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342 9. Diatomic Molecules Fig. 9.28. Valence bond as increased electron charge between the two nuclei E R > Rc Separated atoms Re Rc = rA + rB R Chemical binding Multipole interaction Fig. 9.27. Chemical binding with overlap of atomic orbitals is important for R < Rc . For R > Rc multipole interaction dominates density becomes larger between the two nuclei. This results in an electrostatic attraction between the positive cores of the two atoms (for H2 these are the two protons) and the negative electron charge distribution between them. This effect is emphasized in the valence bond model used in chemistry. In the chemically bound molecule both atoms share one or more valence electrons in a common molecular orbital. This is also described in the LCAO approximation where the molecular orbital is represented by a linear combination of atomic orbitals. The second reason is of quantum mechanically nature and cannot be explained by a classical model. The molecular orbital has a larger spatial extension then the atomic orbitals. This increases the spatial uncertainty for the electrons and therefore decreases their average momentum !| p|" and their kinetic energy !E kin " = ! p2 "/2m, according to Heisenberg’s uncertainty relation. The combination of both effects leads to a minimum in the potential curve E(R), since the potential energy E(R) contains the average kinetic energy of the electrons (see Sect. 9.1). This second contribution to the molecular binding is called the exchange interaction, because the two electrons in the atomic orbitals of the LCAO can be exchanged since they cannot be distinguished in their common molecular orbital. Both effects are important at internuclear distances R # !rA " + !rB " that are smaller than the sum of the Valence bond mean atomic radii !rA " and !rB ", which give the extension of the electron clouds in the separated atoms. For distances smaller than this sum, the orbitals of the two atoms can overlap forming molecular orbitals and sharing electrons (Fig. 9.28). Molecular bonds that are formed due to this effect are called covalent or homopolar. One can also describe the chemical binding by energy conservation. If the energy increase necessary to deform the atomic orbitals when the two atoms approach each other is smaller than the decrease of the total energy (potential energy and mean kinetic energy of the electrons) in the rearranged molecular charge distribution, a stable molecule is formed. The nuclear distance Re and the electron charge distribution always arrange themselves in such a way that the total energy becomes a minimum. 9.4.2 Multipole Interaction For larger internuclear distances R > !rA " + !rB ", where the electron clouds of the two atoms no longer overlap, the chemical binding based on the two effects discussed above looses its importance. Nevertheless stable molecules are possible with such large internuclear distances R, although their binding energy is smaller. The correct treatment of the effects responsible for these interactions at large distances is based on quantum mechanics. However, good physical insight is already provided by the classical model, which starts from the multipole expansion of an arbitrary charge distribution ρ(r) for an observer at a point P at a distance R 9.4. The Physical Reasons for Molecular Binding qi A r ri → A B → → R P Fig. 9.29. Illustration of multipole expansion in (9.46) and (9.47) from the center of the charge distribution, which is large compared to the extension of ρ(r) (Fig. 9.29). We will discuss this model shortly. The potential φ(R) at the point P(R) generated by a distribution of charges qi (ri ) at the locations ri is 1 ! qi (ri ) φ(R) = . (9.46) 4πε0 |R− ri | If we choose the origin of our coordinate system to coincide with the nucleus of atom A, the positive charge q(ri = 0) = +Z A · e is the nuclear charge of atom A and q(ri ) = −e gives the charge of the ith electron in the electron shell. For R % r we can expand (9.46) into the Taylor series and obtain for the potential at a point P(X, Y, Z) R = {X, Y, Z} and r = {x, y, z} "! # 1 X! 1 qi + φ(P) = qi xi 4πε0 R i R R $ Y ! Z! + qi yi + qi z i R R &! #% 2 1 3X + 2 −1 qi xi2 2R R2 % 2 &! 