4. Laplacian of the Electron Density 1592 5. AIM Properties 1593 IV. Topological Analyses of Experimental Densities 1593 A. Experimental versus Theoretical Topology 1593 1. Carbon−Carbon Bond 1593 2. Polar Bonds 1594 B. Reproducibility of the Experimental Topology 1596 1. Amino Acids and Oligopeptides 1596 2. Model Ambiguities 1598 C. Applications to Molecular Crystals 1599 1. Aromaticity in Carbon-Based Ring Systems 1599

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Chemical applications of X-ray charge-density analysis.

The one-electron density function, ρ(r), (in principle deduced from elastically scattered X-ray intensities) is the probability distribution function of an electron, averaged over the positions of all other electrons. A partitioning of ρ(r) into constituent parts is an intellectual exercise that does not lend itself to unique measurement from elastic X-ray scattering experiments. It is shown that in the limit of perfect data and an infinite Ewald sphere, a least-squares fit with a many-centered finite multipole expansion of the charge density about the N nuclei will necessarily satisfy the q-centered multipoles of the molecule for q = 1, 2,…, N. This means that a large number of static-charge physical properties (averages over ρ(r)) are correctly given. Several expressions for averages of ρ(r) or over FH are given. It is shown that outer moments, such as atomic charges, dipole moments and quadrupole moments, always depend on a shape function. On the other hand, inner moments such as potentials, electric forces, and electric field gradients, may be represented by direct Fourier analysis of FH (obs) (suitably phased, of course). Nuclear vibrations have been neglected throughout the discussion.

V. One‐Electron Density Functions and Many‐Centered Finite Multipole Expansions

Accurate, yet simple and efficient, formulae are presented for calculation of the electrostatic potential (ESP), electric field (EF) and electric field gradient (EFG) from the aspherical Hansen–Coppens pseudoatom model of electron density [Hansen & Coppens (1978). Acta Cryst. A34, 909–921]. They are based on the expansion of |r′ − r|−1 in spherical harmonics and the incomplete gamma function for a Slater-type function of the form Rl(r) = rn exp(−αr). The formulae are valid for 0 ≤ r ≤ ∞ and are easily extended to higher values of l. Special treatment of integrals is needed only for functions with n = l and n = l + 1 at r = 0. The method is tested using theoretical pseudoatom parameters of the formamide molecule obtained via reciprocal-space fitting of PBE/6-31G** densities and experimental X-ray data of Fe(CO)5. The ESP, EF and EFG values at the nuclear positions in formamide are in very good agreement with those directly evaluated from density-functional PBE calculations with 6-31G**, aug-cc-pVDZ and aug-cc-pVTZ basis sets. The small observed discrepancies are attributed to the different behavior of Gaussian- and Slater-type functions near the nuclei and to imperfections of the reciprocal-space fit. An EF map is displayed which allows useful visualization of the lattice EF effects in the crystal structure of formamide. Analysis of experimental 100 K X-ray data of Fe(CO)5 yields the value of the nuclear quadrupole moment Q(57Fem) = 0.12 × 10−28 m2 after taking into account Sternheimer shielding/antishielding effects of the core. This value is in excellent agreement with that reported by Su & Coppens [Acta Cryst. (1996), A52, 748–756] but slightly smaller than the generally accepted value of 0.16 ± 5% × 10−28 m2 obtained from combined theoretical/spectroscopic studies [Dufek, Blaha & Schwarz (1995). Phys. Rev. Lett. 25, 3545–3548].

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On the calculation of the electrostatic potential, electric field and electric field gradient from the aspherical pseudoatom model

Recent advances in computational chemistry have produced force fields based on a polarizable atomic multipole description of biomolecular electrostatics. In this work, the Atomic Multipole Optimized Energetics for Biomolecular Applications (AMOEBA) force field is applied to restrained refinement of molecular models against X-ray diffraction data from peptide crystals. A new formalism is also developed to compute anisotropic and aspherical structure factors using fast Fourier transformation (FFT) of Cartesian Gaussian multipoles. Relative to direct summation, the FFT approach can give a speedup of more than an order of magnitude for aspherical refinement of ultrahigh-resolution data sets. Use of a sublattice formalism makes the method highly parallelizable. Application of the Cartesian Gaussian multipole scattering model to a series of four peptide crystals using multipole coefficients from the AMOEBA force field demonstrates that AMOEBA systematically underestimates electron density at bond centers. For the trigonal and tetrahedral bonding geometries common in organic chemistry, an atomic multipole expansion through hexadecapole order is required to explain bond electron density. Alternatively, the addition of inter­atomic scattering (IAS) sites to the AMOEBA-based density captured bonding effects with fewer parameters. For a series of four peptide crystals, the AMOEBA–IAS model lowered Rfree by 20–40% relative to the original spherically symmetric scattering model.

