CE 530 Molecular Simulation

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1 CE 530 Molecular Simulation Lecture 7 Beyond Atoms: Simulating Molecules David A. Kofke Department of Chemical Engineering SUNY Buffalo

2 Review Fundamentals units, properties, statistical mechanics Monte Carlo and molecular dynamics as applied to atomic systems simulating in various ensembles biasing methods for MC Molecular models for realistic (multiatomic) systems inter- and intra- atomic potentials electrostatics Now examine differences between simulations of monatomic and multiatomic molecules

3 Truncating the Potential Many molecular models employ point charges for electrostatic interactions - Potential-truncation schemes must be careful not to split the charges For a 9Å truncation distance, using water-like charges, the interaction energy for a molecule with bare charge is (huge) Always use cutoff based on molecule separation, not atom for large molecules, OK to split molecule but do not split subgroups

4 4 Volume-Scaling Moves Scaling atom displacements leads to large strain on intramolecular bonds Instead perform volume scaling moves using molecule centers-ofmass (or something similar) Let R i be COM of molecule i R i = atoms on i () i irj q j (i) be position of atom j w.r.t. R i () i () i j j For size scaling s, L new = sl old Acceptance based on change in m q = r R () i () i j (new) = s i j r R q i L Number of molecules β ( U PV) N lnv(not atoms) m

5 Rigid vs. Nonrigid Molecules MC and MD can be performed on molecules as already described MD moves advance atom positions based on current forces MC moves translate atoms and accepts based on energy change both are done considering inter- and intra-molecular forces limiting distribution has same form π / N N N β pi mi βu( r ) = dp dr e e N = Number of atoms 3N h 5 if this is all that is done, there is nothing more to say Often it is much more efficient to use a rigid-bond model MD integration then doesn t have to deal with fast intramolecular dynamics, so a larger time step can be used MC can sample configurations more efficiently using rigid-body moves (even if model does not have rigid bonds) but much care is needed to do this properly

6 Molecule Coordinate Frame 6 Molecule-frame coordinates are defined w.r.t. molecule COM with molecule in a reference orientation Simulation-frame coordinate is determined by molecule COM R and orientation ω q j R i For rigid molecules, the molecule-frame coordinates never change

7 Orientation. 7 Orientation described in terms of rotation of molecule frame y Direction cosines can be used to describe rotation Relation between same point in two frames Rotation matrix α = e e α = e e x x x y β = e e β = e e y x y y x α α x = y β β y α α A = β β = r Ar Kinematics of rigid-molecule rotation described in terms of rotation of the molecule coordinate frame (i.e. the direction cosines) r' never changes in a rigid molecule y β y y α x x α y x x x = α x α y x

8 8 Orientation. We also need to invert the relation get the simulation-frame coordinate from the molecule frame value T r = A r = A r Direction cosines are not independent in D, all can be described by just one parameter use rotation angle θ cosθ sinθ A = sinθ cosθ cosθ sinθ = sinθ cosθ inverse can be viewed as replacing θ with -θ A y y θ x x

9 9 x Euler Angles The picture in 3D is similar: (x, y, z) à (x', y', z') Nine direction cosines Three independent coordinates specify orientation Euler angles are the conventional choice ω = φθψ z three rotations give the simulation-frame orientation ζ φ ξ y ζ θ ξ η η z ψ ξ y x η z y x

10 0 3D Rotation Matrix Rotation matrix expressed in terms of Euler angles = r Ar cosψ cosφ cosθsinφsinψ cosψ sinφ cosθ cosφsinψ sinψ sinθ A = sinψ cosφ cosθsinφcosψ sinψ sinφ cosθ cosφcosψ cosψ sinθ sinθsinφ sinθ cosφ cosθ To get space-fixed coordinate, multiply molecule-fixed vector by A - again, A - = A T T r= A r

11 Transforming Coordinates. Consider a simple diatomic positions of two atoms described by x, y, z, x, y, z L z y can instead describe by molecule COM and stretch/orientation coordinates x XYZLθφ,,,,, in molecule frame, each atom position is given by r = Le z r = Le z

12 Transforming Coordinates. To get space-fixed coordinates, use rotation matrix T r = R A r R ( XYZ,, ) T r = Le L r = R A r z r = Le z z y matrix T A cosφ sinφ sinθsinφ = cosθsinφ cosθ cosφ sinθ cosφ 0 sinθ cosθ x the result is x = X Lsinθsinφ x = X Lsinθsinφ y = Y Lsinθcosφ y = Y Lsinθcosφ z = Z Lcosθ z = Z Lcosθ

