Frederick R. Manby

Molpro: a general-purpose quantum chemistry program package

Molpro (available at http://www.molpro.net) is a general‐purpose quantum chemical program. The original focus was on high‐accuracy wave function calculations for small molecules, but using local approximations combined with explicit correlation treatments, highly accurate coupled‐cluster calculations are now possible for molecules with up to approximately 100 atoms. Recently, multireference correlation treatments were also made applicable to larger molecules. Furthermore, an efficient implementation of density functional theory is available.

Fast linear scaling second-order Møller-Plesset perturbation theory (MP2) using local and density fitting approximations

We apply density fitting approximations to generate the 2-electron integrals in local MP2 (LMP2) to produce a method denoted DF-LMP2. The method can equally be seen as a local version of the well-known RI-MP2 method, which in this work is referred to as DF-MP2. Local approximations reduce the asymptotic scaling of computational resources to O(N), and the most expensive step of DF-MP2 [the O(N5] assembly) is rendered negligible in DF-LMP2. It is demonstrated that for large molecules DF-LMP2 is much faster (1–2 orders of magnitude) than either LMP2 or DF-MP2. The availablility of LMP2, DF-MP2 and DF-LMP2 has for the first time made it possible to assess the accuracy of local and density fitting approximations for extended molecules using cc-pVDZ and cc-pVTZ basis sets. The density fitting errors are found to be consistently small, but the errors arising from local approximations are somewhat larger than expected from calculations on smaller systems. It is proposed to apply local density fitting approximatio...

General orbital invariant MP2-F12 theory

A general form of orbital invariant explicitly correlated second-order closed-shell Moller-Plesset perturbation theory (MP2-F12) is derived, and compact working equations are presented. Many-electron integrals are avoided by resolution of the identity (RI) approximations using the complementary auxiliary basis set approach. A hierarchy of well defined levels of approximation is introduced, differing from the exact theory by the neglect of terms involving matrix elements over the Fock operator. The most accurate method is denoted as MP2-F12/3B. This assumes only that Fock matrix elements between occupied orbitals and orbitals outside the auxiliary basis set are negligible. For the chosen ansatz for the first-order wave function this is exact if the auxiliary basis is complete. In the next lower approximation it is assumed that the occupied orbital space is closed under action of the Fock operator [generalized Brillouin condition (GBC)]; this is equivalent to approximation 2B of Klopper and Samson [J. Chem. Phys. 116, 6397 (2002)]. Further approximations can be introduced by assuming the extended Brillouin condition (EBC) or by neglecting certain terms involving the exchange operator. A new approximation MP2-F12/3C, which is closely related to the MP2-R12/C method recently proposed by Kedzuch et al. [Int. J. Quantum Chem. 105, 929 (2005)] is described. In the limit of a complete RI basis this method is equivalent to MP2-F12/3B. The effect of the various approximations (GBC, EBC, and exchange) is tested by studying the convergence of the correlation energies with respect to the atomic orbital and auxiliary basis sets for 21 molecules. The accuracy of relative energies is demonstrated for 16 chemical reactions. Approximation 3C is found to perform equally well as the computationally more demanding approximation 3B. The reaction energies obtained with smaller basis sets are found to be most accurate if the orbital-variant diagonal Ansatz combined with localized orbitals is used for the first-order wave function. This unexpected result is attributed to geminal basis set superposition errors present in the formally more rigorous orbital invariant methods.

R12 methods in explicitly correlated molecular electronic structure theory

The past few years have seen a particularly rich period in the development of the explicitly correlated R12 theories of electron correlation. These theories bypass the slow convergence of conventional methods, by augmenting the traditional orbital expansions with a small number of terms that depend explicitly on the interelectronic distance r 12. Amongst the very numerous discoveries and developments that we will review here, two stand out as being of particular interest. First, the fundamental numerical approximations of the R12 methods withstand the closest scrutiny: Kutzelniggs use of the resolution of the identity and the generalized Brillouin condition to avoid many-electronic integrals remains sound. Second, it transpires that great gains in accuracy can be made by changing the dependence on the interelectronic coordinate from linear (r 12) to some suitably chosen short-range form (e.g., exp(−αr 12)). Modern R12 (or F12) methods can deliver MP2 energies (and beyond) that are converged to chemical accuracy (1 kcal/mol) in triple- or even double-zeta basis sets. Using a range of approximations, applications to large molecules become possible. Here, the major developments in the field are reviewed, and recommendations for future directions are presented. By comparing with commonly used extrapolation techniques, it is shown that modern R12 methods can deliver high accuracy dramatically faster than by using conventional methods. Contents PAGE 1. Introduction 429  1.1. The origin of the problem 429  1.2. Two-electron systems 430  1.3. Explicitly correlated MP2 methods 430  1.4. Gaussian geminals 431  1.5. Exponentially correlated Gaussians 432  1.6. The transcorrelated method 433 2. R12 wavefunctions 433  2.1. Definition 434  2.2. Correlation factors 435  2.3. Projection operators 437  2.4. Levels of theory 439  2.5. Methods for open shells 440 3. Approximations of many-electron integrals 441  3.1. Exact evaluation 442  3.2. Approximations: GBC, EBC and 443  3.3. Resolution of the identity 445  3.4. Numerical quadrature 447  3.5. Density fitting 449  3.6. DF combined with RI 451 4. Examples from second-order perturbation theory 452  4.1. Technical details 453  4.2. R12 results in comparison with extrapolated values 454  4.3. Comparison between R12 and F12 results 458 5. Perspectives 461  5.1. Higher level methods 461  5.2. Local approximations 461  5.3. Conclusions 462   5.3.1. Correlation factor 462   5.3.2. Projection operator 462   5.3.3. Formulation of intermediate B 463   5.3.4. Approximating integrals 463   5.3.5. Efficiency improvements 463 Acknowledgements 463 References 464

