1. MODELING MATTER AT NANOSCALES 3. Empirical classical PES and typical procedures of optimization 3.05. Molecular Dynamics

2. Second Newton’s Law in Nanoworld

3. Second Newton’s law and Hypersurfaces For an isolated system, the translational motion of a given n particle in a container of N particles is described by the second Newton’s law: being the force on the n particle expressed in terms of the potential originated by all particles in the system: where t is the time, m n the mass of n and r n the position vector of any particle.

4. Second Newton’s law and Hypersurfaces For an isolated system, the translational motion of a given n particle in a container of N particles is described by the second Newton’s law: being the force on the n particle expressed in terms of the potential originated by all particles in the system: where t is the time, m n the mass of n and r n the position vector of any particle. Thus, F i is the net force acting on a given particle i, crated by a potential V of interaction with all other particles of the system in an instant dt.

5. Second Newton’s law and Hypersurfaces If we assume that: 1.- The potential can be computed from all pairs of particles and that it is additive;

6. Second Newton’s law and Hypersurfaces If we assume that: 1.- The potential can be computed from all pairs of particles and that it is additive; 2.- There are no external forces

7. Second Newton’s law and Hypersurfaces If we assume that: 1.- The potential can be computed from all pairs of particles and that it is additive; 2.- There are no external forces Then:

8. Second Newton’s law and Hypersurfaces If we assume that: 1.- The potential can be computed from all pairs of particles and that it is additive; where and the force acting on each particle is expressed as: 2.- There are no external forces Then:

9. Second Newton’s law and Hypersurfaces Thus, the problem is reduced to integrate a differential equation of motion as: 9 corresponding to each particle of mass m n and a position in space given by the vector r n

10. Second Newton’s law and Hypersurfaces Thus, the problem is reduced to integrate a differential equation of motion as: 10 corresponding to each particle of mass m n and a position in space given by the vector r n that is developing a potential energy u nl with each one of the other particles m of the system, found at r nl distances.

11. Second Newton’s law and Hypersurfaces Therefore, the requirements for carrying out a simulation of molecular dynamics are: -Selecting a hypersurface or potential energy function, expressing reliable u nl (r nl ) energies between individual particles. 11

12. Second Newton’s law and Hypersurfaces Therefore, the requirements for carrying out a simulation of molecular dynamics are: -Selecting a hypersurface or potential energy function, expressing reliable u nl (r nl ) energies between individual particles. 12 -Solving the equation of motion in a way that given positions and speeds of all and each particles in a given time t, all positions and speeds can be also obtained for a time t +  t where the interval  t is small and limited.

13. Integration of the Equations of Motion and Temporary Parameters 13

14. Equation of motion Requirements: Numerical integration of the equations of motion for hundreds or thousands of atoms or molecules. 14

15. Equation of motion Requirements: Numerical integration of the equations of motion for hundreds or thousands of atoms or molecules. Approximation: Exists a compromise between the quality of results and the time interval size for successive evaluation of integrals, called as  t. 15

16. Equation of motion A common method for integration is the steepest descent procedure where the solution is represented by the values of a finite number of points in the space of each system, named as “grid points” of the equation: 16

17. Equation of motion A common method for integration is the steepest descent procedure where the solution is represented by the values of a finite number of points in the space of each system, named as “grid points” of the equation: 17 Such points are simulated by a polynomial and the complete solution is obtained after the evaluation of the derivative: with different values of variables obtained by the polynomial in each one of the points of the “grid”.

18. Translational motion: The Gear’s Predictor – Corrector Algorithm Integration is carried out in two successive processes: 1.- The position of the center of mass and its derivatives is “predicted” for time t +  t from the actual position and the derivatives in the previous step at t, according to a truncated Taylor’s series of order m : 18

19. Translational motion: The Gear’s Predictor – Corrector Algorithm 19 2.- Derivatives of all orders are “corrected” by the forces F (or negative gradients) that were computed in the predicted positions of the step 1: where the values of the constants depend on the order and can be found in literature.

