On the theory of electron correlation in atoms and molecules: Relation between cluster expansion theory and the correlated wave functions method

A relation between the cluster expansion theory of many electron wave functions and the correlated wave functions method is established. In this way, the theoretical basis of the method is elucidated and the approximations involved in its application become apparent. General forms of the correlated wave function, differing in certain important respects from that form usually assumed, are derived.     

Two-electron integrations in the quantum theory of atoms in molecules with correlated wave functions

A recent method proposed to compute two-electron integrals over arbitrary regions of space [Martin Pendas, A. et al., J Chem Phys 2004, 120, 4581] is extended to deal with correlated wave functions. To that end, we use a monadic factorization of the second-order reduced density matrix originally proposed by E. R. Davidson [Chem Phys Lett 1995, 246, 209] that achieves a full separation of the interelectronic components into one-electron terms. The final computational effort is equivalent to that found in the integration of a one determinant wave function with as many orbitals as occupied functions in the correlated expansion. Similar strategies to extract the exchange and self-interaction contributions from the two-electron repulsion are also discussed, and several numerical results obtained in a few test systems are summarized.     

Hylleraas method for many‐electron atoms. II. Integrals over wave functions with one interelectronic coordinate

A procedure is proposed to evaluate matrix elements containing r linked with angular functions. Using this procedure, the different types of two‐, three‐, and four‐electron radial and angular integrals that appear in a five‐electron atom, in the case of only one r_ij coordinate per basis function, are written in a compact form, separated in radial coordinates of one electron. The general formulas with which to obtain the integrals for powers ν ≥ 1 are developed, based on the orthogonality of the Legendre polynomials. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Ensemble representable densities for atoms and molecules. I. General theory

A method to obtain ensemble representable densities from experimental diffraction data is proposed. The method uses ab initio molecular densities instead of the commonly employed one-electron orbital densities, and as a result, few parameters need to be optimized in the fitting procedure to the experimental structure factors. The optimized coefficients can provide information about intra- and intermolecular electronic correlations, spin-orbit coupling, etc. This work also provides new explicit formulas to determine the rank of a fermionic wave function, i.e., the rank of the one-fermion density matrix. © 1995 John Wiley & Sons, Inc.     

General variation method for the relativistic calculations of atoms and molecules

Use of the general variation method of Weinstein and MacDonald for the relativistic calculation of atoms and molecules is proposed. It is shown from the numerical calculations for hydrogenlike atomic systems that this method is useful in judging an accuracy of energies and wave functions obtained with a relativistic Hamiltonian whose spectra are not bounded. It is also shown that this method can be used to find spurious solutions such as 1p_½ or 2d_3/2 appearing in atomic systems. Problems in extending the method to many-electron atoms and molecules are discussed.     

The matrix theory of electron spin multiplets for atoms and molecules

In this article a matrix method for the construction of spin multiplets (spinconfigurations) is suggested in order to solve the multielectron problem for atoms and mulecules by means of configuration interaction.A simple graphical way is given to enumerate configurations and to break their set into subsets of configurations related to the given projection of the total spin of a system S _z . It is found that all matrices in the theory of spin multiplets are convex and in cases of two, three, and four electrons are broken into blocks of an order no higher than 3.The model of the solution of the multielectron Schrödinger equation, in which the total spin of core electrons is zero, is considered. In this model the construction of linear combinations of configurations is reduced to the construction of those for but valence electrons.     

Benchmark calculations with correlated molecular wave functions XII. Core correlation effects on the homonuclear diatomic molecules B2-F2

Using systematic sequences of the newly developed correlation consistent core-valence basis sets from cc-pCVDZ through cc-pCV6Z, the spectroscopic constants of the homonuclear diatomic molecules containing first row atoms, B–F, are calculated both with and without inclusion of 1s correlation. Internally contracted multireference configuration interaction (IC-MRCI) and singles and doubles coupled cluster (CCSD) theory with a perturbational estimate of connected triple excitations, CCSD(T), have been investigated. By exploiting the convergence of the correlation consistent basis sets, complete basis set (CBS) limits have been estimated for total energies, dissociation energies, equilibrium geometries, and harmonic frequencies. Based on the estimated CBS limits the effects of 1s correlation on D _e (kcal/mol), r _e (?), and ω _e (cm⁻¹) are: +1.1, −0.0070, +10 for B₂; +1.5, −0.0040, +13 for C₂; +0.9, −0.0020, +9 for N₂; +0.3, −0.0020, +6 for O₂; and −0.1, −0.0015, +1 for F₂. Received: 20 January 1997 / Accepted: 6 May 1997     

Locally correlated equation-of-motion coupled cluster theory for the excited states of large molecules

We report an extension of the local correlation concept to electronically excited states via the equation-of-motion coupled cluster singles and doubles (EOM-CCSD) method. We apply the same orbital domain structure used successfully for ground-state CCSD by Werner and co-workers and find that the resulting localized excitation energies are in error generally by less than 0.2 eV relative to their canonical EOM-CCSD counterparts, provided the basis set is flexible and includes Rydberg-like functions. In addition, we account for weak-pair contributions efficiently using a correction to local-EOM-CCSD transition energies based on the perturbative (D) correction used with configuration interaction singles (CIS).     

