Superoperator formalism

It is possible to perform a systematic decoupling of this moment expansion using the superoperator formalism (34,35). An infinite dimensional operator vector space defined by a basis of field operators Xj which supports the scalar product (or metric)... 

The basic idea of the slow-motion theory is to treat the electron spin as a part of the lattice and limit the spin part of the problem to the nuclear spin rather than the IS system. The difficult part of the problem is to treat, in an appropriate way, the combined lattice, now containing the classical degrees of freedom (such as rotation in condensed matter) as well as quantized degrees of freedom (such as the electron Zeeman interaction). The Liouville superoperator formalism is very well suited for treating this type of problems. 

We shall shortly consider such fundamental concepts as density matrices and the superoperator formalism which are convenient to use in a formulation of the lineshape theory of NMR spectra. The general equation of motion for the density matrix of a non-exchanging spin system is formulated in the laboratory (non-rotating) reference frame. The lineshape of a steady-state, unsaturated spectrum is given as the Fourier transform of the free induction decay after a strong radiofrequency pulse. The equations provide a starting point for the formulation of the theory of dynamic NMR spectra presented in Section III. The reader who may be interested in a more detailed consideration of the problems is referred to the fundamental works of Abragam and... 

Before writing an equation for the motion of a spin system, we introduce a convenient notation system, the so-called superoperator formalism. It was used for the first time in a description of NMR phenomena by Banwell and Primas, and later developed by Binsch. (12— 15)... 

The superoperator formalism that has been used in previous publications is outlined here [2, 9, 29], The alternative diagrammatic and algebraic-diagrammatic representations can be found in other works [6],... 

A. THE STOCHASTIC RELAXATION MODEL. The most general theories of magnetic relaxation in Mossbauer spectroscopy involve stochastic models see, for example. Ref. 283 for a review. A formalism using superoperators (Liouville operators) was introduced by Blume, who presented a general solution for the lineshape of radiation emitted (absorbed) by a system whose Hamiltonian jumps at random as a function of time between a finite number of possible forms that do not necessarily commute with one another. The solution can be written down in a compact form using the superoperator formalism. 

This series can be expressed in a more compact form by using the so-called superoperator formalism (Goscinski and Lukman, 1970). We introduce this formalism here, as we had introduced the interaction picture in Section 3.8, in order to facilitate our derivations. The final equations will, however, be written without any superoperators. The superoperator formalism is one level of abstraction higher than the Hilbert vector space of quantum mechanics. In the infinite-dimensional Hilbert space the vectors of the vector space are given as quantum mechanical wavefunctions and the transformations performed on the vectors in the vector space are given by the quantum mechanical operators. The binary product defined in Hilbert space is the overlap integral /) between two wavefunctions, and 4 . In the superoperator formalism we now have an infinite-dimensional vector space, where the quantmn... 

Making use of this superoperator formalism, the moment expansion of the polarization propagator can be written as... 

However, this is only a cosmetic change, because the superoperator resolvent is an inverse operator and is only defined through its series expansion in Eq. (3.148). The way in which we can proceed, is to find a matrix representation of the superoperator resolvent. In order to do that we need a complete set of basis vectors like in the normal Hilbert space. However, the vectors in the superoperator formalism are operators and we therefore need a complete set of operators. Such a set of operators hn consists of a complete set of excitation and de-excitation operators with respect to the reference state This means that all other states of the system or all excited states of... 

Using this relation in Eq. (3.152) leads us to an exact matrix representation of the polarization propagator in the superoperator formalism... 

The formulation of approximate response theory based on an exponential parame-trization of the time-dependent wave function, Eq. (11.36), and the Ehrenfest theorem, Eq. (11.40), can also be used to derive SOPPA and higher-order Mpller-Plesset perturbation theory polarization propagator approximations (Olsen et al., 2005). Contrary to the approach employed in Chapter 10, which is based on the superoperator formalism from Section 3.12 and that could not yet be extended to higher than linear response functions, the Ehrenfest-theorem-based approach can be used to derive expressions also for quadratic and higher-order response functions. In the following, it will briefly be shown how the SOPPA linear response equations, Eq. (10.29), can be derived with this approach. 

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