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APPLICATION OF DIRECT METHODS 133

most reliable when A HK is large, and this occurs 9.3.2 Phase refinement
when the associated normalized magnitudes (|E H |,
Once a set of initial phases has been chosen, it must
|E K |, and |E −H−K |) are large or the number of
be refined against the set of structure invariants
atoms in the unit cell is small. Thus, it is the
whose values are presumed to be known. So far,
largest |E | or |E A |, remaining after the applica-
two optimization methods (tangent refinement and
tion of all appropriate cut-offs, that need to be
parameter-shift reduction of the minimal function)
phased in direct-methods substructure determina-
have proven useful for extracting phase information
tions. The triplet invariants involving these reflec-
from the structure invariants. The tangent formula
tions are generated, and a sufficient number of
(Karle and Hauptman, 1956),
those invariants with the highest A HK values are

retained to achieve an adequate degree of overde- K |E K E H−K | sin(φ K + φ H−K ) (4)
termination of the phases by keeping the invariant- tan(φ H ) = K |E K E H−K | cos(φ K + φ H−K )
to-reflection ratio sufficiently large (e.g. 10:1). The
can be used to compute the value of a phase, φ H ,
inability to obtain a sufficient number of accurate
given a sufficient number of pairs (φ K , φ −H−K )
invariant estimates is the reason why full-structure
of known phases. By treating each reflection in
phasing by direct methods is possible only for
turn as φ H , an entire set of phases can be refined.
the smallest proteins.
Although the tangent formula is a powerful tool
that has been widely used in direct-methods pro-
9.3.1 Multiple trial procedures grams for 40 years, it suffers from the disadvan-
tage that, in space groups without translational
Ab initio phase determination by direct methods
symmetry, it is perfectly fulfilled by a false solu-
requires not only some information about the likely
tion with all phases equal to zero, thereby giving
values of a set of invariants, but also a set of start-
rise to the so-called ‘uranium-atom’ solution with
ing phases. Once the values for some pairs of phases
one dominant peak in the corresponding Fourier
(φ K and φ −H−K ) are available, the triplet structure
synthesis. This problem can be largely overcome
invariants can be used to generate further phases
for small-molecule structures by including carefully
(φ H which, in turn, can be used iteratively to evalu-
selected higher-order invariants (quartets) in a mod-
ate still more phases. The number of cycles of phase
ified tangent formula (Schenk, 1974; Hauptman,
expansion or refinement that must be performed
1974).
to get an adequate number of sufficiently accurate
Constrained minimization of an objective func-
phases depends on the size of the structure to be
tion like the minimal function (Debaerdemaeker
determined. To obtain starting phases, a so-called
and Woolfson, 1983; DeTitta et al., 1994)
multisolution or multiple trial approach is taken in

which the reflections are each assigned many dif- A HK cos HK − I 1 (A HK )
H,K I 0 (A HK )
ferent starting values in the hope that one or more R( ) = (5)
of the resultant phase combinations will lead to a H,K A HK
solution (Germain and Woolfson, 1968). Typically, a provides an alternative approach to phase refine-
random-number generator is used to assign initial ment or phase expansion. R( ) (also known as
values to all phases from the outset (Baggio et al., R MIN ) is a measure of the mean-square difference
1978). Avariant of this procedure employed in SnB is between the values of the cosines of the triplets
to use the random-number generator to assign initial ( HK ) calculated using the current set of phases
coordinates to the atoms in the trial structures and and the expected probabilistic values of the same
then to obtain initial phases from a structure-factor quantities as given by the ratio of modified Bessel
calculation. In SHELXD (Schneider and Sheldrick, functions, I 1 (A HK )/I 0 (A HK ). The minimal function
2002), the percentage of successful trial structures is is expected to have a constrained global minimum
increased by using better-than-random sets of start- when the phases are equal to their correct values
ing coordinates that are, in some way, consistent for some choice of origin and enantiomorph. An
with the Patterson function. algorithm known as parameter shift (Bhuiya and