Contraction integral equation

Therefore, we finally obtain the following form of the contraction integral equation (CIE) with respect to the scaled electric field E ... 

Contraction integral equation as the preconditioned conventional integral equation... 

One can demonstrate that the contraction integral equation (9.210) can be treated as the preconditioned conventional integral equation (9.44). This fact becomes especially important in numerical implementation of the CIE method. 

As mentioned in Section I, Cho [13], Cohen and Frishberg [14, 15], and Nakatsuji [16] integrated the Schrodinger equation and obtained an equation that they called the density equation. This equation was at the time also studied by Schlosser [44] for the 1-TRDM. In 1986 Valdemoro [17] applied a contracting mapping to the matrix representation of the Schrodinger equation and obtained the contracted Schrodinger equation (CSE). In 1986, at the Coleman Symposium where the CSE was first reported, Lowdin asked whether there was a connection between the CSE and the Nakatsuji s density equation. It came out that both... 

The integral equation can be solved by a contraction mapping argument [4,p.222] in the space C[0,T) with T sufficiently small. This local solution can be extended globally (i.e. T -> ) if lu(t) remains finite [4,p.223]. This is the only thing which requires checking in the present case. 

The reference to the exponents is implicit. The vectors c and d will, since we want to compute integrals for more than one integral class, take on the values corresponding to the s, p c, p, and pj functions. Therefore there will be 16 different products cd]. Note that the vector q will take on the values corresponding to s, p, and d functions, i.e., 10 different values. Combining the contraction step with equation (53) the contracted integrals are expressed as... 

This expression would initially appear to require the evaluation of 160 integrals [ab q]. However, if evaluated in the axes-1 coordinate systems there would be only 88 nonzero elements, and all the integrals with an odd sum of the y-indices will be zero by symmetry. This list of three-center integrals can be further reduced by employing the transfer equation on the contracted integrals in the form of equation (18) which is... 

As demonstrated by the Pople-Hehre method it is possible to achieve considerable reduction in the computational expense of contracted ERIs if large parts of the integral manipulation are performed after the contraction step. The McMurchie-Davidson and the Obara-Saika methods utilization of the transfer equation (17) to minimize the operation count has been shown. This idea can, however, be employed to the extent that all manipulations are performed on fully or partially contracted integrals. Recently a number of methods have been presented along those lines.The method of Gill and Pople will be used as an example of the approach because it is currently one of the most commonly used integral methods. Note the concept of early contraction, however, applies to any of the methods presented in the chapter. 

Here, n corresponds to the principal quantum number, the orbital exponent is termed and Ylm are the usual spherical harmonics that describe the angular part of the function. In fact as a rule of thumb one usually needs about three times as many GTO than STO functions to achieve a certain accuracy. Unfortunately, many-center integrals such as described in equations (7-16) and (7-18) are notoriously difficult to compute with STO basis sets since no analytical techniques are available and one has to resort to numerical methods. This explains why these functions, which were used in the early days of computational quantum chemistry, do not play any role in modem wave function based quantum chemical programs. Rather, in an attempt to have the cake and eat it too, one usually employs the so-called contracted GTO basis sets, in which several primitive Gaussian functions (typically between three and six and only seldom more than ten) as in equation (7-19) are combined in a fixed linear combination to give one contracted Gaussian function (CGF),... 

Fig. 13a and b. Intensity contour maps around the 5.9-nm and 5.1-nm actin layer lines (indicated by arrows) a resting state b contracting state. Z is the reciprocal-space axial coordinate from the equator. M5 to M9 are myosin meridional reflections indexed to the fifth to ninth orders of a 42.9-nm repeat, (c) intensity profiles (in arbitrary units) of the 5.9- and 5.1-nm actin reflections. Dashed curves, resting state solid curves, contracting state. Intensity distributions were measured by scanning the intensity data perpendicular to the layer lines at intervals of 0.4 mm. The area of the peak above the background was adopted as an integrated intensity and plotted as a function of the reciprocal coordinate (R) from the meridian... 

Another possibility concerns the resonance integrals /Sab which appear in the Klopman-Salera equation. In a Hiickel picture, these are independent of the orbital energy, but in a double-zeta or better description we would expect the more tightly-bound electrons to have more contracted orbitals, and the higher virtual orbitals to be more diffuse.131 It may be that the HOMO and LUMO have the optimum spatial distribution for strong interaction, and that interactions involving more contracted and more diffuse orbitals are weaker.122... 

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