Direct methods of solving crystal structures
Fan Hai-fu
Institute of Physics, Chinese Academy of Sciences, Beijing 100080, P. R. China
Introduction
Direct methods solve crystal structures by finding phases of a set of structure factors
directly from the corresponding structure-factor magnitudes. The philosophy is that the
phase information, i.e. the value of initial phases of diffraction waves from the sample
crystal, has not lost in the conventional diffraction experiment, but just hidden among the
measured structure-factor magnitudes. Hence from which the phases can be retrieved.
The reason is as follows. Splitting the structure factor expression
F (h)   f j exp  i 2 h  r j 
N
(1)
j 1
into real and imaginary parts, we have
N
F (h) cos (h)   f j cos 2 (hx j  ky j  lz j )
j 1
N
F (h) sin  (h)   f j sin 2 (hx j  ky j  lz j )
.
j 1
This means that, given a set of n measured reflections, we have n structure-factor
magnitudes, |F(h)|. Then we can set up 2n simultaneous equations. The unknowns in
which are the n phases, (h), plus 3 times the number of independent atoms, the number
of positional parameters, (xj, yj, zj). Therefore, at least in theory, we can solve the
simultaneous equations and get the phases as well as the position of atoms, provided n is
equal to or greater than the number of positional atomic parameters. While in practice it is
impossible to solve the above simultaneous equations, we have some other ways to go
round.
Early attempts
Attempts of direct solution for the phase problem by algebraic methods can be
traced back to the late 1920’s (Ott, 1927; Banerjee, 1933; Avrami, 1938; Yu, 1942).
However the work on inequalities by Harker and Kasper (1948) is usually recognized as
the start of direct methods. An elegant application of inequalities to solve a complex
mineral structure was reported by Belov and Rumanowa (1954). For details of HarkerKasper inequalities the reader is referred to the books by Woolfson (1997) and by
Woolfson and Fan (1995).
Unitary and normalized structure factors
A set of structure factors will be easier to manipulate if it is independent of the unitcell contents. Besides we can make mathematical treatment simpler, if crystals are
composed of point atoms. The unitary structure factor and the normalized structure factor
have been thus introduced. The unitary structure factor is defined as
U (h)  F (h)
where
nj  f j
N
f
j 1
N
N
j 1
j 1
 f j   n j exp( 2 h  rj )
is the unitary scattering factor satisfying
j
,
(2)
N
n
j 1
j
 1 . Since
N
| F (h) |   f j it follows
j 1
| U ( h) |  1
.
Assuming all atoms in the unit cell have nearly the same shape, we can write
f j  fˆZ j
.
Here fˆ is the average unitary form factor, which is independent of atomic species. Zj is
the atomic number of the jth atom. Hence we have n  Z
j
j
U (h) 
1
 N

Z j 


 j 1 
N
Z
j 1
j
N
Z
j 1
j
exp( 2 h  r j )
leading to
.
(3)
The normalized structure factor is defined as
 N

E (h)  F (h)    f j2 
 j 1 
Where
N
f
j 1
2
j
12
.
(4)
is the average value of |F(h)|2 for the scattering angle corresponding to h.
The factor is a small positive integer, its value depends on the space group and the type
of reflection. The effect of  is to keep the average value of |E(h)|2 equal to unity for any
particular reflection type. For general-type reflections in all space group we have  = 1.
Assuming all atoms have nearly the same shape, we have
E (h) 
N
1
12
 N 2
  Z j 


 j 1 
Z
j 1
j
exp( 2 h  r j )
.
(5)
Structure invariants and seminvariants
Structure invariants are structural quantities that are independent of the choice of
unit-cell origin. Generally a structure factor is not a structure invariant, since it depends
on the choice of origin. The structure factor is expressed as
F (h)   f j exp  i 2 h  r j   F (h) exp i (h)
N
.
(6)
j 1
Shifting the origin by the vector R leads the structure factor to
F '(h)  F '(h) exp i '(h )
  f j exp i 2 h   r j  R  
j 1
N
.
(7)
 F (h) exp  i 2 h  R 
 F (h) exp i (h)  exp  i 2 h  R 
As is seen,
|F’(h)| = |F(h)|
and