13 Transforming Coordinates 3. 3 The ensemble distribution for the transformed coordinates is obtained via the Jacobian x y z x y z π = Qh Qh 3N 3N the elements of J are the derivatives X Y Z L φ θ For this transformation But we also need to transform the momenta i N N β p / m βu( r ) dp dr e i i dp dq e e = J N N β p / m βu( q ) J i e N = 8L sinθ N J αβ = r α q β 0 0 sinθ sinφ Lsinθ cosφ Lcosθ sinφ 0 0 sinθ cosφ Lsinθ sinφ Lcosθ cosφ 0 0 cosθ 0 Lsinθ 0 0 sinθ sinφ Lsinθ cosφ Lcosθ sinφ 0 0 sinθ cosφ Lsinθ sinφ Lcosθ sinφ 0 0 cosθ 0 Lsinθ x = X Lsinθ sinφ y = Y Lsinθ cosφ z = Z Lcosθ x = X Lsinθ sinφ y = Y Lsinθ cosφ z = Z Lcosθ

14 Transforming Coordinates 4. 4 Begin with the Lagrangian in the original coordinate system L = K U = (!x!y!z!x!y!z ) U (x, y,z,x, y,z ) =!r i U (r) transform to new coordinates assume m=!x = x X X! x Y Y! x Z Z! x L L! x θ θ! x φ φ!!y = etc. in general!r i =!r i = α β r i q α!q α α =!q g i!q!r i =!q G!q i r i r i!q q α q α!q β β r = x y z x y z r = q = X Y Z L θ φ

15 5 Transforming Coordinates 5. Derive momenta L =!q G!q U (q) p α L!q α = The Hamiltonian is β G aβ!q β p = G!q!q = G p!q G!q = p G p Uses G and thus G - are symmetric H = K( pq, ) U( q) = p G p U( q) The Jacobian for the momentum transformation is the reciprocal of the Jacobian for the coordinate transformation π = dp N Qh 3N q dq N J J e β p q G p q e βu (qn ) we don t have to worry about the Jacobian with the full transform

16 6 Integrating Over Momenta If we integrate out the momentum coordinates, the Jacobian again arises π ( pq, ) = π ( q) = = Qh Qh Qh 3N 3N 3N For the diatomic, this term is N N N β p G p βu( q ) dp dq e N N βu( q ) N dq e N dq e N βu( q ) / e dp e β p G p / G = J =cl sinθ c = constant G In MC simulation, the terms must be included in the construction of the transition-probability matrix

17 7 Averages with Constraints. Very stiff coordinates are sometimes treated as rigidly constrained e.g., the bond length L in the diatomic may be held at a constant value MC and MD have different ways to enforce this constraint Regardless of simulation technique, the constrained-system ensemble average may differ from the unconstrained value even when compared to the limit of an infinitely stiff bond! Why the difference? A rigid constraint implies no kinetic energy in vibration Examine the Lagrangian s = soft coordinate L = r i r i q q α q α q β U (q S ; L) β β Lα L =!qs G S!q S U (q S ; L)

18 8 Averages with Constraints. The Jacobian for the coordinate transform is the same as for the unconstrained average But the momentum Jacobian no longer has the term for the constrained coordinate Thus, in general, the distribution of unconstrained coordinates differs N l N l s qs = 3N l s s Qh The difference is N l βu( qs ) N l s = 3N l s s Qh Qh 3N l βu( q ) N s Gs ps βu qs π ( p, ; L) dp dq e e π ( q ; L) dq e dp e = dq N l s e β p N N s π ( qs) Gs = π ( q ; L) G s G / s ( ) β p G p

19 9 Averages with Constraints 3. To get correct (unconstrained-system) averages from a simulation using constraints, averages should be multiplied by this factor M unconstained = M G s G constrained Evaluating this quantity could be tedious but there is a simplification the ratio of determinants (of N-by-N and (N-l)-by-(N-l) matrices) can be given in terms of the determinant of an l-by-l matrix G s G = H a β i ri ri for the diatomic with L constrained, H = H αβ = σ σ

20 0 Rotational Dynamics For completely rigid molecules, only translation and rotation are performed Translational dynamics uses methods described previously, but now applied to the COM Rotational dynamics must consider angular velocities and accelerations Can treat via rotation of the molecule-frame coordinates in the spaced-fixed frame!e s =!ω e s Angular velocity changes in angular velocity are given via torque on molecule

21 Quaternions Rate of change of the Euler angles looks like this A problem arises when θ is near 0 no physical significance, but very inconvenient to integration of equations of motion Quaternions can be used to circumvent the problem describe orientation with 4 (non-independent) variables rotation matrix, equations of motion simply expressed in terms of these quantities note: s sinφ cosθ s cosφ cosθ s!φ = ω x ω sinθ y ω sinθ z 0 3 = q q q q q 0 cos θ φ ψ = cos sin θ φ ψ = cos 3 sin θ φ ψ = sin φ ψ 4 cos θ = sin q q q

22 Monte Carlo Rotations MC simulations of molecules include rotation moves must do this to sample orientations of rigid molecules not strictly necessary for non-rigid molecules, but very helpful very easy to do this incorrectly Trial rotation of a linear molecule Let present orientation be given by vector u Generate a unit vector v with random orientation Let new trial orientation be given by new old u = u γ v where γ is a fixed scale factor that sets the size of the perturbation Nonlinear molecule same procedure, but do perturbation on the 4-dimensional vector of quaternions

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