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Intermolecular Interactions and Potentials Origin of intermolecular force • Is it Gravitational? No, is of the order of 10-52 J and so negligible • Strong force? No, significant in the range of 10-4 nm, but molecular dimensions are typically 5×10-1 nm • Weak force? No, although electromagnetic in origin but exits only between nuclear particles • Electromagnetic? Yes, since they are charged particles, attractive at long range and repulsive at short range Types of intermolecular interaction • Electrostatic • Induction • Dispersion In terms of the coordinates r, θ of the point P and the axial separations of the charge from O this may be written as Q1 Q2 Φ= + 2 2 1 / 2 2 2 1 / 2 4 πε 0 ( r + z 2 − 2 z 2 r cos θ ) ( r + z1 + 2 z1 r cos θ ) 1 r1 r P r2 θ -z1 Q1 O z2 z Q2 expanded in powers of z1/r and z2/r 1 Φ = 4 πε o { Q 1 + Q r 2 + (Q z − Q z ) cos θ 2 2 1 1 2 r + 2 2 2 (Q z − Q z )( 3 cos θ − 1) 1 1 2 2 3 2r + } ... previous eq. may be written as 1 Φ = 4πε 0 { + r 2 µ cos θ Q r 2 θ (3 cos θ − 1) + 2 r 3 } + ... Q = Q1+ Q2 , total charge of the molecule, zeroth moment of the charge distribution µ = Q2Z2-Q1Z1, is the dipole moment of the charge distribution Θ = Q1Z21+Q2Z22 , is the quadrupole moment of the charge distribution the above eq. is valid only for r >> z1, z2 . Since we are interested in the longrange interactions this approximation is valid. Interaction of two linear charge distribution Now we consider the energy of interaction of two charge distribution -Z1 -Z’1 Z2 O Q1 -Z1 Q2 -Z1 Q’1 Q’2 Q2 (a) Q’2 Z’2 -Z’1 P (b) Q’1 Q’2 Z2 O Q1 P Z2 O Q1 Z’2 P (c) Q2 Q’1 θ1 Q1 Q2 θ2 Ф (d) For the configuration (a) the electrostatic energy of the two distribution is Uael(r) = Q’2Ф(Q’2) + Q’1Ф(Q’1) Ф(Q’2) is the electrostatic potential at Q’2 due to the first (unprimed) molecule, Ф(Q’1) is electrostatic potential at Q’1 arising from the same source Since these electrostatic potentials can be expressed in terms of z’2 and z’1 U el { 1 a (r ) = 4 πε 0 ( µQ '− Qµ ' QQ ' + 2 µµ ' − + 2 r r + 3 r (3 Θ ' µ − 3 µ ' Θ ) ( Q Θ ' + ΘQ ' + 3 r 6 ΘΘ ' r } → ( A) + ... 4 5 r various term in the electronic energy is the dipole-dipole interaction which varies as r-3 and is given by Q’ = Q’1+ Q’2 , total charge of the second molecule µ’ = Q’2z’2-Q’1z’1, is the dipole moment Θ’ = Q2’z2’2+Q1’z1’2 , is the quadrupole moment various term in the electronic energy is the dipole-dipole interaction which varies as r-3 and is given by −2 µµ ' U µµ ' (r ) = 4πε 0 r 3 a Thus in the configuration the dipole-dipole contribution to the interaction energy is negative and there is an attractive force between the neutral molecules Similar development can be carried out for configurations (b) and (c) that shown earlier for the dipole-dipole interaction of two neutral molecules and can obtain +2µµ ' U µµ ' (r ) = 4πε 0 r 3 b Which corresponds to repulsive force and Ucμμ’(r) =0. i.e) dipole-dipole interaction energy is zero These illustrations show that the dipole-dipole interaction energy is proportional to r-3 for a fixed configuration of the molecules, and that it is strongly dependent on orientation varying from attractive to repulsive as one molecule is rotated An analysis of the interaction of two linear charge distributions in the general configuration in fig (d) wherein Ф denotes rotation of the second dipole about the line joining them leads to the dipole-dipole interaction energy U Where µµ ' (r , θ1 , θ 2 , φ ) = − µµ ' 4 πε 0 r 3 ζ (θ 1 , θ 2 , φ ) ζ (θ , θ , φ ) = (2 cos θ cos θ − sin θ sin θ cos φ ) 1 2 1 2 1 describes the dependence of the energy on orientation 2 If it is gas phase the two molecules are free to rotate it is often the average of the dipole-dipole energy over all possible orientations <Uel>μμ‘ is required. The probability of observing a configuration of energy U is proportional to the Boltzmann factor exp(-U/KT) U el µµ ' =− 2 µ µ 2 6 '2 3 r kT (4πε 0 ) 2 + ... The leading term of the orientation-averaged dipole-dipole contribution to the electrostatic energy of interaction to two neutral molecules is therefore attractive and inversely proportional to the sixth power of their separation The temperature dependence of <Uel>μμ‘ is a result solely of the orientational averaging with the Boltzmann weighting Eq. A shows the variation of energy as r-4 for dipole-quadrupole interactions r-5 for quadrupole-quadrupole interactions for fixed orientation but for rotational case the