6 The estimation of quartet invariants in P1 and P¯1 via their first representation: Hauptman approach
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The estimation of quartet invariants in P1 and P1¯
2
Z6 = √ (R21 R23 + R22 R24 + 2NC cos )1/2 ,
N
113
(5.18b)
2
Z7 = √ (R22 R23 + R21 R24 + 2NC cos )1/2 .
(5.18c)
N
As for the triplet invariants, distribution (5.16) depends on cos ; therefore
only cos may be estimated, it being impossible to distinguish between +
an − (or, in other words, to distinguish between the two enantiomorphs).
Since L, the scaling factor, has a rather complicated expression, one might
use numerical methods for calculating:
1. the scaling factor L, via the condition
π
P( )d
= 1;
0
2. the mode m of P( );
3. the mean value, given by
P(F)
π
=
P( )d ;
0
4. the variance, V, as given by
π
V=
5. σ =
√
V.
(
−
R1 = 2.27
R2 = 3.01
R3 = 2.49
R4 = 2.16
R5 = 1.85
R6 = 2.84
R7 = 1.90
)2 P( )d .
0
Estimation of | |, via (5.16), depends on an intricate interrelationship among
all the seven magnitudes. However, some working rules can be stated:
1. P( ) is unimodal between 0 and π , and m can, in principle, lie anywhere
between 0 and π;
2. if the cross-magnitudes are large, is expected to be close to zero;
3. if the cross-magnitudes are small, is expected to be close to π ;
4. if the cross-magnitudes are of medium size and N is sufficiently small, then
is expected to be close to ±π/2;
5. the larger N, the larger the overall variance associated with quartet phase
estimation.
Figures 5.6 and 5.7 show (broken curves) the distribution (5.16) for some
values of the seven magnitudes when N = 47. In Fig. 5.6, where all the
cross-magnitudes are large, m = 0.0,
29◦ , σ = V 1/2 = 21.9◦ . In Fig.
5.7 where all the cross-magnitudes are small, m = 180◦ ,
142◦ , σ =
◦
32.7 .
It is clear from the figures that cosines estimated near π will (on average)
be in poorer agreement with the true values than the cosines estimated near
0, because of the relatively larger value of the variance. Even poorer will
be the estimates of the cosines located in the middle range (usually called
enantiomorph sensitive quarters); no useful application has been found for
them.
The three cross-magnitudes are not always in the set of measured reflections.
Then, some marginal joint probability distributions must be considered in order
p/2
p
Fig. 5.6
Distribution (5.16) (broken curve) and
(5.22) (continuous curve) for the indicated |E| values in a structure with
N = 47 atoms in the unit cell.
P(F)
R1 = 2.31
R2 = 2.82
R3 = 1.88
R4 = 2.10
R5 = 0.36
R6 = 0.24
R7 = 0.10
p/2
p
Fig. 5.7
Distribution (5.16) (broken curve) and
(5.22) (continuous curve) for the indicated |E| values in a structure with
N = 47 atoms in the unit cell.
114
The probabilistic estimation of triplet and quartet invariants
to derive useful information in these less favourable cases. In accordance with
equation (M.A.16), marginal distributions may be obtained by performing the
integration of (5.16) with respect to R5, R6, R7, depending on which crossmagnitudes are not measured. The resulting general formula is
P( | . . . .) ≈
1
exp[−2C(n − 1) cos ]I0 (w5 R5 Z5 )I0 (w6 R6 Z6 )I0 (w7 R7 Z7 ),
L
(5.19)
where n is the number of known cross-magnitudes and I 0 (x) is the modified Bessel function of order zero. wi is equal to zero if the cross term Ri is
unknown, otherwise wi = 1.
A last observation concerns the generalization of (5.19) to unequal atom
structures. This can be made by replacing: in equation (5.17), N by σ22 /σ4 ; in
equations (5.18), N by Neq , as given by equation (5.5). It has been verified that,
in practice, one can deal with unequal atom structures by replacing N by Neq
in the full expression (5.19).
For centric space groups, Hauptman and Green (1976) obtained the sign
probability
P± ≈
1
exp(∓2C) cosh(R5 Z5± ) cosh(R6 Z6± ) cosh(R7 Z7± ),
L
(5.20)
where
L = exp(−2C) cosh(R5 Z5+ ) cosh(R6 Z6+ ) cosh(R7 Z7+ )
exp(+2C) cosh(R5 Z5− ) cosh(R6 Z6− ) cosh(R7 Z7− )
and
1
Z5± = √ (R1 R2 ± R3 R4 ),
N
1
Z6± = √ (R1 R3 ± R2 R4 )
N
1
Z7± = √ (R1 R4 ± R2 R3 ).
N
P+ and P− are the probabilities that E1 E2 E3 E4 is positive or negative; they
may lie anywhere between 0 and 1. P− is close to 1 or close to 0 if R5 , R6 , and
R7 are either all relatively small or all relatively large, respectively. In order to
derive useful information, even in the less favourable cases in which some of
the cross-magnitudes are not among the measurements, conditional probability
values may be derived, leading to the general formula,
P± ≈
1
exp[∓C(n − 1)] cosh(w5 R5 Z5± ) cosh(w6 R6 Z6± ) cosh(w7 R7 Z7± ),
L
(5.21)
where n is the number of known cross-magnitudes, wi = 0 if the cross term Ri is
unknown, otherwise wi = 1. As in the acentric case, we can generalize (5.21) to
unequal atom structures by replacing N by Neq , as given by equation (5.5).
