R12 methods in explicitly correlated molecular electronic structure theory

Abstract

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: Kutzelnigg's 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