The sum function

There exist different functions to measure the coincidence or the discrepancy between the observed and the calculated values of the intensitiy data. Two of the most important ones are the Residual value (R) and the Correlation coefficient (C). Their expressions in terms of a collectivity Φ of arbitrary variables are

R(ΦΦΦΦ)= ΣΣΣΣ^H Io(H)² + ΣΣΣΣ^H Ic(H)² –ΣΣΣΣ^H Io(H) ∙ Ic(H) and

C(ΦΦΦΦ) = ΣΣΣΣH Io(H) ∙ Ic(H) / (ΣΣΣΣH Io(H)²)^1/2. ∙ (ΣΣΣΣH Ic(H)²)^1/2

which must be ideally zero and the unity, respectively. Notice that both functions have the summation ΣHIo(H) ∙ Ic(H) in common which must be a large positive quantity for the correct calculated values. This summation is called sum function and since it has no squared terms, it is specially well suited for Fourier analysis. Effectively, since the intensities are the Fourier coefficients of the Patterson function, the physical meaning of the sum function can be easily

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-interpreted as the integral of the product of observed and calculated Patterson functions extended over the whole volume of the unit cell

( ) (

)

In contrast to the R function, the sum function is less sensitive to scaling errors because it only contains products. Strictly speaking, all structure solution methods based either on the R or on the sum function of Patterson type functions should be considered as Patterson search methods. In the practice, however, this name is reserved for those based on the sum function only. This includes not only the rotation and translation functions but also the 'direct methods' sum function which as will be seen later only differs in the selection of the variables.

The rotation function.

Since the internal geometry of the molecular model is known, the corresponding set of intramolecular interatomic vectors can be readily computed.

The calculated Patterson function is obtained by weighting each vector rjk with the product ZjZk (Zj= atomic number of atom j). The rotation ΩΩΩΩ of the vector set can be given either in Eulerian angles or in the axial rotation system. Although the former have considerable advantages in macromolecular crystallography because they allow the introduction of the Fast Fouier Transform algorithm in the computation of the rotation function, this advantage disappears in the case of powder diffraction because the models are rather small. Consequently, the simpler axial rotation system is normally used. It consists on two spherical polar angles (longitude φ and co-latitude ψ) which define the spin axis and a third angle χ specifying the rotation around this axis. To cover all possible model orientations, the following angular interval must be sampled:

0 ≤≤≤≤ φφφφ ≤≤≤≤ 360º 0 ≤≤≤≤ ψψψψ≤≤≤≤ 90º 0 ≤≤≤≤ χχχχ ≤≤≤≤ 360º

If the rotation function (Rossmann & Blow, 1962) is expressed in terms of the Fourier coefficients of the observed and calculated Patterson functions

R(Ω)=ΣΣΣΣ^H I(H) ∙ ΣΣΣΣ^j ΣΣΣΣ^k Zj Zk cos(2ππππHΩrjk)) then it reduces to

R(Ω)= ΣΣΣΣj ΣΣΣΣk Zj Zk P(Ωrjk)

if the H summation is replaced by the value of the Patterson function at point ΩΩΩΩrjk. Consequently, the values of the Patterson function must be stored using a small sampling grid. The expression of R(ΩΩΩΩ) in terms of the interatomic vectors also allows to select in a very comfortable way the region U to be explored around the origin.

In the case of anisometric vector sets, it may occur that the vector set happens to coincide, for an arbitrary orientation ΩΩΩΩ, with the origin of a neighbouring Patterson cell, thus given a wrong solution. To avoid this problem, the origin peak of the Patterson function is normally removed using the coefficients I(H)-<I(H)>.

Normally, the general strategy followed is to perform a coarse search followed by a finer one. The coarse search consists on a χ-scan at each node of a (φ, ψ) lattice (Beurskens et al., 1984). In ROTSEARCH (Rius & Miravitlles, 1987), 406 points with a maximum spherical angular error of 3.5º are employed.

The maximum vector length is 6 Å and ∆χ=5º. The values of R are ranked according to (R(ΩΩΩΩ)-<R>)/σ. After removal of the symmetry-equivalent top-ranked solutions, the surviving solutions are refined by recomputing the R-function using a finer grid (maximum vector length ≈ 9 Å).

