Traditional Wave Function Methods Research Articles

Cage versus Prism: Electronic Energies of the Water Hexamer

Recent experiments show that the cage isomer of the water hexamer is lower in energy than the prism isomer near 0 K, and yet state-of-the-art electronic structure calculations predict the prism to be lower in energy than the cage at 0 K. Here, we study the relative energies of the water hexamers from the parametric two-electron reduced density matrix (2-RDM) method in which the 2-RDM rather than the wave function is the basic variable of the calculations. In agreement with experiment and in contrast with traditional wave function methods, the 2-RDM calculations predict the cage to be more stable than the prism after vibrational zero-point correction. Multiple configurations from the hydrogen bonding are captured by the method. More generally, the results are consistent with our previous 2-RDM applications in that they reveal how multireference correlation in molecular systems is important for resolving small energy differences from hydrogen bonding as well as other types of intermolecular forces, even in systems that are not conventionally considered strongly correlated.

Strong Correlation in Acene Sheets from the Active-Space Variational Two-Electron Reduced Density Matrix Method: Effects of Symmetry and Size

Polyaromatic hydrocarbons (PAHs) are a class of organic molecules with importance in several branches of science, including medicine, combustion chemistry, and materials science. The delocalized π-orbital systems in PAHs require highly accurate electronic structure methods to capture strong electron correlation. Treating correlation in PAHs has been challenging because (i) traditional wave function methods for strong correlation have not been applicable since they scale exponentially in the number of strongly correlated orbitals, and (ii) alternative methods such as the density-matrix renormalization group and variational two-electron reduced density matrix (2-RDM) methods have not been applied beyond linear acene chains. In this paper we extend the earlier results from active-space variational 2-RDM theory [Gidofalvi, G.; Mazziotti, D. A. J. Chem. Phys. 2008, 129, 134108] to the more general two-dimensional arrangement of rings--acene sheets--to study the relationship between geometry and electron correlation in PAHs. The acene-sheet calculations, if performed with conventional wave function methods, would require wave function expansions with as many as 1.5 × 10(17) configuration state functions. To measure electron correlation, we employ several RDM-based metrics: (i) natural-orbital occupation numbers, (ii) the 1-RDM von Neumann entropy, (iii) the correlation energy per carbon atom, and (iv) the squared Frobenius norm of the cumulant 2-RDM. The results confirm a trend of increasing polyradical character with increasing molecular size previously observed in linear PAHs and reveal a corresponding trend in two-dimensional (arch-shaped) PAHs. Furthermore, in PAHs of similar size they show significant variations in correlation with geometry. PAHs with the strictly linear geometry (chains) exhibit more electron correlation than PAHs with nonlinear geometries (sheets).

Strong electron correlation in the decomposition reaction of dioxetanone with implications for firefly bioluminescence

Dioxetanone, a key component of the bioluminescence of firefly luciferin, is itself a chemiluminescent molecule due to two conical intersections on its decomposition reaction surface. While recent calculations of firefly luciferin have employed four electrons in four active orbitals [(4,4)] for the dioxetanone moiety, a study of dioxetanone [F. Liu et al., J. Am. Chem. Soc. 131, 6181 (2009)] indicates that a much larger active space is required. Using a variational calculation of the two-electron reduced-density-matrix (2-RDM) [D. A. Mazziotti, Acc. Chem. Res. 39, 207 (2006)], we present the ground-state potential energy surface as a function of active spaces from (4,4) to (20,17) to determine the number of molecular orbitals required for a correct treatment of the strong electron correlation near the conical intersections. Because the 2-RDM method replaces exponentially scaling diagonalizations with polynomially scaling semidefinite optimizations, we readily computed large (18,15) and (20,17) active spaces that are inaccessible to traditional wave function methods. Convergence of the electron correlation with active-space size was measured with complementary RDM-based metrics, the von Neumann entropy of the one-electron RDM as well as the Frobenius and infinity norms of the cumulant 2-RDM. Results show that the electron correlation is not correctly described until the (14,12) active space with small variations present through the (20,17) space. Specifically, for active spaces smaller than (14,12), we demonstrate that at the first conical intersection, the electron in the σ(∗) orbital of the oxygen-oxygen bond is substantially undercorrelated with the electron of the σ orbital and overcorrelated with the electron of the carbonyl oxygen's p orbital. Based on these results, we estimate that in contrast to previous treatments, an accurate calculation of the strong electron correlation in firefly luciferin requires an active space of 28 electrons in 25 orbitals, beyond the capacity of traditional multireference wave function methods.

