Quartet phase relationships

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. 220-222 | 1 | 2 |

Section 2.2.5.5. Quartet phase relationships

C. Giacovazzo^a ^*

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

2.2.5.5. Quartet phase relationships

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In early papers (Hauptman & Karle, 1953; Simerska, 1956) the phase $[\Phi = \varphi_{\bf h} + \varphi_{\bf k} + \varphi_{\bf l} - \varphi_{{\bf h} + {\bf k} + {\bf l}}]$ was always expected to be zero. Schenk (1973a,b) [see also Hauptman (1974)] suggested that Φ primarily depends on the seven magnitudes: $[R_{\bf h}, R_{\bf k}, R_{\bf l}, R_{{\bf h} + {\bf k} + {\bf l}}]$ , called basis magnitudes, and $[R_{{\bf h} + {\bf k}}, R_{{\bf h} + {\bf l}}, R_{{\bf k} + {\bf l}}]$ , called cross magnitudes.

The conditional probability of Φ in P1 given seven magnitudes $[(R_{1} = R_{\bf h}, \ldots,\ R_{4} = R_{{\bf h} + {\bf k} + {\bf l}},\ R_{5} = R_{{\bf h} + {\bf k}},\; R_{6} = R_{{\bf h} + {\bf l}},\; R_{7} = R_{{\bf k} + {\bf l}})]$ according to Hauptman (1975) is $[\eqalign{P_{7} (\Phi) &= {1\over L} \exp (- 2 B \cos \Phi) I_{0} (2 \sigma_{3} \sigma_{2}^{-3/2} R_{5} Y_{5})\cr &\quad \times I_{0} (2 \sigma_{3} \sigma_{2}^{-3/2} R_{6} Y_{6}) I_{0} (2 \sigma_{3} \sigma_{2}^{-3/2} R_{7} Y_{7}),}]$ where L is a suitable normalizing constant which can be derived numerically, $[\eqalign{B &= \sigma_{2}^{-3} (3 \sigma_{3}^{2} - \sigma_{2} \sigma_{4}) R_{1} R_{2} R_{3} R_{4}\cr Y_{5} &= [R_{1}^{2} R_{2}^{2} + R_{3}^{2} R_{4}^{2} + 2 R_{1} R_{2} R_{3} R_{4} \cos \Phi]^{1/2}\cr Y_{6} &= [R_{3}^{2} R_{1}^{2} + R_{2}^{2} R_{4}^{2} + 2 R_{1} R_{2} R_{3} R_{4} \cos \Phi]^{1/2}\cr Y_{7} &= [R_{2}^{2} R_{3}^{2} + R_{1}^{2} R_{4}^{2} + 2 R_{1} R_{2} R_{3} R_{4} \cos \Phi]^{1/2}.}]$ For equal atoms $[\sigma_{2}^{-3} (3 \sigma_{3}^{2} - \sigma_{2} \sigma_{4}) = 2/N]$ . Denoting $[\displaylines{C = R_{1} R_{2} R_{3} R_{4} / N, \cr Z_{5} = 2 Y_{5} / \sqrt{N}, \quad Z_{6} = 2 Y_{6} / \sqrt{N}, \quad Z_{7} = 2 Y_{7} / \sqrt{N}}]$ gives $[\eqalignno{P_{7} (\Phi) &= {1\over L} \exp (- 4 C \cos \Phi)\cr &\quad \times I_{0} (R_{5} Z_{5}) I_{0} (R_{6} Z_{6}) I_{0} (R_{7} Z_{7}). &(2.2.5.18)}]$ Fig. 2.2.5.3 shows the distribution (2.2.5.18) for three typical cases. It is clear from the figure that the cosine estimated near π or in the middle range will be in poorer agreement with the true values than the cosine near 0 because of the relatively larger values of the variance. In principle, however, the formula is able to estimate negative or enantiomorph-sensitive quartet cosines from the seven magnitudes.

Figure 2.2.5.3 | top | pdf |

Distributions (2.2.5.18) (solid curve) and (2.2.5.20) (dashed curve) for the indicated [|E|] values in three typical cases.

