Chapter 2 Experimental sources of intermolecular potentials 2.1 Overview thermodynamical properties: heat of vaporization (Trouton s rule) crystal structures ionic crystals rare gas solids physico-chemical properties: bulk modulus, phonon spectrum, etc. virial coefficients of real gases viscosity, thermal conductivity (collision integrals spectroscopy: VRT (vibration-rotation tunneling) 2.2 Trouton s rule Empirical relationship between enthalpy of vaporization, H vap and boiling point, T b at atmospheric pressure H vap 10RT b (2.1) This observation can be explained by the fact that at T b the change in Gibbs free energy is zero H vap = T b S vap (2.2) 13
Table 2.1: Pair potential well-depths from Trouton s rule T b /K n (20T b /n)/k (ε/k B )/K ε/k B /kj mol 1 ε lit. He 4.2 12 7 11 0.09 Ar 87.0 12 145 142 1.18 Xe 166.0 12 277 281 2.34 CH 4 111.5 12 186 180-300 1.5-2.50 H 2 O 373.2 4 1866 2400 20 Approximate the entropy change by the liquid/gas volume change: S vap = R ln (V g /V l ) R ln 1000 7R (2.3) The remaining contribution of 3R is due to liquid structure. 2.2.1 Well-depth from Trouton s rule The latent heat of evaporation can be approximated by the energy required to separate the liquid to its constituents is ε (neglect zp energy). The total energy for N molecules, each having n neighbours is H vap 10RT b 1 2 N Anε An estimation of of the well-depth, ε is given in terms of T b ε/k B 20T b /n (2.4) 14
2.3 Theory vs. experiment EXPERIMENT {D exp } {D theor } model model U exp U theor QUANTUM CHEMISTRY 2.4 Lattice energies 2.4.1 Ionic crystal NaCl crystallizes in fcc lattice lattice parameter a = 5.64Å ions are in (0,0,0) and ( 1, 1, 1) 2 2 2 lattice energy U latt = 764.4 kj/mol Cohesion energy is the sum of the Coulomb (Madelung) and repulsion energy: U coh = U C + U R (2.5) where U C is a lattice sum of 1/r interactions, U R is the so-called Born-Mayer potential: U C = Q 2 (±) j r 1 j U R = B exp r i /ρ = 6Be a/2ρ (2.6) j i The lattice sum (Madelung energy) is conditionally convergent, so special techniques (e.g. Ewald summation) are needed to evaluate them. For a NaCl lattice, the energy 15
can be expressed using the Madelung constant, α = 1.7476, U C = Q 2 2α a 2 1.7476 = 1389.9 kj/mol = 861.3kJ/mol (2.7) 5.64 The Born-Mayer potential has two unknown parameters, B and ρ. Repulsion energy U R = U exp coh U C = 764.4 + 861.3 = 96.9kJ/mol (2.8) Condition of equilibrium in the minimum of the lattice energy U coh a 2α = Q2 a 6B 2 2ρ e a/2ρ = U c a U R 2ρ = 0 (2.9) Effective ion radius is obtained from the repulsion and Madelung energies The B parameter of the Born-Mayer potential is B = ρ = 1 U R a = 0.3164 (2.10) 2 U C U R 6e a/2ρ = 1.168 105 kj/mol (2.11) The quality of this potential can be checked by calculating the bulk modulus K = V 2 U coh V 2 (2.12) where the volume of 1 mole NaCl is V = N A a 3 /4. The volume derivative can be calculated as lattice-parameter derivative: ( ) 1 V V = (2.13) a a Bulk modulus for the NaCl structure K = N ( ) Aa 3 2 4 2 U coh = 4 2 U coh (2.14) 4 3N A a 2 a 2 9N A a a 2 Second derivative of the cohesion energy 2 U coh = ( ) Q 2 2α 1 U R a 2 a a 2 2ρ a = 2 a U 2 C + 1 4ρ U 2 R (2.15) Theoretical bulk modulus K = 4 9N A a ( 2UC Conversion to gigapascal 1 kj/mol/å 3 = 1.667 GPa, i.e. a 2 K theor = 24.57 GPa + U ) R = 14.8016 kj/mol/å 3 (2.16) 4ρ 2 16 K exp = 24 GPa
