Accurate quantum-chemical prediction of enthalpies of formation of small molecules in the gas phase

Straightening the Hierarchical Staircase for Basis Set Extrapolations: A Low-Cost Approach to High-Accuracy Computational Chemistry

Because the one-electron basis set limit is difficult to reach in correlated post-Hartree-Fock ab initio calculations, the low-cost route of using methods that extrapolate to the estimated basis set limit attracts immediate interest. The situation is somewhat more satisfactory at the Hartree-Fock level because numerical calculation of the energy is often affordable at nearly converged basis set levels. Still, extrapolation schemes for the Hartree-Fock energy are addressed here, although the focus is on the more slowly convergent and computationally demanding correlation energy. Because they are frequently based on the gold-standard coupled-cluster theory with single, double, and perturbative triple excitations [CCSD(T)], correlated calculations are often affordable only with the smallest basis sets, and hence single-level extrapolations from one raw energy could attain maximum usefulness. This possibility is examined. Whenever possible, this review uses raw data from second-order Møller-Plesset perturbation theory, as well as CCSD, CCSD(T), and multireference configuration interaction methods. Inescapably, the emphasis is on work done by the author's research group. Certain issues in need of further research or review are pinpointed.

Extrapolation of electron correlation energies to finite and complete basis set targets

The electron correlation energy of two-electron atoms is known to converge asymptotically as approximately (L+1)(-3) to the complete basis set limit, where L is the maximum angular momentum quantum number included in the basis set. Numerical evidence has established a similar asymptotic convergence approximately X(-3) with the cardinal number X of correlation-consistent basis sets cc-pVXZ for coupled cluster singles and doubles (CCSD) and second order perturbation theory (MP2) calculations of molecules. The main focus of this article is to probe for deviations from asymptotic convergence behavior for practical values of X by defining a trial function X(-beta) that for an effective exponent beta=beta(eff)(X,X+1,X+N) provides the correct energy E(X+N), when extrapolating from results for two smaller basis sets, E(X) and E(X+1). This analysis is first applied to "model" expansions available from analytical theory, and then to a large body of finite basis set results (X=D,T,Q,5,6) for 105 molecules containing H, C, N, O, and F, complemented by a smaller set of 14 molecules for which accurate complete basis set limits are available from MP2-R12 and CCSD-R12 calculations. beta(eff) is generally found to vary monotonically with the target of extrapolation, X+N, making results for large but finite basis sets a useful addition to the limited number of cases where complete basis set limits are available. Significant differences in effective convergence behavior are observed between MP2 and CCSD (valence) correlation energies, between hydrogen-rich and hydrogen-free molecules, and, for He, between partial-wave expansions and correlation-consistent basis sets. Deviations from asymptotic convergence behavior tend to get smaller as X increases, but not always monotonically, and are still quite noticeable even for X=5. Finally, correlation contributions to atomization energies (rather than total energies) exhibit a much larger variation of effective convergence behavior, and extrapolations from small basis sets are found to be particularly erratic for molecules containing several electronegative atoms. Observed effects are discussed in the light of results known from analytical theory. A carefully calibrated protocol for extrapolations to the complete basis set limit is presented, based on a single "optimal" exponent beta(opt)(X,X+1,infinity) for the entire set of molecules, and compared to similar approaches reported in the literature.

Second order explicitly correlated R12 theory revisited: A second quantization framework for treatment of the operators’ partitionings

Second order R12 theory is presented and derived alternatively using the second quantized hole-particle formalism. We have shown that in order to ensure the strong orthogonality between the R12 and the conventional part of the wave function, the explicit use of projection operators can be easily avoided by an appropriate partitioning of the involved operators to parts which are fully describable within the computational orbital basis and complementary parts that involve imaginary orbitals from the complete orbital basis. Various Hamiltonian splittings are discussed and computationally investigated for a set of nine molecules and their atomization energies. If no generalized Brillouin condition is assumed, with all relevant partitionings the one-particle contribution arising in the explicitly correlated part of the first order wave function has to be considered and has a significant role when smaller atomic orbital basis sets are used. The most appropriate Hamiltonian splitting results if one follows the conventional perturbation theory for a general non-Hartree-Fock reference. Then, no couplings between the R12 part and the conventional part arise within the first order wave function. The computationally most favorable splitting when the whole complementary part of the Hamiltonian is treated as a perturbation fails badly. These conclusions also apply to MP2-F12 approaches with different correlation factors.

A comparison of linear and nonlinear correlation factors for basis set limit Møller-Plesset second order binding energies and structures of He2, Be2, and Ne2

The basis set limit Møller-Plesset second-order equilibrium bond lengths of He2, Be2, and Ne2, accurate to 0.01a0, are computed to be 5.785a0, 5.11a0, and 6.05a0. The corresponding binding energies are 22.4+/-0.1, 2180+/-20, and 86+/-2 muE(h), respectively. An accuracy of 95% in the binding energy requires an aug-cc-pV6Z basis or larger for conventional Møller-Plesset theory. This accuracy is obtained using an aug-cc-pV5Z basis if geminal basis functions with a linear correlation factor are included and with an aug-cc-pVQZ basis if the linear correlation factor is replaced by exp(-gammar(12)) with gamma=1. The correlation factor r(12) exp(-gammar(12)) does not perform as well, describing the atom more efficiently than the dimer. The geminal functions supplement the orbital basis in the description of both the short-range correlation, at electron coalescence, and the long-range dispersion correlation and the values of gamma that give the best binding energies are smaller than those that are optimum for the atom or the dimer. It is important to sufficiently reduce the error due to the resolution of the identity approximation for the three- and four-electron integrals and we recommend the complementary auxiliary basis set method. The effect of both orbital and geminal basis set superposition error must be considered to obtain accurate binding energies with small orbital basis sets. In this respect, we recommend using exp(-gammar(12)) with localized orbitals and the original orbital-variant formalism.

Extensions of r12 corrections to CC2-R12 for excited states

As known since about two decades, R12 methods, which include terms linear in the interelectronic distance r(12) in the wave function, improve substantially the basis set convergence of the ground state correlation energy. In a previous study, however, it was found that the same approach does not give a similar systematic improvement if applied to excited states in the framework of coupled cluster response theory. In the present work, we examine the reason for this behavior and show that the inclusion of additional orbitals in the construction of the r(12) pair functions leads to an enhanced basis set convergence (and thus a balanced description) also for the excited states.

Validation of the M−C/H−C Bond Enthalpy Relationship through Application of Density Functional Theory

Density functional theory has been used to calculate H-C and M-C bond dissociation enthalpies in order to evaluate the feasibility of correlating relative M-C bond enthalpies Delta H(M-C)rel with H-C bond enthalpies Delta H(H-C) via computational methods. This approach has been tested against two experimental correlations: a study of (a) Rh(H)(R)(Tp')(CNCH2CMe3) [R = hydrocarbyl, Tp' = HB(3,5-dimethylpyrazolyl)3] (Wick, D. D.; Jones, W. D. Organometallics 1999, 18, 495) and (b) Ti(R)(silox)2(NHSit-Bu3) (silox = OSit-Bu3) (Bennett, J. L.; Wolczanski, P. T. J. Am. Chem. Soc. 1997, 119, 10696). We show that the observation that M-C bond enthalpies increase more rapidly with different substituents than H-C bond enthalpies is reproduced by theory. Quantitative slopes of the correlation lines are reproduced within 4% of the experimental values with a B3PW91 functional and with very similar correlation coefficients. Absolute bond enthalpies are reproduced within 6% for H-C bonds, and relative bond enthalpies for M-C bonds are reproduced within 30 kJ mol(-1) for Rh-C bonds and within 19 kJ mol(-1) for Ti-C bonds. Values are also calculated with the BP86 functional.

