Journal of ISSN: 2377-4282JNMR

Nanomedicine Research
Research Article
Volume 3 Issue 2 - 2016
Derivation of the Spherical Atom Model for Dispersion Forces
Amy J Austin*
Freelance Tutor for Organic Chemistry, WyzAnt Tutoring, USA
Received: January 28, 2016| Published: March 10, 2016
*Corresponding author: Amy J Austin, Freelance Tutor for Organic Chemistry, WyzAnt Tutoring, USA, Email:
Citation: Austin AJ (2016) Derivation of the Spherical Atom Model for Dispersion Forces. J Nanomed Res 3(2): 00054. DOI: 10.15406/jnmr.2016.03.00054

Abstract

A method for incorporating dispersion forces into density functional theory through the addition of a scaled atom-atom pair wise attractive potential term to the repulsive density functional potential is presented. By considering the deviation from the C 6 R 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba Gaam4qamaaBaaajuaibaGaaGOnaaqcfayabaaabaGaamOuamaaCaaa beqcfasaaiaaiAdaaaaaaaaa@3AD5@ potential arising from the dispersion interaction of two, non-point, spherically distributed dipoles, an analytic function was derived that when multiplied by C 6 R 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba Gaam4qamaaBaaajuaibaGaaGOnaaqcfayabaaabaGaamOuamaaCaaa beqcfasaaiaaiAdaaaaaaaaa@3AD5@ resulted in an attractive potential capable of inducing binding. The scaling function found to give bound Ne and He dimers using Becke’s three parameter hybrid density functional with the Lee-Yang-Parr correlation functional (B3LYP) density functional potential is quite different from the damping functions used in previous studies, causing the C 6 R 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba Gaam4qamaaBaaajuaibaGaaGOnaaqcfayabaaabaGaamOuamaaCaaa beqcfasaaiaaiAdaaaaaaaaa@3AD5@ attraction to be strengthened in the bonding region. Using the scaled attractive potential presented in this work, a dissociation energy 0.4% in error (+0.000015 kcal/mol error) was found for Ne dimer, compared to 60% error (-0.0023 kcal/mol error) when the previous damped attractive potential is used. For He dimer, use of the new scaled attractive potential yields a dissociation energy in 8% error (+0.000074 kcal/mol error); however, the previous damped attractive potential cannot induce binding at all. The interaction potential curves are quite close to the experimental potential curves for both Ne2 and He2. This model has been incorporated into the APF-D (Austin-Petersson-Frisch) Density Functional theory for dispersion forces within the Gaussian 09 Suite under keyword, APFD.

Introduction

Density functional methods have become a popular means of assessing the electronic structure and quantum mechanical properties of molecular systems, owing to their accuracies comparable to Moller-Plesset 2nd Order Perturbation Theory while being more cost-effective [1-10].

The computationally less demanding density functional methods would seem ideal for investigations of large systems. However, dispersion forces, which arise from the interaction of instantaneously fluxuating dipoles [11], are often important for large systems, such as in the stabilization of protein structures and stacked DNA bases [12-16]. Yet, the current density functional methods in use cannot account for these long-range effects since the exchange-correlation energy depends only on the local density and at most its gradient [1]. For example, the use of gradient-corrected exchange functionals has been shown to display purely repulsive potential energy curves for the interaction of rare gases [17]. Although some combinations of exchange and correlation functionals do produce overly bound rare-gas dimers, [18] such as those using exchange functionals rich in local density approximation, the interaction potential does not exhibit the familiar R-6 internuclear dependence given by perturbation theory [19-23].

Background

Following the work of Alrichs et al. [24] on the correction of the Hartree-Fock model, as well as the work of Moorij et al. [25], Wu & Yang [26] introduced an empirical interaction term to the first-principles density functional theory in order to simulate the dispersion interaction energy.

This empirical term, given by Eq. (1), is completely independent of the density functional and depends only on the positions of the nuclei.

U attract ( R )= f d( R ) C 6 R 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadwfada WgaaqcfasaaiaadggacaWG0bGaamiDaiaadkhacaWGHbGaam4yaiaa dshaaKqbagqaamaabmaabaGaamOuaaGaayjkaiaawMcaaiabg2da9m aalaaabaGaeyOeI0IaamOzamaaBaaabaqcfaIaamizaKqbaoaabmaa baGaamOuaaGaayjkaiaawMcaaiaadoeadaWgaaqcfasaaiaaiAdaaK qbagqaaaqabaaabaGaamOuamaaCaaabeqcfasaaiaaiAdaaaaaaaaa @4C86@ (1)

For each atom pair separated a distance of R, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkfaca GGSaaaaa@3800@ U attract MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadwfada WgaaqcfasaaiaadggacaWG0bGaamiDaiaadkhacaWGHbGaam4yaiaa dshaaKqbagqaaaaa@3EC6@ is added to the repulsive density functional potential [26]. The coefficient C 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadoeada WgaaqcfasaaiaaiAdaaKqbagqaaaaa@38DE@ controls the asymptotic behavior of the attractive term for large internuclear separation. f d ( R ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAgada WgaaqcfasaaiaadsgaaKqbagqaamaabmaabaGaamOuaaGaayjkaiaa wMcaaaaa@3B8A@ is a damping function that weakens U attract MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadwfada WgaaqcfasaaiaadggacaWG0bGaamiDaiaadkhacaWGHbGaam4yaiaa dshaaKqbagqaaaaa@3EC6@ for small internuclear separations [24,25]. Thus, the damping function is zero for small R, in the region where R<0.2RM, and becomes one for large R, in the region where R>1RM, where RM is the summation of van der Waals radii. As shown in Figure 1, the effects of two different damping functions were examined, damping function I approaching unity much more gradually than damping function II [26].

Figure 1a: Damping Functions I and II as a Function of Internuclear Separation.

Figure 1b: Lennard Jones attractive Energy term as a function of Internuclear Separation for Ne-Ne dimer.

Figure 1: (a). fd(R/RM) I and II as a function of R/ RM, where RM is the summation of van der Waals radii; fdI approaches unity much more rapidly than fdII.
(b) Uattract=- fd (R/ RM)C6/R6 for fdI and II.

f d I( R R M )= ( 1 e C damp ( R R M ) 3 ) 2 f d II( R R M )= 1 1+ e β( R R M 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaOaam OzamaaBaaajuaibaGaamizaaqcfayabaGaamysamaabmaabaWaaSaa aeaacaWGsbaabaGaamOuamaaBaaajuaibaGaamytaaqcfayabaaaaa GaayjkaiaawMcaaiabg2da9maabmaabaGaaGymaiabgkHiTiaadwga daahaaqabKqbGeaacqGHsislcaWGdbqcfa4aaSbaaKqbGeaajqwba9 VaamizaiaadggacaWGTbGaamiCaKqbaoaabmaajuaibaqcfa4aaSaa aKqbGeaacaWGsbaabaGaamOuaKqbaoaaBaaajqwba9Faaiaad2eaaK qbGeqaaaaaaiaawIcacaGLPaaajuaGdaahaaqcfasabKazfa0=baGa aG4maaaaaKqbGeqaaaaaaKqbakaawIcacaGLPaaadaahaaqabKqbGe aacaaIYaaaaaqcfayaaiaadAgadaWgaaqcfasaaiaadsgaaKqbagqa aiaadMeacaWGjbWaaeWaaeaadaWcaaqaaiaadkfaaeaacaWGsbWaaS baaKqbGeaacaWGnbaajuaGbeaaaaaacaGLOaGaayzkaaGaeyypa0Za aSaaaeaacaaIXaaabaGaaGymaiabgUcaRiaadwgadaahaaqabKqbGe aacqGHsislcqaHYoGyjuaGdaqadaqcfasaaKqbaoaalaaajuaibaGa amOuaaqaaiaadkfajuaGdaWgaaqcKvay=haacaWGnbaajuaibeaaaa GaeyOeI0IaaGymaaGaayjkaiaawMcaaaaaaaaaaaa@7480@

For six rare gas systems (He2, Ne2, Ar2, HeNe, HeAr, and NeAr), the repulsive density functional potential, given in Eq. (2) [21] was added to the attractive term described above in order to obtain the interaction potential, U interact MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadwfada WgaaqcfasaaiGacMgacaGGUbGaaiiDaiaacwgacaGGYbGaaiyyaiaa cogacaGG0baajuaGbeaaaaa@3FAC@ , given in Eq. (3) [26]

U DFT ( R ) E DFT ( R ) E DFT ( 20A ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadwfada WgaaqcfasaaiaadseacaWGgbGaamivaaqcfayabaWaaeWaaeaacaWG sbaacaGLOaGaayzkaaGaeyyyIORaamyramaaBaaabaWaaSbaaKqbGe aacaWGebGaamOraiaadsfaaKqbagqaaaqabaWaaeWaaeaacaWGsbaa caGLOaGaayzkaaGaeyOeI0IaamyramaaBaaajuaibaGaamiraiaadA eacaWGubaajuaGbeaadaqadaqaaiaaikdacaaIWaGaamyqaaGaayjk aiaawMcaaiaacYcaaaa@4ED1@ (2)

U interact ( R )= U DFT ( R )+ U attract ( R ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadwfada WgaaqcfasaaiGacMgacaGGUbGaaiiDaiaacwgacaGGYbGaaiyyaiaa cogacaGG0baajuaGbeaadaqadaqaaiaadkfaaiaawIcacaGLPaaacq GH9aqpcaWGvbWaaSbaaKqbGeaacaWGebGaamOraiaadsfaaKqbagqa amaabmaabaGaamOuaaGaayjkaiaawMcaaiabgUcaRiaadwfadaWgaa qcfasaaiaadggacaWG0bGaamiDaiaadkhacaWGHbGaam4yaiaadsha aKqbagqaamaabmaabaGaamOuaaGaayjkaiaawMcaaiaac6caaaa@55D7@ (3)

Binding was obtained in only three systems (Ne2, Ar2, and NeAr) using the B3LYP hybrid density functional while no binding was produced using the BLYP or BPW91 functionals for any of the six systems [26]. Figures 2 & 3 illustrate the effect of adding the empirical term to the repulsive density functional potentials with damping function I for Ne2 and He2, respectively [26].

For Ne dimer, Figure 2 shows that only the B3LYP functional potential has sufficiently little repulsiveness that binding can occur when the damped attractive term is incorporated [26]. However, the addition of the damped attractive term to the more repulsive BLYP and BPW91 functional potentials cannot yield binding. Indeed, as Figure 4 illustrates, even when the full, undamped attraction, given in Eq. (4), is added to the density functional potentials for Ne2, the more repulsive BLYP and BPW91 functionals cannot produce binding.

Figure 2a: UDFT as a function of Internuclear Separation for Ne Dimer.

Figure 2b: Uinteract=UDFT-fdIC6/R6 for Ne dimer.

Figure 2c: Uinteract=UDFT-fdIC6/R6 (Bonding Region) for Ne dimer.

Figure 2: (a) UDFT as a function of internuclear separation for Ne dimer. All density functional potentials are purely repulsive; (b) Uinteract= UDFT+Uattract as a function of internuclear separation for Ne dimer. Only the B3LYP potential yields binding. De=0.0015eV (experimental*=0.0038eV) Uattract=-fdI(R)C6/R6
(c) Enlargement of bonding region in (b).

U attract ( R )= C 6 R 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadwfada WgaaqaamaaBaaajuaibaGaamyyaiaadshacaWG0bGaamOCaiaadgga caWGJbGaamiDaaqcfayabaaabeaadaqadaqaaiaadkfaaiaawIcaca GLPaaacqGH9aqpcqGHsisldaWcaaqaaiaadoeadaWgaaqcfasaaiaa iAdaaKqbagqaaaqaaiaadkfadaahaaqabKqbGeaacaaI2aaaaaaaaa a@4796@ (4)

For the least repulsive B3LYP potential, addition of the undamped R 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkfada ahaaqabKqbGeaacqGHsislcaaI2aaaaaaa@394D@ attraction actually gives a slight improvement in the dissociation energy compared to the highly accurate values of Oglivie & Wang [27] (1.5 versus 2.0 compared to 3.5 meV). In the case of He dimer, Figure 3 shows that the B3LYP, BLYP, and BPW91 functionals exhibit potential curves that are so repulsive, the addition of the damped attractive term cannot induce binding. Again, as shown in Figure 5, even when the undamped R 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkfada ahaaqabKqbGeaacqGHsislcaaI2aaaaaaa@394D@ attraction is added to these density functional potentials, binding cannot be induced.

