Addition of higher cumulants to the crystallographic structure-factor equation: a generalized treatment for thermal-motion effects

On avoiding negative electron density in Gram-Charlier refinements of anharmonic motion: the example of glutathione

The structure of glutathione, γ-l-Glutamyl-l-cysteinyl-glycine (C10H17N3O6S), was studied by multi-temperature single-crystal X-ray diffraction. Residual density maps from conventional independent atom model refinement gave indication of anharmonic motion in the molecule. This was further investigated by invariom refinement with anisotropic displacement parameters for all atoms, which described asphericity due to chemical bonding and lone pairs; afterwards only the residual-density signal of anharmonic motion remained. Treating anharmonicity with third-order Gram-Charlier displacement parameters led to regions with unphysical negative electron density. In contrast, a maximum entropy method (MEM) determination of the electron density successfully takes the features into account. Respective difference electron density plots (MEM minus prior and [Invariom+GC] minus invariom) agree well with each other. Challenges in treating and understanding the phenomenon are discussed. A procedure is proposed how unphysical negative electron density can be avoided. It is closely related to the free lunch algorithm.

Protonated nucleobases are not fully ionized in their chloride salt crystals and form metastable base pairs further stabilized by the surrounding anions

This paper presents experimental charge-density studies of cytosinium chloride, adeninium chloride hemihydrate and guaninium dichloride crystals based on ultra-high-resolution X-ray diffraction data and extensive theoretical calculations. The results confirm that the cohesive energies of the studied systems are dominated by contributions from intermolecular electrostatic interactions, as expected for ionic crystals. Electrostatic interaction energies (Ees) usually constitute 95% of the total interaction energy. The Ees energies in this study were several times larger in absolute value when compared, for example, with dimers of neutral nucleobases. However, they were not as large as some theoretical calculations have predicted. This was because the molecules appeared not to be fully ionized in the studied crystals. Apart from charge transfer from chlorine to the protonated nucleobases, small but visible charge redistribution within the nucleobase cations was observed. Some dimers of singly protonated bases in the studied crystals, namely a cytosinium-cytosinium trans sugar/sugar edge pair and an adeninium-adeninium trans Hoogsteen/Hoogsteen edge pair, exhibited attractive interactions (negative values of Ees) or unusually low repulsion despite identical molecular charges. The pairs are metastable as a result of strong hydrogen bonding between bases which overcompensates the overall cation-cation repulsion, the latter being weakened due to charge transfer and molecular charge-density polarization.

Ground-State Charge-Density Distribution in a Crystal of the Luminescent ortho-Phenylenediboronic Acid Complex with 8-Hydroxyquinoline

This contribution is devoted to the first electron density studies of a luminescent oxyquinolinato boron complex in the solid state. ortho-Phenylenediboronic acid mixed with 8-hydroxyquinoline in dioxane forms high-quality single crystals via slow solvent evaporation, which allows successful high resolution data collection (sin θ/λ = 1.2 Å^-1) and charge density distribution modeling. Particular attention has been paid to the boron-oxygen fragment connecting the two parts of the complex, and to the solvent species exhibiting anharmonic thermal motion. The experiment and theory compared rather well in terms of atomic charges and volumes, except for the boron centers. Boron atoms, as expected, constitute the most electron-deficient species in the complex molecule, whereas the neighboring oxygen and carbon atoms are the most significantly negatively charged ones. This part of the molecule appears to be very much involved in the charge transfer occurring between the acid fragment and oxyquinoline moiety leading to the observed fluorescence, as supported by the time-dependent density functional theory (TDDFT) results and the generated transition density maps. TDDFT calculations indicated that p-type atomic orbitals contributing to the HOMO-1, HOMO, and LUMO play the major role in the lowest energy transitions, and enabled further comparison with the charge density features, which is discussed in details. Furthermore, the results confirmed the known fact the Q ligand character is most important for the spectroscopic properties of this class of complexes.

Investigating the Resonance in Nitric Acid and the Nitrate Anion Based on a Modern Bonding Analysis

The nitrate anion, NO3−, is often regarded as a textbook example for the very fundamental concept of resonance. Usually, three equivalent resonance structures with one N–O double bond and two N–O single bonds are considered. Consequently, each of the three N–O bonds should have a partial double bond character. In this study, we analyse the resonance in NO3− in comparison with the related species HNO3 and FNO3 by applying a combination of the Quantum Theory of Atoms in Molecules (QTAIM), a natural bond orbital (NBO) analysis, the electron localizability indicator (ELI), and valence bond (VB) calculations. Despite the fundamental importance of nitrate salts and nitric acid for the environment, chemistry, and industry, a bonding analysis is absent from the literature so far. The classical resonance structures are clearly reflected by the bond analysis tools, but are not the only contributions to the bonding situation. The resonance in HNO3 and FNO3 is greatly perturbed by the hydrogen and fluorine atoms. In addition to theoretical calculations, experimental electron density and wave function refinements were carried out on a KNO3 crystal.

