Fragment and localized orbital methods in electronic structure theory

---

Abstract

---

The development of electronic structure methods and rapid advances in computing technology have transformed the role of quantitative numerical calculations in both theoretical and experimental studies of chemistry. The last five decades in which this progress occurred can be roughly divided into three eras. The first era—the 1960s through 80s—established the basic mathematical foundations and computational machinery of today's standard electronic structure methods. They range from Hartree–Fock (HF) theory, configuration-interaction (CI) theory, Møller–Plesset perturbation theory (MP), coupled-cluster (CC) theory, systematic basis sets and Gaussian integral evaluation, to analytical derivative methods as well as excited-state and molecular property calculations. Most notably, density-functional theory (DFT)—the computational workhorse of solid-state physics—was imported and established in chemistry, where it became one of the most popular tools for the study of molecular electronic structures.

The following three decades represented the second era, during which electronic structure theory and its practical software implementations saw acceptance by the broader chemistry community, as reflected by the awarding of the 1998 Nobel Prize in Chemistry to John Pople and Walter Kohn. Electronic structure calculations become a routine or even essential element of experimental research in fields such as synthetic organic and inorganic chemistry, gas-phase physical chemistry and spectroscopy, atmospheric and interstellar chemistry, and many more. As the evidence of the practical utility of these methods mounted, methodological advances continued to be made. For example, excited-state DFT (time-dependent DFT) was established; local-basis linear-scaling algorithms dramatically broadened the applicability of all methods to larger molecules; some of the outstanding weaknesses of DFT functionals were removed; accurate theories and quantitative understanding of weak intermolecular interactions became available; two-electron basis functions were introduced, making MP and CC explicitly correlated and rapidly convergent with basis set size; theories for strong correlation such as multi-reference CI, MP, and CC as well as density matrix renormalization group and other novel approaches were proposed. Concerted efforts made these electronic structure calculations efficiently executable on large numbers of parallel computer processors.

In the past several years, we have entered the third era. There has been a dramatic shift toward using accurate electronic structure methods for large molecular clusters, polymers, solids, surfaces, liquids, and even biomacromolecules such as proteins. While DFT remains the most widely used method for such systems, these newer methods go beyond DFT in the range of problems they can address: optoelectronic properties, excited states, dispersion and other weak intermolecular interactions, and strong correlation, which may be responsible for colossal optoelectronic and magnetic response properties found in many advanced materials and for (photo)catalytic and electrochemical activities. In other words, the electronic structure methods that have traditionally been developed, refined, and used by chemists need to be exported to materials science, solid-state and condensed-matter physics, and biochemistry. The introduction of these new tools will impact those fields in much the same way DFT transformed chemistry after it was introduced in the 1980s.

This special issue, therefore, compiles work from leaders in the field and displays the state-of-the-art in terms of bringing the power of both chemists' electronic structure methods and DFT to macromolecules, solids, and liquids. Making these methods feasible for extended systems relies on, in Kohn's words, the “near-sightedness” of electron-electron correlation. The fundamentally short-range nature of all chemical interactions in the ground state not only makes the total electronic energy extensive, but it also justifies assigning a finite, intensive energy to a fragment or local domain of an extended system which is computable in a finite, size-independent number of arithmetic operations.

The contributions here examine a wide range of issues related to the fragment and localized orbital methods for electronic structures of clusters, solids, surfaces, liquids and biomacromolecules. Within each category, short synopses of the articles are given in alphabetical order according to the first author, although many contributions transcend such simple categorization.

Local correlation theories and algorithms

• Hirata and Ohnishi (DOI: 10.1039/C2CP23958B): That the total energy of a chemical system is an extensive quantity underlies thermodynamics, but proving so mathematically is highly non-trivial. Here, the authors provide a pedagogical, semi-rigorous proof of this principle for electrically neutral, metallic and non-metallic crystals, analyzing the asymptotic decay behavior of all chemical interactions.

• Iwata (DOI: 10.1039/C2CP40217C): Restricting the HF or DFT orbitals to lie on a given fragment has substantial computational benefits, but it inherently omits important interfragment interactions. This article demonstrates one way to re-introduce such interactions using perturbation theory, with examples given for a number of small molecular clusters.

• Kobayashi and Nakai (DOI: 10.1039/C2CP40153C): This Perspective reports the state of the divide-and-conquer methods, with particular emphasis on their ability to handle open-shell and delocalized electron systems, which can be challenging for many local or fragment methods.

