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- A Nk B T 1 = Z 2 1 E Nk B T d = -Z T 2 T 1 E Nk B T d T T . (37) I Integrating the pressure along an isotherm: A Nk B T 2- A Nk B T 1 = Z 2 1 PV Nk B T d = -Z V 2 V 1 PV Nk B T 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 : U = U ( r , ) . I Then A = -k B T ln R d r exp (-U ( r , ) / k B T ) = R d r U exp (-U / k B T ) R d r exp (-U / k B T ) = U . (39) I U needs to be constructed so that for = the answer is accessible through analytic methods (e.g., ideal gas or harmonic lattice). Notes...
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md2011-05-note_Part_6 - I One way is to calculate averages...

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