3Y + −1 qi yi2 R2 % 2 &! $ 3Z 2 + − 1 q z i i R2 ' 1 + 3 [. . . ] + . . . R (! ) (! ) = φM qi + φD pi (! ) *i + . . . QM + φQM (9.47) Fig. 9.30. Deformation and shift of electron charge distribution of atom A by the interaction with atom B The first term φM represents the monopole+contribution, which is zero for neutral atoms where qi = 0. For ions it gives the main contribution. The second term φD describes the potential + of an electric dipole with a dipole moment p = qi · ri , which is the vector sum of the dipole moments pi = ei ri formed by the different electrons and the nucleus at r = 0. The third term φQM gives the contribution of the quadrupole moment to the potential, the next terms the higher moments, such as the octopole or hexadecapole which are here neglected. If another atom B with total charge qB , electric dipole moment pB and quadrupole moment Q MB is placed at the position P(R), the potential energy of the interaction between A and B can be written as the sum E pot (A, B) = E pot (qB ) + E pot ( pB ) * B) + . . . + E pot (QM (9.48) where E pot (qB ) = qB · φ( p) , E pot ( pB ) = + pB · grad φ( p) , * B · grad EA . E pot (QMB ) = QM (9.49) The vector gradient grad EA of the electric field E A , produced by atom A is the tensor ' " ∂E ∂E ∂E , , . (9.50) grad E = ∂x ∂y ∂z From the expression (9.49) we obtain the following contributions to the interaction energy between A and B. Two ions with charge qA and qB have long-range interactions E pot (qA , qB , R) = 1 qA · qB 4πε0 R (9.51) which decrease only with 1/R. An ion with charge qA and a neutral atom or molecule with a permanent dipole moment pB , pointing in a direction with an angle ϑ against the z-axis through 343 344 9. Diatomic Molecules ϑ R → pB qA Fig. 9.31. Interaction between a charge qA and an electric dipole moment pB Fig. 9.33. Interaction between two dipoles A and B (Fig. 9.31) experience the interaction energy where 1 qA pB cos ϑ . E pot ( pA , pB , R) = 4πε0 R2 (9.52) The interaction potential between an ion and a neutral atom with permanent dipole moment p is proportional to 1/R2 and is zero in the direction perpendicular to the dipole axis. Two permanent dipoles pA and pB with angles ϑA and ϑB against the z-axis and angles ϕA and ϕB against the x-axis (Fig. 9.33) have the interaction energy E pot ( pA , pB , R) = − pA · E( pB ) = − pB · E( pA ) (9.53) 1 (3 pB · R̂ · cos ϑ p − pB ) (9.54) 4πε0 R3 is the electric field generated by the dipole pB (Fig. 9.32). The interaction energy then becomes E( pB ) = E pot ( pA , pB , R) (9.55) 1 =− [3 pA pB cos ϑA cos ϑB − pA · pB ] 4πε0 R3 pA · pB [2 cos ϑA cos ϑB =− 4πε0 R3 − sin ϑA sin ϑB · cos(ϕA − ϕB )] . The interaction energy between two permanent electric dipoles is proportional to the product of the two dipole moments and depends on their relative orientation. It decreases as 1/R3 with increasing distance much faster than the 1/R Coulomb interaction between two charges. ER → p Eϑ → +q R → E ϑ S −q ER = ϕ 2p ⋅ cos ϑ 4 πε 0 ⋅ R3 Eϑ = p ⋅ sin ϑ 4 πε 0 ⋅ R3 Eϕ = 0 Fig. 9.32. Electric field of a dipole. It has axial symmetry around the dipole axis 9.4.3 Induced Dipole Moments and van der Waals Potential If a neutral atom without permanent dipole moment is placed in an electric field, the opposite forces on the negative electrons and the positive nucleus shift the electron charge distribution into the opposite direction than the nucleus. The centers of the positive and the negative charges no longer coincide as in a neutral atom without permanent dipole moment, and a dipole moment pind A = αA E (9.56a) is induced by the electric field , which is proportional to the field (Fig. 9.34). The constant αA is the electric 9.4. The Physical Reasons for Molecular Binding e− R → r(t) B + qA Fig. 9.35. Momentary dipole moment of an atom with spherically