In the cs. case (2.2.5.18) is replaced (Hauptman & Green, 1976) by $[\eqalignno{P^{\pm} \simeq &{1\over L} \exp (\mp 2 C) \cosh (R_{5} Z_{5}^{\pm})\cr &\times \cosh (R_{6} Z_{6}^{\pm}) \cosh (R_{7} Z_{7}^{\pm}), &(2.2.5.19)}]$ where $[P^{\pm}]$ is the probability that the sign of $[E_{1} E_{2} E_{3} E_{4}]$ is positive or negative, and $[\eqalign{Z_{5}^{\pm} &= {1\over {N^{1/2}}} (R_{1} R_{2} \pm R_{3} R_{4}),\cr Z_{6}^{\pm} &= {1\over {N^{1/2}}} (R_{1} R_{3} \pm R_{2} R_{4}),\cr Z_{7}^{\pm} &= {1\over {N^{1/2}}} (R_{1} R_{4} \pm R_{2} R_{3}).}]$ The normalized probability may be derived by $[P^{+} / (P^{+} + P^{-})]$ . More simple probabilistic formulae were derived independently by Giacovazzo (1975, 1976): $[P_{7} (\Phi) = [2\pi I_{0} (G)]^{-1} \exp (G \cos \Phi), \eqno(2.2.5.20)]$ where $[G = {2C (1 + \varepsilon_{5} + \varepsilon_{6} + \varepsilon_{7}) \over 1 + Q / (2N)}\eqno (2.2.5.21)]$ $[{Q = (\varepsilon_{1} \varepsilon_{2} + \varepsilon_{3} \varepsilon_{4}) \varepsilon_{5} + (\varepsilon_{1} \varepsilon_{3} + \varepsilon_{2} \varepsilon_{4}) \varepsilon_{6} + (\varepsilon_{1} \varepsilon_{4} + \varepsilon_{2} \varepsilon_{3})} \varepsilon_{7} \eqno (2.2.5.22)]$ and $[\varepsilon_{i} = (|E_{i}|^{2} - 1)]$ . Q is never allowed to be negative.

According to (2.2.5.20) $[\cos\Phi]$ is expected to be positive or negative according to whether $[(\varepsilon_{5} + \varepsilon_{6} + \varepsilon_{7} + 1)]$ is positive or negative: the larger is C, the more reliable is the phase indication. For $[N \geq 150]$ , (2.2.5.18) and (2.2.5.20) are practically equivalent in all cases. If N is small, (2.2.5.20) is in good agreement with (2.2.5.18) for quartets strongly defined as positive or negative, but in poor agreement for enantiomorph-sensitive quartets (see Fig. 2.2.5.3).

In cs. cases the sign probability for $[E_{1} E_{2} E_{3} E_{4}]$ is $[P^{+} = {\textstyle{1 \over 2}} + {\textstyle{1 \over 2}} \tanh (G / 2), \eqno(2.2.5.23)]$ where G is defined by (2.2.5.21).

All three cross magnitudes are not always in the set of measured reflections. From marginal distributions the following formulae arise (Giacovazzo, 1977c; Heinermann, 1977b):

(a) in the ncs. case, if $[R_{7}]$ , or $[R_{6}]$ and $[R_{7}]$ , or $[R_{5}]$ and $[R_{6}]$ and $[R_{7}]$ , are not in the measurements, then (2.2.5.18) is replaced by $[P(\Phi | R_{1}, \ldots, R_{6}) \simeq {1 \over L'} \exp (-2 C \cos \Phi) I_{0} (R_{5} Z_{5}) I_{0} (R_{6} Z_{6}),]$ or $[P (\Phi | R_{1}, \ldots, R_{5}) \simeq {1 \over L''} I_{0} (R_{5} Z_{5}),]$ or $[P (\Phi | R_{1}, \ldots, R_{4}) \simeq {1 \over L'''} \exp (2C \cos \Phi),]$ respectively.
(b) in the same situations, we have for cs. cases $[P^{\pm} \simeq {1 \over L'} \exp (\mp C) \cosh (R_{5} Z_{5}^{\pm}) \cosh (R_{6} Z_{6}^{\pm}),]$ or $[P^{\pm} \simeq {1 \over L''} \cosh (R_{5} Z_{5}^{\pm})]$ or $[P^{\pm} = {1 \over L'''} \exp (\pm C) \simeq 0.5 + 0.5 \tanh (\pm C),]$ respectively.

Equations (2.2.5.20) and (2.2.5.23) are easily modifiable when some cross magnitudes are not in the measurements. If $[R_{i}]$ is not measured then (2.2.5.20) or (2.2.5.23) are still valid provided that in G it is assumed that $[\varepsilon_{i} = 0]$ . For example, if $[R_{7}]$ and $[R_{6}]$ are not in the data then (2.2.5.21) and (2.2.5.22) become $[G = {2C (1 + \varepsilon_{5}) \over 1 + Q / (2N)},\qquad Q = (\varepsilon_{1} \varepsilon_{2} + \varepsilon_{3} \varepsilon_{4}) \varepsilon_{5}.]$ In space groups with symmetry higher than $[P\bar{1}]$ more symmetry-equivalent quartets can exist of the type $[\psi = \varphi_{{\bf h} {\bi R}_{\alpha}} + \varphi_{{\bf k}{\bi R}_{\beta}} + \varphi_{{\bf l} {\bi R}_{\gamma}} + \varphi_{(\overline{{\bf h} + {\bf k} + {\bf l}}) {\bi R}_{\delta}},]$ where $[{\bi R}_{\alpha}, {\bi R}_{\beta}, {\bi R}_{\gamma}, {\bi R}_{\delta}]$ are rotation matrices of the space group. The set $[\{\psi\}]$ is called the first representation of Φ. In this case Φ primarily depends on more than seven magnitudes. For example, let us consider in Pmmm the quartet $[\Phi = \varphi_{123} + \varphi_{\bar{1}5\bar{3}} + \varphi_{\bar{1}\bar{5}8} + \varphi_{1\bar{2}\bar{8}}.]$ Quartets symmetry equivalent to Φ and respective cross terms are given in Table 2.2.5.1.

