Operation count integral transformation

The relativistic transformation must include integral classes (LL LL), LL SS), SS LL), and (SS SS). Without any further efficiencies, this results in a factor of 4 increase in the operation count. 

However, in the first half-transformation, we can exploit the common indices, and add the half-transformed integrals to yield ( > LL), (pq SS), (pq LL), and (p ISS) integrals, with an operation count of 24n. The operation count for the second half-transformation is also 24n, yielding a total of 48n. ... 

The advantage of carrying out this partial summation is that the operation count is brought down from for (9.9.44) to for (9.9.47) and for (9.9.48), clearly a major improvement. Next, we note that, once the first Hermite-to-Cartesian step has been completed, we may carry out the transformation to the contracted spherical-harmonic basis for the first electron before we go on to the second Hermite-to-Cartesian transformation (9.9.48), bringing the number of integrals for the second step (9.9.48) down from to I p. The cost is iJ p for the contraction and for the spherical-harmonic transformation. As a result, the cost of the second step (9.9.48) is reduced from to I jp-. To conclude our calculation, a contraction and transformation to... 

The operation count for the one-index transformation in (10.8.60) is since the transformed integrals needed fra the electronic gradient have one general and three occupied indices. The cost of the evaluation of the gradient from the one-index transformed integrals is nO, the same as for the gradient in Section 10.8.5. Thus, if (10.8.61) is used, then the operation count for the linear transformation in (10.8.8) is - an order of magnitude less than that fra the explicit construction of the Hessian (10.8.53), which requires operations. 

The operation count for setting up the inactive Fock matrices (10.6.17) and (10.8.62) in the AO basis is n. A closed-shell Newton iteration thus scales as when carried out in the AO basis. It is considerably more expensive to carry out the Newton iterations in the MO basis, since the MO transformation of the two-electron integrals scales as n. We also note that, even though the cost of a single AO trial-vector transformation (10.8.8) is the same as for a single Roothaan-Hall SCF iteration (n ), the Newton itraations are still more expensive than the SCF iterations since a number of trial vectors are transformed in each Newton itexation. 

According to the Fourier method, the measured line integral p r,4>) in a sinogram is related to the count density distribution A(x,y) in the object obtained by the Fourier transformation. The projection data obtained in the spatial domain (Fig. 4.2a) can be expressed in terms of a Fourier series in the frequency domain as the sum of a series of sinusoidal waves of different amplitudes, spatial frequencies, and phase shifts running across the image (Fig. 4.2b). This is equivalent to sound waves that are composed of many sound frequencies. The data in each row of an acquisition matrix can be considered to be composed of sinusoidal waves of varying amplitudes and frequencies in the frequency domain. This conversion of data from spatial domain to frequency domain is called the Fourier transformation (Fig. 4.3). Similarly the reverse operation of converting the data from frequency domain to spatial domain is termed the inverse Fourier transformation. 

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