Multi-temperature study of potassium uridine-5'-monophosphate: electron density distribution and anharmonic motion modelling

Uridine, a nucleoside formed of a uracil fragment attached to a ribose ring via a β-N1-glycosidic bond, is one of the four basic components of ribonucleic acid. Here a new anhydrous structure and experimental charge density distribution analysis of a uridine-5'-monophosphate potassium salt, K(UMPH), is reported. The studied case constitutes the very first structure of a 5'-nucleotide potassium salt according to the Cambridge Structural Database. The excellent crystal quality allowed the collection of charge density data at various temperatures, i.e. 10, 100, 200 and 300 K on one single crystal. Crystal structure and charge density data were analysed thoroughly in the context of related literature-reported examples. Detailed analysis of the charge density distribution revealed elevated anharmonic motion of part of the uracil ring moiety relatively weakly interacting with the neighbouring species. The effect was manifested by alternate positive and negative residual density patterns observed for these atoms, which `disappear' at low temperature. It also occurred that the potassium cation, quite uniformly coordinated by seven O atoms from all molecular fragments of the UMPH^- anion, including the O atom from the ribofuranose ring, can be treated as spherical in the charge density model which was supported by theoretical calculations. Apart from the predominant electrostatic interactions, four relatively strong hydrogen bond types further support the stability of the crystal structure. This results in a compact and quite uniform structure (in all directions) of the studied crystal, as opposed to similar cases with layered architecture reported in the literature.

Monte Carlo simulation of an anharmonic Debye-Waller factor to the T4 order

In an increasing number of cases the harmonic approximation is incommensurate with the quality of Bragg diffraction data, while results of the anharmonic Debye-Waller factor are not typically available. This paper presents a Monte Carlo computation of a Taylor expansion of an anharmonic Debye-Waller factor with respect to temperature up to the fourth order, where the lattice was a face-centred cubic lattice and the atomic interaction was described by the Lennard-Jones potential. The anharmonic Debye-Waller factor was interpreted in terms of cumulants. The results revealed three significant points. Firstly, the leading term of anharmonicity had a negative contribution to the Debye-Waller factor, which was confirmed by Green's function method. Secondly, the fourth-order cumulants indicated a non-spherical probability density function. Thirdly, up to the melting point of two different densities, the cumulants up to the fourth order were well fitted by the Taylor expansion up to T⁴, which suggested that the Debye-Waller factor may be calculated by perturbation expansion up to the corresponding terms. In conclusion, Monte Carlo simulation is a useful approach for calculating the Debye-Waller factor.

Invariom modeling of disordered structures: case studies on a dipeptide, an amino acid, and cefaclor, a cephalosporin antibiotic

Routines to facilitate the treatment of disorder in invariom modeling have been implemented in the open-source program MolecoolQt, a visualization program for charge-density work, and InvariomTool, a pre-processor program. Two published structures of an amino acid and a dipeptide and the new structure of cefaclor, a cephalosporin antibiotic, provide examples with increasing amounts of disorder, which can now be successfully modeled with invarioms. Like for ordered structures, these non-spherical scattering factors predicted by density functional theory significantly improve the structural model (figures of merit and standard deviations) also in these cases. Furthermore, they allow rapid calculation and comparison of the electrostatic potential and the molecular dipole moment for the different conformers present in the crystal structures.

First experimental charge density study using a Bruker CMOS-type PHOTON 100 detector: the case of ammonium tetraoxalate dihydrate. Addendum

Addendum to Jarzembskaet al.[Acta Cryst.(2014), B70, 847–855].

First experimental charge density study using a Bruker CMOS-type PHOTON 100 detector: the case of ammonium tetraoxalate dihydrate

The aim of this study was to test the applicability of a Bruker AXS CMOS-type PHOTON 100 detector for the purpose of a fine charge density quality data collection. A complex crystal containing oxalic acid, ammonium oxalate and two water molecules was chosen as a test case. The data was collected up to a resolution of 1.31 Å−1with high completeness (89.1%;Rmrg= 0.0274). The multipolar refinement and subsequent quantum theory of atoms in molecules (QTAIM) analysis resulted in a comprehensive description of the charge density distribution in the crystal studied. The residual density maps are flat and almost featureless. It was possible to derive reliable information on intermolecular interactions to model the anharmonic motion of a water molecule, and also to observe the fine details of the charge density distribution, such as polarization on O and H atoms involved in the strongest hydrogen bonds. When compared with our previous statistical study on oxalic acid data collected with the aid of CCD cameras, the complementary metal-oxide semiconductor (CMOS) detector can certainly be classified as a promising alternative in advanced X-ray diffraction studies.