Thermochemistry of the Hypobromous and Hypochlorous Acids, HOBr and HOCl

The enthalpies of formation of HOBr and HOCl have been estimated by employing coupled cluster theory in conjunction with the correlation consistent basis sets and corrections for core-valence, relativistic, and anharmonic effects. We have employed three different reactions to estimate the DeltaH(o)(f,298)(HOBr), namely, the atomization reaction and two homodesmic reactions. Our best estimation is DeltaH(o)(f,298) (HOBr) = -15.3 +/- 0.6 kcal/mol and is very likely to lie toward the more negative values. The present value is 1.4 kcal/mol lower than the widely used experimental determination of Ruscic and Berkowitz (J. Chem. Phys. 1994, 101, 7795), DeltaH(o)(f,298)(HOBr) > -13.93 +/- 0.42 kcal/mol. However, it is closer to the more recent measurement of Lock et al. (J. Phys. Chem. 1996, 100, 7972), DeltaH(o)(f,298)(HOBr) = -14.8 +/- 1 kcal/mol. In the case of HOCl we have determined DeltaH(o)(f,298)(HOCl) = -18.1 +/- 0.3 kcal/mol, just in the middle of the two experimental values proposed, -17.8 +/- 0.5 kcal/mol (JANAF), obtained from equilibrium constant measurements, and -18.36 +/- 0.03 kcal/mol (Joens, J. A. J. Phys. Chem. A 2001, 105, 11041), determined from the measurements of the Cl-OH bond energy. If our conclusions are correct, several enthalpies of formation that have been determined by experimental chemists, Orlando and Burholder (J. Phys. Chem. 1995, 99, 1143), and theoretical chemists, Lee (J. Phys. Chem. 1995, 99, 15074), need to be revised, since a larger value was used for DeltaH(o)(f,298)(HOBr). Employing the results obtained by Orlando and Burkholder for Br(2)O we propose DeltaH(o)(f,298)(Br(2)O) = 24.9 +/- 0.6 kcal/mol, and employing Lee's enthalpies of reaction we propose the following DeltaH(o)(f,298): for BrBrO, HBrO, ClOBr, ClBrO, BrClO, BrCN, BrNC, BrNO, BrON, FOBr, and FBrO, 39.5 +/- 1, 41.0 +/- 1, 22.7 +/- 1.5, 34.2 +/- 1.5, 40.9 +/- 1.5, 43.7 +/- 1.5, 80.1 +/- 1.5, 22.3 +/- 1, 46.2 +/- 1, 17.3 +/- 1.5, and 6.3 +/- 1.5 kcal/mol, respectively. We expect that this work will stimulate new experimental measurements of the thermodynamic properties of HOBr and HOCl.

Coupled-cluster response theory with linear-r12 corrections: The CC2-R12 model for excitation energies

Coupled-cluster response theory for vertical excitation energies within the second-order approximate coupled-cluster singles-and-doubles model CC2, including linear-r(12) corrections, is derived and implemented for Ansatze 1 and 2 of R12 theory. An orthonormal auxiliary basis set is used for the resolution-of-the-identity approximation in order to calculate the three- and four-electron integrals needed in R12 theory. The basis set convergence is investigated for a selected set of atoms and small molecules and it is found that in many cases the convergence is not improved. An analysis of the different contributions to excitation energies shows that the present scheme for the construction of the R12 pair functions leads in response theory to an unbalanced description of ground- and excited-state wave functions and needs to be generalized to carry the high accuracy of R12 methods over to response theory.

New correlation factors for explicitly correlated electronic wave functions

We have investigated the correlation factors exp(-zetar12), r12 exp(-zetar12), erfc(zetar12), and r12 erfc(zetar12) in place of the linear-r12 term for use in explicitly correlated electronic-structure methods. The accuracy obtained with all of these correlation factors is significantly greater than that obtained with the plain correlation factor r12. Polarization functions that are more diffuse than those of standard basis sets give even better results. The correlation factor exp(-zetar12) is very close to the optimum correlation factor for helium and outperforms the others.