corresponding interaction energy is given by 1 U el µΘ U el 8 r kT (4πε 0 ) ΘΘ ' =− {µ Θ 2 = − 2 2 ΘΘ 14 10 '2 '2 +µ Θ 2 } + ... Dipole-quadrupole '2 5 r kT (4πε 0 ) 2 + ... Quadrupole-Quadrupole Above expressions show that these two contributions to the orientation-averaged electrostatic interaction energy are also attractive Induction energy Any molecule is placed in a uniform, static electric field E there is a polarization of its charge distribution Electric field E ---- ++ ++ ++ Process of induction Induced dipole moment in the molecule is given by μind=αE (a) α is static polarizability of the molecule Energy of a neutral dipolar molecule in an electric field E is π U = − ∫ µ ⋅ dE Using eq (a) in (b) (b) 0 Since μ and E are parallel U ind π 1 0 2 = − ∫ α EdE = − αE 2 (c) Interaction of dipolar molecule with non-polar molecule Neutral molecule possessing permanent dipole moment can induce a dipole moment in a second molecule which is nearby whether the second molecule is itself polar or not Non-polar molecule r O P θ Interaction of a dipolar molecule with non-polar molecule µ µ cos θ 1 Φ = The electrostatic potential due to the dipole at P is E =| E | = ( ) ( ) ∂φ 2 1 ∂φ + ∂r r ∂θ 2 1 2 4πε 0 1 µ = 4πε 0 r 3 [ 4 cos 2 2 θ + sin θ ] 2 r 2 Magnitude of the dipole induced in the molecule at P is thus α'E and the energy is -(1/2) α'E so that the interaction energy of induction is U ind = − 1 2 α ' µ r 2 6 ( 3 cos 2 θ +1 (4πε 0 ) 2 ) (i) which is attractive for all configurations and inversely proportional to the sixth power of the intermolecular separation for a fixed orientation If the above potential energy is averaged over all possible orientations, giving each orientation the Boltzmann weight exp(-U/KT), the average induction energy between a dipolar molecule and a non-polar molecule is, at sufficiently high temperatures 2 U ind = −µ α 2 ' (4πε 0 ) r 6 This term is not temperature dependent unlike the corresponding term for the electrostatic energy Interaction of two polar molecule • • If interaction is between two polar molecules each molecule induces a dipole moment in the other so that there are two contributions to the total interaction energy of the eq. (i). The orientationally-averaged induction energy for two dipolar molecules is U ind • µµ =− 2 { (4πε 0 ) r 6 2 ' ' 2 } µ α + µ α + ... For two identical polar molecules this reduces to U ind • ' 1 µµ ' = −2αµ 2 2 6 + ... (4πε 0 ) r The general treatment of induction forces is very much more complicated than that given here. This because for many molecules the polarizability is not isotropic but is tensorial character. In addition higher-order multipoles in polar molecules can induce higher-order multipoles in other molecules and there can be other contributions to the induced dipole moment. Dispersion energy • So far the we analysed contributions to the long range energy by means of classical electrostatics and classical mechanics • For the interaction of two molecules possessing no permanent electric dipole or higher-order moments, electrostatic and induction contributions are absent and the interaction arises solely from the dispersion energy • The dispersion energy cannot be analysed by classical mechanics since as we shall see its origins are purely quantum mechanical • Consequently dispersion energy is well explained by the Drude Model Drude model Considering each molecule is composed of two charges +Q and –Q. We imagine charge +Q to be stationary and that the negative charge oscillates about the positive charge with an angular frequency ω0 in the Z-direction which is along the line joining the positive charges of the two molecules as show in fig. r 1-d Drude model of the dispersion interaction za +Q zb -Q -Q +Q Animation If we denote the displacement of the negative charge of molecule a from its positive charge by za, we note that at any time t the moelcules possess instantaneous dipole moments μa=Qza(t) and μb=Qza(t) Denoting F as force constant of the harmonic oscillator by k and the mass of the oscillating charge by M, the frequency of the oscillation is ω = k/M thus When the two molecules are infinitely separated, the Schrödinger wave eq. for molecule a is 0 2 1 ∂ Ψa M ∂z a 2 + 2 2 (Ε a − 1 2 which is the equation of SHO, where ½ kZ2a is the potential energy of the oscillator 2 kz a ) Ψ a = 0 The eigen values of the energy for molecules a and b are given by Ε a = ( n a + Ε b = ( n b + 1 2 1 2 ) ω 0 ) ω 0 When the two molecules are infinitely separated