Lattice Energy Minimization Research Articles

Modeling steps and kinks on the surface of calcite

This work presents modeling results on the cleavage face of calcite as well as on steps and isolated kinks on this face. We used static lattice energy minimization and interatomic potentials fitted to bulk properties. The energy needed to cleave a bulk calcite crystal along the [1 0 (-)1 4] plane was calculated to be 0.59 J m(-2) in agreement with previous studies using the same potentials. The perfect surface reconstructs in the top few atomic layers, but its symmetry corresponds to the bulk termination. By contrast, the (1 0 (-)1 4) surface with cleavage steps present reconstructs to form a (2 x 1) super cell. This may help explain experimental observations of (2 x 1) symmetry on calcite surfaces. The energy required to form a monatomic obtuse step is calculated to be 1.3 x 10(-10) J m(-1) and for the acute step, 2.4 x 10(-10) J m(-1), suggesting that obtuse steps dominate on cleaved surfaces. Along the two types of steps, a total of 16 kink geometries exist. We calculated kink defect energy with two different approaches: one where kink pairs were added onto infinitely long steps and one where kinks were placed inside pits on a cleavage surface. Calculations on infinitely long steps show that for vacuum conditions, kink pairs possess roughly identical formation energy, about 1.2-2.2 eV, so based on energetics one cannot expect significant differences in kink site frequency

An Assessment of Lattice Energy Minimization for the Prediction of Molecular Organic Crystal Structures

Lattice energy searches for theoretical low-energy crystal forms are presented for 50 small organic molecules, and we compare the experimentally observed crystal forms to these lists of hypothetical polymorphs. For each known crystal, the relative stability is calculated with respect to the global minimum energy structure, and we determine the number of unobserved structures lower in energy than the experimental form. The distributions of these relative energies and their rankings in the predicted lists are used to determine the efficacy of lattice energy minimization in crystal structure prediction. Although a simple form for the interaction energies has been used, the calculations produce almost a third of the known crystals as the global minimum in energy, and approximately a half of the known structures are within 1 kJ/mol of the global minimum. Molecules with no hydrogen-bonding capacity are most likely to be found close to the global minimum in lattice energy, while increasing the number of possible...

Abinitiostructure determination ofm-toluidine by powder X-ray diffraction

The powder X-ray diffraction pattern of the crystalline phase ofm-toluidine has been recorded with a sensitive curved detector (CPS120) at 150 K. The structure has been solved by real-space methods (simulated annealing) followed by Rietveld refinements with phenyl rings as rigid bodies and with soft constraints on bond lengths for peripheral atoms. The cell is monoclinic with space groupP21/candZ= 8. Equivalent molecules form chains alongc. The crystalline cohesion is achieved by N—H...N hydrogen bonds between neighbouring chains of non-equivalent molecules and by van der Waals interactions of neighbouring chains of equivalent molecules. The hydrogen-bonding network has been confirmed by lattice-energy minimization. Anisotropic strain effects of the cell have been calculated. The directions of the minimal strains correspond to the directions of the hydrogen bonds. An explanation of the difficulty to crystallize the metastable phase is given.

Structure and Dynamics — An Atomic View of Materials

Preface 1. Introduction 2. Structure of materials 3. Formal description of crystal structures 4. The reciprocal lattice 5. Atomic bonding in crystals 6. Diffraction 7. Physical properties 8. Lattice dynamics 10. Experimental methods for measurements of vibrational frequencies 11. Anharmonic interactions 12. Displacive phase transitions A. Real crystals! B. Fourier analysis C. Schoenflies representation of the point groups D. Rhombohedral, trigonal and hexagonal unit cells E. Space groups F. Lattice energy minimization G. Some note on the variational theorem H. Ewald Sphere I. The Wilson Plot J. Diffraction from isotropic materials K. Calculation of physical properties L. Partition function: some key results M. Lattice sums N. Mean-square atomic displacement and temperature factors

Simulations of inorganic crystal structures: Recent advances in structure elucidation

The aim of this paper is to show some recent developments of simulation approaches in the structure elucidation of inorganic crystalline compounds, highlighting at each step their role either as a complementary technique or as a tool to anticipate structure/properties relationships or even to imagine new inorganic architectures. Specific examples are taken in the area of zeolites to illustrate the use of energy minimizations and ( N, V, T) Monte Carlo techniques as a complement to experimental diffraction approaches, typically for the disordered placement of extra-framework species (cations, water molecules). In the still extending area of the synthesis of templated open-framework inorganic structures, we show that lattice energy minimizations may possibly be used to estimate or even anticipate the structures and energetics of the related template-free structures. A third section tackles the development of more sophisticated methods for the computational design of not-yet-synthesized structures using building block concepts.

