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426 chapter 16+ +Figure 16.1 The molecule–molecule effective interaction arises from the many-bodyinteraction of the electrons and nucleus in one electronic cloud with the electrons andnucleus in another electron cloud. (The size of the nucleus at the center is highlyexaggerated relative to the size of the molecule, and the electrons are really just points.)we have assumed a conservative potential, the total energy of the system, that is, thepotential plus kinetic energies summed over all particles, should be conserved overtime. Nonetheless, in a practical computation we “cut the potential off” [assumeu(r ij )=0] when the molecules are far apart. Because the derivative of the potentialproduces an infinite force at this cutoff point, energy will no longer be preciselyconserved. Yet because the cutoff radius is large, the cutoff occurs only when theforces are minuscule, and so the violation of energy conservation should be smallrelative to approximation and round-off errors.In a first-principles calculation, the potential between any two argon atoms arisesfrom the sum over approximately 1000 electron–electron and electron–nucleusCoulomb interactions. A more practical calculation would derive an effectivepotential based on a form of many-body theory, such as Hartree–Fock or densityfunctional theory. Our approach is simpler yet. We use the Lennard–Jones potential,[ (σ ) 12 ( σ) ] 6u(r)=4ɛ − ,r r(16.6)f(r)=− dudr = 48ɛ [ (σ ) 12 1( σ) ] 6−r 2r.r 2 r(16.7)TABLE 16.1Parameter Values and Scales for the Lennard-Jones PotentialQuantity Mass Length Energy Time TemperatureUnit m σ ɛ√mσ2 /ɛɛ/k BValue 6.7 × 10 −26 kg 3.4 × 10 −10 m 1.65 × 10 −21 J 4.5 × 10 −12 s 119 K−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 426