1. D G Kanhere, Center for Simulation and modeling Pune University, Mastani School- IISER Pune, July 2014

2.  What is Molecular Dynamics (An Overview)  Ab Initio MD  Born Oppenheimer Dynamics  Car Parrinello Dynamics  General Comment on “ How to set up dynamics run”  Thermodynamics of clusters  Multiple histogram methods  Results of clusters – As Examples

3. System  Many particles interacting with each other  Solid/liquid : Box of Volume V, Temperature T and Pressure P: Periodic BC  Finite size systems clusters :Free BC  Given the forces acting on all the ions, and initial state at time t=0, Compute  Trajectories of all the particles as a function of time t, by using Newton's laws: Essentially exact  We have entire phase space trajectories! Therefore all the information to compute various statistical quantities from microscopic description.

5. Molecular Dynamics

14.  Consider N particles interacting VIA Lennard Jones Potential (N ~ 10-13)  For given co-ordinates {Ri}write down total energy : V ( R1,R2,…)  Compute Force on atom i.  Write a subroutine which takes coordinates as input and returns total potential energy and forces on each of the atoms.  You are now ready to do simulation experiments after one more step.

15. Verlet’s algorithm How to integrate Newton’s equations

16. Initialize: select starting atomic positions and velocities as close as possible to thermal equilibrium Integrate: compute all forces and determine new positions using Verlet’s algorithm Equilibrate: let the system loose memory of the initial configurations, i.e. let it reach thermal equilibrium Average: accumulate averages of observables of interest (A) Average in Molecular Dynamics = Average in Statistical Mechanics A typical molecular dynamics protocol

25. Sum of pair interactions? How to treat chemical complexity? Where are the electrons? Molecular Dynamics

26. A very simple choice: But the interaction between two atoms is partially mediated by the presence of a third atom… I J K Interatomic potentials

27. “Empirical” potentials Starting point: Pair potentials (Lennard-Jones, Born- Mayer, Coulomb, etc) + three-body corrections + density dependent terms (embedded atom models) + atomic distorsion terms (includes polarization) + charge transfer terms I J K Parameters are determined from “empirical” data, such as experimental EOS, vibrations, phase diagrams, dynamical properties, etc.

28. Quantum simulations: The “standard model” H  E  Schroedinger equation for electrons “Molecular dynamics” for atoms Ma = F = -dE/dR e - -e - interactions: Density Functional Theory e - -nuclei interactions: Pseudopotentials “Ab-initio” molecular dynamics = Classical molecular dynamics in the potential energy surface generated by the electrons in their quantum ground state R. Cohen

29. Electrons respond much faster than nuclei to external forces, because of their lighter mass, therefore: Electrons are always in their instantaneous quantum mechanical ground state, for each given {R}. Consequence: Every MD step requires the calculation of the QM ground state of the electrons Where are the electrons? The adiabatic approximation The electronic Hamiltonian H e depends parametrically on the nuclear positions {R} Nuclei e-e- e-e- e-e- e-e- e-e- e-e-

30. Density functional theory Electron-electron (many-body) interaction Density-functional theory [W. Kohn et al. 1964-1965] states that the e-e interaction can be written as a one-electron “effective” term: However, the exact functional form of V xc is not (yet) known. Current approximations to the exact V xc go under different names (LDA, BLYP, BP, GGA, “hybrid”, et al). While GGA and hybrid functionals provide slightly better results than other approximations, the choice of the V xc is often made in such a way to improve agreement with exps (“ab fine”??).

31. The pseudopotential approximation + Core states are irrelevant + Valence states have nodes as a consequence of orthogonality with core states The 1/r Coulomb potential can be replaced by a suitably constructed effective potential, or pseudopotential Electron-ion interaction Valence state Core state

32. “Ab-initio” forces: the Hellmann-Feynman theorem Forces can be calculated without recalculating the ground state wave function for small atomic displacements

33. The Born Oppenheimer Dynamics

34. How to keep electron in the ground state (1) If we expand wavefunctions in a basis set finding the ground state is equivalent to minimizing a quadratic form in the {c}’s (variational principle). So, standard minimization schemes can be used, e.g. steepest descent:

