Alumni Project
Advanced Methods for Electronic Structure: Local Coupled
Cluster Theory
Martin Head-Gordon, LBNL
In collaboration with the team led by Robert Harrison, ORNL
Summary
High-performance computing offers the promise of being able to model chemical systems of increasing
complexity with increasing realism. At the molecular level (from atoms to thousands of atoms) electronic
structure theory provides the framework to accomplish this goal. But accurate electronic structure theories
have computational costs that rise with the 6th power (or higher) of molecular size, so that 10 times the
computing resources translates into treating a system less than 1.5 times bigger. The goal of this project
is to develop novel "fast" coupled cluster algorithms that are much more scalable with respect to system
size to allow the potential of high performance computing for treating larger systems to be realized.
Our objective is to extend coupled cluster (CC) electronic structure theory to larger molecules. CC
methods are well-established as the wave-function-based electronic structure method of choice. However,
even the simplest CC method, incorporating just correlations between pairs of electrons (double substitutions),
still requires computational costs that scale with the 6th power of the size of the molecule.
Our research is focused on defining new and powerful "local correlation" models that reduce
scaling of coupled cluster calculations, and developing effective algorithms for their implementation.
The basis for these models is to make physically motivated approximations that dramatically reduce the
number of degrees of freedom (the independent variables that describe electron correlations). The number
of pair correlations rises with the 4th power of molecular size without any local model.
Our initial explorations have been focused on testing and developing local CC methods which
operate in a valence active space. This means that a valence electron is described by two functions, one
of which is low energy and typically bonding in character. The other function is higher in energy and is
typically antibonding in character. Pairs of electrons are correlated with each other by jumping from
their bonding to antibonding levels. These correlated fluctuations are normally small to moderate in size,
but become large when a chemical bond is broken. These are the physically most important electron
correlations, but are just a small fraction of all possible pair correlations.
Our first piece of SciDAC-supported work tests the validity of using this valence space
approximations in CC calculations, by comparing predicted molecular properties such as chemical
bond-lengths
and vibrational frequencies to CC theory without this approximation. The results are generally encouraging,
and suggest it is worthwhile to pursue further development of valence space local correlation methods.
We then combine local correlation models with the valence space approximation to yield computationally
efficient CC algorithms. We have reported efficient algorithms for local CC models of this type, under SciDAC
support. Our implementation is atomic-orbital driven and reduces the computational cost of a local coupled
cluster calculation to scale approximately cubically with the size of the molecule.
Compared to the sixth power scaling exhibited by traditional CC methods, this is a substantial
step forward. With a new computer large enough to do a conventional CC calculation on a molecule that is
larger by a factor of x than one could do before, then this same computer would let one do a
molecule x2 larger with this local coupled cluster method! We have been applying this
method to large diradicaloid molecules, including models of a cleaved silicon surface, and a novel
diborodiphosphino species recently isolated experimentally.
We have 3 main areas of interest for new developments over the next year or two. First, is a
novel generalization of the valence space approaches where the new local domain is an atom instead of an
electron pair. Only a linear number of degrees of freedom will be required, making a linear scaling
algorithm possible. At the same time this lifts the valence space approximation, which should yield
higher accuracy. We intend to develop methods for both energies and analytical gradients. The latter
enable prediction of structures and other molecular properties, which is a significant advantage relative
to earlier linear scaling local correlation methods.
The second area of focus is the development of more accurate "fast methods" that still operate
within the valence space approximation. This is motivated by the fact that in a numerical study of the
Cope rearrangement (an organic chemical reaction), we discovered that the form of the potential energy
surface is qualitatively incorrect in the current local models. Thus there is a need for a further level
of sophistication in the valence local correlation modeling. We are starting work on a next level of
modeling, which adds additional ionic interpair excitations, while retaining only a quadratic number
of degrees of freedom. We believe it can give close to 98% of the valence correlation energy, and may
be the most sophisticated two-center valence local correlation model possible.
The third area of focus is combining our simplest valence local correlation model with
Quantum Monte Carlo (QMC) methods. QMC has yielded impressive results for molecular relative energies.
A recent review calls for "integrating quantum chemistry ... packages with QMC methods, and improving
optimization methods and developing more efficient functional forms for variational wavefunctions".
Taking up this challenge, we intend to use our local CC wavefunction as a next generation choice of
guiding function for fixed node diffusion QMC calculations, in collaboration with Prof. W.A. Lester
(Berkeley). This offers the promise of extremely high accuracy.
Our SCIDAC project is in collaboration with other electronic structure groups, led by
R.J. Harrison (ORNL), with whom we share algorithms and ideas. Our algorithmic developments complement
those of Piecuch (Michigan State), Gordon (Iowa State), and Scuseria (Rice), in particular. There are
also interesting issues in optimization and iterative linear solvers relevant to applied mathematics.
Our chemical applications are relevant to those of Gordon (Iowa State) and Schaefer (Georgia) amongst
others. All these electronic structure-based initiatives, provide infrastructure needed for subsequent
kinetic modeling, such as pursued by Wagner (ANL), which in turn feeds into reactive flow models that
are more macroscopic, and capable of modeling challenging problems such as combustion at a systems level.
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