Table 2.2.5.1 | top | pdf |
List of quartets symmetry equivalent to $[\Phi = \Phi_{1}]$ in the class mmm

Quartets	Basis vectors				Cross vectors
$[\Phi_{1}]$	(1, 2, 3)	$[(\bar{1}, 5, \bar{3})]$	$[(\bar{1}, \bar{5}, {8})]$	$[(1, \bar{2}, \bar{8})]$	(0, 7, 0)	$[(0, \bar{3}, 11)]$	$[(\bar{2}, 0, 5)]$
$[\Phi_{2}]$	$[(\bar{1}, 2, 3)]$	$[(1, 5, \bar{3})]$	$[(\bar{1}, \bar{5}, 8)]$	$[(1, \bar{2}, \bar{8})]$	(0, 7, 0)	$[(\bar{2}, \bar{3}, 11)]$	(0, 0, 5)
$[\Phi_{3}]$	$[(1, 2, \bar{3})]$	$[(\bar{1}, 5, 3)]$	$[(\bar{1}, \bar{5}, 8)]$	$[(1, \bar{2}, \bar{8})]$	(0, 7, 0)	$[(0, \bar{3}, 5)]$	$[(\bar{2}, 0, 11]$ )
$[\Phi_{4}]$	$[(\bar{1}, 2, \bar{3})]$	(1, 5, 3)	$[(\bar{1}, \bar{5}, 8)]$	$[(1, \bar{2}, \bar{8})]$	(0, 7, 0)	$[(\bar{2}, \bar{3}, 5)]$	(0, 0, 11)
$[\Phi_{5}]$	$[(\bar{1}, 2, 3)]$	$[(\bar{1}, 5, \bar{3})]$	$[(1, \bar{5}, 8)]$	$[(1, \bar{2}, \bar{8})]$	$[(\bar{2}, 7, 0)]$	$[(0, \bar{3}, 11)]$	(0, 0, 5)
$[\Phi_{6}]$		$[(\bar{1}, \bar{5}, \bar{3})]$	$[(\bar{1}, 5, 8)]$	$[(1, \bar{2}, \bar{8})]$	$[(0, \bar{3}, 0)]$	(0, 7, 11)	$[(\bar{2}, 0, 5)]$
$[\Phi_{7}]$	$[(\bar{1}, 2, 3)]$	$[(1, \bar{5}, \bar{3})]$	$[(\bar{1}, 5, 8)]$	$[(1, \bar{2}, \bar{8})]$	$[(0, \bar{3}, 0)]$	$[(\bar{2}, 7, 11)]$	(0, 0, 5)
$[\Phi_{8}]$	$[(\bar{1}, 2, \bar{3})]$	$[(\bar{1}, 5, 3)]$	$[(1, \bar{5}, 8)]$	$[(1, \bar{2}, \bar{8})]$	$[(\bar{2}, 7, 0)]$	$[(0, \bar{3}, 5)]$	(0, 0, 11)
$[\Phi_{9}]$	$[(1, 2, \bar{3})]$	$[(\bar{1}, \bar{5}, 3)]$	$[(\bar{1}, 5, 8)]$	$[(1, \bar{2}, \bar{8})]$	$[(0, \bar{3}, 0)]$	(0, 7, 5)	$[(\bar{2}, 0, 11)]$
$[\Phi_{10}]$	$[(\bar{1}, 2, \bar{3})]$	$[(1, \bar{5}, 3)]$	$[(\bar{1}, 5, 8)]$	$[(1, \bar{2}, \bar{8})]$	$[(0, \bar{3}, 0)]$	$[(\bar{2}, 7, 5)]$	(0, 0, 11)
$[\Phi_{11}]$	$[(\bar{1}, 2, 3)]$	$[(\bar{1}, \bar{5}, \bar{3})]$	(1, 5, 8)	$[(1, \bar{2}, \bar{8})]$	$[(\bar{2}, \bar{3}, 0)]$	(0, 7, 11)	(0, 0, 5)

Experimental tests on the application of the representation concept to quartets have recently been made (Busetta et al., 1980). It was shown that quartets with more than three cross magnitudes are more accurately estimated than other quartets. Also, quartets with a cross reflection which is systematically absent were shown to be of significant importance in direct methods. In this context it is noted that systematically absent reflections are not usually included in the set of diffraction data. This custom, not exceptionable when only triplet relations are used, can give rise to a loss of information when quartets are used. In fact the usual programs of direct methods discard quartets as soon as one of the cross reflections is not measured, so that systematic absences are dealt with in the same manner as those reflections which are outside the sphere of measurements.

References

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International Tables for Crystallography (2006). Vol. B. ch. 2.2, pp. 220-222