• Krause and Werner (DOI: 10.1039/C2CP40231A): In the pursuit of high-accuracy local correlation methods, the authors examine the performance of explicitly correlated local correlated CC singles and doubles with various ways of handling the virtual orbitals. They propose an efficient hybrid approach which combines the advantages of the pair natural orbital and orbital-specific virtual approaches.

• Krisiloff and Carter (DOI: 10.1039/C2CP23757A): The authors describe the incorporation of screening and Cholesky decomposition techniques into their nearly size-extensive local multi-reference averaged coupled-pair functional model to provide computationally affordable and accurate treatments of large, strongly correlated systems.

• Li, Guo, and Li (DOI: 10.1039/C2CP23916G): The authors present a refined version of their cluster-in-molecule local correlation approach that provides higher quality interactions while also simplifying the determination of the clusters. They demonstrate the performance of the model in a series of water clusters, peptides, and a double-helical foldamer.

• McAlexander, Mach, and Crawford (DOI: 10.1039/C2CP23797K): Coupled cluster methods provide high-quality predictions of chiroptical response properties, but their steep computational cost and gauge-dependence issues can be problematic. Here, the authors present a local, gauge-invariant orbital optimized coupled cluster response theory and examine its performance for several classic chiral molecules.

• Mezey (DOI: 10.1039/C2CP40237H)†: An essential computational element in any fragment method is to have an unambiguous definition of ‘natural’ molecular fragments, which have some levels of chemical autonomy and match the concept of functional groups. Here, the author proposes such a definition based on the shape analysis of electron densities. He also discusses mathematical relationships between fragments and complete systems via the holographic electron density theorem.

• Tang, Nafziger, and Wasserman (DOI: 10.1039/C2CP23994A): The partition density functional theory represents an interesting fragment approach for determining (in principle) the exact ground state energy and density of a system from Kohn–Sham calculations on isolated fragments. This article explores the behavior of this approach for different possible fragmentation schemes, providing nice insights into the near-additivity of fragment densities and the fragment occupations.

• Wagner, Stoudenmire, Burke, and White (DOI: 10.1039/C2CP24118H)†: The authors use the density matrix renormalization group (DMRG) to analyze a series of simple one-dimensional atoms and the hydrogen molecule, focusing on the similarities and differences between one and three dimensions. These one-dimensional models provide a useful environment for examining the performance of density functional theory approximations in strongly correlated systems, for example.

• Yanai, Kurashige, Neuscamman, and Chan (DOI: 10.1039/C2CP23767A): The density matrix renormalization group approach allows one to obtain a complete-active-space self-consistent field (CASSCF) solution for systems with dozens of active orbitals, and canonical transformation (CT) theory augments this description with dynamical correlation. The authors discuss their parallel implementation of CT theory and a level-shifting scheme for eliminating certain intruder-states and potential-energy-curve discontinuities in CT theory.

• Zhang and Valeev (DOI: 10.1039/C2CP40222J): In the pursuit of a straightforward means for constructing an effective Hamiltonian in a diabatic basis, the authors combine an accurate treatment of ionized fragments with an inexpensive one-electron treatment of the couplings between fragment states. They demonstrate how well this inexpensive model reproduces high-level coupled cluster results in benzene and DNA base-pair dimers.

Solids and surfaces

• Karalti, Alfè, Gillan, and Jordan (DOI: 10.1039/C2CP00015F): The authors examine the environmentally important interactions between water molecules and the MgO surface using a variety of state-of-the-art electronic structure methods. They demonstrate that very large basis sets (and/or explicitly correlated wave functions) are necessary to achieve high accuracy. By decomposing the symmetry-adapted perturbation theory (SAPT) interaction energies, they also identify key requirements for successful force field models of this system.

• Müller and Paulus (DOI: 10.1039/C2CP24020C): This Perspective summarizes many of the modern electronic structure techniques used to model solids, with particular emphasis on Stoll's method of increments. The authors highlight the impressive variety of systems and properties successfully predicted with this technique, with specific examples to ground-state crystalline properties and adsorption on surfaces.

• Pisani, Schütz, Casassa, Usvyat, Maschio, Lorenz, and Erba (DOI: 10.1039/C2CP23927B): The authors highlight the capabilities of their periodic local second-order MP (MP2) method and other techniques implemented in the CRYSCOR software package with applications ranging from a boron-nitride nanoscroll to methane clathrates and band gaps in three-dimensional crystals. In addition to enabling fully quantum mechanical treatments of solids, these techniques provide benchmarks against which other, more approximate schemes can be tested.

• Sode and Hirata (DOI: 10.1039/C2CP40236J): Substantial interest lies in understanding materials under high pressures. The authors use their fragment method to study the effect of pressure on various properties of hydrogen fluoride crystals, including the structures, phonon dispersion, and phonon density of states, comparing their predictions with experimental data.

