Appendix C: DICTIONARY

  1. Atom. Smallest particle still characterizing a chemical element. It consists of a nucleus of a positive charge (Z is the number of protons in the atomic nucleus, also known as atomic number, and e the elementary charge) carrying almost all its mass (more than 99.9%) and Z electrons determining its size.
    [Source: IUPAC Compendium of Chemical Terminology. Gold Book, Version 2.3.3, 2014-02-24, p. 121.]

  2. Atomic charge. The charge attributed to an atom A within a molecule defined as ζ = ZA − qA, where ZA is the atomic number of A and qA is the electron density assigned to A. The method of calculation of qA depends on the choice of the scheme of partitioning electron density.
    [Source: IUPAC Compendium of Chemical Terminology. Gold Book, Version 2.3.3, 2014-02-24, p. 122.]

  3. Atomic form factor or atomic scattering factor. The atomic form factor, or atomic scattering factor, is a measure of the scattering amplitude of a wave by an isolated atom.[1"Atomic form factor"
    Wikipedia ("http://en.wikipedia.org/wiki/Atomic_form_factor").    , 2"X-Ray Form Factor, Attenuation, and Scattering Tables"
    Chantler, C. T.
    J. Phys. Chem. Ref. Data 2000, 29(4), 597-1048
    ("http://physics.nist.gov/PhysRefData/FFast/Text2000/contents2000.html").     ]

  4. Deformation density. The deformation electron density is the difference between the actual electron density and the theoretical electron density that would result if the molecule was composed of non-interacting atoms without bonds. The “actual” electron density can be calculated using either an experimental map from crystallography or a quantum calculation.
    [see also: "Chemical deformation densities. 1. Principles and formulation of quantitative determination", W. H. E. SchwarzK. Ruedenberg, L. Mensching J. Am. Chem. Soc. 1989, 111, 18, 6926-6933]

  5. Electron probability density. If ρ(x,y,z) dx dy dz is the probability of finding an electron in the volume element dV = dx dy dz at the point of a molecular entity with coordinates x, y, z then ρ(x,y,z) is the electron density at this point. For many purposes (e.g. X-ray scattering, forces on atoms) the system behaves exactly as if the electrons were spread out into a continuously distributed charge. The term has frequently been wrongly applied to negative charge population. [Source: IUPAC Compendium of Chemical Terminology. Gold Book, Version 2.3.3, 2014-02-24, p. 480.]
    Usually, it is simply referred to as electron density. Sometimes it is also called position electron density at a given point in 3-D space, to point out that it represents the number of electrons contained in an infinitesimally small volume at that point. Its units are e⋅bohr−3.

  6. Independent-atom model (IAM). The independent-atom model assumes that the atomic electron density is well described by the spherically averaged density of the isolated atom. It has been the basis of X-ray structure analysis since its inception. The IAM further assumes the atoms in a molecule (crystal) to be neutral. This assumption is contracticted by the fact that molecules have dipole and higher electrostatic moments.

    https://www.bruker.com/fileadmin/user_upload/8-PDF-Docs/X-rayDiffraction_ElementalAnalysis/SC-XRD/Webinars/Bruker_AXS_Non-Spherical_Electron_Densities_Using_Invarioms_Webinar_20120508.pdf https://www.uni-goettingen.de/en/charge-density-distribution-from-high-resolution-x-ray-diffraction-data/325052.html The model of a superposition of thermally delocalized atoms about mean positions is sometimes called the promolecule (Hirshfeld, F. Acta Crystallogr., Sect.B 1971, B27, 769-781) or the IAM (Fink, M. and Bonham, R. "High Energy Electron Scattering." Van Nostrand-Reihold, Princeton, New Jersey, 1974).

