D. J. Klein

Chemical Sub-Structural Cluster Expansions for Molecular Properties

Texas A&M University at Galveston, Galveston, Texas 77553-1675

The correlation of different molecular sub-structures with various molecular properties has a long history, of over a century, and it still remains of current (perhaps even central) interest in chemistry. E.g., this interest is manifested in the focus on ``functional groups'' in organic chemistry, particularly in connection with reactivities. An example in physical chemistry concerns the use of bond energy or ``group function'' methods to empirically express thermodynamic heats of atomization, though several other properties are often similarly treated. And there are many other examples of such sub-structural analysis. Here a general systematic way to expand rather general properties in terms of molecular sub-structures is described. The most classical such expansion starts with atomic contributions for each type of atom in a molecule, and then incorporates contributions for each type of chemical bond, and after this neighbor bond-pair contributions may be included, and in principle even further larger substructural units may be considered. For example, in expanding enthalpies such contributions may be simply added together. Formally a general type of cluster expansion for a property X taking the value X(G) for structure (or graph) G may be presented thusly


where the tex2html_wrap_inline35 -sum is over a suitable set of substructures of G, tex2html_wrap_inline39 is X-independent depending solely on the manner in which tex2html_wrap_inline35 embedds in G, and the parametric coefficient tex2html_wrap_inline47 is G-independent depending on the property X as well as the type of cluster expansion implemented. As a simple example, tex2html_wrap_inline39 may be 1 or 0 for each specific set of atoms of G depending whether they are connected or not in the substructure gamma in G (e.g., as two particular atoms are connected or not by a bond, and the next higher sub-structures would be triples of atoms with two of them each connected to the third). Quite often the expansion is rephrased in terms of equivalence classes tex2html_wrap_inline61 of structures, whence


where the sum is now over these equivalence classses, and tex2html_wrap_inline63 is a sum over tex2html_wrap_inline39 for all the particular substructures which are so equivalent in G. In our proto-examplar case tex2html_wrap_inline63 then becomes the number of sub-structures of the given type in G (e.g., the number of C-C bonds in G, and in the next higher order the number of C-C-C units). It is emphasized that cluster expansions may be made in a systematic manner including ever larger connected substructures to attain ever higher precision, and different approaches to the choice of the tex2html_wrap_inline75 -coefficients may be made. Within the general cluster- expansion formalism there can be:

Though many authors have discussed cluster-expansion ideas in varying degrees of generality, an outline of the current general formalism is found in ref. [1].

Some differing aspects of making an expansion may be motivated by the particular property under consideration. The rather broad categorization of each property as ``additively'', ``constantively'', ``multipicatively'', or ``derivatively'' bounded is made - and the inter-relations between and utility of this categorization is noted in motivating the manner of combination of the substructural contributions in a cluster expansion.

A wide range of example applications are possible and some may be noted. These include the cluster expansion of: energies, magnetic susceptibilities, boiling points, NO- bioactivities, statistical-mechanical partition functions, quantum-mechanical wave-functions, and model Hamiltonians. Schemes for the choice of the parameters appearing in the expansions are indicated: via least-squares fitting of all data available (whence an optimal data-set-dependent fit is obtained); via ``Möbius'' inversion (whence a data-set-independent fit is enforced); or via an intermediate ``balanced'' scheme. Example applications are found in refs. [2] , [3], & [4]. Overall the cluster-expansion approach can be advocated as a general method not only to analyze a a molecular structure in terms of its substructures, but more generally to analyze a ``whole''in terms of its ``parts''.

[1] D. J. Klein, "Chemical Graph-Theoretic Cluster Expansions", Intl. J. Quantum Chem. 20 S (1986) 153-171.
[2] T. G. Schmalz, T. P. Sivkovic, & D. J. Klein, "Cluster Expansion of the Hckel Molecular Energy of Acyclics: Applications to Pi Resonance Theory", pages 173-190 in MATH/CHEM/COMP 1987, ed. R. C. Lacher (Elsevier, Amsterdam, 1988).
[3] T. G. Schmalz, D. J. Klein, & B. L. Sandleback, "Chemical Graph-Theoretic Cluster Expansion and Diamagnetic Susceptibility", J. Chem. Inf. & Comp. Sci. 32 (1992) 54-57.
[4] D. J. Klein, H. Zhu, R. Valenti, & M. A. Garcia-Bach, "Many-Body Valence-Bond Theory", Intl. J. Quantum Chem. S 65 (1997) 421-438.
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