Explicitly correlated Wn theory: W1-F12 and W2-F12

In an attempt to extend the applicability of the W1 and W2 ab initio computational thermochemistry methods, we propose explicitly correlated versions thereof, denoted W1-F12 and W2-F12. In W2-F12, we can "save" one cardinal number (viz., angular momentum) in the basis set sequences without loss in accuracy; in W1-F12, we can do so for first-row compounds but not for second-row compounds. At a root mean square deviation (RMSD) = 0.19 kcal/mol for the first-row molecules in the W4-11 benchmark dataset, W1-F12 is in fact superior to ordinary W1 theory. For the entire W4-11 set, W2-F12 yields a RMSD = 0.20 kcal/mol, comparable to 0.19 kcal/mol from ordinary W2 theory. The extended applicability ranges of W1-F12 and W2-F12 are not just due to the lower computational cost but also to greatly reduced memory and especially storage requirements. They are illustrated through applications to nucleic acids and to polyacenes (with up to four rings), for which the following revised gas-phase heats of formation are found: Δ(f)H(298)(°) = 19.6 (benzene), 34.94 (naphthalene), 53.9, (anthracene), 73.9 (naphthacene/tetracene), 54.9 (adenine), -16.3 (cytosine), 5.1 (guanine), -80.6 (thymine), and -71.6 (uracil) kcal/mol. Our theoretical values for the DNA/RNA bases largely confirm recent predictions based on much lower-level calculations. The W1-F12 theoretical values for benzene, naphthalene, and anthracene are in very good to reasonable agreement with experiment. However, both W1-F12 and other computational estimates show that the accepted experimental value for naphthacene cannot be reconciled with those for the lower acenes: we suggest that Δ(f)H(298)(°)[naphthacene(g)] = 74.25 ± 1 kcal/mol is a more realistic estimate.     

Communication: Second-order multireference perturbation theory with explicit correlation: CASPT2-F12

An explicitly correlated complete active space second-order perturbation (CASPT2-F12) method is presented which strongly accelerates the convergence of CASPT2 energies and properties with respect to the basis set size. A Slater-type geminal function is employed as a correlation factor to represent the electron-electron cusp of the wave function. The explicitly correlated terms in the wave function are internally contracted. The required density matrix elements and coupling coefficients are the same as in conventional CASPT2, and the additional computational effort for the F12 correction is small. The CASPT2-F12 method is applied to the singlet-triplet splitting of methylene, the dissociation energy of ozone, and low-lying excited states of pyrrole.     

Heats of formation of the amino acids re-examined by means of W1-F12 and W2-F12 theories

We have obtained accurate heats of formation for the twenty natural amino acids by means of explicitly correlated high-level thermochemical procedures. Our best theoretical heats of formation, obtained by means of the ab initio W1-F12 and W2-F12 thermochemical protocols, differ significantly (RMSD = 2.3 kcal/mol, maximum deviation 4.6 kcal/mol) from recently reported values using the lower-cost G3(MP2) method. With the more recent G4(MP2) procedure, RMSD drops slightly to 1.8 kcal/mol, while full G4 theory offers a more significant improvement to 0.72 kcal/mol (max. dev. 1.4 kcal/mol for glutamine). The economical G4(MP2)-6X protocol performs equivalently at RMSD = 0.71 kcal/mol (max. dev. 1.6 kcal/mol for arginine and glutamine). Our calculations are in excellent agreement with experiment for glycine, alanine and are in excellent agreement with the recent revised value for methionine, but suggest revisions by several kcal/mol for valine, proline, phenylalanine, and cysteine, in the latter case confirming a recent proposed revision. Our best heats of formation at 298 K ( $\Delta H_{f,298}^{\circ }$ ) are as follows: at the W2-F12 level: glycine ?94.1, alanine $-$ 101.5, serine $-$ 139.2, cysteine $-$ 94.5, and methionine $-$ 102.4  kcal/mol, and at the W1-F12 level: arginine $-$ 98.8, asparagine $-$ 146.5, aspartic acid $-$ 189.6, glutamine $-$ 151.0, glutamic acid $-$ 195.5, histidine $-$ 69.8, isoleucine $-$ 118.3, leucine $-$ 118.8, lysine $-$ 110.0, phenylalanine $-$ 76.9, proline $-$ 92.8, threonine $-$ 149.0, and valine $-$ 113.6 kcal/mol. For the two largest amino acids, an average over G4, G4(MP2)-6X, and CBS-QB3 yields best estimates of $-$ 58.4 kcal/mol for tryptophan, and of $-$ 117.5 kcal/mol for tyrosine. For glycine, we were able to obtain a “quasi-W4” result corresponding to $\hbox {TAE}_e$  = 968.1, $\hbox {TAE}_0$  = 918.6, $\Delta H_{f,298}^{\circ }=-90.0$ , and $\Delta H_{f,298}^{\circ }=-94.0$  kcal/mol.     

