2 The Theory of Intermolecular Forces
2.9 Intermolecular forces from systematic potentials
As we already mentioned in the previous section, there is a need for more accurate model potentials. One method is to take the systematic approach, where each contribution is represented and quantified separately. Then these are combined to give an overall total potential. This approach has been able to provide reliable potentials for small polyatomics [66,83]. There are many contributions that require to be quantified, and of course the relative importance of each contribution varies with the molecules concerned and their relative separations and orientations. The effort that required for quantifying each contribution is different for each of them. Most of the effort has been put into the development for the electrostatic contribution, which dominates the orientational dependence of the intermolecular potential of most molecules. Since each contribution is different in nature, it is important to consider each possible contribution carefully.
2.9.1 Systematic potentials for water dimer - an illustration of a systematic
potential for small polyatomics
In order to understand the in-depth theory behind a systematic potential, the most direct method is to study a small system whereby the systematic potentials are applied. This should provide us information on the relative importance of each term; hence it can aid us in our development of a New Models for Intermolecular Repulsion and their Application to van der W aals Helen H.Y. Tsui
2 The Theory of Intermolecular F o r c e s _______ ^ model potential for our study of the phenol-water interaction. The water dimer potentials of Millot et al. [48,84] have been chosen for our study because their systematic potential accounted for all the significant contributions to an intermolecular potential. Their water dimer potential was first published in 1992 [48], and it was later improved with modifications made in 1998 [84]. Modifications were made on the ASP-W potential by Millot et al. [84]. The main changes were the updating of dispersion coefficients, an addition of a charge-transfer term and also modification on the distributed multipoles. The two modified potentials are referred to as ASP-W2 and ASP-W4. The only difference between the two potentials was the distributed multipole analysis (DMA). The DMA analysis at the MP2 level used for ASP-W was replaced by a more accurate multireference Cl calculation carried out with the program MOLPRO [85].
The molecules are assumed to be rigid, with an OH bond length of 0.9572 Â and an HOH angle of 104.52 °. In ASP-W4, the atomic distributed multipole expansion is truncated at the hexadecapole on each site, and the energy terms having up to distance dependence. For the potential ASP-W2, the DMA is truncated at the quadrupole on each atom and the hydrogen quadrupoles are shifted to the oxygen atom position in the ASP-W potential [48]. The dispersion coefficients were replaced by more recent and accurate values calculated by Wormer and Hettema [86].
= S S K . O . . Ü ) ) / , (R ) (2.29)
n=6 ^
where {(0^,Q}g,(o) are the normalised real components of orientational S functions, R is the distance between the centres of mass. The Tang-Toennies damping function is used and the damping parameter a is 1.92 a.u.
/ , ( R ) = l - e x p ( - f l R ) X ^ 2 ^ (2.30)
*=0 ^ !
There is also damping model for the induction terms, because the induction may involve more than one site on each molecule, so the distance R is different for the two interactions involved. This is unlike the usual Tang-Toennies damping [57] for the dispersion, which involves the product of two interaction functions from the same distance R .
(/w Ae;7;,“‘/ , ( R . ^ ) ^ a ' (2.31)
^ A B * A
AGi = (R« ( e ! + G“ ) (2.32)
where Q° are permanent multipoles of molecule A , A g/ are induced multipoles of molecule A ,
a°,° is the polarizability tensor and /?^ is the distance between the sites concerned.
The one-site polarizability model is the same as ASP-W, where the one-site polarizabilities up to quadrupole-quadrupole located on the oxygen atom were calculated using the program CADPAC [56]. For the exchange-repulsion-penetration energy, a set of optimised parameters used involving 7 anisotropic S functions for the oxygen atom, and 9 anisotropic S functions for hydrogen atoms.
^ e r p = E [ ^ a b ~ P a b ( ^ ) ) ] (2.33)
New Models for Intermolecular Repulsion and their Application to van d er W aals Helen H.Y. Tsui
2 The Theory of Intermolecular Forces________________________________________________ ^ w here is the distance betw een site a o f m olecule A and site b o f m olecule B , is a hardness param eter depending on the site pair, and (^2) is an orientation-dependent param eter describing the effective size o f the atoms.
P . . (û) = S , S K ,û>..<«)p;È (2.34)
T he charge-transfer term has been added, and its energy was evaluated from IM PT calculations [4,55,87] on the dim er and fitted to an atom -atom functional form . T his form is sim ilar to that o f the repulsion energy, but attractive and involving only the four oxygen-hydrogen pairs for the dimer:
E „ = - K Y . e x p [ - < ; ( R - - P : (£ 2 ))] (2.35) o n pairs
where A' is a constant having the value o f 1 m illihartree.
T he m inim isation o f these potentials was successfully reproduced using O R IE N T [88] and shown in T able 2.1. T he same rigid water m olecule from Stone and co-w orkers [84] is used in the dim er m inim isation calculations. Both o f these m odels for the w ater dim er produced the sam e linear geom etry as show n in F igure 2.3. The m ain difference is that the 0 - H ...O angle o f A SP-W 4 is slightly larger. This show s the sensitivity o f the m inim um and angle to relative sm all changes in DMA.
