Calculation of smooth potential energy surfaces using local electron correlation methods

Pair correlation methods are able to achieve highly accurate solutions for chemical problems. Unfortunately, their applicability is generally restricted to medium-sized molecules due to storage requirements and computational costs. These restrictions can be partly overcome by local correlation methods. These methods use physical and mathematical criteria to decide which interactions are of such a long range that they do not have to be computed and saved. In our new ansatz, we define an alternative way towards local correlation. The range of interactions is strictly bound to the decay of integrals over Gaussian type geminals in the atomic orbital basis. The number of variables is reduced by orders of magnitude applying an efficient contraction scheme, leading to a naturally local representation of correlation effects. This scheme is extended by orbital optimization to describe multi-reference problems and explicit correlation to improve the basis set convergence.

The calculation of accurate reaction energies and barrier heights is essential in computational studies of reaction mechanisms and thermochemistry. To assess methods regarding their ability to predict these two properties, high-quality benchmark sets are required that comprise a reasonably large and diverse set of organic reactions. Due to the time-consuming nature of both locating transition states and computing accurate reference energies for reactions involving large molecules, previous benchmark sets have been limited in scope, the number of reactions considered, and the size of the reactant and product molecules. Recent advances in coupled-cluster theory, in particular local correlation methods like DLPNO-CCSD(T), now allow the calculation of reaction energies and barrier heights for relatively large systems. In this work, we present a comprehensive and diverse benchmark set of barrier heights and reaction energies based on DLPNO-CCSD(T)/CBS called BH9. BH9 comprises 449 chemical reactions belonging to nine types common in organic chemistry and biochemistry. We examine the accuracy of DLPNO-CCSD(T) vis-a-vis canonical CCSD(T) for a subset of BH9 and conclude that, although there is a penalty in using the DLPNO approximation, the reference data are accurate enough to serve as a benchmark for density functional theory (DFT) methods. We then present two applications of the BH9 set. First, we examine the performance of several density functional approximations commonly used in thermochemical and mechanistic studies. Second, we assess our basis set incompleteness potentials regarding their ability to mitigate basis set incompleteness errors. The number of data points, the diversity of the reactions considered, and the relatively large size of the reactant molecules make BH9 the most comprehensive thermochemical benchmark set to date and a useful tool for the development and assessment of computational methods.

This chapter reviews important aspects of local correlation methods, namely the construction of most suitable local orbital spaces, the choice of domains, and pair approximations for weak and distant pairs. The domain sizes which are necessary to reach a certain accuracy (e.g., 99.9% of the canonical correlation energy) depends sensitively on the choice of the virtual orbitals. As is known since the 1960s, fastest convergence is obtained with pair natural orbitals (PNOs), and this has first been fully exploited in the seminal PNO-CI and PNO-CEPA methods of Meyer, and somewhat later also by others. The chapter also discusses local second-order Moller-Plesset perturbation theory (LMP2) methods, and more advanced coupled cluster methods. It focuses on new developments of well-parallelized PNO-LMP2-F12 and PNO-LCCSD-F12 methods. It introduces some benchmark systems and describes some important technical details, such as local density fitting and parallelization.

The two main problems in applying common methods for electron correlation to large systems are (1) the steep dependency of the computational task with the size of the system and (2) the large and often prohibitive number of two‐electron integrals that must be stored and retrieved during the calculation. The direct local correlation method eliminates both these problems. This method is a combination of the local correlation method that eliminates the steep computational dependency with the size of the system and extension of the direct SCF approach to electron correlation methods. The latter method eliminates external storage of the electron repulsion integrals. In this review, the direct local correlation approach and its computer implementation will be described in detail, including computational examples and comparisons to similar methods developed in other groups. The feasibility of future applications to extended systems is discussed.

Molecules of utmost importance like DNA and RNA nucleobases are predicted to be nonplanar by a typical ab initio method, such as second order Møller-Plesset perturbation theory (MP2) combined with standard Pople's basis sets. Similarly to the case of other planar aromatic systems, these pitfalls can be explained in terms of intramolecular basis set superposition error (BSSE) effects, induced by local basis set deficiencies. We demonstrate that conventional BSSE correction techniques such as the Counterpoise method can account for this wrong behavior and provide proper correction whenever spurious results occur, mainly in case of thymine, uracil and guanine but also to lower extent for adenine and cytosine. We also show that special care must be taken when assessing the BSSE by means of ghost-orbital calculations for strongly overlapping fragments. Often molecular orbitals in the isolated fragment calculation have a different orientation as in the ghost-orbital calculation. This can lead to bogus derivatives of the CP-correction term, essential to account for geometry and vibrational BSSE effects.

