Liouville scalar product

This is nothing but the electron-vibration interaction in the chosen notation. The quantity h is the three index supervector acting on the vector of nuclear shifts they form the scalar product (.... ..) giving a 10 x 10 matrix, next forming a Liouville scalar product with matrix V. On the other hand, acting on the variations V of the density matrix by forming the Liouville scalar product h produces a vector to be convoluted with that of nuclear shifts 5q. With use of this set of variables the energy in the vicinity of the symmetric equilibrium point becomes ... 

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]. 

The operation of the Liouville operator can be replaced by time derivative thus the term in the left-hand side of the scalar product can be written as... 

It is easily seen by inspection that the biorthogonal basis set definition (3.55) cmnddes with the definifion (3.18) ven in the discussion of the Lanczos method. We recall that the dynamics of operators (liouville equations) or probabilities (Fokker-Planck equations) have a mathematical structure similar to Eq. (3.29) and can thus be treated with the same techniques (see, e.g., Chapter 1) once an appropriate generalization of a scalar product is performed. For instance, this same formalism has been successfully adopted to model phonon thermal baths and to include, in principle, anharmonicity effects in the interesting aspects of lattice dynamics and atom-solid collisions. ... 

Thus all functions At(r) of finite norm, together with the above definition of the scalar product define a Hilbert space. This space is called Liouville space because the Liouvillian generates the motion in this space. 

In this appendix, we review the main properties of the tetradic linear M= Nx K — N) dimensional space defined by the Liouville operator /,.77.8i,87,300 first introduce the following scalar product of any two interband matrices and t] which are the elements of this space. 

The main reason for introducing this scalar product is that the Liouville operator L defined by eq 2.19 is Hermitian with respect to this scalar product ... 

To start our calculations, we compute the moments 5 and by acting Liouville operator L (2.19) on the source and using the scalar product (Bl). We then solve eqs C7 for the frequencies Qy and oscillator strengths f These nonlinear equations have a simple analytical solution.Once we have Qy and ju we solve eqs C5 and C6 for the modes Py and Qy. The most time-consuming part of the DSMA is the calculation of commutators. Typically only a small number of modes is required and the DSMA greatly reduces the numerical effort involved in solving the complete TDHF equations. 

Because the operator G t) = is unitary, or norm preserving, the vector e A can be regarded as varying in time in such a way that its length (or norm) is preserved. The time evolution of A(t) is thus represented by a rotation in Liouville space. The scalar product of A (t) and A (0) gives the time correlation function Caa (0, and leads to the interpretation of Caa (0 as the component (or projection) of A (f) on A (0). An operator that projects an arbitrary vector onto A is... 

The above identity defines also the scalar product in the extended Liouville space for the present CODDE formulation [Eq. (2.21) and Sec. 3.1]. 

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 ... 

Thus, to solve the problem up to this point we have used the inner product, the eigenproblem and the self-adjoint property of the linear operator. It is recalled that the actions taken so far are identical to the Sturm-Liouville integral transform treated in the last section. The only difference is the element. In the present case, we are dealing with multiple elements, so the vector (rather than scalar) methodology is necessary. 

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