1. 1 Ionic and Molecular Liquids: Long-Range Forces and Rigid-Body Constraints Charusita Chakravarty Indian Institute of Technology Delhi

2. 2 Electrostatic Interactions Ionic liquids: NaCl, SiO 2, NH 4 Cl mobile charge carriers which are atomic or molecular entities Simple ionic melts: model ions as point charges with Coulombic interactions. Short-range repulsions control ionic radii. Molecular Liquids Electronic charge distributions show significant deviations from spherical symmetry Can be modeled by: (a) multipole moment expansions or (b) arrays of partial charges water benzene

3. 3 Multipole Expansion for Electrostatic Potential Represent the electronic charge distribution of a molecule by a set of multipoles:

4. 4 Range of Electrostatic Interactions TypeRangeEnergy (kJ/mol)Comment Ion-Ion 1/r 250 Ion-dipole 1/r 2 15 Dipole-Dipole 1/r 3 2Static dipoles (solid phase) Dipole-quadrupole 1/r 4 Fixed Orientation / Linear Quadrupole-quadrupole 1/r 5 Fixed Orientation / Linear Long-range interactions: Tail correction will diverge for 1/r n interactions with n greater than or equal to 3; therefore minimum image convention cannot be applied

5. 5 Partial Charge Distributions of Some Typical Molecules 109.47  H O H 0.9572  Å   (+0.52e)   (+0.52e) 2   (- 1.04e) 1.0  Å 109.47  O O H H 2   (-0.8472e)   (+0.4238e)   (+0.4238e) TIP4P Water SPC/E Water Dipole + Quadrupole Quadrupole Molecular Nitrogen

6. 6 Array of Point Charges Coulombic contribution to the potential energy for an array of N charges that form a charge-neutral system: Electrostatic potential Particle i interacts with all other charges and their mirror images but not with itself Gaussian units Cannot apply minimum image convention because sum converges very slowly

7. 7 Electrostatic Potential Charge Distribution Poisson Equation Potential Energy and Forces Linear differential equation :

8. 8 Ewald Summation for Point Charges Co Point Charge Distribution: Converges slowly Screened charge distribution: Converges fast in real space Gaussian compensating charge distribution: can be analytically evaluated in real space

9. 9 Ewald Summation Screening a point charge to convert the long-range Coulombic interaction into a short-range interaction Evaluating the real-space contribution due to the screened charges The Poisson equation in reciprocal space for the compensating screened charge distribution Evaluating the reciprocal space contribution Self-interaction correction

10. 10 Screening a point charge

11. 11 Electrostatic potential due to a Gaussian charge distribution

12. 12

13. 13 Real Space Contribution The value of  must be chosen so that the range of interaction of the screened charges is small enough that a real space cutoff of r c < L/2 can be used and the minimum image convention can be applied

14. 14 Poisson Equation in Reciprocal Space  Fourier series representation of a function in a cubic box with edge length L and volume V under periodic boundary conditions: Fourier coefficients Poisson equation in real space Poisson equation in reciprocal space Reminder: Fourier transforms of derivatives

15. 15 Unit Point Charge at origin: Array of point charges Array of Gaussian charges Green’s Function Fourier transform of smearing Function FT of Point charge Array Convolution of Point Charge distribution and smearing function

16. 16 Fourier Part of Ewald Sum Corresponding Electrostatic potential in reciprocal space Electrostatic potential in real space can be obtained using: Array of N Gaussian point charges with periodic images:

17. 17 Fourier Part of Ewald Sum (contd.) Electrostatic potential in real space Reciprocal Space contribution to potential energy System is embedded in a medium with an inifinite dielectric constant

18. 18 Correction for Self-Interaction Must remove potential energy contribution due to a continuous Gaussian cloudof charge q i and a point charge q i located at the centre of the cloud. Electrostatic potential due to Gaussian centred at origin

19. 19 Coulombic Interaction expressed as an Ewald Sum Reciprocal space Self- Interaction Real Space Important: For molecules, the self-interaction correction must be modified because partial charges on the same molecule will not interact

20. 20 Accuracy of Ewald Summation Convergence parameters: – Width of Gaussian in real space,  – Real space cut-off, r c – Cutoff in Fourier or reciprocal space, k c =2  /Ln c

21. 21 Calculating Ewald Sums for NaCl Na + Cl - Liquid Crystal

22. 22 Hands-on Exercise: Calculating the Madelung Constant for NaCl The electrostatic energy of a structure of 2N ions of charge +/- q is where  is the Madelung constant and r nn is the distance between the nearest neighbours. Structure  Sodium Chloride (NaCl)1.747565 Cesium Chloride (CsCl)1.762675 Zinc blende (cubic ZnS)1.6381