20. Translational motion: The Verlet Algorithm It uses positions and accelerations at a time t and in a previous moment t -  t in order to predict positions at time t +  t. The scheme of integration is based in a Taylor expansion to the 3rd. order: 20 and the speeds can be obtained by the basic rule of differentials:

21. Translational motion: The “Leap Frog” Verlet Algorithm It tries the Verlet algorithm for calculations of speeds more precisely: 21

22. Translational motion: The “Leap Frog” Verlet Algorithm It tries the Verlet algorithm for calculations of speeds more precisely: 22  Accelerations at time t are calculated from coordinates in that moment.

23. Translational motion: The “Leap Frog” Verlet Algorithm It tries the Verlet algorithm for calculations of speeds more precisely: 23  Accelerations at time t are calculated from coordinates in that moment.  Speeds are calculated at the mean positive interval of the first equation and are substituted in the second to obtain the positions at time t +  t.

24. Translational motion: The “Leap Frog” Verlet Algorithm It tries the Verlet algorithm for calculations of speeds more precisely: 24  Accelerations at time t are calculated from coordinates in that moment.  Speeds are calculated at the mean positive interval of the first equation and are substituted in the second to obtain the positions at time t +  t.  Speeds at time t are calculated as the mean of the speed at t -  t and t +  t.

25. Selection of time intervals for the simulation If the time intervals  t are “large” (i.e. several femto seconds, or fs), molecular dynamics simulations can be more comprehensive in the exploration of the configuration’s space of the system. 25

26. Selection of time intervals for the simulation If the time intervals  t are “large” (i.e. several femto seconds, or fs), molecular dynamics simulations can be more comprehensive in the exploration of the configuration’s space of the system. 26 However, the integrated equations for translation oblige forces to be constant in each step. This fact compromises the lasting of the simulation with the molecular internal motions originating dynamics in the physical reality, which are mostly molecular vibrations.

27. Selection of time intervals for the simulation The size of the time interval  t is limited by the maximal frequency of vibrations among the component particles, and must be: 27

28. Selection of time intervals for the simulation The size of the time interval  t is limited by the maximal frequency of vibrations among the component particles, and must be: 28 For a typical stretching vibrational mode corresponding to a non – associated O-H bond [~ 3500 cm -1 = 1.05 (10 14 ) s -1 ],  t cannot be larger than --1 = 9.5 (10 -15 ) s, or 9.5 fs. In this case,  t uses to be selected of 1 fs.

29. Selection of time intervals for the simulation The size of the time interval  t is limited by the maximal frequency of vibrations among the component particles, and must be: 29 For a typical stretching vibrational mode corresponding to a non – associated O-H bond [~ 3500 cm -1 = 1.05 (10 14 ) s -1 ],  t cannot be larger than --1 = 9.5 (10 -15 ) s, or 9.5 fs. In this case,  t uses to be selected of 1 fs. Then, if the bond modes [3700 to 400 cm -1 or 1.11(10 14 ) s -1 to 1.20(10 13 ) s -1 ] are excluded and, for example, only the internal rotational modes (torsional) are considered in a certain case [~ 200 cm -1 = 6.00(10 12 ) s -1 ],  t can easily be smaller than --1 = 166.8 (10 -13 ) s, or 166.8 fs. These are the cases of the so – called constrained molecular dynamics and  t could reach 2 fs and even more.

30. Comparisons of methods  Small  t values for each step favors the Gear’s algorithm because presents smaller fluctuations of the total energy than the Verlet’s. 30

31. Comparisons of methods  Small  t values for each step favors the Gear’s algorithm because presents smaller fluctuations of the total energy than the Verlet’s.  Intermediate  t values for each step give similar behaviors for both methods. 31

32. Comparisons of methods  Small  t values for each step favors the Gear’s algorithm because presents smaller fluctuations of the total energy than the Verlet’s.  Intermediate  t values for each step give similar behaviors for both methods.  Large  t values for each step favor the Verlet’s algorithm. 32

33. Integration of molecular rotation Rotational molecular motion can be modeled with the rigid body approximation, where rotation of the main axis is given by the Euler’s equations. They are also integrated with the “leap frog” or Gear’s routines: 33 where I’ s are components of the inertia moment on coordinates x, y, z,  q is the angular speed and  the torque (depending on the applied rotational force and the distance between the point of application and the center) acting on the molecule that originates the motion of such coordinate during the simulation.

