Chapter 2.2. Direct methods

The normalized structure factors E (see also Chapter 2.1 ) are calculated according to (Hauptman & Karle, 1953) where is the squared observed structure-factor magnitude on the absolute scale and is the expected value of .

depends on the available a priori information. Often, but not always, this may be considered as a combination of several typical situations. We mention:

When probability theory is not used, the quasi-normalized structure factors and the unitary structure factors are often used. and are defined according to Since is the largest possible value for represents the fraction of with respect to its largest possible value. Therefore If atoms are equal, then .

N.s.f.'s cannot be calculated by applying (2.2.4.1) to observed s.f.'s because: (a) the observed magnitudes (already corrected for Lp factor, absorption, …) are on a relative scale; (b) cannot be calculated without having estimated the vibrational motion of the atoms.

This is usually obtained by the well known Wilson plot (Wilson, 1942), according to which observed data are divided into ranges of and averages of intensity are taken in each shell. Reflection multiplicities and other effects of space-group symmetry on intensities must be taken into account when such averages are calculated. The shells are symmetrically overlapped in order to reduce statistical fluctuations and are restricted so that the number of reflections in each shell is reasonably large. For each shell should be obtained, where K is the scale factor needed to place X-ray intensities on the absolute scale, B is the overall thermal parameter and is the expected value of in which it is assumed that all the atoms are at rest. depends upon the structural information that is available (see Section 2.2.4.1 for some examples).

Equation (2.2.4.3) may be rewritten as which plotted at various should be a straight line of which the slope (2B) and intercept (ln K) on the logarithmic axis can be obtained by applying a linear least-squares procedure.

Very often molecular geometries produce perceptible departures from linearity in the logarithmic Wilson plot. However, the more extensive the available a priori information on the structure is, the closer, on the average, are the Wilson-plot curves to their least-squares straight lines.

Accurate estimates of B and K require good strategies (Rogers & Wilson, 1953) for:

Once K and B have been estimated, values can be obtained from experimental data by where is the expected value of for the reflection h on the basis of the available a priori information.

Under some fairly general assumptions (see Chapter 2.1 ) probability distribution functions for the variable for cs. and ncs. structures are (see Fig. 2.2.4.1) and respectively. Corresponding cumulative functions are (see Fig. 2.2.4.2)

Figure 2.2.4.1 | top | pdf | Probability density functions for cs. and ncs. crystals.

Figure 2.2.4.2 | top | pdf | Cumulative distribution functions for cs. and ncs. crystals.

Some moments of the distributions (2.2.4.4) and (2.2.4.5) are listed in Table 2.2.4.1. In the absence of other indications for a given crystal structure, a cs. or an ncs. space group will be preferred according to whether the statistical tests yield values closer to column 2 or to column 3 of Table 2.2.4.1.

Table 2.2.4.1 | top | pdf | Moments of the distributions (2.2.4.4) and (2.2.4.5) is the percentage of n.s.f.'s with amplitude greater than the threshold . Criterion Centrosymmetric distribution Noncentrosymmetric distribution 0.798 0.886 1.000 1.000 1.596 1.329 3.000 2.000 6.383 3.323 15.000 6.000 0.968 0.736 2.000 1.000 8.000 2.000 8.691 2.415 R(1) 0.320 0.368 R(2) 0.050 0.018 R(3) 0.003 0.0001

For further details about the distribution of intensities see Chapter 2.1 .

An extensive system of inequalities exists for the coefficients of a Fourier series which represents a positive function. This can restrict the allowed values for the phases of the s.f.'s in terms of measured structure-factor magnitudes. Harker & Kasper (1948) derived two types of inequalities:

Type 1. A modulus is bound by a combination of structure factors: where m is the order of the point group and .

Applied to low-order space groups, (2.2.5.1) gives The meaning of each inequality is easily understandable: in , for example, must be positive if is large enough.

