Operator Liouville space

For a coupled spin system, the matrix of the Liouvillian must be calculated in the basis set for the spin system. Usually this is a simple product basis, often called product operators, since the vectors in Liouville space are spm operators. The matrix elements can be calculated in various ways. The Liouvillian is the conmuitator with the Hamiltonian, so matrix elements can be calculated from the commutation rules of spin operators. Alternatively, the angular momentum properties of Liouville space can be used. In either case, the chemical shift temis are easily calculated, but the coupling temis (since they are products of operators) are more complex. In section B2.4.2.7. the Liouville matrix for the single-quantum transitions for an AB spin system is presented. 

In Liouville space, both the density matrix and the operator are vectors. The dot product of these Liouville space... 

In an alternative formulation of the Redfield theory, one expresses the density operator by expansion in a suitable operator basis set and formulates the equation of motion directly in terms of the expectation values of the operators (18,20,50). Consider a system of two nuclear spins with the spin quantum number of 1/2,1, and N, interacting with each other through the scalar J-coupling and dipolar interaction. In an isotropic liquid, the former interaction gives rise to J-split doublets, while the dipolar interaction acts as a relaxation mechanism. For the discussion of such a system, the appropriate sixteen-dimensional basis set can for example consist of the unit operator, E, the operators corresponding to the Cartesian components of the two spins, Ix, ly, Iz, Nx, Ny, Nz and the products of the components of I and the components of N (49). These sixteen operators span the Liouville space for our two-spin system. If we concentrate on the longitudinal relaxation (the relaxation connected to the distribution of populations), the Redfield theory predicts the relaxation to follow a set of three coupled differential equations ... 

As in Eq. (64), the electron spin spectral densities could be evaluated by expanding the electron spin tensor operators in a Liouville space basis set of the static Hamiltonian. The outer-sphere electron spin spectral densities are more complicated to evaluate than their inner-sphere counterparts, since they involve integration over the variable u, in analogy with Eqs. (68) and (69). The main simplifying assumption employed for the electron spin system is that the electron spin relaxation processes can be described by the Redfield theory in the same manner as for the inner-sphere counterpart (95). A comparison between the predictions of the analytical approach presented above, and other models of the outer-sphere relaxation, the Hwang and Freed model (HF) (138), its modification including electron spin... 

In Liouville space, both the density matrix and the 4 operator become vectors. The scalar product of these Liouville space vectors is the trace of their product as operators. Therefore, the NMR signal, as a function of a single time variable, t, is given by (10), in which the parentheses denote a Liouville space scalar product ... 

Equation (5.38) can be interpreted as the scalar product of a forward-moving density and a backward-moving time-dependent operator. The optimal field at time t is determined by a time-dependent objective function propagated from the target time T backward to time t. A first-order perturbation approach to obtain a similar equation for optimal chemical control in Liouville space has been derived in a different method by Yan et al. [28]. 

Figure 4. TW° ways to describe the time-evolution of the dipole operators single (a) and double (b) Liouville space diagrams.

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... 

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... 

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 ... 

Lexicographically ordered sets of projection operators (Section II.D) which are composed of the vectors of the corresponding sets BK, BL, B(KL), and BKL which span the Liouville spaces LK, Ll, Lk 0 Ll and Lkl are called the basic function sets of a reaction. 

It is convenient to introduce a Liouville space, or double space, that is a direct product of cap and tilde spaces. In Liouville space, operators are considered to be vectors and Hilbert-space commutators are considered to be operators. Equation (48) is then expressed as... 

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... 

We denote linear operators acting on the Hilbert space by capital letters with a hat, such as UAt, whereas linear maps on operators (operators in the Liouville space) are denoted by calligraphic letters such as Uat, where the argument is often placed within braces in order to avoid ambiguity. 

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]. 

The operators which carry the superscript L are written in Liouville-space notation. The most important simplification arising from this notation is that a transformation OpO in Hilbert space can be written as O p in Liouville space. Here exp i tr 1 describes the phase evolution in the applied gradients, and exp -iHJ (r) ti describes the evolution under the internal Hamiltonians of the sample, corresponding to the free-induction signal. Relaxation is neglected in (8.3.4). 

The subscript S signifies that this is a matrix element in the system space only, that is, we perform a partial trace over the system degrees of freedom, and GvX(t) is still a full Liouville space operator in the bath degrees of freedom. The angular brackets < > denote averaging over the bath degrees of freedom, that is,... 

In the previous sections, we derived general correlation function expressions for the nonlinear response function that allow us to calculate any 4WM process. The final results were recast as a product of Liouville space operators [Eqs. (49) and (53)], or in terms of the four-time correlation function of the dipole operator [Eq. (57)]. We then developed the factorization approximation [Eqs. (60) and (63)], which simplifies these expressions considerably. In this section, we shall consider the problem of spontaneous Raman and fluorescence spectroscopy. General formal expressions analogous to those obtained for 4WM will be derived. This will enable us to treat both experiments in a similar fashion and compare their information content. We shall start with the ordinary absorption lineshape. Consider our system interacting with a stationary monochromatic electromagnetic field with frequency w. The total initial density matrix is given by... 

We next briefly survey some properties of Liouville space superoperators that will be useful in the following derivations [49]. The elements of the Hilbert space N XN density matrix, p(t), are arranged as a Liouville space vector (bra or ket) of length N. Operators of N X N dimension in this space are denoted as superoperators. With any Hilbert space operator A, we associate two superoperators Al (left) and Ar (right) defined through their... 

The commutator and anticommutator operations in Hilbert space can thus be implemented with a single multiplication by a and + superoperator, respectively. We further introduce the Liouville space-time ordering operator T. This is a key ingredient for extending NEGFT to superoperators when applied to a product of superoperators it reorders them so that time increases from right to left. We define (A(t)) = Tr A(f)Peq where peq = p(t = 0) represents the equilibrium density matrix of the electron-phonon system. It is straightforward to see that for any two operators A and B we have... 

T is the Liouville space-time ordering operator that rearranges all superoperators in increasing order of time from right to left. 

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