Transition density differences

Fig. 7.57 Plot of the isomer shift 5 of the 36.2 keV Mdssbauer transition of Os versus Dirac-Fock values for the electron density differences at the Os nuclei in free ion 5d configurations. The numbers of the data points refer to the numbering of the compounds in Table 7.9 (from [258])...

The hypothesis that the transition to complex flow is buoyancy driven is supported by a scaling analysis to estimate the bulk density difference A q required to reverse a viscous flow. For the viscous and gravitational contributions to the flow to be comparable, one needs,... 

The interstitial hydrides of transition metals differ from the salt-like hydrides of the alkali and alkaline-earth metals MH and MH2, as can be seen from their densities. While the latter have higher densities than the metals, the transition metal hydrides have expanded metal lattices. Furthermore, the transition metal hydrides exhibit metallic luster and are semiconducting. Alkali metal hydrides have NaCl structure MgH2 has rutile structure. 

Of course, in free-convection mass transfer the transition time is dependent on the density difference generated at the electrode. The dimensionless time variable of the transient process is... 

Table II summarizes surface roughness values measured for PMMA and VMCH samples etched for 1.0 minute at 35mTorr at various power densities. Although the measured values of 80 - 105 A fall within the ranges obtained from the interferometer and transition layer theory, there is no significant variation with power density. Differences in surface roughness between pre-etched films of PMMA and VMCH are also negligible according to the stylus measurements.

This is the usual situation in a high field. It is characterized by the fact that transitions between different M bands of the spectrum are highly improbable because of the vanishingly small relevant density of states or because of the high number of spins that have to be rearranged in order to distribute a Zeeman quantum over... 

The term parameters of the lowest two allowed transitions of ethene calculated with different methods and different choices of computational parameters (48,51,98,105) are summarized in Table I. Included in the table are results obtained with four different basis sets. In combination with these basis sets the MCD parameters were obtained in the transition-based approach through solution of Eq. (60) by direct numerical solution (labeled Direct in Table I) and by expansion in a set of transition densities according to Eq. (72) (labeled SOS ). In some cases approximate forms of the A(1) and B(1) matrices were used (labeled Approx, see Eq. (64) and the discussion following it). MCD parameters derived from a fit to a spectrum obtained by calculation of the imaginary part of the Verdet constant are labeled as Im[V]. The parameters obtained from a fit to the spectrum obtained from the approximate form of Im[V] (see Section... 

A problem does indeed arise when the calculated transition time becomes more than a few seconds. The constant depletion of the electron acceptor makes the electrolyte density near the interface different from that in the bulk. These density differences in different regions of the electrolyte upset the initial condition of hydro-... 

With the inclusion of electronic coupling as in Figure lb, electron transfer between Fe11 and Fe111 becomes a redistribution of electron density within the ground electronic state from Fe11 to Fe111 and not a transition between different states. 

There exists today an alternative approach which has made MCSCF calculations on excited states feasible, also for rather large systems. A method has been developed which makes it easy to obtain orthogonal wave functions and transition densities from CASSCF wave functions optimized independently for a number of excited states of different or the same symmetry as the ground state. The method has been called the CAS State Interaction (CASSI) method. It will be briefly described below. 

As was shown in chapter three we can compute the transition densities from the Cl coefficients of the two states and the Cl coupling coefficients. Matrix elements of two-electron operators can be obtained using similar expresssions involving the second order transition density matrix. This is the simple formalism we use when the two electronic states are given in terms of a common orthonormal MO basis. But what happens if the two states are represented in two different MO bases, which are then in general not oithonormal We can understand that if we realize that equation (5 8) can be derived from the Slater-Lowdin rules for matrix elements between Slater determinants. In order to be a little more specific we expand the states i and j ... 

Fig. 15a-c. Scheme of the side chains arrangement of macromolecule (a), function of distribution of electron density Aq along a normal to the smectic plane (b) and one-dimensional correlation function Yi(x) for polymers V (1) and VI (2) (c),08) a) 1 — main chain 2 — mesogenic groups 3 — alkyl group b) Aq — electron density difference between ordered I, and disordered 12 regions E — width of the transitional region... 

At low temperatures, rj will be unity because all of the Cu atoms will be localized on A sites. 1 But the degree of disorder increases as the temperature increases until the Cu and Zn atoms are mixed randomly on the two sublattices and 77 = 0. This process, called a positional (order + disorder) transition, is often described as a cooperative phenomenon because it becomes easier to produce additional disorder once some disorder is generated. In the vicinity of a critical temperature, the order parameter rj behaves like the density difference (pi — pg) near the gas-liquid critical point. Thus,... 

Electron density difference matrices that correspond to the transition energies in the EP2 approximation may be used to obtain a virtual orbital space of reduced rank [27] that introduces only minor deviations with respect to results produced with the full, original set of virtual orbitals. This quasiparticle virtual orbital selection (QVOS) process provides an improved choice of a reduced virtual space for a given EADE and can be used to speed up computations with higher order approximations, such as P3 or OVGF. Numerical tests show the superior accuracy and efficiency of this approach compared to the usual practice of omission of virtual orbitals with the highest energies [27],... 

Up to this point, our main concern was to reformulate the results of the LD ligand influence theory in the DMM form. Its main content was the symmetry-based analysis of the possible interplay between two types of perturbation substitution and deformation, controlled by the selection rules incorporated in the polarization propagator of the CLS. The mechanism of this interplay can be simply formulated as follows substitution produces perturbations of different symmetries which are supposed to induce transition densities of the same symmetries. In the frontier orbital approximation, only those densities among all possible ones can actually appear, which have the symmetry which enters into decomposition of the tensor product TH TL to the irreducible representations. These survived transition densities then induce the geometry deformations of the same symmetry. 

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