| Inorganic Structure Prediction |
Structure solution
The first involves solution or circumvention of the phase problem so as to derive an initial and approximately correct structural model.
The second stage optimizes and completes the structure so as to minimize the discrepancy between simulated and observed diffraction data.
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Select STRUCTURE SOLVE from the list of modules and click FRAMEWORK ANNEAL to bring the Framework Anneal menu card to the fore. Select STRUCTURE SOLVE from the list of modules and click METAL-OXIDE ANNEAL to bring the Metal-Oxide Anneal menu card to the fore.
Structure solution continues to be a taxing aspect of the characterization of crystalline materials that occur only in polycrystalline form (i.e., in which individual crystallite sizes are smaller than 10µ). The past decade has seen substantial improvements in both neutron and X-ray diffraction methods, notably in the use of synchrotron X-radiation, but initial solutions of the structures of polycrystalline materials remain troublesome.
The Structure Solve method
The methods implemented in Structure Solve are direct space approaches to the structure solution problem (Deem and Newsam 1992; Freeman and Catlow 1992; Newsam et al. 1992; Freeman et al. 1993). The chemical composition, and the dimensions and symmetry of the crystallographic unit cell are taken as input. Simulated annealing is used to adjust an initially random configuration of the required number of atoms of each type within the unit cell so as to minimize the value of a cost, figure of merit, or "energy" function. The cost function is designed to bias the model development process toward structures that are chemically and physically reasonable and which satisfy the available experimental data.
In Structure Solve, either framework structures (e.g., zeolites) or condensed ionic systems can be studied. The following sections describing the theory behind Structure Solve and its implementation are divided appropriately for the two types of systems.
Zeolites are classically crystalline aluminosilicates with framework structures built from silicate and aluminate tetrahedra. Each apical oxygen atom of the tetrahedron is shared with an adjacent tetrahedron, leading to a framework composition TO2, where T is the tetrahedral species (i.e., Si, Al, etc.). Geometrically, the midpoints of the T-O-T vectors are sufficiently close to the actual positions adopted by the apical oxygen atoms to allow least-squares optimization of the coordinates based on distance constraints or diffraction pattern matching. The problem of structure solution for framework structures of this type reduces to that of determining initial T-atom positions. Each T-atom is connected to exactly four first neighbor T-atoms and the T-atom bonding requirements define constraints on the possible T-T distances and T-T-T angles.
General method of framework structure solution
A successful indexing of the powder X-ray diffraction (PXD) pattern measured from a new zeolite material yields the unit cell dimensions and, based on a judgment of which peaks are systematically absent from the PXD pattern, a choice of a single or, more commonly, a small number of possible space groups.
The direct space approach to structure solution relies on quantization of the chemical/geometrical constraints that zeolite structures are known to obey (Deem and Newsam 1992). The method automatically determines ways in which the required number of T-atoms can, subject to the defined space group symmetry, be placed within the unit cell so as to generate viable zeolite models (viability is determined based on the degree to which the model matches the defined chemical and geometrical constraints). The method seeks to determine all of the viable structures. The appropriate framework for the material in question is then selected from the set of viable structures produced based, for example, on the model pore characteristics and the degree to which the simulated powder diffraction pattern matches the experimentally observed pattern. The effectiveness of the method can in fact be improved substantially by using the degree to which the model matches a target powder diffraction pattern as an additional constraint within the structure development process.
The zeolite figure of merit
Given the known, or assumed, unit cell dimensions, symmetry, and the number of framework or T-atoms per unit cell, nT, a figure of merit can be constructed that quantifies the reasonableness of a given arrangement of the unique T-atoms in the unit cell (Deem and Newsam 1992). This figure of merit is used as the basis for adjusting the unique T-atom arrangement to most closely match the required structural characteristics.