symmetric charge distribution → PB + Fig. 9.34. The charge qA produces an induced dipole moment pB → pA ( t) = e ⋅ r( t) polarizability of atom A and is a measure of the restoring forces in the atom against the displacement and deformation of the electron shell. If the electric field is produced by an ion with charge qB , the induced dipole moment of A becomes αA qB pind R̂ (9.56b) A = 4πε0 R2 where R̂ is the unit vector pointing into the direction from B to A. The potential energy of a neutral atom without permanent dipole moment in an electric field E is E pot = − pind A · E = −(αA E) · E . (9.57) If the electric field is produced by an ion with charge qB , the potential becomes E pot = − αA qB2 (4πε0 )2 R4 (9.58) If the field is generated by an atom B with permanent dipole moment pB we obtain from (9.54) and (9.57) the potential energy E pot = − αA p2B (3 cos2 ϑB + 1) . (4πε0 R3 )2 (9.59) In molecular physics the interaction between two neutral atoms is of particular importance. For a charge distribution on the electron shell that is spherically symmetric on the time average (e. g., for 1s electrons in the H atom) the time averaged dipole moment has to be zero. However, there still exists a momentary dipole moment (Fig. 9.35) that produces according to (9.54) a momentary electric field EA = 1 (3 pA R̂ cos ϑA − pA ) , 4πε0 R2 (9.60) which is statistically pointing in all directions and has a time average of zero. However, if we place another → pA =0 atom B in the vicinity of A, this field induces a dipole moment in atom B pind B = +αB EA , (9.61) which in turn generates an electric field at atom A inducing a dipole moment in A. Now the time average of p or E is no longer zero, because the interaction energy between the two induced dipoles depend on their relative orientation and the positions with minimum energy have a larger probability than those with higher energies. This leads to an attraction between A and B which is called a van der Waals interaction and is an interaction between two induced dipoles. We will now treat this more quantitatively. According to (9.55) the negative interaction energy between the two dipoles is a maximum when the two dipoles are either parallel (ϑA = ϑB = 180◦ ) (Fig. 9.36a) with their dipole moments pointing into the −zdirection, or antiparallel (ϑ = 90◦ , ϑ = 270◦ ), pointing in a direction perpendicular to the z-axis (Fig. 9.33). In the case of induced dipole-dipole interactions both dipole moments are directed along the axis through the two nuclei, which we choose as z-axis. This means that cos ϑA = cos ϑB = 180◦ and p ' R. From (9.60) we obtain 2 pA 2 pB EA = R̂ , EB = R̂ . (9.62) 3 4πε0 R 4πε0 R3 The potential energy of the interaction between the two ind induced dipole moments pind A and pB is then ind E pot (R) = − pind B · EA = − pA · EB . (9.63) With pA = αA · E B and pB = αB · E A we get from (9.62) ind 2 E pot (R) ∝ − pind A · pB = −αA · αB · |E| , (9.64) 345 346 9. Diatomic Molecules − − − − + + + + A − + + + → B + − − E R a) → pAind S S b) → pBind + − S → pAind → pBind A S If higher order terms in the multipole expansion are taken into account, the interaction energy between two atoms includes terms with 1/R8 , 1/R10 , 1/R12 , . . . for the induced quadrupole or octupole interactions. For homonuclear molecules only even powers n of 1/Rn can appear for symmetry reasons. − + B Fig. 9.36a,b. Possible orientations of two attraction-induced dipoles with (a) parallel (b) antiparallel orientation which can be written as E pot (R) = −C1 αA αB C6 =− 6 R6 R , interaction energy is also negative if the two induced dipoles are orientated antiparallel but both perpendicular to the z-axis through their centers (Fig. 9.36b). The quantum mechanical treatment of the van der Waals interaction is based on the calculations of the atomic charge distributions, perturbed by the mutual interaction between A and B. Since only this perturbation leads to an attraction between the two atoms one needs a perturbation calculation of second order [9.8, 9], which is beyond the scope of this book. (9.65) αA ·αB with C1 = (4πε1 )2 and C0 = (4πε 2. 