Plotting exciton wavefunction: Difference between revisions

Revision as of 16:16, 29 January 2024

Plotting the wavefunction or charge density corresponding to an exciton can be instrumental for analyzing the symmetry, position, and localization of the excitonic state.

The exciton wavefunction is written as a function of coordinates of two particles (one hole and one electron) $\psi_\lambda(\mathbf{r}_e,\mathbf{r}_h)=\sum_{vc} A_{vc}^\lambda \psi_v(\mathbf{r}_h)\psi_c^*(\mathbf{r}_e)$ . In order to visualize such a function in 3D space we need to "fix" one of the coordinates: the position of the electron $\psi_\lambda(r_e,\mathbf{r}_h)$ or the position of the hole $\psi_\lambda(\mathbf{r}_e,r_h)$ .

How to fix the position of the particle

The position of the fixed particle is provided in direct (fractional) coordinates by tags BSEHOLE or BSEELECTRON for a hole or an electron, respectively. The tag NBSEEIG sets the number of excitons wavefunction that needs to be computed.

When fixing the position of the particle it is important to make sure that it is not fixed exactly at the center of an atom or coincide with a node of the wavefunction. To avoid that, make sure to shift the fixed coordinate slightly away from the center of the atom. Furthermore, the wavefunction of the fixed particle is taken at the nearest $\mathbf{G}$ -vector, whose exact position is written in the OUTCAR file

hole/electron position is fixed at:

How to plot the exciton wavefunction

VASP computes the charge density of a particular excitonic state, i.e., $\rho_\lambda(\mathbf{r})=|\psi_\lambda(r_e,\mathbf{r}_h)|^2$ or $\rho_\lambda(\mathbf{r})=|\psi_\lambda(\mathbf{r}_e,r_h)|^2$ , and writes the resulting charge density into CHGCAR.XX files, where XX stands for the index $\lambda$ of the state. VASP computes the charge density by transforming the unit cell with k-points into a supercell. Thus, the exciton charge density is written for a supercell of dimensions ${\rm NKX}\times{\rm NKX}\times{\rm NKX}$ .

Warning: The size of CHGCAR.XX files can get very large. Estimate the CHGCAR.XX file size as follows

(NGX*NKX)\times(NGX*NKX)\times(NGX*NKX)*18

bytes. Here, NG{X,Y,Z} is the number of grid points and NK{X,Y,Z} are the number of k-points along the axis.

{{NB|warning|The exciton charge density only accounts for the plane-wave part of the wave function and the augmentation terms are neglected.}

Degeneracy

The calculated excitonic states can be degenerate, i.e., multiple eigenvectors have the same energy. For the correct analysis the degenerate states should be added together.