(h) = (h)2 hR
.
This implies that magnitudes of structure factors are structure invariants, while phases are
generally not. The condition for the phase of a structure factor to be a structure invariant
is h = 0.
Now consider the structure-factor product
n
F (h1 )  F (h 2 )  F (h n )   F (h j )
j 1
An origin shift R will bring the product to
.
(8)

 n
  n
F '(h1 )  F '(h 2 )  F '(h n )  exp  i 2   h j   R   F (h j )
 j 1   j 1

. (9)
The condition for this product to be a structure invariant is
n
h
j
0
.
(10)
j1
Some structural quantities though not independent of arbitrary choice of origin, they do
independent of origin assignment to any permitted positions defined by the space group.
Such structural quantities are called structure seminvariant. Examples are structure
factors with indices h, k, and l all even, i.e.(h, k, l) mod(2, 2, 2) = (0, 0, 0) in the space
group P 1 , and structure factors satifying (h, 0, l) mod(2, 2, 2) = (0, 0, 0) in the space
group P21 with the 21 screw axis parallel to the b axis.
The problem of origin and enantiomorph fixing
As is seen from the previous section, not phases of all structure factors are structure
invariants. Hence before start deriving phases, we should fix the origin of the unit cell at
one of the possible positions defined by the space group. This is done by assigning
particular phases to up to three structure factors, which must not be structure
seminvariants. Take the space group P21 for example. Let the 21 screw axis be parallel to
the b axis. According to the space-group symmetry the origin of the unit cell can shift
arbitrarily along the b axis. Assigning an arbitrary phase to a reflection satisfying
(h, k, l)mod(2, 2, 2)  0 and k  0, the y coordinate of the origin will be fixed. Then, on
the projection down the b axis, there will be 4 possible (x, z) positions for the origin, i.e.
(0, 0), (½, 0), (0, ½) and (½, ½). Assign the phase of a reflection satisfying
(h, 0, l)mod(2, 2, 2)  0 and h = 2n + 1 (odd) to zero or . This will respectively set the
origin to x = 0 or ½. Similarly assigning the phase of a reflection satisfying
(h, 0, l)mod(2, 2, 2)  0 and l = 2n + 1 to zero or  will respectively bring the origin to
z = 0 or ½. Now the origin of the unit cell is fixed. However, for a noncentrosymmetric
structure, phases of structure factors could not uniquely defined without specifying the
enantiomorph. Because inversing all phases, i.e. inversing the sign of the imaginary
component of all structure factors, is equivalent of changing the enantiomorph from one
to the other. Hence to fix the enantiomorph, we need to specify the sign of the imaginary
component for one suitable reflection. In the present case, it should be a reflection with
k  0 and, the imaginary part of which should be relatively large. For further description
on fixing origin and enantiomorph the reader is referred to the books by Woolfson & Fan
(1995) and by Giacovazzo (1980). The theory of determining structure
invariant/seminvarant and of origin and enantiomorph fixing to cover all space groups has
been given by Hauptman & Karle (1953, 1956) with some corrections to their work by
Lessinger and Wondratschek (1975).
Sayre's equation
Sayre (1952) derived an exact equation linking structure factors. The equation is
based on the following conditions:
i) positivity;
ii) atomicity;
iii) equal-atom structure.
Given a crystal structure represented by r), we can construct a ‘squared structure’
expressed as
2r)
= r)r)
.
(11)
According to the convolution theorem, the Fourier transform of (11) yields
1
 Fh' Fh h'
V h'
Fhsq 
,
(12)
where
N
Fh   f j exp( i 2 h  r j )
.
(13)
j 1
Since Fsq h is the Fourier transform of 2r), according to (12) the ‘squared structure’ can
be determined through the convolution of structure factors Fh. Now if we can find the
relationship between 2r) and r), then the Fsq h in (12) can be converted to Fh leading
to an equation linking structure factors. If the first two conditions mentioned above are
satisfied, i.e. we have i) r)  0, ii) the electron densities of different atoms do not
overlap, then 2r) and r) will have the same number of maxima (atoms) situated at
exactly the same positions. We can write
N
sq
h
F
  f sq exp( i 2 h  r j )
j 1
(14)
j
with the parameter rj in equation (14) the same as that in (13). If the third condition is
also satisfied, then by dividing (13) with (14) we have F h /Fsq h = f / f sq. This is
substituted in (12) to obtain the Sayre equation
Fh 
f
sq
f V
F F
h' h h'
.
(15)
h'
An important outcome of Sayre’s paper, and two other papers published alongside
that of Sayre by Cochran (1952) and Zachariasen (1952), was the relationship between
the signs of structure factors in centrosymmetric case:
Sh  Sh’ Sh h’
,
(16)
where  means ‘probably equals’. This can be seen from (15); if Fh is large and a
particular product term Fh’Fh-h’ is also large, then it is more likely than not that Fh and
Fh’Fh-h’ will have the same sign (phase). The probability for (16) to be true was given by
Woolfson (1954) and more generally by Cochran and Woolfson (1955)
P( sh ,h ' ) 
 

