A contracted Gaussian basis set (6‐311G**) is developed by optimizing exponents and coefficients at the Mo/ller–Plesset (MP) second‐order level for the ground states of first‐row atoms. This has a triple split in the valence s and p shells together with a single set of uncontracted polarization functions on each atom. The basis is tested by computing structures and energies for some simple molecules at various levels of MP theory and comparing with experiment.

Self‐consistent molecular orbital methods. XX. A basis set for correlated wave functions

An extended basis set of atomic functions expressed as fixed linear combinations of Gaussian functions is presented for hydrogen and the first‐row atoms carbon to fluorine. In this set, described as 4–31 G, each inner shell is represented by a single basis function taken as a sum of four Gaussians and each valence orbital is split into inner and outer parts described by three and one Gaussian function, respectively. The expansion coefficients and Gaussian exponents are determined by minimizing the total calculated energy of the atomic ground state. This basis set is then used in single‐determinant molecular‐orbital studies of a group of small polyatomic molecules. Optimization of valence‐shell scaling factors shows that considerable rescaling of atomic functions occurs in molecules, the largest effects being observed for hydrogen and carbon. However, the range of optimum scale factors for each atom is small enough to allow the selection of a standard molecular set. The use of this standard basis gives theoretical equilibrium geometries in reasonable agreement with experiment.

Self‐Consistent Molecular‐Orbital Methods. IX. An Extended Gaussian‐Type Basis for Molecular‐Orbital Studies of Organic Molecules

A method of ‘‘natural population analysis’’ has been developed to calculate atomic charges and orbital populations of molecular wave functions in general atomic orbital basis sets. The natural analysis is an alternative to conventional Mulliken population analysis, and seems to exhibit improved numerical stability and to better describe the electron distribution in compounds of high ionic character, such as those containing metal atoms. We calculated ab initio SCF‐MO wave functions for compounds of type CH3X and LiX (X=F, OH, NH2, CH3, BH2, BeH, Li, H) in a variety of basis sets to illustrate the generality of the method, and to compare the natural populations with results of Mulliken analysis, density integration, and empirical measures of ionic character. Natural populations are found to give a satisfactory description of these molecules, providing a unified treatment of covalent and extreme ionic limits at modest computational cost.

Natural population analysis

Describes and discusses the use of theoretical models as an alternative to experiment in making accurate predictions of chemical phenomena. Addresses the formulation of theoretical molecular orbital models starting from quantum mechanics, and compares them to experimental results. Draws on a series of models that have already received widespread application and are available for new applications. A new and powerful research tool for the practicing experimental chemist.

AB INITIO Molecular Orbital Theory

Standard sets of supplementary diffuse s and p functions, multiple polarization functions (double and triple sets of d functions), and higher angular momentum polarization functions (f functions) are defined for use with the 6‐31G and 6‐311G basis sets. Preliminary applications of the modified basis sets to the calculation of the bond energy and hydrogenation energy of N2 illustrate that these functions can be very important in the accurate computation of reaction energies.

Self‐consistent molecular orbital methods 25. Supplementary functions for Gaussian basis sets

From the Waller−Hartreee theory, the total x−ray scattering of H2 has been calculated with the first 10 natural orbitals of Davidson and Jones. Vibrational averaging has not been included. The calculated intensities differ by as much as 5% from a similar calculation with the Davidson and Jones SCF orbital. The accuracy of the calculations is demonstrated by agreement of 〈r12−1〉 from Tavard’s sum rule with 〈r12−1〉 values given by Davidson and Jones. The coherent intensities from the first 10 natural orbitals are in close agreement with the SCF results. By correcting the calculated inelastic scattered intensities for relativistic effects, the total scattering intensities from the correlated wavefunction are in excellent agreement with the experimental values of Wollan. Corresponding SCF values are significantly different. The results for molecular hydrogen encourage us to suggest more accurate gas phase measurements of total x−ray scattering by other diatomic molecules in order to determine the two electron...

Total x‐ray scattering by H2

Single crystal x-ray diffraction analysis today has become an essential methodology for scientists in a wide range of disciplines that span solid-state physics, synthetic chemistry and molecular biology. Although x-ray diffraction was first exploited by the Braggs in 1913 for crystal structure determination,1 it has only been the last three decades that the method has become routine for establishing the atomic structure in a crystallographic unit cell. The advent of digital computers has made it possible to implement direct methods, the evaluation of relations between measured diffraction intensities and phases, for the assignment of phases to observed x-ray structure factor moduli. Accurate measurements of the diffracted intensities are now made with diffractometers that operate in an automatic mode with supporting software to select out the photon counts which are mostly due to elastic scattering. For many crystals of organic compounds, which have rather low Debye temperatures, it is routine in many labs to cool the sample to about 100 K with a cold stream of nitrogen and thereby measure Bragg diffraction intensities at these reduced temperatures to larger scattering angles than is possible at room (300 K) temperature. Thus full atomic resolution is often achieved. When the phase estimates of the structure factors are adequate for locating the atoms within ten or so picometers of their mean thermal positions, least squares programs are then used to adjust both atomic positions and mean square amplitudes of vibration by fitting a structure factor model to the “observed” structure factor moduli or their squares. In this latter stage of a crystal structure anlaysis, the observation to parameter ratio is greater than ten to one and, for the case of high resolution data, the ratio may exceed fifty to one. The very large redundancy - the larger number of observations to number of atomic parameters - in x-ray crystallography is what makes the experiment a definitive method for structure determination. Upon completion of a crystal structure determination, it is routine to report atomic positions with estimated standard deviations in the range of tenths of picometers. In some cases a dynamical analysis of the structure, such as a rigid body motion model, is included in an attempt to deduce the equilibrium structure of the atoms in the crystal. The final objective, a complete determination of the atomic arrangement with precise metrics in a crystal unit cell, has been met.

Electrostatic Properties of Molecules from Diffraction Data

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

Within the Waller–Hartree theory it is possible to map out the interelectron density function (the diagonal elements of the intracule matrix) from an analysis of total X-ray scattering intensities. Theoretical studies of He and H2 show that electron correlation effects substantially influence total X-ray scattering intensities, but do not appreciably alter the elastic X-ray scattering intensity. On the other hand, theoretical studies of Be(1S), Ne(1S) and CO(1Σ*) indicate that electron correlation influences both total and elastic X-ray scattering intensities by comparable amounts (∼ 1–2%), but at differing regions of the scattering vector variable.

Robert F. Stewart

Papers

Total x‐ray scattering by H2

Electrostatic Properties of Molecules from Diffraction Data

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

III. Total X-ray Scattering and Two-Electron Density Functions