Completing the square

To understand why integrals over GTOs can be carried out when analogous STO-based integrals are much more difficult, one must only consider the orbital products (XaXc (ri) and XbXd (J l)) which arise in such integrals. For orbitals of the GTO form, such products involve exp(-tta (r-Ra) ) exp(-ac (r-Rc) ). By completing the square in the exponent, this product can be rewritten as follows ... 

The type of cui ve determined by a specific equation of the second degree can also be easily determined by reducing it to a standard form by translation and/or rotation. In the case in which the equation has no xy term, the procedure is merely to complete the squares of the terms in X and y separately. 

In practice, we seldom do the partial fraction expansion of a pair of complex roots. Instead, we rearrange the polynomial p(s) by noting that we can complete the squares ... 

The integral over Ry can be evaluated by converting it into a Gaussian integral (by writing the complex representation of sine and completing the square in the exponent) ... 

This integral is easily evaluated by completing the square and the resulting real space concentration field is... 

Figure 2S.7 Haemoglobin (a) The haem group, composed of the planar FIX molecule and iron, and shown here attached to the globin via an inudazole-nitrogen which completes the square pyramidal coordination of the Fe , and (b) myoglobin showing, diagrammadcally, the haem group in a pocket formed by the folded protein. The globin chain is actually in the form of 8 helical sections, labelled A to H, and the haem is situated between the E and F sections. The 4 subunits of haemoglobin are similar.

This may be regarded as the Fourier transform of the Maxwell distribution with Fourier variable (qt). The integral can be evaluated by completing the squares, giving... 

By completing the squares in the exponential this integral may be evaluated. It is useful to remember that the Fourier transform of a Gaussian is a Gaussian. 

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