Nonempirical wave functions for very large molecules. II. The PRDDO/M/FCP method

A variation of the frozen-core potential (FCP) method is developed and implemented within the modified version of the method of partial retention of diatomic differential overlap (PRDDO/M). The explicit treatment of core electrons is replaced with a potential based on the actual core-valence integrals rather than upon an arbitrary model potential. The core-valence orthogonality requirement is replaced by an energy shift operator. PRDDO/M/FCP calculations exhibit good agreement with ab initio calculations with the same basis set, while reducing the computational cost significantly. © 1997 John Wiley & Sons, Inc.     

Revisiting the calculation of condensed Fukui functions using the quantum theory of atoms in molecules

The analysis of previously reported shortcomings of the condensed Fukui functions obtained making use of the quantum theory of atoms in molecules indicates these drawbacks are due to the inadequacy of the definition employed to compute them and not to the partitioning. A new procedure, which respects the mathematical definition and solves these problems, is presented for the calculation of condensed Fukui functions for atomic basins defined according to the quantum theory of atoms in molecules. It is tested in a set of 18 molecules, which includes the most controversial reported cases.     

Full optimization of Jastrow-Slater wave functions with application to the first-row atoms and homonuclear diatomic molecules

We pursue the development and application of the recently introduced linear optimization method for determining the optimal linear and nonlinear parameters of Jastrow-Slater wave functions in a variational Monte Carlo framework. In this approach, the optimal parameters are found iteratively by diagonalizing the Hamiltonian matrix in the space spanned by the wave function and its first-order derivatives, making use of a strong zero-variance principle. We extend the method to optimize the exponents of the basis functions, simultaneously with all the other parameters, namely, the Jastrow, configuration state function, and orbital parameters. We show that the linear optimization method can be thought of as a so-called augmented Hessian approach, which helps explain the robustness of the method and permits us to extend it to minimize a linear combination of the energy and the energy variance. We apply the linear optimization method to obtain the complete ground-state potential energy curve of the C(2) molecule up to the dissociation limit and discuss size consistency and broken spin-symmetry issues in quantum Monte Carlo calculations. We perform calculations for the first-row atoms and homonuclear diatomic molecules with fully optimized Jastrow-Slater wave functions, and we demonstrate that molecular well depths can be obtained with near chemical accuracy quite systematically at the diffusion Monte Carlo level for these systems.     

On the use of explicitly correlated functions in variational computations for small molecules

The explicitly correlated wave functions used in variational molecular calculations are reviewed. Different types of such functions are considered. The state of art and future perspectives are briefly discussed. © 1994 John Wiley & Sons, Inc.     

Calculation of the average properties of atoms in molecules. II

This article describes an algorithm for the calculation of the average properties of an atom in a molecule. The atom is defined within the topological theory of molecular structure, a theory which defines atoms, bonds, structure, and structural stability in terms of the topological properties of a system's charge distribution. The average properties of the atom so defined are uniquely determined by quantum mechanics. Results for a number of hydrocarbon molecules, obtained by the program PROAIM (properties of atoms in molecules) which implements this algorithm, are given. In general, this program enables one to calculate the average energy of an atom in a molecule to an accuracy of ±1 kcal/mol.     

Shannon information entropy of fractional occupation probability as an electron correlation measure in atoms and molecules

A new method to determine electron correlation energy is presented for atoms and molecules. This method is based on Shannon information entropy that is obtained by fractional occupation probabilities of natural atomic orbitals. It is indicated that the Shannon entropy increases as the number of electrons increases and thus can be considered as a possible measure for the electron correlation in atomic and molecular systems. For neutral atoms and singly charged positive ions we proposed an expression for correlation energy with explicit dependence on the Shannon entropy and atomic number. The obtained correlation energies have been used to compute the first ionization potentials of the ground state of the main group elements from hydrogen through krypton. The calculated ionization potentials are in reasonably good agreement with their corresponding experimental values.We also developed the additivity scheme to find a connection between Shannon entropy and molecular correlation energy. The estimated molecular correlation energies show an excellent agreement with those obtained by elaborate G3 method with R² = 0.990.     

Differential density matrix overlap: an index for assessment of electron correlation in atoms and molecules

Summary A new index, called the differential density matrix overlap (DDMO), is proposed for assessment of the electron correlation effects in atoms and molecules. DDMO can be easily calculated as the negative value of the correlation energy derivative with respect to the relative position of the occupied and virtual orbitals. DDMO is transparent to physical interpretation. It can serve as a tool for analyzing the accuracy of approximate electron correlation methods and the validity of the Hartree-Fock wavefunction as the zeroth-order approximation. The properties of DDMO are discussed using test calculations on 11 atoms and molecules as an example.     

Everyman's derivation of the theory of atoms in molecules

This paper demonstrates that it is straightforward to develop the theory of an atom in a molecule--the extension of quantum mechanics to an open system--by deriving the necessary equations of motion from Schr?dinger's equation, followed by a comparison of the predicted properties with experiment to determine the correct boundary condition. Although less fundamental than the variational derivation of the quantum theory of atoms in molecules, this heuristic approach makes the quantum mechanics of an atom in a molecule accessible to "everyman" possessing a knowledge of Schr?dinger's equation, aiding its general acceptance by experimental chemists.