6512
J. Phys. Chem. 1991, 95, 6512-6514
Electronegatlvlty and Covalent Blnding In Homonuclear Diatomic Mdecules Tapan K.Chanty and Swapan K.Chosb* Heavy Water Division, Bhabha Atomic Research Centre, Bombay 400085, India (Received: September 18, 1990)
We propose a new electronegativity-based approach to covalent binding in homonuclear diatomic molecules in terms of the accumulation of electron density at the bond center during molecule formation by invoking the concepts of bond electronegativity and bond hardness. Expressions for the bond energies that are derived here are numerically tested for selected diatomic molecules.
Introduction The concept of electronegativity is one of the most widely used concepts in chemistry.' For an atomic or molecular system consisting of N electrons, the electronegativityparameter can be defined as the energy derivative2v3
x = -aE/aN
(1)
+
which is evaluated? as (I A ) / 2 , in terms of the ionization potential (I) and the electron affinity ( A ) . A rigorous quantummechanical evaluation is also possible within the framework of density functional theory5l6 following Parr et al.,6s7 who have identified x as the negative of chemical potential ( p ) of the electron cloud, viz.
x = -p = - 6 E / 6 p
(2)
where energy is expressed as a functional of the electron density p(r). This density-functionaldefinition provides not only a means
of formally exact calculation but also a justification for the electronegativity equalization* procedure used widely in chemi~try.'*~*'~ Definition 1 suggests that the electronegativity parameter is linked with the energy involved in charging an atom or molecule, and hence it governs the charge transfer during a molecule formation determined by the equalizationof the final electronegativity of the partners. Consequently, the polarity of a heteronuclear molecule as well as the ionic contribution to the bond energy can be predicted by using electronegativity concepts. The corresponding covalent contribution to the bond energy is generally expressed as a geometric or arithmetic mean of the associated homonuclear bond energies."J2 For a homonuclear diatomic molecule, however, there is no net interatomic charge transfer and the binding is purely covalent. ~~~~
~
(1) Sen, K. D., Jorgensen, C. K., Eds. Electronegutioity, Structure and
Bonding, Springer-Verlag: Berlin, 1987; Vol. 66. (2) Pritchard, H. 0.;Sumner. F. H. Proc. R. Soc. London 1956, A235, 136. (3) Iczkowski, R. P.; Margrave, J. L. J . Am. Chem. Soc. 1%1,83,3547. See also: Allen, L. C. Acc. Chem. Res. 1990. 23, 177. (4) Mulliken, R. S. J. Chem. Phys. 1934, 2, 782. (5) March, N . H., Deb, B. M., Eds. Single Particle Dcnsiry in Physics und Chemistry; Academic Press: New York, 1987. (6) Parr, R. G.; Yang, W. Dcnsity Funcrionul Theory of Atoms and Molecules; Oxford Univ. Press: New York, 1989. (7) Parr, R. G.; Donnelly, R. A.; Levy, M.; Palke, W. E. J . Chem. Phys. 1978,68, 3801. (8) Sanderson, R. T.Chemfcal Bonds and Bond Enerw;Academic: New York, 1976. Sanderson, R. T. Polar Covalencq Academic: New York, 1983. (9) Huheey, J. E. Inorgunic Chemistry, 3rd 4.; Harper & Row: New York, 1983. (10) Mortier, W. J.; Ghosh, S. K.; Shanhr, S. J . Am. Chem. Soc. 19116, 108. 4315 and references therein. (1 1) Pauling, L. The Nature of Chemical Bond, 3rd 4.; Cornel1 Univ. Press: Ithaca, New York, 1960. (12) Ghanty, T. K.; Ghosh, S.K. J . Phys. Chem. Submitted for publication.
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Hence, it has not been possible to account for this binding in terms of the atomic electronegativity parameter. The electron density, however, is reorganized during the molecule formation, and a net accumulation of charge in the bond region has been known to take place.') In this work, we propose to describe covalent binding in terms of this charge transfer from atomic sites to the bond region within the framework of an electronegativity-based picture." Since the density-functionalchemical potential M is defined at each point' and hence also in any region of spacellsone can assign an electronegativityto the bond region, first introduced by Ghosh and Parr.I4 When two atoms are brought to the equilibrium distance, assigning a higher value of electronegativityfor the bond region would account for the charge accumulation in the bond region until the final electronegativities of the atomic and bond regions become equal, and this would provide a formulation of covalent binding. The extent of charge transfer in an electronegativity equalization procedure is governed by another important parameter introduced by Parr and Pearson16 as chemical hardnessI7 (v), which they defined as the second derivative of energy, viz.