Some investigations of the MP2-R12 method

Summary The MP2-R12 method was introduced by Kutzelnigg and Klopper to overcome the problem caused by truncation of the one electron basis set in correlation energy calculations at the Møller-Plesset second order level of approximation. Here, we have evaluated the integrals required by their simplest scheme using the Rys-quadrature procedure. Results are presented for Ne, H₂O, and HF using largespdf gaussian basis sets.     

Approximate molecular orbital theory: the ese MO formalism. I. General theory and notation

A new formalism for simplified molecular orbital calculations is elucidated. The focus of the formalism is the production of a good approximation to the LCAO SCF F matrix of Roothaan's equations, and preoccupation with approximations to individual molecular integrals is avoided. The great majority of multicentre two-electron integrals of the exact formalism are found to be largely inconsequential to the attainment of a good approximation to the F matrix.     

Predicting shielding constants in solution using gauge invariant atomic orbital theory and the effective fragment potential method

A method to approximate ab initio shielding constants is presented, in which the ab initio density matrix is replaced in the gauge invariant atomic orbital formalism with the density matrix resulting from an effective fragment potential calculation. The resulting first-order density matrix is then iterated to self-consistency. The method is compared with fully ab initio gauge invariant atomic orbital restricted Hartree-Fock calculations on hydrogen chloride, water, and ammonia solutes with up to nine solvent water molecules using the 6-31G, 6-31G(d,p), and 6-31+G(d,p) basis sets. Using the 6-31G(d,p) basis sets, the average of the average absolute deviations for the three environments tested is 0.34 ppm. This is sufficiently accurate to allow for the identification of specific (1)H nuclei in a solvated molecule when the chemical shift between nuclei is not less than 1 ppm. The success of the method at this level of approximation is due to a cancellation of errors between the paramagnetic and diamagnetic terms of the shielding constant: the diamagnetic term is underestimated by roughly the same amount that the paramagnetic term is overestimated.     

Møller-Plesset (MP2) perturbation theory for large molecules

Summary A novel formulation of MP2 theory is presented which starts from the Laplace transform MP2 ansatz, and subsequently moves from a molecular orbital (MO) representation to an atomic orbital (AO) representation. Consequently, the new formulation is denoted AO-MP2. As in traditional MP2 approaches electron repulsion integrals still need to be transformed. Strict bounds on the individual MP2 energy contribution of each intermediate four-index quantity allow to screen off numerically insignificant integrals with a single threshold parameter. Implicit in our formulation is a bound to two-particle density matrix elements. For small molecules the computational cost for AO-MP2 calculations is about a factor of 100 higher than for traditional MO-based approaches, but due to screening the computational effort in larger systems will only grow with the fourth power of the size of the system (or less) as is demonstrated both in theory and in application. MP2 calculations on (non-metallic) crystalline systems seem to be a feasible extension of the Laplace transform approach. In large molecules the AO-MP2 ansatz allows massively parallel MP2 calculations without input/output of four-index quantities provided that each processor has in-core memory for a limited number of two-index quantities. Energy gradient formulas for the AO-MP2 approach are derived.Dedicated to Prof. W. Kutzelnigg whose books on theoretical chemistry aroused my interest in this field     

Analytical optimization of orbital exponents in Gaussian-type functions for molecular systems based on MCSCF and MP2 levels of fully variational molecular orbital method

We have analyzed the basis function series in molecular systems by optimization of orbital exponents in Gaussian-type functions (GTFs) including the electron correlation effects with multiconfiguration self-consistent field (MCSCF) and M?ller?CPlesset second-order perturbation (MP2) methods. First, we have derived and implemented the gradient formulas of MCSCF and MP2 energies with respect to GTF exponent, as well as GTF center and nuclear geometry, based on the fully variational molecular orbital (FVMO) method. Second, we have applied these electron-correlated FVMO methods to H₂, LiH, and hydrocarbon (CH₄, C₂H₆, C₂H₄, and C₂H₂) molecules. We have clearly demonstrated that the optimized exponent values with electron-correlated methods are different from those with simple Hartree?CFock method, since adequate basis functions for adequate virtual orbitals are indispensable to describe the accurate wave function and geometry for electron-correlated calculations.     