Triazinone tautomers: solid phase energetics

We investigate the relative stabilities, in the gas and solid phases, of three tautomers of hydroxytriazine. In order to study the solid state energetics, we generate a number of hypothetical crystal structures for each tautomer, which we then subject to lattice energy minimisation. In both the solid and gas phases, we find that hydroxytriazine (TOH) and the ortho-protonated triazinone tautomer (T1) are almost indistinguishable in energy and are significantly more stable than the para-protonated triazinone tautomer (T2). In the crystal structures, both the experimental ones of related molecules and the hypothetical ones generated for hydroxytriazine and its triazinone tautomers, we find the expected hydrogen bonding interactions. In addition, various motifs associated with π-stacking are often found in the experimental structures.

Novel inorganic frameworks constructed from double-four-ring (D4R) units: computational design, structures, and lattice energies of silicate, aluminophosphate, and gallophosphate candidates.

The design of new and interesting inorganic frameworks is an ongoing challenge in materials sciences. New structures containing double-four-ring (D4R) units have recently received particular attention. The present work focuses on the computational design of new three-dimensional frameworks made of D4R units exclusively. In a first step, our simulations explore the possible ways to assemble predefined D4R units in 3D space using a sophisticated cascade of simulated annealing/minimizations steps (autoassembly of secondary building units method). While the existing zeotype topologies were successfully generated, new topologies were predicted including very open frameworks containing new types of cages. In a second step, lattice energy minimizations were performed to estimate the viability of these hypothetical frameworks as silicate, aluminophosphaste, and gallophosphate candidates. When comparing the hypothetical structures to existing compounds, our results raise the challenging question of the appropriate chemical composition that should be aimed at for a given framework topology of interest.

Ordered Phases in n-Heptacosane (C27H56). Structure Determination of the Mdci Phase

n-Heptacosane (C27H56) is known to have a rich polymorphism. With increasing temperature, the following phase transitions are observed: Oi → Odci → Mdci → RIII ···> RIV → L. The Mdci form is known to be isomorphous with the modification B in n-tritriacontane (C33H68). Two possible space groups were given in the literature for the B-form of C33H68: monoclinic (Aa) and orthorhombic (A21am). We have determined the structure of the n-heptacosane at 325 K combining lattice energy minimization calculations and Rietveld refinement using accurate X-ray diffraction data. We have shown that the Mdci form of heptacosane is monoclinic (Aa, Z = 4). Moreover, at each phase transition Oi → Odci → Mdci, a substantial decrease in intensity of the 0 0 l reflections and an increase in the interlayer spacing due to end-gauche defects are observed. In addition, each phase transition is characterized by subtle appearance and disappearance of reflections observed at large angles of diffraction. These characteristics are powerful tools for phase identification in binary or ternary phase diagrams of n-alkanes.

Bridging the gap – structure determination of the red polymorph of tetrahexylsexithiophene by Monte Carlo simulated annealing, first-principles DFT calculations and Rietveld refinement

The crystal structure of the red polymorph of tetrahexylsexithiophene (THST) is solved from X-ray powder diffraction data by a direct-space Monte Carlo simulated-annealing approach. First-principles density functional theory (DFT) calculations are used to distinguish between three nearly identical solutions in the space groupsC2/m,C2 andP\bar{1} and to improve the overall accuracy of the crystal structure. The correct space group is found to beC2/m. In all space groups, the thiophene backbone is planar and the hexyl side chains assume an all-transconformation except for two terminal methyl residues, which adopt agaucheorientation. The ability of first-principles DFT calculations to provide atomic coordinates of single-crystal quality is demonstrated by lattice-energy minimization of the known crystal structure of the yellow polymorph of THST. The combination of Monte Carlo simulated annealing, first-principles DFT calculations and Rietveld refinement presented in this paper is generally applicable. It provides a powerful alternative to standard approaches in cases where the information content of the powder diffraction pattern alone is insufficient to distinguish between different structure solutions. DFT calculations can also provide invaluable guidance in Rietveld refinement.

Computational prediction of the phase transformation of two as-synthesized oxyfluorinated compounds into the zeotype CHA forms.