Since the addition of the full, undamped R 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkfada ahaaqabKqbGeaacqGHsislcaaI2aaaaaaa@394D@ attraction cannot induce binding for those density functional potentials which are more repulsive than U B3LYP MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadwfada WgaaqcfasaaiaadkeacaaIZaGaamitaiaadMfacaWGqbaajuaGbeaa aaa@3C38@ of Ne dimer [26] and the addition of the undamped R 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkfada ahaaqabKqbGeaacqGHsislcaaI2aaaaaaa@394D@ attraction improves the dissociation energy of the bound cases compared to that produced by the damped attractive term, then damping the attractive energy cannot provide a general solution to the problem of simulating dispersive interaction potentials. Two apparent solutions thus present themselves: we may either (a) strengthen the attractive term, or (b) damp the repulsive term.

Figure 3a: UDFT as a function of internuclear separation for He dimer.

Figure 3b: Uinteract=UDFT-fdIC6/R6 for He dimer.

Figure 3c: Uinteract=UDFT-fdIC6/R6 for He dimer.

Figure 3: (a) UDFT as a function of internuclear separation for He dimer. All density functional potentials are purely repulsive; (b) Uinteract=UDFT+Uattract as a function of internuclear separation for He dimer. experimental* curve also plotted. None of the density functional potentials yield
binding, Uattract=-fdI(R)C6/R6 (c) Enlargement of bonding region in (b).

Figure 4a: Uinteract=UDFT-C6/R6 for Ne Dimer.

Figure 4b: Uinteract=UDFT-C6/R6 for Ne Dimer (Bonding Region).
Figure 4: Uinteract=UDFT+Uattract as a function of internuclear separation for Ne dimer. Even using
the full, undamped attractive potential, only the B3LYP potential yields binding.; De=0.0020eV
(experimental*=0.0038eV) Uattract=-C6/R6, (c) Enlargement of bonding region in (b).

Figure 5a: Uinteract=UDFT-C6/R6 for He dimer.

Figure 5b: Uinteract=UDFT-C6/R6 for He Dimer           
Figure 5: Uinteract=UDFT+Uattract as a function of internuclear separation for He dimer. Even using the full, undamped attractive potential, none of the density functional potentials yield binding. Experimental* potential also plotted. Uattract=-C6/R6, (c) Enlargement of bonding region in (b).

It is not desirable to damp the repulsive density functional potentials, since the energy lowering would cause a reduction in the force constant and hence frequencies of bound vibrational modes. Indeed, if this energy were damped, the dispersion forces responsible for weak interactions could be introduced only at the expense of the covalent chemistry, the latter of which current density functional methods describe quite well. Therefore, the goal of this work will be to simulate the dispersive interaction potential, typified by the rare gases, through the addition of the density functional potential to a scaled attractive potential. Since the density functional methods studied in the present work are shown to be too repulsive to induce binding, as discussed above, the effect of this scaling will be to strengthen the attractive term.

Proposed Methods

If the objective is to scale the attractive potential, U attract MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadwfada WgaaqcfasaaiaadggacaWG0bGaamiDaiaadkhacaWGHbGaam4yaiaa dshaaKqbagqaaaaa@3EC6@ in order to induce binding, one plausible approach is to simulate the behavior of a scaling function, f s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAgada WgaaqcfasaaiaadohaaKqbagqaaaaa@3939@ that when multiplied by C 6 R 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba Gaam4qamaaBaaajuaibaGaaGOnaaqcfayabaaabaGaamOuamaaCaaa beqcfasaaiaaiAdaaaaaaaaa@3AD5@ will result in an attractive potential capable of inducing binding, as given by Eq. (5), below.

U interact = U DFT f s C 6 R 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadwfada WgaaqcfasaaiGacMgacaGGUbGaaiiDaiaacwgacaGGYbGaaiyyaiaa cogacaGG0baajuaGbeaacqGH9aqpcaWGvbWaaSbaaKqbGeaacaWGeb GaamOraiaadsfaaKqbagqaaiabgkHiTiaadAgadaWgaaqcfasaaiaa dohaaKqbagqaamaalaaabaGaam4qamaaBaaajuaibaGaaGOnaaqcfa yabaaabaGaamOuamaaCaaabeqcfasaaiaaiAdaaaaaaaaa@4CDF@ (5)

In general, the errors associated with the use of various density functional potentials could be quite different, resulting in perhaps drastically dissimilar scaling functions. It has been shown that interaction potentials obtained using the B3LYP density functional method are closer to the experimental potentials than those obtained with either the BLYP or BPW91 methods [26]. Therefore, the B3LYP density functional potential is an obvious choice for comparison with the known, experimental [27] interaction potential in order to obtain the value of the scaling function, C 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadoeada WgaaqcfasaaiaaiAdaaKqbagqaaaaa@38DE@ at any particular internuclear separation, as shown in Eq. (6)

f s pp ( R )= U B3LYP ( R ) U exact ( R ) C 6 R 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOza8aadaWgaaqcfasaa8qacaWGZbaajuaGpaqabaWaaWba aeqajuaibaWdbiaadchacqGHsislcaWGWbaaaKqba+aadaqadaqaai aadkfaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiaadwfadaWgaaqc fasaaiaadkeacaaIZaGaamitaiaadMfacaWGqbaajuaGbeaadaqada qaaiaadkfaaiaawIcacaGLPaaacqGHsislcaWGvbWaaSbaaKqbGeaa caWGLbGaamiEaiaadggacaWGJbGaamiDaaqcfayabaWaaeWaaeaaca WGsbaacaGLOaGaayzkaaaabaWaaSaaaeaacaWGdbWaaSbaaKqbGeaa caaI2aaajuaGbeaaaeaacaWGsbWaaWbaaeqajuaibaGaaGOnaaaaaa aaaaaa@56FE@ (6)

Once the point-by-point values of the scaling function, f f s pp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOza8aadaWgaaqcfasaa8qacaWGZbaajuaGpaqabaWaaWba aeqajuaibaWdbiaadchacqGHsislcaWGWbaaaaaa@3CBE@ have been determined, it may be possible to derive an analytic function that will simulate its shape by considering the deviation from the C 6 R 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba Gaam4qamaaBaaajuaibaGaaGOnaaqcfayabaaabaGaamOuamaaCaaa beqcfasaaiaaiAdaaaaaaaaa@3AD5@ potential arising from the dispersion interaction of two non-point, spherically distributed dipoles. This analytic function would depend on the position of the nuclei and a fitting parameter to the experimental potential. If the shape of the scaling function is relatively constant among atom pairs of varying size, then the simulated scaling function could potentially provide a viable means of obtaining the interaction potentials for other systems (ie: rare gas dimers) using the B3LYP hybrid density functional method. Since the scaling function is fit to the experimental potential through parameterization, its predictive utility is compromised; however, considering the interaction between two spherically distributed dipoles may prove insightful for developing a more general, predictive theory.

Computational Details

Two examples were selected from the literature in order to determine the appropriate behavior for the scaling function, fs, using the B3LYP density functional potential. He and Ne dimers were obvious choices. For Ne dimer, addition of the damped attractive potential to the density functional potential binds only with the least repulsive B3LYP functional potential of Ne dimer [26]. For He dimer, this approach fails to give binding for any of the density functional potentials [26].

For both He2 and Ne2, single point energy calculations were performed using Becke’s three parameter hybrid density functional [28,29] with the Lee-Yang-Parr correlation functional [30] (B3LYP) density functional method between 2.10 to 4.80 angstroms at 0.02 angstrom intervals. The molecular energy at 20 angstroms was subtracted from each of these total molecular energies in order to obtain the B3LYP density functional potential, given in equation (2). This method is similar to that of Perez-Jorda et al. [18] as well as that of Wu & Yang [26] recently. The coefficients of Ne and He used in this work have been taken from the work of Kumar and Meath [31-37]. Although Wu and Yang use the cc-pV5Z basis set (augmented with diffuse functions for Ne2) in their single point calculations, use of the cc-pVQZ basis set yielded density functional potentials reported to be in good agreement with those found with the pentuple zeta basis set [26]. As a timesaving measure, therefore, the quadruple zeta basis set (again, augmented with diffuse functions for Ne2. Diffuse functions for the correlation consistent basis sets are not available at this time for He) was used in all single point energy calculations. Tight convergence criteria were used in the SCF calculations of Ne, since diffuse functions were used. A pruned (99590) grid was used for integration [26]. The Gaussian-03 program was used to perform all density functional calculations [38].

Results and Discussion

The point-by-point scaling functions, f s pp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOza8aadaWgaaqcfasaa8qacaWGZbaajuaGpaqabaWaaWba aeqajuaibaWdbiaadchacqGHsislcaWGWbaaaaaa@3CBE@ necessary to induce binding in Ne and He, given by equations (5) and (6) above, are presented in Figures 6a & 6b for Ne and He dimers, respectively.

For each of the scaling functions, the C 6 R 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba Gaam4qamaaBaaajuaibaGaaGOnaaqcfayabaaabaGaamOuamaaCaaa beqcfasaaiaaiAdaaaaaaaaa@3AD5@ attraction is attenuated for R σ <1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaamOuaaqaaiabeo8aZbaacqGH8aapcaaIXaaaaa@3AE2@ , where σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo8aZb aa@383C@ is the atomic “hard-sphere” diameter. As the internuclear separation approaches the hard-sphere diameter, the scaling functions gradually rise to unity such that the C 6 R 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba Gaam4qamaaBaaajuaibaGaaGOnaaqcfayabaaabaGaamOuamaaCaaa beqcfasaaiaaiAdaaaaaaaaa@3AD5@ term is no longer being attenuated when R=σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkfacq GH9aqpcqaHdpWCaaa@3A19@ . Thus, in this region, the scaling functions behave quite similarly to the damping functions used in previous studies [24-26]. However, in the region where R>σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkfacq GH+aGpcqaHdpWCaaa@3A1B@ , the scaling functions each become larger than one, such that the C 6 R 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba Gaam4qamaaBaaajuaibaGaaGOnaaqcfayabaaabaGaamOuamaaCaaa beqcfasaaiaaiAdaaaaaaaaa@3AD5@ attractive term is being strengthened; yet, purely damping functions cannot account for these effects. Finally, in the limit of infinite internuclear separation, the scaling functions for both Ne and He dimers smoothly decline to unity. Although Ne and He differ by eight electrons, the shape of the scaling functions necessary to induce binding remains roughly constant in these systems.

The oscillatory behavior of the scaling functions at large internuclear separations may be mere artifacts of the size of the grid used for integration. For example, when a smaller pruned (75302) grid was used for numerical integration, more oscillatory behaviour was observed. It seems reasonable that such unphysical behaviour could be eliminated with the use of larger grid sizes. Therefore, these oscillations were ignored when deriving an analytical function that simulates the curvature of the scaling function.

Figure 6a: Point-by-point Scaling Function (fs, p-p) as a function of internuclear distance for Ne dimer

Figure 6b: Point-by-Point Scaling Fcn (fs, p-p) as a function of Internuclear Separation for He dimer
Figure 6: (a) The point-by-point scaling function, fsp-p as a function of R/σ for Ne dimer, (b) The point-by-point scaling function, fsp-p as a function of R/σ for He dimer.

Diagram 1: Dispersion interaction between a spherically distributed dipole and a point dipole.

Diagram 2: Dispersion interaction between two spherically distributed dipoles.

Unlike Gauss’s law, where Coulomb’s inverse square force is at work, the dispersion interaction between two atoms cannot be modeled as point dipoles centered at the nuclei. Indeed, the point-by-point scaling functions, f s pp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOza8aadaWgaaqcfasaa8qacaWGZbaajuaGpaqabaWaaWba aeqajuaibaWdbiaadchacqGHsislcaWGWbaaaaaa@3CBE@ , necessary to induce binding in Ne and He show that the C 6 R 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba Gaam4qamaaBaaajuaibaGaaGOnaaqcfayabaaabaGaamOuamaaCaaa beqcfasaaiaaiAdaaaaaaaaa@3AD5@ attraction between two point charges must be strengthened in order to induce binding, indicating that the interaction distance is shorter than the internuclear distance. This strengthening of the C 6 R 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba Gaam4qamaaBaaajuaibaGaaGOnaaqcfayabaaabaGaamOuamaaCaaa beqcfasaaiaaiAdaaaaaaaaa@3AD5@ attraction may be more important than the expansion in higher-order atomic polarizabilities used in traditional treatments [11]. Therefore, the potential energy function describing the dispersion interaction between two spherically distributed dipoles must be derived.

First consider the dispersion interaction between a spherically distributed dipole and a point dipole, as depicted in Diagram 1, below.

