Hermitian superoperator

In the renormalized composite Liouville space the superoperator F , defined by equation (121), commutes with all the (Hermitian) superoperators Hf, R, and X. Therefore, upon the proper rearrangement of the basis set in this space, one can obtain a factorization of the equation (137) of motion into blocks which are connected with individual eigenvalues of the superoperator F°. This resembles the analogous procedure in the case of static NMR spectra, i.e. those for non-exchanging spin systems (Section II.E.2). The equations for the free induction decay M ID and for the lineshape of an unsaturated steady-state spectrum, in terms of quantities from composite Liouville space, are therefore obtained for exchanging spin systems in a way which is analogous to that for non-exchanging systems (Section II.F). 

The matrix Hfj would be the transpose of Hf, if it were Hermitian. The Hermiticity of the superoperator Hamiltonian has been a concern since the beginnings of the electron propagator theory (46,129). For a Hermitian spin ftee Hamiltonian (// ) the following relation can be written describing the Hermiticity problem,... 

It is evident that if the operator A is Hermitian, then the superoperators AD, AL, and AR are also Hermitian. 

The matrices in equation (35) for a system of n spins of 1/2 have dimensions of 22n. This means that, for example, a four-spin system must be considered within a space of 256 dimensions. If we deal with the motion of a spin system in a static magnetic field (as in pulse-type experiments), significant simplifications are possible owing to the rules of commutation. Namely, if the Hermitian operators A and 6 commute in Hilbert space, then all the corresponding superoperators AL, AR, AD, BL, BR, and BD in Liouville space also commute. The proof of this is given in reference (12). In Hilbert space, the following commutation takes place ... 

Any derivation superoperator AD which is constructed of a Hermitian operator A obeys the relation ... 

It follows from equation (84) that the superoperators Q in equations (107a) to (107d) are represented by unitary matrices. Thus, their inverses can be replaced by their Hermitian adjoints ... 

Equation (116) has a form which is similar to that of the equation (35) of motion for non-exchanging spin systems. The analogy is even closer, as is shown later, since a judicious renormalization of the vectors in the composite Liouville space can convert equation (116) into one in which all the superoperators become Hermitian. Firstly we wish to draw attention to some of the properties of the exchange superoperator X. ... 

The superoperator X may always be reduced to diagonal form by a proper similarity transformation, because there is always one which can convert X into a Hermitian form X ... 

In order to prove the Hermitian character of the superoperator X it is sufficient to show that ... 

Equation (149) is valid provided that the matrix T and that within the brackets are non-singular. It has been shown that this is always true in the case considered. (47) Since Newmark and Sederholm have used the exchange superoperator in its non-Hermitian form, such as that given in equation (117), their method is rather time-consuming. Probably, this is the reason why the method has been largely forgotten. 

The dissipative superoperator Wp can also be constructed from a dissi-pative potential operator Wp which describes the interaction of the p-region with the electrons in the s-region followed by a back interaction, [8] and therefore depends on the initial equilibrium temperature T of the s-region. This leads, for a general non-hermitian Wpel to the expression... 

As a Gnal remark before dosing this section, we emphasize that everything that has been said for Hermitian and relaxation operators also applies to Hermitian or relaxation superoperators (see also Chapters I and IV). Hie formal changes to be performed are trivial the state of interest /q) is to be replaced by the operator of interest. /4o)> operator H by the superoperator (— L) where L = [H,...], and the scalar product by a suitable average on an appropriate equilibrium distribution. The moments now have the form... 

All the mathematical apparatus of Hankel determinants and continued fractions expansion apply also to Hermitian or relaxation superoperators. 

Earlier in this chapter, we noted that the question of the hermiticity of f) T ) had to be examined in individual cases (i.e., it was not automatically valid). When a perturbation expansion is used to determine the reference slate, we may more explicitly state the conditions under which the matrix is hermitian by examining the difference between the (k/)th and the complex conjugate of the (/k)th element of the superoperator Hamiltonian. When this difference... 

---