and both are in their ground states the total energy of the two-molecule system is Ε ( ∞ ) = Ε a + Ε b = ω 0 When the molecules are separated by a finite distance r, which is still considered to be large by comparison with the dimensions of the molecules, there is an energy of interaction between the two dipoles at any instant and this interaction energy is included in the Schrödinger wave eq. 2 1 ∂ Ψ M ∂z a 2 2 + 1 ∂ Ψ M ∂z b 2 + 2 2 (Ε − 1 2 2 kz a − 1 2 2 kz b − 2 za zb Q 4 πε 0 r 2 3 )Ψ = 0 Where ψ is the wave function for the two-molecule system If we make the transformations Z 1 = za + zb 2 Z2 = za − zb 2 After transformation Schrödinger eq. becomes 2 1 ∂ Ψ M ∂Z 1 2 2 + 1 ∂ Ψ M ∂Z 2 2 + 2 2 [Ε − 1 2 2 k1 Z 1 − 1 2 2 k 2 Z 2 ]Ψ = 0 where k1 = k − 2Q 2 4 πε 0 r k2 = k + 3 2Q 2 4 πε 0 r 3 Above Schrödinger wave equation is for two independent SHO in the coordinates Z1 and Z2. Thus the eigenvalues for the total energy of the system are 1 1 ) ω1 + ( n 2 + Ε ( r ) = ( n1 + 2 ) ω 2 2 If we consider the two molecules to be in their ground states then 1 ( ω1 + ω 2 ) Ε( r ) = 2 or ω 1 = ω 0 {1 − 1/ 2 ω1 = ( k1 / M ) where 2Q 2 } 1 / 2 ω 2 = ω 0 {1 + 3 4πε 0 r k 1/ 2 ω2 = ( k2 / M ) 2Q 2 } 1/ 2 3 4 πε 0 r k We are interested in the long-range interaction for which the perturbation potential is small so we can expand ω1,ω2 by the binomial theorem which allows to write E(r) as 4 Ε ( r ) = ω 0 − Q ω 0 2 6 2 ( 4 πε 0 ) r k + ... 2 The energy of interaction of the two molecules for our model is 4 U disp = Ε ( r ) − Ε (∞ ) = − Q ω0 2 3 4(4πε 0 ) r k 2 + ... The motion of the oscillating charge can be resolved into three oscillations of identical frequency along three Cartesian coordinates centered on the positive charge. Thus 4 U disp = Ε( r ) − Ε (∞ ) = − Q ω0 2 3 4(4πε 0 ) r k 2 + ... The force constant, k is related to the polarizability of the molecules If we place the single Drude molecule to an external electric field E a force of magnitude QE acts on each charge to produce a displacement z’a which attains a static value when the restoring force kz’a is equal to the imposed ' electrical force. Then Q = kz n / Ε so that static dipole moment induced in the molecule by the field is μind= Qz’n=Q2E/ k 2 α = Q /k Now, Polarizability So that we can write the dispersion energy for two identical molecules in their ground states for this model as Udisp =C6/r6 where 2 C 6 = − 3 α ω 0 4 (4πε 0 ) 2 The dipole-dipole dispersion energy for two molecules in their ground states is therefore attractive and inversely proportional to the sixth power of the intermolecular separation If we treat the same problem classically the interaction energy is zero. The existence of the dispersion energy is therefore a consequence of the zero-point energy of the oscillators, a purely quantum mechanical concept. Considering additional contributions to the dispersion energy arising from instantaneous dipole-quadrupole, quadrupole-quadrupole interactions, etc. we get U disp = C6 r 6 + C8 r 8 + C10 r 10 + ... where, for interactions between ground-state molecules, each coefficient is negative so that each contributions is attractive London first estimated the magnitude of the coefficient C6 by means of the oscillator model and he obtained for argon atoms as -5 × 10 -78 J m6 which is only 30 percent smaller than the value obtained by the most refined calculations Long-range energy: Summary Order of magnitude For the most general case of the interaction of two polar molecules the total long-range intermolecular energy is U=Uel+Uind+Udisp For two identical, neutral molecules, free to rotate with a dipole moment μ and static polarizability α, the leading contributions are −1 U = (4πε 0 ) 2 { 2 µ 4 2 3 + 2µ α + 3 kT 2 α ω 0 4 } r −6 Table 1.1 contains a list of the magnitudes of the three coefficients of the r-6 term for the different contributions to the interaction of like pairs of simple molecules for a temperature of 300K In order to compare the magnitude of the various attractive contributions to the intermolecular energy, calculation were made for 1 mole of a gas on the assumption that the molecules interact in pairs at a separation, σ, where the total potential energy U(σ) = 0 and the values are tabulated in table 1.2 Limitations The description of the electrostatic energy has been confined to axially-symmetric charge distributions The discussion of induction energy has been restricted to molecules with an isotropic polarizability The treatment of the