Here, the point dipole is located a distance τ away from the center of the sphere, and ρ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg8aYn aaBaaajuaibaGaaGymaaqcfayabaaaaa@39D1@ is the interaction distance between a point on the surface of the sphere, which is described in spherical-polar coordinates ( r 1 , θ 1 , ϕ 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaamOCamaaBaaajuaibaGaaGymaaqcfayabaGaaiilaiabeI7aXnaa BaaajuaibaGaaGymaaqcfayabaGaaiilaiabew9aMnaaBaaajuaiba GaaGymaaqcfayabaaacaGLOaGaayzkaaaaaa@429F@ and the point dipole. The potential energy between any point on the surface of the sphere and the point dipole is just the familiar 1 R 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaaGymaaqaaiaadkfadaahaaqabKqbGeaacaaI2aaaaaaaaaa@392B@ attraction between two point dipoles. The component of this interaction that contributes to the dispersion attraction between the sphere and the point dipole must be in the direction of the axis adjoining them. However, since the potential energy itself is a scalar quantity, we cannot merely sum up the potential energies from all individual interactions between points on the sphere and the point dipole to obtain the total potential. Rather, we must sum up the electric fields, vector quantities, from all the individual interactions in order that the perpendicular components will cancel. To this end, consider Diagram 1, above. The electric field from the interaction of any point on the sphere’s surface and the point dipole has a magnitude that is the derivative of the potential between two point dipoles, as given by Eq. 7,

E 1 = V ρ 1 = ( γ 6 ρ 6 ) ρ 1 = 6 γ 6 ρ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaFiaaba WaauWaaeaacaWGfbWaaSbaaKqbGeaacaaIXaaajuaGbeaaaiaawMa7 caGLkWoaaiaawEniaiabg2da9maalaaabaGaeyOaIyRaamOvaaqaai abgkGi2kabeg8aYnaaBaaajuaibaGaaGymaaqcfayabaaaaiabg2da 9maalaaabaGaeyOaIy7aaeWaaeaadaWcaaqaaiabeo7aNnaaBaaaju aibaGaaGOnaaqcfayabaaabaGaeqyWdi3aaWbaaeqajuaibaGaaGOn aaaaaaaajuaGcaGLOaGaayzkaaaabaGaeyOaIyRaeqyWdi3aaSbaaK qbGeaacaaIXaaajuaGbeaaaaGaeyypa0ZaaSaaaeaacqGHsislcaaI 2aGaeq4SdC2aaSbaaKqbGeaacaaI2aaajuaGbeaaaeaacqaHbpGCda WgaaqcfasaaiaaigdaaKqbagqaaaaaaaa@5E9F@ (7)

Where γ 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeo7aNn aaBaaajuaibaGaaGOnaaqcfayabaaaaa@39BD@ is a proportionality constant between the surface area of a sphere and the C 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadoeada WgaaqcfasaaiaaiAdaaKqbagqaaaaa@38DE@ coefficient. Note here that Eq. (7) only holds for a point dipole that is outside the spherically distributed dipole. Eq. 8 gives the interaction distance, ρ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg8aYn aaBaaajuaibaGaaGymaaqcfayabaaaaa@39D1@ between the point on the surface of the sphere and the point dipole

ρ 1 2 = r 1 2 Sin θ 1 2 + ( τ r 1 Cos θ 1 ) 2 = r 1 2 + τ 2 2τ r 1 Cos θ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg8aYn aaBaaajuaibaGaaGymaKqbaoaaCaaajuaibeqaaiaaikdaaaaajuaG beaacqGH9aqpcaWGYbWaa0baaKqbGeaacaaIXaaabaGaaGOmaaaaju aGcaGGtbGaaiyAaiaac6gacqaH4oqCdaqhaaqcfasaaiaaigdaaeaa caaIYaaaaKqbakabgUcaRmaabmaabaGaeqiXdqNaeyOeI0IaamOCam aaBaaajuaibaGaaGymaaqcfayabaGaam4qaiaad+gacaWGZbGaeqiU de3aaSbaaKqbGeaacaaIXaaajuaGbeaaaiaawIcacaGLPaaadaahaa qabKqbGeaacaaIYaaaaKqbakabg2da9iaadkhadaqhaaqcfasaaiaa igdaaeaacaaIYaaaaKqbakabgUcaRiabes8a0naaCaaabeqcfasaai aaikdaaaqcfaOaeyOeI0IaaGOmaiabes8a0jaadkhadaWgaaqcfasa aiaaigdaaKqbagqaaiaadoeacaWGVbGaam4CaiabeI7aXnaaBaaaju aibaGaaGymaaqcfayabaaaaa@6A58@ (8)

Note that the only point on the sphere’s surface that will produce an electric field totally in the direction of the axis adjoining the center of the sphere and the point dipole is the one directly in between them, at ( r 1 ,θ=0,ϕ=1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaaiikaiaadkhapaWaaSbaaKqbGeaapeGaaGymaaqcfa4daeqa a8qacaGGSaGaeqiUdeNaeyypa0JaaGimaiaacYcacqaHvpGzcqGH9a qpcaaIXaWdaiaacMcaaaa@432D@ . All other points on the sphere’s surface will produce an electric field that has a perpendicular component which does not contribute to the dispersion attraction between the sphere and the point dipole. However, rotating ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabew9aMb aa@3841@ from 0 to 2π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaikdacq aHapaCaaa@38F2@ for any fixed value of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@ sweeps out a circle (outlined in yellow in Diagram 1 above) that is perpendicular to the axis adjoining the centre of the sphere and the point dipole. Due to the cylindrical symmetry, for every point on the circle that produces an electric field given by Eq. (9),

E a =( 6 γ 6 Cos λ 1 ρ 1 7 ) e 1 +( 6 γ 6 Sin λ 1 ρ 1 7 ) e 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaFiaaba GaamyraaGaay51GaWaaSbaaKqbGeaacaWGHbaajuaGbeaacqGH9aqp daqadaqaamaalaaabaGaeyOeI0IaaGOnaiabeo7aNnaaBaaajuaiba GaaGOnaaqcfayabaGaam4qaiaad+gacaWGZbGaeq4UdW2aaSbaaKqb GeaacaaIXaaajuaGbeaaaeaacqaHbpGCdaWgaaqcfasaaKqbaoaaDa aajuaibaGaaGymaaqaaiaaiEdaaaaajuaGbeaaaaaacaGLOaGaayzk aaWaa8HaaeaacaWGLbaacaGLxdcadaWgaaqcfasaaiaaigdaaKqbag qaaiabgUcaRmaabmaabaWaaSaaaeaacqGHsislcaaI2aGaeq4SdC2a aSbaaKqbGeaacaaI2aaajuaGbeaacaWGtbGaamyAaiaad6gacqaH7o aBdaWgaaqcfasaaiaaigdaaKqbagqaaaqaaiabeg8aYnaaBaaajuai baqcfa4aa0baaKqbGeaacaaIXaaabaGaaG4naaaaaKqbagqaaaaaai aawIcacaGLPaaadaWhcaqaaiaadwgaaiaawEniamaaBaaajuaibaGa aGOmaaqcfayabaaaaa@6816@ (9)

there exists a second point on the circle that produces an electric field given by Eq. (10), which cancels the component of the field perpendicular to the axis adjoining the sphere’s center and the point dipole.

E b =( 6 γ 6 Cos λ 1 ρ 1 7 ) e 1 +( 6 γ 6 Sin λ 1 ρ 1 7 ) e 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaFiaaba GaamyraaGaay51GaWaaSbaaKqbGeaacaWGIbaajuaGbeaacqGH9aqp daqadaqaamaalaaabaGaeyOeI0IaaGOnaiabeo7aNnaaBaaajuaiba GaaGOnaaqcfayabaGaam4qaiaad+gacaWGZbGaeq4UdW2aaSbaaKqb GeaacaaIXaaajuaGbeaaaeaacqaHbpGCdaWgaaqcfasaaKqbaoaaDa aajuaibaGaaGymaaqaaiaaiEdaaaaajuaGbeaaaaaacaGLOaGaayzk aaWaa8HaaeaacaWGLbaacaGLxdcadaWgaaqcfasaaiaaigdaaKqbag qaaiabgUcaRmaabmaabaWaaSaaaeaacqGHsislcaaI2aGaeq4SdC2a aSbaaKqbGeaacaaI2aaajuaGbeaacaWGtbGaamyAaiaad6gacqaH7o aBdaWgaaqcfasaaiaaigdaaKqbagqaaaqaaiabeg8aYnaaBaaajuai baqcfa4aa0baaKqbGeaacaaIXaaabaGaaG4naaaaaKqbagqaaaaaai aawIcacaGLPaaadaWhcaqaaiaadwgaaiaawEniamaaBaaajuaibaGa aGOmaaqcfayabaaaaa@6817@ (10)

Therefore, the average electric field felt by the point dipole from these two points is given by Eq. (11)

E 1 =( 6 γ 6 Cos λ 1 ρ 1 7 ) e 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaFiaaba GaamyraaGaay51GaWaaSbaaKqbGeaacaaIXaaajuaGbeaacqGH9aqp daqadaqaamaalaaabaGaeyOeI0IaaGOnaiabeo7aNnaaBaaajuaiba GaaGOnaaqcfayabaGaam4qaiaad+gacaWGZbGaeq4UdW2aaSbaaKqb GeaacaaIXaaajuaGbeaaaeaacqaHbpGCdaqhaaqcfasaaiaaigdaae aacaaI3aaaaaaaaKqbakaawIcacaGLPaaadaWhcaqaaiaadwgadaWg aaqcfasaaiaaigdaaKqbagqaaaGaay51Gaaaaa@506F@ (11)

Substituting into Eq. (11) for the cosine of angle λ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSn aaBaaajuaibaGaaGymaaqcfayabaaaaa@39C5@ as presented in Diagram 1, and for the interaction distance ρ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg8aYn aaBaaajuaibaGaaGymaaqabaaaaa@3943@

, gives Eq. (12), below. E 1 = 6 γ 6 ( τ r 1 Cosθ ) ρ 1 8 e 1 = 6 γ 6 ( τ r 1 Cosθ ) ( r 1 2 + τ 2 2τ r 1 Cos θ 1 ) 4 e 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaFiaaba GaamyramaaBaaajuaibaGaaGymaaqcfayabaaacaGLxdcacqGH9aqp daWcaaqaaiabgkHiTiaaiAdacqaHZoWzdaWgaaqaamaaBaaajuaiba GaaGOnaaqcfayabaWaaeWaaeaacqaHepaDcqGHsislcaWGYbWaaSba aKqbGeaacaaIXaaajuaGbeaacaWGdbGaam4BaiaadohacqaH4oqCai aawIcacaGLPaaaaeqaaaqaaiabeg8aYnaaDaaajuaibaGaaGymaaqa aiaaiIdaaaaaaKqbaoaaFiaabaGaamyzamaaBaaajuaibaGaaGymaa qcfayabaaacaGLxdcacqGH9aqpdaWcaaqaaiabgkHiTiaaiAdacqaH ZoWzdaWgaaqaamaaBaaajuaibaGaaGOnaaqcfayabaWaaeWaaeaacq aHepaDcqGHsislcaWGYbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaWG dbGaam4BaiaadohacqaH4oqCaiaawIcacaGLPaaaaeqaaaqaamaabm aabaGaamOCamaaDaaajuaibaGaaGymaaqaaiaaikdaaaqcfaOaey4k aSIaeqiXdq3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHsislcaaIYa GaeqiXdqNaamOCamaaBaaajuaibaGaaGymaaqcfayabaGaam4qaiaa d+gacaWGZbGaeqiUde3aaSbaaKqbGeaacaaIXaaajuaGbeaaaiaawI cacaGLPaaadaahaaqabKqbGeaacaaI0aaaaaaajuaGdaWhcaqaaiaa dwgadaWgaaqcfasaaiaaigdaaKqbagqaaaGaay51Gaaaaa@807F@ (12)

Assuming a hollow sphere with an infinitely thin shell (such that there is no need to integrate radially), the total electric field produced by the entire sphere can be obtained by rotating ϕ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabew9aMb aa@3841@ from 0 to 2π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaikdacq aHapaCaaa@38F2@ for all values of θ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXb aa@382F@ between 0 and π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@ , and is given by Eq. (13), below. Integration was performed with Mathematica [39]. The output is appended to this work (Appendix I).

