Comment on Probability density functions of the average and difference intensities of Friedel opposites by Shmueli & Flack (2010)

Olczak, A.

doi:10.1107/S0108767311007896

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Comment on Probability density functions of the average and difference intensities of Friedel opposites by Shmueli & Flack (2010)

Andrzej Olczak ^a ^*

^aInstitute of General and Ecological Chemistry, Technical University of Łódź, Żeromskiego 116, 90-924 Łódź, Poland
^*Correspondence e-mail: olczakan@p.lodz.pl

(Received 4 November 2010; accepted 2 March 2011; online 30 March 2011)

A comment is made on the discussion of probability density functions (p.d.f.s) and simulations of Friedel intensity differences in Shmueli & Flack [Acta Cryst. (2010), A66, 669–675].

Keywords: probability density functions; Friedel opposites.

Similar articles

An interesting article by Shmueli & Flack (2010 ) (referred to as S&F in the following) was published recently [Acta Cryst. A66, 669–675]. Part of this article is devoted to probability density functions (p.d.f.s) and simulations of Friedel intensity differences, upon which I would like to share some remarks:

(1) Simulations presented in S&F are based on the assumption that $[{\bf h}\cdot{\bf R}_{jk}]$ ( $[{\bf R}_{jk} = {\bf r}_{j} - {\bf r}_{k}]$ , where $[{\bf r}_{j}]$ denote atom positions, $[j,k = 1, \ldots, N]$ and N is the number of atoms in the unit cell) are random, statistically independent variables whose fractional parts are uniformly distributed in the [0, 1] range. This assumption made by the authors (hereafter referred to assumption A) is inconsistent with the commonly held assumption that atoms are independently and uniformly distributed in the unit cell. This second assumption (hereafter referred to as assumption B) differs from assumption A in that fractional parts of $[{\bf h}\cdot{\bf r}_{j}]$ (not $[{\bf h}\cdot{\bf R}_{jk}]$ ) are random, independent variables uniformly distributed in the [0, 1] range.

Under assumption B one can define $[{\bf R}_{jk}]$ vectors as the $[{\bf r}_{j} - {\bf r}_{k}]$ differences ( $[{\bf R}_{jk} \equiv {\bf r}_{j} - {\bf r}_{k}]$ ). These symbols ( $[{\bf R}_{jk}]$ and $[{\bf r}_{j} - {\bf r}_{k}]$ ) are different ways of denoting the same thing.

Under assumption A, however, $[{\bf R}_{jk}]$ vectors can be generated independently. This means that what was the identity $[{\bf R}_{jk} \equiv {\bf r}_{j} - {\bf r}_{k}]$ now becomes the system of equations $[{\bf R}_{jk} = {\bf r}_{j} - {\bf r}_{k}]$ (with $[{\bf r}_{j}]$ as unknowns), which can be fulfilled or not. It is worth noting that there are a lot more `degrees of freedom' on the left-hand side of this system [N(N-1)/2] than on the right-hand side (N), which suggests that the solution to this system need not exist.

Let us consider a simple example showing the difference between these two approaches. The structure factor $[F_{{\bf h}} = \textstyle\sum_{j = 1}^{N} f_j \exp(2\pi {\bf h}\cdot {\bf r}_j)]$ under assumption B is a function of $[{\bf r}_{j}]$ vectors, $[F_{{\bf h}}({\bf r}_{1}, \ldots, {\bf r}_{N})]$ . The square of its modulus (and the modulus itself) is always a non-negative real number, $[|F_{{\bf h}}|^2 \ge 0]$ . $[|F_{{\bf h}}|^2]$ can be expressed in the form

$[|F_{{\bf h}}|^2 = \textstyle\sum\limits_{j = 1}^{N} f_j^2 + 2 \textstyle\sum\limits_{j \gt k}^N f_j\, f_k \cos(2\pi {\bf h}\cdot({\bf r}_j - {\bf r}_k)).]$

If we replace $[{\bf r}_j - {\bf r}_k]$ by $[{\bf R}_{jk}]$ (which are independent under assumption A) in this expression, we get a completely new function [let us call it $[M({\bf R}_{12}, {\bf R}_{13}, \ldots, {\bf R}_{N-1,N})]$ ]:

$[M = \textstyle\sum\limits_{j = 1}^{N} f_j^2 + 2 \textstyle\sum\limits_{j \gt k}^N f_j\, f_k \cos(2\pi {\bf h}\cdot{\bf R}_{jk}),]$

which can take negative values, if for example all $[\cos(2\pi {\bf h}\cdot{\bf R}_{jk})]$ = -1. Therefore, this function can no longer be interpreted as the square of the modulus of the familiar structure factor $[F_{{\bf h}}]$ .