35. Carr- Parrinello dynamics

36. How to keep electron in the ground state (2) {R} The Car-Parrinello algorithm down to the minimum for each {R} ???

37. (i) integrate the equations of motion on the (long) time scale set by the nuclear motion but nevertheless (ii) take intrinsically advantage of the smooth time evolution of the dynamically evolving electronic subsystem as much as possible. The second point allows to circumvent explicit diagonalization or minimization to solve the electronic structure problem for the next molecular dynamics step as it is done in Born-Oppenheimer Molecular dynamics. Car-Parrinello molecular dynamics is an efficient method to satisfy requirement (ii) in a numerically stable fashion and makes an acceptable compromise concerning the length of the time step (i). Car-Parrinello Molecular Dynamics

38. In CPMD a two-component quantum / classical problem is mapped onto a two- component purely classical problem with two separate energy scales at the expense of loosing the explicit time dependence of the quantum subsystem dynamics. Now, in classical mechanics the force on the nuclei is obtained from the derivative of a Lagrangian with respect to the nuclear positions. This suggests that a functional derivative with respect to the orbitals, which are interpreted as classical fields, might yield the force on the orbitals, given a suitable Lagrangian. In addition, possible constraints within the set of orbitals have to be imposed, such as e.g. orthonormality (or generalized orthonormality conditions that include an overlap matrix).

39. Car-Parrinello Lagrangian Car and Parrinello (1985) have postulated the following Lagrangian where  i are fictitious masses and  i are classical fields. Classical action needs to be minimized which results in equation of motions Kinetic EnergyPotential EnergyConstraints

40. Car-Parrinello Equations of Motions Car and Parrinello (1985) have derived equations of motions which are obviously transformed back to Born-Oppenheimer molecular dynamics if fictitious masses for the electrons  i  At the eqilibrium, there are no forces on electrons, therefore Ground state of density functional theory is reached with the eigenvalues being the Kohn Sham eigenstates.

41. Why does the Car-Parrinello Method works? Conserved Energy in CPMD is not a physical energy but supplemented with a small fictitious kinetic term

42. Various Energies extracted from CPMD for a model system. T fict E DFT The fictitious kinetic energy of the electrons is found to perform bound oscillations around a constant, i.e. the electrons do not heat up “ systematically “ in the presence of the nuclei; note that T fict is a measure for deviations from the exact Born- Oppenheimer surface. Closer inspection shows actually two time scales of oscillations: the one visible in the Figure stems from the drag exerted by the moving nuclei on the electrons and is the mirror image of the E DFT fluctuations. As a result the physical energy (the sum of the nuclear kinetic energy and the electronic total energy which serves as the potential energy for the nuclei) is essentially constant on the relevant energy and time scales.

43. Given the adiabatic separation and the stability of the propagation, the central question remains if the forces acting on the nuclei are actually the “correct" ones in Car-Parrinello molecular dynamics. As a reference serve the forces obtained from full self-consistent minimizations of the electronic energy at each time step, i.e. Born- Oppenheimer molecular dynamics with extremely well converged wavefunctions.

44. How to control adiabaticity? Since the electronic degrees of freedom are described by much heavier masses than the electronic masses, time step to perform CPMD simulations needs not to be too small as compared to Ehrenfest molecular dynamics. For a system with a gap in the spectrum, the lowest possible frequency of “fictitious” electronic oscillations To guarantee adiabatic separation this frequency should be much larger than the typical phonon energy and/or the gap in the spectrum which would make sure that the electrons follow the nuclei adiabatically. Hence fictitious mass .

45. At the same time, small fictitious mass would imply smaller and smaller time step because maximum fictitious electronic frequency is proportional to the plane-wave cutoff energy As a result a compromise fictitious mass needs to be found in CPMD simulations. For metals gap is zero and zero frequency “fictitious” electronic modes occur in the spectrum overlapping with the phonon spectrum. Thus, a well-controlled Born-Oppenheimer approach can only be recommended

46. Consider variational principle which can be used to find an upper bound for the lowest eigenstates of the hamiltonian using basis set expansion CP Method as dynamical solution of DFT equations CP Method invented a new way to solve Kohn-Sham equations alternative to diagonalization. CPMD offers a way to determine the coefficients without reduction to the eigenvalue problem.