34. Stages of the Simulation 34

35. Initialization  Each participating center is initially placed according a trial or guess geometry. The simulation begins with random speeds, although corresponding to the input simulation temperature. Temperatures in molecular dynamics are usually depending on the equipartition principle. The calculated temperature of each step is used for the energetic control of the system. 35

36. Initialization  Each participating center is initially placed according a trial or guess geometry. The simulation begins with random speeds, although corresponding to the input simulation temperature. Temperatures in molecular dynamics are usually depending on the equipartition principle. The calculated temperature of each step is used for the energetic control of the system.  The speed of all and each particle is normalized initially to avoid a resulting linear translation of the whole system, that is essentially meaningless. 36

37. Initialization  Each participating center is initially placed according a trial or guess geometry. The simulation begins with random speeds, although corresponding to the input simulation temperature. Temperatures in molecular dynamics are usually depending on the equipartition principle. The calculated temperature of each step is used for the energetic control of the system.  The speed of all and each particle is normalized initially to avoid a resulting linear translation of the whole system, that is essentially meaningless.  During the initialization step of the simulation, renormalization of speeds is repeated as much as needed to maintain the system at the input temperature. Initialization is considered done when the temperature becomes stable (the energy of the system becomes stationary and equilibrium is reached) and, therefore, renormalizations are no longer necessary. 37

38. Development  The “development” stage means a simulation during a given virtual time given by a significant number of  t steps. This time of simulation is virtual because is NOT the time elapsed by the computer to carry it out, but an imposed parameter to the molecular dynamics modeling. 38

39. Development  The “development” stage means a simulation during a given virtual time given by a significant number of  t steps. This time of simulation is virtual because is NOT the time elapsed by the computer to carry it out, but an imposed parameter to the molecular dynamics modeling.  During this stage the position and speed of all particles in selected time steps are registered and stored for further processing and property calculations. 39

40. Analysis The analysis stage means the calculation of all static and dynamic properties of the system from data stored during the previous step of “development”. The treatment of positions and speeds is performed statistically from such information collected during the simulation. 40

41. Thermodynamics and molecular dynamics  The common practice is performing molecular dynamics simulations in isolated systems, although such system meant several interacting molecules. It means that “temperature”, being a typical macroscopic property, only got interest as a consequence and result of the calculation of kinetic energies of the molecular set on the grounds of the classical principle of equipartition of energy. 41

42. Thermodynamics and molecular dynamics  The common practice is performing molecular dynamics simulations in isolated systems, although such system meant several interacting molecules. It means that “temperature”, being a typical macroscopic property, only got interest as a consequence and result of the calculation of kinetic energies of the molecular set on the grounds of the classical principle of equipartition of energy.  Any kind of calculation of macroscopic termodynamic properties will be on the grounds that the system’s energy is maintained as constant during the development stage, in equilibrium conditions, although such energy is expressed in terms of temperature. 42

43. Thermodynamics and molecular dynamics  The common practice is performing molecular dynamics simulations in isolated systems, although such system meant several interacting molecules. It means that “temperature”, being a typical macroscopic property, only got interest as a consequence and result of the calculation of kinetic energies of the molecular set on the grounds of the classical principle of equipartition of energy.  Any kind of calculation of macroscopic termodynamic properties will be on the grounds that the system’s energy is maintained as constant during the development stage, in equilibrium conditions, although such energy is expressed in terms of temperature.  Thus, molecular dynamics systems are usually treated as microcanonical statistical ensembles (NVE). 43

