The Small Displacement Methd for Phonon Calculations Theory 小位移方法的声子计算理论.pptVIP

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The Small Displacement Methd for Phonon Calculations Theory 小位移方法的声子计算理论.ppt

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The Small Displacement Methd for Phonon Calculations Theory 小位移方法的声子计算理论

The Small Displacement Method for Phonon Calculations Theory and Implementation Adam Shai May 2008 Quick Review of Linear Response Hellmann-Feynman Theorem: Make a linear perturbation in your system - a perturbation in the electron density - and get a second order perturbation in your energy We can express the linear perturbation of the electron density in terms of the unperturbed ground state and get the dynamical matrix This requires no supercell Small Displacement Method Algorithm for Small Displacement Method Thermodynamic Properties We will derive an equation for heat capacity as an example of how to go from phonons to thermodynamic properties Heat Capacity: Total energy of phonons in a crystal is the sum of the energy over all of the phonon modes Heat Capacity We can replace the summation by an integral over the frequencies with the density of states Density of states is the number of vibrational modes per frequency The energy can now be written as an integral: Thermodynamic Properties Small Displacement Method In Practice To calculate force constant matrix we make a supercell In Quantum Espresso we are using periodic boundary conditions in our calculations; we have to make sure the supercell is big enough so that the forces go to zero at the edges of the supercell Use symmetry to reduce the number of calculations At most the number of calculations must be three times the atoms in the primitive cell For any other symmetry operation which takes one atom to the place of another, and the whole supercell invariant, than only one of those has to be calculated Small Displacement Method In Practice Also, if there is a point group symmetry operation on a displacement which leaves the new displacement linearly independent from the original one, there is a form for the new force field, so both do not have to be done So for an hcp crystal: 2 atom basis, 3 coordinates - 6 calculations Translation then inversion leaves crystal the same So displacements are only need