Mary P. McCourt

Disorder and the molecular packing of C60 buckminsterfullerene: a direct electron-crystallographic analysis

The crystal structure of C60 buckminsterfullerene was determined at room temperature by a direct phasing analysis of single-crystal electron-diffraction intensity data. The initial electrostatic potential map is well fit by a regular icosahedron of C atoms but with an average rotational disorder corresponding to Fm3m symmetry. The static appearance of this directly determined map, however, does not refute the notion of uncorrelated molecular positions in the crystal lattice, indicated earlier by nuclear-magnetic resonance spectroscopy and neutron scattering. Although the direct determination of crystallographic phases is correct, the occurrence of strong axial h00 reflections in the electron diffraction patterns appears to be a result of secondary scattering. Correction for this perturbation produces a good fit of the intensities to an isotropic spherical shell of C atoms. In fact, the static appearance of the initial potential map is artificial, owing to the use of only a limited set of phased structure factors in the Fourier transform carried out after the ab initio structure analysis.

Electron Crystallography of Small Organic Molecules: Criteria for Data Collection and Strategies for Structure Solution

By crystallization onto an organic substrate such as naphthalene, thin microcrystals of a small organic molecule, such as triphenylene, can be grown. These crystals are less perturbed by shear and erratic bend disorder than the samples produced by rapid growth from dilute solutions. Selected-area electron diffraction intensities are consistent from specimen to specimen, show the symmetry expected for the crystalline projection and, furthermore, correspond to the Fourier transform of the entire unit cell. However, under sampling of the three-dimensional reciprocal lattice by goniometry can frustrate structure determination if conventional direct methods are used. Nevertheless, the crystal structure may still be solved quite accurately by energy minimization.

Direct phase determination for polymer fibre X-ray data—the structure of poly(tetramethyl-p-silphenylene siloxane)

The X-ray crystal structure of poly(tetramethyl-p-silphenylene siloxane) has been re-determined from fibre diffraction data by direct methods (Sayre equation via a multisolution approach). Despite the somewhat low resolution of the observed data set, this approach is sufficient for distinguishing the correct space group as well as producing an electron density envelope that fits the outlines of the monomer repeat. The resultant structural solution is virtually identical to the one found earlier by a model refinement procedure.

Phase determination in protein x-ray crystallography at low resolution : 6.0 Å Data from rubredoxin via the pseudoatom glob approximation

With a re-scaled pseudoatom scattering factor approximation to the Fourier transform of globular density units in the protein, the crystal structure of rubredoxin was determined from experimental x-ray diffraction data by direct methods at 6 A resolution. The Sayre equation expanded a relatively small basis set and screening criteria, such as density flatness and a Patterson correlation coefficient, selected the best of several possible solutions. In the two-dimensional determination, principal peak positions were located in the maps, giving a mean phase error of 57.6° for all 17 hk0 reflections after two cycles of Fourier refinement. In the three-dimensional determination, the initial phase set could be compared to the original x-ray model by a mean deviation of 85.3° for all 127 hkl reflections but only 54.3° for the 23 most intense maxima. Refinement improved the overall fit (mean error: 77.4° but only 62.3 for the 23 most intense reflections). Because the phase value of the most intense reflection is greatly improved, the electron density map is closer to that of the actual structure. However, a priori imposition of globular substructure causes the 3-D density map to become discontinuous. The success of this phasing attempt is quite surprising for an example where the solvent space contains a high concentration of ammonium sulfate where the contrast at low resolution should be reversed.

An alternative parameterization of electron repulsion terms in mndo and amI

Abstract An alternative method of parameterizing the two electron coulomb integrals in MNDO and AMI is described Although the method is used to closely approximate the Dewar and Thiel integrals, the formalism and the resulting formulas are much simpler and computationally more efficient The method strictly obeys rotational invariance, gives integrals and their derivatives directly in space fixed cartesian coordinates, can be used with arbitrary basis sets and gives the correct behavior both at the large and small (united atom) limits of the inter-atomic distance The integrals are numerically compared with those of Dewar and Thiel

A molecular orbital study of a model of the Mg2+ coordination complex of the self splicing reaction of ribosomal RNA

Recent discoveries have established the fact that RNA is capable of acting as an enzyme. In this study two different types of molecular orbital calculations, INDO and ab initio, were used in an attempt to assess the structural/functional role of the Mg2+ hydrated complex in ribozyme reactions. Preliminary studies indicate that the reaction is multistep and that the Mg2+ complex exerts a stabilizing effect on the intermediate or midpoint of the reaction.

Fiber diffraction as a useful supplement to single-crystal electron diffraction data in the direct three-dimensional structure analysis of polymers

Abstract When zonal electron diffraction data from chain-folded lamellae are combined with the three-dimensional (3-D) information from a microfiber, it is possible to determine a polymer crystal structure by direct methods to visualize the atomic arrangement in the unit cell. An analysis of the polyethylene sulfide) structure (space group Pbcn, a = 8.51, b = 4.94, c = 6.69 A) is presented as an example of such a determination. The crystallographic phases—including 17 of 20 unique zonal hkO data collected earlier from chain-folded lamellae, as well as two other phases permitted for origin definition in three dimensions—served as a basis set for extension into three dimensions by the Sayre equation. The phase extension into the 3-D indices of the fiber diffraction set (39 unique amplitudes including 32 estimated from 14 overlapped maxima) was quite accurate, especially for the most intense reflections. Combination of these phases with amplitude values to generate the three-dimensional potential map allowed...

New formulas for integrals in semiemperical molecular orbital methods: Part 2. Coulomb integrals and their first and second Cartesian derivatives in the NDDO theory

Abstract Formulas for the two center coulomb integrals that occur in the Neglect of Diatomic Differential Overlap (NDDO) methods, together with their first and second analytical derivatives with respect to nuclear displacements are given. The formulas are derived using an operator representation of the charge distributions. The resulting formulas are expressed directly in the Cartesian coordinates of the two nuclei, thus avoiding the transformation from local diatomic to molecular coordinates.

New formula for integrals in semiempirical molecular orbital methods: Part 1. General theory and results for two-center overlap integrals and their first and second derivatives with respect to Cartesian nuclear displacements

Abstract A very simple method for deriving formulas for integrals over Slater—Zener type atomic orbitals is given. Essentially, the method obtains integrals over orbitals having higher n and l quantum numbers from the integral over 1 s orbitals by repeated application of operators involving the orbital exponents and the positions of the orbital centers. The method is made practicable by using a symbolic algebra expert system to derive the formulas. Formulas for overlap integrals (and their derivatives) over 1 s , 2 s , and 2 p atomic orbitals are given directly in Cartesian coordinates, thus avoiding the transformation from a local to a molecular coordinate system.