(3) A density-functional definition has also been suggested in terms of the integral of a local hardness,18-20which would enable one to provide a rigorous definition of hardness for the bond region. In the present work, we aim at providing a formulation of covalent binding (in homonuclear diatomic molecules for simplicity) through the charge accumulation in the bond region, invoking these new concepts of bond electronegativity and bond hardness. The magnitude of the bond charge is determined by equalization of final electronegativities of the atomic and bond regions. The bond energy consists of contributionsfrom the energy involved in the charge-transfer process as well as the Coulombic interaction among these charges.
Bond Electronegativity-Bod Hardness Model for Covalent Binding Consider two atoms of an element A placed at a distance R, the equilibrium bond length of the molecule A2. Let the electrone ativity and hardness of the neutral atom be denoted by x i and wA and the bond electronegativit and bond hardness for the bond midpoint region by xoM and vwr respectively. If xow > x i , electrons will flow from the two atomic sites to the bond site located at the bond midpoint.
f
P
(13) Bamjai, A. S.;Deb, B. M. Rev. Mod. Phys. 1981.53, 95. (14) Ghosh, S. K.; Parr, R. G. Theor. Chim. Acru. 1987, 72. 379. (15) Weinstein, H.; Politzer, P. J . Chem. Phys. 1979, 71. 4218. (16) Parr, R. G.; P e a m . R. G. J . Am. Chem. Soc. 1983, 105, 7512. (1 7) Pearson, R. G. Hurd and Soft Acids und Base& Dowden, Hutchinson and Ross: Stroudsville, PA, 1973. (18) Berkowitz, M.; Ghwrh, S. K.; Parr, R. G. J . Am. Chem. Soc. 1985, 107, 6811. (19) Ghoah, S. K.; Berkowitz, M. J . Chem. Phys. 1985, 83, 2976. (20) Ghosh, S.K. Chem. Phys. Lett. 1990, 172, 77.
0 1991 American Chemical Society
The Journal of Physical Chemistry, Vol. 95, No. 17, 1991 6513
Electronegativity and Covalent Binding in Diatomics Let MA be the number of electrons transferred from each of the atoms, leaving them charged with a positive charge qA(= -UA) and resulting in an accumulation of 2 m A electrons at the bond site, i.e. a negative bond charge qM(= 2", = -2qA). The electronegativity of the charged atom A is given by XA
x i + 2&qA + qA/R + 2qbnd/R
(4)
where the second term on the right denotes the change in chemical potential due to the charge transfer, while the last two terms arise from the potential due to the charges present at the other atom and the bond site, respectively. Analogously, the resulting electronegativity of the bond site after the charge transfer can be expressed as xbond
= xknd + 2dnndqbond + 4qA/R
(5)
At equilibrium, the electronegativities xA and xbnd will be equalized, and hence one obtains after equating (4) and (3, the expression for q A given by qA = (Xkd
- XX)/(2& + 4&nd - 7/R)
(6)
The total energy change involved in this charge reorganization process consists of contributions from the charge transfer as well as the Coulomb interaction of the resulting charges and is given by
DZ
F w 1. Plot of the experimental bond energies against the calculated covalent bond energies for selected homonuclear diatomic molecules. Dark circles correspond to single-bonded molecules.
TABLE I: Atomic Ele~troaeg8tivity,~ a d Hardnw@ PMmctem d a d Energies of CorresponfUog Homollwku Diatomic
Bond Mokcules
Using (6), one obtains the bond energy expression
Equations 8 and 6 enable one to calculate the homonuclear bond energy and the bond char e ( q b n d = -2qA), provided suitable values for the quantities Xbnd and ?$nd are used. The chemical potential of an atom has been shown2I to be approximately equal to the electrostatic potential at the covalent radius. Due to the additivity of the electrostatic potential, the chemical potential at the point of contact of two atomic spheres can be assumed to be close to the sum of the individual atomic chemical potentials. Thus, the bond electronegativity x k n d can be modeled as
f
with kl a proportionality constant. Alternative modeling by including R dependence of xLd is possible," but since we are here concerned with only the equilibrium properties, this simplified model is employed. The modeling of the parametera,&,, is arrived at from two important results. Since the molecular hardness is proportional22to VHM, the harmonic mean of the atomic hardnesses, one can assume that the bond hardness is proportional (or equal) to vHM (&/2 in the homonuclear case). The quantity (I - A), which defines the hardness, is known in the PPPZ3theory to be a measure of the intraatomic electron repulsion integral. Analogously, the bond hardness here can be assumed to be a measure of the interatomic electron repulsion, and consequently, as in the PPP theory, this term can be modeled to have an R I d ~ p e n d e n c e .A~ ~similar correlation of hardness with covalent radius for atomic systems has been indicated earlier." Motivated (21) Politzer, P.; Parr, R. G.; Murphy, D. R. fhys. Rev. 1985,631,6809. (22) Yang, W.;Lee, C.; Ghosh, S . K.J. fhys. Chem. 1985, 89, 5412. (23) Parr, R. G. Quantum Theory of Molecular Electronic Structure; Benjamin: New York, 1963. (24) Ray, N.K.;Samuels, L.; Parr, R. G. J . Chem. fhys. 1979.10.3680. (25) Peanon, R. G. Inorg. Chem. 1988, 27, 734.
