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Explicitly Correlated Local Electron Correlation Methods

Hans-JoachimWerner, Christoph Köppl, Qianli Ma, and Max Schwilk

Institute for Theoretical Chemistry, University of Stuttgart, Germany

1.1 Introduction

Accurate wave function methods for treating the electron correlation problem are indispensable in quantum chemistry. A well-defined hierarchy of such methods exists, and in principle, these methods allow to approach the exact solution of the non-relativistic electronic Schrödinger equation to any desired accuracy. A much simpler alternative is density functional theory (DFT), which is probably most often used in computational chemistry. However, its failures and uncertainties are well known, and there is no way for systematically improving or checking the results other than comparing with experiment or with the results of accurate wave function methods.

Due to the steep scaling of the computational resources (CPU-time, memory, disk space) with the molecular size, conventional wave function methods such as CCSD(T) (coupled-cluster with single and double excitations and a perturbative treatment of triple excitations) can only be applied to rather small molecular systems. For example, the CPU-time of CCSD(T) scales as , where is a measure of the molecular size (e.g., the number of correlated electrons) and even the simplest electron correlation method, MP2 (second-order Møller-Plesset perturbation theory) scales as . This causes a "scaling wall" that cannot be overcome. Even with massive parallelization and using the largest supercomputers, this wall can only be slightly shifted to larger systems. However, it is well known that electron correlation in insulators is a short-range effect. The pair correlation energies decay at long-range with R- 6, where R is the distance between two localized spin orbitals. Therefore, the steep scaling is unphysical. It results mainly from the use of canonical molecular orbitals, which are usually delocalized over larger parts of the molecule.

The scaling problem can be much alleviated by exploiting the short-range character of electron correlation using local orbitals and by introducing local approximations. This was first proposed in the pioneering work of Pulay et al. [1-6], and in the last 20 years enormous progress has been made in developing accurate local correlation methods. There are two different approaches, both of which are based on the use of local orbitals. The traditional one is to treat the whole molecule in one calculation and to apply various approximations that are based on the fast decay of the correlation energy. We will denote such methods "local correlation methods." A large variety of such approaches has been published in the past [7-59].

The second approach is the so-called "fragmentation methods" [60-87], in which the system is split into smaller pieces. These pieces are treated independently, mostly using conventional methods (although the use of local correlation methods is also possible). The total correlation energy of the system is then assembled from the results of the fragment calculations. Various methods differ in the way in which the fragments are chosen and the energy is assembled. A special way of assembling the energy using a many-body expansion is used in the so-called incremental methods [88-96], but these also belong to the group of fragmentation methods. Fragmentation methods will be described in other chapters of this volume and are therefore not the subject of this chapter. However, in Section 1.13, we will comment on the relation of local correlation and fragmentation methods.

Another problem of the CCSD(T) method is the slow convergence of the correlation energy with the basis set size. Very large basis sets are needed to obtain converged results, and this makes conventional high-accuracy electron correlation calculations extremely expensive. This problem is due to the fact that the wave function has a cusp for r12 0, where r12 is the distance between two electrons. The cusp is due to the singularity of the Coulomb operator , and cannot be represented by expanding the wave function in antisymmetrized products of molecular orbitals (Slater determinants). This leads to the very slow convergence of the correlation energy with the size of the basis set, and in particular with the highest angular momentum of the basis functions. This problem can be solved by including terms into the wave function that depend explicitly on the distance r12, and these methods are known as "explicit correlation methods" [97-155].

The combination of explicit correlation methods with local approximations has been particularly successful [140-153]. As will be explained and demonstrated later in this chapter, this does not only drastically reduce the basis set incompleteness errors, but also strongly reduces the errors caused by local approximations. Local correlation methods employ two basic approximations. The first is based on writing the total correlation energy as a sum of pair energies. Each pair describes the correlation of an electron pair (in a spin-orbital formulation), or, more generally, the correlation of the electrons in a pair of occupied local molecular orbitals (LMOs). Depending on the magnitude of the pair energies, it is possible to introduce a hierarchy of "strong," "close," "weak," or "distant" pairs [7,18,31,32]. Different approximations can be introduced for each class, ranging from a full local coupled-cluster (LCCSD) treatment for strong pairs to a non-iterative perturbation correction for distant pairs, which can be evaluated very efficiently using multipole approximations [12, 13]. We will denote such approximations as "pair approximations." The second type of local approximations is the "domain approximation," which is applied to each individual pair. A domain is a subset of local virtual orbitals which is spatially close to the LMO pair under consideration. Asymptotically, the number of orbitals in each pair domain (the "domain sizes") become independent of the molecular size. Also the number of pairs in each class (except for the distant pairs) becomes independent of the molecular size. This leads to linear scaling of the computational effort as a function of molecular size, as has already been demonstrated for LMP2 and up to the LCCSD(T) level of theory more than 25 years ago [12-18].

The critical question is, of course, how quickly the correlation energy as well as relative energies (e.g., reaction energies, activation energies, intermolecular interaction energies, and electronic excitation energies) converge with the domain sizes and how they depend on the pair approximations. 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) [156], and this has first been fully exploited in the seminal PNO-CI and PNO-CEPA methods of Meyer [157, 158], and somewhat later also by others [159-163]. The problem with this approach is that the PNOs are different for each pair and non-orthogonal between different pairs. This leads to complicated integral transformations and prevented the application of PNO methods to large molecules for a long time. The method was revived by Neese and coworkers in 2009 and taken up also by others (including us) later on [32,33,48-57,146-150]. The problem of evaluating the integrals was overcome by using local density-fitting approximations [22]. Furthermore, the integrals are first computed in a basis of projected atomic orbitals (PAOs), which are common to all pairs, and subsequently transformed to the pair-specific PNO domains [54,146,147,150]. Also, hybrid methods, in which so-called orbital-specific virtuals (OSVs) [164-167] are used at an intermediate stage, have been proposed [53, 146, 147]. Later sections of this chapter will explain these approaches in some detail.

Local approximations have also been developed for multi-reference wave functions [168-176]. The description of these methods is beyond the scope of the current article, but we mention that recently very efficient and accurate PNO-NEVPT2 [175] (N-electron valence state perturbation-theory) and PNO-CASPT2 [176] (complete active space second-order perturbation theory) methods have been described.

In the current article, we will focus on new developments of well-parallelized PNO-LMP2-F12 and PNO-LCCSD-F12 methods recently developed in our laboratory. These methods also have a close relation to the methods of Neese et al. [54-57, 153]. After introducing some benchmark systems, which will be used later on, we will first outline the principles of local correlation and describe the choice of the local occupied and virtual orbitals as well as of the domains. The convergence of the correlation energy as a function of the domain sizes will be demonstrated for various types of virtual orbitals for LMP2. Subsequently, based on these foundations, we will discuss more advanced approximations for distant pairs and close/weak pair approximations used in local coupled...