Abstract
Quantum chemistry was a compute intensive field from the beginning. It was also an early adopter of parallel computing, and hence, has more than twenty years of experience with parallelism. However, recently parallel computing has seen dramatic changes, such as the rise of multi-core architectures, hybrid computing, and the prospect of exa-scale machines requiring 1 billion concurrent threads. It is doubtful that current approaches can address the challenges ahead. As a result, the field finds itself at a crossroads, facing the challenge to successfully identify the way forward. This chapter tells a story in two parts. First, the achievements to date are considered, offering insights learned so far. Second, we look at paradigms based on directed acyclic graphs (DAG). The computer science community is strongly advocating this paradigm, but the quantum chemistry community has no experience with this approach. Therefore recent developments in that area will be discussed and their suitability for future parallel quantum chemistry computing demands considered.
TopIntroduction
Quantum chemistry was formulated with the Schrödinger equation in1926 as the science that seeks to explain chemistry, chemical processes and properties of molecules from solutions of this equation. As a result this field of science was a compute intensive enterprise from the start. Since the beginning, when the word “computer” referred to a job description rather than a machine (Kopplin, 2002), there have been dramatic developments both on the theory side and as well as the practical side. On the theory side the formulation of the problem in a basis set, the formulation of theories to deal with the complex electron-electron interaction, such as Density Functional Theory (DFT) (Hohenberg & Kohn, 1964), Many Body Perturbation Theory (MBPT) (Møller & Plesset, 1934), and Coupled Cluster (CC) (Čížek, 1966), provided important handles on the problem. On the practical side the rapid development of computer technology and the design of efficient algorithms delivered access to the amount of computation needed to deliver useful answers.
In particular the development of computers followed initially a straightforward path of ever increasing compute power from single processing units. Most recently this trend has been broken as practical limits were hit. As it became impossible to simply crank up the clock speed even higher processor engineers opted instead for building chips with multiple processing units, the so called multi-core processors. This is a significant change. Although parallelism, as we currently know it, was introduced in quantum chemistry as early as 1988 (Clementi et al.,1988) and continuous and significant investment in parallel codes was made there was always the excuse that most machines were actually single processor machines to avoid becoming embroiled in code parallelization. However, now that every new machine is effectively a parallel machine it is clear that only parallel applications can utilize the available hardware effectively. But the changes go far beyond this point. Alongside the development of multi-core processors the development of Graphics Processing Units (GPU) towards General Purpose computing (GPGPU) created a new level of parallelism (Ufimtsev et al., 2008) (Vogt et al., 2008) (Genovese et al., 2009). The GPGPU’s of today offer the potential of using 240 cores concurrently within a single physical machine. This is paired with the development of off-the-shelf solutions to connect multiple machines to build compute clusters. As a result relatively powerful machines can be built for low costs at present.
Nevertheless, the development towards exa-scale computers is likely going to be disruptive. The reason for this lies in the energy requirements for such a machine. Having that the performance is specified and thereby the CPU power consumption to a large extent fixed it will be essential to save on memory and network power consumption to deliver these machines. This, however, raises the bar for writing scalable applications as there will be less space to keep data but it will also become more expensive to move data.
Against this back-drop of computer developments let us consider how close we have come to delivering a dream that was once captured in the title of a Europort2 project “Interactive Molecular Modeling through Parallelism” (Colbrook et al., 1995). What has been learned about parallelism in quantum chemistry and do we currently have a way to tackle the next level of parallelism?
Key Terms in this Chapter
Density Matrix: The electron density is given by multiplying a pair of basis functions with a weight and summing this over all pairs. The weights in this case actually form a matrix, the density matrix. As the basis functions themselves are often combined with operators to give matrix representations of energy terms, the density matrix also has acquired an important role of its own in current matrix based methods.
Atomic Basis Functions: The equations that govern the behavior of electrons in quantum chemistry cannot be solved exactly. To arrive at approximate methods that can be used on a computer the solutions of the equations are formally expressed as linear combinations (summations) of known functions. These known functions are referred to as the basis set. From basic physical considerations we know that for atoms these known functions should decay exponentially with the distance to the nucleus. If one choose functions with (approximately) this characteristic for the basis set one is using atomic basis functions.
MP2: This is short for “second order Møller-Plesset perturbation theory”. It is an attempt to correct for the assumption that electrons only feel where other electrons are on average in Hartree-Fock theory. Formulated by C. Møller and M.S. Plesset in 1934 it starts from Hartree-Fock theory and adds a perturbative estimate of the energy associated with the electrons actually feeling eachothers precise positions. This energy is referred to as the correlation energy. Apart from DFT this method is the computationally cheapest way to account for electron correlation and still frequently used.
Density Functional Theory (DFT): This a theory that states that the exact energy of a many electron system depends only on the electron density. The assumptions underlying the theory were proven by Hohenberg and Kohn in 1964 (in 1998 Kohn received the Noble prize for this). In 1965 Kohn and Sham proposed a method for solving the corresponding equations which is almost the same as the Hartree-Fock method but with one term replaced by the so called exchange-correlation functional. Today it is the most commonly used ab-initio method.
Fock Matrix: The Fock matrix results from differentiating the Hartree-Fock energy expression (or other similar energy expressions often referred to as effective one-electron models) with respect to the density matrix elements. Hence is also a measure of the change of the energy with respect to changes in the density matrix. As such it often used in optimization algorithms.
Orbital: An orbital is a function that describes the behavior of a single electron. The word stems from the word “orbit” which is the path a macroscopic body follows around an attractor, which shows clear similarities with the behavior of an electron near an atomic nucleus.
Hartree-Fock: This is the simplist ab-initio energy expression to describe many electron systems such as molecules. Essentially it assumes that a given electron only “feels” where the other electrons are on average, rather than being influenced by the precise positions of all other electrons all of the time. It was formulated in its present form as early as 1935 by D. R. Hartree. In 1955 it was reformulated by Roothaan to facilitate solving the equations on a computer introducing the concept of Linear Combinations of Atomic Orbitals (LCAO). Today every quantum chemistry calculation still starts with a Hartree-Fock level calculation.