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simulating matter with molecular dynamics 42710Lennard-Jonesu(r)repulsive00.8 1 1.2attraction1.4 1.6 1.8 2rFigure 16.2 The Lennard-Jones potential. Note the sign change at r = 1 and the minimum atr ≃ 1.1225 (natural units). Note too that because the r axis does not extend to r = 0, the veryhigh central repulsion is not shown.Here the parameter ɛ governs the strength of the interaction, and σ determinesthe length scale. Both are deduced by fits to data, which is why this is called a“phenomenological” potential.Some typical values for the parameters, and corresponding scales for the variables,are given in Table 16.1. In order to make the program simpler and to avoidunder- and overflows, it is helpful to measure all variables in the natural unitsformed by these constants. The interparticle potential and force then take the forms[ 1u(r)=4r 12 − 1 ]r 6 , f(r)= 48 [ 1r r 12 − 1 ]2r 6 . (16.8)The Lennard-Jones potential is seen in Figure 16.2 to be the sum of a long-rangeattractive interaction ∝ 1/r 6 and a short-range repulsive one ∝ 1/r 12 . The changefrom repulsion to attraction occurs at r = σ. The minimum of the potential occursat r =2 1/6 σ =1.1225σ, which would be the atom–atom spacing in a solid bound bythis potential. The repulsive 1/r 12 term in the Lennard-Jones potential (16.6) ariseswhen the electron clouds from two atoms overlap, in which case the Coulombinteraction and the Pauli exclusion principle keep the electrons apart. The 1/r 12term dominates at short distances and makes atoms behave like hard spheres. Theprecise value of 12 is not of theoretical significance (although it’s being large is)and was probably chosen because it is 2 × 6.The 1/r 6 term dominates at large distances and models the weak van der Waalsinduced dipole–dipole attraction between two molecules. 3 The attraction arisesfrom fluctuations in which at some instant in time a molecule on the right tendsto be more positive on the left side, like a dipole ⇐. This in turn attracts the negativecharge in a molecule on its left, thereby inducing a dipole ⇐ in it. As long as3 There are also van der Waals forces that cause dispersion, but we are not considering thosehere.−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 427