Operator second moment

The first term in the brackets is the expectation value of the square of the dipole moment operator (i.e. the second moment) and the second term is the square of the expectation value of the dipole moment operator. This expression defines the sum over states model. A subjective choice of the average excitation energy As has to be made. 

The second-order terms give the magnetizability. The first term is known as the diamagnetic part and it is particularly easy to calculate since it is just the expectation value of the second moment operators. The second term is called the paramagnetic part. 

It is also useful to define the transformed operator L whose operation on a function f is L f = L[Peqf). This operator coincides with the time reversed backward operator, further details on these relationships may be found in Refs. 43,44. L operates in the Hilbert space of phase space functions which have finite second moments with respect to the equilibrium distribution. The scalar product of two functions in this space is defined as (f, g) = (fgi q. It is the phase space integrated product of the two functions, weighted by the equilibrium distribution P The operator L is not Hermitian, its spectrum is in principle complex, contained in the left half of the complex plane. 

As Figure 2 shows, the two models differ only by a vertical displacement, which is a measure of the difference in the mean square interparticle dipolar fields (second moments) operative in the two models. The values of these mean square local fields can be calculated easily from the minimum value of Ti or from the rigid lattice values of T2. Because of the 1000-fold ratio between electronic and nuclear magnetic moments, even... 

Computed Second Moment Operators and Relaxation Energies for the First Four MOs of (rr-C5H5)NiNOa... 

Using ab initio wave functions and the full operators, dipole and second moments (cf. Section 4.11.3.2.2) as well as electronic charge distributions have been calculated for... 

These results seem to be fully satisfactory, both the equations of the first and second moments agree with the classical results, and the eventual equilibrium agrees with thermal considerations. However, the form (26) is not acceptable, as we can see from the following considerations [Hakim 1985 Ambegaokar 1991 Munro 1996], For a summary of the situation see Ref. [Stenholm 1994], In the case of the harmonic oscillator, we introduce the customary annihilation and creation operators by setting... 

However, there are two common operationally defined types of structure that are determined from effective moments of inertia. The more common, the so-called effective or r0 structure, is somewhat loosely defined. In practice, any structural parameter that requires for its determination fitting one or more of the second moment relations is designated as r0. r0 structures are not uniquely defined since, for any over-determined system, the value of structural parameters obtained depends somewhat on the manner in which the data are treated and the values are isotopically dependent. This problem is examined in more detail by Schwendeman (this volume). 

For example, rs coordinates are defined operationally as functions of ground-state planar second moments. The only uncertainty in an r% coordinate as an estimate of an rs coordinate is from experimental uncertainty in the planar second moments, and this is easy to compute from Eq. (10). As estimates of equilibrium values of the coordinates, however, rs coordinates contain an additional contribution to their uncertainty from the neglected pseudoinertial defects. This contribution may be estimated from the Costain rule [Eq. (14)]. 

The second moment of the charge distribution is a measure of the absolute size of the charge distribution in each direction. The molecular quadrupole moment, on the other hand, is a measure of the shape or deviation from spherical symmetry of the charge distribution.29 Thus the accuracy of the expectation value of the quadrupole moment operator is a rather sensitive test of the quality of the calculated charge distri-... 

H NMR measurements were performed at 20°C using a Maran NMR spectrometer (Resonance Instruments, UK) operating at resonance frequency of 23 MHz. The second moment of the FIDs was calculated using Microsoft Excel Solver. 

Clearly, these two quantities can be obtained from a direct computation of the first and second moments of the TB position operator, and we demonstrate below that one can indeed reach the long-time limit required to extract these transport quantities numerically from Eqs. (4.6) and (4.7). Here we confine ourselves to the linear case = 0, where the Einstein relation provides a simple check for our data. The Einstein relation, which has recently been proved for the general quantum case [72], connects the diffusion coefficient with the linear mobility. 

A low order QMF that minimally extends beyond the ordinary MF approximation is used, where the heavy particle subsystem is described in terms of the first and second moments of the position and momentum operators, and the coupling potential is Taylor expanded to the first order, Eq. (23). The resulting EOM for the light... 

The spectra at various temperatures obtained by a broad-line NMR spectrometer, JEOL-JES-BE-1, operating at 40 MHz were processed by computer in order to calculate second moments. Variations of the second moment with observation temperature are shown in Fig. 7.2. for three kinds of materials. All parts of bulk polyethylene... 

Scalar column vector rule Consider X to be a 1 xn vector of random variables, for which it is desired to compute the second moment, m. In such a case, the correct formula is given as m2 = E, where T is the transpose operator. It is easy to verify that this will give a scalar value. 

This says that the dipole moment operator will be needed for the derivative Schrodinger equations involving derivatives with respect to V (the -com-ponent of a uniform field), or that the second moment operator will be needed for derivative Schrodinger equations involving differentiation with respect to field gradient components such as V. In general, there will be operators combined with parameters in the Hamiltonians, and then the derivative Hamiltonians will be operators of some sort. These must be constructed. 

It is evident from Eq. (6-17) or (6-19) that each transport step ves a separate and additive contribution to the second moment. Then, for instance, from second moments for different gas velocities, the contribution of axial dispersion can be separated from the contribution of the other transport steps. Similarly, from the data for different particle sizes, the contribution of intraparticle diffusion can be separated from the contribution of adsorption. By choosing proper operating conditions, the contribution of the particular step can be maximized so that the rate parameter may be determined with good accuracy. 

Accordingly, invariance of second-, third-, and fourth-rank tensors in a change of coordinate system is fulfilled if the state functions are exact eigenfunctions to a model Hamiltonian, and satisfy the hypervirial theorems for position and second-moment operators. 

Dipole Moments.— Electronic charge distribution dipole and second moments have been calculated for thiophen using ab initio wavefunctions and the full operators. The structure of thiophen-2-aldehyde in the gas phase has been calculated from dipole-moment data and principal moments of... 

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