Phonons: Theory: Difference between revisions

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:<math>
:<math>
C_{I\alpha J\beta} (\{\mathbf{R}^0\}) =  
\Phi_{I\alpha J\beta} (\{\mathbf{R}^0\}) =  
\left. \frac{\partial E(\{\mathbf{R}\})}{\partial R_{I\alpha} \partial R_{J\beta}} \right|_{\mathbf{R} =\mathbf{R^0}}
\left. \frac{\partial E(\{\mathbf{R}\})}{\partial R_{I\alpha} \partial R_{J\beta}} \right|_{\mathbf{R} =\mathbf{R^0}}
=
=
Line 33: Line 33:
E(\{\mathbf{R}^0\})+
E(\{\mathbf{R}^0\})+
\sum_{I\alpha} -F_{I\alpha} (\{\mathbf{R}^0\}) u_{I\alpha}+
\sum_{I\alpha} -F_{I\alpha} (\{\mathbf{R}^0\}) u_{I\alpha}+
\sum_{I\alpha J\beta} C_{I\alpha J\beta} (\{\mathbf{R}^0\}) u_{I\alpha} u_{J\beta} +
\sum_{I\alpha J\beta} \Phi_{I\alpha J\beta} (\{\mathbf{R}^0\}) u_{I\alpha} u_{J\beta} +
\mathcal{O}(\mathbf{R}^3)
\mathcal{O}(\mathbf{R}^3)
</math>
</math>
Line 41: Line 41:
H =  
H =  
\frac{1}{2} \sum_{I\alpha} M_I \dot{u}^2_{I\alpha} +  
\frac{1}{2} \sum_{I\alpha} M_I \dot{u}^2_{I\alpha} +  
\frac{1}{2} \sum_{I\alpha J\beta} C_{I\alpha J\beta} u_{I\alpha} u_{J\beta},
\frac{1}{2} \sum_{I\alpha J\beta} \Phi_{I\alpha J\beta} u_{I\alpha} u_{J\beta},
</math>
</math>
and the equation of motion
and the equation of motion
:<math>
:<math>
M_I \ddot{u}^2_{I\alpha} = -  
M_I \ddot{u}^2_{I\alpha} = -  
C_{I\alpha J\beta} u_{J\beta}
\Phi_{I\alpha J\beta} u_{J\beta}
</math>
</math>


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:<math>
:<math>
D_{I\alpha J\beta} (\mathbf{q}) =  
D_{I\alpha J\beta} (\mathbf{q}) =  
\frac{1}{\sqrt{M_I M_J}} C_{I\alpha J\beta} e^{i\mathbf{q} \cdot (\mathbf{R}_J-\mathbf{R}_I)}
\frac{1}{\sqrt{M_I M_J}} \Phi_{I\alpha J\beta} e^{i\mathbf{q} \cdot (\mathbf{R}_J-\mathbf{R}_I)}
</math>
</math>


Line 85: Line 85:
The second-order force constants are then computed using
The second-order force constants are then computed using
:<math>
:<math>
C_{I\alpha J\beta}=
\Phi_{I\alpha J\beta}=
\frac{\partial^2E}{\partial u_{I\alpha} \partial u_{J\beta}}=
\frac{\partial^2E}{\partial u_{I\alpha} \partial u_{J\beta}}=
-\frac{\partial F_{I\alpha}}{\partial u_{J\beta}}
-\frac{\partial F_{I\alpha}}{\partial u_{J\beta}}
Line 153: Line 153:


:<math>
:<math>
C_{I\alpha J\beta}=
\Phi_{I\alpha J\beta}=
\frac{\partial^2E}{\partial u_{I\alpha} \partial u_{J\beta}}=
\frac{\partial^2E}{\partial u_{I\alpha} \partial u_{J\beta}}=
-\frac{\partial F_{I\alpha}}{\partial u_{J\beta}}
-\frac{\partial F_{I\alpha}}{\partial u_{J\beta}}

Revision as of 14:50, 1 August 2022

To understand them we start by looking at the Taylor expansion of the total energy () around the set of equilibrium positions of the nuclei ()

where the positions of the nuclei. The first term in the expansion corresponds to the forces

,

and the second to the second-order force-constants


We can define a variable that corresponds to the displacement of the atoms with respect to the equilibrium position which leads to

If we take the system to be in equilibrium, the forces on the atoms are zero and then the Hamiltonian of the system is

and the equation of motion

Using the following ansatz

where are the phonon mode eigenvectors. Replacing we obtain the following eigenvalue problem

with

the dynamical matrix in the harmonic approximation. Now by solving the eigenvalue problem above we can obtain the phonon modes and frequencies at any arbitrary q point.

Finite differences

The second-order force constants are computed using finite differences of the forces when each ion is displaced in each independent direction. This is done by creating systems with finite ionic displacement of atom in direction with magnitude , computing the orbitals and the forces for these systems. The second-order force constants are then computed using

where corresponds to the displacement of atom in the cartesian direction and retrieves the set of forces acting on all the ions given the orbitals.

Similarly, the internal strain tensor is

where computes the strain tensor given the orbitals.

Density functional perturbation theory

Density-functional-perturbation theory provides a way to compute the second-order linear response to ionic displacement, strain, and electric fields. The equations are derived as follows.

In density-functional theory, we solve the Kohn-Sham (KS) equations

where is the DFT Hamiltonian, is the overlap operator and, and are the KS eigenstates.

Taking the derivative with respect to the ionic displacements , we obtain the Sternheimer equations

Once the derivative of the KS orbitals is computed from the Sternheimer equations, we can write

The second-order response to ionic displacement, i.e., the force constants or Hessian matrix can be computed using the same equation used in the finite differences case

where yields the forces for a given set of KS orbitals.

Similarly, the internal strain tensor is computed using

The Born effective charges are then computed using Eq. (42) of Ref. [1].

where is the atom index, the direction of the displacement of atom and the polarization direction. The results should be equivalent to the ones obtained using LCALCEPS and LEPSILON.

References