E total =6 γ 6 ϕ 1 =0 2π θ 1 =0 π ( τ r 1 Cos θ 1 ) ( r 1 2 + τ 2 2τ r 1 Cos θ 1 ) 4 ( r 1 2 Sin θ 1 )d ϕ 1d θ 1 = γ 6 8π r 1 2 τ( 5 r 1 2 +3 τ 2 ) ( r 1 τ ) 5 ( r 1 +τ ) 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada WgaaqcfasaaiaadshacaWGVbGaamiDaiaadggacaWGSbaajuaGbeaa cqGH9aqpcqGHsislcaaI2aGaeq4SdC2aaSbaaKqbGeaacaaI2aaaju aGbeaadaWdXbqaamaapehabaWaaSaaaeaadaqadaqaaiabes8a0jab gkHiTiaadkhadaWgaaqcfasaaiaaigdaaKqbagqaaiaadoeacaWGVb Gaam4CaiabeI7aXnaaBaaajuaibaGaaGymaaqabaaajuaGcaGLOaGa ayzkaaaabaWaaeWaaeaacaWGYbWaa0baaKqbGeaacaaIXaaabaGaaG OmaaaajuaGcqGHRaWkcqaHepaDdaahaaqabKqbGeaacaaIYaaaaKqb akabgkHiTiaaikdacqaHepaDcaWGYbWaaSbaaKqbGeaacaaIXaaaju aGbeaacaWGdbGaam4BaiaadohacqaH4oqCdaWgaaqcfasaaiaaigda aKqbagqaaaGaayjkaiaawMcaamaaCaaabeqcfasaaiaaisdaaaaaaa qaaiabeI7aXLqbaoaaBaaajuaibaGaaGymaaqabaGaeyypa0JaaGim aaqaaiabec8aWbqcfaOaey4kIipaaKqbGeaacqaHvpGzjuaGdaWgaa qcfasaaiaaigdaaeqaaiabg2da9iaaicdaaeaacaaIYaGaeqiWdaha juaGcqGHRiI8amaabmaabaGaamOCamaaDaaajuaibaGaaGymaaqaai aaikdaaaqcfaOaam4uaiaadMgacaWGUbGaeqiUde3aaSbaaKqbGeaa caaIXaaabeaaaKqbakaawIcacaGLPaaacaWGKbGaeqy1dy2aaSbaae aajuaicaaIXaqcfaOaamizaiabeI7aXnaaBaaajuaibaGaaGymaaqa baaajuaGbeaacqGH9aqpdaWcaaqaaiabgkHiTiabeo7aNnaaBaaaju aibaGaaGOnaaqcfayabaGaaGioaiabec8aWjaadkhadaqhaaqcfasa aiaaigdaaeaacaaIYaaaaKqbakabes8a0naabmaabaGaaGynaiaadk hadaqhaaqcfasaaiaaigdaaeaacaaIYaaaaKqbakabgUcaRiaaioda cqaHepaDdaahaaqabKqbGeaacaaIYaaaaaqcfaOaayjkaiaawMcaaa qaamaabmaabaGaamOCamaaBaaajuaibaGaaGymaaqcfayabaGaeyOe I0IaeqiXdqhacaGLOaGaayzkaaWaaWbaaeqajuaibaGaaGynaaaaju aGdaqadaqaaiaadkhadaWgaaqcfasaaiaaigdaaKqbagqaaiabgUca Riabes8a0bGaayjkaiaawMcaamaaCaaabeqcfasaaiaaiwdaaaaaaa aa@B51D@ (13)

The potential energy from the interaction between a spherically distributed dipole and a point dipole, then, is the integral of the total electric field with respect to the distance, τ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabes8a0b aa@383E@ between the center of the sphere and the point dipole, as given in Eq. (14), below. Again, integration was performed with Mathematica [39]. The output is appended to this work (see Appendix I).

U attract ( r 1 ,τ)=8π γ 6 r 1 2 τ(5 r 1 2 +3 τ 2 ) ( r 1 τ) 5 ( r 1 +τ ) 5 dτ= γ 6 4π r 1 2 ( τ 2 r 1 2 ) 3 ( τ 2 + r 1 2 τ 2 r 1 2 ) = C 6 ( τ 2 r 1 2 ) 3 ( τ 2 + r 1 2 τ 2 r 1 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaOaam yvamaaBaaajuaibaGaamyyaiaadshacaWG0bGaamOCaiaadggacaWG JbGaamiDaaqcfayabaGaaiikaiaackhadaWgaaqcfasaaiaaigdaaK qbagqaaiaacYcacqaHepaDcaGGPaGaeyypa0JaeyOeI0IaaGioaiab ec8aWjabeo7aNnaaBaaajuaibaGaaGOnaaqcfayabaWaa8qaaeaada WcaaqaaiaadkhadaqhaaqcfasaaiaaigdaaeaacaaIYaaaaKqbakab es8a0jaacIcacaaI1aGaaiOCamaaDaaajuaibaGaaGymaaqaaiaaik daaaqcfaOaey4kaSIaaG4maiabes8a0naaCaaajuaibeqaaiaaikda aaqcfaOaaiykaaqaaiaacIcacaGGYbWaaSbaaKqbGeaacaaIXaaabe aajuaGcqGHsislcqaHepaDcaGGPaWaaWbaaeqajuaibaGaaGynaaaa juaGdaqadaqaaiaadkhadaWgaaqaaiaaigdaaeqaaiabgUcaRiabes 8a0bGaayjkaiaawMcaamaaCaaabeqcfasaaiaaiwdaaaaaaaqcfaya beqacqGHRiI8aiaadsgacqaHepaDcqGH9aqpdaWcaaqaaiabgkHiTi abeo7aNnaaBaaabaGaaGOnaaqabaGaaGinaiabec8aWjaadkhadaqh aaqcfasaaiaaigdaaeaacaaIYaaaaaqcfayaamaabmaabaGaeqiXdq 3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHsislcaWGYbWaa0baaKqb GeaacaaIXaaabaGaaGOmaaaaaKqbakaawIcacaGLPaaadaahaaqcfa sabeaacaaIZaaaaaaajuaGdaqadaqaamaalaaabaGaeqiXdq3aaWba aeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaWGYbWaa0baaKqbGeaaca aIXaaabaGaaGOmaaaaaKqbagaacqaHepaDdaahaaqabKqbGeaacaaI YaaaaKqbakabgkHiTiaadkhadaqhaaqcfasaaiaaigdaaeaacaaIYa aaaaaaaKqbakaawIcacaGLPaaaaeaacqGH9aqpdaWcaaqaaiabgkHi TiaadoeadaWgaaqcfasaaiaaiAdaaeqaaaqcfayaamaabmaabaGaeq iXdq3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHsislcaWGYbWaa0ba aKqbGeaacaaIXaaabaGaaGOmaaaaaKqbakaawIcacaGLPaaadaahaa qcfasabeaacaaIZaaaaaaajuaGdaqadaqaamaalaaabaGaeqiXdq3a aWbaaeqajuaibaGaaGOmaaaajuaGcqGHRaWkcaWGYbWaa0baaKqbGe aacaaIXaaabaGaaGOmaaaaaKqbagaacqaHepaDdaahaaqabKqbGeaa caaIYaaaaKqbakabgkHiTiaadkhadaqhaaqcfasaaiaaigdaaeaaca aIYaaaaaaaaKqbakaawIcacaGLPaaaaaaa@B676@ (14)

It is important to not that this potential energy function is only valid for a point dipole that is outside of a spherically distributed dipole, and does not apply to the region where R r 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadkfacq GHKjYOcaWGYbWaaSbaaKqbGeaacaaIXaaajuaGbeaaaaa@3B94@ .

In order to derive the potential energy function for the dispersion interaction of two spherically distributed dipoles, let the point dipole of Diagram 1 be a point on the surface of a second sphere, as presented in Diagram 2, below.

Here, the internuclear distance is given by R, and the second interaction distance, ρ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg8aYn aaBaaajuaibaGaaGOmaaqcfayabaaaaa@39D2@ is the distance between the center of one sphere and a point on the surface of the second sphere (note that ρ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeg8aYn aaBaaajuaibaGaaGOmaaqcfayabaaaaa@39D2@ is the same distance as τ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabes8a0b aa@383E@ in Diagram 1). Again, the points on the second sphere are described in spherical, polar coordinates ( r 2 , θ 2 , ϕ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaaba GaamOCamaaBaaajuaibaGaaGOmaaqcfayabaGaaiilaiabeI7aXnaa BaaajuaibaGaaGOmaaqcfayabaGaaiilaiabew9aMnaaBaaajuaiba GaaGOmaaqabaaajuaGcaGLOaGaayzkaaaaaa@42A2@ . From the previous discussion concerning the sphere and the point dipole, it was found that the electric field, labeled E 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada WgaaqcfasaaiaaigdaaKqbagqaaaaa@38DB@ in Diagram 2, felt by any point on the surface of the second sphere due to interaction with the other sphere is given by Eq (13) above.

Substituting the second interaction distance for , we have Eq. (15)

E 1 = γ 6 8π r 1 2 ρ 2 ( 5 r 1 2 +3 ρ 2 2 ) ( r 1 ρ 2 ) 5 ( r 1 + ρ 2 ) 5 e 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaFiaaba GaamyramaaBaaajuaibaGaaGymaaqcfayabaaacaGLxdcacqGH9aqp daWcaaqaaiabgkHiTiabeo7aNnaaBaaajuaibaGaaGOnaaqcfayaba GaaGioaiabec8aWjaadkhadaqhaaqcfasaaiaaigdaaeaacaaIYaaa aKqbakabeg8aYnaaBaaajuaibaGaaGOmaaqabaqcfa4aaeWaaeaaca aI1aGaamOCamaaDaaajuaibaGaaGymaaqaaiaaikdaaaqcfaOaey4k aSIaaG4maiabeg8aYnaaDaaajuaibaGaaGOmaaqaaiaaikdaaaaaju aGcaGLOaGaayzkaaaabaWaaeWaaeaacaWGYbWaaSbaaKqbGeaacaaI XaaajuaGbeaacqGHsislcqaHbpGCdaWgaaqcfasaaiaaikdaaeqaaa qcfaOaayjkaiaawMcaamaaCaaajuaibeqaaiaaiwdaaaqcfa4aaeWa aeaacaWGYbWaaSbaaKqbGeaacaaIXaaabeaajuaGcqGHRaWkcqaHbp GCdaWgaaqcfasaaiaaikdaaKqbagqaaaGaayjkaiaawMcaamaaCaaa juaibeqaaiaaiwdaaaaaaKqbaoaaFiaabaGaamyzaaGaay51GaWaaS baaKqbGeaacaaIXaaabeaaaaa@6BD6@ (15)

This electric field vector can be written as the sum of two components, one parallel to the internuclear axis and the other perpendicular to it, as given by Eq. (16),

E 1 = E 2 + E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaFiaaba GaamyramaaBaaajuaibaGaaGymaaqcfayabaaacaGLxdcacqGH9aqp daWhcaqaaiaadweadaWgaaqcfasaaiaaikdaaKqbagqaaaGaay51Ga Gaey4kaSYaa8HaaeaacaWGfbWaaSbaaKqbGeaacqGHLkIxaKqbagqa aaGaay51Gaaaaa@459A@ (16)

where E 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada WgaaqcfasaaiaaikdaaKqbagqaaaaa@38DC@ and E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada WgaaqcfasaaiabgwQiEbqcfayabaaaaa@39D1@ are given by Eq. (17) and Eq. (18), respectively.

E 2 =( Cos λ 2 ) E 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaFiaaba GaamyramaaBaaajuaibaGaaGOmaaqcfayabaaacaGLxdcacqGH9aqp daqadaqaaiaadoeacaWGVbGaam4CaiabeU7aSnaaBaaajuaibaGaaG OmaaqabaaajuaGcaGLOaGaayzkaaWaa8HaaeaacaWGfbWaaSbaaKqb GeaacaaIXaaajuaGbeaaaiaawEniaaaa@4736@ (17)

E =(Sin λ 2 ) E 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaFiaaba GaamyramaaBaaajuaibaGaeyyPI4fajuaGbeaaaiaawEniaiabg2da 9iaacIcacaGGtbGaaiyAaiaac6gacqaH7oaBdaWgaaqcfasaaiaaik daaKqbagqaaiaacMcadaWhcaqaaiaacweadaWgaaqcfasaaiaaigda aKqbagqaaaGaay51Gaaaaa@47FC@ (18)

The only component of the E 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaacweada WgaaqcfasaaiaaigdaaKqbagqaaaaa@38DA@ that contributes to the dispersion attraction is the component in the direction of the internuclear axis E 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada WgaaqcfasaaiaaikdaaKqbagqaaaaa@38DC@ . However, rotating ϕ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabew9aMn aaBaaajuaibaGaaGymaaqcfayabaaaaa@39D9@ from 0 to 2π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaikdacq aHapaCaaa@38F2@ or any fixed value of θ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXn aaBaaajuaibaGaaGymaaqcfayabaaaaa@39C7@ sweeps out a circle (outlined in yellow in Diagram 2 above) that is perpendicular to the internuclear axis. As illustrated in Diagram 2, for any point on this circle that produces an electric field, E 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaacweada WgaaqcfasaaiaaigdaaKqbagqaaaaa@38DA@ given by Eq. (16), on the other sphere there exists another point on the circle that produces a field given by Eq. (19), where E 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada WgaaqcfasaaiaaikdaaKqbagqaaaaa@38DC@ and E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada WgaaqcfasaaiabgwQiEbqcfayabaaaaa@39D1@ are given by Eq. (17) and Eq. (18), respectively.

