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md2011-11-notes

# md2011-11-notes - Molecular Dynamics simulations Lecture 11...

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Molecular Dynamics simulations Lecture 11: Energy Minimization Dr. Olli Pakarinen University of Helsinki Fall 2011 Original lecture notes by Dr. Jani Kotakoski, 2010 Optimization Problem I Optimization problems are in the heart of many physical applications I In short, optimization is about finding the set of parameters x , y , . . . for a given objective function f ( x , y , . . . ) which will yield either the minimum or maximum value I The extreme values searched for can be either local or global I In one-dimensional case: local minimum global minimum

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I The brute force method would be to map the complete configuration space (all possible parameters) and thereby find all possible values of the function I However, this is obviously not a very effective approach to a multi-dimensional problem I The computational approach is, as so often, dictated by the computational cost I This leads to a simple rule: evaluate f ( x , y , . . . ) as few times as possible I In general, it is a very difficult problem to effectively find the global extreme value I There are two traditional ways to do this: (1) Start minimization from several initial conditions and find the local extreme (2) Perturb a local extreme and optimize the function to see if you are returned to the initial extreme I Depending on the exact problem, some constraints may apply I In the context of this course, we are mostly interested in energy minimization (for other methods, see f.ex. [http://www.nrbook.com/c/] ) I This is important in two different regards: I Local energy minimization for structure relaxation, e.g., after a defect is created I Global energy minimization for finding minimum energy structures under certain constraints ( P , T )
Energy Minimization I The first of these is obviously a local minimization problem (without changing the lattice around the defect) whereas the second one is about finding the global minimum (under constraints) I Correspondingly, different methods are often used for these different problems I Local energy minimization is carried out using traditional optimization techniques I Global optimization, on the other hand, has exploded during the recent years with the introduction of evolutionary structure optimization algorithms (allowed by fast enough computers) I In any case, energy minimization is a surprisingly complicated task to solve efficiently I The set of parameters is the degrees of freedom of the system [i.e., three spatial coordinates for each of the particles x = ( x 1 , x , x 1 , y , x 1 , z , x 2 , x , x 2 , y , x 2 , z , . . . ) ] I (Energy minimization is a T = 0 K problem) I Obviously, the quantity to minimize is the potential energy of the system U ( x ) I One limiting factor for the energy minimization techniques is the memory requirement; for a very large set of atoms, e.g., O ( N 2 ) methods can be impossible ( N is the number of parameters)

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Examples of Methods Used in Atomistic Energy Minimization (1) Monte Carlo simulation This can be carried out by setting the temperature T 0 K in a MC simulation (system will naturally reach for the lowest energy) +
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md2011-11-notes - Molecular Dynamics simulations Lecture 11...

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