The estimation of quartet invariants in P1 and P1¯
5.7 The estimation of quartet invariants
¯ via their first representation:
in P1 and P1
Giacovazzo approach
Giacovazzo expressions for estimating quartet invariants via their first phasing shell are simpler than Hauptman formulas, but equally efficient. His final
expression for an acentric space group is a von Mises formula,
P( |R1 , R2 , . . . , R7 ) = (2πI0 (G))−1 exp(G cos ),
(5.22)
where
G=
2C(1 + ε5 + ε6 + ε7 )
,
1+Q
(5.23)
Q = [(ε1 ε2 + ε3 ε4 )ε5 + (ε1 ε3 + ε2 ε4 )ε6 + (ε1 ε4 + ε2 ε3 )ε7 ]/2N,
and εi = R2i − 1.
It should be noted that:
1. It is convenient to set Q = 0 when Q ≤ 0.
2. G is positive if ε5 + ε6 + ε7 > 1, and negative in the opposite case. If G
is negative, the most probable value of
is π , and the quartet cosine is
estimated to be negative.
3. Large values of the cross-magnitudes (no matter whether the basis magnitudes are large or small) will correspond to positive estimated quartets;
small values of the cross-magnitudes will mark negative estimated quartets.
4. The marginal distributions of , corresponding to cases in which some of
the cross-magnitudes are unknown, can be obtained by setting the corresponding terms εi to zero (this corresponds to making E2 equal to 1, its
expected value). In mathematical notation, a general expression may be
used for G, covering all cases,
G=
2C(1 + w5 ε5 + w6 ε6 + w7 ε7 )
,
1+Q
(5.24)
where
Q = [w5 (ε1 ε2 + ε3 ε4 )ε5 + w6 (ε1 ε3 + ε2 ε4 )ε6 + w7 (ε1 ε4 + ε2 ε3 )ε7 ]/2N.
wi is equal to one, but for the case in which the ith cross-reflection is not
measured. In this last case, wi = 0.
5. The generalization of (5.23) to unequal atom structures can be done by
replacing N with Neq (as given by (5.5)).
Distribution (5.22) is drawn (full curve) in Figs. 5.6 and 5.7, for the same
values for which distribution (5.16) is calculated.
If an analogous approach is applied in P1, we have
P+
0.5 + 0.5 tanh(G/2),
(5.25)
where G is given by (5.23) (or (5.24)) and P+ is the probability that the sign of
Eh1 Eh2 Eh3 Eh4 is positive. If G > 0, then P+ >1/2.
115
116
The probabilistic estimation of triplet and quartet invariants
The accuracy of the Hauptman and Giacovazzo formulations is discussed in
Section 5.8.
Quartet estimates may improve if full use is made of the symmetry. Readers
interested in this topic will find preliminary information in Appendix 5.D and
a more general description in Giacovazzo (1976d).
5.8 About quartet reliability
There are three basic questions concerning the use of quartet invariants in
phasing procedures. Let us consider these in the following order:
(i) How many quartets can be found among a selected (and sufficiently large)
number of reflections? Is this number larger than the number of triplet
invariants? The answer is yes; the number of quartets is usually much
larger than the number of reflections. The practical aspects concerning
triplet and quartet identification are discussed in Appendix 6.A, but, even
intuitively, the reader can easily understand why the quartet number is
much larger than the triplet number (i.e. quartets have a one degree of
freedom more).
(ii) In its first representation a triplet invariant depends on three diffraction
amplitudes only, while a quartet invariant depends on (at least) seven
magnitudes. Does this mean that quartets should be preferable to triplets
in the phasing process? The answer is no; indeed, quartets are phase rela√
tionships of order N, while triplets are phase relationships of order N.
Thus, for medium-sized or large crystal structures it may be expected
that the number of reliable quartets may be a small percentage of the
large total number of estimated quartets. The number of reliable quartets
decreases with structural complexity much more rapidly than the number of reliable triplets. Thus, in spite of the large number of quartets, the
number of reliable ones is usually smaller than the corresponding number
of triplet invariants.
(iii) Can quartets and triplets be used together in phasing procedures? To
answer this question, some preliminary considerations should be made.
As soon as a phasing procedure progresses, the number of estimated
phases increases and at a certain step, the seven phases of the reflections
belonging to the first phasing shell of the quartet are estimated. The
following tripoles may then be established:
⎧
⎨ 1=
t1 = −φh1 − φh2 + φh1 +h2
⎩
t2 = −φh3 + φh1 +h2 +h3 − φh1 +h2
⎧
⎨ 1=
t3 = −φh1 − φh3 + φh1 +h3
⎩
t4 = −φh2 + φh1 +h2 +h3 − φh1 +h3
⎧
⎨ 1=
t5 = −φh2 − φh3 + φh2 +h3
⎩
t2 = −φh1 + φh1 +h2 +h3 − φh2 +h3 .