Connected-moments expansion

J. Cioslowski, Connected-moments expansion—a new tool for quantum many-body theory. Phys. Rev. Lett. 58, 83 (1987). 

Acceptable comprehensive methods of analysis are analytical, model-test, and chart methods, which evaluate for the entire piping system under consideration the forces, moments, and stresses caused by bending and torsion from a simultaneous consideration of terminal and intermediate restraints to thermal expansion and include all external movements transmitted under thermal change to the piping by its terminal and intermediate attachments. Correction factors, as provided by the details of these rules, must be applied for the stress intensification of curved pipe and branch connections and may be applied for the increased flexibihty of such component parts. 

In addition, very few observations are pristine and basic measurements such as angular deviation of a needle on a display, linear expansion of a fluid, voltages on an electronic device, only represent analogs of the observation to be made. These observations are themselves dependent on a model of the measurement process attached to the particular device. For instance, we may assume that the deviation of a needle on a display connected to a resistance is proportional to the number of charged particles received by the resistance. The model of the measurement is usually well constrained and the analyst should be in control of the deterministic part through calibration, working curves, assessment of non-linearity, etc. If the physics of the measurement is correctly understood, the residual deviations from the experimental calibration may be considered as random deviates. Their assessment is an integral part of the measurement protocol and the moments of these random deviations should be known to the analyst and incorporated in the model. 

Having derived the symmetry relations between the expansion parameters in equation (55), we can proceed to fit the expansions through the ab initio dipole moment values. The expansion parameters in the expressions for and fiy are connected by symmetry relations since these two quantities have E symmetry in and so and fiy must be fitted together. The component ji, with A" symmetry, can be fitted separately. The variables p in equation (55) are chosen to reflect the properties of the potential surface, rather than those of the dipole moment surfaces. Therefore, the fittings of fi, fiy, fifi require more parameters than the fittings of the MB dipole moment representations. We fitted the 14,400 ab initio data points using 77 parameters for the component and 141 parameters for fi, fiy. The rms deviations attained were 0.00016 and 0.0003 D, respectively. 

Since momentum densities are unfamiliar to many. Section II outlines the connection between the position and momentum space representations of wavefunctions and reduced-density matrices, and the connections among one-electron density matrices, densities, and other functions such as the reciprocal form factor. General properties of momentum densities, including symmetry, expansion methods, asymptotic behavior, and moments, are described in... 

The expressions on the right sides of Eq. (6.1-5) now involve only vibrational wavefunc-tions (vi, Vf, Ve) whereas the electronic wave functions appear in M, which is the pure electronic transition moment connecting the ground with the excited electronic state e. M is a function of nuclear coordinate and can be expanded into a Taylor series about the equilibrium position (Herzberg-Teller expansion) ... 

The Eckart conditions play an important role in this connection. We shall discuss this in more detail below, since the arguments presented apply equally well to the treatment of nonrigid molecules. Hence, to study the basis of introducing Eckart conditions, let us for a moment go back to an earlier stage where axis conventions were not yet formulated. We recapitulate that we are looking for the conditions required in order that the atomic position coordinates, rag, can be given as unique functions of 3 N-6 internal coordinates, or equivalently stated, in order that the expansion [Eq. (3.6)] can be determined as a unique inverse of Eq. (3.5). 

For a very short time interval, one can determine the time evolution of the correlation function by a Taylor expansion. This is an application of the time differentiation theorem already mentioned in connection with Eq. (31). What the theorem provides is a relation between the coefficients in the Taylor expansion and the frequency moments of the... 

Example 11.17. A 12-in. NPS Schedule 160 branch and run pipe are attached to one another. The design pressure is 2200 psi. The allowable stress at ambient temperature is = 17,5 ksi and at design temperature is S = 12.0 ksi. In addition to the internal pressure, the branch is subjected to externally applied forces and moments from thermal expansion of connecting piping. These moments and force are M = 600,000 in.-lb Mg = 900,000 in.-Ib M, = 750,000 in.-lb and Fma = 90,000 lb. The nozzle is designed for 20,000 cycles. Using the design procedure of the ASME-ANSI B31.1 Code, what is the total applied stress and what is the allowable stress ... 

We note that in the expansion of the induced dipole moment, higher-order corrections may also be important, in particular in connection with electric-field perturbations. For instance, considering only electric dipole contributions, it is customary to also include contributions to H arising from frequency-dependent hyperpolarizabilities (see O Eq. 11.84 below). 

Owing to the coherence, we need to consider the macroscopic evolution of the field in a medium that shows a macroscopic polarization induced by the field-matter interaction. This will be done in three steps. First, the polarization induced by an arbitrary field will be calculated and expanded in power series in the field, the coefficients of the expansion being the material susceptibilities (frequency domain) or response function (time domain) of wth-order. Nonlinear Raman effects appear at third order in this expansion. Second, the perturbation theory derivation of the third-order nonlinear susceptibility in terms of molecular eigenstates and transition moments will be outlined, leading to a connection with the spontaneous Raman scattering tensor components. Last, the interaction of the initial field distribution with the created polarization will be evaluated and the signal expression obtained for the relevant techniques of Table 1. 

---