W. J. Hehre

Self—Consistent Molecular Orbital Methods. XII. Further Extensions of Gaussian—Type Basis Sets for Use in Molecular Orbital Studies of Organic Molecules

Two extended basis sets (termed 5–31G and 6–31G) consisting of atomic orbitals expressed as fixed linear combinations of Gaussian functions are presented for the first row atoms carbon to fluorine. These basis functions are similar to the 4–31G set [J. Chem. Phys. 54, 724 (1971)] in that each valence shell is split into inner and outer parts described by three and one Gaussian function, respectively. Inner shells are represented by a single basis function taken as a sum of five (5–31G) or six (6–31G) Gaussians. Studies with a number of polyatomic molecules indicate a substantial lowering of calculated total energies over the 4–31G set. Calculated relative energies and equilibrium geometries do not appear to be altered significantly.

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

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. I. Use of Gaussian Expansions of Slater‐Type Atomic Orbitals

Least‐squares representations of Slater‐type atomic orbitals as a sum of Gaussian‐type orbitals are presented. These have the special feature that common Gaussian exponents are shared between Slater‐type 2s and 2p functions. Use of these atomic orbitals in self‐consistent molecular‐orbital calculations is shown to lead to values of atomization energies, atomic populations, and electric dipole moments which converge rapidly (with increasing size of Gaussian expansion) to the values appropriate for pure Slater‐type orbitals. The ζ exponents (or scale factors) for the atomic orbitals which are optimized for a number of molecules are also shown to be nearly independent of the number of Gaussian functions. A standard set of ζ values for use in molecular calculations is suggested on the basis of this study and is shown to be adequate for the calculation of total and atomization energies, but less appropriate for studies of charge distribution.

Self‐Consistent Molecular Orbital Methods. IV. Use of Gaussian Expansions of Slater‐Type Orbitals. Extension to Second‐Row Molecules

Least‐squares representations of the 3s and 3p Slater‐type atomic orbitals by a small number of Gaussian functions are presented. The use of these Gaussian representations in self‐consistent molecular orbital calculations extends our previous study to molecules containing second row elements. Calculated atomization energies, electric dipole moments, and atomic charges are shown to rapidly converge (with increasing number of Gaussians) to their Slater limits. Results of valence shell optimization studies on a series of second‐row compounds are nearly independent of the level of the Gaussian approximation, and they allow a set of standard molecular ξ exponents to be proposed.

Self‐Consistent Molecular Orbital Methods. V. Ab Initio Calculation of Equilibrium Geometries and Quadratic Force Constants

Ab initio calculation of equilibrium geometries and quadratic force constants for a large group of first‐row polyatomic molecules has been carried out, using the previously described [J. Chem. Phys. 51, 2657 (1969)] STO–3G approximation for STO basis functions. The average deviation of calculated and experimental bond lengths and angles is 0.035 A and 1.7°, respectively. Nearly all important experimental trends are reproduced. For a few cases involving bonds between electronegative atoms, significant discrepancies are found. Quadratic force constants are evaluated for symmetric stretching and bending modes and are found to be overestimated, typically by 20%–30%. Nearly all experimental trends are satisfactorily accounted for. It is concluded that a minimal STO basis with properly chosen orbital exponents offers a useful and computationally efficient model for potential surface studies.

Self‐Consistent Molecular Orbital Methods. XIV. An Extended Gaussian‐Type Basis for Molecular Orbital Studies of Organic Molecules. Inclusion of Second Row Elements

We have recently proposed an extended basis of atomic functions expressed as fixed linear combinations of Gaussian functions for hydrogen and the first row atoms [Ditchfield, Hehre, and Pople, J. Chem. Phys. 54, 724 (1971); Hehre and Pople, J. Chem. Phys., 56, 4233 (1972)]. We now extend our treatment to consider atoms of the second row. This basis set is close to minimal in that each inner shell atomic orbital is represented by a single function and only valence shell orbitals (3s, 3p) are extended, these being split into inner and outer parts. No functions of higher quantum number than are necessary in the ground state atom are included. We have considered the effect of molecular environment on the size of the valence shell orbitals and have proposed a ``standard set of scaling factors.

Self‐Consistent Molecular Orbital Methods. VI. Energy Optimized Gaussian Atomic Orbitals

Minimal basis atomic orbitals expressed as sums of N Gaussian functions are presented for hydrogen and for the first row atoms boron to fluorine. The expansion coefficients and Gaussian exponents are determined by minimizing the total calculated energy of the atomic ground state. For expansion lengths of up to six Gaussians, two sets of atomic orbitals are reported. In the first set, which we describe as unconstrained, different Gaussian exponents are used for the 2s and 2p atomic orbitals. In the second set, the 2s and 2p atomic orbitals are constrained to share the same Gaussian exponents. It is shown that this constraint, which produces a significant gain in computational speed in molecular calculations, does not seriously reduce the quality of the atomic orbitals for given N. A comparison of the contracted sets presented here with previous studies on uncontracted basis sets for the first row atoms, shows that the uncontracted Gaussian exponents are a poor approximation to those of the contracted funct...

Self‐Consistent Molecular‐Orbital Methods. III. Comparison of Gaussian Expansion and PDDO Methods Using Minimal STO Basis Sets

Self‐consistent‐field molecular‐orbital calculations over a minimal basis set of Slater‐type atomic orbitals are presented for a set of organic molecules and positive ions containing up to eight first‐row atoms. The necessary molecular integrals are calculated by two previously introduced schemes: the Gaussian expansion (STO–KG) method and the projection of diatomic‐differential‐overlap (PDDO) method. Atomization energies, electric dipole moments, density matrices, optimum STO ζ exponents, and computation times are compared for the PDDO, STO‐3G, and STO‐4G methods, the latter of which has previously been shown to closely reproduce the full STO results. Relative to the STO‐4G values, the PDDO method leads to errors of up to 0.22 a.u. in the atomization energy, 0.16D for the dipole moment, and 0.05 for the optimum ζ exponents. The corresponding limits for the STO‐3G method are 0.06 a.u., 0.07D, and 0.02. Two electron integrals are evaluated at rates of 125–175, 25–140, and 10–70 integrals per second for the...

Self‐Consistent Molecular‐Orbital Methods. VIII. Molecular Studies with Least Energy Minimal Atomic Orbitals

A minimal set of energy‐optimized contracted Gaussian orbitals is used for single‐determinant molecular‐orbital studies of a set of small polyatomic molecules. Optimized scale factors for valence atomic orbitals are determined by energy variation and are found to be significantly greater than unity for hydrogen and carbon. The convergence of calculated properties with increasing size of Gaussian expansion is studied and found to be less rapid than in corresponding studies using a Slater‐type minimal basis. Theoretical equilibrium geometries are determined. The agreement with experimental geometries is resonable but somewhat less satisfactory than with a Slater‐type set.

Small Gaussian Expansions of Atomic Orbitals: Second‐Row Atoms

Gaussian expansions of Clementis SCF atomic orbitals for the second‐row atoms have been obtained by the method of least squares. Expansion lengths vary from two through six GTOs. Total energy evaluations of the least‐squares expansions reveals a rather slow convergence rate with increasing number of GTOs to the Hartree–Fock limit. The GTO AOs in this work have optimal space filling properties and are useful for rapid evaluation of generalized x‐ray scattering factors of second‐row atoms.