Hamiltonian magnetic constant

Basic changes in theory, definition, and designation of internal and external magnetic coupling parameters have not occurred in the past ten years. Therefore the overall-disposal and arrangement of the table of magnetic constants of polyatomic molecules in Vol. 11/19 is retained. A leading factor of -1 was, however, introduced in the spin-rotation interaction hamiltonian, see eqs. (1 a) and (2a) below. The reason for this will be outlined later in connection with eq. (4a). 

A simple, non-selective pulse starts the experiment. This rotates the equilibrium z magnetization onto the v axis. Note that neither the equilibrium state nor the effect of the pulse depend on the dynamics or the details of the spin Hamiltonian (chemical shifts and coupling constants). The equilibrium density matrix is proportional to F. After the pulse the density matrix is therefore given by and it will evolve as in equation (B2.4.27). If (B2.4.28) is substituted into (B2.4.30), the NMR signal as a fimction of time t, is given by (B2.4.32). In this equation there is a distinction between the sum of the operators weighted by the equilibrium populations, F, from the unweighted sum, 7. The detector sees each spin (but not each coherence ) equally well. 

The Hamiltonian function for an electron in a constant magnetic field of strength H parallel to the s axis is1... 

The spin Hamiltonian for the hydrogen atom will be used to determine the energy levels in the presence of an external magnetic field. As indicated in Section II.A, the treatment may be simplified if it is recognized that the g factor and the hyperfine constant are essentially scalar quantities in this particular example. An additional simplification results if the z direction is defined as the direction of the magnetic field. For this case H = Hz and Hx = Hv = 0 hence,... 

Wangsness and Bloch16>17 were the first to give a quantum mechanical treatment of spin relaxation using the density matrix formalism. The system considered is a spin interacting with an external magnetic field (which we suppose here to be constant) and with a heat bath. The corresponding Hamiltonian is... 

From the classical electrodynamics, the Dirac Hamiltonian of a hydrogen molecule moving in a constant magnetic field B is [102]... 

On matrix form the non-unitary transformations (27) and (30) of the previous section are easily extended to the complete Hamiltonian and have therefore allowed relativistic and non-relativistic spin-free calculations of spectroscopic constants and first-order properties at the four-component level (see, for instance. Refs. [45 7]). In this section, we consider the elimination of spin-orbit interaction in four-component calculations of second-order electric and magnetic properties. Formulas are restricted to the Hartree-Fock [48] or Kohn-Sham [49] level of theory, but are straightforwardly generalized. 

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. 

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... 

This spin Hamiltonian is solved in Appendix D for the S= spin system. Comparing the solution of Eq. (48) to Eqs. (41) and (42) we find that the behavior of the two ground-state functions in the presence of a magnetic field can be represented by the solution of the spin Hamiltonian of Eq. (48) in which g, and gL are simply constants to be evaluated by experiment. 

In such a system, the external magnetic field defines the molecular z axis. If we rotate the molecule with respect to Bo, the spin and its magnetic moment are not affected (Fig. 1.14B). However, in the molecule of Fig. 1.14A, a molecular z axis can be defined. When rotating the molecule, the orbital contribution to the overall magnetic moment changes, whereas the spin contribution is constant. The total Zeeman Hamiltonian is... 

---