Type 2. The modulus of the sum or of the difference of two structure factors is bound by a combination of structure factors: where stands for `real part of'. Equation (2.2.5.2) applied to P1 gives

A variant of (2.2.5.2) valid for cs. space groups is After Harker & Kasper's contributions, several other inequalities were discovered (Gillis, 1948; Goedkoop, 1950; Okaya & Nitta, 1952; de Wolff & Bouman, 1954; Bouman, 1956; Oda et al., 1961). The most general are the Karle–Hauptman inequalities (Karle & Hauptman, 1950): The determinant can be of any order but the leading column (or row) must consist of U's with different indices, although, within the column, symmetry-related U's may occur. For and , equation (2.2.5.3) reduces to which, for cs. structures, gives the Harker & Kasper inequality For , equation (2.2.5.3) becomes from which where If the moduli , , are large enough, (2.2.5.4) is not satisfied for all values of . In cs. structures the eventual check that one of the two values of does not satisfy (2.2.5.4) brings about the unambiguous identification of the sign of the product .

It was observed (Gillis, 1948) that `there was a number of cases in which both signs satisfied the inequality, one of them by a comfortable margin and the other by only a relatively small margin. In almost all such cases it was the former sign which was the correct one. That suggests that the method may have some power in reserve in the sense that there are still fundamentally stronger inequalities to be discovered'. Today we identify this power in reserve in the use of probability theory.

For any space group (see Section 2.2.3) there are linear combinations of phases with cosines that are, in principle, fixed by the magnitudes alone (s.i.'s) or by the values and the trigonometric form of the structure factor (s.s.'s). This result greatly stimulated the calculation of conditional distribution functions where , is an s.i. or an s.s. and is a suitable set of diffraction magnitudes. The method was first proposed by Hauptman & Karle (1953) and was developed further by several authors (Bertaut, 1955a,b, 1960; Klug, 1958; Naya et al., 1964, 1965; Giacovazzo, 1980a). From a probabilistic point of­view the crystallographic problem is clear: the joint distribution , from which the conditional distributions (2.2.5.5) can be derived, involves a number of normalized structure factors each of which is a linear sum of random variables (the atomic contributions to the structure factors). So, for the probabilistic interpretation of the phase problem, the atomic positions and the reciprocal vectors may be considered as random variables. A further problem is that of identifying, for a given Φ, a suitable set of magnitudes on which Φ primarily depends. The formulation of the nested neighbourhood principle first (Hauptman, 1975) fixed the idea of defining a sequence of sets of reflections each contained in the succeeding one and having the property that any s.i. or s.s. may be estimated via the magnitudes constituting the various neighbourhoods. A subsequent more general theory, the representation method (Giacovazzo, 1977a, 1980b), arranges for any Φ the set of intensities in a sequence of subsets in order of their expected effectiveness (in the statistical sense) for the estimation of Φ.

In the following sections the main formulae estimating low-order invariants and seminvariants or relating phases to other phases and diffraction magnitudes are given.

The basic formula for the estimation of the triplet phase given the parameter is Cochran's (1955) formula where , is the atomic number of the jth atom and is the modified Bessel function of order n. In Fig. 2.2.5.1 the distribution is shown for different values of G.

Figure 2.2.5.1 | top | pdf | Curves of (2.2.5.6) for some values of .

The conditional probability distribution for , given a set of and , is given (Karle & Hauptman, 1956; Karle & Karle, 1966) by where is the most probable value for . The variance of may be obtained from (2.2.5.7) and is given by which is plotted in Fig. 2.2.5.2.

Figure 2.2.5.2 | top | pdf | Variance (in square radians) as a function of α.

Equation (2.2.5.9) is the so-called tangent formula. According to (2.2.5.10), the larger is α the more reliable is the relation .

For an equal-atom structure .

The basic conditional formula for sign determination of in cs. crystals is Cochran & Woolfson's (1955) formula where is the probability that is positive and k ranges over the set of known values . The larger the absolute value of the argument of tanh, the more reliable is the phase indication.

An auxiliary formula exploiting all the 's in reciprocal space in order to estimate a single Φ is the formula (Hauptman & Karle, 1958; Karle & Hauptman, 1958) given by where C is a constant which differs for cs. and ncs. crystals, is the average value of and p is normally chosen to be some small number. Several modifications of (2.2.5.12) have been proposed (Hauptman, 1964, 1970; Karle, 1970a; Giacovazzo, 1977b).

A recent formula (Cascarano, Giacovazzo, Camalli et al., 1984) exploits information contained within the second representation of Φ, that is to say, within the collection of special quintets (see Section 2.2.5.6): where k is a free vector. The formula retains the same algebraic form as (2.2.5.6), but where , is assumed to be zero if it is experimentally negative. The prime to the summation warns the reader that precautions have to be taken in order to avoid duplications in the contributions.

