Acta Chim. Slov. 2003, 50, 539-546. 539 INFORMATION ENERGY AND INFORMATION TEMPERATURE FOR MOLECULAR SYSTEMS Costinel Lepădatu and Emilia Nitulescu Faculty of Pharmacy, University of Medicine and Pharmacy, Bulevard Maresal I. Antonescu, 94, Craiova, Romania Received 15-10-2002 Abstract The information energy and information temperature are proposed as new quantum descriptors for the characterization of molecular systems. Ab initio and semiempirical (PM3) procedures are used to create molecular probability fields on which the aforementioned information quantities are applied. The connection with other structural descriptors is discussed in the čase of linear condensed aromatic rings and linear hydrocarbons with alternate double bonds. Introduction The first mathematical function capable to describe the information content of a complex system has been proposed by Shannon related to the information content transmitted through different communication systems. This quantity named information entropy is actually a convex function / = -plog2 p, defined onto the closed interval f : [0, 1] —> R, for which Jensen’s inequality 1 -^ 1 -^ r^ nn --- X J \Pn ) — J (--- X Pn )> Pn G \y,\\ N „=1 N „=1 should be valid. Shannon proved that the function N S = -y,/>„ l0g2 P„ (1) named information entropy may characterize quantitatively very well the structure of a discrete set of “pn” values from the [0, 1] domain which may be considered as a probability field N YjPn =L n=\ C. Lepadatu, E. Nitulescu: Information Energy And Information Temperature For Molecular Systems... 540 Acta Chim. Slov. 2003, 50, 539-546. One may easily demonstrate that Shannon’s entropy assumes a minimum value Smin = 0 for pi = 1; p2 = P3 = ... = 0 and a maximum value for equal probabilities pi = P2 = • • • = 1/N. These extreme two cases reflect the total order and total disorder respectively. This matter of facts justifies the name of entropy for the S function successfully applied to various domains, including chemistry and biology. ' The aim of this paper is to analyze two new information quantities capable to characterize the molecular systems, and namely, the information energy and temperature respectively. Like the information entropy, the information energy introduced in mathematical statistics by Octav Onicescu is a convex function defined onto [0, 1] interval, which may be a discrete or continuous probability field. N E = 'S\p2n,pn? [0,1] (2) n=l This function assumes for the two extreme cases total order and total disorder the maximum (E,^ = 1) and minimum (Em;n = 1/N) values respectively: Pi = 1, P2 = P3 = • • • = 0 => Emax = 1 (total order) Pi = P2 = • • • = Pn = 1/N => Emin = 1/N (total disorder). Because the function (2) reaches a minimum value for equal probabilities (total disorder), by analogy with thermodynamics, it has been called information energy. Also, Shannon’s information entropy has to do with thermodynamic entropy. Jaynes has demonstrated the connection between Boltzmann entropy from statistical physics and information entropy. By analogy with thermodynamics, one may define the information temperature given by the ratio between Onicescu information energy and Shannon information entropy. E 1 = — (3) S We shall try in the following, to analyze these three information quantities S, E and T applied onto a probability field generated by the quantum molecular wave functions. The scope of this attempt is to see if these quantities may be used for the characterization of the molecular systems. Another goal is the possible connection betvveen these quantities and another physico-chemical descriptors characterizing the molecular systems. C. Lepadatu, E. Nitulescu: Information Energy And Information Temperature For Molecular Systems... Acta Chim. Slov. 2003, 50, 539-546. 541 Results and discussion One knows from the quantum theory of the chemical bond (LCAO-MO approach) that the molecular wavefunctions (Molecular Orbitals) \j/; are built from atomic orbitals (pj y/'=Hcv(pj' (4> Cij being the mixing coefficients. The probability field can be generated for each molecular electronic state, based on dt = \ and vfPA dr = \. The mixing coefficients Cy represent the measure of the atomic orbital contributions (weight) to the construction of the molecular orbitals. They are useful for the partition of electron population of the molecule usually made using two approximations: Mulliken’s one where the electron densities from interatomic regions are equally shared to the two atoms participating to the chemical bonding (a kind of molecular democracy) and L6wdin’s one which considers the baricenter of the interatomic electron density which is accordingly distributed to the two atoms. In L6wdin’s approximation the more electronegative atom receives more interatomic electron density than the less electronegative atom (a kind of molecular liberalism). The probability field characterizing the molecular system can be obtained in different ways depending on the partition of the electron density: (1) For each atomic orbital { where electron densities distributed on the atomic orbital of “j” atom, pj represents the electron density of that atom, so that ^Pj = N, where N is the number of electrons in