Klopman-Salem equation

The Klopman-Salem equation describes the energetic change that occurs when two chemical species react and their associated molecular orbitals interact with each other. First described independently by Gilles Klopman^[1] and Lionel Salem^[2] in 1968, this relationship provides a mathematical basis for the key assumptions of frontier molecular orbital theory (i.e., theory of HOMO-LUMO interactions) and hard soft acid base (HSAB) theory.

Formulation and interpretation

In modern form,^[3] the Klopman-Salem equation is commonly given as

$\Delta E={\Big (}-\sum _{a,b}(q_{a}+q_{b})\beta _{ab}S_{ab}{\Big )}+{\Big (}\sum _{k<\ell }{\frac {Q_{k}Q_{\ell }}{\varepsilon R_{k\ell }}}{\Big )}+{\Big (}\sum _{r}^{\mathrm {occ.} }\sum _{s}^{\mathrm {unocc.} }-\sum _{s}^{\mathrm {occ.} }\sum _{r}^{\mathrm {unocc.} }{\frac {2(\sum _{a,b}c_{ra}c_{sb}\beta _{ab})^{2}}{E_{r}-E_{s}}}{\Big )}$ ,

where

$q_{a}$ is the electron population in atomic orbital a,

$\beta _{ab}$ , $S_{ab}$ are the resonance and overlap integrals for the interaction of atomic orbitals a and b,

$Q_{k}$ is the total charge on atom k,

$\varepsilon$ is the local dielectric constant,

$R_{k\ell }$ is the distance between the nuclei of atoms k and l,

$c_{ra}$ is the coefficient of atomic orbital a in molecular orbital r,

and $E_{r}$ is the energy of molecular orbital r.

Broadly speaking, the first term describes to the closed-shell repulsion of the occupied molecular orbitals of the reactants (four-electron filled-filled interactions). The second term describes the Coulombic attraction or repulsion between the atoms of the reactants (ionic contribution). Finally, the third term accounts for all possible interactions between the occupied and unoccupied molecular orbitals of the reactants (two-electron filled-unfilled interactions).

Because of the difference in MO energies appearing in the denominator of the third term, energetically close orbitals make the biggest contribution. Hence, approximately speaking, analysis can often be simplified by considering only the highest occupied and lowest unoccupied molecular orbitals of the reactants (the HOMO-LUMO interaction in frontier molecular orbital theory).^[4] The relative contributions of the second (ionic) and third (covalent) terms play an important role in justifying hard soft acid base theory (HSAB), with hard-hard interactions governed by the ionic term and soft-soft interactions governed by the covalent term.^[5]

References

↑ Klopman, Gilles (1968-01-01). "Chemical reactivity and the concept of charge- and frontier-controlled reactions". Journal of the American Chemical Society. 90 (2): 223–234. doi:10.1021/ja01004a002. ISSN 0002-7863.
↑ Salem, Lionel (1968-01-01). "Intermolecular orbital theory of the interaction between conjugated systems. I. General theory". Journal of the American Chemical Society. 90 (3): 543–552. doi:10.1021/ja01005a001. ISSN 0002-7863.
↑ Fleming, Ian (1976). Frontier Orbitals and Organic Chemical Reactions (Reprinted 2006 ed.). Chichester, UK: Wiley. p. 27. ISBN 0471018201.
↑ Fukui, Kenichi (1982). "Role of Frontier Orbitals in Chemical Reactions". Science. 218 (4574): 747–754. Bibcode:1982Sci...218..747F. doi:10.1126/science.218.4574.747. JSTOR 1689733.
↑ Pearson, Ralph G. (1997). Chemical Hardness. Wiley-VCH Verlag GmbH & Co. KGaA. pp. 1–27. doi:10.1002/3527606173.ch1/summary. ISBN 9783527606177.

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[1] Klopman, Gilles (1968-01-01). "Chemical reactivity and the concept of charge- and frontier-controlled reactions". Journal of the American Chemical Society. 90 (2): 223–234. doi:10.1021/ja01004a002. ISSN 0002-7863.

[2] Salem, Lionel (1968-01-01). "Intermolecular orbital theory of the interaction between conjugated systems. I. General theory". Journal of the American Chemical Society. 90 (3): 543–552. doi:10.1021/ja01005a001. ISSN 0002-7863.

[3] Fleming, Ian (1976). Frontier Orbitals and Organic Chemical Reactions (Reprinted 2006 ed.). Chichester, UK: Wiley. p. 27. ISBN 0471018201.

[4] Fukui, Kenichi (1982). "Role of Frontier Orbitals in Chemical Reactions". Science. 218 (4574): 747–754. Bibcode:1982Sci...218..747F. doi:10.1126/science.218.4574.747. JSTOR 1689733.

[5] Pearson, Ralph G. (1997). Chemical Hardness. Wiley-VCH Verlag GmbH & Co. KGaA. pp. 1–27. doi:10.1002/3527606173.ch1/summary. ISBN 9783527606177.