E 1 = E 2 E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaFiaaba GaamyramaaBaaajuaibaGaaGymaaqcfayabaaacaGLxdcacqGH9aqp daWhcaqaaiaadweadaWgaaqcfasaaiaaikdaaKqbagqaaaGaay51Ga GaeyOeI0Yaa8HaaeaacaWGfbWaaSbaaKqbGeaacqGHLkIxaKqbagqa aaGaay51Gaaaaa@45A5@ (19)

Thus, the average electric field, E 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadweada WgaaqcfasaaiaaikdaaKqbagqaaaaa@38DC@ from these two points is given by Eq. (17), above. As shown in Diagram 2, the cosine of angle λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeU7aSb aa@382D@ is given by Eq. (20), below.

Cos λ 2 = R r 2 Cos θ 2 ρ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadoeaca WGVbGaam4CaiabeU7aSnaaBaaajuaibaGaaGOmaaqcfayabaGaeyyp a0ZaaSaaaeaacaWGsbGaeyOeI0IaamOCamaaBaaajuaibaGaaGOmaa qcfayabaGaam4qaiaad+gacaWGZbGaeqiUde3aaSbaaKqbGeaacaaI YaaajuaGbeaaaeaacqaHbpGCdaWgaaqcfasaaiaaikdaaKqbagqaaa aaaaa@4B40@ (20)

Substituting Eq. (20) in to Eq. (17) for the component of the electric field at any point on the surface of the second sphere contributing to the dispersion interaction, E2 gives Eq. (21), below.

E 2 = γ 6 8π r 1 2 ρ 2 ( 5 r 1 2 +3 ρ 2 2 ) ( r 1 ρ 2 ) 5 ( r 1 + ρ 2 ) 5 ( R r 2 Cos θ 2 ρ 2 ) e k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaFiaaba GaamyramaaBaaajuaibaGaaGOmaaqcfayabaaacaGLxdcacqGH9aqp daWcaaqaaiabgkHiTiabeo7aNnaaBaaajuaibaGaaGOnaaqcfayaba GaaGioaiabec8aWjaadkhadaqhaaqcfasaaiaaigdaaeaacaaIYaaa aKqbakabeg8aYnaaBaaajuaibaGaaGOmaaqcfayabaWaaeWaaeaaca aI1aGaamOCamaaDaaajuaibaGaaGymaaqaaiaaikdaaaqcfaOaey4k aSIaaG4maiabeg8aYnaaDaaajuaibaGaaGOmaaqaaiaaikdaaaaaju aGcaGLOaGaayzkaaaabaWaaeWaaeaacaWGYbWaaSbaaKqbGeaacaaI XaaajuaGbeaacqGHsislcqaHbpGCdaWgaaqcfasaaiaaikdaaKqbag qaaaGaayjkaiaawMcaamaaCaaabeqcfasaaiaaiwdaaaqcfa4aaeWa aeaacaWGYbWaaSbaaKqbGeaacaaIXaaajuaGbeaacqGHRaWkcqaHbp GCdaWgaaqcfasaaiaaikdaaKqbagqaaaGaayjkaiaawMcaamaaCaaa beqcfasaaiaaiwdaaaaaaKqbaoaabmaabaWaaSaaaeaacaWGsbGaey OeI0IaamOCamaaBaaajuaibaGaaGOmaaqcfayabaGaam4qaiaad+ga caWGZbGaeqiUde3aaSbaaKqbGeaacaaIYaaajuaGbeaaaeaacqaHbp GCdaWgaaqcfasaaiaaikdaaeqaaaaaaKqbakaawIcacaGLPaaadaWh caqaaiaadwgadaWgaaqcfasaaiaadUgaaKqbagqaaaGaay51Gaaaaa@7BE3@ (21)

As before, assuming a hollow sphere with an infinitely thin shell (such that there is no need to integrate radially), the total electric field felt due to the interaction of the two spherically distributed dipoles can be obtained by rotating ϕ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabew9aMn aaBaaajuaibaGaaGOmaaqcfayabaaaaa@39DA@ from 0 to 2π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaaikdacq aHapaCaaa@38F2@ for all values of θ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabeI7aXn aaBaaajuaibaGaaGOmaaqcfayabaaaaa@39C8@ between 0 and π MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabec8aWb aa@3836@ and is given by Eq. (22), below. Integration was performed with Mathematica [39]. The output is appended to this work (see Appendix I).

E total = γ 6 8π ϕ 2 =0 2π θ 2 =0 π r 1 2 ρ 2 ( 5 r 1 2 +3 ρ 2 2 ) ( r 1 ρ 2 ) 5 ( r 1+ ρ 2 ) 5 ( R r 2 Cos θ 2 ρ 2 )( r 2 2 Sin θ 2 )dϕd ϕ 2 =16 π 2 r 1 2 r 2 2 ( 15 R 4 +10 R 2 r 1 2 r 1 4 +48 R 3 r 2 +8R r 1 2 r 2 +54 R 2 r 2 2 2 r 1 2 r 2 2 +24R r 2 3 +3 r 2 4 (24 R 2 r 2 ( R 2 r 1 2 2R r 2 + r 2 2 ) 4 15 R 4 +10 R 2 r 1 2 r 1 4 48 R 3 r 2 8R r 1 2 r 2 +54 R 2 r 2 2 2 r 1 2 r 2 2 24R r 2 3 +3 r 2 4 (24 R 2 r 2 ( R 2 r 1 2 +2R r 2 + r 2 2 ) 4 ) = C 6 ( 15 R 4 +10 R 2 r 1 2 r 1 4 +48 R 3 r 2 +8R r 1 2 r 2 +54 R 2 r 2 2 2 r 1 2 r 2 2 +24R r 2 3 +3 r 2 4 24 R 2 r 2 ( R 2 r 1 2 2R r 2 + r 2 2 ) 4 15 R 4 +10 R 2 r 1 2 r 1 4 48 R 3 r 2 8R r 1 2 r 2 +54 R 2 r 2 2 2 r 1 2 r 2 2 24R r 2 3 +3 r 2 4 (24 R 2 r 2 ( R 2 r 1 2 +2R r 2 + r 2 2 ) 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaOaam yramaaBaaajuaibaGaamiDaiaad+gacaWG0bGaamyyaiaadYgaaKqb agqaaiabg2da9iabgkHiTiabeo7aNnaaBaaajuaibaGaaGOnaaqcfa yabaGaaGioaiabec8aWnaapehabaWaa8qCaeaadaWcaaqaaiaadkha daqhaaqcfasaaiaaigdaaeaacaaIYaaaaKqbakabeg8aYnaaBaaaju aibaGaaGOmaaqcfayabaWaaeWaaeaacaaI1aGaamOCamaaDaaajuai baGaaGymaaqaaiaaikdaaaqcfaOaey4kaSIaaG4maiabeg8aYnaaDa aajuaibaGaaGOmaaqaaiaaikdaaaaajuaGcaGLOaGaayzkaaaabaWa aeWaaeaacaWGYbWaaSbaaeaajuaicaaIXaqcfaOaeyOeI0IaeqyWdi 3aaSbaaKqbGeaacaaIYaaajuaGbeaaaeqaaaGaayjkaiaawMcaamaa Caaabeqcfasaaiaaiwdaaaqcfa4aaeWaaeaacaWGYbWaaSbaaeaaju aicaaIXaqcfaOaey4kaSIaeqyWdi3aaSbaaKqbGeaacaaIYaaajuaG beaaaeqaaaGaayjkaiaawMcaamaaCaaabeqcfasaaiaaiwdaaaaaaa qaaiabeI7aXLqbaoaaBaaajuaibaGaaGOmaaqabaGaeyypa0JaaGim aaqaaiabec8aWbqcfaOaey4kIipaaKqbGeaacqaHvpGzjuaGdaWgaa qcfasaaiaaikdaaeqaaiabg2da9iaaicdaaeaacaaIYaGaeqiWdaha juaGcqGHRiI8amaabmaabaWaaSaaaeaacaWGsbGaeyOeI0IaamOCam aaBaaajuaibaGaaGOmaaqcfayabaGaam4qaiaad+gacaWGZbGaeqiU de3aaSbaaKqbGeaacaaIYaaajuaGbeaaaeaacqaHbpGCdaWgaaqcfa saaiaaikdaaKqbagqaaaaaaiaawIcacaGLPaaadaqadaqaaiaadkha daqhaaqcfasaaiaaikdaaeaacaaIYaaaaKqbakaadofacaWGPbGaam OBaiabeI7aXnaaBaaajuaibaGaaGOmaaqcfayabaaacaGLOaGaayzk aaGaamizaiabew9aMjaadsgacqaHvpGzdaWgaaqcfasaaiaaikdaaK qbagqaaaqaaiabg2da9iaaigdacaaI2aGaeqiWda3aaWbaaeqajuai baGaaGOmaaaajuaGcaWGYbWaa0baaKqbGeaacaaIXaaabaGaaGOmaa aajuaGcaWGYbWaa0baaKqbGeaacaaIYaaabaGaaGOmaaaajuaGdaqa daabaeqabaWaaSaaaeaacaaIXaGaaGynaiaadkfadaahaaqabKqbGe aacaaI0aaaaKqbakabgUcaRiaaigdacaaIWaGaamOuamaaCaaabeqc fasaaiaaikdaaaqcfaOaamOCamaaDaaajuaibaGaaGymaaqaaiaaik daaaqcfaOaeyOeI0IaamOCamaaDaaajuaibaGaaGymaaqaaiaaisda aaqcfaOaey4kaSIaaGinaiaaiIdacaWGsbWaaWbaaeqajuaibaGaaG 4maaaajuaGcaWGYbWaaSbaaKqbGeaacaaIYaaabeaajuaGcqGHRaWk caaI4aGaamOuaiaadkhadaqhaaqcfasaaiaaigdaaeaacaaIYaaaaK qbakaadkhadaWgaaqcfasaaiaaikdaaKqbagqaaiabgUcaRiaaiwda caaI0aGaamOuamaaCaaabeqcfasaaiaaikdaaaqcfaOaamOCamaaDa aajuaibaGaaGOmaaqaaiaaikdaaaqcfaOaeyOeI0IaaGOmaiaadkha 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aaaajuaGcaWGYbWaa0baaKqbGeaacaaIXaaabaGaaGOmaaaajuaGcq GHsislcaWGYbWaa0baaeaacaaIXaaabaGaaGinaaaacqGHsislcaaI 0aGaaGioaiaadkfadaahaaqabKqbGeaacaaIZaaaaKqbakaadkhada WgaaqcfasaaiaaikdaaKqbagqaaiabgkHiTiaaiIdacaWGsbGaamOC amaaDaaajuaibaGaaGymaaqaaiaaikdaaaqcfaOaamOCamaaBaaaju aibaGaaGOmaaqcfayabaGaey4kaSIaaGynaiaaisdacaWGsbWaaWba aeqajuaibaGaaGOmaaaajuaGcaWGYbWaa0baaKqbGeaacaaIYaaaba GaaGOmaaaajuaGcqGHsislcaaIYaGaamOCamaaDaaajuaibaGaaGym aaqaaiaaikdaaaqcfaOaamOCamaaDaaajuaibaGaaGOmaaqaaiaaik daaaqcfaOaeyOeI0IaaGOmaiaaisdacaWGsbGaamOCamaaDaaajuai baGaaGOmaaqaaiaaiodaaaqcfaOaey4kaSIaaG4maiaadkhadaqhaa qcfasaaiaaikdaaeaacaaI0aaaaaqcfayaaiaacIcacaaIYaGaaGin aiaadkfadaahaaqabKqbGeaacaaIYaaaaKqbakaadkhadaWgaaqcfa saaiaaikdaaKqbagqaaiaacIcacaWGsbWaaWbaaeqajuaibaGaaGOm aaaajuaGcqGHsislcaWGYbWaa0baaKqbGeaacaaIXaaabaGaaGOmaa aajuaGcqGHRaWkcaaIYaGaamOuaiaadkhadaWgaaqcfasaaiaaikda aKqbagqaaiabgUcaRiaadkhadaqhaaqcfasaaiaaikdaaeaacaaIYa aaaKqbakaacMcadaahaaqabKqbGeaacaaI0aaaaaaaaaqcfaOaayjk aiaawMcaaaaaaa@0A6C@ (22)

Finally, the potential energy from the interaction between two spherically distributed dipoles is the integral of the total electric field with respect to the internuclear distance, R, as given in Eq. (23), below. Again, this attractive potential is not valid when the spheres touch or as they unite. Integration was performed with Mathematica [39]. The output is appended to this work (Appendix I).