molecule (ab initio) or the number of valence electrons (semiempirical approximations). One may define a probability field for each “j” atom in molecule, p — — — p =—;p =1-p;p + p =1, where p means the probability that pj electron iV density does not exist on “j” atom. The S, E, T functions applied onto such a probability field will be called atomic information entropy (S), energy (E) and temperature (T). Let us discuss in the following, the first two possibilities, and namely the molecular and orbital information S, E, T functions as applied to a series of linear with alternate double bonds hydrocarbons and to a series of condensed aromatic hydrocarbons. The quantum molecular calculations were performed by means of GAMESS (ab initio) and MOPAC 7 (semiempirical) offered as shareware programs for academic researches. ' The molecular energy levels, electron population distributed on AO’s, MO’s have been calculated, as well as dipole moments, polarizabilities etc. have been calculated by means of these programs. The analysis of the results and the calculation of information entropy, energy and temperature have been done using specially dedicated programs developed by us. C. Lepadatu, E. Nitulescu: Information Energy And Information Temperature For Molecular Systems... Acta Chim. Slov. 2003, 50, 539-546. 543 As may be seen in Table 1, the values for the molecular information entropy continuously decreases with the increase of the number of carbon atoms. In exchange, the orbital temperature increases separately in the sets of hydrocarbons with odd and even number of carbon atoms. It is worth to note that the sums of orbital entropies and energies notably increase with the increase of carbon atoms of the hydrocarbons. In exchange, the corresponding orbital temperature values remain almost constant. The correlation between the information quantities S, E and T and other quantum molecular descriptors characterizing the molecular structures have also been performed. Such a correlation has been done in order to see the possible connection between the information quantities herein defined and other physical and chemical quantities representing the molecules. Table 1. Information quantities S, E and T calculated from ab initio results (GAMESS, RHF, MP4, STO-6G, L6wdin) for linear hydrocarbons. olecule S E T ?S ?E T=?E/?S (molecular) (orbital) 7.98449 C2H4 3.75085 0.07800 0.02080 12.08473 0.66071 C3H6 4.33590 0.05200 0.01199 18.13502 11.99466 0.66031 C4H6 4.64125 0.04215 0.00908 22.17254 14.96649 0.67500 C5H8 4.98579 0.03318 0.00666 28.22144 18.95725 0.67173 CeH8 5.18777 0.02887 0.00557 32.25860 21.94943 0.68042 C7H10 5.43228 0.02436 0.00448 38.30763 25.93976 0.67716 CsH10 5.58323 0.02196 0.00393 42.34547 28.93134 0.68322 C9H12 5.77273 0.01925 0.00333 48.40075 32.92108 0.68018 CioH12 5.89330 0.01771 0.00301 52.44267 35.9140 0.68477 C20H22 6.86921 0.00901 0.00131 102.8644 gies. 70.82584 0.68854 ?E = sum of orbital information entropies or ener As may be seen in Table 2, the linear regression between various quantum molecular descriptors and information quantities S, E, T shows that the molecular information temperature correlates well with the average electrophilic reaction index for the carbon atoms in the series of studied hydrocarbons (R2 = 0.9954) and fairly good with the ionization potentials which are the highest occupied molecular orbital (HOMO) energy level (R2 = 0.8777). C. Lepadatu, E. Nitulescu: Information Energy And Information Temperature For Molecular Systems... 544 Acta Chim. Slov. 2003, 50, 539-546. Table 2. Linear correlation y=ao+bix between the information quantities S, E, T and quantum molecular descriptors for linear hydrocarbons. y R2 = correlation coefficient. R2 T (molecular) Average electrophilic reaction index Ec for the carbon atoms 0.9954 HOMO energy levels 0.8777 T (orbital) Average electrophilic reaction index Ec for the carbon 0.9999 atoms(only for hydrocarbons with odd number of carbon atoms) Average nucleophilic reaction index Nc for the carbon 0.9999 atoms(only for hydrocarbons with even number of carbon atoms) E (molecular) Average electrophilic reaction index Ec for the carbon atoms 0.9994 LUMO energy levels 0.9188 ?E (orbital) HOMO - 1 energy levels 0.9568 The orbital temperature correlates very well with the average electrophilic reaction index of the carbon atoms in the hydrocarbons with odd number of carbon atoms and with average nucleophilic reaction index Nc in the hydrocarbons with even number of carbon atoms. The reactivity indices are defined as EA = i?A 2 i HOMO and N j? A 2 j LUMO (electrophilic) (nucleophilic), where the summation are done over ali AO’s of a given atom species, c;homo and Cj lumo being the AO coefficients on the HOMO and LUMO respectively. These reactivity indices are related to the activation energy of the corresponding chemical reaction (electrophilic and nucleophilic). Onicescu molecular information energy correlates with the electrophilic reaction index Ec of the carbon atoms for ali linear hydrocarbons (R2 = 0.9994) and also with affinity (LUMO energy level) (R2 = 0.9188). The sum of orbital information energies correlate with the HOMO-1 energy levels which are the second ionization potential of the molecule (R2 = 0.9568). It is interesting to note that the sum of orbital information energies values EE(orbital) are close to the number of occupied energy levels in the molecule. As an example, for C7H10 every carbon atom participates with 6 electrons and every hydrogen x C. Lepadatu, E. Nitulescu: Information Energy And Information Temperature For Molecular Systems... Acta Chim. Slov. 2003, 50, 539-546. 