The X-band Delta-Differential One-way Range is first formally used in Chang’E-3 spacecraft, and will be continually used for other Chinese future deep space navigation. We have carried out researches on each DOR tone using local correlation method which is quite different from the conventional VLBI correlator. The CE-3 signal interferometry processing is completed with this method. High precision Delta-DOR group delay and radio source correction are obtained for orbit determination. The phase noise of local correlation is significantly smaller than that of the VLBI conventional correlator. The ephemeris from local correlation is about 20 m different from the precise ephemeris of CE-3 project, and the corresponding equivalent residual delay is about 0.4 ns. Experiment result shows that local correlation is more suitable for weak signal processing, and it can be used for the future Chinese lunar and other deep space exploration projects.

All traditional methods for electron correlation share a steep power law dependence on the molecular size. This high scaling prohibits the use of these methods to large systems in spite of the very impressive advances in computer technology over the past decades. Clearly, this problem cannot be solved with improvements of computers alone, and new methods reducing the power law scaling to one or near one must be developed. In this chapter some linear of low scaling methods for electron correlation will be reviewed. The focus will be on the linear scaling MP2 methods, but other more accurate correlation methods will also be briefly discussed. In addition, the very efficient RI-MP2 will be discussed even though the high power law scaling of conventional MP2 has not been reduced. A discussion of the RI-MP2 method has been included since it is perhaps an order of magnitude more efficient than other efficient MP2 methods. The RI or density fitting approach has now been combined with the local correlation method, and the RI-LMP2 method exhibits linear scaling with the size of the system. Most of the methods discussed herein are based on the local correlation method introduced by Pulay and Saebo in the early eighties and developed further by Schutz, Werner and co-workers. The topic was reviewed in 2002 and this review will focus on the more recent advances in this field. A new linearly scaling LMP2 approach yielding essentially identical results to conventional canonical MP2 will be described, and MP2 calculations with around 5,000 contracted basis functions have been performed without density fitting using this approach.

A hybrid scheme for the computation of reaction energies in large molecular systems is proposed. The approach is based on localized orbitals and allows for the treatment of different parts of a molecule at different computational levels. The localized orbitals are assigned to regions, and then different local correlation methods, such as local MP2 or local CCSD(T), can be applied to different regions. In contrast to previous hybrid schemes, the molecule does not have to be split into parts and, therefore, it is not necessary to saturate dangling bonds using link atoms. For fixed region sizes, the cost of the high-level calculation becomes independent of the molecular size, and it is demonstrated that O(1) scaling can be achieved. Illustrative applications are presented and the convergence of the results with respect to the size of the regions is investigated for reaction energies, barrier heights, and weakly bound complexes.

We use Cholesky decomposition of the density matrix in atomic orbital basis to define a new set of occupied molecular orbital coefficients. Analysis of the resulting orbitals ("Cholesky molecular orbitals") demonstrates their localized character inherited from the sparsity of the density matrix. Comparison with the results of traditional iterative localization schemes shows minor differences with respect to a number of suitable measures of locality, particularly the scaling with system size of orbital pair domains used in local correlation methods. The Cholesky procedure for generating orthonormal localized orbitals is noniterative and may be made linear scaling. Although our present implementation scales cubically, the algorithm is significantly faster than any of the conventional localization schemes. In addition, since this approach does not require starting orbitals, it will be useful in local correlation treatments on top of diagonalization-free Hartree-Fock optimization algorithms.

The structures and stabilities of small water clusters are studied by local electron correlation methods. It is demonstrated that the local treatment eliminates basis set superposition errors (BSSEs) to a large extent and thus allows BSSE-free geometry optimizations. Results for various basis sets are presented which show that the interaction energies and structural parameters obtained by local second-order Møller−Plesset perturbation theory (LMP2) without counterpoise correction are in close agreement with counterpoise-corrected conventional MP2 results. Furthermore, a partitioning of the LMP2 energies of (H2O)n, n = 2−4, into different excitation classes is reported, which underlines the importance of ionic contributions as well as intramolecular correlation for hydrogen-bonded clusters. The results of this analysis are compared with previous data obtained by symmetry-adapted perturbation theory (SAPT).