44. Computing Properties 44

45. Phase space restrictions As molecular dynamics is a simulation considering that the behavior of each particle is classical, knowing positions and instant speeds at such points must do possible the exact calculation of any related collective property. 45

46. Phase space restrictions As molecular dynamics is a simulation considering that the behavior of each particle is classical, knowing positions and instant speeds at such points must do possible the exact calculation of any related collective property. As it has been previously noticed, the main constraint is that the phase space can no account very large numbers, being them of the Avogadro’s number order. 46

47. Phase space restrictions As molecular dynamics is a simulation considering that the behavior of each particle is classical, knowing positions and instant speeds at such points must do possible the exact calculation of any related collective property. As it has been previously noticed, the main constraint is that the phase space can no account very large numbers, being them of the Avogadro’s number order. Therefore, only significant phases of one or several molecules in the system are considered as they are selected to be at minima of the hypersurface. 47

48. Types of Properties  Properties are statical when they are not related with a time dependent path (i.e. heat capacity at constant pressure), or dynamical when they are (i.e. self diffusion coefficients). 48

49. Types of Properties  Properties are statical when they are not related with a time dependent path (i.e. heat capacity at constant pressure), or dynamical when they are (i.e. self diffusion coefficients).  Properties are unimolecular when they depend only from the behavior of a single particle (i.e. self diffusion coefficients), or collective when depend from an ensemble of particles (i.e. dielectric constants). 49

50. Static Properties Translational kinetic energy: Potential energy: Where the expression between means and average over all  t time steps of the simulation during the development stage, N is the number of atoms or molecules in the system under study, u n is the energy calculated over each atom or molecule n in the system, and m n is the atomic mass of such n body. 50

51. Static Properties Translational temperature: It is the main contributing component to system’s calculated T. 51

52. Static Properties Translational temperature: It is the main contributing component to system’s calculated T. 52 Rotational temperature can also be calculated by replacing force by torque in the previous kinetic energy expressions and provides the complementary value.

53. Static Properties Translational temperature: It is the main contributing component to system’s calculated T. 53 If internal vibrations and torsions are to be accounted, their temperature can be obtained from their corresponding expressions of kinetic energy. Rotational temperature can also be calculated by replacing force by torque in the previous kinetic energy expressions and provides the complementary value.

54. Static Properties Translational temperature: It is the main contributing component to system’s calculated T. 54 If internal vibrations and torsions are to be accounted, their temperature can be obtained from their corresponding expressions of kinetic energy. The total temperature of the system is the sum of all these contributions from the equipartition principle: Rotational temperature can also be calculated by replacing force by torque in the previous kinetic energy expressions and provides the complementary value.

55. Static Properties The heat capacity at constant volume can be obtained from the fluctuations of energy during the simulation development: where f n is the number of degrees of freedom per particle, R is the ideal gas constant and N the number of atoms or molecules of the system. 55

56. Static Properties Pressure is calculated with a truncated virial ecuation, as is normally managed in the case of gases at low pressure: 56 where V is the system’s volume and w is the second virial coefficient as given by:

57. Static Properties The radial distribution function of pairs is the probability of finding a pair of bodies  and b separated by a distance r between them. 57

58. Static Properties The radial distribution function of pairs is the probability of finding a pair of bodies  and b separated by a distance r between them. 58 Being n(r) the number of body pairs  that are found separated at distances between r n and r n +  r, then: where V is the volume of simulation.

59. Static Properties The radial distribution function of pairs is the probability of finding a pair of bodies  and b separated by a distance r between them. 59 Being n(r) the number of body pairs  that are found separated at distances between r n and r n +  r, then: where V is the volume of simulation. X ray or neutron diffraction occurs because irregularities of radial distribution with respect to uniformity. Thanks to that, the obtained values in simulations can be compared with experiments to test the reliability of a potential or hypersurface used in it.

