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Size-Extensivity and Size-Consistency

    Two important concepts in electronic structure theory are size-consistency and size-extensivity. Though these terms are sometimes used interchangeably in the literature, there are very important distinctions to be made between them.

There are two primary definitions of size-consistency in use. The first was employed by Pople gif as one criterion for a well-constructed quantum chemical method. If we imagine two molecules, separated by a large distance (large enough that we may consider them to be non-interacting) then the energy calculated for both molecules simultaneously should be exactly twice that calculated for only one, isolated molecule of , just like the exact energy. This ``non-interacting limit'' description is the original concept of size-consistency. From this perspective, size-consistency describes what has been referred to as the ``additive separability''  of the wavefunction. However, a more recently imposed definition[27] requires that the method not only correctly describe the fragmentation limit, but the entire process (in a qualitative sense). That is, the entire potential energy curve mapped out when we bring our two non-interacting molecules close together must be correctly described as well. For example, both spin-unrestricted Hartree-Fock (UHF)   and spin-restricted Hartree-Fock (RHF)   wavefunctions are size-consistent for the separated dimer system described above. However, for a closed-shell molecule dissociating into open-shell fragments, a RHF wavefunction does not conform to the second definition of size-consistency, as we will discuss further below.

Size-extensivity, on the other hand, is a more mathematically formal characteristic which refers to the correct (linear) scaling of a method with the number of electrons. The term was introduced to electronic structure theory by Bartlett[25]  gif, and is based on analogous ``extensive'' thermodynamic properties. All Hartree-Fock   methods qualify as size-extensive, as well as many-body perturbation theory    and coupled-cluster theories . Truncated configuration interaction   methods, however, are not size-extensive. An important advantage of a size-extensive method is that it allows straightforward comparisons between calculations involving variable numbers of electrons, e.g. ionization processes or calculations using different numbers of active electrons. Lack of size-extensivity implies that errors from the exact energy increase as more electrons enter the calculation.

Size-extensivity and size-consistency are not mutually exclusive properties, by any means. At the non-interacting limit, size-extensivity of a method is a necessary and sufficient condition to ensure size-consistency, implying that the former is more general than the latter. However, size-extensivity does not ensure correct fragmentation. For example, we may consider two different fragmentation processes for :

and

The first process is correctly described by both RHF and UHF wavefunctions, and hence, both methods are size-consistent. However, the second process is not correctly described by a RHF wavefunction (and, therefore, perturbation theory and coupled-cluster theory methods which use this as a reference will not be size-consistent.) Both RHF and UHF are always size-extensive, though. This implies, then, that size-consistency is more general than size-extensivity, but this is also incorrect. At non-interacting limits, size-extensivity is a more general property, and its existence implies that of size-consistency. However, size-consistency has the additional requirement of correct fragmentation that is not necessarily dependent on the mathematical scaling of the energy. Chapter 1 of the review by Taylor[27]  deals explicitly with these concepts, and the interested reader is urged to study this reference.    



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Next: Formal Coupled-Cluster Theory Up: Fundamental Concepts Previous: Other Cluster-Based Wavefunctions



© 1996 by T. Daniel Crawford  / crawdad@otanes.ccqc.uga.edu
Last modified: Fri Jan 19 17:13:01 EST 1996