47. Born-Oppenheimer –versus- Car-Parrinello AIMD Born-OppenheimerCar-Parrinello

48. Born-Oppenheimer –versus- Car-Parrinello forces

49. from M Sprik Born-Oppenheimer –versus- Car-Parrinello: summary

50. R. Car and M. Parrinello, Phys. Rev. Lett. 55, 2471 (1985) M. Payne, M. Teter, D. Allan, T. Arias, J. Joannopoulos, Rev. Mod. Phys. 64, 1045 (1992). D. Marx, J. Hutter, "Ab Initio Molecular Dynamics: Theory and Implementation", in "Modern Methods and Algorithms of Quantum Chemistry" (p. 301-449), Editor: J. Grotendorst, (NIC, FZ Jülich 2000) http://www.theochem.ruhr-uni-bochum.de/research/marx/cprev.en.html R. Rousseau and S. Scandolo, “Car-Parrinello Molecular Dynamics”, in “Encyclopedia of Condensed Matter Physics”, edited by G. Bassani, G. Liedl, and P. Wyder, Elsevier, Amsterdam (2005) Ab-initio Molecular Dynamics: bibliography

51. Finite Temperature Properties of finite size systems D G KanhereD G Kanhere Pune UniversityPune University

52. Lindemann Criteria

53. Mean Square Displacements

54. Extracting ionic density of States Constant Temperature Constant Volume Phase space trajectories The Multiple Histogram Method To Extract Density of States

63. Sodium Clusters ● Simplest of atomic clusters ● Jellium model works ( Depends…) ● Nice delocalized charge density ● Magic Clusters at N=8,20,40,58,92,138,… ● Icosahedra for N=13,55,147,… Chacko et al., Phys. Rev. B 71,155407 (2005) Lee et al., J. Chem. Phys. 123,164310 (2005) Lee et al., Phys. Rev. B (2007) Shahab et al., Phys Rev B ( 2007) Gazi et al J Chem Phys ( 2008)

64. The heat Capacities N=40 : peaked N=50 : flat N=55 : very sharp N=58 : peaked but broad BUT Highest melting Point ( T m = 375 K )

65. Sodium Clusters ● Simplest of atomic clusters ● Jellium model works ( Depends…) ● Nice delocalized charge density ● Magic Clusters at N=8,20,40,58,92,138,… ● Icosahedra for N=13,55,147,… Chacko et al., Phys. Rev. B 71,155407 (2005) Lee et al., J. Chem. Phys. 123,164310 (2005) Lee et al., Phys. Rev. B (2007) Shahab et al., Phys Rev B ( 2007) Gazi et al J Chem Phys ( 2008)

66. Application II Thermodynamics Size sensitive specific heats Magic melters Higher than Bulk Melting temperatures

67. The Case of Gallium and Aluminum clusters ● Gallium and aluminum clusters show extreme size sensitivity ● Gallium clusters melt at Higher than their Bulk Tm ● Magic Melters ----- Non - Melter

68. Gallium Clusters: Heat Capacity Experimental Data from Indiana group * N=30 Flat N=31 Peaked * Tm Shifts by 200K @ N=45 And by 350K across the series

69. Calculations Density functional with LDA/GGA Born Oppenheimer MD Delta T 100 au Total simulation time per temp 100 ps or more Soft pseudo potentials, plane waves Extensive search for equilibrium geometries. Simulated annealing, Basin Hopping …

70. The MD generated data

71. The Heat Capacity : Ga 30 - Ga 31 Joshi et al. Phys. Rev. Lett. 96,135703 (2006)

72. The MSD : Ga 30 and Ga 31

73. Finite Size Effect : Amorphous Clusters In amorphous cluster each atom may have different environment. Atoms may be bonded with the rest of the cluster with different strength. When heated, they will begin diffusive motion at different temperatures This may result in continuous phase change. No sharp peak in specific heat, very broad transition. Cluster with large island having local order will show a peak in the heat capacity..Most of the atoms “melt” together.

74. Thermodynamics: Ga27Si3 Ghazi and Kanhere J Phys Chem (2012)

75. Ga 13 – Ga 12 C and Al 13 - Al 12 C Ga 13 is Decahedra Ga 12 C is a Perfect Icosahedra

76. Calculated Specific Heat Ga

77. Specific Heat