Computer calcination: The phase transformation of two as-synthesized fluorinated inorganic compounds into their related microporous structures upon calcination (see scheme) is predicted by using lattice energy minimizations, successfully capturing in one step dehydration, defluorination, and template elimination.

Framework stability of nanoporous inorganic structures upon template extraction and calcination: a theoretical study of gallophosphate polymorphs.

A systematic computational study of gallophosphates was undertaken. First, lattice energy minimization calculations using a formal-charge shell model potential have been carried out on a series of hypothetical gallium phosphates derived from their metallogallophosphate, aluminophosphate, or aluminosilicate analogues through atomic substitution. The minimized structures show the typical features in terms of bond angles and distances as expected in zeolitic gallophosphates. Second, the crystal structures of several gallophosphates in their calcined forms have been predicted, using for each compound lattice energy minimization and an initial model derived from its as-synthesized templated form. All the modified structures thus have the same GaPO(4) composition. The lattice energies of all the simulated gallophosphate structures were compared to that of GaPO(4)-quartz as a reference structure. Interestingly, among all predicted calcined structures, various zeolitic topologies were found. The study of the energetics of these zeotypic structures showed a linear dependence of lattice energy upon density. Strikingly, a few simulated structures showed unrealistic structural features, such as important framework distortions, often associated with the occurrence of a hexameric unit in the original as-synthesized structures. Also, those gallophosphates with structural faults were found in the upper part of the energy/density plot. To address the validity of our force field calculations in these special cases, first principles calculations were undertaken on ULM-4, chosen as a typical representative structure. Indeed, the qualitative agreement found between our results and those obtained with the nonlocal density functional theory demonstrates the robustness of our force field. Further minimization also showed that the inclusion of polarizability is crucial for yielding results comparable with those obtained using first principles methods.

A study of the known and hypothetical crystal structures of pyridine: why are there four molecules in the asymmetric unit cell?

The oldest crystal structure of pyridine is unusually complex, with four molecules in the asymmetric unit cell of Pna21 symmetry. In an attempt to understand why pyridine crystallises with 16 molecules in the unit cell, we have considered its thermodynamic stability relative to hypothetical pyridine structures. These were generated by a search for minima in the lattice energy of pyridine amongst the more common space groups, using the crystal structure prediction procedure MOLPAK followed by lattice energy minimisation using a distributed multipole-based intermolecular potential. We find over two dozen distinct crystal structures in the energy gap of less than 6 kJ mol−1 between the corresponding models for the observed and most stable (hypothetical) structure. Adding harmonic phonon estimates of the intermolecular zero point energy and entropy at the melting point of pyridine slightly improves the relative stability of the observed Z = 16 structure. Several of these hypothetical structures can be eliminated as only just mechanically stable, or because the growth rate of the crystal is estimated to be very slow by the attachment energy model. Nevertheless, there are still over a dozen structures that appear competitive with the known structure as polymorphs of pyridine. Following these predictions, an intense experimental search has found a new polymorph of perdeutero-pyridine (form II), which was not found in the search. This structure is also predicted to be metastable with a similar energy to form I. Although there is some evidence for kinetic factors favouring the observed structures, the metastable Z = 16 structure and the new form II remain a challenge for our understanding of crystallisation.

Nanocluster Epitaxy by Annealing: Ag on H-terminated Si (111) Surfaces

ABSTRACTWe report an experimental investigation on the morphology and orientation of Ag nanoclusters by RT deposition and subsequent annealing. We show that epitaxial Ag clusters of 2 ∼ 6 nm in diameter can be synthesized in this way. The RT self-assembled Ag clusters grow as mostly single-crystal crystallites with Ag(111)//Si(111), but the in-plane orientation has a dispersion of ∼ 9° centering at Si[110] direction. Upon annealing, the Ag clusters drastically rotated to the epitaxial configuration with the in-plane orientation aligned to the Ag[110] //Si[110] direction. The rotation and epitaxy of the Ag nanoclusters are explained based on a coincident site lattice model and interface energy minimization.

