Basis operator

Fig. 70.3 En ergy level diagram of a two spin J4 system / and S showing the identification of com ponents of the 2D multiplet expressed via single transition basis operators, Z13, I24, S12, S34. Contour plots of a N- H backbone moiety with...

Li is the matrix representation of the lattice Liouvillian in the space of the basis operators, 1 is a unit (super)operator and Ci are projection vectors representing the operators of Eq. (32) in the same space. The... 

The indices k in the Ihs above denote a pair of basis operators, coupled by the element Rk. - The indices n and /i denote individual interactions (dipole-dipole, anisotropic shielding etc) the double sum over /x and /x indicates the possible occurrence of interference terms between different interactions [9]. The spectral density functions are in turn related to the time-correlation functions (TCFs), the fundamental quantities in non-equilibrium statistical mechanics. The time-correlation functions depend on the strength of the interactions involved and on their modulation by stochastic processes. The TCFs provide the fundamental link between the spin relaxation and molecular dynamics in condensed matter. In many common cases, the TCFs and the spectral density functions can, to a good approximation, be... 

Let G be a local transformation group that acts on M and is the symmetry group of system (5). Next, let the basis operators of the Lie algebra g of the group G be of the form... 

Note that the conformal group C(l,3) generated by the infinitesimal operators (27) acts in the space of independent variables R1 3 only. That is why the basis operators of the algebra c l, 3) act in the space of dependent variables R4 as zero operators. 

Figure 11.7 PFR-CSTR train sized to produce 400000kg/year of celecoxib (Basis Operating 350 days/year).

A systematic investigation of the application of Lie algebra to NMR was presented.29 The symmetry properties of the nuclear spin systems were naturally included in selection of the sets of the basis operators. With this theoretical framework, the existing sets of basis operators used for various specific purposes can be treated in a unified manner and their respective advantages and disadvantages can be evaluated. A number of 2H MAS spectra calculated on the basis of that theoretical framework are shown in Fig. 2. The... 

Having expressed the Hamiltonian in an appropriate basis, the elements of the L-matrix in Equation (4) can be determined using the form of Equation (1). Arranging the basis operators according to their order and rank ... 

One specific sub-matrix, Lo, operating on the basis operators Tg, ..., T0 for an N-site jump during free precession can be evaluated analytically because of its high degree of symmetry. That is,... 

Implementation of the method of Spent Nuclear Fuel (SNF) unloading from unwatered NS reactors represents a crucial solution of the nuclear safety challenge because removal of moderator eliminates the risk for reactor core to attain critical condition under any design-basis/beyond-the design-basis operations with reactor control systems. 

Ernst, 1979 Chingas et al., 1981 Cavanagh et al., 1990). Similarly, the diagonalization of the mixing Hamiltonian superoperator can be facilitated if a set of basis operators is used that is completely reduced with respect to rotation, permutation, and particle number (Listerud and Drobny, 1989 Listerud et al., 1993). 

Design basis — operating conditions 3tid performance... 

This important equation governs, like an equation of motion, the time development of the system under a Hamiltonian H. It yields eight coupled differential equations for the coefficients c,(r). The resulting solutions for various Hamiltonians resemble rotations in an eight-dimensional space spanned by the eight nontrivial basis operators. With the convenient basis set of p, [14, 16, 68], the time evolution under Zeeman interaction can be visualized as a precession in the pi-pi, Ps-Pe Pi P planes. The axially symmetric quadrupolar interaction, on the other hand, mixes between pi and pe and between p and ps. Therefore, evolution under quadrupolar interaction does not lead to precession of magnetization within the x-y plane, but rotates it out of this plane into a not directly accessible order and back again. 

Quantum mechanics can also be formulated in terms of an alternate Hilbert space whose elements are operators, with the density operator p and the typical measurables F among them. A variety of complete sets of basis operators in the space may be constructed, for example, as the tensor product of a basis set of the Hilbert space vectors > and its dual, yielding elements of the form... 

The alternative of exclusively using the AO basis operationally in CC calculations, as the most local basis possible (avoiding any kind of local MO description for the occupied orbitals), has also been considered [192,193], but has not yet been shown to pay dividends. In fact, unpublished work on polymers [194,195] suggests that AO methods actually do not compare very well with the standard delocal, Bloch periodic symmetry treatment. By the way CCSD has now been applied with periodic symmetry to simple polymers at the CCSD level [196,197]. 

B. Linear Independence of Basis Operators with Respect to EOM Equations. 16... 

These attempts have considered simple problems like the anharmonic oscillator problem in a harmonic oscillator basis or the hydrogen atom in a Gaussian-type basis, generally with rather poor results. The reason for these difficulties is rather clear. Given N basis functions i>, there are basis operators iX7 - Consequently, the equations for the eigenvalues and eigenvectors of L represent equations of rank as compared with... 

One possible means of circumventing the problem of handling both X > and 0> is to expand Ol in an appropriate set of basis operators. 

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