  7. Molecular entity. Any constitutionally or isotopically distinct atom, molecule, ion, ion pair, radical, radical ion, complex, conformer etc., identifiable as a separately distinguishable entity. Molecular entity is used in the "IUPAC Compendium of Chemical Terminology" as a general term for singular entities, irrespective of their nature, while chemical species stands for sets or ensembles of molecular entities. Note that the name of a compound may refer to the respective molecular entity or to the chemical species, e.g. methane, may mean a single molecule of CH4 (molecular entity) or a molar amount, specified or not (chemical species), participating in a reaction. The degree of precision necessary to describe a molecular entity depends on the context. For example 'hydrogen molecule' is an adequate definition of a certain molecular entity for some purposes, whereas for others it is necessary to distinguish the electronic state and/or vibrational state and/or nuclear spin, etc. of the hydrogen molecule.
    [Source: IUPAC Compendium of Chemical Terminology. Gold Book, Version 2.3.3, 2014-02-24, p. 950.]

  8. Molecule. An electrically neutral entity consisting of more than one atom (n > 1).
    [Source: IUPAC Compendium of Chemical Terminology. Gold Book, Version 2.3.3, 2014-02-24, p. 958.]

  9. Planar electron density. The planar electron density at a given point on a line is the number of electrons in a plane perpendicular to that line in that point. Its units are e⋅bohr−1.

  10. Position electron density. The position electron density at a given point in 3-D space is the number of electrons contained in an infinitesimally small volume at that point. Its units are e⋅bohr−3. Usually, it is simply referred to as electron density. Sometimes it is also called volumetric electron density

  11. Promolecule.The promolecule is an ideal reference system made up by non-interacting atoms held fixed at the same positions they have in the corresponding real molecule. As the charge density of an isolated atom is spherically symmetric, the corrisponding promolecular density is just the sum of spherically-averaged atomic charge densities, each centred on the coordinates of the corresponding nucleus. In other words, the promolecular density is a reference electron density prior to molecule formation [Spackman, M. A.; Maslen, E. N. J. Phys. Chem. 1986, 90, 2020-2027]. It is a well-defined quantum mechanical entity, derived from a trial wavefunction consisting of non-interacting atomic wavefunctions. The term promolecule was originally used by Hirshfeld and Rzotkiewicz [Hirshfeld, F. L.; Rzotkiewicz, S. Mol. Phys. 1974, 27, 1319-1343]. It is equivalent to the IAM model in electron scattering [Bonham, R. A,; Fink, M. High Energy Electron Scattering; Van Nostrand-Reinhold: New York, 1974.] and the spherical atom approximation in X-ray crystallography [Hirshfeld, F. L. Isr. J. Chem. 1977, 16, 87-229.].

  12. Pseudoatoms or Stewart atoms. "Stewart atoms are the unique nuclear-centered spherical functions whose sum best fits a molecular electron density in a least-squares sense".[3 (a) "Extraction of Stewart Atoms from Electron Densities"
    Gill, P. M. W. J. Phys. Chem. 1996, 100, 15421-15427.
     
    (b) "Methods for constructing Stewart atoms"
    Gilbert, A. T. B.; Lee, A. M.; Gill, P. M. W. J. Mol. Struct. (Theochem) 2000, 500, 363-374.     ]

    In Stewart's generalized scattering factor approach, [4 (a) "Generalized X‐Ray Scattering Factors"
    Stewart, R. F. J. Chem. Phys. 1969, 51, 4569-4577;
    (b) "Generalized x‐ray scattering factors in diatomic molecules"
    Stewart, R. F.; Bentley, J.; Goodman, B. J. Chem. Phys. 1975, 63, 3786-3793;
    (c) "Testing aspherical atom refinements on small-molecule data sets"
    Hansen, N. K.; Coppens, P. Acta Cryst. 1978, A34, 909-921;
    (d) Coppens, P. X-ray Charge Densities and Chemical Bonding; Oxford University Press: New York, 1997.     ] the ED about each atomic center is represented by a finite multipole expansion, called pseudoatom:[5 "Electron population analysis with rigid pseudoatoms"
    Stewart, R. F. Acta Cryst. 1976, A32, 564-574.    ]

    ρa(ra) = fa(ra)†pa

    The elements of vector pa are the multipole populations of the pseudoatom a and the elements of vector fa(ra) are nuclear-centered solid harmonics functions which account explicitly for the dependence of the ED on the spacial coordinates ra =  |xa, ya, za|†. The nuclear weight functions, wa(r), are implicitly assumed to be equal 1 everywhere if the multipole expansion is centered on atom a and zero otherwise.