(2) Simulations based on assumptions A and B lead to completely different results, which can be illustrated by using simulation results for the hypothetical structure U₂C₁₉. These simulations are based on the following formula for the anomalous differences [equation (14) in S&F]:

$[D_E({\bf h}) = 4\textstyle\sum\limits_{L_{jk}} d_{jk} \sin (2\pi{\bf h}\cdot {\bf R}_{jk}). \eqno (1)]$

Assumption A produces the histogram presented in Fig. 2 in S&F, while assumption B leads to the histogram presented in Fig. 1 of this letter.

Figure 1
Histogram calculated by the random-structure model (5 ×10⁴ simulations) and p.d.f. (under assumption B) of D_E for the compound U₂C₁₉. R = 0.03. The abscissa is on an arbitrary scale.

Which of these two assumptions is more appropriate for describing the structure with atoms randomly distributed in the unit cell?

Assumption B demonstrates that $[{\bf h}\cdot{\bf r}_{j}]$ are uniformly distributed. From a statistical point of view there is no difference whether $[{\bf h}]$ or $[{\bf r}_{j}]$ is varying in the $[{\bf h}\cdot{\bf r}_{j}]$ expression while the other parameter is fixed. Consequently, a very similar histogram to that presented in Fig. 1 of this letter should be obtained by the following procedure: using a random-number generator we generate independently atom positions for two U atoms and 19 C atoms, then we calculate structure factors, normalize them, calculate all anomalous differences and finally build the respective histogram out of these differences.

Let us try to apply this procedure in the case of assumption A. First, using the random-number generator we generate independently a set of $[{\bf R}_{jk}]$ interatomic vectors, then in order to calculate structure factors we need to determine the atom positions, $[{\bf r}_{j}]$ , for all atoms, or at least the fractional parts of the $[{\bf h}\cdot{\bf r}_{j}]$ scalar products. Unfortunately, in most cases this is not possible, because of the reason given below.

(3) Let us consider N(N-1)/2 independent random variables $[{\bf h}\cdot{\bf R}_{jk}]$ , each of which is uniformly distributed over the [0, 1] range.

Using a random-number generator one can generate a set of N(N-1)/2 values (let us call them T_jk) of these variables.

To determine $[{\bf h}\cdot{\bf r}_{j}]$ corresponding to the generated set of values T_jk, we need to solve the system of linear equations

$[T_{jk} = {\bf h}\cdot {\bf r}_{j} - {\bf h}\cdot {\bf r}_{k}.]$

This system is, however, overdetermined (for $[N \,\gt\, 3]$ ) because it contains N(N-1)/2 equations and only N unknowns $[{\bf h}\cdot{\bf r}_{k}]$ . Usually, there is no solution to such a system and consequently the values T_jk are purely abstract numbers, which do not correspond to any configuration of atoms in the unit cell (see Fig. 2 for more geometrical explanation). In such a case D_E calculated on the basis of the formula

$[D_E({\bf h}) = 4\textstyle\sum\limits_{L_{jk}} d_{jk} \sin (2\pi T_{jk})]$

loses its physical meaning. Hence the histograms are built of these `unphysical states'.

Figure 2
(a) Three atoms represented by small circles and their interatomic vectors, $[{\bf R}_{jk}]$ . Movement of atom 3 to position $[{3^\prime}]$ causes a change of two vectors: $[{\bf R}_{13}]$ and $[{\bf R}_{23}]$ , which means that interatomic vectors $[{\bf R}_{13}]$ and $[{\bf R}_{23}]$ are not independent. They always have to fulfil the relation $[{\bf R}_{12} + {\bf R}_{23} + {\bf R}_{31} = 0]$ . (b) Possible collection of three randomly generated vectors. These vectors fulfil the condition $[{\bf R}_{12} + {\bf R}_{23} + {\bf R}_{31} = 0]$ and, consequently, there exists an arrangement of three atoms (the structure) for which these vectors can be interpreted as interatomic vectors [see part (a)]. (c) Another possible collection of three vectors generated randomly. These vectors do not fulfil the relation $[{\bf R}_{12} + {\bf R}_{23} + {\bf R}_{31} = 0]$ and consequently there is no configuration of three atoms for which these vectors could be interpreted as interatomic vectors. This means that they cannot be expressed as $[{\bf R}_{12} = {\bf r}_{1} - {\bf r}_{2}]$ , $[{\bf R}_{23} = {\bf r}_{2} - {\bf r}_{3}]$ , $[{\bf R}_{31} = {\bf r}_{3} - {\bf r}_{1}]$ , where $[{\bf r}_j]$ represent atom positions. These vectors should not contribute to D_E and A_E histograms, because they are not associated with any arrangement of atoms in the unit cell.

In summary, the results concerning the simulations of Friedel intensity differences and the respective p.d.f.s presented in the S&F paper are interesting from a mathematical point of view, but do not appear to relate to any crystal structure.