dispersion energy limited to that for the interaction of spherically symmetric atoms in their ground states Finally, the retardation effect, which means by the time field acts at the second molecule the dipole in the first molecule will have changed Short-range energy Short-range repulsive forces between molecules is much more complicated the the long-range forces Here we discuss briefly the HeitlerLondon or valence-bond method, a method for the evaluation of shortrange forces The interaction of two hydrogen atoms Fig show the two hydrogen atoms at a separation r and defines the coordinates of the system The Hamiltonian, H for the system can be written H = Ha+Hb+Ve where Ve is the electrostatic energy which arises from the interaction of the two atoms and is given by Ve = − e 2 4πε 0 1 r a2 + 1 rb 1 − 1 r 12 − 1 r The integral S, the overlap integral, measures the degree of overlap of the wave functions of the two atoms J describes the columbic interaction between the electron 1 in the orbital A(1) with nucleus b J’ describes the columbic interaction between the two electrons K and K’ are exchange integrals Contours of equal electron density, in a plane containing the two nuclei for the interaction of two hydrogen atoms In fig a the two positively charged protons are attracted towards negatively charged region and hence there is a net attractive force on them according to classical electrostatics. This wave function ψ+ therefore corresponds to that of a bonding orbital for a hydrogen molecule. In fig b there is decreased electron density between the two nuclei and thus two nuclei are incompletely shielded from each other and an electrostatic repulsion results. The wave function ψ- is therefore anti-bonding. Potential energy for the interaction of two hydrogen atoms U(r) e U+ = Uo r U+ ' ' + 4πε 0 r e U− = J + K − 2 ( J + SK ) 2 1+ S 2 J − K − 2 ( J − SK ) 2 ' ' − 4πε 0 r 1+ S 2 Evaluating the energies U+ and U- and plotting for U(r) confirms the qualitative discussion that done before The energy U+ possesses an attractive minimum region and corresponds to a stable hydrogen molecule The energy U- is repulsive and because of the spin symmetry of the wave function is similar to the interaction energy for closed shell atoms. At short range the energy U- varies as 1/r owing to the internuclear repulsion At larger separations the energy decays as e-2r/a, where a is the radius of the Bohr orbit of the hydrogen atom. This final exponential form has been used as an analytic representation of the behavior of short –range forces Representation of the intermolecular pair potential energy function • The earliest and simplest representation was molecule viewed as a hard sphere of diameter σ so that the intermolecular potential energy is written U (r ) = ∞ r ≤ σ U (r ) = 0 r > σ • This form of potential has the single disposable parameter σ. In view of our description of the nature of intermolecular forces this model is evidently unrealistic • The most frequently used model potential is that due to Lennard-Jones(LJ) 6 U (r ) = ε n − 6 () rm r n − n n−6 () rm r 6 • This function possesses the general features of the true intermolecular potential energy in tat it has a repulsive short-range region joined to a long-range attractive region by a single minimum which occurs at rm where the energy is –ε • The attractive component of the function is theoretically based on the dispersion energy contribution. But the form of the repulsive term has no theoretical justification. In the above form the LJ potential has one disposable parameter n in addition to ε and rm • Most often the repulsive exponent has been given the value n=12 and the potential is then written U ( r ) =∈ ( ) ( ) 12 rm −2 r rm r 6 • Or, equivalently, U (r ) = 4 ∈ () () σ r 12 − σ r 6 • Where σ (=2-1/2rm) is the intermolecular separation for which the energy is zero • L-J(12-6) potential has no adjustable parameters other than σ and ε, whose values can be determined by forcing agreement between experimental data for a physical property and calculated values for the potential model LJ potential σ 12 σ 6 U (r ) = 4 ∈ − r r Interatomic pair potential for argon molecules Interatomic pair potential for Cesium molecules Other pair potential functions are Buckingham potential (buck) (6-exp) Born-Huggins-Meyer potential (bhm) Hydrogen-bond (12 - 10) potential (hbnd) Other topics required for MD • Intermolecular interactions • Classical Mechanics • Statistical Mechanics Thank you