CE 530 Molecular Simulation
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1 1 CE 530 Molecular Simulation Lecture 14 Molecular Models David A. Kofke Department of Chemical Engineering SUNY Buffalo
2 2 Review Monte Carlo ensemble averaging, no dynamics easy to select independent variables lots of flexibility to improve performance Molecular dynamics time averaging, yields dynamical properties extended Lagrangians permit extension to other ensembles Models atomic systems only hard sphere, square well Lennard-Jones
3 3 Modeling Molecules Quantitative calculations require more realistic treatment of molecular interactions Quantum mechanical origins Intermolecular forces Intramolecular forces Effects of long-range interactions on properties Multibody interactions
4 4 Quantum Mechanical Origins Fundamental to everything is the Schrödinger equation Ψ HΨ= ih Nuclear coordinates t wave function Ψ( Rrt,, ) H = Hamiltonian operator time independent form HΨ= EΨ h 2 2m i H = K + U = + U Electronic coordinates Born-Oppenheimer approximation electrons relax very quickly compared to nuclear motions nuclei move in presence of potential energy obtained by solving electron distribution for fixed nuclear configuration it is still very difficult to solve for this energy routinely usually nuclei are heavy enough to treat classically
5 5 Force Field Methods Too expensive to solve QM electronic energy for every nuclear configuration Instead define energy using simple empirical formulas force fields or molecular mechanics Decomposition of the total energy N (1) (2) (3) i i i j< i i j i j< i k< j i j k U( r ) = u ( r) + u ( r, r ) + u ( r, r, r ) + K Single-atom energy (external field) Atom-pair contribution 3-atom contribution Neglect 3- and higher-order terms Force fields usually written in terms of pairwise additive interatomic potentials with some exceptions
6 6 Contributions to Potential Energy Total pair energy breaks into a sum of terms N U( r ) = U + U + U + U + U + U + U str bend tors cross vdw el pol Intramolecular only U str stretch U bend bend U tors torsion U vdw van der Waals U el electrostatic U pol polarization U cross cross
7 7 Contributions to Potential Energy Total pair energy breaks into a sum of terms N U( r ) = U + U + U + U + U + U + U str bend tors cross vdw el pol Intramolecular only U str stretch U bend bend U tors torsion U vdw van der Waals U el electrostatic U pol polarization U cross cross
8 8 Contributions to Potential Energy Total pair energy breaks into a sum of terms N U( r ) = U + U + U + U + U + U + U str bend tors cross vdw el pol Intramolecular only U str stretch U bend bend U tors torsion U vdw van der Waals U el electrostatic U pol polarization U cross cross
9 9 Contributions to Potential Energy Total pair energy breaks into a sum of terms N U( r ) = U + U + U + U + U + U + U str bend tors cross vdw el pol Intramolecular only U str stretch U bend bend U tors torsion U vdw van der Waals U el electrostatic U pol polarization U cross cross
10 10 Contributions to Potential Energy Total pair energy breaks into a sum of terms N U( r ) = U + U + U + U + U + U + U str bend tors cross vdw el pol Intramolecular only U str stretch U bend bend U tors torsion U vdw van der Waals U el electrostatic U pol polarization U cross cross
11 11 Contributions to Potential Energy Total pair energy breaks into a sum of terms N U( r ) = U + U + U + U + U + U + U str bend tors cross vdw el pol Intramolecular only U str stretch U bend bend U tors torsion U vdw van der Waals U el electrostatic U pol polarization U cross cross
12 12 Contributions to Potential Energy Total pair energy breaks into a sum of terms N U( r ) = U + U + U + U + U + U + U str bend tors cross vdw el pol Intramolecular only U str stretch U bend bend U tors torsion U vdw van der Waals U el electrostatic U pol polarization Repulsion U cross cross Mixed terms
13 13 Contributions to Potential Energy Total pair energy breaks into a sum of terms N U( r ) = U + U + U + U + U + U + U str bend tors cross vdw el pol Intramolecular only U str stretch U bend bend U tors torsion U vdw van der Waals U el electrostatic U pol polarization Repulsion U cross cross Mixed terms
14 14 Contributions to Potential Energy Total pair energy breaks into a sum of terms N U( r ) = U + U + U + U + U + U + U str bend tors cross vdw el pol Intramolecular only U str stretch U bend bend U tors torsion U vdw van der Waals U el electrostatic U pol polarization Repulsion Attraction U cross cross Mixed terms
15 15 Contributions to Potential Energy Total pair energy breaks into a sum of terms N U( r ) = U + U + U + U + U + U + U str bend tors cross vdw el pol Intramolecular only Repulsion U str stretch U vdw van der Waals Attraction U bend bend U el electrostatic U tors torsion U pol polarization U cross cross Mixed terms
16 16 Contributions to Potential Energy Total pair energy breaks into a sum of terms N U( r ) = U + U + U + U + U + U + U str bend tors cross vdw el pol Intramolecular only Repulsion U str stretch U vdw van der Waals Attraction U bend bend U tors torsion U el electrostatic U pol polarization - + U cross cross Mixed terms
17 17 Contributions to Potential Energy Total pair energy breaks into a sum of terms N U( r ) = U + U + U + U + U + U + U str bend tors cross vdw el pol Intramolecular only Repulsion U str stretch U vdw van der Waals u (2) Attraction U bend bend U el electrostatic u (2) U tors torsion U pol polarization u (N) U cross cross Mixed terms