60. Static Properties As it is well known from elementary sciences, correlation functions have been defined for establishing the relationship or dependence between, i.e. any two variables, being x and y, over M measurements. 60

61. Static Properties As it is well known from elementary sciences, correlation functions have been defined for establishing the relationship or dependence between, i.e. any two variables, being x and y, over M measurements. 61 A common expression is:

62. Static Properties As it is well known from elementary sciences, correlation functions have been defined for establishing the relationship or dependence between, i.e. any two variables, being x and y, over M measurements. 62 A common expression is: The correlation function outputs a number between -1 and 1 from anti-correlated to fully correlated variables. A null value means that they are independent (uncorrelated).

63. Static Properties The so – called Liouville´s theorem establishes that the phase- space distribution function  0 (x n,p n ) of a given system in equilibrium is constant along the trajectories of the system:

64. Static Properties The so – called Liouville´s theorem establishes that the phase- space distribution function  0 (x n,p n ) of a given system in equilibrium is constant along the trajectories of the system: It is, itself, a statistical definition of equilibrium in a given system, and can be applied to the stage of development in a molecular dynamics simulation.

65. Static Properties The so – called Liouville´s theorem establishes that the phase- space distribution function  0 (x n,p n ) of a given system in equilibrium is constant along the trajectories of the system: It is, itself, a statistical definition of equilibrium in a given system, and can be applied to the stage of development in a molecular dynamics simulation. Then, when correlated variables of a system in equilibrium are time – dependent, the correlation function also becomes time – dependent.

66. Static Properties Autocorrelation is the correlation of a signal or calculation result with itself in different conditions, i.e. the similarity between a function evaluated at different separated points of any independent variable x among M samples:

67. Static Properties Autocorrelation is the correlation of a signal or calculation result with itself in different conditions, i.e. the similarity between a function evaluated at different separated points of any independent variable x among M samples: If the variable is time, it is a mathematical tool for finding repeating patterns, such as the presence of a periodic signal which has been buried under noise.

68. Static Properties Obviously, time autocorrelation relates values of the same function A(t) calculated at times t apart and the convergence of C AA (t) denotes equilibrium or stationary states.

69. Static Properties The mathematical expression of time autocorrelation of a given A(t) quantity in molecular dynamics is defined as: being the ensemble average of the product of the initial value of a certain A property and the values of A at several t times. Obviously, time autocorrelation relates values of the same function A(t) calculated at times t apart and the convergence of C AA (t) denotes equilibrium or stationary states.

70. Static Properties In just the same way we may define time-correlation functions of different variables as cross-correlation:

71. Static Properties In just the same way we may define time-correlation functions of different variables as cross-correlation: If the choice of origin of the time scale is entirely arbitrary and the equilibrium ensemble distribution function is invariant to changes of time, according the Liouville´s theorem, we have the obvious general identity: both for cross- and auto- correlation.

72. Static Properties In just the same way we may define time-correlation functions of different variables as cross-correlation: If the choice of origin of the time scale is entirely arbitrary and the equilibrium ensemble distribution function is invariant to changes of time, according the Liouville´s theorem, we have the obvious general identity: both for cross- and auto- correlation. This is a desirable ergodic condition of molecular dynamics simulations.

73. Static Properties In these conditions the time correlation formalism to the trajectory in a molecular dynamics simulation implies that we can use many time origins provided that they are sufficiently separated that there is no correlation between them.

74. Static Properties In these conditions the time correlation formalism to the trajectory in a molecular dynamics simulation implies that we can use many time origins provided that they are sufficiently separated that there is no correlation between them. It means that molecular dynamics can use many separate time frames instead of many ensembles (as is normal in the usual statistical mechanical approach) in order to obtain useful time decays during separate simulation times  that can be analyzed for given  t time steps.

75. Static Properties For samples in the sum to be independent.  should be chosen so that: For the same variable (autocorrelation), also expressed as a summation of steps:

76. Static Properties The initial value of the time autocorrelation value in any starting reference time is:

77. Static Properties The initial value of the time autocorrelation value in any starting reference time is: If the evolution is random, very separated times t and t’ (the value of  ) must give uncorrelated values of the variable:

78. Static Properties Using as a normalization factor, a graphic of decay of autocorrelation of the A quantity during a molecular dynamics simulation is:

79. Static Properties Using as a normalization factor, a graphic of decay of autocorrelation of the A quantity during a molecular dynamics simulation is: The time required for a full relaxation of the autocorrelation is called relaxation time (  ) and simulations must be longer than the one orresponding to the property of interest.

