A least-squares procedure is described for modeling an empirical transmission surface as sampled by multiple symmetry-equivalent and/or azimuth rotation-equivalent intensity measurements. The fitting functions are sums of real spherical harmonic functions of even order, Ylm(-UO) + Ylm(Ul), 2 < l = 2n _< 8. The arguments of the functions are the components of unit direction vectors, -u0 for the reverse incident beam and Ul for the scattered beam, referred to crystal-fixed Cartesian axes. The procedure has been checked by calculations against standard absorption test data.

Experience with a variety of diffraction data-reduction problems has led to several strategies for dealing with mismeasured outliers in multiply measured data sets. Key features of the schemes employed currently include outlier identification based on the val-

For crystal structures analyzed by both X-ray and neutron diffraction, the anisotropic mean-square displacement parameters of the non-H atoms are sometimes found to differ significantly. The differences can usually be adjusted by either: (1) an isotropic factor q, defined by UijX = qUijN, to correct for a temperature difference between the two experiments; (2) anisotropic factors qij, defined by UijX = qijUijN, to correct for a temperature difference and different anisotropic diffraction effects of absorption, extinction, thermal diffuse scattering, multiple reflection, or systematic measuring errors in the two experiments; (3) anisotropic diffraction correction terms delta Uij, defined by UijX = UijN + delta Uij; (4) the sum of an isotropic temperature correction and anisotropic diffraction corrections, defined by UijX = qUijN + delta Uij. Correction parameters q, qij and delta Uij are easily calculated by linear least-squares fit, and the corrections from (3) or (4) seem to be the most reliable. Corrections calculated from X-ray and neutron Uij's of the non-H atoms of a crystal can be useful for adjusting the neutron Uij's of the H atoms for adoption, along with the neutron coordinates of the H atoms, as fixed parameters in an X-ray analysis of the electron density distribution.

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The charge density distribution of a protein has been refined experimentally. Diffraction data for a crambin crystal were measured to ultra-high resolution (0.54 Å) at low temperature by using shortwavelength synchrotron radiation. The crystal structure was refined with a model for charged, nonspherical, multipolar atoms to accurately describe the molecular electron density distribution. The refined parameters agree within 25% with our transferable electron density library derived from accurate single crystal diffraction analyses of several amino acids and small peptides. The resulting electron density maps of redistributed valence electrons (deformation maps) compare quantitatively well with a high-level quantum mechanical calculation performed on a monopeptide. This study provides validation for experimentally derived parameters and a window into charge density analysis of biological macromolecules.T he electronic charge density distribution of a molecule carries information (1) that determines its intermolecular interactions. For example, the charge distribution of an enzyme complements that of the substrate it recognizes and binds. The electrostatic potential and electric moments derivable from the charge density (1-3) provide maps that can guide the design of molecules for specified interactions. Furthermore, powerful insights into the nature and strength of hydrogen bonding and ionic interactions result from analysis of the electron density gradient and Laplacian (4-6). Extension of such analyses to proteins would permit a unique understanding of the driving forces between biomolecules as well as the subtleties of enzymatic reactions (7).Experimental electron density distributions are obtained by analysis of single-crystal x-ray diffraction data measured to ultra-high resolution, typically to a diffraction resolution limit d min Ϸ 0.5 Å (1,8,9). The crystallographic studies usually map and analyze the deformation density, which is the difference between the actual electron density of the molecule and the density calculated for the promolecule, a molecular superposition of spherical, neutral, i.e., free, atoms. The deformation density thus reveals the redistribution of valence electron density caused by chemical bonding and intermolecular interactions and also is used to calibrate theoretical electron density calculations (10). However, a difficulty in crystallography is the separation of the anisotropic atomic mean-square displacements from the static molecular electron distribution (11). Proper experimental deconvolution requires very accurate diffraction data to ultrahigh resolution. Thus, charge density studies have so far been limited to small-unit-cell crystals, and proteins still await study.We have shown (12, 13) that effective thermal displacement deconvolution and meaningful deformation density distributions can be achieved for larger structures at lower resolution by transferring average experimental electron density parameters. We have built a database of such parameters derived from ultra-high ...