3/22/2005 CHEM 408 - Sp05L10 - 1Intermolecular InteractionsIntermolecular Interactions• the fact that all molecules aggregate at low enough T reflects the universal presence of attractive interactions between molecules or between parts of a single molecule• estimate the strength of intermolecular attractions á la Stone* using only the boiling point, Tb: εzNRTHAbvap2110 ≅≅∆bBTkz20≅εenthalpy of vaporization# neighborsTrouton’s ruleAvog. #which predicts a pairinteraction strength εof:System Tb / K z (20Tb/z) / K εobs/kB / K He 4.2 12 7 11 Xe 166 12 277 281 CH4 111.5 12 186 144 C6H6 352.3 12 587 ~428 H2O 373.2 4 1866 ~2400 Some Values of ε*A. J. Stone, The Theory of Intermolecular Forces (Oxford, 1996)3/22/2005 CHEM 408 - Sp05L10 - 2• intermolecular interactions can be approximately grouped into four categories:3. short-range repulsion (+)- interactions due to the overlap of filled electronic orbitals & atomic cores1. electrostatic (+/-) - interactions between the permanent charge distributions of molecules (i.e. dipole-dipole, dipole-quadrupole, etc.)4. inductive (-) - interactions between the permanent charge distribution of one molecule and the electronic polarizability of a second molecule (e.g. dipole-induced dipole, etc.)2. dispersion (-) - interactions due to the instantaneous correlations between the electronic motions in two molecules3/22/2005 CHEM 408 - Sp05L10 - 31. Electrostatic Interactions; Multipole Expansion⎟⎟⎠⎞⎜⎜⎝⎛−++++=⎟⎟⎠⎞⎜⎜⎝⎛+=θθπεπεφcos2cos24141)(2222212121022110rzzrqrzzrqrqrqBelBAq1q2r1 r2rθ⎟⎟⎠⎞⎜⎜⎝⎛+++−++≅ ...2)()(41322221121122210rzqzqrzqzqrqqπε• Leech considers the electrical potential φelat a distant point B that is created by a pair of charges near A:z⎟⎟⎠⎞⎜⎜⎝⎛+−Θ++= ...2)1cos3(cos413220rrrqθθµπε21, zzr >z1z2• the last expression shows that φel(B) can be expressed in terms of the multipole moments of the A charge distribution: the net charge q, the dipole moment µ, the dipole moment, the quadrupole moment Θ, etc.3/22/2005 CHEM 408 - Sp05L10 - 4BA• more generally, for a distribution of charge due to a molecule centered at A:RraRrr−ar∑−=aaAelaRqB||41)(0rπεφ...15131+Ω−+−≅αβγαβγαβαβααµµTTTTqaRrr>where α, β, γare cartesian coordinates (x, y, z), repeated indicies imply summation, and the Ts are the spatial derivatives:νβαναβπεRRRTn∂∂∂∂∂∂= ...410)(...041πε=TyxTxy∂∂∂∂=041πεi. e.etc.• this expansion enables the electrostatic interaction between twomolecules A and B to be expressed:...+∂∂∂∂∂∂Ω+∂∂∂∂Θ+∂∂+=AelBAelBAelBAelBABelqVφγβαφβαφαµφαβγαβα1stmoment of B charge1stderiv. of el. potential from A baRrrr,>no overlap of A and B charge dists.3/22/2005 CHEM 408 - Sp05L10 - 5∑−=Θaaaaaq charges22123)(αββααβδ∑++−=Ωaaaaaaaaaq charges22125)}({αβγαγββγαγβααβγδδδquadrupole momentoctopole moment• the multipole moments of a molecule are given by:∑=aaqq charges∑=aaaq chargesααµchargedipole moment∫∑== rxdrxqaaaxrr)(ρµrdrzrrrzqaaaazzrr∫∑−=−=Θ )()()(2212232221223ρ• For a continuous charge distribution ρ(r) the summations in these equations are