Accurate properties for open-shell states of large molecules

Peter R. Taylor, University of Warwick

Summary

Our project involves the development and implementation of new computational methods for treating molecules containing unpaired electrons. These methods are designed form the outset to provide results of high accuracy and to perform well on parallel high-performance computer systems. We aim to extend the range of computations on open-shell systems to handle more atoms, more unpaired electrons, and to achieve higher accuracy.

Chemists are usually introduced to molecular electronic structure in terms of filled shells of molecular orbitals: functions that describe the behaviour of each electron in the molecule. In these filled shells or closed-shell systems, each molecular orbital is occupied by a pair of electrons which differ in their spin states: one having spin up and the other spin down. The overall spin angular momentum of such a system is zero. The electronic structure of many molecules is of this general type, but there are also many molecules in which not all of the electrons are paired: the molecular orbitals are not all doubly occupied. Simple examples of such open-shell systems include radical species, excited electronic states, and the majority of transition-metal compounds.

In general, open-shell systems are more difficult to describe using electronic structure methods than are closed-shell systems. Nevertheless, existing software packages allow us to treat reliably open-shell systems of up to forty or fifty atoms, and with perhaps a dozen or more unpaired electrons. The particular technique is termed multireference perturbation theory and a common acronym is CASPT2. However, extending the limits of CASPT2 is much more difficult than extending the range of treatments for closed-shell systems. First, for all but the crudest approaches, the work required to treat a given number of open shells rises steeply (sometimes factorially) with the number of shells. Second, there is often more delocalization of the electrons in open-shell systems, which makes it harder to develop methods for treating large numbers of atoms. In particular, the ability to localize the molecular orbitals that can be exploited in closed-shell systems is more constrained in open-shell systems, because the calculated energy is not necessarily invariant to such transformations.

We should perhaps note that the availability of high-performance computing resources is not, in itself, a solution to these problems: new methods must be developed and implemented in order to use these resources effectively to treat open-shell systems. Although the first phase of this work primarily involves developments in computational chemistry, an efficient implementation will profit from SciDAC Integrated Software Infrastructure Centers (ISIC) activities. For example, we have already had discussions with members of the Algorithmic and Software Framework for Applied Partial Differential Equations ISIC about improved methods for matrix diagonalization.

Our first priority is to be able to calculate molecular properties such as the equilibrium geometry, excitation energies, and electric and magnetic properties for open-shell systems at a high level of accuracy. Mathematically, all of these properties can be related to differentiation of the energy, and we will therefore be deriving and implementing formulas for the first derivative of the CASPT2 energy. This has not been done previously and has been considered a very complicated task. We are collaborating with the groups in Tromso Norway and Ferrara Italy to reformulate the CASPT2 equations in a way that lends itself more readily to differentiation and allows for more efficient parallelization.

Subsequent priorities involve two different directions. The first involves scalable methods for treating large systems accurately. We have developed a technique for closed-shell systems that provides much higher accuracy that conventional methods and is much more scalable (even small systems involving half a dozen atoms can be run efficiently on 1000 processors!). As part of another project we are coupling this technique to the CASPT2 method to treat open shells, and our plan is to examine fitting methods that will extend this to molecules of perhaps hundreds of atoms.

The other subsequent direction is targeted at systems with many open shells. Most open-shell systems involve only a few unpaired spins, and existing computational techniques for accurate properties typically handle up to twelve or fourteen unpaired electrons. However, there are very important systems with many more unpaired electrons than this. One example is provided by biology: there are enzymes such as the ferredoxins (crucial electron transfer species) that have multiple metal ions with unpaired spins. These enzymes contain clusters of iron and sulfur atoms with up to four iron atoms, as depicted in Fig. 1.

Fig.1 Iron-sulfur cluster from ferredoxin

Depending on the oxidation state of the irons, such a cluster may have up to twenty unpaired spins. Another example is provided by work on "molecular magnets": molecules that can act as tiny magnets and may feature up to fifty unpaired spins. Such molecular magnets offer promise as new materials for data storage.

At present we have no accurate methods for studying such systems. Methods that can be applied efficiently to, e.g., twelve unpaired spins scale disastrously with the number of spins: the work increases so rapidly that even twenty spins would be out of the question. We plan to take advantage of the rather weak coupling between the metal atoms to develop a factorization of the problem into a set of sub-problems. The sub-problems will be small enough for us to treat them using conventional techniques and we only have to implement new schemes for the coupling between the sub-problems.

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