1 1
 tanh 33/ 2 Eh Eh ' Ehh ' 
2 2
 2

.
(17)
During the early days of direct methods, the sign relationships were used in various ways
to solve centrosymmetric structures or centrosymmetric projections of noncentrosymmetric structures. For details of the practical procedure that can be manipulated
by hand calculations, the reader is referred to papers by Zachariasen (1952), Woolfson
(1957) and by Grant, Howelles & Rogers (1957). A very successful technique of applying
Sayre’s equation in ab initio phasing is the SAYTAN method, which will be described
later. On the other hand, Sayre’s equation and its variations have also been successfully
used in phase extension and refinement for a wide variety of structures from proteins to
aperiodic crystals (see Woolfson & Fan, 1995).
It should be noticed that, while the three conditions mentioned above are necessary
for deriving the Sayre equation, they are not satisfied exactly in practice. It is useful to
know what would happen when one or more conditions do not hold. In theory, any
violation of the three conditions would lead to the collapse of equation (15). However
even in this case equation (12) is still valid. Consequently results of applying Sayre’s
equation would tend to2r) rather than r). The problem is that, to what extent will
2r) and r) resemble each other? For example, in neutron diffraction the scattering
factor of some elements is negative resulting in ‘negative atoms’ on the density map r).
When Sayre’s equation is used with neutron diffraction data, the result, approximately
2r), will differ from r) mainly in that the negative atoms are changed to positive ones.
In another case, if the crystal contains heavy atoms together with light atoms, the
resultant map from Sayre’s equation will have heavy atoms heavier and light atoms
lighter than that in r). Finally when dealing with low-resolution data, the condition of
atomicity may be broken, i.e. the electron density of adjacent atoms may overlap. This
causes rj in equation (14) to be different to that in equation (13) but not inevitably lead to
the failure of structure solution. It is concluded that, in many cases Sayre’s equation may
still be applicable even when the three conditions are not completely fulfilled.
Cochran's distribution
The three-phase relationship
 h + h’ + h h’  0
and the probability distribution of
(modulo 2)
(18)
 3  h  h'  hh'
,
(19)
were given by Cochran (1955)
P(3 )   2 I 0 ( h ,h ' )  exp h ,h ' cos  3  ,
1
(20)
where
 h ,h'  2323/2 Eh Eh' Ehh'
(21)
for non-equal-atom structures and
 h,h'  2 N 1/2 Eh Eh' Ehh'
(22)
for equal-atom structures.
The derivation of (20) is based on the central limit theorem: Given a set of n
independent random variables, xi , with means < xi > and variances ithe function
n
y   ai xi
(23)
i 1
has a probability distribution that tends, as n becomes large, to a normal (Gaussian)
distribution
P( y ) 
with mean
1
 y 2 1/2
  y   y  2 

exp
2 y2


n
n
i 1
i 1
(24)
 y    ai  xi  and variance  2y   ai i2 .
It has been shown by Kitaigorodskii (1957) that, taking cos(2 h.rj) as xi and taking Fh as
y , for a crystal in space group P1 containing more than 10 atoms in the unit cell, the
probability distribution of Fh tends, to a good approximation, to Gaussian distribution.
Now let the trigonometric factors
sin
sin
sin
cos[2 ( h)  r j ]cos (2 h'  r j ) cos[2 (h  h' )  r j ]
,
s in means ‘sin’ or ‘cos’, be the independent random variables xi and the product
where cos
E-h Eh’ Eh-h’ be the function y. Assuming the probability distribution of both the real and
the imaginary component of y tend to a Gaussian distribution, and assuming the
amplitudes E-h, Eh’ and Eh-h’ are known, we will finally obtain the Cochran distribution
(20).
The tangent formula
From equations (19) and (20), if there are more than one pair of known phases,  h’
andh h’ , associated with the same  h , then the total probability distribution for  h
will be