The zeolite figure of merit, H, is defined as:
Eq. 1
where the 
's are the various contributors and 
's are the corresponding weights used in forming the energy sum. By definition, the lower the value of the total figure of merit, H, the more physically reasonable the model structure. This figure of merit can be applied to any set of T-atom positions within a unit cell, whether or not it resembles a zeolite. It is, in fact, initially applied to random positions. By adjusting an initial, random set of unique T-atom positions to minimize the figure of merit, we produce viable structures that have the defined unit cell dimensions and symmetry.
The distance, angle, and average angle terms (the first three terms in Eq. 1) are derived from the geometries observed in known zeolite structures (Deem and Newsam 1992). The simulated annealing optimization process, discussed further below, reproduces Boltzmann statistics. Potential energy curves are defined which, interpreted in the Boltzmann sense of probabilities being proportional to exp (-E/KBT), reproduce the observed geometry histograms; continuous curves have been fitted to the discrete curves predicted by this Boltzmann interpretation of the distribution histograms (Deem and Newsam 1992).
The coordination number term,
coordination, accounts for the four-connectedness of zeolite frameworks. The neighborhood of each T-atom is inspected to determine which T-atoms are linked to it, that is, those that are at less than a defined cutoff distance (typically 5.0 Å). The coordination number term provides bias in favor of the desired coordination number(s) by adding a repulsive contribution to the energy for coordination numbers that are not desired. Typically, values of 1000, 650, 300, 100, 0, 300, and 5000 are used for coordinations of 0, 1, 2, 3, 4, and 5 or more. The correspondence between coordination number and repulsive energy needs to be adjusted for other coordination environments, such as in the interrupted frameworks in which some T-atoms are only 3-connected, or in framework structures containing both tetrahedrally- and octahedrally-coordinated framework cations.
The merging term facilitates the handling of T-atoms that must lie on symmetry elements (termed special positions), a common occurrence in zeolite structures. The total number of atoms per unit cell, nT, is equal to the product of the number of crystallographically unique atoms, nunique, and the number of symmetry operators, nsymm, only when all T-atoms occupy general positions; the coordinates of crystallographically unique T-atoms alone are the independent variables.
merge is given a negative contribution. Such merging is permitted while the total number of atoms within the unit cell remains equal to or greater than nT. This merging term therefore requires definition of the number of unique T-atoms, nunique, as an input parameter. While nT can be measured, nunique must often be assumed based on the known number of symmetry operations and value of nT, and it usually necessary, in practice, to try several values of nunique.
The geometrical zeolite structural constraints can be satisfied by large numbers of possible structures in low symmetry cases. The figure of merit can also include contributions based on the degree of match between powder X-ray or neutron diffraction data computed based on the model versus that measured (Deem and Newsam 1992). The experimental powder diffraction data are input as a series of integrated intensities with associated Miller indices, weights and multiplicities. For powder data such a list will, in general, include groups of reflections for which overlap prevents separate intensity estimations and which must therefore be treated as a combined intensity sum. The calculated powder diffraction pattern is first scaled to have identical total intensity to that observed, and
PXD (or PND) is then defined as the weighted sum of the squares of the differences between the observed and calculated intensities. The information content of the diffraction data is sufficiently high that even an approximate simulation is valuable.
The weights associated with each separate term included in the full figure of merit are typically set to unity, except for the values
T-T-T = 3.0 and T-T-T = 6.0. The weights can be adjusted, if necessary, to facilitate convergence in particular cases. In practice, for problems that require more than one pair of symmetry-related atoms to merge, merge, is best set to approximately 2.5.
The Ionic Solids Figure of Merit
For ionic solids that are relatively close-packed, and for which isotropic rigid ion potentials are viable, less constraining input data and a modified structure development procedure prove appropriate (Freeman and Catlow 1992). The starting point is again a defined unit cell, typically one derived experimentally by indexing a powder diffraction pattern, and a complete chemical composition that defines how many atoms or ions of each type are contained within the unit cell. No model symmetry is assumed and models generated based on this triclinic P1 symmetry may be inspected post facto to indicate which symmetry elements are present (see the "Crystal Builder" chapter in the Cerius2 Builders manual).
