Determinantal formulae

Giacovazzo, C.

doi:10.1107/97809553602060000555

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 2.2, pp. 223-224 | 1 | 2 |

Section 2.2.5.7. Determinantal formulae

C. Giacovazzo^a ^*

^aDipartimento Geomineralogico, Campus Universitario, I-70125 Bari, Italy
Correspondence e-mail: c.giacovazzo@area.ba.cnr.it

2.2.5.7. Determinantal formulae

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In a crystal structure with N identical atoms the joint probability distribution of n normalized s.f.'s $[E_{{\bf h}_{1} + {\bf k}}, E_{{\bf h}_{2} + {\bf k}}, \ldots, E_{{\bf h}_{n} + {\bf k}}]$ under the following conditions:

(a) the structure is kept fixed whereas k is the primitive random variable;
(b) $[E_{{\bf h}_{i} - {\bf h}_{j}},\ i, j = 1, \ldots, n]$ , have values which are known a priori; is given (Tsoucaris, 1970) [see also Castellano et al. (1973) and Heinermann et al. (1979)] by $[P(E_{1}, E_{2}, \ldots, E_{n}) = (2\pi)^{-n/2} D_{n}^{-1/2} \exp (-{\textstyle{1 \over 2}} Q_{n}) \eqno(2.2.5.27)]$ for cs. structures and $[P(E_{1}, E_{2}, \ldots, E_{n}) = (2\pi)^{-n} D_{n}^{-1/2} \exp (-Q_{n}) \eqno(2.2.5.28)]$ for ncs. structures. In (2.2.5.27) and (2.2.5.28) we have denoted $[\displaylines{D_{n} = \lambda,\qquad Q_{n} = \textstyle\sum\limits_{p, \, q = 1}^{n} \Lambda_{pq} E_{p} E_{q}^{*}\cr E_{j} = E_{{\bf h}_{j} + {\bf k}},\qquad U_{pq} = U_{{\bf h}_{p} - {\bf h}_{q}},\qquad j, p, q = 1, \ldots, n.}]$ $[\Lambda_{pq}]$ is an element of $[\boldlambda ^{-1}]$ , and $[\boldlambda ]$ is the covariance matrix with elements $[\displaylines{\langle E_{{\bf h}_{p} + {\bf k}} E_{{\bf h}_{q} + {\bf k}}\rangle = U_{{\bf h}_{p} - {\bf h}_{q}}\cr \boldlambda = \left|\matrix{1 &U_{12} &\ldots &U_{1q} &\ldots &U_{1n}\cr U_{21} &1 &\ldots &U_{2q} &\ldots &U_{2n}\cr \vdots &\vdots &\ddots &\vdots &\ddots &\vdots\cr U_{p1} &U_{p2} &\ldots &U_{pq} &\ldots &U_{pn}\cr \vdots &\vdots &\ddots &\vdots &\ddots &\vdots\cr U_{n1} &U_{n2} &\ldots &U_{nq} &\ldots &1\cr}\right|.}]$ $[\lambda]$ is a K–H determinant: therefore $[D_{n} \geq 0]$ . Let us call $[\Delta_{n + 1} = {1 \over N} \left|\matrix{1 &U_{12} &\ldots &U_{1n} &E_{{\bf h}_{1} + {\bf k}}\cr U_{21} &1 &\ldots &U_{2n} &E_{{\bf h}_{2} + {\bf k}}\cr \vdots &\vdots &\ddots &\vdots &\vdots\cr U_{n1} &U_{n2} &\ldots &1 &E_{{\bf h}_{n} + {\bf k}}\cr E_{-{\bf h}_{1} - {\bf k}} &E_{-{\bf h}_{2} - {\bf k}} &\ldots &E_{-{\bf h}_{n} - {\bf k}} &N\cr}\right|\hbox{;}]$ the K–H determinant obtained by adding to $[\boldlambda ]$ the last column and line formed by $[E_{1}, E_{2}, \ldots, E_{n}]$ , and $[E_{1}^{*}, E_{2}^{*}, \ldots, E_{n}^{*}]$ , respectively. Then (2.2.5.27) and (2.2.5.28) may be written $[\eqalignno{&P(E_{1}, E_{2}, \ldots, E_{n})\cr &\quad = (2\pi)^{-n/2} D_{n}^{-1/2} \exp \left[N {\Delta_{n + 1} - D_{n}\over 2D_{n}}\right] &(2.2.5.29)}]$ and $[\eqalignno{&P(E_{1}, E_{2}, \ldots, E_{n})\cr &\quad= (2\pi)^{-n} D_{n}^{-1/2} \exp \left[N {\Delta_{n + 1} - D_{n}\over D_{n}}\right], &(2.2.5.30)}]$ respectively. Because $[D_{n}]$ is a constant, the maximum values of the conditional joint probabilities (2.2.5.29) and (2.2.5.30) are obtained when $[\Delta_{n + 1}]$ is a maximum. Thus the maximum determinant rule may be stated (Tsoucaris, 1970; Lajzérowicz & Lajzérowicz, 1966): among all sets of phases which are compatible with the inequality $[\Delta_{n + 1} (E_{1}, E_{2}, \ldots, E_{n}) \geq 0]$ the most probable one is that which leads to a maximum value of $[\Delta_{n + 1}]$ .

If only one phase, i.e. $[\varphi_{q}]$ , is unknown whereas all other phases and moduli are known then (de Rango et al., 1974; Podjarny et al., 1976) for cs. crystals $[P^{\pm} (E_{q}) \simeq 0.5 + 0.5 \ \tanh \ \left\{\pm |E_{q}| \textstyle\sum\limits_{p = 1 \atop p \neq q}^{n} \Lambda_{pq} E_{p}\right\}, \eqno(2.2.5.31)]$ and for ncs. crystals $[P(\varphi_{q}) = [2\pi I_{0} (G_{q})]^{-1} \exp \{G_{q} \cos (\varphi_{q} - \theta_{q})\}, \eqno(2.2.5.32)]$ where $[G_{q} \exp (i\theta_{q}) = 2 |E_{q}| \textstyle\sum\limits_{p \neq q = 1}^{n} \Lambda_{pq} E_{p}.]$ Equations (2.2.5.31) and (2.2.5.32) generalize (2.2.5.11) and (2.2.5.7), respectively, and reduce to them for . Fourth-order determinantal formulae estimating triplet invariants in cs. and ncs. crystals, and making use of the entire data set, have recently been secured (Karle, 1979, 1980).

Advantages, limitations and applications of determinantal formulae can be found in the literature (Heinermann et al., 1979; de Rango et al., 1975, 1985). Taylor et al. (1978) combined K–H determinants with a magic-integer approach. The computing time, however, was larger than that required by standard computing techniques. The use of K–H matrices has been made faster and more effective by de Gelder et al. (1990) (see also de Gelder, 1992). They developed a phasing procedure (CRUNCH) which uses random phases as starting points for the maximization of the K–H determinants.

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International Tables for Crystallography (2006). Vol. B. ch. 2.2, pp. 223-224