Isomerization of nitrosomethane to formaldoxime: Energies, geometries, and frequencies from the parametric variational two-electron reduced-density-matrix method

The isomerization of nitrosomethane to trans-formaldoxime is treated with the parametric variational two-electron reduced-density-matrix (2-RDM) method. In the parametric 2-RDM method, the ground-state energy is minimized with respect to a 2-RDM that is parameterized to be both size extensive and nearly N-representable. The calculations were performed with an efficient version of the 2-RDM method that we developed as an extension of the PSI3 ab initio package. Details of the implementation, which scales like configuration interaction with single and double excitations, are provided as well as a comparison of two optimization algorithms for minimizing the energy functional. The conversion of nitrosomethane to trans-formaldoxime can occur by one of two pathways: (i) a 1,3-sigmatropic hydrogen shift or (ii) two successive 1,2-sigmatropic hydrogen shifts. The parametric 2-RDM method predicts that the reaction channel involving two sequential 1,2-shifts is about 10 kcal/mol more favorable than the channel with a single 1,3-shift, which is consistent with calculations from other ab initio methods. We computed geometric parameters and harmonic frequencies for each stationary point on the reaction surfaces. Transition-state energies, geometries, and frequencies from the 2-RDM method are often more accurate than those from traditional wave function methods of a similar computational cost. Although electronic-structure methods generally agree that the 1,2-shift is more efficient, the energy ordering of the reactant nitrosomethane and the 1,2-shift intermediate formaldonitrone is unresolved in the literature. With an extrapolation to the complete-basis-set limit the parametric 2-RDM method predicts formaldonitrone to be very slightly more stable than nitrosomethane.

Activation energies of sigmatropic shifts in propene and acetone enolate from the anti-Hermitian contracted Schrödinger equation

The hydrogen [1,3]-sigmatropic shift in propene is predicted by the Woodward-Hoffman rules to occur by an antarafacial pathway, yet the lack of experimental evidence suggests that this pathway is not favorable. Two natural questions arise: (i) can the [1,3]-shift be made more favorable by a symmetry-forbidden multistep pathway, and (ii) can the energetics be influenced by a substituent on propene? As in many chemical reactions, describing the energetics of these reactions requires a balanced treatment of both single-reference and multireference electron correlations, and yet traditional wave function methods often excel in treating only one kind of correlation. An equitable description of correlation effects, however, can be achieved, at a cost similar to efficient single-reference methods, by computing the two-electron reduced density matrix (2-RDM) from the anti-Hermitian part of the contracted Schrödinger equation (ACSE) [D. A. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006)]. As with the contracted Schrödinger equation, the indeterminacy of the ACSE is removed without the many-electron wave function by reconstructing the 3-RDM from the 2-RDM via cumulant theory [D. A. Mazziotti, Chem. Phys. Lett. 289, 419 (1998)]. In this paper we apply the ACSE to study sigmatropic shifts in both propene and acetone enolate while extending its formalism to treat doublet spin states. In the 6-311G(**) basis set the ACSE predicts the activation energy of the trimethylene-to-propene rearrangement to be 8.8 kcal/mol while multireference perturbation theory yields a smaller barrier of 2.2 kcal/mol and coupled cluster singles-doubles predicts a negative barrier. We further find that the [1,3]-shift in acetone enolate is more favorable by approximately 30 kcal/mol than the [1,3]-shift in propene, which is consistent with a prior theoretical investigation as well as experimental observations of these shifts in 2-butanone enolate.

