Chapter 4 On the Selection of Domains and Orbital Pairs in Local Correlation Treatments

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This chapter discusses and compares different strategies to improve the accuracy by domain extensions at fixed geometries. This allows approaching the non-local solution to arbitrary precision. However, the computational effort increases rather quickly with the domain sizes, in particular for local coupled cluster with single and double excitations and perturbative treatment of triple excitations [LCCSD(T)], and therefore, it is important to find a best compromise between accuracy and cost. The accuracy of local coupled-cluster calculations depends on (1) the domain sizes and (2) the definition of the strong-pair list included in the LCCSD. These approximations can be controlled using connectivity or distance criteria. In most cases, the domain and the weak-pair approximations in LCCSD(T0)|LMP2 calculations compensate each other to a large extent. Therefore, rather accurate results are obtained with the standard approximation in which the domains are determined by the BP procedure and strong orbital pairs are determined by the condition that the domains of the two orbitals overlap.

Section snippets

1. Introduction

Local correlation treatments, as originally proposed by Pulay [1] and first implemented by Pulay and Saebø[2], [3], [4], [5], [6] employ local orbital spaces to restrict the number of excited configurations in the wave function. The occupied orbitals are localized using standard procedures (e.g. Boys [7] or Pipek-Mezey [8] localization), while the virtual space is spanned by non-orthogonal projected atomic orbitals (PAOs). By fully exploiting the locality, our group has achieved linear scaling

Method

In the following sections, we summarize the criteria used to define the local orbital spaces and domains as implemented in the MOLPRO package of ab initio programs [35]. The keywords for thresholds are the same as used in this program.

Standard domains

For each correlated occupied orbital φiLMO an orbital domain [i] is generated according to the procedure proposed by Boughton and Pulay [17]. Some modifications were made in our program, which are outlined in the following. The purpose of the method is to include all PAOs in the domain, which arise from AOs (basis functions), that significantly contribute to the considered LMO. The PAOs arising from AOs at a given atom are considered as a group. Therefore, the first step is to select a set of

Pair classes

The orbital pairs (ij) are classified according to the closest distance R(ij) between atoms in the primary domains [i] and [j]. This classification is independent of domain extensions. Furthermore, only atoms in the primary domains are considered for the pair classification if the atomic Löwdin charge is larger than CHGMIN_PAIRS (default value 0.2). This criterion was introduced in order to reduce the dependence of the pair selection on localization tails. The strong pairs (0≤R(ij)<RCLOSE)

Triple excitations

As for the double excitations, the triple excitations are restricted to domains (cf. Section 3.3) and by a triples list (ijk) of LMOs, so that the total number of triple excitations scales linearly with molecular size. As discussed in detail in Ref. [14], the triples list (ijk) is defined by the condition that the pairs (ij), (ik), or (jk) must be either strong or close pairs. Additionally, at least one of these pairs must be strong. Second, the close pair amplitudes as determined in the

Dependence of the correlation energy on the domain approximation

In order to demonstrate the effect of the domain approximation on correlation and reaction energies, we have studied 52 chemical reactions involving 59 molecules [38]. Here we present only a representative subset of the results. The geometries of all molecules have been optimized at the MP2/aug-cc-pV(T+d)Z level. Table 1, Table 2, Table 3 show the computed correlation energies of 22 molecules for LMP2, LCCSD, and LCCSD(T0). In each case, the results are compared to the full non-local

Conclusions

We have reviewed the approximations made in the linear scaling local correlation methods developed in our group. The accuracy of local coupled-cluster calculations depends on (i) the domain sizes and (ii) the definition of the strong-pair list included in the LCCSD. These approximations can be controlled using connectivity or distance criteria. The domain sizes can be determined by a single parameter (IEXT or REXT), and we have shown that domain extensions lead to rapid and systematic

Acknowledgements

This work has been supported by the Deutsche Forschungsgemeinschaft and the Fonds der Chemischen Industrie. The authors thank Dr. A. Schäfer for providing the experimental data as well as zero-point and thermodynamic corrections for the reaction enthalpies.

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