Interaction energies for the water dimer by supermolecular methods and symmetry-adapted perturbation theory: the role of bond functions and convergence of basis subsets

 Using a systematic series of basis sets in supermolecular and symmetry-adapted intermolecular perturbation theory calculations it is examined how interaction energies of various water dimer structures change upon addition and shifting of bond functions. Their addition to augmented double- and triple-zeta basis sets brings the sum of the electron correlation contributions to the second-order interaction energy nearly to convergence, while accurate first-order electrostatic and exchange contributions require better than augmented quadruple-zeta quality. A scheme which combines the different perturbation energy contributions as computed in different basis subsets performs uniformly well for the various dimer structures. It yields a symmetry-adapted perturbation theory value of −21.08 kJ/mol for the energy of interaction of two vibrationally averaged water molecules compared to −21.29 kJ/mol when the full augmented triple-zeta basis set is used throughout. Received: 4 November 1999 / Accepted: 8 February 2000 / Published online: 12 May 2000     

Cyanovinyl radical: an illustration of the poor performance of unrestricted perturbation theory and density functional theory procedures in calculating radical stabilization energies

Stabilization energies for the 1-cyanovinyl radical (CH₂=CCN) have been calculated using a variety of conventional ab initio (M?ller–Plesset, quadratic configuration interaction and coupled-cluster) and density functional theory (B-LYP, B3-LYP) procedures, as well as with a range of compound methods. Compared with a high-level benchmark value (that predicts a stabilization energy of 17.1 kJ mol⁻¹), UMP2 and UMP4 give the wrong sign and magnitude of the stabilization energy (both methods predicting desta- bilization instead of stabilization), while B-LYP and B3-LYP overestimate the degree of stabilization. The RMP2, RMP4, QCISD(T) and CCSD(T) techniques, and several, but not all, variants of G2 and CBS theories give radical stabilization energies in good agreement with the benchmark value. Received: 15 June 1998 / Accepted: 19 August 1998 /  Published online: 15 February 1999     

Molecular integrals by numerical quadrature. I. Radial integration

 This article presents a numerical quadrature intended primarily for evaluating integrals in quantum chemistry programs based on molecular orbital theory, in particular density functional methods. Typically, many integrals must be computed. They are divided up into different classes, on the basis of the required accuracy and spatial extent. Ideally, each batch should be integrated using the minimal set of integration points that at the same time guarantees the required precision. Currently used quadrature schemes are far from optimal in this sense, and we are now developing new algorithms. They are designed to be flexible, such that given the range of functions to be integrated, and the required precision, the integration is performed as economically as possible with error bounds within specification. A standard approach is to partition space into a set of regions, where each region is integrated using a spherically polar grid. This article presents a radial quadrature which allows error control, uniform error distribution and uniform error reduction with increased number of radial grid points. A relative error less than 10⁻¹⁴ for all s-type Gaussian integrands with an exponent range of 14 orders of magnitude is achieved with about 200 grid points. Higher angular l quantum numbers, lower precision or narrower exponent ranges require fewer points. The quadrature also allows controlled pruning of the angular grid in the vicinity of the nuclei. Received: 30 August 2000 / Accepted: 21 December 2000 / Published online: 3 April 2001     

Convergence of multipole expanded intermolecular interaction energies for Gaussian-type-function and Slater-type-function basis sets

 It is shown that the multipole expansion of each order of the polarization series converges for large enough intermolecular distances when finite basis sets of Gaussian or Slater-type functions are used to approximate molecular response properties. Convergence of the multipole expansion for each order of the polarization series does not imply convergence of the polarization series itself. A corresponding convergence condition is extracted from the general perturbation theory in a finite-dimensional space and is applied to the H + H⁺ problem. Received: 29 September 1999 / Accepted: 22 May 2000 / Published online: 18 August 2000     

Inexpensive and accurate predictions of optical excitations in transition-metal complexes: the TDDFT/PBE0 route

Time-dependent density functional theory (TDDFT) is applied to calculate vertical excitation energies of three representative transition metal complexes. The computational model (PBE0) is obtained by combining the Perdew-Burke-Erzenrhof (PBE) generalized gradient functional with a predetermined amount of exact exchange. Our results show that the TDDFT/PBE0 model represents a cheap and reliable tool for the computation of optical excitations for transition metal complexes. Received: 8 August 2000 / Accepted: 7 September 2000 / Published online: 23 November 2000     

