# Reduced Spin Hamiltonian in High-Field NMR Spectroscopy

In high-field NMR the Zeeman term *H** _{Z}* dominates the spin Hamiltonian and in the absence of the high-frequency B

_{rf}. field, the energy levels of the total Hamiltonian can be determined using first-order perturbation theory, considering

*H*

*as the unperturbed Hamiltonian H*

_{Z}_{0}=

*H*

*and the remaining interactions as the perturbation H*

_{Z}_{1}=

*H*

_{a}*+ H*j

*+ H*

_{D}*+ H*q. In this case the different terms in the perturbation Hamiltonian H

_{1}can be simplified, remaining only those parts that contribute in first-order perturbation theory thus commuting with H

_{0}, they are called the adiabatic or secular part of H

_{1}(Ernst et al., 1992). To find the secular part of H

_{1}we note that all the terms in

*H*1 correspond to a contraction of two second-rank tensors, one a tensor operator derived from the spin operators 5

_{!;}-,

_{lm}*= I*

_{i}*t*

*l*

*I*

*j*

_{i}*m*and another tensor

*R*

*fj*dependent upon the interaction considered

_{lm}*a:*

Every second-rank tensor can be decomposed in a sum of three terms respectively a symmetric traceless tensor, an antisymmetric tensor and a scalar the trace:

I

where *Si _{Im}* = f

^{1}

*(%*

_{Im}+*%*3

_{M}) —*8i*kk] is a symmetric traceless tensor,

_{m}J2l_{=x}Sij,*Si*[1 (S

_{Im}=*is an antisymmetric tensor and*

_{i;}-,_{Im}— Sij_{tmI})]*Sj = k=x Sij,*is a scalar, the trace of

_{kk}*Sij*With the

_{iIm}.same decomposition applied to the tensor *R ^{a} _{Im},* Hi

*,*becomes:

_{a}

Considering without loss of generality the external magnetic induction Bo along the *Z* axis of the laboratory frame, B_{0} = *B _{0}e_{z}* and introducing the raising and lowering spin operators

*I*and I

_{k}+ = I_{kx}+H_{k},_{y }_{k},

_{—}= I

_{k},

*it is possible to show that in first-order perturbation theory the different contributions to H*

_{x}— iI_{ky}_{i}become: