Application of recurrent relations in chemistry

Abstract

As it was previously demonstrated, mathematical properties of simplest first-order recurrent relations A(n R 1) ¼ aA(n) R b provide important advantages at their applications in chemistry, namely in the approximation of monotonous variations of different physicochemical properties of homologues of various series of organic compounds. Besides that it was empirically revealed the unique 'chemical' property of recurrences that is the application of single equations of this kind to constants of different series. To explain this high 'approximating ability' of recurrences it was proved first time that numerical sequences that obey high-order recurrence in a limiting case can be approximated by first-order recurrence. Consequently, this fact explains us the existence of close analogy between first-order recurrent approximation of recursive numerical sequences (Fibonacci, Padovan, Lucas and Perrin sequences are considered as examples) and physicochemical constants of homologues. This analogy is considered as not 'chemically' correct proof, but reasonable explanation of the applicability of single recurrent equations to constants of different homologous series. The mentioned properties of recurrences seem to be the reason of their high approximating ability in relation to different chemical variables. Recurrent relations can be recommended as universal approach in processing of not only various sets of physicochemical constants of homologues, but also properties or processes characterized by equidistant values of arguments.

Approximation of physicochemical properties of homologs using recurrent and related non‐recurrent relations

“…The recurrent approximation of physicochemical properties was suggested first for any kinds of homologs (normal linear, multi-row, cyclic, insertion, etc.) However, later it was shown [5,8,9] that the coefficients of recurrent relations (2) are very close to each other for homologs of different series with the same homologous differences (for each property separately). However, later it was shown [5,8,9] that the coefficients of recurrent relations (2) are very close to each other for homologs of different series with the same homologous differences (for each property separately).…”

“…The mathematical properties of recurrences were considered in the previously published papers [1][2][3][4][5][6][7][8][9]. Here, it seems reasonable to discuss shortly only the main of them, associated with their chemical applications in the following discussion.…”

“…This feature of recurrent relations in chemistry was interpreted by considering analogy between physicochemical properties of homologs and members of numerical sequences [5,9].…”

“…Recent studies [1][2][3][4][5][6][7][8][9] showed that monotonic variation of most of values of physicochemical properties (A) of homologs within various taxonomic groups of organic compounds can be approximated with recurrent (recursive) relations:…”

“…As for recurrences, this problem was briefly discussed [1,8] Temperature dependence of solubility, S(T) [6,9] Critical pressure (P crit ) [1,8] Vapor pressure and its temperature dependence, P(T) [1,8] Dynamic viscosity (, 20 C) [1,8] Dependence of boiling point on pressure, T(P) [10] Surface tension (w, 20 C) [1,8] Temperature dependence of GC retention times, t R (T) [7,8] Dielectric permittivity (e) [1,5,8] Dependence of HPLC retention times on the composition of an eluent, t R (C) [7] First adiabatic ionization potential (J) [1,8] Others [5,8,9] in [9], but it remains important. However, the most time-consuming and labor-consuming step in the application of any quantitative structure-property relationship QSPR approaches for approximation and/or prediction of physicochemical properties of organic compounds is, apparently, the selection and, if necessary, additional statistical processing of their reference values.…”