md2011-05-note_Part_6

md2011-05-note_Part_6 - I One way is to calculate averages...

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I Allowing the cell shape to change is critical in the case of some phase transitions. I In the original paper, the authors used this method to model a fcc hcp phase transition in Ni. Notes
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M±²³´µ¶ ·´¸ μ VT E¹¶±º»¼±¶ I Within these methods, the μ stays constant while the number of atoms fluctuates. I These methods are used more often in MC simulations because they tend to lead to unphysicalities resulting from adding atoms to random places within the system. I However, it has been done also within MD simulations. I Two examples are presented in 8594 (1997)] and [Heffelfinger, J. Chem. Phys. 100, 7548 (1994)] . Notes
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H±² ³± C´±±µ¶? I If simple scaling is needed, the Berendsen methods are the ones to be used because they are simple but efficient. I For accurate T control or NVT thermodynamic averaging, the Nosé-Hoover methods are most likely to work the best. I For orthogonal NPT simulations, the Andérsen method works nicely. I If shear pressure changes in crystal structure are of interest, one should use the Parrinello-Rahman method. Notes
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C±²³´²±µ¶·¸ T¹º»¼½¾¿·±¼¶³ P½µº·µ¶±²À
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Unformatted text preview: I One way is to calculate averages for a path between two states (1 and 2) and to integrate over the reversible path. I Integrating the internal energy along a line of constant density: ± A Nk B T ² 2-± A Nk B T ² 1 = Z β 2 β 1 ± E Nk B T ² d β β = -Z T 2 T 1 ± E Nk B T ² d T T . (37) I Integrating the pressure along an isotherm: ± A Nk B T ² 2-± A Nk B T ² 1 = Z ρ 2 ρ 1 ± PV Nk B T ² d ρ ρ = -Z V 2 V 1 ± PV Nk B T ² d V V . (38) I This has to be done accurately for many closely spaced points, and is hence rather expensive. Notes I In Frenkel-Ladd method [Frenkel-Ladd, J. Chem. Phys. 81, 3188 (1984)] absolute internal energy is calculated by constructing a potential function which depends on parameter λ : U = U ( r , λ ) . I Then ∂ A ∂λ = -k B T ∂ ∂λ ± ln R d r exp (-U ( r , λ ) / k B T ) ² = R d r ∂ U ∂λ exp (-U / k B T ) R d r exp (-U / k B T ) = ³ ∂ U ∂λ ´ . (39) I U needs to be constructed so that for λ = λ the answer is accessible through analytic methods (e.g., ideal gas or harmonic lattice). Notes...
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This note was uploaded on 02/14/2012 for the course CSE 6590 taught by Professor Kotakoski during the Winter '12 term at York University.

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md2011-05-note_Part_6 - I One way is to calculate averages...

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