P( h )   Ph' ( 3 )  N exp   h,h' cos  3 
 h'

h'
,
(25)
where N is a normalising factor.
Let
sin =
and
h’  h, h’ sin (h’ +hh’)
(26)
cos =
h’  h, h’ cos (h’ +hh’)
(27)
(25) becomes
P(h )  2 I 0 ( ) exp[ cos(h   )] .
1
(28)
By maximising P(h) in(28) we have h =  Then from (26) and (27) we obtain
tan h 
  h,h' sin(h'  hh' )
h'
  h,h' cos(h'  hh' )
.
(29)
h'
with
2
2



 
     h ,h ' sin(h '  hh ' )     h ,h ' cos(h '  hh ' ) 

 h'
 
 h '
1/2
(30)
indicating the reliability of the estimation of h. This is the tangent formula introduced by
Karle and Hauptman (1956), which is the most widely used formula in direct methods.
The form of the tangent formula given here differs a little from, but is equivalent to, that
of the original one.
It is easily obtained from Sayre’s equation a formula similar to (29) but with quite
different meaning. By splitting equation (15) into the real and the imaginary parts and by
dividing the imaginary part with the real part, it follows
tan h 
 Fh' Fhh' sin(h'  hh' )
h'
 Fh' Fhh' cos(h'  hh' )
.
(31)
h'
Equation (31) may be regarded as the angular portion of Sayre’s equation. It differs from
(29) in that the summation in (31) should include all available h’ terms, while that in(29)
may just include a few or even only one term of h’. Besides, (31) is an exact equation,
while (29) gives the most probable value of h. The tangent formula is much easier to use
for ab initio phasing. A historical breakthrough on the application of direct methods was
made by Karle and Karle (1964) when they solved the non-centrosymmetric crystal
structure of L-arginine dihydrate by the symbolic-addition procedure using the tangent
formula. A few years later a systematic procedure to use the tangent formula and a
computer program MULTAN (MULtisolution TANgent-formula method) (Germain &
Woolfson, 1968). were introduced by Woolfson and his colleagues. The development and
application of the MULTAN and related procedures led to the domination of direct
methods in solving small molecular structures.
SAYTAN
The basic idea behind SAYTAN is to use not only the relationship among phases but also
that among amplitudes implied in Sayre’s equation. The philosophy is that a good set of
phases should satisfy a system of Sayre equations.
Eh 
K
gh
 E h 'E h h '
,
(32)
h'
where gh is the scattering factor for ‘squared’ atoms and K is an overall scaling constant,
which allows for the fact that only structure factors with large magnitude are included on
the right-hand side. The derivation started with the following residual for a system of
Sayre equations:
R   g h E h  K  E h ' E hh '
h
2
.
(33)
h'
As a condition that R should be a minimum it is necessary that
R
0
h
for all h
and this leads to the Sayre-equation tangent formula (Debaerdemaeker, Tate & Woolfson,
1988).


 h  phase of   gh  gh"  ghh" Eh"Ehh"  2 K  Ehh"  Eh 'Eh" h '  .
 h"
h"
h'

(34)
A distinctive feature of the Sayre-equation tangent formula is that it can use the
information from Sayre equations, for which the values of Eh are small, ideally zero. As
is well known in powder-method crystal-structure analysis, a good structure model should
satisfy the weakest reflections as well as the strongest. The Sayre-equation tangent
formula tends to develop phase sets, which satisfy the smallest magnitudes as well as the
largest. Since it uses extra information SAYTAN is more effective than MULTAN, either
giving a solution in fewer trials or giving a solution where MULTAN would not.
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