ASP-W2 ASP-W4
Figure 2.3 Geometry o f the water dimer minimum, using ASP-W 2 and ASP-W 4 model potential.
Full M odel ASP-W 2 ASP-W 4
d ( 0 . . . 0 ) / Â 2.96 2.97 d ( O ...H ) / Â 2.01 2.02 Angle 0 - H . . . 0 / ° 170.87 176.95 Energy / kJ/mol Electrostatic -25.74 -25.19 Induction -4.07 -3.44 Dispersion -8.40 -8.49 Exchange-Repulsion 19.45 18.50 Charge-Transfer -2.32 -2.24 Total -21.08 -20.87
Table 2 . 1 Energy contributions for A SP-W 2 ad A SP-W 4 water dimer potentials.
2.9.2 Sensitivity of potential to each contribution
The sensitivity o f the potential to each contribution has been tested, by studying the energy and structural changes at the m inim um energy structure o f the full A SP-W 4 potential. Each term has been om itted or altered to create a variety o f potentials, and their energies and structures have been re calculated using O R IE N T [88]. The results are presented in T able 2.2 and T able 2.3. It can be observed clearly that atom ic anisotropy plays an im portant role in interm olecular potential. The energy and structural changes are significant, influenced by the om ission o f anisotropy term s, in
N e w M o (jels for Interm olecular Repulsion and their Application to van d er W a a ls H e len H .Y . Tsui
2 The Theory of Intermolecular Forces_________________________________________________ # particular the repulsion and electrostatic energy are affected significantly when the anisotropic terms are removed. This indicates in order to construct an accurate intermolecular potential, one must incorporate the anisotropy terms. The omission of charge-transfer and induction terms shows only slight deviation in energy and in structure.
Full Model Energy / kJ/mol Modified Models Energy / kJ/mol Electrostatic -25.19 Only charges on atoms -3.86
Induction -3.44 Only charges on atoms -0.53
Dispersion -8.49 Only Cg isotropic -3.56
Exchange-Repulsion 18.50 Only isotropic terms 22.16 Charge-Transfer -2.24 Only isotropic terms -3.17
Total -20.87 Without long range damping -25.66
Table 2.2 Energy contributions at minimum energy structure, upon variations o f potential.
Energies on the left are energies for the contributions using the full model, with the corresponding energies upon modifications on the full model on the right. Where appropriate, higher multipoles were omitted to give charges only electrostatic model. Similarly for dispersion, exchange-repulsion and charge-transfer, anisotropic terms were omitted.
d ( 0 .. .0 ) / A d (O ...H )/À Angle 0 - H . . . 0 / °
Full potential 2.97 2.02 176.95
A if charge only electrostatic model +0.33 +0.80 -64.78 A if charge-transfer omitted +0.07 +0.07 -0.02
A if induction omitted +0.07 +0.09 -8.20
A if only isotropic repulsion +0.09 +0.09 -2.84 A if only isotropic dispersion +0.12 +0.11 + 1.51
Table 2.3 Sensitive of minimum energy structure of (HzO); to various approximations in the model potentials. A = the difference of (structure for modified potential - structure for full potential). Each modified model is the full potential with the parameters and model of one contribution modified.
The Millot et al. water potential illustrates the state-of-the-art on the application theory of intermolecular forces, and shows the accuracy of the theory is important in order to derive a realistic model potential. Millot et al. [84] have tested their model potentials by determining the second virial coefficient B ( T ) , and both the ASP-W2 and ASP-W4 potentials give values that are close to experiment in the range 373-973 K when first order quantum corrections are included. Systematic models of the water dimer have shown their capability in reproducing experimental data to a reasonable level. This has been demonstrated by the ASP-W4 potential, that gave good results for the tunnelling splittings in the water dimer spectrum [89,90]. The ASP-W potential has also been modified by Fellers et al. [91], by fitting it to microwave, terahertz, and mid-infrared (D20)2 spectra through a rigorous calculation of the water dimer eigenstates. This modified potential (VRT-ASP-W) reproduced most ground state vibration-rotation-tunnelling spectra and yielded excellent second virial coefficients. The calculated dimer structure and dipole moment are in good agreement to experiments. This shows that the importance of an accurate systematic potential can aid interpretation of spectroscopic data, and may shed light on explaining unidentifiable experimental features. This type of potential is far too complicated for other simulations, since the number of terms will require too much effort computational effort to achieve good results, in particular for organic systems.
The water dimer studies have shown that there are some terms that are just too complex to model for large organic systems. We will only consider the most important contributions, electrostatic, dispersion and exchange-repulsion, and also include penetration and charge-transfer in N ew M odels for Intermolecular Repulsion and their Application to van der W aals Helen H.Y. Tsui
2 The Theory of Intermolecular Forces_________________________________________________4