Nature and importance of three-body interactions in the (H2O)2HCl trimer

 The nature and importance of nonadditive three-body interactions in the (H₂O)₂HCl cluster have been studied by the supermolecule coupled-cluster method and by symmetry-adapted perturbation theory (SAPT). The convergence of the SAPT expansion was tested by comparison with the results obtained from the supermolecule coupled-cluster calculations including single, double, and noniterative triple excitations [CCSD(T)]. It is shown that the SAPT results reproduce the converged CCSD(T) results within 3% at worst. The SAPT method has been used to analyze the three-body interactions for various geometries of the (H₂O)₂HCl cluster. It is shown that the induction nonadditivity is dominant, but it is partly quenched by the first-order Heitler–London-type exchange and higher-order exchange–induction/deformation terms. This implies that the classical induction term alone is not a reliable approximation to the nonadditive energy and that it will be difficult to approximate the three-body potential for (H₂O)₂HCl by a simple analytical expression. The three-body energy represents as much as 21–27% of the pair CCSD(T) intermolecular energy. Received: 15 September 1999 / Accepted: 3 February 2000 / Published online: 2 May 2000     

Harmonic frequency scaling factors for Hartree-Fock, S-VWN, B-LYP, B3-LYP, B3-PW91 and MP2 with the Sadlej pVTZ electric property basis set

Scaling factors for obtaining fundamental vibrational frequencies from harmonic frequencies calculated at six of the most commonly used levels of theory have been determined from regression analysis for the polarized-valence triple-zeta (pVTZ) Sadlej electric property basis set. The Sadlej harmonic frequency scaling factors for first- and second-row molecules were derived from a comparison of a total of 900 individual vibrations for 111 molecules with available experimental frequencies. Overall, the best performers were the hybrid density functional theory (DFT) methods, Becke's three-parameter exchange functional with the Lee–Yang–Parr fit for the correlation functional (B3-LYP) and Becke's three-parameter exchange functional with Perdew and Wang's gradient-corrected correlation functional (B3-PW91). The uniform scaling factors for use with the Sadlej pVTZ basis set are 0.9066, 0.9946, 1.0047, 0.9726, 0.9674 and 0.9649 for Hartree–Fock, the Slater–Dirac exchange functional with the Vosko–Wilk–Nusair fit for the correlation functional (S-VWN), Becke's gradient-corrected exchange functional with the Lee–Yang–Parr fit for the correlation functional (B-LYP), B3-LYP, B3-PW91 and second-order M?ller–Plesset theory with frozen core (MP2(fc)), respectively. In addition to uniform frequency scaling factors, dual scaling factors were determined to improve the agreement between computed and observed frequencies. The scaling factors for the wavenumber regions below 1800 cm⁻¹ and above 1800 cm⁻¹ are 0.8981 and 0.9097, 1.0216 and 0.9857, 1.0352 and 0.9948, 0.9927 and 0.9659, 0.9873 and 0.9607, 0.9844 and 0.9584 for Hartree–Fock, S-VWN, B-LYP, B3-LYP, B3-PW91 and MP2(fc), respectively. Hybrid DFT methods along with the Sadlej pVTZ basis set provides reliable theoretical vibrational spectra in a cost-effective manner. Received: 22 May 2000 / Accepted: 30 August 2000 / Published online: 28 February 2001     

Second-order energy components in basis-set-superposition-error-free intermolecular perturbation theory

 Recently a basis-set-superposition-error-free second-order perturbation theory was introduced based on the “chemical Hamiltonian approach” providing the full antisymmetry of all wave functions by using second quantization. Subsequently, the “Heitler–London” interaction energy corresponding to the sum of the zero- and first-order perturbational energy terms was decomposed into different physically meaningful components, like electrostatics, exchange and overlap effects. The first-order wave function obtained in the framework of this perturbation theory also consists of terms having clear physical significance: intramolecular correlation, polarization, charge transfer, dispersion and combined polarization–charge transfer excitations. The second-order energy, however, does not represent a simple sum of the respective contributions, owing to the intermolecular overlap. Here we propose an approximate energy decomposition scheme by defining some “partial Hylleraas functionals” corresponding to the different physically meaningful terms of the first-order wave functions. The sample calculations show that at large and intermediate intermolecular distances the total second-order intermolecular interaction energy contribution is practically equal to the sum of these “physical” terms, while at shorter distances the overlap-caused interferences become of increasing importance. Received: 18 June 2001 / Accepted: 28 August 2001 / Published online: 16 November 2001     

