Remarkably simple relationship connecting the calculated geometries of isomolecular states of three different multiplicities

We argue that for a large class of molecules, M, there exists an unexpectedly simple relationship among the calculated values of bond lengths in M-0 (its ground singlet state), M-T (its first excited triplet state), M.- (the ground-state doubler of its radical anion), and M.+ (the ground-state doublet of the related radical cation). An estimate of the equilibrium bond length (R) over bar for given bond in M-T may be obtained as follows: (R) over bar(M-T) = (R) over bar(M.-) + R(M.+) - (R) over bar(M-0). The accuracy of the R(M-T) so calculated is usually 0.01-0.03 Angstrom, less than 10% of whole range of distances for the species considered. The error is usually larger for small basis sets and for semiempirical parametrizations but is almost basis-set-independent for medium (D95, 3-21G) and large basis sets (6-31G, 6-31G*, 6-31G**, 6-311G**). The above equation may be qualitatively understood recognizing the "paired" properties of HOMO and LUMO in alternant hydrocarbons. We approach a quantitative rationalization of the relation from a general perspective of one-electron operators. Any property that can be represented by a one-electron operator should be subject to such a simple relationship. However, equilibrium bond lengths are not represented by one-electron operators. Instead, upon introduction of the empirical notion that equilibrium bond lengths are linear in bond order, the simple equation can be justified as an excellent approximate form. Several other relationships, still reasonably rooted in the shape of potential energy surfaces, do not fit as well. The simple relation applies exclusively to bonds constituting the chromophore part of a molecule and works best for systems with conjugated double bonds.