Magnetic Hamiltonian with nuclear spin

5 INCLUSION OF THE NUCLEAR SPIN 3.5.1 Magnetic Hamiltonian with nuclear spin 

Let us consider the second important case when the energy of the system E(B, jlN) depends upon the external magnetic field and the magnetic moments of nuclei due to their spin. The total electromagnetic potential consists of two terms 

While the first term is due to the external magnetic field, the second one arises from the contribution of nuclei. The external electromagnetic potential is expressed through the magnetic induction and the position vector measured relative to the gauge origin 

When several contributions to the vector potential are simultaneously present, then squares and cross-terms exist 

Let us deal, for clarity, with a single electron and a single magnetoactive nucleus first. Then the kinetic energy operator for one electron (qt = —e, Anuc Afi) is 

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INCLUSION OF THE ELECTRON AND NUCLEAR SPINS 3.6.1 Magnetic Hamiltonian with electron and nuclear spins... 

An atomic nucleus, either in the ground or excited state, with nuclear spin quantum number />0 possesses a magnetic dipole moment and interacts with a magnetic field at the nuclear site. The magnetic field may arise from the electronic environment or from an external magnet. This interaction is called magnetic dipole interaction and is described by the Hamiltonian... 

If one starts from a formally nonrelativistic Hamiltonian, third-order perturbation theory has to be used, as the spin-orbit operator has to be included in addition to the perturbations due to the nuclear magnetic moments and to the external magnetic field. As the spin-orbit operator permits spin polarization, a Fermi contact (FC) term and a spin-dipolar (SD) term also appear in the perturbed Hamiltonian and couple nuclear magnetic moment with electronic spin. 

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... 

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

A systematic development of relativistic molecular Hamiltonians and various non-relativistic approximations are presented. Our starting point is the Dirac one-fermion Hamiltonian in the presence of an external electromagnetic field. The problems associated with generalizing Dirac s one-fermion theory smoothly to more than one fermion are discussed. The description of many-fermion systems within the framework of quantum electrodynamics (QED) will lead to Hamiltonians which do not suffer from the problems associated with the direct extension of Dirac s one-fermion theory to many-fermion system. An exhaustive discussion of the recent QED developments in the relevant area is not presented, except for cursory remarks for completeness. The non-relativistic form (NRF) of the many-electron relativistic Hamiltonian is developed as the working Hamiltonian. It is used to extract operators for the observables, which represent the response of a molecule to an external electromagnetic radiation field. In this study, our focus is mainly on the operators which eventually were used to calculate the nuclear magnetic resonance (NMR) chemical shifts and indirect nuclear spin-spin coupling constants. 

H is the total Hamiltonian (in the angular frequency units) and L is the total Liouvillian, divided into three parts describing the nuclear spin system (Lj), the lattice (Ll) and the coupling between the two subsystems (L/l). The symbol x is the density operator for the whole system, expressible as the direct product of the density operators for spin (p) and lattice (a), x = p <8> ci. The Liouvillian (Lj) for the spin system is the commutator with the nuclear Zeeman Hamiltonian (we thus treat the nuclear spin system as an ensemble of non-interacting spins in a magnetic field). Ll will be defined later and Ljl... 

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 spins give rise to additional terms in the Breit-Pauli Hamiltonian due to the interaction of the electrons with the magnetic moment of the nuclei and the electrostatic interaction with the electric quadrupole interaction of the nuclei. The magnetic interaction term of the spins with the nuclei is of the same type as the spin-spin interaction and following Abragam and Pryce (61) can be written as... 

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. 

The calculation of magnetic parameters such as the hyperfine coupling constants and g-factors for oligonuclear clusters is of fundamental importance as a tool for the evaluation of spectroscopic data from EPR and ENDOR experiments. The hyperfine interaction is experimentally interpreted with the spin Hamiltonian (SH) H = S - A-1, where S is the fictitious, electron spin operator related to the ground state of the cluster, A is the hyperfine tensor, and I is the nuclear spin operator. Consequently, it is... 

For a molecule with several nuclei with nonzero spin, magnetic moment interacts with B0 and the Hamiltonian is each nuclear... 

From the three operators obtained by differentiating the Hamiltonian with respect to the nuclear magnetic moments (Equation (2.12)) only the singlet paramagnetic spin-orbital (PSO) operator... 

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