J. Phys. Chem. 1994,98, 1840-1843
1840
A New Simple Density Functional Approach to Chemical Binding Tapan K. Ghanty and Swapan K. Ghosh' Heavy Water Division, Bhabha Atomic Research Centre, Bombay 400 085, India Received: August 17, 1993; I n Final Form: November 19, 1993"
A new simple density functional approach to chemical binding in molecules is developed by representing the electron density reorganization of the constituent atoms in terms of the atomic charges and dipoles. The potential and capacity parameters for the atomic dipole are introduced, and a new equation for binding energy is derived. The calculated numerical results on the bond energies of a number of homo- and heteronuclear diatomics are shown to agree quite well with available data.
Introduction is concerned with a quantum Density functional mechanical description of atomic and molecular system in terms of the electron density and has provided a rigorous foundation for the widely used chemical concepts like electr~negativity,~ hardness: softness,s frontier orbital theory? etc. In a density functional description, chemical binding can be viewed7-"J as resultingfrom reorganization and redistribution of electrondensity among the atoms in a molecule. While there have been studied' on the representation of the molecular electron density with reference to the atomic ones through difference density plots, a simplifiedrepresentation of the density redistribution in molecule formation in terms of the first few members of a multipolar expansion12 of the total charge density around each atom of the molecule is also of immense importancefor easy conceptualization. The objective of the present work is to explore the possibility of a density functional theory in terms of these atom-centered multipole representations of the density for an understanding and prediction of chemical binding in simple molecular systems. yr"
In density functional theory, the energy of a many-electron system characterized by an external potential u(r) is a unique functional of its electron density p(r) and can be written as
where F [ p ] is a universal functional of density. The energy functional E[p] assumes a minimum value for the true density which satisfies the Euler equation p = (6E/6p) = u(r)
+ (6F/6p)
(2) withp as the Lagrange multiplier for the normalizationconstraint Jdr p(r) = N, where Nis the number of electrons. The parameter p is the chemical potential*for the electron cloud and has led to many chemical applications of density functional theory. Thus, it has been identified3 with chemical electronegativity x (=(aE/aN) = -(6E/6p) = - p ) ; its derivative defines4the hardness T ( = ( 1 / 2 ) ( a z E / a w = (1/2)(acc/aN)). We consider the formation of a diatomic molecule AB from the isolated atoms A and B. The energy change associated with the density reorganization of atom A can be expressed as
mA
j(bE/6pA)ApA(r)dr
dependent7in a general nonequilibrium situation. For the present case of two atoms placed at the equilibrium separation (which is not an equilibrium situation before charge transfer), one can thus include a weak position dependence and write
~ E / ~ P=A + ~*VPA(O) which on substitution into eq 3 leads to the result m A
~AuA
Abstract published in Advance ACS Abstracts, January 1 , 1994.
0022-3654/94/2098-1840%04.50/0
+ ~ A I ~ +A JPA(r) I ~ v A ( r )dr +
higher-order terms ( 5 ) where ANArepresents the change in the number of electrons of A, dA is the dipole moment induced in the atom A, and thequantity is proportional to ) V p ~ l .Since we are interested here in representing the atomic electron densitiesin the molecule in terms of the first two moments, viz. the atomic charges and dipoles, we express the higher-order contributions of eq 5 through a direct expansion in terms of these quantities. The total energy change associated with the formation of the molecule AB (with internuclear distance RAB)can thus be expressed (retaining terms up to second order) as
= pAUA
+ pBUB + qA(UA)2 + q B ( N B ) * +