80. Static Properties Using as a normalization factor, a graphic of decay of autocorrelation of the A quantity during a molecular dynamics simulation is: The time required for a full relaxation of the autocorrelation is called relaxation time (  ) and simulations must be longer than the one orresponding to the property of interest. Resulting structures and energy values after each period of correlation times is a “shot” or “frame” of the simulation.

81. Dynamic Properties We consider as dynamic properties of the system those that depend on time advance during the simulation.

82. Dynamic Properties We consider as dynamic properties of the system those that depend on time advance during the simulation. Position, velocity, dipole moment, density, etc. are dynamic variables.

83. Dynamic Properties We consider as dynamic properties of the system those that depend on time advance during the simulation. In general, their sampling during the development stage in a given simulation provides a statistical description of the time-evolution of such variables or a correlated pair of them for an ensemble at thermal equilibrium. Position, velocity, dipole moment, density, etc. are dynamic variables.

84. Dynamic Properties A molecular dynamics simulation produces a substantial amount of useful information, and it is normal to store: vectors of the positions of each particle, velocities (as angular velocities), forces (as torques) for each molecule, as well as the instantaneous values of all the calculated properties.

85. Dynamic Properties A molecular dynamics simulation produces a substantial amount of useful information, and it is normal to store: vectors of the positions of each particle, velocities (as angular velocities), forces (as torques) for each molecule, as well as the instantaneous values of all the calculated properties. They use to be sampled after a certain number of steps, say 5 or 10, to save data storage.

86. Dynamic Properties Let us consider a set of n particles (atoms or molecules) in a unit of volume and n 1 the number of certain labeled particles among them in such volume. If J z is the mean number of the labeled particles crossing or diffusing through a unit area of a plane dividing and inside such volume per unit of time in the positive z direction, we can write:

87. Dynamic Properties Let us consider a set of n particles (atoms or molecules) in a unit of volume and n 1 the number of certain labeled particles among them in such volume. If J z is the mean number of the labeled particles crossing or diffusing through a unit area of a plane dividing and inside such volume per unit of time in the positive z direction, we can write: This relation establishes that the diffusion is proportional to the gradient of the number of labeled particles in the flow direction if the n 1 particles are not uniformly distributed.

88. Dynamic Properties Let us consider a set of n particles (atoms or molecules) in a unit of volume and n 1 the number of certain labeled particles among them in such volume. If J z is the mean number of the labeled particles crossing or diffusing through a unit area of a plane dividing and inside such volume per unit of time in the positive z direction, we can write: This relation establishes that the diffusion is proportional to the gradient of the number of labeled particles in the flow direction if the n 1 particles are not uniformly distributed. If the n 1 particles are uniformly distributed in the volume there are no gradient and then J z = 0.

89. Dynamic Properties Let us consider a set of n particles (atoms or molecules) in a unit of volume and n 1 the number of certain labeled particles among them in such volume. If J z is the mean number of the labeled particles crossing or diffusing through a unit area of a plane dividing and inside such volume per unit of time in the positive z direction, we can write: This relation establishes that the diffusion is proportional to the gradient of the number of labeled particles in the flow direction if the n 1 particles are not uniformly distributed. If the n 1 particles are uniformly distributed in the volume there are no gradient and then J z = 0. self – diffusion coefficient D is called as the self – diffusion coefficient of the substance, and can be considered as the particle flow number when the gradient is unitary.

90. Dynamic Properties A particular case is the time correlation function of particle velocities, known as velocity autocorrelation function, defined as before:

91. Dynamic Properties A particular case is the time correlation function of particle velocities, known as velocity autocorrelation function, defined as before: It can be deduced that the velocity autocorrelation function describes how long velocities of the involved particles persist until they averaged out by microscopic motions and interactions with its surroundings.

