Contraction integral equation method

Equation (9.203) can be rewritten with respect to the product of a and the total electric field E, using simple algebraic transformations  

In this equation operator C is a contraction operator for any lossy medium (see section 9.3.2)  

Using the original Green s operator given by expression (9.37) and taking into account formula (9.205), one can rewrite equation (9.208) as follows  

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

This is the basic equation of the CIE method of electromagnetic modeling. 

---

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 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),... 

Flow experiments were carried out at LRMP for axisymmetric contractions and at CEMEF for plane geometries. Numerical simulations were performed at Laboratoire de Rheologie, with Wagner memory-integral constitutive equations, with the stream-tube method (sub-section 5.1) and at CEMEF, where the finite-... 

Skinner and Wolynes came back to the problem of Eq. (1.8) and solved it with a projection method in Laplace space. An interesting aspect of their work is the development of the contracted distribution function a(a t) (see also Section II) inside the time-convolution integral. They pointed out that this provides perturbation terms neglected erroneously by Brinkman. This interesting feature of their approach is included in the AEP illustrated in this chapter. They explicitly evaluated correction terms up to order The projection technique has also been used by Chaturvedi and Shibata, who used a memoiyless equation as the starting point of their treatment. 

---