atomic atomic electroneg hardness molecules x,,, eV qA, eV 7.18 6.43 H2 2.39 Li2 3.01 2.30 2.85 Na2 1.92 K2 2.42 1.85 Rb2 2.34 2.18 1.71 CS2 4.68 c12 8.30 4.22 7.59 Br2 3.69 I2 6.76 7.23 7.30 N2 4.88 p2 5.62 4.50 5.30 As2 3.80 Sb2 4.85 6.27 5.00 c2 3.40 Ge2 4.60 6.08 0 2 7.54 4.14 s2 6.22 3.87 Se2 5.89 3.52 5.49 Te2
bond exptl bond calcd bond length energyC PA, energyd PA, kcal/mol R, A kcal/mol 0.742 103.25 103.83 2.672 25.00 21.20 3.078 17.30 16.97 3.923 11.80 13.59 4.950 10.80 10.98 5.309 10.40 10.38 1.988 57.30 53.95 2.284 45.45 48.75 2.666 35.60 43.60 1.098 225.07 236.60 1.893 115.00 108.35 2.440 91.00 71.39 2.900 70.60 70.37 1.340 144.00 139.58 2.410 65.00 59.06 1.207 117.97 117.27 1.887 101.50 98.76 2.150 79.50 82.45 2.700 62.10 63.10
Values of x and q calculated from ionization potential and electron affinity data from ref 25. bValues of R from ref 9 and from CRC Handbook of Chemistry and Physics, 70th ed.; CRC Press: Boca Raton, FL, 1989. CExperimentalbond dissociation energies from ref 9. The values for Se and Te are from the CRC Handbook. dBond dissociation energies calculated by using q 12.
by these facts, we propose the following model for the bond hardness parameter, viz. &md
THM
+ k2/R
=
+ k2/R
(10)
where k2 is an empirical proportionality constant. If x and 7 are expressed in electronvolts and the internuclear distance R in angstroms, the R-dependent terms in all these equations are to be multiplied by a factorf = 14.14 for conversion to proper units. By use of the modeling of (9) and (lo), the final expressions for the charge (qA) and the bond energy (DM) are given by qA DAA
= (zkl
- 1)&/(4&
+ (4k2 - 7)f/R)
= ( 2 h - 1 ) 2 ( x i ) 2 / ( 4 ~ i + (4k2 - 71f/R)
(11) (12)
where qA is expressed in atomic units while DM is in electronvolts.
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J. Phys. Chem. 1991, 95, 6514-6519
Results a d Discussion We have calculated the bond energies of several homonuclear diatomic molecules through the proposed scheme, using the values of electronegativity, hardness, and covalent radius of the constituent atoms. The parameters kl and k2 introduced through the modeling of bond electronegativity and bond hardness quantities are evaluated empirically by minimizing the error in predicted bond energies of nine single-bonded molecules, as compared to the experimental results. For molecules involving multiple bonds, the bond order is larger in magnitude and one expects the bond charge to be higher, which corresponds to higher bond electronegativity and/or lower bond hardness. To mimic this, we propose an empirical relation for the constants k, and k2 for multiple-bonded molecules given by
(klk2)mult
(klkhsingle
In Figure 1, we have plotted the calculated bond energies, using the parameters listed above, against the experimental results, which show good agreement. In Table I, the atomic electronegativity, hardness, and homonuclear diatomic bond length parameters used in the calculation are reported along with the calculated values of bond dissociation energies as well as the experimental ones. The average percentage error for bond energies for 19 molecules is 6.52, which reduces to 4.7 if the two molecules iodine and arsenic, for which the error is maximum, are excluded. Considering the fact that in certain cases the experimental error is high (above 5%), the prediction is very good. For the single-bonded systems (except hydrogen), the bond charges compare rather well with the usual bond order although the deviation is greater for the multiple-bonded systems.