Polytypism and mixed layering in minerals

Polytypes are stacking variations that occur in crystals, forming periodic extended defects, without changing the overall composition. Mineral examples are the different polytypes of micas, kaolinite, talc, graphite. None of the theories that have been proposed to explain the occurrence of short and long period superstructures have met with universal success in explaining their occurrences. Polysomatism is defined as the stacking of compositionally different modules, similar to polytypism. Minerals belonging to a polysomatic series have compositions that are linear combinations of the end members. Structures derived from the bixbyite (Mn2O3) structure are used as examples of a polysomatic series and a number of polysomes in single crystals have been described using HRTEM. The relative stabilities of different polysomes have been modelled with a lattice energy minimization programme. The results show that certain polysomes are significantly more stable than others, and that the contribution of next-nearest neighbour interactions seems to be an important factor in determining polysome stability.

Raman spectra and lattice-dynamical calculations of natrolite

Polarized single-crystal Raman scattering and powder infrared absorption spectra of Fdd2 orthorhombic natural natrolite (Na1.88 K0.02 Ca0.04 )(Al1.96 Si3.03 O10)⋅2.03 H2O from Khibiny, Kola peninsula, were measured. Using short-range models, lattice-dynamical calculations were performed for an idealized natrolite structure Na4(Al4Si6O20)4H2O containing 46 atoms in the primitive unit cell (Z = 2). By varying the valence force constants, the calculated frequencies in the Raman and IR spectra were fitted to the observed frequencies. On considering their calculated intensities as well, nearly all the observed bands (especially those corresponding to the A1 modes) could be unambiguously assigned and interpreted. The external vibrations of H2O could be correctly assigned using deuter- ated samples. The strongest Raman band at 534 cm-1 corresponds to a breathing mode of the four-membered alumi- nosilicate ring. The calculated bulk modulus (52.7 GPa at zero pressure) is close to the experimental value of 47 ± 6 GPa. The natrolite structure has some advantages upon other zeolites to understand the amorphization mechanism, because samples of this mineral surrounded by a non-penetrating medium show no crystal phase transitions with increasing pressure. Lattice energy minimization calculated with variable unit-cell dimensions shows the crystal structure to become unstable at about 5.5 GPa, thereby apparently explaining the amorphization process at 4-7 GPa. This instability is connected with shear acoustic modes coupled with soft internal framework vibrations.

Three conformational polymorphs of di-mu-chlorotetrakis(1-methylboratabenzene)diyttrium: synthesis, x-ray structures, quantum chemical calculations, and lattice energy minimizations.

The reaction of yttrium trichloride with lithium 1-methylboratabenzene (1/2) in toluene (110 degrees C, 3 days) afforded the donor-free dinuclear sandwich complex [(C(5)H(5)BMe)(2)Y(mu-Cl)](2) (1) in 85% yield as pale-yellow crystals. By means of single crystal and powder diffraction methods, three conformational polymorphs, alpha-1 [P2(1)/n (No. 14), monoclinic, a = 6.6124(8) A, b = 14.352(9) A, c = 14.120(1) A, beta = 95.57(1) degrees, V = 1333.7(9) A(3), Z = 2], beta-1 [P2(1)/a (No. 14), monoclinic, a = 8.542(2) A, b = 13.712(6) A, c = 11.76(1) A, beta = 102.60(4) degrees, V = 1344.5(13) A(3), Z = 2], and gamma-1 [Pbca (No. 61), orthorhombic, a = 20.091(5) A, b = 13.527(3) A, c = 9.976(2) A, V = 2711.2(11) A(3), Z = 4], were characterized in the solid state of 1. The molecules in the three phases vary remarkably in the rotational position of boratabenzene ligands with differences of 91.1, 133.1, and 24.9 degrees between each pair. DFT calculations at the B3LYP/LanL2DZ level reveal that the three molecular structures observed in the solid state correspond closely to three minima on the gas-phase potential energy surface. The beta conformation is 2.8 and 7.2 kJ/mol more stable than the alpha and gamma conformations, respectively. Lattice energy minimizations predict that the alpha-1 phase is about 5.5 and 18.7 kJ mol(-)(1) more stable than the beta-1 and gamma-1 modifications, in agreement with the packing coefficients and the molecular volumes of the three crystal structures. While the alpha-1 and beta-1 modifications have comparable total energies, the gamma-1 form is less stable. The total energy differences among the polymorphs are greater than generally expected.