18 18 Contributions to Potential Energy Total pair energy breaks into a sum of terms N U( r ) = U + U + U + U + U + U + U str bend tors cross vdw el pol Intramolecular only Repulsion U str stretch U vdw van der Waals u (2) Attraction U bend bend U el electrostatic u (2) U tors torsion U pol polarization u (N) + - U cross cross Mixed terms
19 19 Stretch Energy Expand energy about equilibrium position 2 o du o d U o dr o r= r dr o r= r Ur ( ) = Ur ( ) + ( r r ) + ( r r ) +K define minimum 2 (neglect) o 12 = Ur ( ) kr ( r ) Model fails in strained geometries 2 harmonic better model is the Morse potential 12 ( ) Ur ( ) = D1 e αr 12 2 Energy (kcal/mole) Morse dissociation energy force constant Stretch (Angstroms)
20 20 Bending Energy Expand energy about equilibrium position 2 o du o d U o dθ o 2 θ= θ dθ o θ= θ U( θ) = U( θ ) + ( θ θ ) + ( θ θ ) +K define minimum 2 θ (neglect) U( θ) = k( θ θ ) improvements based on including higher-order terms Out-of-plane bending o 2 harmonic u (4) o U( χ) = k( χ χ ) 2 χ
21 21 Two new features periodic Torsional Energy weak (Taylor expansion in f not appropriate) Fourier series U ( φ ) = U n 1 n cos( nφ ) = terms are included to capture appropriate minima/maxima depends on substituent atoms e.g., ethane has three mimum-energy conformations n = 3, 6, 9, etc. depends on type of bond e.g. ethane vs. ethylene usually at most n = 1, 2, and/or 3 terms are included φ
22 22 Van der Waals Attraction Correlation of electron fluctuations Stronger for larger, more polarizable molecules CCl 4 > CH 4 ; Kr > Ar > He Theoretical formula for long-range behavior U att vdw : C r O( r ) Only attraction present between nonpolar molecules reason that Ar, He, CH 4, etc. form liquid phases a.k.a. London or dispersion forces
23 Van der Waals Repulsion Overlap of electron clouds Theory provides little guidance on form of model Two popular treatments inverse power rep A exponential typically n ~ 9-12 Combine with attraction term Lennard-Jones model U vdw : r n Exp-6 two parameters A C 10 Br C LJ U = U = Ae r 12 r 6 Exp-6 6 r 8 U rep vdw : Ae Br a.k.a. Buckingham or Hill x Beware of anomalous Exp-6 short-range attraction Exp-6 repulsion is slightly softer
24 Electrostatics Interaction between charge inhomogeneities Modeling approaches point charges point multipoles Point charges assign Coulombic charges to several points in the molecule total charge sums to charge on molecule (usually zero) Coulomb potential qq i j Ur () = 4πε r 0 very long ranged Lennard-Jones Coulomb 4
25 Electrostatics 2. At larger separations, details of charge distribution are less important Multipole statistics capture basic features! Vector Dipole Quadrupole Octopole, etc. Point multipole models based on long-range behavior dipole-dipole u dd i i µ = q i r i Q = dipole-quadrupole µ 1Q2 ˆ µ 2 ˆ µ ˆ µ quadrupole-quadrupole q i r i r i Tensor µµ = 3 r 1 2 [ 3( ˆ µ rˆ)( ˆ µ rˆ) ( ˆ µ ˆ µ )] ( ) µ 0, Q = 0 µ = 0, Q 0 µ µ Q u 3 dq = ( 4 1 rˆ) 5( Q ˆ 2 rˆ) 1 2( 1 2)( Q ˆ 2 rˆ) 2 r 3 QQ u 1 2 QQ = 1 5c 5c + 2c + 35c c 20cc c 4 5 r Axially symmetric quadrupole Q 25
26 Electrostatics 3. Some Experimental/Theoretical Values Molecule µ, Debye Q, B α, A 3 He Ar O N Cl HF CO H 2 O (xx) (yy) (zz) 1.5 (xx) 1.43 (yy) 1.45 (zz) CH CCl C 6 H NH C 2 H
27 27 Polarization Charge redistribution due to influence of surrounding molecules dipole moment in bulk different from that in vacuum Modeled with polarizable charges or multipoles Involves an iterative calculation evaluate electric field acting on each charge due to other charges adjust charges according to polarizability and electric field re-compute electric field and repeat to convergence Re-iteration over all molecules required if even one is moved
28 28 Explicit Multibody Interactions Axilrod-Teller u (3) consider response of atoms 2 and 3 to fluctuation in dipole moment of atom 1 average over all fluctuations in 1 3 Eαα α u( r, r, r ) = 3cos cos cos + 1 ( θ θ θ ) r12r23r13 1 θ 1 2 3
29 29 Unlike-Atom Interactions Mixing rules give the potential parameters for interactions of atoms that are not the same type Ur () = no ambiguity for Coulomb interaction 4πε0r for effective potentials (e.g., LJ) it is not clear what to do Lorentz-Berthelot is a widely used choice 1 12 = σ ( σ σ ) ε = ε ε qq i j Treatment is a very weak link in quantitative applications of molecular simulation
30 Common Approximations in Molecular Models 30 Rigid intramolecular degrees of freedom fast intramolecular motions slow down MD calculations Ignore hydrogen atoms united atom representation Ignore polarization expensive n-body effect Ignore electrostatics Treat whole molecule as one big atom maybe anisotropic Model vdw forces via discontinuous potentials Ignore all attraction Model space as a lattice especially useful for polymer molecules Qualitative models
31 31 Summary Intermolecular forces arise from quantum mechanics too complex to include in lengthy simulations of bulk phases Empirical forms give simple formulas to approximate behavior intramolecular forms: bend, stretch, torsion intermolecular: van der Waals, electrostatics, polarization Unlike-atom interactions weak link in quantitative work
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