For these ionic systems, a starting configuration is first generated by loading the unit cell sequentially with the required complement of ions. Each new ion is added at a random position. If this position causes overlaps with any existing ion the insertion is attempted at a different position and the process repeated until unit cell filling is complete. The simulated annealing driver described below is then applied to first melt and then anneal the system.
In these initial stages the figure of merit contains only an interaction term of Hij = Aij/rij12, with the constant Aij being for similarly charged ions, twice that for oppositely charged species. This simple figure of merit was found to be effective in developing evenly dispersed atomic arrangements with locally preferred cation-anion ordering (Freeman and Catlow 1992). At the end of the simulated annealing schedule, a switch is made to the quadratically convergent conjugate gradient minimization method to optimize the model with respect to this simple figure of merit. A final optimization step can then entail the application of the OFF or Discover tools.
Implementation
Simulated annealing applied to the figure of merit
The figures of merit described above (The zeolite figure of merit and The Ionic Solids Figure of Merit) provide quantitative measures of how close a given arrangement of T-atoms or ions in the known unit cell is to being a viable model for a zeolite or ionic structure, respectively.
The method of adjusting the T-atom or ionic coordinates to produce the most reasonable models (i.e., those that have the lowest energy values) is simulated annealing, a proven algorithm for minimizing multidimensional functions.
Starting at a point in the multidimensional space with calculated energy Eold, another point is generated by perturbing the original point. The new point is accepted if its energy, Enew, is less than or equal to Eold or, if the energy difference
E = Enew - Eold is positive, with a specified transition probability that depends on E and a temperature, T. In practice, the new configuration is accepted if exp (-E/T) is greater than a random number picked between 0 and 1. In the simulated annealing procedure, the simulation commences at a high temperature where most attempted moves are accepted and the temperature is then slowly reduced. The transition probability is reduced in concert so that the average energy of the sampled points in the space also diminishes. At the conclusion of the annealing, the resulting point will, in general, be near the global energy minimum for the system.
Typical values for the temperature decrement factors, Tfactn(T), are 0.7 throughout the annealing. The perturbation step moves a T-atom in a random direction by a random amount within a sphere in crystallographic coordinates. The size of this sphere is arranged to decrease with temperature, typically starting at 1.4 Å and ending at 0.1 Å. The decrement is chosen to be either linear or quadratic in the temperature. The routine acceptance-rejection protocol described above, defined by min{1, exp (-
E/T)}, is the standard Metropolis transition probability for Boltzmann statistics. Throughout the framework annealing process, low-energy frameworks that have the target connectivity are stored so that the (single) structure produced at the finish of the annealing is not the only data gathered from one run. For ionic structures only the final structure is saved.
For framework structures, the procedure described above can provide a range of plausible models for a prescribed set of experimental constraints. The generation of reasonable models for more condensed structures can be achieved by exploiting a combination of simulated annealing and lattice energy minimization (Freeman et al. 1993).
Ions are introduced successively and a straightforward proximity criterion is used to avoid excessive steric overlap; overlapping positions are rejected and repeated random insertions made until the cell is filled with the required number of ions of each type. No symmetry assumptions are made, and the procedure uses triclinic symmetry, P1, with translational periodicity implicitly included.
Given a sufficiently accurate potential model, direct energy minimization using standard techniques of numerical gradient minimization could be used to lower the energy of the system with respect to the coordinates of the ions within the unit cell. However, such a procedure is only successful if the starting point is close to the global energy minimum, as gradient minimization methods progress only in energy decreasing directions: atomic displacements which might temporarily increase energy (encountered, for example, in the migration of a cation from an unfavorable to favorable site) are not permitted. Gradient minimization procedures, therefore, generally converge to the closest local minimum from an arbitrary starting point. However, as described above, simulated annealing implicitly allows energy increasing moves on the path towards the minimum.
annealing
Simulated annealing using Metropolis (Metropolis et al. 1953) Monte Carlo is applied to the initial starting point to relieve any unreasonably close interatomic contacts. The energy function employed in this stage is an extremely simple r-12 form, repulsion between ions of similar charge being twice that between pairs of formally dissimilar charges.
