week2.ppt - Week 2 Basics of MD Simulations Lecture 3 Force...

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Week 2 Basics of MD Simulations Lecture 3: Force fields (empirical potential functions among atoms) Boundaries and computation of long-range interactions Lecture 4: Molecular mechanics (T=0, energy minimization) MD simulations (integration algorithms, constrained dynamics, ensembles)

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Empirical force fields (potential functions) Problem: Reduce the ab initio potential energy surface among the atoms j i k j i k j i j i i U U U ) , , ( ) , ( 3 2 R R R R R R To a classical many-body interaction in the form i e j i j i j i i E e z z U R R R R 2 Such a program has been tried for water but so far it has failed to produce a classical potential that works. In strongly interacting systems, it is difficult to describe the collective effects by summing up the many-body terms. Practical solution: Truncate the series at the two-body level and assume (hope!) that the effects of higher order terms can be absorbed in U 2 by reparametrizing it.
i i i i i i i j i j i U R R R q q U ) 0 ( 3 2 1 ˆ ) ˆ ( 3 1 , E E p R R p E E p R R pol pol Coul Interaction of two atoms at a distance R can be decomposed into 4 pieces 1. Coulomb potential (1/R) 2. Induced polarization (1/R 2 ) 3. Dispersion (van der Waals) (1/R 6 ) 4. Short range repulsion (e R/a ) The first two can be described using classical electromagnetism 1. Coulomb potential: 2. Induced polarization: Dipole field Total polarization int. (Initial and final E fields in iteration) Non-bonded interactions

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a R Ae U / sr 2 / ) ( j i ij j i ij 6 2 4 3 R U  disp Here corresponds to the depth of the potential at the minimum; 2 1/6 σ Combination rules for different atoms: The last two interactions are quantum mechanical in origin. Dispersion is a dipole-dipole interaction that arises from quantum fluctuations (electronic excitations to non-spherical states) Short range repulsion arises from Pauli exclusion principle (electron clouds of two atoms avoid each other), so it is proportional to the electron density The two interactions are combined using a 12-6 Lennard-Jones (LJ) potential 6 12 4 R R U LJ
12-6 Lennard-Jones potential (U is in kT, r in Å) 3 3.5 4 4.5 5 -0.5 0 0.5 1 1.5 2 2.5 r U(r) A kT 3 , 4 1

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Because the polarization interaction is many-body and requires iterations, it has been neglected in most force fields. The non-bonded interactions are thus represented by the Coulomb and 12-6 LJ potentials. Model R O-H (Å) HOH q H (e) (kT) (Å) (D) (T=298 C) SPC 1.0 109.5 0.410 0.262 3.166 2.27 65±5 TIP3P 0.957 104.5 0.417 0.257 3.151 2.35 97±7 Exp. Gas 1.86 80 Ab initio Water 3.00 SPC: simple point charge TIP3P: transferable intermolecular potential with 3 points Popular water models (rigid)
There are hundreds of water models in the market, some explicitly including polarization interaction. But as yet there is no model that can describe all the properties of water successfully.

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• Spring '14
• SaibalMitra

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