Density fitting in second-order linear-r12 Møller–Plesset perturbation theory

Density fitting is used to approximate all of the 4-index 2-electron integrals in the explicitly correlated MP2-R12 theory of Kutzelnigg and Klopper. The resulting method—DF-MP2-R12—requires only 2- and 3-index integrals over various 2-electron operators, and is extremely efficient. The errors arising from the fitting process can be made small by using robust fitting formulas throughout, such that the error in each fitted integral is quadratic in the error of the fitted orbital product densities. Sample calculations on glycine reveal that for large basis sets DF-MP2-R12 is faster than a standard MP2 calculation and takes only a small fraction of the time for the Hartree–Fock calculation.

Fast Hartree-Fock theory using local density fitting approximations

Density fitting approximations are applied to generate the Fock matrix in Hartree–Fock calculations. By localizing the orbitals in each iteration and performing separate fits for each orbital the scaling of the computational effort for the exchange can be reduced to . We also use the Poisson method to replace almost all Coulomb integrals with simple overlaps, an efficient alternative to diagonalization, and dual basis sets such that the Hartree–Fock calculation is performed in a smaller basis than the subsequent treatment of electron correlation. The accuracy and efficiency of the method is demonstrated in calculations with almost 4000 basis functions. The errors introduced by the local approximations on HF and MP2 energies are small compared to those that arise from the density fitting, and the fitting errors themselves (typically 1–10 microhartree per atom) are very small compared, for example, to the effect of basis set variations.

Analytical energy gradients for local second-order Møller–Plesset perturbation theory using density fitting approximations

An efficient method to compute analytical energy derivatives for local second-order Møller-Plesset perturbation energy is presented. Density fitting approximations are employed for all 4-index integrals and their derivatives. Using local fitting approximations, quadratic scaling with molecular size and cubic scaling with basis set size for a given molecule is achieved. The density fitting approximations have a negligible effect on the accuracy of optimized equilibrium structures or computed energy differences. The method can be applied to much larger molecules and basis sets than any previous second-order Møller-Plesset gradient program. The efficiency and accuracy of the method is demonstrated for a number of organic molecules as well as for molecular clusters. Examples of geometry optimizations for molecules with 100 atoms and over 2000 basis functions without symmetry are presented.

A Simple, Exact Density-Functional-Theory Embedding Scheme

Density functional theory (DFT) provides a formally exact framework for quantum embedding. The appearance of nonadditive kinetic energy contributions in this context poses significant challenges, but using optimized effective potential (OEP) methods, various groups have devised DFT-in-DFT methods that are equivalent to Kohn–Sham (KS) theory on the whole system. This being the case, we note that a very considerable simplification arises from doing KS theory instead. We then describe embedding schemes that enforce Pauli exclusion via a projection technique, completely avoiding numerically demanding OEP calculations. Illustrative applications are presented using DFT-in-DFT, wave-function-in-DFT, and wave-function-in-Hartree–Fock embedding, and using an embedded many-body expansion.

Local explicitly correlated second-order perturbation theory for the accurate treatment of large molecules

A local explicitly correlated LMP2-F12 method is described that can be applied to large molecules. The steep scaling of computer time with molecular size is reduced by the use of local approximations, the scaling with respect to the basis set size per atom is improved by density fitting, and the slow convergence of the correlation energy with orbital basis size is much accelerated by the introduction of terms into the wave function that explicitly depend on the interelectronic distance. The local approximations lead to almost linear scaling of the computational effort with molecular size without much affecting the accuracy. At the same time, the domain error of conventional LMP2 is removed in LMP2-F12. LMP2-F12 calculations on molecules of chemical interest involving up to 80 atoms, 200 correlated electrons, and 2600 contracted Gaussian-type orbitals, as well as several reactions of large biochemical molecules are reported.

Explicitly correlated second-order perturbation theory using density fitting and local approximations.

Three major obstacles in electronic structure theory are the steep scalings of computer time with respect to system size and basis size and the slow convergence of correlation energies in orbital basis sets. Three solutions to these are, respectively, local methods, density fitting, and explicit correlation; in this work, we combine all three to produce a low-order scaling method that can achieve accurate MP2 energies for large systems. The errors introduced by the local approximations into the R12 treatment are analyzed for 16 chemical reactions involving 21 molecules. Weak pair approximations, as well as local resolution of the identity approximations, are tested for molecules with up to 49 atoms, over 100 correlated electrons, and over 1000 basis functions.