Grid-free DFT implementation of local and gradient-corrected XC functionals

Following an approach to density functional theory calculations based on the matrix representation of operators, we implemented a scheme as an alternative to traditional grid-based methods. These techniques allow integrals over exchange-correlation operators to be evaluated through matrix manipulations. Both local and gradient-corrected functionals can be treated in a similar way. After deriving all the required expressions, selected examples with various functionals are given. Received: 7 March 1998 / Accepted: 21 May 1998 / Published on line: 6 August 1998     

Degenerate symmetry-adapted perturbation theory of weak interactions between closed- and open-shell monomers: application to Rydberg states of helium hydride

Symmetry-adapted perturbation theory is extended to the (quasi) degenerate, open-shell case. The new formalism is tested in calculations of the interaction energies for a helium atom in the ground state interacting with an excited hydrogen atom. It is shown that the method gives satisfactory results if the coupling with higher Rydberg states of the dimer is small, as is the case for the A²Σ⁺,B²Π,E²Π,3²Π, and 1²Δ states of HeH. For the C²Σ⁺ state convergence of the method is very slow, but it can be improved by including the n=3 states in the model space. Received: 3 June 1998 / Accepted: 9 September 1998 / Published online: 7 December 1998     

Recognizing a triple bond between main group atoms

The electron pair density, in conjunction with the theory of an atom in a molecule, enables one to unambiguously determine the nature of the bonding between the gallium atoms in bent [HGa-GaH]²⁻. The Ga-Ga bonding in the dianion at the experimental bond length is found to be the result of the sharing of two electron pairs at the Hartree-Fock level of theory, the level consistent with the Lewis model of the electron pair. Received: 27 July 2000 / Accepted: 4 October 2000 / Published online: 19 January 2001     

A density functional theory test study on the N2··· He dimer

 In the present contribution we report a study of the weakly bound van der Waals N₂–He molecule in the framework of the supermolecule approach by means of the PWPW and mPW1PW exchange–correlation functionals, using density functional theory local-spin-optimized atom-centered basis sets complemented with bond functions optimized at the mPW1PW level of theory. Calculations show that the mPW1PW functional using bond functions gives a realistic representation of the interaction-energy potentials for this van der Waals dimer, comparable to reference M?ller–Plesset perturbation theory calculations. In contrast, the PWPW functional is unable to describe the bonding properties of this system and all values of the bonding properties obtained at different geometries with this functional are considered out-of-scale compared with the rest of the calculations presented in this study. Received: 30 October 2000 / Accepted: 3 January 2001 / Published online: 3 April 2001     

Generalization of coupled-cluster response theory to multireference expansion spaces: application of the coupled-cluster singles and doubles effective Hamiltonian

Employing separate cluster ansatz in time-independent and time-dependent wave-operators, coupled-cluster (CC) response theory is generalized to multireference (MR) expansion spaces. For state energies, this corresponds to the MR secular problem with an arbitrary similarity-transformed effective Hamiltonian, H˜=Ω⁻¹ HΩ. The effective Hamiltonian can be generated via size-extensive CC methods. Thus the states in MR linear response theory (MRLRT) maintain the usual CC core-extensive properties. We have used the Gelfand unitary group basis of the spin-adapted configurations to construct the matrix of H˜ in the MR excitation space. As a preliminary application, the CC singles and doubles effective Hamiltonian is applied to excitation and photoionization energies of the CH⁺ and N₂ molecules, and is compared with experimental results and results from other numerical procedures including conventional CC linear response theory (CC-LRT), MR and full configuration interaction (MRCI and FCI) methods. The numerical results indicate that MRLRT reproduces valence and external excited states quantitatively, combining the best features of CC-LRT and MRCI. Received: 2 July 1998 / Accepted: 28 August 1998 / Published online: 11 November 1998     

Some fundamental problems with zero flux partitioning of electron densities

A number of fundamental problems with the topological analysis of molecular electron densities using the atoms in molecules (AIM) theory developed by Bader and coworkers are highlighted. In particular, contrary to statements made in the literature, we show that the local zero flux condition used in the AIM theory to define an atom in a molecule does not follow from the Schwinger variation principle, nor does it define unambiguously the atomic domains. Serious limitations of the definition of an atom in AIM theory also arise due to vibrational effects. A general definition of the structure of a molecular isomer based on a generalisation of the Born-Oppenheimer potential energy surface allow these limitations to be overcome. Received: 9 March 2000 / Accepted: 13 August 2000 / Published online: 21 December 2000     