G may be positive or negative. In particular, if the triplet is estimated negative.

The accuracy with which the value of Φ is estimated strongly depends on . Thus, in practice, only a subset of reciprocal space (the reflections k with large values of ɛ) may be used for estimating Φ.

(2.2.5.13) proved to be quite useful in practice. Positive triplet cosines are ranked in order of reliability by (2.2.5.13) markedly better than by Cochran's parameters. Negative estimated triplet cosines may be excluded from the phasing process and may be used as a figure of merit for finding the correct solution in a multisolution procedure.

A strength of direct methods is that no knowledge of structure is required for their application. However, when some a priori information is available, it should certainly be a weakness of the methods not to make use of this knowledge. The conditional distribution of Φ given and the first three of the five kinds of a priori information described in Section 2.2.4.1 is (Main, 1976; Heinermann, 1977a) where stand for h, , , and for . The quantities have been calculated in Section 2.2.4.1 according to different categories: is a suitable average of the product of three scattering factors for the ith atomic group, p is the number of atomic groups in the cell including those related by symmetry elements. We have the following categories.

In early papers (Hauptman & Karle, 1953; Simerska, 1956) the phase was always expected to be zero. Schenk (1973a,b) [see also Hauptman (1974)] suggested that Φ primarily depends on the seven magnitudes: , called basis magnitudes, and , called cross magnitudes.

The conditional probability of Φ in P1 given seven magnitudes according to Hauptman (1975) is where L is a suitable normalizing constant which can be derived numerically, For equal atoms . Denoting gives 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 values in three typical cases.

In the cs. case (2.2.5.18) is replaced (Hauptman & Green, 1976) by where is the probability that the sign of is positive or negative, and The normalized probability may be derived by . More simple probabilistic formulae were derived independently by Giacovazzo (1975, 1976): where and . Q is never allowed to be negative.

According to (2.2.5.20) is expected to be positive or negative according to whether is positive or negative: the larger is C, the more reliable is the phase indication. For , (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 is 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):

Equations (2.2.5.20) and (2.2.5.23) are easily modifiable when some cross magnitudes are not in the measurements. If is not measured then (2.2.5.20) or (2.2.5.23) are still valid provided that in G it is assumed that . For example, if and are not in the data then (2.2.5.21) and (2.2.5.22) become In space groups with symmetry higher than more symmetry-equivalent quartets can exist of the type where are rotation matrices of the space group. The set 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 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 in the class mmm Quartets Basis vectors Cross vectors (1, 2, 3) (0, 7, 0) (0, 7, 0) (0, 0, 5) (0, 7, 0) ) (1, 5, 3) (0, 7, 0) (0, 0, 11) (0, 0, 5) (0, 7, 11) (0, 0, 5) (0, 0, 11) (0, 7, 5) (0, 0, 11) (1, 5, 8) (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.

A quintet phase may be considered as the sum of three suitable triplets or the sum of a triplet and a quartet, i.e. or It depends primarily on 15 magnitudes: the five basis magnitudes and the ten cross magnitudes In the following we will denote Conditional distributions of Φ in P1 and given the 15 magnitudes have been derived by several authors and allow in favourable circumstances in ncs. space groups the quintets having Φ near 0 or near π or near to be identified. Among others, we remember:

For cs. cases (2.2.5.24) reduces to Positive or negative quintets may be identified according to whether G is larger or smaller than zero.

If is not measured then (2.2.5.24) and (2.2.5.25) are still valid provided that in (2.2.5.25) .

If the symmetry is higher than in then more symmetry-equivalent quintets can exist of the type where are rotation matrices of the space groups. The set is called the first representation of Φ. In this case Φ primarily depends on more than 15 magnitudes which all have to be taken into account for a careful estimation of Φ (Giacovazzo, 1980a).

A wide use of quintet invariants in direct methods procedures is prevented for two reasons: (a) the large correlation of positive quintet cosines with positive triplets; (b) the large computing time necessary for their estimation [quintets are phase relationships of order , so a large number of quintets have to be estimated in order to pick up a sufficient percentage of reliable ones].

In a crystal structure with N identical atoms the joint probability distribution of n normalized s.f.'s under the following conditions:

Advantages, limitations and applications of determinantal formulae can be found in the literature (Heinermann et al., 1979; de Rango et al., 1975, 1985). Taylor et al. (1978) combined K–H determinants with a magic-integer approach. The computing time, however, was larger than that required by standard computing techniques. The use of K–H matrices has been made faster and more effective by de Gelder et al. (1990) (see also de Gelder, 1992). They developed a phasing procedure (CRUNCH) which uses random phases as starting points for the maximization of the K–H determinants.

