Density functions contracted

The distribution is still normal, so the contracted density function... 

Interlude 3.1 Contracted Density Functions and the Loss of Information ... 

Now, the probability of finding our sprocket with only a certain diameter is given by the contracted density function P(a i)... 

Conditional probability, 267 density function, 152 Condon, E. U., 404 Configuration space amplitude, 501 Heisenberg operator, 507 operators, 507, 514, 543 Conservation laws for light particles (leptons), 539 for heavy particles (baryons), 539 Continuous memoryless channels, 239 Contraction symbol for two time-labelled operators, 608 Control of flow, 265 Converse to coding theorem, 215 Convex downward function, 210 Convex upward function, 209 Cook, L. F 724... 

A number of different atom-centered multipole models are available. We distinguish between valence-density models, in which the density functions represent all electrons in the valence shell, and deformation-density models, in which the aspherical functions describe the deviation from the IAM atomic density. In the former, the aspherical density is added to the unperturbed core density, as in the K-formalism, while in the latter, the aspherical density is superimposed on the isolated atom density, but the expansion and contraction of the valence density is not treated explicitly. 

The nature of the charge density parameters to be added to those of the structure refinement follows from the charge density formalisms discussed in chapter 3. For the atom-centered multipole formalism as defined in Eq. (3.35), they are the valence shell populations, PLval, and the populations PUmp of the multipolar density functions on each of the atoms, and the k expansion-contraction parameters for... 

The molecular electron density function needed for EP calculation can be obtained through ab initio as well as various semi-empirical methods. Since ab initio calculations are not economical for large molecules (several hundred atoms), the use of well-parameterized semi-empirical methods are still justified. When semi-empirical methods are used the three-center potential integrals usually disappear, and therefore the electronic contribution can be easily calculated by Slater-type orbitals. In ab initio methods (primitive or contracted) Gaussian-type orbitals are used for calculating the three-center integrals because their calculations are clumsy with Slater-type orbitals. 

Also in 2-substituted ethanesulphonates,35 the 33S chemical shift has a reverse substituent effect and correlates with both Taft substituent constants and the chemical shift of the carboxylic carbon in related carboxylic acids. It seems that the substituent effect does not depend on conformation, but prevailingly on intramolecular electronic effects. Density functional theory (DFT) calculations of 33S nuclear shielding constants and natural bond orbital (NBO) analysis made it possible to conclude that substituents cause a variation in the polarization of the S-C and S-O bonds and of the oxygen lone pairs of the C — S03 moiety. This affects the electron density in the surroundings of the sulphur nucleus and consequently the expansion or contraction of 3p sulphur orbitals. 

According to decoherence theory the human observer perceives only the system of interest. Thus, we have to evaluate first the total density matrix corresponding to the wave function of Eq. (259) and to make a contraction over the environmental degrees of freedom, to derive out of it the contracted density matrix out, p(f). As a result of this procedure we get... 

Alternatively, the deformation ED can be described by a series of multipoles, that is, spherical harmonic density functions, the parameters of which can be refined by the LS technique. The core ED is described as spherically symmetrical, using the so-called k formalism. The k parameter expresses the isotropic expansion ( < > 1) or contraction (/c < 1) of a valence shell as a whole. The higher-order multipoles describe the deviations of the ED from spherical symmetry. 

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