0 0) This is the van der Waals interaction potential between two neutral atoms with the polarizabilities αA and αB . The constant C6 , which is proportional to the product αA · αB of the atomic polarizabilities, is called the van der Waals constant. The multipole interaction between two neutral atoms is only important for internuclear distances R > !rA " + !rB ". For smaller values of R the overlap of the electron shells of A and B has to be taken into account, which results in the above-mentioned exchange interaction and the electrostatic interaction due to the increased electron density between the two nuclei. The total range of R-values can, however, be covered by the empirical Lenard–Jones potential LJ E pot (R) = a b − R12 R6 (9.66) Epot Epot (R) = The interaction potential between two neutral atoms scales for large separations R as R−6 . The attraction is much weaker than between charged particles. a R12 − b R6 ∝ R −12 R Note that the interaction is attractive (because of the negative sign) and decreases as 1/R6 with increasing distance R. It is therefore a short range interaction compared with the Coulomb-interaction that is ∝ 1/R, but has a longer range than the interaction of the chemical bond, which falls of exponentially with increasing R. The R0 EB = b2 2a ∝ R− 6 Fig. 9.37. Lenard– Jones potential 9.4. The Physical Reasons for Molecular Binding where a and b are two parameters that depend on the two atoms A and B and which are adapted to fit best the potential curve obtained either experimentally or by accurate and extensive calculations (Fig. 9.37). From (9.66) it follows that E pot (R) = 0 for R = (a/b)1/6 . The minimum of E pot (R) is obtained for dE/ dR = 0. This gives the distance Re = 2(a/b)1/6 = 21/6 R0 (9.67) for the minimum. The binding energy at Re is then E B = −E pot (Re ) = b2 /2a . (9.68) 9.4.4 General Expansion of the Interaction Potential The potential energy E pot (R) of a diatomic molecule can be expanded for |R − Re |/Re < 1 into a Taylor series around the equilibrium distance Re of the potential minimum: % & ∞ ! 1 ∂ n E pot E pot (R) = (R − Re )n . (9.69a) n n! ∂R Re n=0 Because (∂ 0 E/∂R0 ) Re = E pot (Re ) and (∂E/∂R) Re = 0, this gives % & 1 ∂ 2 E pot E pot (R) = E pot (Re ) + (9.69b) 2 ∂R2 Re × (R − Re )2 + . . . . In molecular physics the minimum of the ground state potential is generally chosen as E pot (Re ) = 0. Instead of the negative binding energy E B (which is used if the zero point is chosen as the energy of the separated ground state atoms) the positive energy E D = −E B is now used, which gives the energy necessary to dissociate the molecule from its energy minimum at R = Re to the separated atoms at R = ∞. For |R − Re |/Re # 1 the higher order terms with n > 2 can be neglected and we obtain a parabolic potential in the vicinity of the potential minimum. The potential energy of a diatomic molecule can be approximated in the vicinity of the potential minimum by a parabolic potential (Fig. 9.38). Fig. 9.38. Comparison of parabolic and Morse potentials with the real (experimental) potential 9.4.5 The Morse Potential In 1929 P.M. Morse had already proposed an empirical potential form , -2 E pot (R) = E D 1 − e−a(R−Re ) , (9.70) which represents the attractive part of the potential with a much better approximation to the experimental values than the parabolic potential. This potential converges for R → ∞ correctly towards the dissociation energy E D , while the parabolic potential goes to infinity for R → ∞ (Fig. 9.38). The repulsive part of the potential for R < Re deviates more severely from the experimental data. We obtain from (9.70) , -2 lim E pot (R) = E D 1 − e+aRe R→0 (9.71) while the experimental potential must converge towards the energy levels of the united atom (see Fig. 9.25). The Morse potential has the great advantage that the Schrödinger equation of two atoms vibrating in this potential can be solved exactly (see Sect. 9.6). 347