According to the representations method (Giacovazzo, 1977a, 1980a,b):

The more general expressions for the s.s.'s of first rank are

In other words:

The set of special quartets (2.2.5.35a) and (2.2.5.35b) constitutes the first representations of Φ.

Structure seminvariants of the second rank can be characterized as follows: suppose that, for a given seminvariant Φ, it is not possible to find a vectorial index h and a rotation matrix such that is a structure invariant. Then Φ is a structure seminvariant of the second rank and a set of structure invariants ψ can certainly be formed, of type by means of suitable indices h and l and rotation matrices and . As an example, for symmetry class 222, or or are s.s.'s of the first rank while is an s.s. of the second rank.

The procedure may easily be generalized to s.s.'s of any order of the first and of the second rank. So far only the role of one-phase and two-phase s.s.'s of the first rank in direct procedures is well documented (see references quoted in Sections 2.2.5.9 and 2.2.5.10).

Let be our one-phase s.s. of the first rank, where In general, more than one rotation matrix and more than one vector h are compatible with (2.2.5.36). The set of special triplets is the first representation of . In cs. space groups the probability that , given and the set , may be estimated (Hauptman & Karle, 1953; Naya et al., 1964; Cochran & Woolfson, 1955) by where In (2.2.5.37), the summation over n goes within the set of matrices for which (2.2.5.35a,b) is compatible, and h varies within the set of vectors which satisfy (2.2.5.36) for each . Equation (2.2.5.36) is actually a generalized way of writing the so-called relationships (Hauptman & Karle, 1953).

If is a phase restricted by symmetry to and in an ncs. space group then (Giacovazzo, 1978) If is a general phase then is distributed according to where with a reliability measured by The second representation of is the set of special quintets provided that h and vary over the vectors and matrices for which (2.2.5.36) is compatible, k over the asymmetric region of the reciprocal space, and over the rotation matrices in the space group. Formulae estimating via the second representation in all the space groups [all the base and cross magnitudes of the quintets (2.2.5.40) now constitute the a priori information] have recently been secured (Giacovazzo, 1978; Cascarano & Giacovazzo, 1983; Cascarano, Giacovazzo, Calabrese et al., 1984). Such formulae contain, besides the contribution of order provided by the first representation, a supplementary (not negligible) contribution of order arising from quintets.

Denoting formulae (2.2.5.37), (2.2.5.38), (2.2.5.39) still hold provided that is replaced by where m is the number of symmetry operators and is the Hermite polynomial of order four.

is assumed to be zero if it is computed negative. The prime to the summation warns the reader that precautions have to be taken in order to avoid duplication in the contributions.

Two-phase s.s.'s of the first rank were first evaluated in some cs. space groups by the method of coincidence by Grant et al. (1957); the idea was extended to ncs. space groups by Debaerdemaeker & Woolfson (1972), and in a more general way by Giacovazzo (1977e,f).

The technique was based on the combination of the two triplets which, subtracted from one another, give If all four 's are sufficiently large, an estimate of the two-phase seminvariant is available.

Probability distributions valid in according to the neighbourhood principle have been given by Hauptman & Green (1978). Finally, the theory of representations was combined by Giacovazzo (1979a) with the joint probability distribution method in order to estimate two-phase s.s.'s in all the space groups.

According to representation theory, the problem is that of evaluating via the special quartets (2.2.5.35a) and (2.2.5.35b). Thus, contributions of order will appear in the probabilistic formulae, which will be functions of the basis and of the cross magnitudes of the quartets (2.2.5.35) . Since more pairs of matrices and can be compatible with (2.2.5.34), and for each pair more pairs of vectors and may satisfy (2.2.5.34), several quartets can in general be exploited for estimating Φ. The simplest case occurs in where the two quartets (2.2.5.35) suggest the calculation of the six-variate distribution function which leads to the probability formula where is the probability that the product is positive, and It may be seen that in favourable cases .

For the sake of brevity, the probabilistic formulae for the general case are not given and the reader is referred to the original papers.