Related tags and sections

CHGCAR, NBSEEIG, BSEHOLE, BSEELECTRON, BSE

References

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-Plotting the wavefunction of an exciton can be instrumental for analyzing the symmetry, position, and localization of the excitonic state.
+Plotting the wavefunction or charge density corresponding to an exciton can be instrumental for analyzing the symmetry, position, and localization of the excitonic state.
-The exciton wavefunction <math>\psi_\lambda(r_e,r_h)=\sum_{vc} A_{vc}^\lambda \psi_v(r_h)\psi_c^*(r_e)</math> is written as a function of coordinates of two particles: one hole and one electron {{cite|gatti:prb:2013}}.  In order to visualize this such function in 3D space we need to "fix" one of the coordinates: the position of the electron <math>\psi_\lambda(r^*_e,r_h)</math> or  the position of the hole <math>\psi_\lambda(r_e,r^*_h)</math>.
+The exciton wavefunction is written as a function of coordinates of two particles (one hole and one electron) <math>\psi_\lambda(\mathbf{r}_e,\mathbf{r}_h)=\sum_{vc} A_{vc}^\lambda \psi_v(\mathbf{r}_h)\psi_c^*(\mathbf{r}_e)</math> {{cite|gatti:prb:2013}}.  In order to visualize such a function in 3D space we need to "fix" one of the coordinates: the position of the electron <math>\psi_\lambda(r_e,\mathbf{r}_h)</math> or  the position of the hole <math>\psi_\lambda(\mathbf{r}_e,r_h)</math>.
-# How to fix the position of the particle:
+=== How to fix the position of the particle ===
-When fixing the position of the particle it is important to make sure that the hole or electron is not fixed exactly at the center of an atom as that can coincide with a node of the wavefunction and yield wrong results. Thus, make sure to shift the fixed coordinate slightly away from the center of the atom. Furthermore, the wavefunction of the fixed particle is taken at the nearest <math>\mathbf{G}</math>-vector, whose exact position is written in the {{FILE|OUTCAR}} file ('hole/electron position is fixed at:').
+The position of the fixed particle is provided in direct (fractional) coordinates by tags {{TAG|BSEHOLE}} or {{TAG|BSEELECTRON}} for a hole or an electron, respectively. The tag {{TAG|NBSEEIG}} sets the number of excitons wavefunction that needs to be computed.
-The position of the fixed particle is provided in direct (fractional) coordinates by tags {{TAG|BSEHOLE}} or {{TAG|BSEELECTRON}} for a hole or an electron, respectively. The tag {{TAG|NBSEEIG}} sets the number of excitons wavefunction that are computed.
-VASP can compute the charge density of a particular excitonic state, i.e., <math>\rho_\lambda(r)=\psi_\lambda(r^*_e,r_h)\psi^*_\lambda(r^*_e,r_h)</math> or <math>\rho_\lambda(r)=\psi_\lambda(r_e,r^*_h)\psi^*_\lambda(r_e,r^*_h)</math>, and writes the resulting charge density into {{FILE|CHGCAR}}.XX files, where XX stands for the index <math>\lambda</math> of the state. VASP computes the charge density by transforming the unit cell with k-points into a supercell. Thus, the exciton charge density is written for a supercell of dimensions <math>{\rm NKX}\times{\rm NKX}\times{\rm NKX}</math>.
-{{NB|warning| The size of CHGCAR.XX files can get very large. Estimate the CHGCAR.XX file size as follows: <math>(NGX*NKX)*(NGX*NKX)*(NGX*NKX)*18</math> bytes.}}
+When fixing the position of the particle it is important to make sure that it is not fixed exactly at the center of an atom or coincide with a node of the wavefunction. To avoid that, make sure to shift the fixed coordinate slightly away from the center of the atom. Furthermore, the wavefunction of the fixed particle is taken at the nearest <math>\mathbf{G}</math>-vector, whose exact position is written in the {{FILE|OUTCAR}} file
+ hole/electron position is fixed at:
+[[File:HBN exciton.png|500px|thumb|Charge density of the first bright exciton in hBN.]]
-{{NB|warning|The exciton charge density only accounts for the plane-wave part of the wave function, thus, all the augmentation terms are not neglected.}
-The calculated excitonic states can be degenerate, i.e., multiple eigenvectors have the same energy, thus, it is important to add together all the degenerate states to be able to analyze the exction.
+=== How to plot the exciton wavefunction ===
+VASP computes the charge density of a particular excitonic state, i.e., <math>\rho_\lambda(\mathbf{r})=|\psi_\lambda(r_e,\mathbf{r}_h)|^2</math> or <math>\rho_\lambda(\mathbf{r})=|\psi_\lambda(\mathbf{r}_e,r_h)|^2</math>, and writes the resulting charge density into {{FILE|CHGCAR}}.XX files, where XX stands for the index <math>\lambda</math> of the state. VASP computes the charge density by transforming the unit cell with k-points into a supercell. Thus, the exciton charge density is written for a supercell of dimensions <math>{\rm NKX}\times{\rm NKX}\times{\rm NKX}</math>.
-# Example:
+{{NB|warning| The size of {{FILE|CHGCAR}}.XX files can get very large. Estimate the {{FILE|CHGCAR}}.XX file size as follows <math>(NGX*NKX)\times(NGX*NKX)\times(NGX*NKX)*18</math> bytes. Here, NG{X,Y,Z} is the number of grid points and NK{X,Y,Z} are the number of k-points along the axis. }}
-In the figure, we provide an example, of the exciton wavefunction of the first bight excitonic state in hexagonal BN.
+{{NB|warning|The exciton charge density only accounts for the plane-wave part of the wave function and the augmentation terms are neglected.}
+=== Degeneracy ===
+The calculated excitonic states can be degenerate, i.e., multiple eigenvectors have the same energy. For the correct analysis  the degenerate states should be added together.
-[[File:HBN exciton.png|500px|thumb|Charge density of the first bright exciton in hBN.]]