• Taylor, Bygrave, Hart, Allan, and Manby (DOI: 10.1039/C2CP24090D): It was recently demonstrated that many traditional density functionals have difficulty predicting the relative energetics of the two experimentally known polymorphs of diiodobenzene as compared to experiment and high-level diffusion Monte Carlo results. Here, the authors show that excellent results are obtained when the DFT results are corrected using pairwise interaction energies calculated at the MP2 level.

• Wen, Nanda, Huang, and Beran (DOI: 10.1039/C2CP23949C): Molecular crystal structure affects the properties of pharmaceuticals, organic semiconductors, energetic materials, and many other systems. This Perspective reviews the various fragment methods used to model molecular crystals and employs a series of examples to highlight the theoretical and computational challenges in achieving predictive accuracy in crystal simulations.

Liquids and clusters

• Herbert, Jacobson, Lao, and Rohrdanz (DOI: 10.1039/C2CP24060B): The authors provide a detailed account of their recently proposed XPol+SAPT method for treating intermolecular interactions accurately and efficiently. They examine molecular clusters and ion-water complexes as models of condensed-phase systems.

• Hu, Jin, Zeng, Hu, and Yang (DOI: 10.1039/C2CP23714H): The authors demonstrate how the density fragment interaction approach can be reformulated to substantially reduce the computational costs such that several experimental properties of liquid water can be predicted accurately using 256-molecule molecular dynamics simulations.

• Leverentz, Maerzke, Keasler, Siepmann, and Truhlar (DOI: 10.1039/C2CP24113G): Fragment methods typically focus on calculating energetics and structures. This contribution demonstrates that these methods also provide useful insights into the electrostatic behavior of systems by accurately predicting dipole moments, partial atomic charges, and charge transfer in a variety of clusters.

• Pruitt, Addicoat, Collins, and Gordon (DOI: 10.1039/C2CP00027J): The authors compare the fragment molecular orbital (FMO) and systematic fragmentation methods on a series of water clusters. Despite their inherent differences, both approaches are capable of accurately reproducing water cluster interactions at greatly reduced costs.

• Yeole, Sahu, and Gadre (DOI: 10.1039/C2CP23761J): The authors use their divide-and-conquer-style molecular tailoring approach to examine the structures of carbon dioxide clusters containing up to 13 molecules. They find several new low-energy structures, and they reproduce the experimentally observed blue shift in the vibrational spectra of larger clusters.

• Zhang, Truhlar, and Gao (DOI: 10.1039/C2CP23758J): The X-Pol method has proved useful for modeling large molecules such as biomolecules. Here, the authors extend the X-Pol approach with an implementation of Ewald summation for treating long-range electrostatics in periodic systems, with a view toward biomolecular and condensed-phase simulations.

Biomolecules

• Aoki and Gu (DOI: 10.1039/C2CP24033E): This Perspective article reviews the development of the elongation method for treating polymers and biopolymers with linear-scaling effort. They discuss applications of the technique to proteins, DNA, and conjugated polymers.

• Collins (DOI: 10.1039/C2CP23832B): The author describes an improved version of his group's systematic fragmentation method which enables a more efficient and physically reasonable fully-automated fragmentation of large molecules such as proteins. He demonstrates how this can be used to perform electronic structure calculations on a protein with over 2000 atoms.

• Faver, Zheng, and Merz (DOI: 10.1039/C2CP23715F): Intramolecular basis-set superposition error (BSSE) plays an increasingly important role in large systems such as proteins. The authors demonstrate the utility of their simple statistical model for estimating BSSE corrections with trivial computational costs.

• Fedorov, Nagata, and Kitaura (DOI: 10.1039/C2CP23784A): The fragment molecular orbital method developed by the authors inspired much of the modern work in fragment methods. Here, the Perspective by them highlights both the theory and the high diversity of systems to which FMO has been successfully applied.

• Hughes, Harvey, and Friesner (DOI: 10.1039/C2CP40220C): Coordinated transition metals play an important role in many biological systems, and the authors develop improvements to their empirical d-block localized orbital correction scheme which enables accurate and inexpensive calculations of ligand removal enthalpies in transition metal complexes.

• Zhu, He, and Zhang (DOI: 10.1039/C2CP23746F): Nuclear magnetic resonance (NMR) has become an important tool in determining protein structures. The authors examine how well fragment-based protein NMR chemical shift calculations reproduce experimental chemical shifts with a view to providing computational assistance for future protein structure determinations.

---

Footnote

† These papers will be published in a separate, later issue of PCCP.

---