Configuration contributions

It is simplest to consider these factors as they are reflected in the entropy of the solution, because it is easy to subtract from the measured entropy of solution the configurational contribution. For the latter, one may use the ideal entropy of mixing, — In, since the correction arising from usual deviation of a solution (not a superlattice) from randomness is usually less than — 0.1 cal/deg-g atom. (In special cases, where the degree of short-range order is known from x-ray diffuse scattering, one may adequately calculate this correction from quasi-chemical theory.) Consequently, the excess entropy of solution, AS6, is a convenient measure of the sum of the nonconfigurational factors in the solution. 

For the A FI state, considerably more configurations contribute the above threshold coefficient and in order to increase the value of the sum of the squares of the Cl... 

We now return to the issue of configurational density of states. In the simulation of molecular systems, we are interested only in the calculation of their configurational properties, or more explicitly, the configurational contribution to their partition functions. This is because the kinetic component is analytic, and, hence, there is no need to measure it via simulation. For conventional MC simulations in the... 

These ideas have been employed to compute configurational entropy and hence test the AG relation via molecular simulation of several model systems.91-94,101,102 The approach used in those studies is conceptually simple. First, the total entropy of the fluid S is calculated by integration of standard thermodynamic relationships, for example, as discussed below. Then, the configurational contribution to the entropy Sc = S — Sv b, is approximated by subtracting from the total entropy an estimate for the vibrational contribution, Svib. 

The behavior of self-diffusivity for these state points is displayed in Figure 12(d). It is clearly visible that the non-monotonic dependence of D on density p is directly reflected by the configurational contribution to the entropy. In fact, the quantitative relationship between D and Sc predicted by Eq. [10] (i.e., the AG relationship) holds remarkably well for SPC/E water over this wide range of thermodynamic conditions, as is shown in Figure 13. 

Figure 5.18. Eight possible configurations of ligands occupying three sites. Only four configurations contribute to the interaction energy.

Figure 1. Initial and final configurations contributing to the mean value (c Cj<7)g-...

The <5-function guarantees that only those configurations contribute to the integral (2.6),. for which rf — - r holds. Thus P(r, n) gives the proba ... 

However, a number of tris-chelate mixed ligand Cr(III) complexes do exhibit a correlation of the same signs between the lowest frequency 2E component and the major 4E component.119,120,122,123 This relation has been verified by examining the CD for diastereomers of [Cr(en)x(diamine)3 x]3 + (x = 0, 1, 2), c/i-[Cr(NH3)2 (diamine)2]3 +, where the diamines used were (R)- or (5)-propylenediamine and (IR,2R)- or (15,250-1,2-traras-cyclohexanediamine. The differences in CD patterns between each pair of the diastereomers were accounted for by separability and additivity of the configurational and conformational CD contributions. The CD in the spin-allowed 4T2 <— 4A2 transitions shows a positive major component (4E) for the configurational contribution and a minor negative (4Ai) and major positive (4E) components from the lower frequency side for the conformational contribution as shown in Figure 5.30. 

The experimental CD spectra of these donor-acceptor cyclophanes were very well reproduced by the theoretical calculations at the TD-DFT-BH-LYP/TZV2P level [32]. The amount of the exact exchange in the functional to be used turned out to be important for better agreement with experiment. The analysis of the configuration contributions showed that the observed CD spectra essentially resulted from simple overlap of the Cotton effects of pure n-n transitions in the donor and acceptor parts, in addition to the CT transitions. Thus, the exciton chirality method cannot be applied to the absolute configuration determination for these cyclophanes. The experimental CD spectrum of more congested donor-acceptor cyclophane 22 was reproduced in a less satisfactory manner compared to the cases of 20 and 21. The absolute configuration was safely determined by a comparison of the experimental spectrum with the theoretical data at the TD-DFT level however, the satisfactory reproduction of the whole CD spectrum of 22 was only possible by... 

The three first-mentioned configurations contribute (see p. 268) all (respectively 39.4 %, 30.3 % and 30.3 % ) to the stationary state, so that the ground has disappeared for a discussion on which formula is preferable. 

The structure of ozone 21, after a long period of uncertainty, is now probably as indicated, with a bond angle of 126° and a bond length 1.26 A ozone has a dipole moment of 0.49 D. The following four configurations contribute so that the O—O distance lies between those of a single and of a double bond (1.46 A and 1.10 A). 

Note that 1/ is independent of c. For a two-electron system, three states have to be considered in the calculation of p. The configuration of the ground state g) is and there are two excited states, a) and b), with configurations ijj 1/ and ij/l tpi. Again, no doubly excited configurations contribute to (3 since the transition dipole to the ground state is zero. The transition a) <— g) is polarized perpendicularly to the C2 axis (z axis), while the transition b) g)... 

Kraka and Cremer have estimated the biradical character of the benzynes in two ways. The first involves the occupation number of the natural orbitals ( j) obtained from the CCSD(T) wavefunction. The biradical character is then defined as S , - S j(reference) where the reference here is benzene. With this method, the biradical character is 11 percent, 20 percent, and 65 percent for 41, 42, and 43, respectively. Alternatively, one can use the largest amplitude for a doubles excitation within the CCSD(T) wavefunction to indicate how strongly the next most important configuration (after the HF configuration) contributes to the total wave-function. This ampUrnde is 0.24 for 41, 0.34 for 42, and 0.71 for 43. Yet another measure of biradical character is the ratio where these are the coeffi-... 

In aciueous solutions containing high concentrations of LiBr or CaCb, the specific rotation of poly-E-proline II approaches [a] = —240°. Under these CH)nditions the intrinsic viscosity of the highly a.symmetric form II structure decreases markedly to values in the range of the globular proteins and it can be assumed that the configurational contribution to rotation of form II has been eliminated. 

Certain other features of the optical rotatory properties are worthy of comment. First, as mentioned previously and as might be expected on X-ray grounds, the structural portion of the specific rotation of collagen is very close to that obtained with poly-L-proline II. Thus (see Table IX) at the sodium D-line the specific rotation of collagen is —400°, while the residue rotation is about —125°, yielding a configurational contribution... 

Estimation of the Configurational Contribution to Optical Rotation for Various Gelatins... 

For an absorption band with a Gaussian band shape and involving single configurational contributions, Eq. 16 may be used to obtain values Hqe [1, 19, 20]... 

If 0 = I is adopted as the standard state for the adsorbed phase, s" = s") ), the last term in Eq. (54) (configurational contribution to entropy of adsorbed phase) disappears. As a result, the standard entropy of adsorption is shown to be... 

Tlie (5-function guarantees that only those configurations contribute to the... 

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