Superoperator

In the presence of some fomi of relaxation the equations of motion must be supplemented by a temi involving a relaxation superoperator—superoperator because it maps one operator into another operator. The literature on the correct fomi of such a superoperator is large, contradictory and incomplete. In brief, the extant theories can be divided into two kinds, those without memory relaxation (Markovian) Tp and those with memory... 

It is more convenient to re-express this equation in Liouville space [8, 9 and 10], in which the density matrix becomes a vector, and the commutator with the Hamiltonian becomes the Liouville superoperator. In tliis fomuilation, the lines in the spectrum are some of the elements of the density matrix vector, and what happens to them is described by the superoperator matrix, equation (B2.4.25) becomes (B2.4.26). 

This may be further transformed by an inner projection onto a complete set of excitation and de-excitation operators, h this is equivalent to inserting a resolution of the identity in the operator space (remember that superoperators work on operators). 

In (7.90) a slightly modified notation is introduced for convenience for the bra and ket vectors in the Liouville space for the resolvent superoperator... 

Through introduction of superoperators and a corresponding metric [13], the propagator may be represented more compactly [2, 6]. Superoperators act on field operator products, X, where the number of annihilators exceeds the number of creators by one. The identity superoperator, I, and the Hamiltonian superoperator, H, are defined by... 

Thus the matrix elements of the electron propagator are related to field operator products arising from the superoperator resolvent, El — H), that are evaluated with respect to N). In this sense, electron binding energies and DOs are properties of the reference state. 

Poles of the propagator therefore occur at values of E that are equal to eigenvalues, u>, of the superoperator Hamiltoniem matrix ... 

The usual choice of superoperator metric starts from a HF wavefunction plus perturbative corrections [4, 5] ... 

This choice produces asymmetric superoperator matrices. A simplified final form for the self-energy matrix that does not require optimization of cluster amplitudes is sought for large molecules the approximation... 

According to equation 15, eigenvalues of the superoperator Hamiltonian matrix, H, are poles (electron binding energies) of the electron propagator. Several renormalized methods can be defined in terms of approximate H matrices. The... 

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)... 

In this form, it is necessary to perform the inversion of a superoperator which is very cumbersome, so an inner p ojection technique (79-81) or fte resolution of the superoperator identity (49),... 

This e qnession for the propagators is still exact, as long as, the principal sub-manifold h and its complement sub-manifold h arc complete, and the characteristics of the propagator is reflected in the construction of these submanifolds (47,48). It should be noted that a different (asymmetric) metric for the superoperator space, Eq. (2.5), could be invoked so that another decoupling of the equations of motion is obtained (62,63,82-84). Such a metric will not be explored here, but it just shows the versatility of the propagator methods. 

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,... 

The matrix H 3 can be obtained fiom the general ex)x ession for the superoperator Hamiltonian matrix H33 recently derived (126). For the non-diagonal one-electron Hamiltonian it can be written as... 

The action of the Hamiltonian, H, can be expressed as a superoperator mapping the Hilbert space 2 (iK j into itself by... 

H The Hamiltonian superoperator (the Hamiltonian represented as an operator acting on Hilbert Schmidt operators)... 

For in-situ studies of reaction mechanisms using parahydrogen it is desirable to compare experimentally recorded NMR spectra with those expected theoretically. Likewise, it is advantageous to know, how the individual intensities of the intermediates and reaction products depend on time. For this purpose a computer simulation program DYPAS2 [45] has been developed, which is based on the density matrix formalism using superoperators, implemented under the C++ class library GAMMA. 

The kinetics of hydrogenation transfer is covered by the use of an exchange superoperator assuming a pseudo first-order reaction. Thereby, competing hydrogenations of the substrate to more than one product can also be accommodated. In addition, the consequences of relaxation effects or NOEs can be included into the simulations if desired. Furthermore, it is possible to simulate the consequences of different types of pulse sequences, such as PH-INEPT or INEPT+, which have previously been developed for the transfer of polarization from the parahydrogen-derived protons to heteronuclei such as 13C or 15N. The... 

MSN. 184. G. Ordonez, T. Petrosky, E. Karpov, and I. Prigogine, Exphcit construction of a time superoperator for quanmm unstable systems. Chaos, SoUtons and Fractals 12, 2591-2601 (2001). 

The stochastic Liouville equation, in the form relevant for the ESR line shape calculation, can be written in a form reminiscent of the Redfield equation in the superoperator formulation, Eq. (19) (70-73) ... 

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. 

Bertini and co-workers 119) and Kruk et al. 96) formulated a theory of electron spin relaxation in slowly-rotating systems valid for arbitrary relation between the static ZFS and the Zeeman interaction. The unperturbed, static Hamiltonian was allowed to contain both these interactions. Such an unperturbed Hamiltonian, Hq, depends on the relative orientation of the molecule-fixed P frame and the laboratory frame. For cylindrically symmetric ZFS, we need only one angle, p, to specify the orientation of the two frames. The eigenstates of Hq(P) were used to define the basis set in which the relaxation superoperator Rzpsi ) expressed. The superoperator M, the projection vectors and the electron-spin spectral densities cf. Eqs. (62-64)), all become dependent on the angle p. The expression in Eq. (61) needs to be modified in two ways first, we need to include the crossterms electron-spin spectral densities, and These terms can be... 

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