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

Generalized x‐ray scattering factors have been extended to include a 3d Slater‐type atomic orbital The new scattering functions include deformations that represent octapole and hexadecapole atomic charge distributions These terms are suitable for charge density analysis of atoms with high site symmetry For a reasonable exponential parameter, the 3d orbital does not introduce significantly different dipole and quadrupole radial scattering factors which have previously been constructed with 2s and 2p Slater‐type orbital products A new monopole term (M‐shell scattering factor) is significantly different from the L‐shell scattering factor, but it may be difficult to measure the corresponding population parameter from x‐ray diffraction data The valence scattering model is extended to second row atoms by restricting deformation terms to M‐shell Slater‐type orbitals Neglect of L‐shell deformation in the x‐ray scattering analysis of second row atoms may not be appropriate; the matter warrants further study

Electron population analysis with generalized x‐ray scattering factors: Higher multipoles

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

It is assumed that the one‐electron density function for a molecular crystal can be determined from x‐ray diffraction data with the use of generalized x‐ray scattering factors. The relationship between several physical properties and electron population parameters are given. These include molecular dipole moments and atomic electric field gradients, which can be compared to results from other experiments. The physical properties that follow from electron population analysis can serve as criteria in judging the reliability of x‐ray diffraction data for charge density information.

Valence Structure from X‐Ray Diffraction Data: Physical Properties

Generalized x‐ray scattering factors, in a simplified form, have been applied to several organic molecular crystals. Atomic charge parameters for L‐shell scattering factors have been determined from x‐ray structure factors by the method of least squares. L‐shell scattering factors have been computed for products of SCF AO's and of standard molecular STO's (Slater‐type orbitals). Standard STO's are more satisfactory for point‐charge analysis than are SCF AO's. Bias from several sets of thermal parameters is rather small and does not alter general conclusions on valence structure. The experimental x‐ray charges are compared with Mulliken gross atomic populations from INDO and STO‐3G calculations. Agreement with theory is good for the molecular crystals of s‐triazine, cyanuric acid, and uracil. Within the point‐charge approximation, the experimental x‐ray dipole moment for uracil is 4.0 ± 1.3 D. The direction is at an angle 71° ± 12° from the N(1)–C(4) interatomic vector towards atom N(3).

Valence Structure from X‐Ray Diffraction Data: An L‐Shell Projection Method

The detailed distribution of electrons in molecules can be broken down into atomic orbital populations if an atomic orbital basis is used for a molecular orbital wave function (LCAO). A total assignment of all electrons in a molecule to atomic orbitals (and hence to particular atoms) can be made by finding gross populations as suggested by Mulliken. These are of some interest in discussing various molecular properties associated with charge density, particularly if a minimal basis is used consisting of only atomic orbitals in the valence and inner shells. It has now become possible to evaluate atomic orbital populations for a range of molecules both by semi-empirical and ab initio methods. The semi-empirical methods (based on zero differential overlap) utilize some experimental atomic data, but can be applied to large systems with little difficulty. Ab initio methods generally require the evaluation of a large number of difficult integrals, but they are becoming available for many small systems. The main aim of this paper is to compare atomic electron populations obtained by the semi-empirical INDO molecular orbital theory with some values from full SCF calculations using a minimal basis set of exponential-type functions. Agreement is moderately good, but only if the exponents in the full calculations are chosen carefully to minimize the total energy.

Robert F. Stewart

Papers

Electron population analysis with generalized x‐ray scattering factors: Higher multipoles

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

Valence Structure from X‐Ray Diffraction Data: Physical Properties

Valence Structure from X‐Ray Diffraction Data: An L‐Shell Projection Method

Atomic electron populations by molecular orbital theory