U attract =( r 1 , r 2 ,R)= E total dR= C 6 ( [ R 2 ( r 1 r 2 ) 2 ] 3 + 2 3 δ( r 1 , r 2 ,R) ) [ R 2 ( r 1 r 2 ) 2 ] 3 [ R 2 ( r 1 + r 2 ) 2 ] 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadwfada WgaaqcfasaaiaadggacaWG0bGaamiDaiaadkhacaWGHbGaam4yaiaa dshaaKqbagqaaiabg2da9iaacIcacaWGYbWaaSbaaKqbGeaacaaIXa aajuaGbeaacaGGSaGaamOCamaaBaaajuaibaGaaGOmaaqcfayabaGa aiilaiaackfacaGGPaGaeyypa0Zaa8qaaeaacaWGfbWaaSbaaKqbGe aacaWG0bGaam4BaiaadshacaWGHbGaamiBaaqcfayabaaabeqabiab gUIiYdGaamizaiaadkfacqGH9aqpdaWcaaqaaiaadoeadaWgaaqcfa saaiaaiAdaaKqbagqaamaabmaabaWaamWaaeaacaWGsbWaaWbaaeqa juaibaGaaGOmaaaajuaGcqGHsislcaGGOaGaamOCamaaBaaajuaiba GaaGymaaqcfayabaGaeyOeI0IaamOCamaaBaaajuaibaGaaGOmaaqc fayabaGaaiykamaaCaaabeqcfasaaiaaikdaaaaajuaGcaGLBbGaay zxaaWaaWbaaeqajuaibaGaaG4maaaajuaGcqGHRaWkdaWcaaqaaiaa ikdaaeaacaaIZaaaaiabes7aKjaacIcacaWGYbWaaSbaaKqbGeaaca aIXaaajuaGbeaacaGGSaGaamOCamaaBaaajuaibaGaaGOmaaqabaqc faOaaiilaiaadkfacaGGPaaacaGLOaGaayzkaaaabaWaamWaaeaaca WGsbWaaWbaaeqajuaibaGaaGOmaaaajuaGcqGHsislcaGGOaGaamOC amaaBaaajuaibaGaaGymaaqabaqcfaOaeyOeI0IaamOCamaaBaaaju aibaGaaGOmaaqabaqcfaOaaiykamaaCaaajuaibeqaaiaaikdaaaaa juaGcaGLBbGaayzxaaWaaWbaaKqbGeqabaGaaG4maaaajuaGdaWada qaaiaadkfadaahaaqcfasabeaacaaIYaaaaKqbakabgkHiTiaacIca caWGYbWaaSbaaKqbGeaacaaIXaaabeaajuaGcqGHRaWkcaWGYbWaaS baaKqbGeaacaaIYaaabeaajuaGcaGGPaWaaWbaaKqbGeqabaGaaGOm aaaaaKqbakaawUfacaGLDbaadaahaaqcfasabeaacaaIZaaaaaaaaa a@946F@ (23)

This potential energy function, Eq. (23) is written as the sum of two terms, the first involving a simplified deviation from the C 6 R 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba Gaam4qamaaBaaajuaibaGaaGOnaaqcfayabaaabaGaamOuamaaCaaa beqcfasaaiaaiAdaaaaaaaaa@3AD5@ attraction of two point-dipoles, and the second containing the error terms arising from this simplification, where δ(r1,r2,R) is given by Eq. (24), below.

δ( r 1 , r 2 ,R)=3 R 4 ( r 1 2 3 r 1 r 2 + r 2 2 )+3 ( r 1 r 2 ) 2 ( r 1 4 r 1 3 r 2 +4 r 1 2 r 2 2 r 2 3 r 1 + r 2 4 ) 2 R 2 (3 r 1 4 9 r 1 3 r 2 +10 r 1 2 r 2 2 9 r 2 3 r 1 +3 r 2 4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOabaeqabaqcfaOaeq iTdqMaaiikaiaadkhadaWgaaqcfasaaiaaigdaaKqbagqaaiaacYca caWGYbWaaSbaaKqbGeaacaaIYaaabeaajuaGcaGGSaGaamOuaiaacM cacqGH9aqpcaaIZaGaamOuamaaCaaajuaibeqaaiaaisdaaaqcfaOa aiikaiaackhadaqhaaqcfasaaiaaigdaaeaacaaIYaaaaKqbakabgk HiTiaaiodacaWGYbWaaSbaaKqbGeaacaaIXaaajuaGbeaacaWGYbWa aSbaaKqbGeaacaaIYaaabeaajuaGcqGHRaWkcaWGYbWaa0baaKqbGe aacaaIYaaabaGaaGOmaaaajuaGcaGGPaGaey4kaSIaaG4maiaacIca caGGYbWaaSbaaKqbGeaacaaIXaaabeaajuaGcqGHsislcaWGYbWaaS baaKqbGeaacaaIYaaabeaajuaGcaGGPaWaaWbaaKqbGeqabaGaaGOm aaaajuaGcaGGOaGaamOCamaaDaaajuaibaGaaGymaaqaaiaaisdaaa qcfaOaeyOeI0IaamOCamaaDaaajuaibaGaaGymaaqaaiaaiodaaaqc faOaamOCamaaBaaajuaibaGaaGOmaaqabaqcfaOaey4kaSIaaGinai aadkhadaqhaaqcfasaaiaaigdaaeaacaaIYaaaaKqbakaadkhadaqh aaqcfasaaiaaikdaaeaacaaIYaaaaKqbakabgkHiTiaadkhadaqhaa qcfasaaiaaikdaaeaacaaIZaaaaKqbakaadkhadaWgaaqcfasaaiaa igdaaeqaaKqbakabgUcaRiaadkhadaqhaaqcfasaaiaaikdaaeaaca aI0aaaaKqbakaacMcacqGHsislaeaacqGHsislcaaIYaGaamOuamaa CaaajuaibeqaaiaaikdaaaqcfaOaaiikaiaaiodacaWGYbWaa0baaK qbGeaacaaIXaaabaGaaGinaaaajuaGcqGHsislcaaI5aGaamOCamaa DaaajuaibaGaaGymaaqaaiaaiodaaaqcfaOaamOCamaaBaaajuaiba GaaGOmaaqabaqcfaOaey4kaSIaaGymaiaaicdacaWGYbWaa0baaKqb GeaacaaIXaaabaGaaGOmaaaajuaGcaWGYbWaa0baaKqbGeaacaaIYa aabaGaaGOmaaaajuaGcqGHsislcaaI5aGaamOCamaaDaaajuaibaGa aGOmaaqaaiaaiodaaaqcfaOaamOCamaaBaaajuaibaGaaGymaaqaba qcfaOaey4kaSIaaG4maiaadkhadaqhaaqcfasaaiaaikdaaeaacaaI 0aaaaKqbakaacMcaaaaa@A54B@ (24)

In the case of homonuclear diatomics, like Ne2 and He2, Eq. (23) reduces to Eq. (25), where δ(r,R) is given by Eq. (26)

U attract = C 6 [ R 2 (2r) 2 ] 3 + 2 3 C 6 δ(r,R) R 6 [ R 2 (2r) 2 ] 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadwfada WgaaqcfasaaiaadggacaWG0bGaamiDaiaadkhacaWGHbGaam4yaiaa dshaaKqbagqaaiabg2da9maalaaabaGaam4qamaaBaaajuaibaGaaG OnaaqcfayabaaabaWaamWaaeaacaWGsbWaaWbaaeqajuaibaGaaGOm aaaajuaGcqGHsislcaGGOaGaaGOmaiaadkhacaGGPaWaaWbaaeqaju aibaGaaGOmaaaaaKqbakaawUfacaGLDbaadaahaaqabKqbGeaacaaI ZaaaaaaajuaGcqGHRaWkdaWcaaqaamaalaaabaGaaGOmaaqaaiaaio daaaGaam4qamaaBaaajuaibaGaaGOnaaqcfayabaGaeqiTdqMaaiik aiaadkhacaGGSaGaamOuaiaacMcaaeaacaWGsbWaaWbaaeqajuaiba GaaGOnaaaajuaGdaWadaqaaiaadkfadaahaaqabKqbGeaacaaIYaaa aKqbakabgkHiTiaacIcacaaIYaGaamOCaiaacMcadaahaaqabKqbGe aacaaIYaaaaaqcfaOaay5waiaaw2faamaaCaaabeqcfasaaiaaioda aaaaaaaa@65A6@ (25)

δ(r,R)=4 R 2 r 4 3 R 4 r 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabes7aKj aacIcacaWGYbGaaiilaiaadkfacaGGPaGaeyypa0JaaGinaiaadkfa daahaaqabKqbGeaacaaIYaaaaKqbakaadkhadaahaaqabKqbGeaaca aI0aaaaKqbakabgkHiTiaaiodacaWGsbWaaWbaaeqajuaibaGaaGin aaaajuaGcaWGYbWaaWbaaeqajuaibaGaaGOmaaaaaaa@48DD@ (26)

The corresponding scaling function, f s sim MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadAgada qhaaqcfasaaiaadohaaeaacaWGZbGaamyAaiaad2gaaaaaaa@3B84@ for strengthening the C 6 R 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba Gaam4qamaaBaaajuaibaGaaGOnaaqcfayabaaabaGaamOuamaaCaaa beqcfasaaiaaiAdaaaaaaaaa@3AD5@ attraction of two point dipoles, is given by Eq. (25), where δ(r,R) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakabes7aKj aacIcacaWGYbGaaiilaiaadkfacaGGPaaaaa@3BF5@ is given by Eq. (26)

U attract = f s sim C 6 R 6 f s sim = R 6 [ R 2 ( 2r ) 2 ] 3 + 2/3δ( r,R ) [ R 2 ( 2r ) 2 ] 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyvam aaBaaajuaibaGaamyyaiaadshacaWG0bGaamOCaiaadggacaWGJbGa amiDaaqcfayabaGaeyypa0JaamOzamaaDaaajuaibaGaam4Caaqaai aadohacaWGPbGaamyBaaaajuaGdaWcaaqaaiaadoeadaWgaaqcfasa aiaaiAdaaKqbagqaaaqaaiaadkfadaahaaqabKqbGeaacaaI2aaaaa aajuaGcqGHshI3caWGMbWaa0baaKqbGeaacaWGZbaabaGaam4Caiaa dMgacaWGTbaaaKqbakabg2da9maalaaabaGaamOuamaaCaaabeqcfa saaiaaiAdaaaaajuaGbaWaamWaaeaacaWGsbWaaWbaaeqajuaibaGa aGOmaaaajuaGcqGHsisldaqadaqaaiaaikdacaWGYbaacaGLOaGaay zkaaWaaWbaaeqajuaibaGaaGOmaaaaaKqbakaawUfacaGLDbaadaah aaqabKqbGeaacaaIZaaaaaaajuaGcqGHRaWkdaWcaaqaaiaaikdaca GGVaGaaG4maiabes7aKnaabmaabaGaamOCaiaacYcacaWGsbaacaGL OaGaayzkaaaabaWaamWaaeaacaWGsbWaaWbaaeqajuaibaGaaGOmaa aajuaGcqGHsisldaqadaqaaiaaikdacaWGYbaacaGLOaGaayzkaaWa aWbaaeqajuaibaGaaGOmaaaaaKqbakaawUfacaGLDbaadaahaaqabK qbGeaacaaIZaaaaaaaaaa@7599@ (27)

The behavior of the second, error term, above, is presented in Figure 7a for Ne dimer. Here, the radius of Ne is taken to be half its hard-sphere diameter.

Note that the error term approaches negative infinity as the internuclear distance, R, approaches the sum of the radii for the two interacting Ne atoms. However, the potential energy should not diverge when the spheres touch or as they unite. This divergence can be prevented by adding a constant to the denominator of the error term in Eq. (27), to give Eq. (28), again where δ( r,R ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqiTdq 2aaeWaaeaacaWGYbGaaiilaiaadkfaaiaawIcacaGLPaaaaaa@3C30@ is given by Eq. (26)

Error_term= 2/3δ( r,R ) [ R 2 ( 2r ) 2 ] 3 +γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyrai aadkhacaWGYbGaam4BaiaadkhacaGGFbGaamiDaiaadwgacaWGYbGa amyBaiabg2da9maalaaabaGaaGOmaiaac+cacaaIZaGaeqiTdq2aae WaaeaacaWGYbGaaiilaiaadkfaaiaawIcacaGLPaaaaeaadaWadaqa aiaadkfadaahaaqabKqbGeaacaaIYaaaaKqbakabgkHiTmaabmaaba GaaGOmaiaadkhaaiaawIcacaGLPaaadaahaaqabKqbGeaacaaIYaaa aaqcfaOaay5waiaaw2faamaaCaaabeqcfasaaiaaiodaaaqcfaOaey 4kaSIaeq4SdCgaaaaa@570E@ (28)

Figure 7a: Error_Term of Scaling Function of Eq. (27).

Figure 7b: Error_Term of Scaling Function of Eq. (27).
Figure 7: (a) Second term (Error Term) of Eq. (27), unphysical divergence as spheres touch; (b) Error term of Eq. (28), preventing unphysical divergence.

By increasing the value of the constant, γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4SdC gaaa@382B@ this error term becomes negligible for values of the internuclear distance surrounding the bonding region ( R=1.12σ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGsbGaeyypa0JaaGymaiaac6cacaaIXaGaaGOmaiabeo8aZbGa ayjkaiaawMcaaaaa@3E91@ as shown in Figure 7b, below.

