A new algorithm is presented for obtaining points on a steepest descent path from the transition state of the reactants and products. In mass‐weighted coordinates, this path corresponds to the intrinsic reaction coordinate. Points on the reaction path are found by constrained optimizations involving all internal degrees of freedom of the molecule. The points are optimized so that the segment of the reaction path between any two adjacent points is given by an arc of a circle, and so that the gradient at each point is tangent to the path. Only the transition vector and the energy gradients are needed to construct the path. The resulting path is continuous, differentiable and piecewise quadratic. In the limit of small step size, the present algorithm is shown to take a step with the correct tangent vector and curvature vector; hence, it is a second order algorithm. The method has been tested on the following reactions: HCN→CNH, SiH2+H2→SiH4, CH4+H→CH3+H2, F−+CH3F→FCH3+F−, and C2H5F→C2H4+HF. Reaction paths calculated with a step size of 0.4 a.u. are almost identical to those computed with a step size of 0.1 a.u. or smaller.

An improved algorithm for reaction path following

A new internally contracted direct multiconfiguration–reference configuration interaction (MRCI) method is described which allows the use of much larger reference spaces than any previous MRCI method. The configurations with two electrons in the external orbital space are generated by applying pair excitation operators to the reference wave function as a whole, while the singly external and internal configurations are standard uncontracted spin eigenfunctions. A new efficient and simple method for the calculation of the coupling coefficients is used, which is well suited for vector machines, and allows the recalculation of all coupling coefficients each time they are needed. The vector H⋅c is computed partly in a nonorthogonal configuration basis. In order to test the accuracy of the internally contracted wave functions, benchmark calculations have been performed for F−, H2O, NH2, CH2, CH3, OH, NO, N2, and O2 at various geometries. The deviations of the energies obtained with internally contracted and uncontracted MRCI wave functions are mostly smaller than 1 mH and typically 3–5 times smaller than the deviations between the uncontracted MRCI and the full CI. Dipole moments, electric dipole polarizabilities, and electronic dipole transition moments calculated with uncontracted and contracted MRCI wave functions also are found to be in close agreement. The efficiency of the method is demonstrated in large scale calculations for the CN, NH3, CO2, and Cr2 molecules. In these calculations up to 3088 reference configurations and up to 154 orbitals were employed. The biggest calculation is equivalent to an uncontracted MRCI with more than 78 million configurations.

An efficient internally contracted multiconfiguration–reference configuration interaction method

A modified conjugate gradient algorithm for geometry optimization is outlined for use with ab initioMO methods. Since the computation time for analytical energy gradients is approximately the same as for the energy, the optimization algorithm evaluates and utilizes the gradients each time the energy is computed. The second derivative matrix, rather than its inverse, is updated employing the gradients. At each step, a one-dimensional minimization using a quartic polynomial is carried out, followed by an n-dimensional search using the second derivative matrix. By suitably controlling the number of negative eigenvalues of the second derivative matrix, the algorithm can also be used to locate transition structures. Representative timing data for optimizations of equilibrium geometries and transition structures are reported for ab initioSCF–MO calculations.

Optimization of equilibrium geometries and transition structures

A scheme that reduces the calculations of ground-state properties of systems of interacting particles exactly to the solution of single-particle Hartree-type equations has obvious advantages. It is not surprising, then, that the density functional formalism, which provides a way of doing this, has received much attention in the past two decades. The quality of the energy surfaces calculated using a simple local-density approximation for exchange and correlation exceeds by far the original expectations. In this work, the authors survey the formalism and some of its applications (in particular to atoms and small molecules) and discuss the reasons for the successes and failures of the local-density approximation and some of its modifications.

The density functional formalism, its applications and prospects

A contracted Gaussian basis set capable of describing about 63% of the correlation energy of N2 has been used in a detailed configuration-interaction calculation. Second-order perturbation theory overestimated the correlation energy by 23–50% depending on how H0 was chosen. Pair-pair interaction affects the correlation energy by about 20% while quadruple excitations have an 8% effect.

Configuration interaction calculations on the nitrogen molecule

Molecular properties of the first-row diatomic hydrides, calculated with accurate configuration-interaction wave functions, are presented Ground-state functions, constructed from approximate natural orbitals, were determined at the equilibrium internuclear separation for each molecule The basis sets used, were capable of reproducing recently published self-consistent-field energies to within 0003 hartrees A minimum of 70% of the correlation energy and an error in the dipole moment of less than 3% were obtained in a systematic study of each molecule To attain this accuracy, large numbers of singly and doubly excited configurations were used in the wave functions

Studies in Configuration Interaction: The First-Row Diatomic Hydrides

A Natural Orbital Based Energy Calculation for Helium Hydride and Lithium Hydride

As a prelude to the study of energy transfer in the HF–HF system, the potential energy surface for the interaction of two rigid HF molecules has been calculated within the ab initio self‐consistent‐field framework. An H(4s 1p/2s 1p), F(9s 5p 1d/4s 2p 1d) basis set of contracted Gaussian function was employed. The number of unique points on the surface is greatly reduced by symmetry, and only 294 points were required to give a fairly complete description of the four‐dimensional surface. Parts of the surface are illustrated by a series of contour maps. Some preliminary attempts to fit the surface to an analytic form are described. The equilibrium geometry of (HF)2 is predicted.

/pdf/interaction-potential-between-two-rigid-hf-molecules-3ag4r86bex.pdf

Interaction potential between two rigid HF molecules

Rigorous quantum mechanical calculations have been carried out for about 150 linear and 200 non‐linear geometries for the FH2 system. The contracted Gaussian basis set used consisted of four s and two p functions on fluorine and two s functions on hydrogen. The barrier height and exothermicity are poorly predicted by single configuration self‐consistent‐field calculations. However, the 214‐configuration correlated results are in qualitative agreement with experiment (low barrier height and substantial exothermicity). The reaction coordinate is discussed, and pictures of the potential surface are presented. A second series of calculations is being carried out with a larger basis set. These latter calculations yield nearly quantitative agreement with experiment for both the barrier height and exothermicity.

Potential Energy Surface Including Electron Correlation for the Chemical F + H2 → FH + H I. Preliminary Surface

The low‐lying electronic states of the Be2 molecule have been examined using MO SCF CI with a basis set of elliptical orbitals. The ground state of the molecule was found to be repulsive in the best calculation. The excited states all seem to be bound with potential curves similar to that predicted for Be2+. Two new techniques for improving the convergence of configuration interaction are introduced and applied to the ground‐state calculation.

Charles F. Bender

Papers

Studies in Configuration Interaction: The First-Row Diatomic Hydrides

A Natural Orbital Based Energy Calculation for Helium Hydride and Lithium Hydride

Interaction potential between two rigid HF molecules

Potential Energy Surface Including Electron Correlation for the Chemical F + H2 → FH + H I. Preliminary Surface

Theoretical Calculation of the Potential Curves of the Be2 Molecule