    Atomic populations pa are the main outcome of least-squares multipole analysis programs, like VALRAY[6 Stewart, R. F.; Spackman, M. A.; Flensburg, C.
    VALRAY User's Manual (Version 2.1),Carnegie-Mellon University, Pittsburg, and University of Copenhagen, Copenhagen, 2000.    ] and XD.[7 Koritsanszky, T.; Mallison, P.; Macchi, P.; Volkov, A.; Gatti, C.; Richter, T.; Farrugia, L.:
    XD2006. A Computer Program Package for Multipole Refinement, Topological Analysis of Charge Densities and Evaluation of Intermolecular Energies from Experimental or Theoretical Structure Factors (2007).    ] PAMoC is interfaced to both VALRAY and XD and, as a consequence, it can use their multipole analysis of experimental ED's to evaluate atomic and molecular properties, including, for instance, inner and outer moments and electrostatic interaction energies. The variance-covariance matrix of the best values of the least-squares populations is used to estimate standard uncertainties on all the derived quantities, as a measure of the reliability of the pseudoatom model.

    Though Gill has shown that Stewart atoms can be extracted directly from theoretical ED's,[3a "Extraction of Stewart Atoms from Electron Densities"
    Gill, P. M. W. J. Phys. Chem. 1996, 100, 15421-15427.    ] most applications of pseudoatom partitioning of theoretical ED's are based on the least-squares multipole refinement of theoretical structure factors.[4b"Generalized x‐ray scattering factors in diatomic molecules"
    Stewart, R. F.; Bentley, J.; Goodman, B. J. Chem. Phys. 1975, 63, 3786-3793.    , 8 (a) "A simple quantitative model of hydrogen bonding" Spackman, M. A.
    J. Chem. Phys. 1986, 85, 6587-6601.
    (b) "Influence of intermolecular interactions on multipole-refined electron densities"
    Spackman, M. A.; Byrom, P. G.; Alfredsson, M.; Hermansson, K. Acta Cryst. 1999, A55, 30-47.
    (c) "Aspherical-atom scattering factors from molecular wave functions. 1. Transferability and conformation dependence of atomic electron densities of peptides within the multipole formalism"
    Koritsanszky, T.; Volkov, A.; Coppens, P. Acta Cryst. 2002, A58, 464-472.
    (d) "Ab Initio Quality Electrostatic Atomic and Molecular Properties Including Intermolecular Energies from a Transferable Theoretical Pseudoatom Databank"
    Volkov, A.; Li, X.; Koritsanszky, T.; Coppens, P. J. Chem. Phys. A 2004, 108, 4283-4300.    ] PAMoC is able to perform least-squares multipole refinement of theoretical ED's in 3D space. This is the default procedure, whenever the IDF is a 3D grid file. Otherwise, when the IDF is an AIMPAC wavefunction file, the keyword lsq is needed, in order to instruct PAMoC to build a 3D grid of ED values and then to perform a least-squares refinement.

    Least-squares refinement of experimental and theoretical ED's provide a set of pseudoatom populations in the form of spherical harmonics tensors, that can be easily converted into traceless cartesian moments by a linear transformation. However a volume integration is needed to evaluated unabridged cartesian moments. This can be achieved by means of keyword stw.

  13. Radial electron density. The radial electron density at a given radius is the number of electrons in a infinitesimally thin spherical shell at that radius. Its units are e⋅bohr−1.

  14. Spherical Atom Kappa Formalism. See Coppen's book, page 55.

References