Structure solution methods

From the knowledge of a complete set of X-ray reflections, a Fourier summation of the structure factors, Y hid, leads to the electron density p at any position % y z in the unit cell.

p { x , = exp[-/2;r(Ax + ^ + /z)] (3.10)

Yq h k !

To calculate we require values for Yhki, however as Equation 3.9 suggests we can only measure Y^hki and obtain | Y hid I • This is referred to as the phase problem, and invokes the necessity to make assumptions for structure solution methods.

3.1.4.1 Patterson synthesis

The Fourier transform of the squared amplitude, IF P, with all phases set equal to zero (all waves in phase) produces a Patterson map. This is a map of vectors representing distances between pairs of atoms in the structure. For pairs of atoms at position x\, y\,z\

and X2, yi,%2 there is a peak in the Patterson map at X2-x\, y2-yj, Z2-Zi. The Patterson peaks show the atom positions relative to each other, not where they lie relative to the cell origin. Patterson peaks are proportional in size to the product of the atomic numbers of the two atoms concerned, these methods are therefore most useful in solving structures which contain heavy atoms.

3.1.4.2 Direct methods

Direct methods is a statistical approach to structure solution. Such methods use the fact that the amplitude and phase of a structure factor are related to one another by electron density. Structures with centrosymmetric symmetry possess structure factors with phase angles of 0 ° or 180 therefore if N reflections are observed, 2^ electron density maps could be generated. By making several assumptions about the nature of the electron density it is possible to derive relations among the phases of different reflections and generate the best electron density maps and trial structures.

The most important assumptions in describing the electron density function are: • Electron densities cannot be negative {p{x) > 0)

• Electron density is near zero except for spherical peaks at atomic positions • Atoms are identical

The first assumption allows elimination of phases which lead to p{x) being negative for some value of x. The second assumption is used to give exact relationships between structure factors. The third assumption removes the effects of atomic shape and gives a predicted electron density map of the structure as point atoms. In practice direct methods phase determination is carried out on normalised structure factors, I £ I. I Ehki I is related to | Fhki I as shown in equation 3.11, 1^1 values are compensated for the fall-off in the individual temperature corrected atomic scattering factors, g, with increasing 20. The distribution of | ^ | values holds useful information about the space group of a crystal, particularly the presence of a centre of symmetry.

Chapter 3: Theory and Experimental Details

% L (3.11)

I y=i

• 8 takes account of the fact that some reflections in certain reciprocal lattice zones may have intensities different to those for general reflections

Reflections which posses large values of \e \ can be applied to the Sayre

probability relationship, shown in Equation 3.12.

S(hkl).S(h'k'n ~ S ( h - h \ k - k \ l - r ) (3.12)

• S represents, ‘sign o f

• ~ is used to show the relationship is statistically probable rather than exact

This relation is used to determine a series of single phase reflections. Equation 3.13 is then used to determine the probability of Ehki being positive.

P+i^hkl) — % 1/2 (3.13)

z Z y j y h'kr J

• AT is the number of atoms in the unit cell.

From the resultant list of positive Ehk values, electron density maps can be calculated and trial structures proposed.

3.1.4.3 Electron density mapping

The trial structure (obtained from direct or Patterson methods) is often a partial structure. A reverse Fourier transform can be carried out using |F^|or |F^|-|F^| to calculate a difference electron density map.

^Pxy. = -& Z I - |)exp('«. ) exp[- + ^ + /z)] (3.14)

^ hkl

3.1.5 Structure refinement

3.1.5.1 Least squares refinement

The trial structure can be refined to obtain enhanced atomic positions and other physical properties, such as atomic types, temperature factors, and positional occupancies. The least squares method is commonly used to find the best fit of a calculated model to experimentally observed diffraction data. The least squares method is a statistical method for solving simultaneous equations. In the linear case, the line of best fit through a number of points (%i, y{) has the equation:

yij = mx,. +c or f = y. - mx^ - c (3.15 a,b)

Chapter 3; Theory and Experimental Details

For diffraction data, the best fit is defined as the solution which minimises the function:

or (3.16 a,b)

• is the observed structure factor

• Fc is the calculated structure factor

• w is a weight given to each reflection (see below)

For a least squares solution to be possible it is necessary to have an excess of observed equations over unknowns; an overdetermined system. Consequently, certain statistical techniques are utilized to improve the least squares fit. Increased significance can be leant to more reliable observations through weighting. The number of observations can be increased by applying restraints (e.g. restricting a bond distance or angle) and constraints. The former uses a weight to define how severely the restraint is applied whereas the latter must be fulfilled exactly.

3.1.5.2 Reliability factors

There are several goodness-of-fit indices (or reliability or ‘R ’ factors) that are used to assess the quality of the model and the progress of the refinement. These will further be discussed in section 3.1.6.

hkl hid

wR takes account of the reflection weightings and is the reliability factor most closely related to the minimisation procedure