Atomistic simulation of the crystal structures and bulk moduli of TiO2 polymorphs

The crystal structures and bulk moduli of the known or proposed TiO2 polymorphs (which include the low-pressure forms rutile, anatase, brookite, TiO2(B), TiO2(H) and TiO2(R), and the high-pressure columbite-, baddeleyite-, cotunnite-, pyrite-, and fluorite-structured forms) have been simulated using static lattice energy minimisation and two independent forcefields, namely a variable charge Morse potential and a fixed charge rigid ion potential. The results obtained demonstrate that the variable charge model is capable of describing the crystal structures of all the polymorphs with Ti–O coordination ranging from six to eight, but fails for the nine-coordinated cotunnite structure. The fixed-charge model is successful in predicting the crystal structures of most of the polymorphs including the cotunnite-type, but fails for the ramsdellite-structured TiO2(R) and is poor for baddeleyite-type TiO2. The variable charge model is more successful in predicting the bulk moduli of the low-pressure polymorphs, whereas the fixed charge model is marginally better in predicting the bulk moduli of the high-pressure phases. The relative phase stabilities of the low-pressure phases and the high-pressure phase diagram obtained with the variable charge model are also in broad qualitative agreement with available calorimetric, phase equilibrium, and simulation data.

NaZn(H2O)2[BP2O8].H2O: a novel open-framework borophosphate and its reversible dehydration to microporous sodium zincoborophosphate N.

Crystals of NaZn(H2O)2[BP2O8].H2O were grown under mild hydrothermal conditions at 170 degrees C. The crystal structure (solved by X-ray single-crystal methods: hexagonal, P6(1)22 (no. 178), a = 946.2(2), c= 1583.5(1) pm, V= 1227.8(4).10(6) pm3, Z = 6) exhibits a chiral octahedral-tetrahedral framework related to the CZP topology and contains helical ribbons of corner-linked borate and phosphate tetrahedra. Investigation of the thermal behavior up to 180 degrees C shows a (reversible) dehydration process; this leads to the microporous compound Na[ZnBP2O8].H2O, which has the CZP topology. The crystal structure of Na[ZnBP2O8].H2O was determined by X-ray powder diffraction by using a combination of simulated annealing, lattice-energy minimization, and Rietveld refinement procedures (hexagonal, P6(1)22 (no. 178), a = 954.04(2), c = 1477.80(3) pm, V= 164.88(5).10(6) pm3, Z = 6). The essential structural difference caused by the dehydration concerns the coordination of Zn2- changing from octahedral to tetrahedral arrangement.

Which organic crystal structures are predictable by lattice energy minimisation?

A survey of the molecules which have been used in crystal structure prediction studies is presented. The results of these studies have been analysed in terms of whether the experimentally observed crystal structures are found at or near the global minimum in the lattice energy. The results suggest that whilst some crystal structures can be predicted just on the basis of lattice energy searches, there is yet insufficient experience to judge for which molecules this energetic criterion is sufficient, within the limitations of current force-field accuracy. The molecules chosen to test crystal structure prediction methods appear to be biased away from the types that would be expected to be readily predictable and suitable for crystal engineering. The survey highlights the need for more theoretical and experimental collaboration to understand what determines whether a molecule's crystal structure will be so favourable that other polymorphs are unlikely.

Cation-vacancy ordering in dehydrated Na6[AlSiO4

The low-temperature cation-ordered superstructure of anhydrous sodium sodalite, a zeolite with composition Na6[AlSiO4]6, has been determined through the use of both density functional theory (DFT) and classical force-field lattice energy minimizations. The charge-balancing Na+ cations are assumed to occupy their characteristic locations within the cubic alumino–silicate framework near the centers of the 6-ring windows. Within the constraints of the volume-doubled pseudotetragonal supercell reported in a previous x-ray diffraction study [B. Campbell, S. R. Shannon, H. Metiu, and N. P. Blake (submitted)], all possible arrangements of cations and vacancies amongst the 6-ring window sites were considered. Force-field calculations employing the ab initio based potential energy function derived by Blake, Weakliem, and Metiu [J. Phys. Chem. B 102, 67 (1998)] and the empirical shell-model potential of Catlow et al. [J. Chem. Soc. Commun. 1984, 1271; Mol. Simul. 1, 207 (1988)], were used to perform full lattice-energy minimizations of each configuration, and to assess their relative stabilities both before and after minimization. The most stable configurations were then examined in more detail via ab initio density functional calculations in the generalized gradient approximation. The lowest-energy supercell ordering proved more stable than the lowest-energy parent cell ordering, and also yielded a pseudotetragonal distortion (space group Pnc2) and a calculated diffraction pattern that qualitatively match experimental results. The structural influences that contribute to the low energy of the correct vacancy ordering are described in detail.