2.4.2 Rare gas crystal Cohesion energy is the sum of pair potentials U coh = N 2 Ar crystallizes in fcc lattice lattice parameter a = 5.3109Å ions are in (0,0,0) and ( 1 2, 1 2, 1 2 ) lattice energy U latt = 8.4732 kj/mol (after zpe correction) U LJ (r ij ) (2.17) ij where N = 4, number of atoms in the unit cell, and U LJ potential: [ (σ ) 12 ( σ ) ] 6 U LJ (r) = 4ε r r is the Lennard-Jones (2.18) Lattice sums can be calculated from the number of neighbours at the multiples of the nearest-neighbour distance d = a/ 2: General form of the lattice sums shells ( ) n σ m i = f i d i shell 1 2 3 4 5 i r d 2d 3d 2d 5d fi d m i 12 6 24 12 24 m i shells Lattice sum for the 6- and 12-potentials i m i f n i ( σ d ) n = pn ( σ d i 1 5 fcc( ) hcp ( ) p 12 = i m if 12 i 12 12.13114 12.13188 12.13229 p 6 = i m if 12 i 12 14.01839 14.45392 14.45489 Cohesion energy of the rare-gas lattice in terms of lattice sums U coh = Nε [ ( σ ) 12 ( σ ) ] 6 p 12 p6 2 d d Stability condition ) n (2.19) (2.20) U ( coh σ ) 12 d = 12p 1 ( σ ) 6 12 d d + 6p 1 6 d d = 0 (2.21) 17
yields the σ parameter in the function of d, nearest-neighbour distance d = 6 2p12 p 6 σ = 1.09026σ (2.22) Lattice constant a = 1.54186σ. Cohesion energy can be expressed in the function of ε. [ U coh = Nε ( ) 2 ( p6 p6 p 12 p 6 2 2p 12 can be expressed in the function of ε U coh = N 8 2p 12 ) ] (2.23) p 2 6 p 12 ε = 2.1525 N ε (2.24) Experimental lattice parameters from best Lennard-Jones potentials: Ne Ar Kr Xe ε σ a U coh a U coh 2.4.3 Three-body forces and the structure of rare gas solids 2.5 Equation of state virial coeffiicients 2.5.1 van der Waals equation of state Equation of state for an ideal gas (noninteracting point masses) P V = RT (2.25) where P is the pressure, V is molar volume, and R = 8.31J K 1 mol 1 is the gas constant. 18
The standard molar volume V s (1 atm, T=273.15 K o of real gases are different from the ideal value (22414 cm 3 ). Considerable deviations can be observed in the behaviour of the compression factor, z = P V. RT These deviations can be accounted for by the van der Waals equation of state, by introducing two empirical constants, a and b ( P + a ) (V b) = RT (2.26) V excluded volume in a binary collision 4 3 πσ3 b = 2 3 πn Aσ 3 (2.27) reduction of pressure due to intermolecular attraction: the number of binary interactions is proportional to the square of the density, (a/v 2 ) Virial equation of state power series of (1/V ) z = 1 + B 2 V + B 3 V 2 + B 4 V 3 +... (2.28) 2.5.2 Equations of state - generalities The total energy is a function of x α external parameters E = E(x 1, x 2,... x n ) (2.29) Define generalized forces corresponding to external parameters X α = E x α (2.30) 19