Concluding Remarks The present work has been concerned with a description of covalent binding in homonuclear diatomic molecules within the framework of an electronegativity-based picture through the electronegativity and hardness parameters of the constituent atoms. The novel feature has been the prediction of the bond energy by using the concepts of bond electronegativity and bond hardness. Further studies on improved R-dependent modeling and hence prediction of the potential energy curve as well as extension to polyatomic molecules are in progress. Acknowledgment. It is a pleasure to thank H. K. Sadhukhan for his kind interest and encouragement.
(14)
where n denotes the bond order and a is an empirical parameter determined by minimization of the overall error in the predicted bond energies of multiple-bonded molecules. In the present calculation, the values of the coefficients are (kl)sinple= 0.801, (k2),inglc= 1.467, and a = 0.0786. The value of kl is close to unity which indicates an approximate additivity of the chemical potentials (electrostatic potential at the Wigner-Seitz radius) in the bond region.
Dldehydropyrldlnes (Pyrldynes): An ab Inltlo Study H. H. Nam, G. E. Leroi,* and J. F. Harrison* Department of Chemistry, Michigan State University, East Lansing, Michigan 48824- I322 (Received: October 22, 1990)
We report the first ab initio studies on the six DHP (didehydropyridine) isomers at the RHF, ROHF, and GVB levels, obtained with a 3-21G basis set. The geometries of all DHPs are fully optimized at each level of theory to determine their relative stability sequence as 3,4-(S) > 2,3-(S) = 2,4-(S) > 3 3 4 s ) = 2.6-(T) = 2,5-(S), where (S) and (T)represent singlet and triplet, respectively. At the RHF level the hexagonal structures of the 2,n-DHPs (n = 3,4, 5 6 ) are severely distorted (for n = 3, 4, 6) or broken (for n = 5 ) by strong Nl-C2 bonding, facilitated through a delocalized MO between the nitrogen lone pair orbital and its adjacent radical (C2) lobe orbital. At the GVB level the limited electron correlation between the two radical electrons lowers the RHF energies of the DHPs by 30-48 kcal/mol and the optimized geometries are much less strained hexagonal structures because the two radical electrons and the nitrogen lone pair tend to be confined to different regions. Only 2,6-DHP has a triplet ground state; it may be explained in terms of hindered through-space interaction by the nitrogen lone pair orbital, which stands between the two radical centers.
Introduction Didehydropyridines (DHPs, commonly called pyridynes) have been the most studied of all known didehydroheteroarenes (or heteroarynes).I-’ They have been proposed as likely intermediates in many organic reactions, principally those involving cyclo( 1 ) den Hertog, H. J.; van der Plas, H. C. Ado. Heterocycl. Chem. 1965, 4, 121. (2) Kauffmann, T. Angew. Chem., Int. Ed. Engl. 1965,4, 543. ( 3 ) Hoffmann, R. W. &hydrobenzene and Cycloulkynes; Academic Ress:
New York, 1967. (4) Kauffmann, T.; Wirthwein, R. Angew. Chem., Inr. Ed. Engl. lWl,IO, 20. __ (5) Reinecke, M. G. In Reactive Intermediate; Abramovitch, R. A,, Ed.; Plenum Preas: New York. 1981; Vol. 2. (6) Reinecke, M. G. Tetrahedron 1982, 38, 427. (7) van der Plas, H. C.; Roeterdink, F. In The Chemistry of Functional Groups, Supplement C The Chemistry of Triple Bonded Functionu~Groups; Patai, S., Rappoport, Z., Eds.; Wiley: New York. 1983.
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addition, cine-substitution, or tele-substitution.6 DHP has six possible isomers denoted as ij-DHP, where i and j indicate the two dehydrogenated carbon centers indexed counterclockwise from the n i t r ~ g e n .Among ~ the six DHPs, the 3,4isomer is the most firmly established.Id For example, diazabiphenylene, the dimer of 3,4-DHP, was identified in the timeof-flight mass spectrometric and kinetic UV spectroscopic analysis of the products formed by flash photolysis of pyridine-3-diazonium-4-carboxylate! Recently, we have reported the infrared spectrum of 3,4-DHP generated via mild photolysis of 3,4pyridinedicarboxylic anhydride (3,4-PDA) in N2 or Ar matrices at 13 K.9 These experiments provide convincing evidence that 3,4-DHP exists as a true reactive intermediate, not just as a transition state. On the other hand, the evidence for other DHP (8) Krammer, J.; Rerry, R. S. J. Am. Chem. Soc. 1972, 91. 8336. (9) Nam, H. H.; Leroi, G. E. J . Am. Chem. Soc. 1988, 110,4096.
0 1991 American Chemical Society