+ d d B l + $AIdAl2 + $BldB12 + qABUAUB + J/ABldAlldBl+ S ' A A U A V A I + r B B m B l d B l + lABmAldBl+ rBAWldAI ( 6 ) where cp and $ are the potential and capacity parameters for the atomic dipole and { represents the capacity parameter for the cross term involving ANand the dipole. Since ANA= -ANB, the last four terms in eq 6 are of alternating sign and cancel exactly in homonuclear diatomic molecules. Even for heteronuclear diatomics,the sumof these terms whichdependson thedifference in atomic dipole moments is small13 and will be neglected henceforth. On using the charge conservation condition ANA+ ANB= 0, the energy expression (6) after simplification contains three unknown quantities ANA,ldAl, and Idel, which can be determined from the three equations resulting from minimization of hE (equivalent to chemical potential e q ~ a l i z a t i o nwith ~ ~ ) respect to these quantities. The resulting expressions for these quantities at equilibrium are given by vAldAl
+ j(6E/6uA)AuA(r)dr +
higher-order terms (3) where A ~ and A AVAdenote respectively the change in the density and external potential for the electrons of atom A. Although the quantity (6E/6p) is constant at equilibrium, it can be position
(4)
ANA
= ( ~ l g- PA)/~(TA+ TB - TAB)
+ 'PB$AB)/(4$A$B + $ A t )
ldAl
= -(2vA$B
ldBl
= -(2%$A + 'PA$AB)/(4$A$B + $A ):
(74 (7b) (7c)
Substituting eqs 7 into the energy expression (6), one obtains 0 1994 American Chemical Society
Chemical Binding in Molecules
The Journal of Physical Chemistry, Vol. 98, No. 7, 1994 1841
the bond energy given by DAB
TABLE 1: Atomic Parameters Used in Bond Energy Calculation
= (pB - c1A)2/4(0A + TB - TAB) + [((PA)'$B (%)'$A
+ 'PA%$ABl/(4$A#B
+ + $ A t ) (8)
For a homonuclear diatomic molecule Az, there is no net interatomic charge transfer ( h l v ~ = 0), and eqs 7 and 8 simplify respectively to IdAlfi
= I d B I u = -(PA/(WA + $ f i )
(9)
and
potential parameter atom (A) Li Na
K
Rb cs F
c1 D , = (v..J'/(wA
+ $fi)
(10) The binding energy can thus be calculated for a molecule AB (or Az) once the parameters x (=-p), 8, cp, and +are known. While the values of atomic x and v governing the ionic contribution are easily available, the corresponding parameters cp and for the dipole, determining the covalent part, are introduced here for the first time and are to be modeled. Modeling is also required for the atom-atom cross coefficients VAB and +AB. The induced dipole moment in an atom due to an electric field Fcl is given by Id1 = d e l , where a is the atomic polarizability. The corresponding interaction energy depends on aFet,which is equal to ldl2/a. Comparing this with the quadratic term +Id12 in the energy expression, we are prompted to model the parameter +A for atom A as inversely proportional to the atomic polarizability (YA, viz.
+
$A 2/ffA (11) (It may be noted that an analogous inverse proportionality to polarizability has been shown15.16 to exist even for the hardness parameter.) We also propose, mainly on the basis of dimensional considerations, a model for (PA directly in terms of the atomic parameters XA, VA, and the covalent radius ( R u / 2 ) , viz.
= -(X:/VA)(Rfi)-l (12) where the negative sign is demanded by the positive values of ldlA in eq 9. The atom-atom parameters TAB and +AB are now modeled here using the Ohno-type formula" used in semiempirical electronic structure theories, viz. (dA
where RABis the bond length of the molecule AB. Here, the different dependenceson RABin the two expressionsare motivated by dimensional considerations. The quantities a, and a+ can be expressed through atomic T and parameters, which can be further approximated in terms of RAB. Thus, u,, and u+ turn out to be proportional to RABand R A B ~respectively. , For simplicity, we consider18 the proportionality constants to be unity which lead to the results
+
Thus, the bond energies of a heteronuclear diatomic molecule AB can be obtained from eq 8 using the parametrization of PA, +A, TAB,and +AB from eqs 12,11, 14a, and 14b, respectively. One can also obtain the parameter (PA from the corresponding homonuclear bond energy values using eq 10, i.e.
instead of using eq 12.
Br
I
D A A , ~ RAA:
XA,'
?A,"
eV
eV
kcal/mol
A
3.01 2.85 2.42 2.34 2.18 10.41 8.30 7.59 6.76
2.39 2.30 1.92 1.85 1.71 7.01 4.68 4.22 3.69
25.00 17.30 11.80 10.80 10.40 37.00 57.30 45.45 35.60
2.672 3.078 3.923 4.209 4.470 1.418 1.988 2.284 2.666
a~ld
*A/
A3
au
au
24.3 23.6 43.4 47.3 59.6 0.557 2.18 3.05 4.7
0.0275 0.0244 0.0150 0.0138 0.0121 0.2462 0.1537 0.1160 0.0835
0.0276 0.0223 0.0151 0.0137 0.0121 0.2121 0.1441 0.1163 0.0904
~~
a Thevaluesof x and 7 calculated from ionizationpotential and electron affinity data from ref 23. Values of DM are from ref 24. Values of RM are from refs 24,25 (for Rbz), and 26 (for Csz). Values of a A are from ref 27. e Values of +OAcalculated using eq 15 with homonuclear bond energies. IValues of +OAcalculated using the modeling of eq 12.
