Examples overlapping Gaussians

The functions put into the determinant do not need to be individual GTO functions, called Gaussian primitives. They can be a weighted sum of basis functions on the same atom or different atoms. Sums of functions on the same atom are often used to make the calculation run faster, as discussed in Chapter 10. Sums of basis functions on different atoms are used to give the orbital a particular symmetry. For example, a water molecule with symmetry will have orbitals that transform as A, A2, B, B2, which are the irreducible representations of the C2t point group. The resulting orbitals that use functions from multiple atoms are called molecular orbitals. This is done to make the calculation run much faster. Any overlap integral over orbitals of different symmetry does not need to be computed because it is zero by symmetry. 

Fig. 6.5. Peak spreading strongly affects enrichment ratio at fixed probability of retention. The coefficient of variance CV is equal to the ratio of the standard deviation to the mean, and is a measure of peak breadth. For example, in both curves shown in Fig. 6.3 the CV is 1.0. The enrichment ratio was calculated for a situation in which mutant fluorescence intensity was double wild-type fluorescence intensity, the mutant was initially present at 1 in 106 cells, and the probability of retention was fixed at 95 %. The logarithmic fluorescence intensity was assumed to follow a Gaussian distribution. Fixing the probability of retention defines the cutoff fluorescence value for screening at a given CV. Enrichment ratio drops precipitously with increasing CV, as the mutant and wild-type fluorescence distributions begin to overlap. At a CV of 0.2, the enrichment factor is 600. However, at a CV of 0.4, the enrichment factor has dropped to 3 Clearly, every effort should be expended to minimize peak spreading and subsequent overlap of the mutant and wild-type fluorescence distributions.

We show two numerical examples for a 64 x 64 random matrix Hamiltonian One is the relatively short-time case with T = 20 and a = 1 shown in Fig. 1, and the other is the case with T = 200 and ot = 10 shown in Fig. 2. In both cases, we obtain the optimal field s(f) after 100 iterations using the Zhu-Botina-Rabitz (ZBR) scheme [13] with s(f) = 0 as an initial guess of the field. The initial and the target state is chosen as Gaussian random vectors as mentioned above. The final overlaps are Jo = 0.971 and 0.982, respectively. 

A detailed analysis of the UV-VIS spectrum of (spinach) plasto-cyanin in the Cu(II) state has been reported (56). A Gaussian resolution of bands at 427, 468, 535, 599, 717, 781, and 926 nm is indicated in Fig. 7. Detailed assignments have been made from low-temperature optical absorption and magnetic circular dichroic (MCD) and CD spectra in conjunction with self-consistent field Xa-scattered wave calculations. The intense blue band at 600 nm is due to the S(Cys) pvr transition, which is intense because of the very good overlap between ground- and excited-state wave functions. Other transitions which are observed implicate, for example, the Met (427 nm) and His (468 nm) residues. These bonds are much less intense. The low energy of the d 2 orbital indicates a reasonable interaction between the Cu and S(Met), even at 2.9 A. It is concluded that the S(Cys)—Cu(II) bond makes a dominant contribution to the electronic structure of the active site, which is strongly influenced by the orientation of this residue by the... 

Early data analysis attempted to extract values of the individual structure factors from peak envelopes and then apply standard single crystal methods to obtain structural information. This approach was severely limited because the relatively broad peaks in a powder pattern resulted in substantial reflection overlap and the number of usable structure factors that could be obtained in this way was very small. Consequently, only very simple crystal structures could be examined by this method. For example, the neutron diffraction study of defects in CaF2-YF3 fluorite solid solutions used 20 reflection intensities to determine values for eight structural parameters. To overcome this limitation, H. M. Rietveld realized that a neutron powder diffraction pattern is a smooth curve that consists of Gaussian peaks on top of a smooth background... 

The calculated uncertainty = 0.2 appears relatively small but is statistically correct, for the values are assumed to follow a Gaussian distribution. As a consequence of Eq. (C.7), will always come out smaller than the smallest a,. Assuming 04 = 0.10 instead of 0.25 would yield X = (25.0 + 0.1) and 04 = 0.60 would result in X = (25.6 0.2). In fact, the values (A, a,) in this example are at the limit of consistency, that is, the range (A4 + 04) does not overlap with the ranges (X2 + (J2) and (A3 CT3). There might be a better way to solve this problem. Three possible alternatives seem more reasonable ... 

An expansion of the Morse potential, for example, in a set of Gaussian functions is given by eq (C4) in Appendix C. Matrix elements of the Morse potential in terms of the Gaussian primitive basis functions are therefore simply three center overlap integrals [49], These matrix elements can be evaluated for each term in the sum and then converted to the final expression in a straightforward manner. 

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