Therefore, it is reasonable to ignore this second, error term in the scaling function for the interaction of two spherically distributed dipoles. The simplified scaling function is given by Eq. (29), below.

f s sim = R 6 [ R 2 ( 2r ) 2 ] 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaDaaajuaibaGaam4CaaqaaiaadohacaWGPbGaamyBaaaajuaGcqGH 9aqpdaWcaaqaaiaadkfadaahaaqabKqbGeaacaaI2aaaaaqcfayaam aadmaabaGaamOuamaaCaaabeqcfasaaiaaikdaaaqcfaOaeyOeI0Ya aeWaaeaacaaIYaGaamOCaaGaayjkaiaawMcaamaaCaaabeqcfasaai aaikdaaaaajuaGcaGLBbGaayzxaaWaaWbaaeqajuaibaGaaG4maaaa aaaaaa@4ADB@ (29)

Note that this simplified scaling function will diverge when the two spheres touch. To prevent this divergence, a constant can be added to the denominator, as shown in Eq. (30)

f s sim = R 6 [ R 2 ( 2r ) 2 ] 3 +γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaDaaajuaibaGaam4CaaqaaiaadohacaWGPbGaamyBaaaajuaGcqGH 9aqpdaWcaaqaaiaadkfadaahaaqabKqbGeaacaaI2aaaaaqcfayaam aadmaabaGaamOuamaaCaaabeqcfasaaiaaikdaaaqcfaOaeyOeI0Ya aeWaaeaacaaIYaGaamOCaaGaayjkaiaawMcaamaaCaaabeqcfasaai aaikdaaaaajuaGcaGLBbGaayzxaaWaaWbaaeqajuaibaGaaG4maaaa juaGcqGHRaWkcqaHZoWzaaaaaa@4DF2@ (30)

By allowing the radius, r to be a multiple (1/n) of σ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaSaaae aacqaHdpWCaeaacaaIYaaaaaaa@3913@ , where σ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Wdm haaa@3847@ is the hard-sphere diameter, f s sim MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaDaaajuaibaGaam4CaaqaaiaadohacaWGPbGaamyBaaaaaaa@3B8F@ can be parameterized. The radius, r, is then given by Eq.

r= σ 2n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOCai abg2da9maalaaabaGaeq4WdmhabaGaaGOmaiaad6gaaaaaaa@3C03@ (31)

The constant, Υ is parameterized by setting fssim at R=σ equal to fsp-p at R=σ and solving Υ as shown in Eq. (32)

f s sim ( R=σ )= f s pp ( R=σ )= σ 6 [ σ 2 ( σ n ) 2 ] 3 +γ σ 6 = f s pp ( R=σ ) [ σ 2 ( σ n ) 2 ] 3 +γγ=[ 1 f s pp ( R=σ ) ( n 2 1 n 2 ) 3 ] σ 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaDaaajuaibaGaam4CaaqaaiaadohacaWGPbGaamyBaaaajuaGdaqa daqaaiaadkfacqGH9aqpcqaHdpWCaiaawIcacaGLPaaacqGH9aqpca WGMbWaa0baaKqbGeaacaWGZbaabaGaamiCaiabgkHiTiaadchaaaqc fa4aaeWaaeaacaWGsbGaeyypa0Jaeq4WdmhacaGLOaGaayzkaaGaey ypa0ZaaSaaaeaacqaHdpWCdaahaaqabKqbGeaacaaI2aaaaaqcfaya amaadmaabaGaeq4Wdm3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHsi sldaqadaqaamaalaaabaGaeq4WdmhabaGaamOBaaaaaiaawIcacaGL PaaadaahaaqabKqbGeaacaaIYaaaaaqcfaOaay5waiaaw2faamaaCa aabeqcfasaaiaaiodaaaqcfaOaey4kaSIaeq4SdCgaaiabgkDiElab eo8aZnaaCaaabeqcfasaaiaaiAdaaaqcfaOaeyypa0JaamOzamaaDa aajuaibaGaam4CaaqaaiaadchacqGHsislcaWGWbaaaKqbaoaabmaa baGaamOuaiabg2da9iabeo8aZbGaayjkaiaawMcaamaadmaabaGaeq 4Wdm3aaWbaaeqajuaibaGaaGOmaaaajuaGcqGHsisldaqadaqaamaa laaabaGaeq4WdmhabaGaamOBaaaaaiaawIcacaGLPaaadaahaaqabK qbGeaacaaIYaaaaaqcfaOaay5waiaaw2faamaaCaaabeqcfasaaiaa iodaaaqcfaOaey4kaSIaeq4SdCMaeyO0H4Taeq4SdCMaeyypa0Zaam WaaeaadaWcaaqaaiaaigdaaeaacaWGMbWaa0baaKqbGeaacaWGZbaa baGaamiCaiabgkHiTiaadchaaaqcfa4aaeWaaeaacaWGsbGaeyypa0 Jaeq4WdmhacaGLOaGaayzkaaaaaiabgkHiTmaabmaabaWaaSaaaeaa caWGUbWaaWbaaeqajuaibaGaaGOmaaaajuaGcqGHsislcaaIXaaaba GaamOBamaaCaaabeqcfasaaiaaikdaaaaaaaqcfaOaayjkaiaawMca amaaCaaabeqcfasaaiaaiodaaaaajuaGcaGLBbGaayzxaaGaeq4Wdm 3aaWbaaeqajuaibaGaaGOnaaaaaaa@A3D9@ (32)

Substituting Eq. (31) for the radius, r, and Eq. (32) for the constant, Υ in Eq. (30), gives Eq. (33), below.

f s sim = R 6 [ R 2 ( σ n ) 2 ] 3 +[ 1 f s pp ( R=σ ) ( n 2 1 n 2 ) 3 ] σ 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOzam aaDaaajuaibaGaam4CaaqaaiaadohacaWGPbGaamyBaaaajuaGcqGH 9aqpdaWcaaqaaiaadkfadaahaaqabKqbGeaacaaI2aaaaaqcfayaam aadmaabaGaamOuamaaCaaabeqcfasaaiaaikdaaaqcfaOaeyOeI0Ya aeWaaeaadaWcaaqaaiabeo8aZbqaaiaad6gaaaaacaGLOaGaayzkaa WaaWbaaeqajuaibaGaaGOmaaaaaKqbakaawUfacaGLDbaadaahaaqa bKqbGeaacaaIZaaaaKqbakabgUcaRmaadmaabaWaaSaaaeaacaaIXa aabaGaamOzamaaDaaajuaibaGaam4CaaqaaiaadchacqGHsislcaWG WbaaaKqbaoaabmaabaGaamOuaiabg2da9iabeo8aZbGaayjkaiaawM caaaaacqGHsisldaqadaqaamaalaaabaGaamOBamaaCaaabeqcfasa aiaaikdaaaqcfaOaeyOeI0IaaGymaaqaaiaad6gadaahaaqabKqbGe aacaaIYaaaaaaaaKqbakaawIcacaGLPaaadaahaaqabKqbGeaacaaI ZaaaaaqcfaOaay5waiaaw2faaiabeo8aZnaaCaaabeqcfasaaiaaiA daaaaaaaaa@6892@ (33)

The Levenberg-Marquardt method [39] of non-linear least squares (please refer to the Fortran code presented in Appendix I) was used to obtain the optimized value for the parameter, n, by fitting Eq. (34), below, for R>σ to the experimental potentials [27] for both He and Ne. These values for n were found to be just above unity, indicating that the optimized radii for He and Ne, as given by Eq. (31), are smaller than their corresponding van der Waals radii, as expected since most of the electron density is contained therein.

U interact = U B3LYP f s sim C 6 R 6 = U B3LYP C 6 [ R 2 ( σ n ) 2 ] 3 +[ 1 f s pp ( R=σ ) ( n 2 1 n 2 ) 3 ] σ 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyvam aaBaaajuaibaGaciyAaiaac6gacaGG0bGaaiyzaiaackhacaGGHbGa ai4yaiaacshaaKqbagqaaiabg2da9iaadwfadaWgaaqcfasaaiaadk eacaaIZaGaamitaiaadMfacaWGqbaajuaGbeaacqGHsislcaWGMbWa a0baaKqbGeaacaWGZbaabaGaam4CaiaadMgacaWGTbaaaKqbaoaala aabaGaam4qamaaBaaajuaibaGaaGOnaaqcfayabaaabaGaamOuamaa BaaajuaibaGaaGOnaaqcfayabaaaaiabg2da9iaadwfadaWgaaqcfa saaiaadkeacaaIZaGaamitaiaadMfacaWGqbaajuaGbeaacqGHsisl daWcaaqaaiaadoeadaWgaaqcfasaaiaaiAdaaKqbagqaaaqaamaadm aabaGaamOuamaaCaaabeqcfasaaiaaikdaaaqcfaOaeyOeI0YaaeWa aeaadaWcaaqaaiabeo8aZbqaaiaad6gaaaaacaGLOaGaayzkaaWaaW baaeqajuaibaGaaGOmaaaaaKqbakaawUfacaGLDbaadaahaaqabKqb GeaacaaIZaaaaKqbakabgUcaRmaadmaabaWaaSaaaeaacaaIXaaaba GaamOzamaaDaaajuaibaGaam4CaaqaaiaadchacqGHsislcaWGWbaa aKqbaoaabmaabaGaamOuaiabg2da9iabeo8aZbGaayjkaiaawMcaaa aacqGHsisldaqadaqaamaalaaabaGaamOBamaaCaaabeqcfasaaiaa ikdaaaqcfaOaeyOeI0IaaGymaaqaaiaad6gadaahaaqabKqbGeaaca aIYaaaaaaaaKqbakaawIcacaGLPaaadaahaaqabKqbGeaacaaIZaaa aaqcfaOaay5waiaaw2faaiabeo8aZnaaCaaabeqcfasaaiaaiAdaaa aaaaaa@84FC@ (34)

The point-by point scaling functions, simulated scaling functions, as given by Eq. (33), above, and the damping functions employed in previous studies are presented in Figures 8a & 8b for Ne and He dimers, respectively.

Figures 8a & 8b, above, reveal that Ne requires roughly 1.5 times more attraction in the bonding region ( R=1.12σ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfa4aaeWaae aacaWGsbGaeyypa0JaaGymaiaac6cacaaIXaGaaGOmaiabeo8aZbGa ayjkaiaawMcaaaaa@3E91@ in order to induce binding than purely damping functions can provide, while He requires nearly 2 times this amount of attraction. Figures 9 & 10 below, show the interaction potentials obtained using the simulated scaling functions, given by Eq. (33), compared to the experimental [27] interaction potentials for Ne and He, respectively.

Figure 8a: Point-by-Point Scaling fcn (fs,p-p), Simulated Scaling Function (fs,sim), and Damping fcn (fdI) for Ne Dimer

Figure 8b: Point-by-Point Scaling Fcn (fs,p-p), Simulated Scaling Fcn (fs,sim), and Damping Function (fdI) for He Dimer
Figure 8: (a) The point-by-point scaling function, fsp-p the simulated scaling function, fssim and Wu and Yang’s damping function I, fdI as functions for R>σ. n=1.35605931.
(b) The point-by-point scaling function, fsp-p the simulated scaling function, fssim and Wu and Yang’s damping function I, fdI as functions for R>σ. n=1.16882509.

Figure 9a: Uinteract=UB3LYP-fC6/R6 for Ne Dimer.

Figure 9b: Uinteract=UB3LYP-fC6/R6 for Ne Dimer (Bonding Region).
Figure 9: Uinteract=UB3LYP-fssimC6/R6, Uinteract=UB3LYP-fdIC6/R6 and experimental* potentials for Ne dimer, (b) Enlargement of bonding region in (a) for R>σ. n=1.35605931.

Figure 10a: Uinteract=UB3LYP-fC6/R6 for He Dimer.
Figure 10b: Uinteract=UB3LYP-fC6/R6 for He Dimer (Bonding Region).
Figure 10: Uinteract=UB3LYP-fssimC6/R6, Uinteract=UB3LYP-fdIC6/R6, and experimental* potentials for He dimer, (b) Enlargement of bonding region in (a) for R>σ. n=1.16882509.

The equilibrium internuclear distances and dissociation energies obtained using the simulated scaling function, Wu and Yang’s damping function I, and the experimental [27] values are summarized in Table 1 below.

 Although the dissociation energies obtained using the simulated scaling functions for Ne and He dimers are 0.4% (+0.000015 kcal/mol error) and 8% (+0.000074 kcal/mol error) in error, respectively, the dissociation energy of Ne dimer obtained with Wu and Yang’s damping function I is over 60% in error and this damping function cannot give binding for He dimer at all. The equilibrium geometries for Ne and He dimers obtained using the simulated scaling function are within 0.3% (10 mA) of the experimental bond lengths. Therefore, a scaling function of the type presented in Eq. (33) shows promise for improving interaction potentials compared to those obtained using a purely damping function for the B3LYP density functional method.