While eq 15 is useful for obtaining the bond energy of a heteronuclear diatomic molecule AB from the corresponding homonuclear bond energies, modeling of (PA by eq 12 provides a route to the bond energies of homonuclear diatomics themselves (in addition to the heteronuclear ones). Thus, the bond energy of a homonuclear diatomic molecule A2 can be calculated using eq 10 and the models of eqs 11, 12, and 14 for +A, 'PA, VAB, and +AB, respectively. For a heteronuclear diatomic molecule AB, one can use in the bond energy expression (8) either eq 12 or eq 15 for (PA. The heteronuclear bond energy expression of eq 8 consists of ionic (the first term) and covalent (the second term) contributions. The covalent term can be reexpressed, under some approximations, as the average of arithmetic and geometric means of D u and DBB-a result which has already been derivedg-10 by us earlier using a different approach. In the energy expression (6)the term involving the parameter $AB which is negative as modeled by eq 13b can be interpreted as the dipoldipole interaction energy in an effective dielectric medium19 and is analogous to the expression for dispersion interaction. Although the dipoleinduced dipole (dispersion) interaction is well-known20in the calculation of long-range forces, in the context of chemical binding of homonuclear diatomics, the role of this type of term was first emphasized by Pitzer.21 This attractive dipoledipole dispersion-typeterm is interpreted there as representing the interaction of induced instantaneous dipoles22 resulting from quantum mechanical charge fluctuations and interatomic electron correlations. The charge fluctuations have already been known to play an important role in the context of local softness introduced by Yang and Parr.5 Unified link thus seems to exist between the quantities and approaches of interest to the study of chemical binding and reactivity.
Results and Discussion The bond energy of a heteronuclear diatomic molecule AB can thus be calculated using eqs 8, 11, 14, and 15 using the electronegativity, hardness, and polarizability of A and B atoms and the bond energy of the correspondinghomonuclear molecules A2 and B2. Alternatively, one can also use eqs 8, 11, 12, and 14 where the d a t a for homonuclear bond energies are not required. This second scheme provides a means to the calculation of bond energies of not only heteronucleardiatomicsbut also homonuclear molecules. The bond energies of a number of heteronuclear diatomic molecules have been calculated using both these schemes. The atomic electronegativity (x), hardness (v), and polarizabilities (a)values used in the calculation are given in Table 1 along with the values of bond energies and bond lengths of the corresponding homonuclear diatomics.
Ghanty and Ghosh
1842 The Journal of Physical Chemistry, Vol. 98, No. 7, 1994
TABLE 2 Bond Dissociation Energies (Dm)and Bond Lengths (Rm)of Heteronuclear Diatomic Molecules
175
molecule D;;: Dg: Dg: (AB) kcal/mol kcal/mol kcal/mol R A BA ,~ 145.1 1.547 139.8 137.0 LiF 122.5 2.020 118.9 111.0 LiCl 99.5 2.170 99.8 100.0 LiBr 75.2 2.392 78.7 83.0 LiI 115.1 1.840 108.8 114.0 NaF 101.9 97.0 2.361 97.5 NaCl 84.7 2.502 83.4 86.7 NaBr 66.2 2.710 67.9 72.7 NaI 113.6 2.130 108.8 117.0 KF 106.4 103.1 2.667 101.0 KCl 89.9 90.1 2.821 90.5 KBr 71.9 75.1 3.048 78.0 KI 110.2 105.2 2.266 117.0 RbF 101.7 105.3 2.787 106.0 RbCl 89.2 2.945 89.2 92.0 RbBr 71.7 74.6 3.177 79.0 RbI 113.2 2.345 108.4 120.0 CsF 108.7 2.906 105.3 104.5 CSCl 92.8 92.6 3.072 99.5 CsBr 75.1 78.2 3.315 80.0 CSI a Experimental bond energies are from ref 24. b Bond energies calculated using eqs 8, 11, 12, and 14. Bond energies calculated using eqs 8, 11, 14, and 15. Bond lengths used for the calculation are from ref 24.