Protein structures cannot be solved ab initio by traditional direct methods (i.e., by application of the tangent formula alone). Accordingly, the first applications were focused on two tasks:

The application of standard tangent techniques to (a) and (b) has not been found to be very satisfactory (Coulter & Dewar, 1971; Hendrickson et al., 1973; Weinzierl et al., 1969). Tangent methods, in fact, require atomicity and non-negativity of the electron density. Both these properties are not satisfied if data do not extend to atomic resolution . Because of series termination and other errors the electron-density map at presents large negative regions which will appear as false peaks in the squared structure. However, tangent methods use only a part of the information given by the Sayre equation (2.2.6.5). In fact, (2.2.6.5) express two equations relating the radial and angular parts of the two sides, so obtaining a large degree of overdetermination of the phases. To achieve this Sayre (1972) [see also Sayre & Toupin (1975)] suggested minimizing (2.2.10.1) by least squares as a function of the phases: Even if tests on rubredoxin (extensions of phases from 2.5 to 1.5 Å resolution) and insulin (Cutfield et al., 1975) (from 1.9 to 1.5 Å resolution) were successful, the limitations of the method are its high cost and, especially, the higher efficiency of the least-squares method. Equivalent considerations hold for the application of determinantal methods to proteins [see Podjarny et al. (1981); de Rango et al. (1985) and literature cited therein].

A question now arises: why is the tangent formula unable to solve protein structures? Fan et al. (1991) considered the question from a first-principle approach and concluded that:

Sheldrick (1990) suggested that direct methods are not expected to succeed if fewer than half of the reflections in the range 1.1–1.2 Å are observed with (a condition seldom satisfied by protein data).

The most complete analysis of the problem has been made by Giacovazzo, Guagliardi et al. (1994). They observed that the expected value of α (see Section 2.2.7) suggested by the tangent formula for proteins is comparable with the variance of the α parameter. In other words, for proteins the signal determining the phase is comparable with the noise, and therefore the phase indication is expected to be unreliable.

Section 2.2.10.1 suggests that the mere use of the tangent formula or the Sayre equation cannot solve ab initio protein structures of usual size. However, even in an ab initio situation, there is a source of supplementary information which may be used. Good examples are the `peaklist optimization' procedure (Sheldrick & Gould, 1995) and the SIR97 procedure (Altomare et al., 1999) for refining and completing the trial structure offered by the first E map.

In both cases there are reasons to suspect that the correct structure is sometimes extracted from a totally incorrect direct-methods solution. These results suggest that a direct-space procedure can provide some form of structural information complementary to that used in reciprocal space by the tangent or similar formulae. The combination of real- and reciprocal-space techniques could therefore enlarge the size of crystal structures solvable by direct methods. The first program to explicitly propose the combined use of direct and reciprocal space was Shake and Bake (SnB), which inspired a second package, half-bake (HB). A third program, SIR99, uses a different algorithm.

The SnB method (DeTitta et al., 1994; Weeks et al., 1994; Hauptman, 1995) is the heir of the cosine least-squares method described in Section 2.2.8, point (4). The function where is the triplet phase, and .

is expected to have a global minimum, provided the number of phases involved is sufficiently large, when all the phases are equal to their true values for some choice of origin and enantiomorph. Thus the phasing problem reduces to that of finding the global minimum of (the minimum principle).

SnB comprises a shake step (phase refinement) and a bake step (electron-density modification), the second step aiming to impose phase constraints implicit in real space. Accordingly, the program requires two Fourier transforms per cycle, and numerous cycles. Thus it may be very time consuming and it is not competitive with other direct methods for the solution of the crystal structures of small molecules. However, it introduced into the field the tremendous usefulness of intensive computations for the direct solution of complex crystal structures.

Owing to Sheldrick (1997), HB does most of its work in direct space. Random atomic positions are generated, to which a modified peaklist optimization process is applied. A number of peaks are eliminated subject to the condition that remains as large as possible (only reflections with are involved, where ). The phases of a suitable subset of reflections are then used as input for a tangent expansion. Then an E map is calculated from which peaks are selected: these are submitted to the elimination procedure.

Typically 5–20 cycles of this internal loop are performed. Then a correlation coefficient (CC) between and is calculated for all the data. If the CC is good (i.e. larger than a given threshold), then a new loop is performed: a new E map is obtained, from which a list of peaks is selected for submission to the elimination procedure. The criterion now is the value of the CC, which is calculated for all the reflections. Typically two to five cycles of this external loop are performed.

