**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 -sum is over a suitable set of substructures of *G*,
is *X*-independent depending
solely on the manner in which embedds in *G*, and the parametric coefficient
is *G*-independent depending on the property *X* as well as the type of cluster expansion implemented.
As a simple example, 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 of structures, whence

where the sum is now over these equivalence classses, and is a sum over for all the
particular substructures which are so equivalent in *G*. In our proto-examplar case 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 -coefficients may be
made. Within the general cluster- expansion formalism there can be:

- different types of weightings for the substructural units (i.e., even for the same
*G*& different choices for can be made, e.g., for the i & j atoms which are connected by a C-C bond one could take with & the degrees of atoms i & j in the C-network); - different interpretations of ``sub-structures'' (e.g., including some geometrical aspects as
whether two additional heavy atoms attached to two carbons of a C=C double bond are
connected in a
*cis*or*trans*manner); - the incorporation of different sets of connected substructures in the expansion (e.g., excluding C-C-C substructures whenever the 1 and 3 of these C atoms are also directly connected, and allowing their contribution to be subsumed in a separate term for 3 mutually interconnected C atoms);
- the combination of sub-structural contributions in a different manner (e.g., with the different
disjoint substructures combined in the
*a*-coefficients as products, so that one may instead view the -sums in the two displayed equations as being over disconnected spanning graphs in*G*, possibly with one of the components distinguished).

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.

[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

[3] T. G. Schmalz, D. J. Klein, & B. L. Sandleback, "Chemical Graph-Theoretic Cluster Expansion and Diamagnetic Susceptibility", J. Chem. Inf. & Comp. Sci.

[4] D. J. Klein, H. Zhu, R. Valenti, & M. A. Garcia-Bach, "Many-Body Valence-Bond Theory", Intl. J. Quantum Chem. S

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