150
The numerical results on bond energies using both the schemes are compared in Table 2 with the experimental data," which show a very good overall agreement. The percentage error for the first scheme is 5.0 while the same for the second scheme is 4.3. It is important to note that the bond energy calculation of heteronuclear diatomic molecules using the second scheme does not involve the experimental homonuclear bond energies, in contrast to the approaches used earlier9*28in electronegativitybased procedures. Moreover, using this scheme, we have been able to make quite good prediction for the homonuclear bond energies as well. This is demonstrated in Figure 1 through a plot of calculated vs experimental bond energies of the homonuclear diatomics corresponding to Table 1 and also the heteronuclear bond energies of Table 2. Since the difference between the two schemes lies in the method of obtaining the value of 9, for comparison we have tabulated the value of this quantity for different atoms corresponding to both the schemes in Table 1. It is interesting to note that these values of (PA correlate quite well with the third-derivative parameter y ( = a 3 E / d P ) studied by Fuentealba and Parr.29 Although the two quantities appear as different coefficients, (PA shows a proportionality to the product (NATA) for the same group of atoms.
Concluding Remarks The present work has been concerned with a new density functional description of chemical binding in homonuclear as well as heteronuclear diatomic molecules within the framework of an electronegativity-based picture through the electronegativity and hardness parameters of the constituent atoms and the potential and capacity parameters for the creation of an atomic dipole. The novel feature has been the prediction of the bond energy using the concepts of charge and dipole a t the atomic sites, with the former representing ionic binding and the latter the covalent contribution. The formalism is based on atomic parameters which are clearly transferable from molecule to molecule. Better modeling of the atom-atom cross coefficients might however be conceived through further studies of a large class of molecules. It is of interest tocompare the present model of covalent binding with the ones proposed'-"JJo earlier using the concept of bond charge. While in the earlier approaches the covalent binding has been accounted for in terms of an accumulation of negative charge in the bond region, in the present case this is viewed as a
125
100
75
50 25
'0
25
50
75
100
125
150
1 '5
CALCULATED BOND ENERGY (KCU/HOL)
Figure 1. Plot of the experimentalbond energies against the calculated bond energies for selected homonuclear and heteronuclear diatomic molecules: (A) homonuclear bond energies; (0)heteronuclear bond
energies using VA from eq 12; (0)heteronuclear bond energies using VA from eq 15.
consequence of quantum mechanical charge fluctuations and interatomic electron correlation resulting in dispersion-type interaction. While for simple diatomics both the models lead to reasonably good predictions, for polyatomics (especially nonlinear molecules), as the earlier work of Gazquez et al.31 suggests, a model using the bond charge as well as the atomic dipole might prove to be more successful in predicting other aspects of chemical binding. Although the method proposed here for the prediction of both homonuclear and heteronuclear bond energies from a single bond energy equation (eq 8) using no adjustable parameter has been illustrated with only the simple diatomic molecules as examples, the formalism is general and the extension to polyatomics is straightforward. However, one has to consider for this purpose new modeling of the additional atom-atom cross coefficients involving nonbonded atoms. Another aspect that requires consideration is that at present the molecular bond distances are used as inputs, whereas ultimately one would like to predict the bond lengths and geometries of the molecules. Work in this direction is in progress, and even a partial success would be of considerable importance.
Acknowledgment. It is a pleasure to thank T. G. Varadarajan and H. K.Sadhukhan for their kind interest and encouragement. We are grateful to the referees for their valuable suggestions for improvement of the original version of the manuscript. References and Notes ( 1 ) March, N. H., Deb, B. M., Eds. Single Particle Density in Physics and Chemistry; Academic Press: New York, 1987. (2) Parr, R. G.; Yang, W . Density Functional Theory of Atoms and Molecules; Oxford University Press: New York, 1989. (3) Parr, R. G.; Donnelly, R. A.; Levy, M.; Palke, W .E.J. Chem. Phys. 1978,68,3801. Sen,K.D., Jorgensen,C. K.,Eds.Electronegatiui~y,Srructure and Bonding, Springer-Verlag: Berlin, 1987; Vol. 66. (4) Parr, R. G.; Pearson, R. G. J. Am. Chem. SOC.1983, 105, 7512. Berkowitz, M.; Ghosh,S. K.; Parr, R. G. J. Am. Chem. Soc. 1985,107,681 1. Ghosh, S. K. Chem. Phys. Lett. 1990,172,77. Datta, D. Inorg. Chem. 1992, 31, 2797. (5) Yang, W.; Parr, R. G. Proc. Natl. Acad. Sci. U.S.A. 1985,82,6723. (6) Parr, R. G.; Yang, W . J. Am. Chem. SOC.1984,106,4049. (7) Ghosh, S.K.; Parr, R. G. Theor. Chim. Acta 1987, 72, 379. (8) Ghanty, T. K.; Ghosh, S. K. J . Phys. Chem. 1991,95,6S12. (9) Ghanty, T. K.; Ghosh, S. K. Inorg. Chem. 1992, 31, 1951. (10) Ghanty, T. K.; Ghosh, S.K. J. Chem. Soc., Chem. Commun. 1992, 1502. (11) Bamjai, A. S.; Deb, B. M.Rev. Mod. Phys. 1981, 53, 9s.
