Transcript R m
MODELING MATTER AT NANOSCALES 3. Empirical classical PES and typical procedures of optimization 3.03. Monte Carlo and other heuristic procedures 1 Exploring n-dimensional space Exploration of energy landscapes of n-dimensional hypersurfaces Fair raffles must allow equal opportunities for everyone in any contest. Exploration of energy landscapes of n-dimensional hypersurfaces Fair raffles must allow equal opportunities for everyone in any contest. A random selection of numbers related with points in a given space allow a fair guided visit of all “regions” or subspaces of such real or virtual space regardless of its position and simulates a complete exploration. Exploration of energy landscapes of n-dimensional hypersurfaces Fair raffles must allow equal opportunities for everyone in any contest. A random selection of numbers related with points in a given space allow a fair guided visit of all “regions” or subspaces of such real or virtual space regardless of its position and simulates a complete exploration. A quite comprehensive energy landscape of a given space can be obtained by evaluating the corresponding hypersurface in several spatial points provided by a random distribution. Monte Carlo simulations Monte Carlo simulations are considered those that can be used to numerically evaluate energy functions of molecular systems by means of randomly explored points of the corresponding space. Monte Carlo simulations Monte Carlo simulations are considered those that can be used to numerically evaluate energy functions of molecular systems by means of randomly explored points of the corresponding space. It allows the simulation of even macroscopic system departing from the atomic and molecular “bricks” that appear as more probable. Monte Carlo simulations Monte Carlo simulations are considered those that can be used to numerically evaluate energy functions of molecular systems by means of randomly explored points of the corresponding space. It allows the simulation of even macroscopic system departing from the atomic and molecular “bricks” that appear as more probable. It is based on the principle that any static configuration of a system of particles determine its state functions. Monte Carlo simulations Let us consider a classical system with N particles, being the ith of them associated with a ri generalized coordinate in the Euclidean space. Monte Carlo simulations Let us consider a classical system with N particles, being the nth of them associated with a rn generalized coordinate in the Euclidean space. The more probable or expected value of a given state function, as the internal energy, will be: E E (R m ) P (R m )drnm where E(Rm) is the evaluation of the configurational energy and P(Rm) is the probability (between 0 and 1) of the system in a given Rm point of the configuration space of N of n particles. Monte Carlo simulations The evaluation of this integral in the hyperspace by the Monte Carlo method consists in the random generation of Rm points, evaluating E(Rm) and taking into account a “validity criterion” depending on the P probability at this point. Monte Carlo simulations The evaluation of this integral in the hyperspace by the Monte Carlo method consists in the random generation of Rm points, evaluating E(Rm) and taking into account a “validity criterion” depending on the P probability at this point. As large was the number of M “visited” points, the “stability” of <E> improves, because the average is more realistic. Monte Carlo simulations The evaluation of this integral in the hyperspace by the Monte Carlo method consists in the random generation of Rm points, evaluating E(Rm) and taking into account a “validity criterion” depending on the P probability at this point. As large was the number of M “visited” points, the “stability” of <E> improves, because the average is more realistic. However, the optimal criterion is to limit the exploration of the configuration space to the minimal number of structures that could give appropriate and stable (non fluctuating) results. Monte Carlo simulations The following conditions are necessary to solve the problem: • Availability of an algorithm for random number generation serving to select every new point in the space given by Rm. Monte Carlo simulations The following conditions are necessary to solve the problem: • Availability of an algorithm for random number generation serving to select every new point in the space given by Rm. • Knowing or establishing conventions for probability criteria P(Rm) of each point. Monte Carlo simulations The following conditions are necessary to solve the problem: • Availability of an algorithm for random number generation serving to select every new point in the space given by Rm. • Knowing or establishing conventions for probability criteria P(Rm) of each point. • Availability of functional forms of E(Rm) to be evaluated in each point (an appropriate hypersurface). Monte Carlo simulations The Metropolis algorithm takes N particles (atoms or molecules) in a given configuration, with an appropriate V volume and a given T temperature. Monte Carlo simulations The Metropolis algorithm takes N particles (atoms or molecules) in a given configuration, with an appropriate V volume and a given T temperature. This case is treated as a canonical ensemble (NVT) although other types of ensembles can also be used (as the microcanonical, maintaining constant the internal energy, in place of temperature). Procedure The total energy of the system in a starting guess geometry is evaluated by means of the selected potential. Then, the following iterative steps are followed: 1 Random selection of a n component particle. Procedure The total energy of the system in a starting guess geometry is evaluated by means of the selected potential. Then, the following iterative steps are followed: 1 Random selection of a n component particle. 