Although use of the simulated scaling function given in Eq. (33) has been shown to give significantly improved dissociation energies and equilibrium bond lengths (Table 1) compared to the use of a purely repulsive damping function, it is important to note that the model used to derive it not completely general. For example, while the optimized radii for He and Ne dimers are less than their respective van der Waals radii, as is expected since most of the electron density is contained therein, the optimized value of the radius for He, 1.095 A (n=1.16882509), is greater than that of Ne, 1.0136 A (n=1.35605931). Despite this discrepancy, however, the simplicity of the model presented herein affords field and potential integrals that are analytically tractable, and do provide improved dispersion potentials compared to experiment [27].

Re (A)

De (kcal/mol)

% Error in Re

% Error in De

Ne

Scaling attraction

3.08

0.003815

0.3

0.4

Damping attraction

3.00

0.0015

2.9

60.5

Experimental

3.09

0.0038

He

Scaling attraction

2.96

0.001019

0.0

7.8

Damping attraction

No Binding

No Binding

No Binding

No Binding

Experimental

2.96

0.000945

Table 1: Equilibrium geometry, Re and dissociation energy, De values obtained from addition of the scaled attractive potential to the repulsive B3LYP density functional potential for Ne and He dimers, according to the equation:

U interact = U B3LYP f s sim C 6 R 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadwfada WgaaqcfasaaiGacMgacaGGUbGaaiiDaiaacwgacaGGYbGaaiyyaiaa cogacaGG0baajuaGbeaacqGH9aqpcaWGvbWaaSbaaKqbGeaacaWGcb GaaG4maiaadYeacaWGzbGaamiuaaqcfayabaGaeyOeI0IaamOzamaa DaaajuaibaGaam4CaaqaaiaadohacaWGPbGaamyBaaaajuaGdaWcaa qaaiaadoeadaWgaaqcfasaaiaaiAdaaKqbagqaaaqaaiaadkfadaah aaqabKqbGeaacaaI2aaaaaaaaaa@5153@ (Figures 9 &10 for graphs).

If the shape of the scaling function required to give binding is invariant to differing atom-atom pairs, as is suggested by the systems studied here, then it seems reasonable that an attractive scaling function like that presented in Eq. (33) could be useful for extending these ideas to other systems. In general, however, the exact interaction potentials are only known for the most trivial of chemical systems and since the potential energy function derived in this work must be fit to a known potential, its predictive value is compromised. The potential energy function derived herein is the solution to the six-dimensional volume integral over electron densities, as presented in Eq. (35),

U attract = χ 1 ( r 1 ) χ 2 ( r 2 ) ρ 1 ( r 1 ) ρ 2 ( r 2 ) R 6 d V 1 d V 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyvam aaBaaajuaibaGaamyyaiaadshacaWG0bGaamOCaiaadggacaWGJbGa amiDaaqcfayabaGaeyypa0Zaa8GaaeaadaWcaaqaaiabeE8aJnaaBa aajuaibaGaaGymaaqcfayabaWaaeWaaeaacaWGYbWaaSbaaKqbGeaa caaIXaaajuaGbeaaaiaawIcacaGLPaaacqaHhpWydaWgaaqcfasaai aaikdaaKqbagqaamaabmaabaGaamOCamaaBaaajuaibaGaaGOmaaqc fayabaaacaGLOaGaayzkaaGaeqyWdi3aaSbaaKqbGeaacaaIXaaaju aGbeaadaqadaqaaiaadkhadaWgaaqcfasaaiaaigdaaKqbagqaaaGa ayjkaiaawMcaaiabeg8aYnaaBaaajuaibaGaaGOmaaqcfayabaWaae WaaeaacaWGYbWaaSbaaKqbGeaacaaIYaaajuaGbeaaaiaawIcacaGL PaaaaeaacaWGsbWaaSbaaKqbGeaacaaI2aaajuaGbeaaaaaabeqabi abgUIiYlabgUIiYdGaamizaiaadAfadaWgaaqcfasaaiaaigdaaKqb agqaaiaadsgacaWGwbWaaSbaaKqbGeaacaaIYaaajuaGbeaaaaa@6A75@ (35)

 for the simplified case in which the electron density, ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyWdi haaa@3844@ is distributed over the surface of a sphere. Here, χ( r i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Xdm 2aaeWaaeaacaWGYbWaaSbaaKqbGeaacaWGPbaajuaGbeaaaiaawIca caGLPaaaaaa@3C86@ is the electric polarization response of the electron density at position i to an applied electric field, such that the atomic polarizability is given as Eq. (36),

α i ( r i )= χ i ( r i )d V i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaWKajuaGcq aHXoqydaWgaaqcfasaaiaadMgaaKqbagqaamaabmaabaGaamOCamaa BaaajuaibaGaamyAaaqcfayabaaacaGLOaGaayzkaaGaeyypa0Zaa8 qaaeaacqaHhpWydaWgaaqcfasaaiaadMgaaKqbagqaaaqabeqacqGH RiI8amaabmaabaGaamOCamaaBaaajuaibaGaamyAaaqcfayabaaaca GLOaGaayzkaaGaamizaiaadAfadaWgaaqcfasaaiaadMgaaKqbagqa aaaa@4CCF@ (36)

and the C 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbakaadoeada WgaaqcfasaaiaaiAdaaKqbagqaaaaa@38DE@ term is defined by Eq. (37)

C 6 = 3 π 0 α 1 ( r 1 ) α 2 ( r 2 )dr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaam4qam aaBaaajuaibaGaaGOnaaqcfayabaGaeyypa0ZaaSaaaeaacqGHsisl caaIZaaabaGaeqiWdahaamaapehabaGaeqySde2aaSbaaKqbGeaaca aIXaaajuaGbeaaaKqbGeaacaaIWaaabaGaeyOhIukajuaGcqGHRiI8 amaabmaabaGaamOCamaaBaaabaWaaSbaaKqbGeaacaaIXaaajuaGbe aaaeqaaaGaayjkaiaawMcaaiabeg7aHnaaBaaajuaibaGaaGOmaaqc fayabaWaaeWaaeaacaWGYbWaaSbaaKqbGeaacaaIYaaajuaGbeaaai aawIcacaGLPaaacaWGKbGaamOCaaaa@533F@ (37)

The next logical step towards extending these concepts to a complete, predictive theory is to solve the six-dimensional volume integral over electron densities for the general case where ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyWdi haaa@3844@  is the electron density of an atom or molecular fragment [40]. Andersson et al. [41] have recently shown that the long range C 6 R 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba Gaam4qamaaBaaajuaibaGaaGOnaaqcfayabaaabaGaamOuamaaCaaa beqcfasaaiaaiAdaaaaaaaaa@3AD5@ attractive potential becomes Eq. (38),

U attract = 6e 4 ( 4π ) 3/2 m 1/2 1 | r 1 r 2 | 6 V 1 d 3 r 1 V 2 ρ 1 ( r 1 ) ρ 2 ( r 2 ) ρ 1 ( r 1 ) + ρ 2 ( r 2 ) d 3 r 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamyvam aaBaaajuaibaGaamyyaiaadshacaWG0bGaamOCaiaadggacaWGJbGa amiDaaqcfayabaGaeyypa0ZaaSaaaeaacaaI2aGaamyzaaqaaiaais dadaqadaqaaiaaisdacqaHapaCaiaawIcacaGLPaaadaahaaqabKqb GeaacaaIZaGaai4laiaaikdaaaqcfaOaamyBamaaCaaabeqcfasaai aaigdacaGGVaGaaGOmaaaaaaqcfa4aaSaaaeaacaaIXaaabaWaaqWa aeaacaWGYbWaaSbaaKqbGeaacaaIXaaajuaGbeaacqGHsislcaWGYb WaaSbaaKqbGeaacaaIYaaajuaGbeaaaiaawEa7caGLiWoadaahaaqa bKqbGeaacaaI2aaaaaaajuaGdaWdrbqaaiaadsgadaahaaqabKqbGe aacaaIZaaaaaqaaiaadAfajuaGdaWgaaqcfasaaiaaigdaaeqaaaqc fayabiabgUIiYdGaamOCamaaBaaajuaibaGaaGymaaqcfayabaWaa8 quaeaadaWcaaqaamaakaaabaGaeqyWdi3aaSbaaKqbGeaacaaIXaaa juaGbeaadaqadaqaaiaadkhadaWgaaqcfasaaiaaigdaaKqbagqaaa GaayjkaiaawMcaaiabeg8aYnaaBaaajuaibaGaaGOmaaqcfayabaaa beaadaqadaqaaiaadkhadaWgaaqcfasaaiaaikdaaKqbagqaaaGaay jkaiaawMcaaaqaamaakaaabaGaeqyWdi3aaSbaaKqbGeaacaaIXaaa juaGbeaadaqadaqaaiaadkhadaWgaaqcfasaaiaaigdaaKqbagqaaa GaayjkaiaawMcaaaqabaGaey4kaSYaaOaaaeaacqaHbpGCdaWgaaqc fasaaiaaikdaaKqbagqaamaabmaabaGaamOCamaaBaaajuaibaGaaG OmaaqcfayabaaacaGLOaGaayzkaaaabeaaaaaajuaibaGaamOvaKqb aoaaBaaajuaibaGaaGOmaaqabaaajuaGbeGaey4kIipacaWGKbWaaW baaeqajuaibaGaaG4maaaajuaGcaWGYbWaaSbaaKqbGeaacaaIXaaa juaGbeaaaaa@8C01@ (38)

where the electron densities, ρ i ( r i ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeqyWdi 3aaSbaaKqbGeaacaWGPbaajuaGbeaadaqadaqaaiaadkhadaWgaaqc fasaaiaadMgaaKqbagqaaaGaayjkaiaawMcaaaaa@3E5A@ have been extracted from Hartree-Fock wave functions with Slater basis sets, and leads to the calculation of significantly improved C6 coefficients [41]. The results of the simplified case presented in this work, in which the electron density is distributed over the surface of a sphere, indicate that the strengthening of the C 6 R 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba Gaam4qamaaBaaajuaibaGaaGOnaaqcfayabaaabaGaamOuamaaCaaa beqcfasaaiaaiAdaaaaaaaaa@3AD5@ dispersion potential may be achieved by including the 1 r 1 r 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba GaaGymaaqaaiaadkhadaWgaaqcfasaaiaaigdaaKqbagqaaiabgkHi TiaadkhadaWgaaqcfasaaiaaikdaaKqbagqaaaaaaaa@3D50@ separation term in the integrand, in accord with Eq. (35). Thus, further studies toward this end are worthwhile.

Conclusion

A method for incorporating dispersion forces into density functional theory through the addition of a scaled atom-atom pairwise attractive potential term to the repulsive density functional potential is presented. By considering the deviation from the C 6 R 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba Gaam4qamaaBaaajuaibaGaaGOnaaqcfayabaaabaGaamOuamaaCaaa beqcfasaaiaaiAdaaaaaaaaa@3AD5@ potential arising from the interaction of two spherical, non-point, dipoles, an analytic function was derived that when multiplied by C 6 R 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba Gaam4qamaaBaaajuaibaGaaGOnaaqcfayabaaabaGaamOuamaaCaaa beqcfasaaiaaiAdaaaaaaaaa@3AD5@ resulted in an attractive potential capable of inducing binding. The scaling function found to give bound Ne and He dimers using the repulsive B3LYP density functional potential is quite different from the damping functions used in previous studies, causing the C 6 R 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaaba Gaam4qamaaBaaajuaibaGaaGOnaaqcfayabaaabaGaamOuamaaCaaa beqcfasaaiaaiAdaaaaaaaaa@3AD5@ attraction to be strengthened in the bonding region. The attraction needed to give bound Ne and He dimers was found to be 1.5 and 2 times that which the damped attractive potential could provide, respectively. When the interaction potential for Ne dimer is constructed through the addition of the B3LYP density functional potential to the scaled attractive term, a dissociation energy 0.4% (+0.000015 kcal/mol error) in error is obtained, compared with 60% (-0.0023 kcal/mol error) error using the damped attraction. Similarly, for He dimer, use of the scaled attractive term results in a dissociation energy 8% (0.000074 kcal/mol error) in error; however, binding could not be induced using the damped attraction. These results, obtained by considering the dispersion interaction of two spherically distributed electron densities, are have proven insightful for developing a completely general density functional theory for predicting van der Waals interactions involving the dispersive interaction of electron distributions in atoms and molecular fragments. Through the introduction of dispersion forces, density functional theory, which has become an important tool for studies of covalent chemistry, may be extended to studies of weakly interacting systems. This model has been incorporated into the APF-D (Austin-Petersson-Frisch) Density Functional theory for dispersion forces within the Gaussian 09 Suite under keyword, APFD [42].

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