Chemical Binding in Molecules (12) Fernandez, R. J.; Alvarez-collado, J. R.; Paniagua, M. Mol. Phys. 1985, 56, 1145. (13) Kong, J.; Yan, J. M. Int. J . Quantum Chem. 1992, 42, 489. (14) Mortier, W. J.; Ghosh, S.K.; Shankar, S. J . Am. Chem. Soc. 1986, 108,431 5 and references therein. Ghanty, T. K.; Ghosh, S. K. J. Mol. Srrucr. (THEOCHEM) 1992,276,83. (15) Politzer, P. J. Chem. Phys. 1987, 86, 1072. See also: Vela, A.; Gazquez, J. L. J . Am. Chem. Soc. 1990,112, 1490. Fuentealba, P.; Reyes, 0. J. Mol. Srrucr. (THEOCHEM) 1993,282,65. (16) Ghanty, T. K.; Ghosh, S. K. J. Phys. Chem. 1993,97,4951. (17) Ohno, K. Theor. Chim. Acta (Berlin) 1968, 10, 111. See also: Nalewajski, R. F.; Korchowiec, J.; Zhou, 2.Inr. J. Quantum Chem. Symp. 1988, 22,349. (18) For an early paper indicating the proportionality of ?A with Ru-1, see: Ray, N. K.; Samuels, L.; Parr, R. G. J . Chem. Phys. 1979, 70, 3680.
While a proportionality constant can be employed, it would introduce an adjustable parameter which we avoid in the present prescription. (19) Rappe, A. K.; Goddard 111, W. A. J. Phys. Chem. 1991,95,3358. (20) See for example: Gray, C. G.; Gubbins, K. E. Theory of Molecular Fluids; Clarendon Press: Oxford, 1984; Vol. 1.
The Journal of Physical Chemistry, Vol. 98, No. 7, I994 1843 (21) Pitzer, K. S. J . Chem. Phys. 1955,23,1735. See also: Pitzer, K. S. Adv. Chem. Phys. 1959,2, 59. (22) See for example: Hirschfelder, J. 0.; Linnett, J. W. J. Chem. Phys. 1950, 18, 130. (23) Pearson, R. G. Inorg. Chem. 1988, 27,734. (24) Huheey, J. E. Inorgunic Chemistry, 3rd 4.Harper ; & Row: New York, 1983. (25) Zavitsas, A. A. J. Am. Chem. SOC.1991, 113, 4755. (26) Huber, K. P.; Herzberg, G. Molecular Spectra and Molecular Structure; Van Nostrand Reinhold: New York, 1979; Vol. 4. (27) CRC Handbook of Chemistry and Physics, 70th ed.; CRC Press: Bwa Raton, FL, 1989. (28) Pauling, L. The Nature Of ChemicalBond, 3rded.;CornellUniversity Press: Ithaca, NY, 1960. Reddy, R. R.; Rao, T. V. R.; Biswanath, R. J. Am. Chem. Soc. 1989, 111, 2914 and referenccs therein. (29) Fuentealba, P.; Parr, R. G. J. Chem. Phys. 1991, 94, 5559. (30) Borkman, R. F.; Simons, G.; Parr, R.G. J. Chem. Phys. 1969,50, 58. (31) Gazquez, J. L.; Ray, N. K.; Parr, R. G. Theor. Chim. Acta 1978,49, 1; see also Chapter 10 of ref 2.