Synthesis of the eight enantiomerically pure diastereomers of the 12-F(2)-isoprostanes

Syntheses of the eight enantiomerically pure diastereomers of the 12-F(2)-isoprostanes (4-11) are described. The key steps included rhodium-mediated intramolecular cyclopropanation and enzymatic resolution of the racemic diol 12.     

Iterative diagonalization for orbital optimization in natural orbital functional theory

A challenging task in natural orbital functional theory is to find an efficient procedure for doing orbital optimization. Procedures based on diagonalization techniques have confirmed its practical value since the resulting orbitals are automatically orthogonal. In this work, a new procedure is introduced, which yields the natural orbitals by iterative diagonalization of a Hermitian matrix F . The off‐diagonal elements of the latter are determined explicitly from the hermiticity of the matrix of the Lagrange multipliers. An expression for diagonal elements is absent so a generalized Fockian is undefined in the conventional sense, nevertheless, they may be determined from an aufbau principle. Thus, the diagonal elements are obtained iteratively considering as starting values those coming from a single diagonalization of the matrix of the Lagrange multipliers calculated with the Hartree‐Fock orbitals after the occupation numbers have been optimized. The method has been tested on the G2/97 set of molecules for the Piris natural orbital functional. To help the convergence, we have implemented a variable scaling factor which avoids large values of the off‐diagonal elements of F . The elapsed times of the computations required by the proposed procedure are compared with a full sequential quadratic programming optimization, so that the efficiency of the method presented here is demonstrated. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2009     

Comparative study of BSSE correction methods at DFT and MP2 levels of theory

A comparative study of intermolecular potential energy curves is performed on the complexes H₂O(SINGLE BOND)HF, H₂O(SINGLE BOND)H₂O, H₂O(SINGLE BOND)H₂S, and H₂S(SINGLE BOND)H₂S using nine different basis sets at the MP2 and DFT (BLYP and B3LYP) levels of theory. The basis set superposition error is corrected by means of the counterpoise scheme and based on the “chemical Hamiltonian approach.” The counterpoise and CHA-corrected DFT curves are generally close to each other. Using small basis sets, the B3LYP functional cannot be favored against the BLYP one because the BLYP results sometimes get closer to the MP2 values than those of B3LYP. From the results—including the available literature data—we suggest that one has to use at least polarized-valence triple-zeta-quality basis sets (TZV, 6-311G) for the investigation of hydrogen-bonded complexes. Special attention must be paid to the physical nature of the binding. If the dispersion forces become significant DFT methods are not able to describe the interaction. Proper correction for the basis set superposition error is found to be mandatory in all cases. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 575–584, 1998     

Localized orbital theory and ammonia triborane

In a previous paper [J. Subotnik, Y. Shao and W. Liang, and M. Head-Gordon, J. Chem. Phys., 2004, 121, 9220], we proposed a new and efficient method for computing localized Edmiston-Ruedenberg (ER) orbitals, which are those localized orbitals that maximize self-interaction. In this paper, we improve upon our previous algorithm in two ways. First, we incorporate the resolution of the identity (RI) and atomic resolution of the identity (ARI) approximations when generating the relevant integrals, which allows for a drastic reduction in computational cost. Second, after convergence to a stationary point, we efficiently calculate the lowest mode of the Hessian matrix in order to either (i) confirm that we have found a minimum, or if not, (ii) move us away from the current saddle point. This gives our algorithm added stability. As a chemical example, in this paper, we investigate the electronic structure (including the localized orbitals) of ammonia triborane (NH(3)B(3)H(7)). Though ammonia triborane is a very electron-deficient compound, it forms a stable white powder which is now being investigated as a potential hydrogen storage material. In contrast to previous electronic structure predictions, our calculations show that ammonia triborane has one localized molecular orbital in the center of the electron-deficient triborane ring (much like the single molecular orbital in H(3)(+)), which gives the molecule added energetic stability. Furthermore, we believe that NH(3)B(3)H(7) is the smallest stable molecule supporting such a closed, three-center BBB bond.     

Integral approximations for molecular orbital theory

The electron repulsion integrals arising in LCAO-MO theory are approximated by replacement of the product of two orbitals on different centers by linear combinations of one-center products. The approximation differs from those previously proposed in that the coefficients of the various terms are determined by requiring agreement for certain integrals, and in the emphasis of the role of symmetry in selecting the one-center products. For two-center integrals, the new approximation is significantly better than older approximate methods. Reasons are given for expecting this improvement to extend also to multi-center integrals.