A full nine-dimensional potential-energy surface for hydrogen molecule-water collisions

The hydrogen and water molecules are ubiquitous in the Universe. Their mutual collisions drive water masers and other line emission in various astronomical environments, notably molecular clouds and star-forming regions. We report here a full nine-dimensional interaction potential for H2O-H2 calibrated using high-accuracy, explicitly correlated wave functions. All degrees of freedom are included using a systematic procedure transferable to other small molecules of astrophysical or atmospherical relevance. As a first application, we present rate constants for the vibrational relaxation of the upsilon2 bending mode of H2O obtained from quasiclassical trajectory calculations in the temperature range of 500-4000 K. Our high-temperature (T > or = 1500 K) results are found compatible with the single experimental value at 295 K. Our rates are also significantly larger than those currently used in the astrophysical literature and will lead to a thorough reinterpretation of vibrationally excited water emission spectra from space.

Coupled-cluster theory with simplified linear-r12 corrections: The CCSD(R12) model

A simplified singles-and-doubles linear-r(12) corrected coupled-cluster model, denoted CCSD(R12), is proposed and compared with the complete singles-and-doubles linear-r(12) coupled-cluster method CCSD-R12. An orthonormal auxiliary basis set is used for the resolution-of-the-identity approximation to calculate three-electron integrals needed in the linear-r(12) Ansatz. Basis-set convergence is investigated for a selected set of atoms and small molecules. In a large basis, the CCSD(R12) model provides an excellent approximation to the full linear-r(12) energy contribution, whereas the magnitude of this contribution is significantly overestimated at the level of second-order perturbation theory.

Improved Reaction and Activation Energies of [4+2] Cycloadditions, [3,3] Sigmatropic Rearrangements and Electrocyclizations with the Spin-Component-Scaled MP2 Method

A new quantum mechanical scheme to calculate electronic correlation energies, spin-component-scaled MP2, was tested as a tool to predict reaction energies and barriers in computational organic chemistry. Three common pericyclic reactions with known unsatisfactory MP2 descriptions were reinvestigated with the modified MP2 approach, in which the parallel and anti-parallel spin components of the correlation energy are scaled separately. The SCS-MP2 calculated reaction and activation energies of nine Diels-Alder reactions, four [3,3] sigmatropic rearrangements, and ten electrocyclization reactions are compared to those of the MP2, B3 LYP, QCISD(T), and G3 methods. For each, the SCS-MP2 results are in excellent agreement with the experimental data and compare far more favorably to G3 than both MP2 and B3 LYP. Careful evaluation of the effect of the size of the atomic orbital (AO) basis set shows that the larger expansions improve the agreement with experiment for the SCS-MP2 method, while they get worse for both MP2 and B3 LYP.

Improved third-order Møller-Plesset perturbation theory

Based on a partitioning of the total correlation energy into contributions from parallel- and antiparallel-spin pairs of electrons, a modified third-order Møller-Plesset (MP) perturbation theory is developed. The method, termed SCS-MP3 (SCS for spin-component-scaled) continues previous work on an improved version of MP2. A benchmark set of 32 isogyric reaction energies, 11 atomization energies, and 11 stretched geometries is used to assess to performance of the model in comparison to the standard quantum chemical approaches MP2, MP3, and QCISD(T). It is found, that the new method performs significantly better than usual MP2/MP3 and even outperforms the more costly QCISD method. Opposite to the usual MP series, the SCS third-order correction uniformly improves the results. Dramatic enhancements are especially observed for the more difficult atomization energies, some of the stretched geometries, and reaction and ionization energies involving transition metal compounds where the method seems to be competitive or even superior to the widely used density functional approaches. Further tests performed for other complex systems (biradicals, C(20) isomers, transition states) demonstrate that the SCS-MP3 model yields often results of QCISD(T) accuracy. The uniformity with which the new approach improves for very different correlation problems indicates significant robustness, and suggests it as a valuable quantum chemical method of general use.