Cryo-crystallography: diffraction at low temperature and more

This chapter comments on the motivations and the methods of crystallographic studies at low temperature. Cry-crystallography is a brunch of Crystallography, a science that is too often confused with a technique. On the other hand, the scientific background to study crystal phases at low temperature is here provided, together with a survey of many possible techniques that provide complementary or supplementary information. Several applications are discussed, in particular in relation with highly accurate studies like electron density determination or phase transition mechanisms.

Using molecular-dynamics simulations to understand and improve the treatment of anharmonic vibrations. I. Study of positional parameters

Molecular-dynamics-derived numerical probability density functions (PDFs) have been used to illustrate the effect of different models for thermal motion on the parameters refined in a crystal structure determination. Specifically, anharmonic curved or asymmetric PDFs have been modelled using the traditional harmonic approximation and the anharmonic Gram-Charlier series treatment. The results show that in cases of extreme anharmonicity the mean and covariance matrix of the harmonic treatment can deviate significantly from physically meaningful values. The use of a Gram-Charlier anharmonic PDF gives means and covariance matrices closer to the true (numerically determined) anharmonic values. The physical significance of the maxima of the anharmonic distributions (the most probable or mode positions) is also discussed. As the data sets used for the modelling process are theoretical in origin, these most probable positions can be compared to equilibrium positions that represent the system at the bottom of its potential-energy surface. The two types of position differ significantly in some cases but the most probable position is still worthy of report in crystal structure determinations.

Using molecular-dynamics simulations to understand and improve the treatment of anharmonic vibrations. II. Developing and assessing new Debye-Waller factors

Two new anharmonic forms for the Debye-Waller factor, aimed at modelling curvilinear and asymmetric motion, have been introduced. These forms permit the refinement of structures with these types of anharmonic motion using a small number of additional parameters. Molecular-dynamics-derived numerical probability density functions (PDFs) have been used to assess the merit of these new functions in real space. The comparison is favourable particularly for the curvilinear PDF based on a parabolic coordinate system change of a trivariate Gaussian distribution. The initial results also suggest that high-order even terms from the Gram-Charlier series may be important for modelling methyl-group libration. The molecular-dynamics data sets provide useful insights into the nature of anharmonic thermal motion. Addressing the problem in real space allows intuitive PDFs to be developed but numerical methods may be necessary for these methods to be implemented in refinement programs as an analytical Debye-Waller factor cannot always be obtained.

Limitations of apparent diffusion coefficient-based models in characterizing non-gaussian diffusion

Diffusion in complex heterogeneous structures, for example, the neural fiber system, is non-gaussian. Recently, several methods have been introduced to address the issue of non-gaussian diffusion in multifiber systems. Some are based on apparent diffusion coefficient (ADC) analysis; and some are based on q-space analysis. Here, using a simple mathematic derivation, ADC-based models are shown to be mathematically self-inconsistent in the presence of non-gaussian diffusion. Monte Carlo simulation on restricted diffusion is applied to demonstrate the poor data fitting that can result from ADC-based models. Specific comparisons are performed between two generalized diffusion tensor imaging methods: one of them is based on ADC analysis, and the other is shown to be consistent with q-space formalism. The issue of imaging asymmetric microstructures is also investigated. Signal phase and spin exchange are necessary to resolve multiple orientations of an asymmetric structure.

Anisotropy and anharmonicity of atomic fluctuations in proteins: analysis of a molecular dynamics simulation

Positional probability density functions (pdf) for the atomic fluctuations are determined from a molecular dynamics simulation for hen egg-white lysozyme. Most atoms are found to have motions that are highly anisotropic but only slightly anharmonic. The largest deviations from harmonic motion are in the direction of the largest rms fluctuations in the local principal axis frame. Backbone atoms tend to be more nearly harmonic than sidechain atoms. The atoms with the largest anharmonicities tend to have pdfs with multiple peaks, each of which is close to harmonic. Several model pdfs are evaluated on the basis of how well they fit probability densities from the dynamics simulations when parameterized in terms of the moments of the distribution. Gram-Charlier and Edgeworth perturbation expansions, which have been successful in describing the motions of small molecules in crystals, are shown to be inadequate for the distributions found in the dynamics of proteins. Multipeaked distribution functions are found to be more appropriate.

Crystallographic studies of the dynamic properties of lysozyme

The patterns of atomic displacements in the crystals of hen and human lysozyme derived from independent crystallographic refinement are broadly similar. Analysis of the pattern indicates a close correlation with molecular structure, strongly suggestive of intramolecular motion. The active site of lysozyme is located in a region of high displacement. It is concluded that protein mobility may play a significant part in biological activity and that X-ray crystallography can contribute to its analysis.