simply replaced by integrals. For example:∫∑==Θ rxydryxqaaaaxyrr)(2323ρ)or ,,( zyx=α•“2n-poles”:+q (n=0)µ(n=1)++--+-++++----Θ(n=2) Ω(n=3)3/22/2005 CHEM 408 - Sp05L10 - 60.40.40.40.40.40.40.40.40.40.60.60.60.60.60.60.80.80.80.80.81.01.01.01.01.01.01.01.01.01.00.60.4-3 -2 -1 0 1 2 3-3-2-10123-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 0.00.0-0.2-0.4-0.60.2-1.0-0.80.20.4-1.0-1.00.40.60.80.60.81.01.01.01.01.0-1.00.80.60.40.20.0-0.2-0.4-0.6-0.8-1.0-1.00.40.40.20.20.00.0-3 -2 -1 0 1 2 3-3-2-10123-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 0.00.00.20.2-0.2-0.20.40.4-0.4-0.40.6-0.60.8-0.81.0-1.01.01.0-1.0-1.00.6-0.6-3 -2 -1 0 1 2 3-3-2-10123-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 Electrical Potentials of Multipoles n=0-2qΘµrq1∝φ2cosrzθφµ∝3231cos3rzz−∝Θθφ3/22/2005 CHEM 408 - Sp05L10 - 7rdrrzzrr)1cos()(2232−=Θ∫θρ• some examples of quadrupole moments of molecules Fig. 2.1 from A. J. Stone, The Theory of Intermolecular Forces(Oxford, 1996)• some important units:charge q: 1e = 1.6022×10-19 C dipole moment µ: 1 ea0= 8.478×10-30 C m = 2.5418 Dquadrupole moment Θ: 1 ea02= 4.487×10-40 C m2= 1.354 D Ånode-3.3+1.8+5.6-6.73/22/2005 CHEM 408 - Sp05L10 - 8• some explicit forms for interaction energy: }cossinsincoscos2{430ϕθθθθπεµµµµBABABArV −=}cos2sinsin)1cos3({cos234240ϕθθθθπεµµBABABArV −−Θ=Θ)}cos2sinsin-coscos2(4 coscos15cos5cos51{434222250ϕθθθθθθθθπεµBABABABABArV+−−−ΘΘ=Θ• quadrupole forms assume a linear quadrupole with the symmetry axis chosen as z with:Θ−=Θ=ΘΘ≡Θ21 and yyxxzz• the factor: 4πε= 1.11265×10-10 C2 N-1m-2is appropriate for the SI unit systemfigure.in ;, :Note2121ϕϕϕθθθθ−===BA3/22/2005 CHEM 408 - Sp05L10 - 92. Dispersion Interactions• also “London” dispersion forces after F. London (1930) who first gave QM explanation• long-range attraction, universally present• dominant in nonpolar systems like He, Xe, CH4, CCl4…• due to correlated electron motions on two molecules• understand using Drude model of harmonically bound es: +--+charge e, force constant k, frequency ω()622041)4(43rkerVπεωh−≅∞→r• London formula is:()6212121123rIIIIrV+−≅ααmolecules 1 & 2, α= polarizability, I = ionization energy• leading term in expansion in 1/r:()...10108866+++≅rCrCrCrV3/22/2005 CHEM 408 - Sp05L10 - 103. Short-Range Repulsion• also called “exchange-repulsion” or just “exchange” energy• 2 effects: electron exchange between two molecules (-) and Pauli repulsion between e of like spin (+) • net is repulsive and of exponential form (like overlap of Ψ)Fig. 6.2 from A. J. Stone, The Theory of Intermolecular Forces (Oxford, 1996)H-HArArH-H())exp( rArVsrβ−≅3/22/2005 CHEM 408 - Sp05L10 - 114. Inductive Interactions • result from permanent charge moments on one molecule inducing an electronic polarization in another molecule• attractive Fig. 4.1 from A. J. Stone, The Theory of Intermolecular Forces (Oxford, 1996)()42021)4( rqrVqπεαα−=()62rrVαµµα−∝()82rrVααΘ−∝Θinduction is non-additive:3/22/2005 CHEM 408 - Sp05L10 - 12• although the interaction