Long-range–short-range separation of the electron-electron interaction in density-functional theory

By splitting the Coulomb interaction into long-range and short-range components, we decompose the energy of a quantum electronic system into long-range and short-range contributions. We show that the long-range part of the energy can be efficiently calculated by traditional wave function methods, while the short-range part can be handled by a density functional. The analysis of this functional with respect to the range of the associated interaction reveals that, in the limit of a very short-range interaction, the short-range exchange-correlation energy can be expressed as a simple local functional of the on-top pair density and its first derivatives. This provides an explanation for the accuracy of the local density approximation (LDA) for the short-range functional. Moreover, this analysis leads also to new simple approximations for the short-range exchange and correlation energies improving the LDA.

On the importance of electron correlation effects for the pi-pi interactions in cyclophanes.

Correlated ab initio quantum chemical methods based on second-order perturbation theory and density functional theory (DFT) together with large atomic orbital (AO) basis sets are used to calculate the structures of four cyclophanes with two aromatic rings and one sulphur-containing phane with one aromatic ring. The calculated geometrical data for [2.2]paracyclophane, cyclophane (superphane), 8,16-dimethyl[2.2]metacyclophane, 16-methyl[2.2]metaparacyclophane, and 2,6,15-trithia[3(4,10)][7]metacyclophane are compared to experimental data from X-ray crystal structure determinations. In all cases, very accurate theoretical predictions are obtained from the recently developed spin-component-scaled MP2 (SCS-MP2) method, in which the deviations are within the experimental accuracy and expected crystal-packing or vibrational effects. Especially the inter-ring distances, which are determined by a detailed balance between attractive van der Waals (dispersive) and repulsive (Pauli) contributions, are very sensitive to the level of theory employed. While standard MP2 theory in general overestimates the dispersive interactions (pi-pi correlations) between the two aromatic rings leading to too short distances (between 3 and 8 pm), the opposite is observed for DFT methods (errors up to 15 pm). This indicates that an explicit account of dispersive-type electron correlation effects between the clamped aromatic units is essential for a quantitative description of cyclophane structures. In order to distinguish these effects from "normal" van der Waals interactions, the term "overlap-dispersive" interaction may be employed. The popular B3 LYP hybrid density functional offers no advantage over the pure PBE functional that at least qualitatively accounts for some of the dispersive effects. The use of properly polarized AO basis sets of at least valence-triple-zeta quality is strongly recommended to obtain quantitative predictions with traditional wave function methods.

Relative Stabilities of Three Low-Energy Tautomers of Cytosine: A Coupled Cluster Electron Correlation Study

High-level quantum chemical calculations have been carried out in an effort to reinvestigate the relative stabilities of the three lowest-lying tautomers of cytosine. Geometries were optimized at the CCSD/TZP level and electronic energies calculated at CCSD(T)/cc-pVTZ and vibrational frequencies at MP2/TZP. Comparative DFT calculations were also performed. From these data Gibbs free energies and equilibrium mole ratios were calculated. In agreement with most previous theoretical and experimental results, the amino−hydroxy tautomer 2b was found to be the most stable structure. As a new result, the amino−oxo form 1 and the imino−oxo form 3a have very nearly the same electronic energy, about 1.5−1.7 kcal/mol above 2b. The calculated ΔG values at standard temperature are ∼0.8 kcal/mol relative to 2b, again for both 1 and 3a. These results about the stability of the oxo form 1 are in quantitative agreement with experimental estimates in the literature, both from matrix isolation infrared and from molecular beam microwave spectroscopy. However, the calculated stability of the imino form is much higher than suggested by experiment. Our tentative reanalysis of the IR spectrum did not resolve this discrepancy. The widely used MP2 method gives significant deviations from the coupled cluster results and may be not accurate enough for determining close-lying energies of tautomers. Also, density functional theory gives qualitatively different results from traditional wave function methods.