92. Dynamic Properties The integral value of C vv in the total time of simulation serves to find the self - diffusion coefficient:

93. Dynamic Properties The integral value of C vv in the total time of simulation serves to find the self - diffusion coefficient: It can be defined as the integral of all velocity self – correlation functions in a given system during the whole simulation time. It is a good measure of translational mobility.

94. Dynamic Properties Vibrational spectra simulations are a natural result of molecular dynamics as the motion in nanoscopic dimensions dealing with them are directly modeled with this method.

95. Dynamic Properties Vibrational spectra simulations are a natural result of molecular dynamics as the motion in nanoscopic dimensions dealing with them are directly modeled with this method. Therefore, the Fourier transform of D self – difussion coefficient is a measure of the spectral density of states.

96. Dynamic Properties Vibrational spectra simulations are a natural result of molecular dynamics as the motion in nanoscopic dimensions dealing with them are directly modeled with this method. Therefore, the Fourier transform of D self – diffution coefficient is a measure of the spectral density of states. Left: velocity autocorrelation functions for oxygen and hydrogen atoms in a water MD simulation; Right: Corresponding spectral density for the liquid and the gas.

97. Dynamic Properties Vibrational spectra simulations are a natural result of molecular dynamics as the motion in nanoscopic dimensions dealing with them are directly modeled with this method. Therefore, the Fourier transform of D self – diffution coefficient is a measure of the spectral density of states. Left: velocity autocorrelation functions for oxygen and hydrogen atoms in a water MD simulation; Right: Corresponding spectral density for the liquid and the gas.

98. Dynamic Properties Vibrational spectra intensities depend on transition dipoles.

99. Dynamic Properties Vibrational spectra intensities depend on transition dipoles. They can be obtained for systems in which the time evolution of the dipole moment and polarizability can be described with the classical potentials used in the molecular dynamics simulation.

100. Dynamic Properties Vibrational spectra intensities depend on transition dipoles. They can be obtained for systems in which the time evolution of the dipole moment and polarizability can be described with the classical potentials used in the molecular dynamics simulation. The absorption cross section (giving absorption intensities) of IR spectra is proportional to the line shape obtained after a Fourier transform of the dipole moment autocorrelation function:

101. Dynamic Properties Vibrational spectra intensities depend on transition dipoles. They can be obtained for systems in which the time evolution of the dipole moment and polarizability can be described with the classical potentials used in the molecular dynamics simulation. The absorption cross section (giving absorption intensities) of IR spectra is proportional to the line shape obtained after a Fourier transform of the dipole moment autocorrelation function: where is the dipole moment autocorrelation that can be averaged over single molecule dipole moment autocorrelation functions for localized vibrations.

102. More Common Potentials and Computer Programs

103. AMBER “Assisted Model Builder with Energy Refinement” (AMBER) by the former Peter Kollman, in UC San Francisco, is used for simulations of molecular dynamics. It uses only five bonding and no bonding terms, as well as a special treatment for electrostatic potentials. No cross terms are included. It gives good results with proteins and nucleic acids although could be erratic with other systems if the appropriate hypersurface is not selected. 103

104. AMBER 1. Weiner, P. K.; Kollman, P. A., AMBER: Assisted model building with energy refinement. A general program for modeling molecules and their interactions. J. Comput. Chem. 1981, 2 (3), 287-303.

