A consistent set of ab initio effective core potentials (ECP) has been generated for the main group elements from Na to Bi using the procedure originally developed by Kahn. The ECP’s are derived from all‐electron numerical Hartree–Fock atomic wave functions and fit to analytical representations for use in molecular calculations. For Rb to Bi the ECP’s are generated from the relativistic Hartree–Fock atomic wave functions of Cowan which incorporate the Darwin and mass–velocity terms. Energy‐optimized valence basis sets of (3s3p) primitive Gaussians are presented for use with the ECP’s. Comparisons between all‐electron and valence‐electron ECP calculations are presented for NaF, NaCl, Cl2, Cl2−, Br2, Br2−, and Xe2+. The results show that the average errors introduced by the ECP’s are generally only a few percent.

Ab initio effective core potentials for molecular calculations. Potentials for main group elements Na to Bi

A fifth-order perturbation comparison of electron correlation theories

A general procedure is introduced for calculation of the electron correlation energy, starting from a single Hartree–Fock determinant. The normal equations of (linear) configuration interaction theory are modified by introducing new terms which are quadratic in the configuration coefficients and which ensure size consistency in the resulting total energy. When used in the truncated configuration space of single and double substitutions, the method, termed QCISD, leads to a tractable set of quadratic equations. The relation of this method to coupled‐cluster (CCSD) theory is discussed. A simplified method of adding corrections for triple substitutions is outlined, leading to a method termed QCISD(T). Both of these new procedures are tested (and compared with other procedures) by application to some small systems for which full configuration interaction results are available.

Quadratic configuration interaction. A general technique for determining electron correlation energies

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 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

It is shown that the convergence of the configuration expansion of a many‐electron wavefunction may be drastically improved without significantly complicating the energy matrix elements by using partially non‐orthogonal orbitals. The wavefunction thus obtained is the many‐electron analog to the natural orbital expansion of a two‐electron wavefunction. The orbitals involved may be approximated by pseudonatural orbitals (``PNO–CI''), and an efficient way of calculating these orbitals is presented. An approximate treatment of multiple substitutions is discussed yielding a coupled electron pair theory formulated within the CI scheme. These methods are applied to the ground state and three ionized states of methane. Including only double substitutions, the computed upper bound to the energy of the ground state is −40.4584 hartree, i.e. 0.057 above the experimental nonrelativistic energy. About 83% of the total correlation energy is accounted for variationally, whereas the coupled‐pair approximation yields 89%....

PNO–CI Studies of electron correlation effects. I. Configuration expansion by means of nonorthogonal orbitals, and application to the ground state and ionized states of methane

In the present approach the high reliability of ab initio techniques is combined with the easily amenable phenomenological core polarization concept for an efficient treatment of intershell correlation effects in all‐electron SCF and valence CI calculations. By use of only a single adjustable atomic parameter, which is related to the radius of the core and determines the cutoff at short range, our effective core polarization potential (CPP) accounts quantitatively for dynamical intershell correlation as well as exclusion effects on the correlation energy of the core. The applications refer to alkali and alkaline earth atoms (Li to K and Be to Ca) and a detailed analysis is performed for core polarization effects on ionization energies, electron affinities, oscillator strengths, polarizabilities, van der Waals coefficients, the valence electron density, and spin densities. Very accurate results are obtained for well‐known energetic properties and spin densities at the nucleus. With respect to the other app...

Treatment of intershell correlation effects in ab initio calculations by use of core polarization potentials. Method and application to alkali and alkaline earth atoms

Dipole moments and static dipole polarizabilities are calculated for neon and the molecules HF, H2O, NH3, CH4 and CO from SCF and correlated wavefunctions. The construction of appropriate gaussian-type basis sets is discussed and the convergence of the correlation contributions to the polarizability is analysed. The effect of vibrational averaging is also investigated. The polarizabilities as obtained from the coupled electron pair approximation (CEPA) with the most extended basis sets differ from experimental values by less than 1·5 per cent in all cases. The calculated polarizability anisotropies appear to be correct to about 5–15 per cent. The correlation contributions to the polarizabilities are found to vary from 3 to 12 per cent.

PNO-CI and PNO-CEPA studies of electron correlation effects

A systematic investigation of the ground state potential curves and dipole moment functions has been performed for the diatomic hydrides LiH to HCl on the basis of variational configuration interaction wavefunctions PNO–CI and the coupled electron pair approximation CEPA. The basis sets of Gaussian‐type orbitals are derived by simple rules from optimized atomic sets available in the literature. Between 95% (LiH) and 85% (HCl) of the valence shell correlation energies are accounted for in the CEPA calculations. Core–valence intershell correlation has been included for several members of each row, and its effect on the potential curves is analyzed. Comparison of the spectroscopic constants derived from the CEPA potential curves with experiment shows a high reliability of the theoretical values. The standard deviations over both rows are as follows: re:0.003 A, ωe:14 cm−1, αe:0.005 cm−1, and ωexe:1.5 cm−1. The errors of the calculated dissociation energies go up to 0.3 eV but behave very regularly. They are believed to allow for predictions which are correct to about ±0.05 eV. The following new D0 values are recommended (empirical values in parenthesis): NH:3.40 (3.2±0.16); NaH:1.88 (2.05±0.2); MgH:1.23 (2.0±0.5); PH:3.02 (3.5±0.3). Various vibrational matrix elements have been calculated from the dipole moment curves. The μ0 values show errors of 0.02–0.04 D. The differences between diagonal elements as well as the transition elements for the first few levels deviate by only about 10% from the available experimental data.

PNO–CI and CEPA studies of electron correlation effects. III. Spectroscopic constants and dipole moment functions for the ground states of the first‐row and second‐row diatomic hydrides

A quadratically convergent MCSCF method is described which allows one to optimize an energy average of several states with arbitrary weight factors. An analysis of the problems connected with the variational determination of excited states is given and it is concluded that the averaging method is a natural solution to these problems. In the energy expansion minimized in each iteration, certain cubic and higher order terms can be included. It is demonstrated that this greatly facilitates convergence in cases where the Hessian matrix of second energy derivatives has many negative eigenvalues. Several approximations to the exact quadratically convergent scheme, which are useful when calculating potential surfaces, are discussed.

Wilfried Meyer

Papers

PNO–CI Studies of electron correlation effects. I. Configuration expansion by means of nonorthogonal orbitals, and application to the ground state and ionized states of methane

Treatment of intershell correlation effects in ab initio calculations by use of core polarization potentials. Method and application to alkali and alkaline earth atoms

PNO-CI and PNO-CEPA studies of electron correlation effects

PNO–CI and CEPA studies of electron correlation effects. III. Spectroscopic constants and dipole moment functions for the ground states of the first‐row and second‐row diatomic hydrides

A quadratically convergent MCSCF method for the simultaneous optimization of several states