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Polarizable atomic multipole X-ray refinement: application to peptide crystals.

X-ray study of molecular oxygen adsorbed on graphite

The total and elastic x‐ray scattering intensities from the diatomic molecules CO, N2, and O2 have been calculated within the Waller–Hartree theory. Near Hartree–Fock quality wavefunctions with extensive basis sets at experimental Re values have been employed. The calculated intensities reflect the full accuracy of the published wavefunctions. The electron–nuclear interference terms have also been computed and tabulated so that studies of high energy electron scattering intensities can be made. An independent atom model has been constructed from atomic wavefunctions, which served as a starting basis for calculations of the molecular wavefunctions. It is found in the scattering range for which 4π sinϑ/λ?10 A−1, neither the total nor the elastic x‐ray intensities can be sensibly neglected compared to the electron–nuclear interference terms for the three diatomic molecules studied in this work.

X-ray and electron scattering from diatomic molecules in the first Born approximation

Generalized x‐ray scattering factors for H2 (1Σg+) and N2 (1Σg+) are reported for [2‖2] and [3‖3] expansions. Results for O2 (3Σg−) include [2‖2], [3‖3], and [4‖4] expansions. The multipole analysis of the molecular x‐ray form factor of these diatomics is a guide to a rational choice for the pseudoatom multipoles and for the radial basis functions in the Kohl–Bartell model [J. Chem. Phys. 51, 2891 (1969)]. It appears that by using more multipoles, relatively simple radial functions for the pseudoatoms are needed for a Kohl–Bartell model in the analysis of electron diffraction intensities for charge density information.

On the use of generalized x‐ray scattering factors for analysis of charge density from gas‐phase electron diffraction intensities

X‐ray and high energy electron scattering intensities by CO(1Σ+) have been calculated for an MCSCF wavefunction. These results are compared to previous studies with an SCF wavefunction. It is found that both one‐electron and two‐electron parts of the scattering intensities undergo small, but comparable changes with the introduction of electron correlation by the MCSCF wavefunction. Generalized x‐ray scattering factors for a [2‖2] expansion are also compared with a previous SCF result.

A study of calculated x‐ray and electron diffraction scattering intensities by CO(1Σ+): Electron correlation effects

Ground‐state vibrational averages of the intensities of x‐rays and high‐energy electrons elastically scattered by N2 (1Σ+g), CO (1Σ+), BF (1Σ+), and FH (1Σ+) have been calculated When nonrigid contributions are included, the vibrational averages are accurately computed from generalized x‐ray scattering factors at the [2‖2] level Maximum anharmonic contributions to the average intensities are 03%–05% Contributions from nonrigid pseudoatom components are generally smaller by an order of magnitude and are negligible for S greater than 10 A−1

Vibrational averages of x‐ray and high energy electron scattering intensities from diatomic molecules

The electric field gradients at the two distinct nitrogen nuclei of imidazole have been calculated from the charge density derived from Bragg diffraction data for crystalline inidazole at 103 K. The resulting 14N coupling constants and asymmetry parameters were N1: e 2 q Q/h = −2.6(0.3) MHz, η = 0.4(0.2); N3: e 2 q Q/h = −2.2(0.3) MHz, η = 0.4(0.2). In contrast with the corresponding results obtained from NQR experiments, these calculations could not distinguish between the two nitrogen sites. At each site the parameters were approximately equal to the mean of the NQR values for N1 and N3. Model calculations for the diatomic molecules N2 and BF suggest that the accuracy with which field gradients are obtained directly from the Bragg structure factors depends critically on the elimination of quadrupole components induced by vibrational averaging. Estimates of the coupling constants and asymmetry parameters for imidazole obtained directly from the Bragg structure factors were N1: e 2 q Q/h = −1.4(0.3) MHz, η = 0.2(0.3); N3: e 2 q Q/h = −0.9(0.3) MHz, η = 0.7(0.5). Detailed analysis showed that the magnitudes of the coupling constants estimated in this way are too small due to residual vibrationally induced quadrupole components.

Joel Epstein

Papers

X-ray and electron scattering from diatomic molecules in the first Born approximation

On the use of generalized x‐ray scattering factors for analysis of charge density from gas‐phase electron diffraction intensities

A study of calculated x‐ray and electron diffraction scattering intensities by CO(1Σ+): Electron correlation effects

Vibrational averages of x‐ray and high energy electron scattering intensities from diatomic molecules

Electric field gradients in imidazole at 103 K from x-ray diffraction