Very often, besides the value of the rotation function, the value of the correlation function for that particular orientation is also calculated. The reason for this is its higher discriminating power for small models.

In most cases, the rotation function only explores the orientation of the model. However, it can also be expressed in terms of additional variables like the torsion angles. If the two parts of the model which are connected by the torsion angle are of similar weight, one can refine the torsion angle without positioning the molecule first. Notice that if the torsion angle is varied gradually and the value of the rotation function remains unchanged, this very probably indicates a wrong solution.

The translation function.

According to Crowther & Blow (1967) , Harada et al. (1981) and Rius et al. (1986) the translation function is defined by

( )

( )

( )

T _cross

V

cross ⋅ ⋅

=

∫

wherein r is the position vector of the model to be placed. Since T(r) is a sum function, it can be expressed as the sum of the product of the Fourier coefficients of the observed and calculated cross-Patterson functions. In view of the above definition of T(r) and of expressions (2) and (3) it follows,

T(r) = ΣΣΣΣH {I(H)-ΣΣΣΣj||||Sj(H)||||² } ∙ {ΣΣΣΣjΣΣΣΣk≠≠≠≠j Sj(H)* Sk(H) exp(i2ππππH(tk-tj))

x exp(-i2ππππ H(Rj-Rk) r)}

Since the coordinates of the vector H(Rj-Rk) are always integers, T(r) can be calculated using the Fast Fourier Transform algorithm.

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-The translation function can be generalized to molecular crystals with more than one symmetry-independent molecule in the unit cell in order to take advantage of the molecular orientations found in the rotation search. If the symmetry-independent molecules are similar, a single rotation search furnishes all correct orientations. Once the first molecular fragment is positioned, another symmetry-independent oriented molecular fragment can be placed with respect to it.

If Fp denotes the known part of the structure, the observed and calculated Fourier coefficients in the translation function are

I(H) - ||||Fp(H)||||² - ΣΣΣΣ^j||||Sj(H)||||² and

ΣΣΣΣ^j ΣΣΣΣ^k≠≠≠≠j Sj(H)* Sk(H) exp(i2ππππH(tk-tj)) exp(-i2ππππ H(Rj-Rk) r) + +ΣΣΣΣ^j Fp(H) Sj(H)* exp(-i2ππππHtj) exp(-i2ππππ HRj r)

Notice that the last summation causes the convolution of the inverted search fragment with the modified α function (Rius & Miravitlles, 1988).

If the search model has no internal symmetry higher than the identity, the asymmetrical unit of T coincides with the unit cell of the normalizer of the space group (International Tables of Crystallography, 1987). However, as soon as one fragment has been located, the asymmetrical unit of the T function becomes the whole unit cell (except for non-primitive lattices).

Unlike rotation function searches which are safer, translation function searches are less reliable. To be sure that the correct solution is not overlooked, a rather large number of maxima of T must be checked.

The direct methods sum function

The sum function can also be used to explore other Patterson-type synthesis. In the case of direct methods the explored Patterson-type function is the modulus synthesis which is calculated with the moduli of the structure factors as coefficients. In direct methods, the most important preliminary information about the compound which is exploited is the atomicity, i.e. the knowledge that the electron density is concentrated forming maxima. Consequently, the experimental resolution of the intensity data must be high enough to produce separate maxima in the corresponding electron density Fourier synthesis. The variables which are refined in direct methods, are the set Φ of phases associated with the large structure factors. The expression of the direct methods sum function is then

where Po' represents the observed modulus function with removed origin peak (Rius, 1993). In the practice, the coefficients used to compute Po' are corrected for the form factor decay as well as for the atomic thermal vibration. The substraction of the origin peak, i.e. the introduction of the coefficients |F(H)|

-<|F(H)|> causes the weak structure factors to play an active role in the maximization of S(Φ). The set of refined phases which maximizes the direct methods sum function is normally considered as the most probably correct solution and is used to compute the final electron density map.

Nowadays, active research fields in this area are the application of the direct methods sum function to data from patterns with systematic overlap (Rius et al., 1999), and also the simultaneous solution of the crystal structure and the accidental peak overlap. This is achieved by considering the partitioning coefficients as additional variables in the sum function.