23. 23 Rotational and Vibrational Modes of Water Symmetric Stretch 3657cm -1 http://chsfpc5.chem.ncsu.edu/~franzen/CH795N/lecture/XIV/XIV.html Rotational Constants (cm -1 ) A40.1 B20.9 C13.4 Bend 1595cm -1 Asymmetric Strech 3756cm -1 http://www1.lsbu.ac.uk/water/vibrat.htm Intermolecular vibrations (cm -1 ) Librations800 OO stretch  200 OOO bend  60

24. 24 Molecules: Multiple Time-scales Bonded interactions are much stronger than non-bonded interactions Intramolecular vibrations have frequencies that are typically an order magnitude greater than those of intermolecular vibrations MD/MC: time step will be dictated by fastest vibrational mode Fast, intramolecular vibrational modes do not explore much of configuration space- rapid, essentially harmonic, small amplitude motion about equilibrium geometry Require efficient sampling of orientational and intermolecular vibrational motions

25. 25 Simulation Methods for Molecules Freeze out all or some intramolecular modes: – Serve to define vibrationally averaged molecular structure and are completely decoupled from intermolecular vibrations, librations or rotations – Rigid-Body Rotations: Characterize the mass distribution of the molecule by its moment of inertia tensor MC: Use orientational moves MD: Propagate rigid-body equations of motion Will not work if there are low-frequency vibrational modes – Apply Geometrical Constraints MD: SHAKE MD: RATTLE Multiple Time-Scale Algorithms

26. 26 Rotations of Rigid-Bodies H2H2 N2N2 CH 4 SF 6 NH 3 lCl 5 H2OH2O I x  I y  I z Space-fixed (SF) and Body-fixed (BF) axes (Goldstein, Classical Mechanics) Moments of Inertia of Molecules: I a < I b < I c Linear: Spherical polar angles:  Non-linear: Euler angles:  (Atkins, Physical Chemistry)

27. 27 Monte Carlo Orientational Moves for Linear Molecules Orientation of a linear molecule is specified by a unit vector u. To change it by a small amount: 1.Generate a unit vector v with a random orientation. See algorithm to generate random vector on unit sphere 2.Multiply vector v with a scale factor g, which determines acceptance probability of trial orientational move 3.Create a sum vector: t = u + gv 4.Normalise t to obtain trial orientation

28. 28 Euler Angles (  http://stackoverflow.com/questions http://mathworld.wolfram.com/EulerAngles.html Euler’s rotation theorem: Any rotation of a rigid-body may be described by a set of three angles Rotation, A: Initial orientation of body-fixed axes (X,Y,Z) to final orientation (X’’,Y’’’, Z’)

29. 29 Euler angles (  (contd.) Rotate the X-, Y-, and Z-axes about the Z-axis by  resulting in the X'-, Y'-, and Z-axes. Rotate the X'-, Y'-, and Z-axes about the X'-axis by  resulting in the X'-, Y''-, and Z'-axes. Rotate the X'-, Y''-, and Z'-axes about the Z'-axis by  resulting in the X''-, Y'''-, and Z'-axes. Rotation A = BCD, therefore new coordinates are:

30. 30 Monte Carlo Orientational Moves for Non-linear Molecules Specify the orientation of a rigid body by a quaternion of unit norm Q that may be thought of as a unit vector in four-dimensional space.

31. 31 Applying Geometrical Constraints Lagrangian Equations of Motion Kinetic Energy Potential Energy Cartesian coordinates Geometrical Constraints Define constraint equations and require that system moves tangential to the constraint plane.

32. 32 Introduce a new Lagrangian that contains all the constraints: The equations of motion of the constrained system are: Constraint force acting along coordinate q i

33. 33 Bond Length Constraint for Diatomic Molecule m1m1 m2m2 Bond constraint Constraint forces: lie along bond direction are equal in magnitude opposite in direction do no work Verlet algorithm: New position with constraint Unconstrained position Constraint forces on atom i

34. 34 Diatomic molecule (contd) Three unknowns In the case of a diatomic molecule, can obtain quadratic equation in. However, cannot be conveniently generalised for larger molecules with more constraints.

35. 35 Multiple Constraints Verlet algorithm for N atoms in the presence of l constraints: Constraint force acting on i-th atom due to k-th constraint Taylor expansion of the constraints with respect to unconstrained positions. Impose the requirement that after one time step, the constraints must be satisfied. Substitute from Taylor expansion above

36. 36 Multiple Constraints (contd.) Solution by Matrix Inversion Need to find the unknown Lagrange multipliers: Since the Taylor expansion was truncated at first-order iterative scheme will be required to obtain convergence.

37. 37 SHAKE Algorithm Iterative scheme is applied to each constraint in turn: RATTLE: Similar approach within a velocity Verlet scheme

38. 38 References D. Frenkel and B. Smit, Understanding Molecular Simulations: From Algorithms to Applications A. R. Leach, Molecular Modelling: Principles and Applications D. C. Rapaport, The Art of Molecular Dynamics Simulation (Details of how to implement algorithms for molecular systems) M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids (SHAKE, RATTLE, Ewald subroutines)