105. CHARMM “Chemistry at HARvard Macromolecular Mechanics” (CHARMM) by Martin Karplus, in Harvard, was originally designed for proteins and nucleic acids, although now has been applied to various types of biomolecules, to solvation energies and structures, vibrational analyses, and QM/MM (quantum mechanics / molecular mechanics) studies. The general terms are: 105

106. GROMOS “GROnigen MOlecular Simulations” (GROMOS) by van Gunsteren, ETHZ, is popular for dynamics simulations of molecules and pure liquids, as well as biomolecules. 106 bonding angular improper angles torsions non bonding electrostatics

107. Some examples 107

108. Water dimer with the NCC potential The water dimer model is represented by O (centers 5 and 6, in figure) and H nuclei (centers 1, 2, 3 and 4) although fiction “centers” are also created to represent electric dipoles (centers 7 and 8). 108

109. Water dimer with the NCC potential The employed potential is called as Nieser-Corongiu-Clementi (NCC) and consider two parts: one for two body interactions and other for polarization: 109

110. Water dimer with the NCC potential The two body component is: 110

111. Water dimer with the NCC potential The polarization component is required for simulating quantum effects related with interactions between charge densities between molecules: 111 where N m is the number of molecules N d is the number of dipoles in molecules is the induced dipole moment in molecule i by dipole. is the electric field in the site of the induced dipole i originated by the distribution of point charges in the surroundings.

112. Water dimer with the NCC potential Results for water dimer: 112

113. Water dimer with the NCC potential Radial distribution: 113

114. Water dimer with the NCC potential Radial distribution: 114

115. Water dimer with the NCC potential Radial distribution: 115

116. Example of water simulation in a myoglobin crystal Here water is studied as solvent. Coordinates of 1261 heavy atoms of myoglobin were obtained from X ray diffraction data of the crystalized protein as the starting system. Hydrogen atoms were absent because the experimental technique cannot resolve them and were introduced by considering typical geometries, up to 2532 atoms. 856 molecules of water were used in a virtual unit cell, giving a crystal with a density of 1.2705 g cm -3. 116

117. Example of water simulation in a myoglobin crystal Water - protein interactions were calculated by a potential based in ab initio quantum calculations and water – water were potentials were standard. 117

118. Example of water simulation in a myoglobin crystal Water - protein interactions were calculated by a potential based in ab initio quantum calculations and water – water were potentials were standard. 118 The simulation was carried out in 50 ps, with an interval or time step of 0.5 fs. After 20 ps equilibrium was reached and the following 30 ps were used for the “development stage”. Average temperature of simulation was 304.4 ± 8.15 K and the total potential energy of –17.46 ± 0.03 kcal mol -1 with maximal fluctuations of 0.001 kcal mol -1.

119. Example of water simulation in a myoglobin crystal 119 Radial distribution function water – water and water – protein

120. Example of water simulation in a myoglobin crystal 120 Comparison between velocity autocorrelation functions of bulk water with respect to water in myoglobin

121. Molecular dynamics simulation of salt dissolving in water Chlorine ions are shown in yellow, sodium in blue. The large chlorine dissociates from the salt crystal. 121 http://www.chem.ucl.ac.uk/ice Group leader Prof. Angelos Michaelides University College London London Centre for Nanotechnology 17-19 Gordron Street London, WC1H 0AH Tel: +44 (0) 207 679 0647 (Ext: 30647) Email: angelos.michaelides@ucl.ac.uk Prof. Angelos Michaelidesangelos.michaelides@ucl.ac.uk

122. References Alder, B. J.; Wainwright, T. E., Studies in Molecular Dynamics. I. General Method. J. Chem. Phys. 1959, 31 (2), 459-466. Zwanzig, R., Time-Correlation Functions and Transport Coefficients in Statistical Mechanics. Annu. Rev. Phys. Chem. 1965, 16 (1), 67-102. Clementi, E.; Corongiu, G.; Bahattacharya, D.; Feuston, B.; Frye, D.; Preiskorn, A.; Rizzo, A.; Xue, W., Selected topics in ab initio computational chemistry in both very small and very large chemical systems. Chem. Revs. 1991, 91 (5), 679-699. Clementi, E.; Corongiu, G.; Aida, M.; Niesar, U.; Kneller, G., Monte Carlo and Molecular Dynamics Simulations. In MOTECC. Modern Techniques in Computational Chemistry, Clementi, E., Ed. ESCOM: Leiden, 1990; pp 805-888. 122