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Zusammenfassung Es wird ein Verfahren zur näherungsweisen Berechnung von Elektronenwechselwirkungsintegralen der LCAO-MO-Theorie beschrieben, bei welchem das Produkt zweier Zustandsfunktionen an verschiedenen Zentren durch eine Linearkombination von Produkten am gleichen Zentrum ersetzt wird. Der Unterschied zu ähnlichen Ansätzen liegt in der Justierung der Koeffizienten. Für Zweizentrenintegrale liefert die hier vorgeschlagene Methode bedeutend bessere Ergebnisse als das Mulliken-Verfahren.

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Résumé Les intégrales de répulsion électroniques intervenant dans la théorie LCAO MO sont calculées d'une manière approchée en remplaçant le produit de deux orbitales sur des centres différents par des combinaisons linéaires de produits à un centre. Cette approximation diffère de celles proposées auparavant par la détermination des coefficients des différents termes au moyen de l'ajustement de certaines intégrales et par l'importance du rôle de la symétrie dans le choix des produits monocentriques. Cette nouvelle approximation est bien meilleure que les anciennes en ce qui concerne les intégrales bi-centriques. Nous donnons des raisons d'espérer que cette amélioration s'étendra aux intégrales polycentriques.

     

A simple and efficient CCSD(T)-F12 approximation

A new explicitly correlated CCSD(T)-F12 approximation is presented and tested for 23 molecules and 15 chemical reactions. The F12 correction strongly improves the basis set convergence of correlation and reaction energies. Errors of the Hartree-Fock contributions are effectively removed by including MP2 single excitations into the auxiliary basis set. Using aug-cc-pVTZ basis sets the CCSD(T)-F12 calculations are more accurate and two orders of magnitude faster than standard CCSD(T)/aug-cc-pV5Z calculations.     

Approximate molecular orbital theory for inorganic molecules

Electron correlation is known to have an important influence on the results of molecular orbital calculations, but is not usually directly included in such calculations. An analysis of the general theory of electron correlation leads to a pair correlation hypothesis, which serves as a basis for the subsequent derivation of a way of explicitly allowing for electron correlation in the LCAOMO energy. The derived expressions carry significant implications for the semi-empirical parameter schemes of all-valence-electron methods, implying that they cannot be regarded as incorporating the correct form of correlation correction. This points to the advantage of aiming at a theoretically founded parameter scheme in approximate molecular orbital calculations, designed to produce approximate Hartree-Fock molecular wave functions. The electron correlation correction can then be applied to the expression for the total valence electron energy as developed in the present paper.

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Zusammenfassung Bekanntlich hat die Elektronenkorrelation einen wichtigen Einfluß auf die Resultate von MO-Berechnungen. Sie wird jedoch gewöhnlich nicht direkt in solche Berechnungen mit einbezogen. Eine Analyse der allgemeinen Theorie der Elektronenkorrelationen führt zu einer Paar-Korrelationshypothese, die als Grundlage für die Ableitung einer Methode dient, um die Elektronenkorrelation explizit in der LCAOMO-Energie zu berücksichtigen. Die abgeleiteten Ausdrücke wirken sich stark auf semi-empirische Parameter-Schemata von Methoden unter Einschluß aller Valenzelektronen aus, woraus folgt, daß diese keine korrekte Berücksichtigung der Korrelationskorrekturen enthalten. Es ist also vorzuziehen, theoretisch begründete Parameter-Schemata in Näherungs-MO Berechnungen anzustreben, die zu Näherungs-Hartree-Fock-Molekül-Funktionen führen sollen. Die Elektronenkorrelationskorrektur kann dann bei dem Ausdruck für die totale Valenzelektronen-Energie angewendet werden, wie es in der vorliegenden Arbeit dargestellt wird.

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Résumé La corrélation électronique a une influence importante sur les résultats des calculs d'orbitales moléculaires mais n'est pas d'ordinaire incluse dans ces calculs. Une analyse de la théorie générale de la corrélation électronique conduit à une hypothèse de corrélation de paire que l'on utilise pour trouver un moyen explicite de tenir compte de la corrélation électronique dans l'énergie LCAOMO. Les expressions que l'on obtient contiennent des implications significatives pour les schéma semiempiriques des méthodes à électrons de valence, montrant que ces méthodes ne contiennent pas la corrélation électronique sous une forme correcte. Ceci indique l'intérêt d'un schéma paramétrique théoriquement fondé dans les méthodes Hartree-Fock approchées. La correction de corrélation électronique peut alors être appliquée à l'énergie totale des électrons de valence ainsi que cela est exposé dans cet article.

     

New developments in CNDO molecular orbital theory

A CNDO-level all-valence electron method is presented which yields both good heats of formation and good bond distances for hydrocarbon molecules. The success stems from a careful analysis of the total energy of molecular systems. A new versatile formula is also proposed for the computation of Coulomb integrals.