2 Moving the n particle or reference body to a randomly selected new coordinate of the given space. All coordinates of the particle are independently displaced in all and each dimension with respect to the coordinate origin of the whole system. The selected particle is also randomly rotated along the corresponding internal axis. Procedure 3 Evaluation of the system’s total energy in the new Rm geometry If the new energy value is lower that that of the previous step the new geometry is accepted with a total probability, P(Rm) = 1. Procedure 3 Evaluation of the system’s total energy in the new Rm geometry If the new energy value is lower that that of the previous step the new geometry is accepted with a total probability, P(Rm) = 1. If the obtained energy is higher than the previous value, an additional random number x is generated (between 0 and 1) Procedure Then, if x e Em kT the new geometry is rejected and the corresponding Rm is not considered in the final average. Procedure Then, if x e Em kT the new geometry is rejected and the corresponding Rm is not considered in the final average. But, if x e En kT the new Rm is then accepted and the energy accounted for the final average, even conducting to a higher point in the hyperspace. Procedure Then, if x e Em kT the new geometry is rejected and the corresponding Rm is not considered in the final average. But, if x e En kT the new Rm is then accepted and the energy accounted for the final average, even conducting to a higher point in the hyperspace. The process is then repeated from step 1 until a desired ending Procedure Experience indicates that the probability of acceptance of moves must be adjusted to a 50 %. It is currently achieved with limits to displacements of each particle. Procedure Experience indicates that the probability of acceptance of moves must be adjusted to a 50 %. It is currently achieved with limits to displacements of each particle. This procedure is confusedly named in literature: It could appear either as “simulated annealing” or as “Monte Carlo – Metropolis” procedures, with small variants Practical steps Thermalization The previously described procedure is performed until an acceptable level of fluctuation of <E> around a determined mean value, generally lower than the initial state. Practical steps Thermalization The previously described procedure is performed until an acceptable level of fluctuation of <E> around a determined mean value, generally lower than the initial state. Sampling At the end of thermalization the process continues as described, although properties are recorded at each valid step to be stored in memory for further calculations, mostly energy and geometry related values. Such new accounting excludes geometries “visited” during thermalization. Computing properties Average configurational energy The average internal energy originated in the hypersurface results during the sampling step is computed by: 1 M E E (R m ) M m1 where M is the number of valid counts during the sampling step. Computing properties Average configurational energy The average internal energy originated in the hypersurface results during the sampling step is computed by: 1 M E E (R m ) M m1 where M is the number of valid counts during the sampling step. This kind of average can be used for any other kind of recorded property. Computing properties It must be observed that the statistical weighting of this average energy is always unitary. It means the same probabilities for each point although annealing randomly allows uphill moves. 1 M P (R m ) 1 E E (R m ) M m1 Computing properties It must be observed that the statistical weighting of this average energy is always unitary. It means the same probabilities for each point although annealing randomly allows uphill moves. 1 M P (R m ) 1 E E (R m ) M m1 This condition can be established only when the system reaches a steady state after thermalization, although very fluctuating systems require alternative weighting procedures. Computing properties Heat capacity at constant volume Heat capacity is computed after statistical considerations for a classical Boltzmann distribution: 3 CV (T ) [ E E ] / kT Nk 2 2 2 where N is the number of particles. 2 Computing properties Radial distribution function Molecular, atomic or element layering is given by radial distribution functions g giving the probabililty to find an kind body at an r distance from other kind body and it is normalized to be 1 when r is large: V (r ) g (r ) 2 N 4r r where (r) is the number of kind bodies appearing netween ri and ri+r, and r is the interval dividing the segment between the kind body by a certain given cutoff distance. Computing properties Radial distribution function are evaluated for each one of the concerning bodies (atoms, elements, molecular fragments, etc.) with respect to other in each sampling step and the final value is averaged over all steps, in the same way as internal configurational energy. Computing properties Radial distribution function are evaluated for each one of the concerning bodies (atoms, elements, molecular fragments, etc.) with respect to other in each sampling step and the final value is averaged over all steps, in the same way as internal configurational energy. This property can be related with X ray or neutron dispersion intensities in diffraction experiments. Monte Carlo simulation of aspartic acid in water Radial distributions in liquid water PES Parameters Jorgensen, W. L.; Chandrasekhar, J.; Madura, J. D.; Impey, R. W.; Klein, M. L., Comparison of simple potential functions for simulating liquid water. The Journal of Chemical Physics 1983, 79, (2), 926-935. Other zero order treatments Other heuristic methods • Homology modeling – Use geometry of similar molecules as a guess for predicting geometries of very complex nanoscopic systems Other heuristic methods • Homology modeling – Use geometry of similar molecules as a guess for predicting geometries of very complex nanoscopic systems • Fragment Approach – Fix/Constrain part of the system while optimizing others by appropriate methods Other heuristic methods • Homology modeling – Use geometry of similar molecules as a guess for predicting geometries of very complex nanoscopic systems • Fragment Approach – Fix/Constrain part of the system while optimizing others by appropriate methods • Rule-Based – Use regularities in structural behaviors of molecules or fragments to guess initial geometries. It is the case of proteins fixing tertiary structure according to statistically likelihood of amino acid sequence to adopt such a structure Simplex algorithm If only the energy is known of a given system, the probably simplest way to obtain an optimized structure is one called the simplex algorithm. This is just a systematic way of trying larger and smaller variables for the coordinates and keeping the changes that result in a lower energy by an inter and extrapolation procedure. Simplex optimizations are rarely used because the huge computational requirements.