md2011-12-notes

md2011-12-notes - Molecular Dynamics simulations Lecture...

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Molecular Dynamics simulations Lecture 12: Path Optimization Calculations Dr. Olli Pakarinen University of Helsinki Fall 2011 Original lecture notes by Dr. Jani Kotakoski, 2010 Reaction Paths I Calculations for minimum energy paths are widely used in chemistry, physics and materials science I Typical examples are migration paths for defects in a solid or on a surface and chemical reactions I When the minimum energy path is known, the rate of occurrence for the related events can be easily calculated from the energy by the Arrhenius equation: Γ = ν 0 exp [- E / k B T ] (1) where ν is an attempt frequency, such as a typical frequency for lattice vibrations in the case of defect migration (see Swanson, Phys. Rev. 121, 1668–1674 (1961)] for a theoretical analysis on ν ) and E is the activation energy
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I In a two-dimensional case, the most simple example is given by two energy minima with a saddle point in the middle I In principle, such processes can be studied with the MD method, as will be explained below I However, if the barrier is too high, the processes are so slow that they won’t be observed during MD time scales at reasonable temperatures I Therefore, we need a method which allows calculating these energy barriers without carrying out a dynamical simulation Transition State Theory I In the case of rare events, one can use the transition state theory (TST) to study reaction rates I According to TST, the reaction rate is Γ = N Y j = 1 ν j N - 1 Y j = 1 ν 0 j - 1 e - E / k B T (2) where ν j and ν 0 j are the vibrational frequencies at the initial potential minimum and at the saddle point, respectively
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I The saddle point is associated with one less frequency, because there is one direction in which the system is driven away from the saddle point rather than towards it I It is typical, at least in solid state physics calculations for defect migration to simple assume that the frequencies can be described by only one material-dependent vibrational frequency ν 0 I Values are often in the range of ν 0 10 12 s - 1 I Now, if we can follow the system over a period of time and observe the reaction rate, we can calculate the reaction path energy barrier E I On the other hand, if we know the barrier, we can calculate the reaction rate without carrying out the (possible impossibly long) dynamical simulation I The barrier energy is defined as the potential energy difference between the saddle point and the initial energy minimum Schematic presentation of the energy barrier A B Potential Energy Reaction Coordinate saddle point E
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Locating the minimum energy reaction path (MEP) I There are several methods for finding the minimum energy path and calculating thus the energy barrier I These methods include at least the following [Wikipedia] I Synchronous Transit The linear synchronous transit (LST) method generates an estimate of the transition state by finding the highest point along shortest line connecting two minima. A
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md2011-12-notes - Molecular Dynamics simulations Lecture...

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