In statistical mechanics the equations of state describe the relationship of the external parameters, the generalized forces and the temperature: X α = kt ln Z x α (2.31) where Z is the partition function. For the special case of volume (x α = V ) and pressure (X α = P ) P = kt ln Z (2.32) V The semi-classical partition function Z = 1 N!h 3N... e (K+U)/kT d 3 p 1 d 3 p 2... d 3 p N d 3 r 1 d 3 r 2... d 3 r N (2.33) is a product of two terms, depending on the kinetic and the potential energy, respectively: K = 1 p 2 2m j U = 1 u jk (2.34) 2 jk Z = 1 N!h 3N... e K/kT d 3 p 1 d 3 p 2...d 3 p N } {{ } (2πmkT ) 3N/2... e U/kT d 3 r 1 d 3 r 2...d 3 r N } {{ } Z U (2.35) Partition function ( ) 3N/2 2πmkT Z U (2.36) Z = 1 N! h 2 2.5.3 Equations of state - ideal gas For an ideal gas u ij 0 (or if T is high kt ) e U/kT 1 Z U = V N (2.37) and the corresponding partition function ln Z id = ln 1 [ N! + N ln V + 3 2 ln kt + 3 ( )] 2πm 2 ln h 2 (2.38) Equation of state for an ideal gas or P = kt ln Z id V = NkT V (2.39) P V = NkT (2.40) 20
2.5.4 Equations of state - real gas For a real gas with low number density (n = N/V small) an approximate expression can be obtained for the configurational partition function. Take the average potential energy with β = 1/kT, which satisfies the following relationship Ue βu d 3 r 1 d 3 r 2... d 3 r N U = = e βu d 3 r 1 d 3 r 2... d 3 r N β ln Z U (2.41) The logarithm of the partition function is ln Z U (β) = N ln V β 0 U(β )dβ (2.42) In a low-density system the U, the average potential energy of 1/2N(N 1) molecule pairs is equal N 2 /2-fold of the average potential energy of an arbitrary pair of molecules: U = 1 2 N(N 1) 1 2 N 2 u (2.43) The average potential energy of a pair of molecules u(r)e βu(r) d 3 R u = e βu(r) d 3 R = β ln e βu(r) d 3 R (2.44) Since u(r) = 0 almost everywhere, excepted the small distances, it is worthwhile to transform the integral as e βu(r) d 3 R = [1 + ( e βu(r) 1 )] d 3 R = V + I(β) (2.45) The quantity in parentheses is the Mayer-function f(r) = e βu(r) 1, and its integral over the intermolecular separation, R is I(β) = 4π 0 ( e βu(r) 1 ) R 2 dr (2.46) Calculate the average potential energy for a pair of molecules u = [ ( ln [V + I(β)] = ln V + ln 1 + I(β) )] 1 β β V V I(β) β which is substituted in the expression of the configurational partition function The equation of state is (2.47) ln Z U (β) = N ln V + 1 N 2 [I(0) I(β)] (2.48) 2 V P kt = ln Z V = ln Z = N V U V 1 N 2 I(β) (2.49) 2 V 2 21
or P V RT = P V NkT = 1 4π 2 N V This is the first term of the virial equation of state: and the virial coefficient is B 2 (T ) = 2πN z = P V RT = 1 + 0 0 n=1 ( e βu(r) 1 ) R 2 dr (2.50) B n+1 V n (2.51) ( e u(r)/kt 1 ) R 2 dr (2.52) 2.5.5 Temperature dependence of the virial coefficient T \R small (u > 0) large (u < 0) low + B 2 < 0 kt < u 0 high kt > u 0 + B 2 > 0 22
2.5.6 Third virial coefficient For a strictly additive potential B add 3 = N 3 8π3 B 3 (T ) = B add 3 + B 3 (2.53) f 12 f 13 f 23 r 12 r 13 r 23 dr 12 dr 12 dr 12 (2.54) The non-additivity correction depends on the 3-body potential U 3 B 3 = N 3 8π3 (e U 3 /kt 1 ) e U 3/kT r 12 r 13 r 23 dr 12 dr 12 dr 12?? (2.55) 2.6 VRT spectroscopy - water dimer Calculated splitting due to tunneling is very sensitive to the height and shape of the barrier stringent test of the PES. 23
2.6.1 VRT spectroscopy - water dimer Experimental vs. calculated VRT splittings with different water-water potentials. None of the 14 potentials was found to be fully satisfactory. Fellers, Braly, Saykally, Leforestier, J. Chem. Phys. 110 (1999) 6306) 24