Perturbation calculation of the interaction energy using orthogonalized orbitals

The diagrammatic Rayleigh-Schr?dinger perturbation theory for the interaction of two closed-shell systems is developed up to the third order of pertur-bation using orthogonalized orbitals. The interaction energy is expressed by the Rayleigh-Schr?dinger perturbation expansion. A simple approach for the estimation of basis set superposition error is introduced. The preliminary calculations of the intermolecular interactions for the He dimer within the augmented cc-pVTZ basis set are compared with the supermolecular approach, perturbation calculation in biorthogonal basis sets and symmetry adapted perturbation theory results. Received: 17 December 1996 / Accepted: 5 November 1997     

A theoretical study of the copper–cysteine bond in blue copper proteins

 The accuracy of theoretical calculations on models of the blue copper proteins is investigated using density functional theory (DFT) Becke's three-parameter hybrid method with the Lee–Yang–Parr correlation functional (B3LYP) and medium-sized basis sets. Increasing the basis set to triple-zeta quality with f-type functions on all heavy atoms and enlarging the model [up to Cu(imidazole-CH₃)₂(SC₂H₅) (CH₃SC₂H₅)^0/+] has only a limited influence on geometries and relative energies. Comparative calculations with more accurate wave-function–based methods (second-order M?ller–Plesset perturbation theory, complete-active-space second-order perturbation theory, coupled-cluster method, including single and double replacement amplitudes and in addition triple replacement perturbatively) and a variety of basis sets on smaller models indicate that the DFT/B3LYP approach gives reliable results with only a small basis set dependence, whereas the former methods strongly depend on the size of the basis sets. The effect of performing the geometry optimizations in a continuum solvent is quite small, except for the flexible Cu-S_Met bond. The results of this study confirm the earlier results that neither the oxidized nor the reduced copper site in the blue proteins is strained to any significant degree (in energy terms) by the protein surrounding. Received: 7 July 2000 / Accepted: 17 November 2000 / Published online: 21 March 2001     

The equivalence of bond-energy schemes and the determination of resonance energies

 It is argued that the preservation of algebraic equivalence between the Allen and Laidler bond-energy schemes for nonconjugated alkenes logically determines that the Allen scheme should apply to a classical structure of a conjugated hydrocarbon exactly as it stands, i.e. no additional parameters are needed. Extending the requirement of equivalence to conjugated alkenes implies that, in the Laidler scheme, the bond energy of the pure single CC bond in a conjugated system is a combination of the bond-energies of the semiconjugated and normal CC single bonds: E(C_d—C_d)=2E(C_d—C)−E(C—C). This result is a deduction and is not an independent hypothesis. The equivalence of the two schemes for conjugated hydrocarbons is demonstrated numerically, by calculating the resonance energies of some selected molecules by both methods. Received: 5 December 1999 / Accepted: 5 March 2000 / Published online: 5 June 2000     

A theoretical study of the N8 cubane to N8 pentalene isomerization reaction

The isomerization reaction of cubic N₈ to the planar bicyclic structure analogous to pentalene has been investigated using multiconfigurational self-consistent field and second-order perturbation theory (CASPT2). Comparative calculations using density functional theory have also been performed. Five local minima on the energy surface have been found, and the transition states between each two consecutive minima have been determined. The results show that all steps in the isomerization process, except one, can proceed via a set of transition states with moderately high energy barriers (10–20 kcal/mol). Received: 9 December 1996 / Accepted: 18 February 1997     

Theoretical study of the valence π → π* excited states of polyacenes: anthracene and naphthacene

The valence π → π ^* excited states of anthracene and naphthacene are studied with multireference perturbation theory with complete active space self-consistent field reference functions. The predicted spectra provide a consistent assignment of all one- and two-photon spectra and T-T spectra of low-lying valence π → π ^* excited states of anthracene and naphthacene. The present theory predicts the valence π → π ^* excitation energies with an accuracy of 0.15 eV for anthracene and of 0.25 eV or better for naphthacene. The excited states of anthracene and naphthacene are compared with those of benzene and naphthalene studied previously. The present calculations predict that, going from anthracene to naphthacene, there is a symmetry reversal of the two lowest singlet state transitions, but not for the triplet, just as indicated by the experimental data. Some general trends of polyacene excited states are discussed based on the calculated results for benzene to naphthacene. Conclusive results obtained for anthracene and naphthacene can be used as a model for understanding the excited states of larger polyacenes. Received: 22 April 1998 / Accepted: 6 July 1998 / Published online: 28 September 1998