The program works indefinitely, restarting from random atoms until interrupted. It may work either by applying the true space-group symmetry or after having expanded the data to P1.

The SIR99 procedure (Burla et al., 1999) may be divided into two distinct parts: the tangent section (i.e., a double tangent process using triplet and quartet invariants) is followed by a real-space refinement procedure. As in SIR97, the reciprocal-space part is followed by the real-space refinement, but this time this last part is much more complex. It involves three different techniques: EDM (an electron-density modification process), the HAFR part (in which all the peaks are associated with the heaviest atomic species) and the DLSQ procedure (a least-squares Fourier refinement process). The atomicity is gradually introduced into the procedure. The entire process requires, for each trial, several cycles of EDM and HAFR: the real-space part is able to lead to the correct solution even when the tangent formula does not provide favourable phase values.

The modulus of the isomorphous difference may be assumed at a first approximation as an estimate of the heavy-atom s.f. . Normalization of 's and application of the tangent formula may reveal the heavy-atom structure (Wilson, 1978).

The theoretical basis for integrating the techniques of direct methods and isomorphous replacement was introduced by Hauptman (1982a). According to his notation let us denote by and atomic scattering factors for the atom labelled j in a pair of isomorphous structures, and let and denote corresponding normalized structure factors. Then where The conditional probability of the two-phase structure invariant given and is (Hauptman, 1982a) where Three-phase structure invariants were evaluated by considering that eight invariants exist for a given triple of indices h, k, l : So, for the estimation of any , the joint probability distribution has to be studied, from which eight conditional probability densities can be obtained: for .

The analytical expressions of are too intricate and are not given here (the reader is referred to the original paper). We only say that may be positive or negative, so that reliable triplet phase estimates near 0 or near π are possible: the larger , the more reliable the phase estimate.

A useful interpretation of the formulae in terms of experimental parameters was suggested by Fortier et al. (1984): according to them, distributions do not depend, as in the case of the traditional three-phase invariants, on the total number of atoms per unit cell but rather on the scattering difference between the native protein and the derivative (that is, on the scattering of the heavy atoms in the derivative).

Hauptman's formulae were generalized by Giacovazzo et al. (1988): the new expressions were able to take into account the resolution effects on distribution parameters. The formulae are completely general and include as special cases native protein and heavy-atom isomorphous derivatives as well as X-ray and neutron diffraction data. Their complicated algebraic forms are easily reduced to a simple expression in the case of a native protein heavy-atom derivative: in particular, the reliability parameter for is where indices P and H warn that parameters have to be calculated over protein atoms and over heavy atoms, respectively, and Δ is a pseudo-normalized difference (with respect to the heavy-atom structure) between moduli of structure factors.

Equation (2.2.10.2) may be compared with Karle's (1983) qualitative rule: if the sign of is plus then the value of is estimated to be zero; if its sign is minus then the expected value of is close to π. In practice Karle's rule agrees with (2.2.10.2) only if the Cochran-type term in (2.2.10.2) may be neglected. Furthermore, (2.2.10.2) shows that large reliability values do not depend on the triple product of structure-factor differences, but on the triple product of pseudo-normalized differences. A series of papers (Giacovazzo, Siliqi & Ralph, 1994; Giacovazzo, Siliqi & Spagna, 1994; Giacovazzo, Siliqi & Platas, 1995; Giacovazzo, Siliqi & Zanotti, 1995; Giacovazzo et al., 1996) shows how equation (2.2.10.2) may be implemented in a direct procedure which proved to be able to estimate the protein phases correctly without any preliminary information on the heavy-atom substructure.

Combination of direct methods with the two-derivative case is also possible (Fortier et al., 1984) and leads to more accurate estimates of triplet invariants provided experimental data are of sufficient accuracy.

If the frequency of the radiation is close to an absorption edge of an atom, then that atom will scatter the X-rays anomalously (see Chapter 2.4 ) according to . This results in the breakdown of Friedel's law. It was soon realized that the Bijvoet difference could also be used in the determination of phases (Peerdeman & Bijvoet, 1956; Ramachandran & Raman, 1956; Okaya & Pepinsky, 1956). Since then, a great deal of work has been done both from algebraic (see Chapter 2.4 ) and from probabilistic points of view. In this section we are only interested in the second.

We will mention the following different cases: