249) into (3.248) leads in a straightforward manner to the result  

The second term in (3.250) yields the spin-rotation and spin-vibration interactions, whilst the third term leads to the orbit-rotation and orbit-vibration interactions. We will examine these terms in more detail in the next section. 

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A more general theory for outer-sphere paramagnetic relaxation enhancement, valid for an arbitrary relation between the Zeeman coupling and the axial static ZFS, has been developed by Kruk and co-workers (96 in the same paper which dealt with the inner-sphere case. The static ZFS was included, along with the Zeeman interaction in the unperturbed Hamiltonian. The general expression for the nuclear spin-lattice relaxation rate of the outer-sphere nuclei was written in terms of electron spin spectral densities, as ... 

The form of the nuclear electric quadrupole interaction in the effective Hamiltonian for a diatomic molecule is given in equations (7.158) and (7.161), with the latter applying only to molecules in n electronic states. The two parameters which can be determined from a fit of the experimental data are eqo Q and et/i Q respectively. Since the electric quadrupole moment eQ is known for most nuclei, an experimental observation gives information on q0 (and perhaps qi), the electric field gradient at the nucleus. This quantity depends on the electronic structure of the molecule according to the expression... 

In all of the preceding discussion, we have simply assumed that we have some kind of one-particle potential V, which is used to set up the one-particle Hamiltonian matrix. We should now consider the effect of the electron-electron interaction in the discussion. Intuition tells us that since the solutions are mostly dominated by the nuclear Coulomb attraction, things will not change much, if at all. 

In this chapter, we discussed the permutational symmetry properties of the total molecular wave function and its various components under the exchange of identical particles. We started by noting that most nuclear dynamics treatments carried out so far neglect the interactions between the nuclear spin and the other nuclear and electronic degrees of freedom in the system Hamiltonian. Due to... 

Electron spin resonance (ESR), or electron paramagnetic resonance (EPR) as it is sometimes known, shares many similarities with its cousin, NMR. The origin of the phenomenon is the spin of the electron (rather than the nuclear spin) coupling with the nuclear spins of the atoms in the polymer, but much of the physics of their interactions are similar. The usual spin Hamiltonian, which is used to determine the energies of the interactions, can be written as... 

The summation index n has the same meaning as in Eq. (31), i.e., it enumerates the components of the interaction between the nuclear spin I and the remainder of the system (which thus contains both the electron spin and the thermal bath), expressed as spherical tensors. are components of the hyperfine Hamiltonian, in angular frequency units, expressed in the interaction representation (18,19), with the electron Zeeman and the ZFS in the zeroth order Hamiltonian Hq. The operator H (t) is evaluated as ... 

In general, fluctuations in any electron Hamiltonian terms, due to Brownian motions, can induce relaxation. Fluctuations of anisotropic g, ZFS, or anisotropic A tensors may provide relaxation mechanisms. The g tensor is in fact introduced to describe the interaction energy between the magnetic field and the electron spin, in the presence of spin orbit coupling, which also causes static ZFS in S > 1/2 systems. The A tensor describes the hyperfine coupling of the unpaired electron(s) with the metal nuclear-spin. Stochastic fluctuations can arise from molecular reorientation (with correlation time Tji) and/or from molecular distortions, e.g., due to collisions (with correlation time t ) (18), the latter mechanism being usually dominant. The electron relaxation time is obtained (15) as a function of the squared anisotropies of the tensors and of the correlation time, with a field dependence due to the term x /(l + x ). 

The nuclear Zeeman term describes the interaction of the nuclear spins with the external magnetic field. Just as the hyperfine splitting, this term is not incorporated in the original purely electronic Breit-Pauli Hamiltonian as presented in Eqs. (59) and (60) but becomes relevant for ESR spectroscopy. 

Here He is the electronic Hamiltonian determining the wave functions and the eigenvalues of the Fe-(NO) fragment in the fixed nuclear configuration. Reading off the coordinate qA of the full symmetric vibration from its equilibrium position in the state 1Ai(Fe2+(d6)), one can write down the operator of the linear electron-vibrational interaction in the following form ... 

Here, He(j) is Hamiltonian of a free electron, V,-(r) is Coulomb s interaction of the electron with the donor ion residue, Hlv( q ) is Hamiltonian of the vibration subsystem depending on the set of the vibration coordinates qj that corresponds to the movement of nuclei without taking into account the interaction of the electron with the vibrations. The short-range (on r) potential Ui(r, q ) describes the electron interaction with the donor ion residue and with the nuclear oscillations. The wave function of the system donor + electron may be represented in MREL in the adiabatic approach (see Section 2 of Chapter 2) ... 

The unpaired electron with its spin S = 1/2 in a sample disposed into the resonator of the EPR spectrometer interacts magnetically a) with the external magnetic field H (Zeeman interaction) b) with the nuclear spin of the host atom or metal ion / (hyperfine interaction) c) with other electron spins S existing in the sample (dipole-dipole interaction). In the last case, electrons can be localized either at the same atom or ion (the so called fine interaction), for example in Ni2+, Co2+, Cr3+, high-spin Fe3+, Mn2+, etc., or others. These interac-tions are characterized energetically by the appropriate spin-Hamiltonian... 

The last stage is now to replace tv, by P, + eA, and to use explicit expressions for the potentials A, and (pi. Our previous expressions for A, and electron interactions have now been derived more naturally by starting with the Breit Hamiltonian and the vector and scalar potentials therefore contain only terms describing external fields or electrostatic interactions involving the nuclear charge. Hence we make the substitutions... 

We now show how the many-electron Hamiltonian developed in the previous chapter may be extended to include magnetic interactions which arise from the presence of nuclear spin magnetic moments. Equation (3.140) represents the Hamiltonian for electron i in the presence of other electrons we present it again here ... 

The rotational and Zeeman perturbation Hamiltonian (X) to the electronic eigenstates was given in equation (8.105). It did not, however, contain terms which describe the interaction effects arising from nuclear spin. These are of primary importance in molecular beam magnetic resonance studies, so we must now extend our treatment and, in particular, demonstrate the origin of the terms in the effective Hamiltonian already employed to analyse the spectra. Again the treatment will apply to any molecule, but we shall subsequently restrict attention to diatomic systems. 

There is a further term which should be included in the effective Hamiltonian, derived in chapter 7, describing the electron spin-nuclear rotation interaction. This may be written in the form... 

The total energy has an explicit geometry dependence in the nuclear-electron and nuclear-nuclear interaction terms, and an implicit geometry dependence in the wave function. In approximate calculations where finite nuclear-fixed basis sets are used, the total energy has an explicit dependence also in the basis set. Using the technique of second quantization, the geometry dependence of the basis set may be transferred to the Hamiltonian. In Section II we describe how the Hamiltonian at X0 + p may be expanded around X0... 

Nuclear relaxation in paramagnetic complexes occurs due to the time dependent terms in the nuclear spin Hamiltonian. The amount of relaxation effect is dependent on the intensity of electron-nuclear interaction and the rate at which this interaction is interrupted. Thus the relaxation rates of ligand nuclei are determined by the two factors, namely, molecular structure and molecular dynamics in solution. Thus the relaxation rates of ligand nuclei shed light on molecular structure and dynamics in solution. 

The chemical shift can be defined as the shielding interaction between the electron clouds and the nuclear spin angular momenta it is isotropic in liquids but has rotational anisotropy in solids. The spin Hamiltonian describing the interaction is therefore tensorial in nature ... 

The through-space interaction is a dipolar coupling between the electron and nuclear magnetic moments. When the Zeeman interaction for both the electron and nuclear spins is the dominant term in the spin Hamiltonian of the system, the energy of the dipole-dipole interaction is inversely proportional to the third power of the dipole-dipole distance fis according to... 

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