Extensive ab initio Gaussian-3-type calculations of potential energy surfaces (PES), which are expected to be accurate within 1-2 kcal/mol, combined with statistical theory calculations of reaction rate constants have been applied to study various possible pathways in the hydrogen abstraction acetylene addition (HACA) mechanism of naphthalene and acenaphthalene formation as well as Diels-Alder pathways to acenaphthalene, phenanthrene, and pyrene. The barrier heights; reaction energies; and molecular parameters of the reactants, products, intermediates, and transition states have been generated for all types of reactions involved in the HACA and Diels-Alder mechanisms, including H abstraction from various aromatic intermediates, acetylene addition to radical sites, ring closures leading to the formation of additional aromatic rings, elimination of hydrogen atoms, H disproportionation, C2H2 cycloaddition, and H2 loss. The reactions participating in various HACA sequences (e.g., Frenklach's, alternative Frenklach's, and Bittner and Howard's routes) are demonstrated to have relatively low barriers and high rate constants under combustion conditions. A comparison of the significance of different HACA mechanisms in PAH growth can be made in the future using PES and molecular parameters obtained in the present work. The results show that the Diels-Alder mechanism cannot compete with the HACA pathways even at high combustion temperatures, because of high barriers and consequently low reaction rate constants. The calculated energetic parameters and rate constants have been compared with experimental and theoretical data available in the literature.

Low doses radiation in mammography results in low contrast images. In this paper we propose a method to enhance the contrast in mammography; it combines the modification of the coefficients of the Logarithmic Discrete Wavelet Transform using the Local Correlation method and Symmetric Logarithmic Image Processing model. Experimental results shown the better performance for the anomalies known as calcifications and masses. This paper also presents a methodology to select a combination of decomposition levels to be processed for good contrast improvement according to the values of measures based on region of interest. This procedure relies on Principal Components Analysis of the data. The experiments show that the chosen combination of levels can improve the contrast in mammograms, and that the regions of interest definition is an important factor to explain the poor contrast improvement of some anomalies.

Divide-Expand-Consolidate (DEC) is a local correlation method where the inherent locality of the electron correlation problem is used to express the correlated calculation on a large molecular system in terms of small independent fragment calculations employing small subsets of local HF orbitals. A crucial feature of the DEC scheme is that the sizes of the local orbital spaces are determined in a black box manner during the calculation. In this way it is ensured that the correlation energy has been determined to a predefined precision compared to a conventional calculation. In the present work we apply the DEC scheme to calculate the correlation energy as well as the electron density matrix for the insulin molecule using second order Møller-Plesset (MP2) theory. This is the first DEC calculation on a molecular system which is too large to be treated using a conventional MP2 implementation. The fragmentation errors for the insulin DEC calculation are carefully analyzed using internal consistency checks. It is demonstrated that size-intensive properties are determined to the same precision for small and large molecules. For example, the percentage of correlation energy recovered and the error per electron in the correlated density matrix depend only on the predefined precision and not on the molecular size.

I use coupled-cluster theory and a modest basis set, aug-cc-pVDZ, to calculate structures and harmonic vibrational frequencies of local minima and transition states on the C(3)H(5)O potential energy surface. Accurate energies are computed using explicitly correlated coupled-cluster methods and a large basis set, cc-pVQZ-F12, to approach the one-particle basis set limit. My computations characterize eight additional stable radical structures on the global potential energy surface for this system. Additionally, this study encompasses many more isomerization and dissociation pathways, both between previously known intermediates and ones first characterized here. Analysis of the transition states and statistical transition-state theory results shows that energetically small barriers connect many of the alkenol and epoxide intermediates to the straight-chain alkoxy isomers, leading to significant branching to these alkoxy radical intermediates. Facile isomerization to these alkoxy intermediates is significant because the barrier heights leading to H + acrolein and HCO + C(2)H(4) product channels are energetically accessible even at low vibrational energies. The low dissociation barrier heights and loose transition states of these pathways result in unimolecular dissociation as opposed to isomerization to a different C(3)H(5)O intermediate.

A global ground-state potential energy surface for the dissociation reactions of difluoroethylenes (DFEs) was computed by B3LYP and QCISD calculations, using the standard 6-311G(2d,2p) basis set. RRKM calculations were performed to compute relative abundances of hydrogen fluoride (HF) and molecular hydrogen produced from 1,1-DFE and from 1,2-DFE (cis and trans) at energies ranging from 110 to 180 kcal mol-1 relative to the zero point energy of 1,1-DFE. Thermal rate constants were also evaluated by the variational transition state theory for temperatures in the range 1250−1500 K. Both theoretical methods agree that, at the energies and temperatures studied, the main channel for HF elimination from 1,1-DFE is through a four-center transition state, whereas for 1,2-DFE the process occurs through a direct three-center elimination. At the energies studied, the RRKM method predicts that the main channel for molecular hydrogen elimination from the DFEs goes through a three-center transition state that connects 1,1-DFE with products.