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2.2 The Application of Direct Methods to Centrosymmetric Structures Containing Heavy Atoms⁴

It is assumed that the positions of the heavy atoms are known and that there is a sufficient number of reflections whose signs are determined by the heavy atoms. These reflections do not obey the probability relation (2.6).

$$S_{h+h^{\prime}} {\sim} S_h . S_{h^{\prime}}$$ (2.6)

On subtracting the heavy atom contribution from the observed structure factors of these reflections, one obtains the sign of the light atom contributions for these reflections. Thereafter one can solve the remaining light atom structure by applying equation (2.6) to obtain the signs of the reflections that do not have contributions from the heavy atoms.

The procedure was used to solve the structure of the complex Au[S₂C₂(CN)₂]₂Au[S₂CN(C₄H₉)₂]₂. The space group was found to be P2₁/c, with two formula units per unit cell. The reflections hkl (h = 2n, k + l = 2n) were all very strong and the gold atoms were placed at the (special) position 000, $\frac{1}{2}$00, $\frac{1}{2}$$\frac{1}{2}$$\frac{1}{2}$, and 0$\frac{1}{2}$$\frac{1}{2}$. 1337 observed `strong' reflections (with equal positive contributions from the gold atoms) and 538 observed `weak' reflections (without any contributions from the gold atoms) were used.

The first step was a calculation of the Wilson plot. The following expression was used:

$$\langle{I}\rangle_h = K_L \langle\textstyle\sum^L_i f^2_i \e... ...ert F_H\vert^2 \exp(-2B_H \sin^2 {\theta}/{\lambda}^2)\rangle_h$$ (2.7)

where $I = (K\vert F_{\mbox{obs}}\vert^2_{\perp})$ is the observed intensity on a relative scale, K = K_L = K_H is the scale factor, $\sum^L$ denotes a summation over all light atoms in the unit cell, F_H is the heavy atom contribution to the structure factor and B_L and B_H are the overall temperature factor parameters of the light and heavy atoms respectively. The average is taken over reflections h within a given sin $\theta$ interval.

For the `weak' reflections (F<sub>H</sub> = 0) the second term in equation (2.7) vanishes and a Wilson plot for these reflections gave the scale factor K<sub>L</sub> (1.29) and the value of B<sub>L</sub> (3.24 Å<sup>2</sup>). On substituting these results in equation (2.7) a Wilson plot for the `strong' reflections gave the scale factor K_H (1.26) and the value of B_H (2.91 Ų). A small difference in K_L and K_H will not affect the following steps.

The second step is the calculation of the normalized structure factors E. The formulae normally used for the calculation of E values do not make sense for a structure containing heavy atoms. For the corresponding light atom structure the E values, E_L, are defined by:

$$E_L = F_L ({\varepsilon} \sum^L_i f^2_i)^{-1/2} \exp (B_L \sin^2 {\theta}/{\lambda}^2)$$ (2.8)

where F_L is the light atom contribution to the structure factor and, for space group $P2_1/c, \varepsilon$ = 2 for h01 and 0k0 reflections and $\varepsilon$ = 1 for all other reflections.⁶

The `strong' reflections have positive structure factors and we have $F_1 = F_{\mbox{obs}} - F_H$; the magnitude and the sign of the E<sub>L</sub> value is obtained by equation (2.8). This resulted in 365 signed E<sub>L</sub> values, with |E<sub>L</sub>| > 1.3. For the `weak' reflections we have |F_L| = $\vert F_{\mbox{obs}}\vert$ and only the magnitude of the E_L value is obtained. This resulted in 270 reflections with |E_L| > 1.3.

The third step is the application of equation (2.6) to obtain the signs of the `weak' reflections. When several interactions of the type ($h + h^{\prime}$) = (h) + ($h^{\prime}$) occur for |E<sub>L</sub>| > 1.3, where both S<sub>h</sub> and $S_{h^{\prime}}$ are known, several predictions of the sign $S_{h+h^{\prime}}$are obtained by application of (2.6). These predictions should be reasonably consistent before $S_{h+h^{\prime}}$ is considered to be determined and singly occurring interactions should never be trusted. We have followed a procedure similar to the sign correlation procedure. The origin is partly fixed by the choice of the gold atom positions and further determined by assigning arbitrary signs to two `weak' reflections: 221 (|E_L| = 4.0) and 34$\overline{8}$(|E_L| = 2.9). We define the following sets of reflections, all |E_L| > 2.0:

• h₁ are `strong' reflections, hkl(h = 2n, k + 1 = 2n).
• h₂ are the two origin determining choices.
• h₃ are the reflections h₁ + h₂ and $h_2 + h^{\prime}_2$.
• h₄ are the reflections h₁ + h₃, h₂ + h₃ and $h_3 + h^{\prime}_3$.

The application of the equation (2.6) on only reflections h₁ cannot give new signs; together with the reflections h₂ probable signs for 36 reflections h₃ were calculated. Upon entering h₃ in equation (2.6), many reflections take part in the calculations and consequently the sign of one reflection h₄ will often be found from several independent sign relations (2.6). Signs were calculated for 48 reflections h₄; of these the signs of 24 reflections were determined by at least five consistent relations (2.6) and accepted to be correct. Although some of the signs for reflections h₃ may be incorrectly determined, it is highly improbable that all reflections h₃ used for the signs determination of one reflection h₄ are incorrect. The intermediate results for h₃ and the rest of h₄ were rejected.

Continued application of equation (2.6) on 365 `strong' reflections, 2 reflections h<sub>2</sub> and 24 reflections h<sub>4</sub> resulted in the sign determination of 158 more `weak' reflections with |E_L| > 1.3. A Fourier synthesis revealed the positions of all of the light atoms, except the hydrogen atoms.

The above described procedure may be generalized for heavy atoms on general positions. In this case there also exist reflections with intermediate heavy atom contributions. For these reflections |F_L| = |$\vert F_{\mbox{obs}}\vert$ $\pm$ $\vert F_H\Vert$ | and the lowest F_L value is taken to avoid incorrect sign indications. In our opinion this procedure is well suited to an automatic solution of structures containing heavy atoms.

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