545 atom with one electron in ab initio interactions, the number of occupied molecular energy levels is equal to 26. In this case ?E = 25.939 (see Table 1). For the second series of linearly condensed aromatic hydrocarbons (benzene, naphthalene, anthracene etc.) the semiempirical results a summarized in Table 3. Table 3. Information quantities S, E and T calculated from semiempirical results MOPAC 7 (RHF, MNDO, Mulliken) for condensed aromatic hydrocarbons. Molecule S E (molecular) T=E/S ?S ?E (orbital) T=?E/?S CY1 4.89868 0.03373 0.00689 29.74158 15.17808 0.51033 CY2 5.57657 0.02109 0.00378 47.57882 24.29032 0.51053 CY3 6.03592 0.01534 0.00254 65.41570 33.40282 0.51062 CY4 6.38379 0.01205 0.00189 83.25190 42.51576 0.51069 CY5 6.66386 0.00992 0.00149 101.0886 51.62837 0.51072 CY6 6.89830 0.00844 0.00122 118.9243 60.74168 0.51076 CY1 = benzene; CY2 = naphthalene; CY3 = anthracene etc. As may be seen in Table 3, the studied information quantities vary almost monotonously irrespective of the parity of the condensed rings number. As in the čase of linear hydrocarbons with alternate double bonds, the sum of orbital information energies EE(orbital) approximates fairly well the number of occupied molecular levels. As an example, for the molecule with six linearly condensed aromatic rings (CY6), the total number of valence electrons is equal to 120 and hence the number of occupied molecular levels 60, comparable with EE = 60.74. This observation suggest us a way to approximate EE(orbital) for both ab initio and PM3 procedures with the number of occupied molecular levels in the molecule. The linear correlation of the analyzed information quantities S, E, T with different quantum molecular descriptors is given in Table 4. Table 4. Linear correlation y=ao+biX between the information quantities S, E, T and quantum molecular descriptors for condensed aromatic hydrocarbons. y X R2 T (molecular) LUMO + 1 0.9271 T (orbital) FPSA 0.9779 E (molecular) Average electrophilic reaction index for C atoms 0.9999 LUMO + 1 0.9487 ?E (orbital) HOMO - 1 0.9627 R2 = correlation coefficient. C. Lepadatu, E. Nitulescu: Information Energy And Information Temperature For Molecular Systems... 546 Acta Chim. Slov. 2003, 50, 539-546. As may be seen in Table 4, the molecular information energy correlates very well with the average electrophilic reaction index for the carbon atoms in molecule (R2 = 0.999). The same situation has been observed with the linear hydrocarbons (Table 4). The molecular information energy as well as the molecular information temperature correlate fairly good with LUMO+1 energy level, which is the second affmity energy of the molecule (R2 = 0.9627 and 0.9271 respectively). The orbital information temperature correlates very well (R2 = 0.9779) with an important molecular descriptor FPSA equal to the fraction betvveen the Partial Positive Surface Area of the molecule (PPSA) and the Total Molecular Surface Area (TMSA). This descriptor belongs to the charged partial surface area descriptors ' may describe the polar interactions betvveen molecules or receptor-ligand interactions for biomolecules or drugs. This kind of researches regarding the information entropy, energy and temperature and their correlations with other physical and chemical descriptors will be extended to other classes of substances. References 1. C. Shannon, W. Weaver, Mathematical Theory of Communication, Illinois University, Illinois Press, Urbana, 1949. 2. N. Rashevsky, Buli. Math. Biophysics 1955, 17, 229-235. 3. G. Karreman, Buli. Math. Biophysics 1955, 17, 279-285. 4. O. Onicescu V. Stefanescu, Elemente de statistica informationala cu aplicatii, Ed. Tehnica, Bucharest, 1979. 5. E. T. Jaynes, Phys. Rev. 1957, 106, 620-630. 6. M. W. Schmidt, K. K. Baldrige, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguven, S. J. Su, T. L. Windus, M. Dupuis, J. A. Montgomery, J. Comput. Chem. 1993, 14, 1347-1363. 7. J. J. P. Stewart, J. Comp. Chem. 1989, 10, 209-220. 8. K. Fukui, Theory of Orientation and Stereoselection, Springer Verlag, Berlin, 1975. 9. D. T. Stanton, P.CJurs, Anah Chem.; 1990, 62, 2323-2329. 10. D. T. Stanton, L. M. Egalf, P. C. Jurs, M. G. Hicks, J. Chem. Inf. Comput. Sci. 1992, 32, 306-316. Povzetek V delu predlagamo informacijsko energijo in informacijsko temperaturo kot nova kvantnokemij ska deskriptorja za karakterizacijo molekularnih sistemov. Ab initio in semiempirično (PM3) metodologijo smo uporabili za izračun molekularnih verjetnostnih polj, na katerih sta bili določeni omenjeni količini. Povezanost z drugimi strukturnimi deskriptorji je v delu obravnavana na primeru linearno kondenziranih aromatskih obročev in linearnih ogljikovodikov z alternirajočimi dvojnimi vezmi. C. Lepadatu, E. Nitulescu: Information Energy And Information Temperature For Molecular Systems...