The final stage of the Structure Solve procedure for condensed oxides employs lattice energy minimization (Norgett and Fletcher 1970) using the Born model of the solid and the METAPOCS simulation code developed by Parker and coworkers (Parker et al. 1984). The starting point is derived using the annealing and minimization procedure described above and again the cell dimensions are maintained at the experimental values. Relaxation of the unit cell is possible but crystal topologies are not significantly affected by this extra degree of freedom.
The lattice energy calculations are based on the Born model of the solid (Born and Huang 1954). The lattice energy, V, of the crystal is written as:
Eq. 2
with appropriate potential parameters Aij, Pij and Cij chosen for the pairwise interatomic interactions. The Structure Solve procedure for condensed materials makes appropriate automatic choices for the potential parameters for a particular material and composition. Once the procedure has completed you can use the Discover, GULP and OFF tools to further refine your structure.
The Structure Solve procedure for condensed materials produces, in general, several structural predictions for a particular set of input parameters. It is useful, therefore, to perform a sequence of calculations to sample the structural possibilities for a given system. The conditions of formation of the crystal may be strongly influenced by kinetic, deposition, or templating effects which are not explicitly included in the lattice energy function. The use of multiple runs, therefore, allows the calculation to compensate for two inherent limitations of the approach: the reliance on a gradient optimization method and the inability of the energy function to account for the dynamics of crystal formation.
Before you can apply the Structure Solve tools:
Methodology
The unit cell information for structure types that can be reasonably described by the isotropic ionic model is entered using the Lattice Parameters in the Metal-Oxide Anneal control panel.
Use of the simulated annealing protocol to develop models of framework structures is possible in a quite analogous manner. In this case, however, the appropriate space group symmetry information should be also entered using the Space Group parameter on the Framework Anneal control panel. This information is input as a space group label and qualifier. Even for relatively simple zeolite structures, if the symmetry is specified as being triclinic (P1) as is routinely used for condensed ionic systems, with the number of unique atoms equal to the total number of atoms contained within the unit cell, a huge number of viable structures will be produced. Given the complexity of the figure of merit surface in this case, it is highly likely that the correct structure will not be generated even within a very large number of produced structures. Although based on powder diffraction data alone, it is usually impossible to restrict the choice of space group to a single possibility, often the choice can be narrowed to a small number of space groups. A structure solution strategy then entails sequential use of the various space group symmetry choices, with a detailed evaluation of the results obtained for each case.
For framework structures, the relative weighting of the various geometrical terms and that of the diffraction data can be altered using the Scale Factors in the Framework Controls control panel. The terms for which weights can be set by this command are described above in the Theory section. A wide range of known zeolite and aluminophosphate structures were validated with the Structure Solve tools using the single set of weights provided as defaults.
Performing the simulated annealing procedure
With appropriate specification of the structure development constraints and the relative weights of the various contributors to the figure of merit, the simulated annealing procedure can be started again using the CALCULATE button in the Framework Anneal control panel. As for ionic systems, simple cases may require only a single iteration using the default simulated annealing schedule. For cases in which the desired result is not produced by a single iteration of the procedure, it is advisable first to attempt a much larger number of repetitions with the same set of default weights and cooling schedule parameters and only at that stage to begin exploring changed cooling schedule parameters, or perhaps altered weights for the various terms in the framework figure of merit.
Structure completion
Once framework structures have been generated, they can be completed by adding oxygen atoms between the T-atoms. This is achieved using the Bridging Atom command (on the Framework Anneal menu card click Bridging Atom to bring up the Bridging Atom control panel).
Evaluating the results
Once the simulated annealing run has completed successfully, the first phase evaluation of the output is best performed visually by displaying all the structures.