(4) One exception exists when independence of the $[{\bf h}\cdot{\bf R}_{jk}]$ variables is consistent with assumption B: all but one of the atoms in the structure have the same atomic factor phase $[\delta_Q]$ { $[f_Q = f_{oQ} \exp({i\delta_Q})]$ , where $[f_{oQ} = [(f_Q + f^\prime_Q)^2 + (f^{\prime\prime}_Q)^2]^{1/2}]$ } (Olczak et al., 2003 ). In such a case we can choose the origin of the coordinate system at this particular atom, which allows for the transformation of all $[\sin (2\pi{\bf h}\cdot{\bf R}_{jk})]$ components in equation (1) into the $[\sin (2\pi {\bf h}\cdot{\bf r}_k)]$ form. It can be shown using the central limit theorem that in this case the p.d.f. for D_E is a Gaussian function if the number of atoms with phase $[\delta_Q]$ is large enough (in practice greater than six or seven) (Olczak et al., 2003; Olczak, 2004 ).

(5) Under assumption B, the p.d.f. of the Friedel differences for the U₂C₁₉ structure (in which two atoms are distinguished) takes a quite different form from the Gaussian shape. In this case D_E can be represented in the form

$[D_E = -8 d_{\rm UC} \cos (2\pi {\bf h}\cdot {\bf r}_{\rm U_1}) \textstyle\sum\limits_{j = 1}^{N-2} \sin (2\pi {\bf h} \cdot {\bf r}_{{\rm C}_{j}})]$

if one puts the origin of the coordinate system exactly in the middle between the two U atoms ( $[{\bf r}_{\rm U_1} = - {\bf r}_{\rm U_2}]$ ). D_E is a product of two independent random variables: $[\cos (2\pi {\bf h}\cdot {\bf r}_{\rm U_1})]$ and $[\textstyle\sum_{j = 1}^{N-2} \sin (2\pi {\bf h}\cdot {\bf r}_{{\rm C}_j})]$ . Now we can follow the reasoning of Parthasarathy & Srinivasan (1964 ) to get a p.d.f. in the form

$[{2}^{1/2}/(\pi^{3/2} a) K_{0}(D_E^2/a^2) \exp{(-D_E^2/a^2)},]$

where a is a constant and K₀ is a modified Bessel function of the second kind. This function fits perfectly the simulations of D_E based on assumption B (Fig. 1) and is a generalization of results obtained by Parthasarathy and Srinivasan, who considered structures containing P resonant atoms (in this case P = 2) of the same chemical element along with Q non-resonant atoms where Q has to be large. The generalization is that Q atoms can also be resonant ones.

Some other cases considered by Parthasarathy and Srinivasan can be generalized as well, but it is much more difficult (if at all possible) to construct a universal p.d.f. embracing all possibilities (any chemical composition of the structure) at once, especially when partially occupied sites are considered (Olczak, 2008 ) (Fig. 3).

Figure 3
Histograms of D_E for the compound C₁₅H₂₃NO₂ (Cu Kα radiation) by the random-structure model (10⁵ simulations under assumption B) containing in addition one Cl atom (thick dotted line), two Cl atoms (solid line), or one Cl atom with an occupancy factor 0.5 and one with an occupancy factor 1 (thin dotted line). The abscissa is on an arbitrary scale and is not common to all plots. All plots are normalized to 10⁵.

(6) There is also a good side to this diversity of p.d.f. shapes, which opens some new opportunities. In principle, it should be possible to determine from diffraction data whether the structure contains one or more strong anomalous scatterers. It is necessary, however, to remember that experimental differences, along with anomalous differences, also contain measurement uncertainties (see §3.6 in Olczak et al., 2010 ), which should be deconvoluted before the experimental differences are compared to the theoretical ones.

References

Olczak, A. (2004). 46th Polish Crystallographic Meeting, p. 16. Wrocław, Poland. Google Scholar
Olczak, A. (2008). 50th Polish Crystallographic Meeting, p. 191. Wrocław, Poland. Google Scholar
Olczak, A., Cianci, M., Hao, Q., Rizkallah, P. J., Raftery, J. & Helliwell, J. R. (2003). Acta Cryst. A59, 327–334. Web of Science CrossRef CAS IUCr Journals Google Scholar
Olczak, A., Główka, M. L., Szczesio, M., Bojarska, J., Wawrzak, Z. & Duax, W. L. (2010). Acta Cryst. D66, 874–880. Web of Science CrossRef CAS IUCr Journals Google Scholar
Parthasarathy, S. & Srinivasan, R. (1964). Acta Cryst. 17, 1400–1407. CrossRef CAS IUCr Journals Web of Science Google Scholar
Shmueli, U. & Flack, H. D. (2010). Acta Cryst. A66, 669–675. Web of Science CrossRef CAS IUCr Journals Google Scholar

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Comment on Probability density functions of the average and difference intensities of Friedel opposites by Shmueli & Flack (2010)

References

letters to the editor