Convolutions Patterson function

The phase problem can be solved, that is, phases of the scattered waves determined, either by Patterson function or by direct methods. The Patterson function P is a self-convolution of the electron density p, and its magnitude at a point u, v, w can be obtained by multiplying p (x, y, z) hy p (x + u, y + V, z + w) and summing these products for every point of the unit cell. In practice, it is calculated as... 

The interpretation of the Patterson function is based on a specific property of Fourier transformation (denoted as 3[...]) when it is applied to convolutions (<8>) of functions ... 

As follows from Eq. 2.137, the multiplication of functions in the reciprocal space (e.g. structure amplitudes) results in a convolution of functions (e.g. electron or nuclear density) in the direct space, and vice versa. Since Eq. 2.136 contains the structure amplitude multiplied by itself, the resultant Patterson function, P yy, represents a self-convolution of the electron (nuclear) density. Hence, it may be described as follows ... 

If the selfconvolution of the electron density of a molecule (projected onto z) is denoted M(z) and the point-distribution function prescribing how the molecules are repeated along z by D(z), the desired Patterson function is their convolution... 

The Patterson function is yet another example of a convolution function (see Chapter 3), and it maps vector relationships in real space into a second coordinate system, which is Patterson space. It will be instructive here to examine the Patterson function s relationship to a real atom distribution by asking how it may be physically generated if the distribution of atoms in real space is known. Examples are illustrated in Figures 9.3 and 9.4. The Patterson function of a known structure is formed in the following way. 

The Patterson function is the convolution of the electron density p(r) with itself inverted with respect to the origin ... 

We have seen that the intensities of diffraction are proportional to the Fourier transform of the Patterson function, a self-convolution of the scattering matter and that, for a crystal, the Patterson function is periodic in three dimensions. Because the intensity is a positive, real number, the Patterson function is not dependent on phase and it can be computed directly from the data. The squared stmcture amplitude is... 

Based on the three-dimensional fimction proposed by Patterson in 1934, a new Fourier series that could be calculated directly from the measured intensities. This function is defined as the self-convolution of the electron density, p r), and has the same periodicity as the electron density ... 

Patterson synthesis by itself cannot solve even moderately large molecular structures (five or six equally heavy atoms) unless concrete structure information already exists so that the vector space can be systematically searched for a presumed structure or substructure. A search of this kind might be done with a computer, perhaps by the method of pattern-seeking functions, or the convolution molecule method [57]. [75]. These techniques, however, are not so often used today, having been supplanted by direct phase-determination methods (Section 15.2.2.2), which make it possible to reach the goal quickly on the basis of relatively little prior knowledge [76], Patterson synthesis may return to prominence when used to identify sets of starting phases (Section 15.2.2.2). 

An important theorem relates the Fourier transform and convolution operations. The Convolution Theorem (8,9) states that the Fourier transform of a convolution is the product of the Fourier transfomis, or F (f g) = F(u)G(u). Applying this to the autocorrelation yields F Kx) f(-x)] = F(u)F(-u). If f(x) is real, F(u)F(-u)=F(u)F (u)= F(u)p. Thus, "the Fourier transform ofthe autocorrelation of a function frx) is the squared modulus of its transform" (Ref. 9, p. 81). Application to scattering replaces frx) with the electron density profile, p(x). We have then the important result that the Fourier transform of the autocorrelation of the electron density profile is exactly equal to the intensity in reciprocal space, F(u). The autocomelation function cf the electron density has a special name it is called the generalized Patterson fimction(8), P(x), given by ... 

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