Liouville superoperator

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). 

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. 

In this approach, the diffusion constant, Di, is related to the corresponding characteristic time, x, describing the distortions of the normal coordinate, Westlund et al. (85) used the framework of the general slow-motion theory to incorporate the classical vibrational dynamics of the ZFS tensor, governed by the Smoluchowski equation with a harmonic oscillator potential. They introduced an appropriate Liouville superoperator ... 

This result is the Redfield-Liouville-von Neumann equation of motion or, simply, the Redfield equation [29,30,49-53]. Here the influence of the bath is contained entirely in the Redfield relaxation tensor, 3i, which is added to the Liouville operator for the isolated subsystem to give the dissipative Redfield-Liouville superoperator (tensor) that propagates (T. Expanded in the eigenstates of the subsystem Hamiltonian, H, Eq. (9) yields a set of coupled linear differential equations for the matrix... 

Bj = radiofrequency field = total magnetization h = Planck s constant = perturbation RF field along x i = 4 k = rate of exchange = kinetic superoperator matrix L = Liouville superoperator ... 

In this case the associated Liouville superoperator L t) with 36x36 total matrix elements is diagonal and the expression for the transverse magnetisation can be cast into three contributions with formal solutions in a scalar form ... 

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... 

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) ... 

In this regard, we should notice that the time evolution of a quantum system is ruled by two different types of eigenvalues corresponding to the wave function and the statistical descriptions. On the one hand, we have the eigenenergies of the Hamiltonian within the wave function description. On the other hand, we have the eigenvalues of the Landau-von Neumann superoperator in the Liouville formulation of quantum mechanics. These quantum Liouvillian eigenvalues j are related to the Bohr frequencies according to... 

These challenges can be dealt with the powerful mathematical tools of quantum chemistry, as advocated by Per-Olov Lowdin.[l, 2, 3, 4] In our studies, linear algebras with matrices,[4] partitioning techniques,[3] operators and superoperators in Liouville space, and the Liouville-von Neumann... 

The powerful mathematical tools of linear algebra and superoperators in Li-ouville space can be used to proceed from the identification of molecular phenomena, to modelling and calculation of physical properties to interpret or predict experimental results. The present overview of our work shows a possible approach to the dissipative dynamics of a many-atom system undergoing localized electronic transitions. The density operator and its Liouville-von Neumann equation play a central role in its mathematical treatments. 

P. O. Lowdin. On operators, superoperators, hamiltonians and liouvil-lians. Intern. J. Quantum Chem., QCS 16 485, 1982. 

F. Properties of the exchange superoperator in composite Liouville space 254... 

Every linear transformation of a set of linear operators onto the same set (e.g. premultiplication or postmultiplication of the operators by a given operator) may be represented in Liouville space as a matrix (a superoperator). We shall calculate, for example, the representation of the commutator [A, ], the so-called derivation superoperator, and its... 

Superoperators in Liouville space are denoted by capital Latin or Greek letters, in boldface. The representation of the derivation superoperator AD within the basis set of v vectors is given by ... 

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 ... 

Hence, in Liouville space, the energy superoperator HP commutes with all the mutually commuting superoperators Fp, Ff, and Fp ... 

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 exchange superoperator X commutes with the superoperator Ff in composite Liouville space ... 

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). 

In equation (143), the subscript (—1) denotes the subspace of the composite Liouville space which is concerned with the eigenvalue — 1 of the superoperator Ff (the subspace of single-quantum transitions). The vector fx is normalized according to equation (135) and the q, (t) function has the same meaning as in equation (50). The coefficient C is given by [equation (139)1 ... 

By analogy with non-exchanging spin systems the superoperators which commute with both the super-Hamiltonian and the superoperator T in composite Liouville space may be called the constants of the motion. In some instances there may be additional constants of the motion which result from the conservation of some molecular symmetry in the exchange, from the magnetic equivalence of some nuclei, and from weak spin-spin coupling. (15, 52) For example,... 

It is obvious that the superoperator II acts as a projection operator in the Liouville space, cutting out those components of the 10 x 10 transition density matrices which mix 7h state with the 7l state, which is only possible if the symmetries of the perturbations of both the symmetry of deformation rdef and the symmetry of substitution Ts satisfy the selection rule ... 

When considering relaxation, a Liouville space representation is typically used in which the Hamiltonian and density matrix are represented as superoperators in addition to the relaxation operator being represented as a superoperator. Once a... 

It is possible to perform more precise calculations that simultaneously account for the coherent quantum mechanical spin-state mixing and the diffusional motion of the RP. These employ the stochastic Liouville equation. Here, the spin density matrix of the RP is transformed into Liouville space and acted on by a Liouville operator (the commutator of the spin Hamiltonian and density matrix), which is then modified by a stochastic superoperator, to account for the random diffusive motion. Application to a RP and inclusion of terms for chemical reaction, W, and relaxation, R, generates the equation in the form that typically employed... 

Because T operates on each element of a matrix it is called a superoperator. In fact, the Hilbert-space formulation of quantum mechanics leading to the von Neumann equation of motion of the density matrix can be simplified considerably by introduction of a superoperator notation in the so-called Liouville space. Furthermore, for the analysis of NMR experiments with complicated pulse sequences it is of great help to expand the density matrix into products of operators, where each product operator exhibits characteristic transformation properties under rotation [Eml]. 

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