Plotting exciton wavefunction: Difference between revisions

Latest revision as of 10:52, 19 March 2024

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

The exciton wavefunction is written as a function of coordinates of two particles (one hole and one electron) $\psi_\lambda(\mathbf{r}_e,\mathbf{r}_h)=\sum_{vc} A_{vc}^\lambda \phi_v^*(\mathbf{r}_h)\phi_c(\mathbf{r}_e)$ . 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 the tags BSEHOLE or BSEELECTRON for a hole or an electron, respectively. The tag NBSEEIG sets the number of exciton wavefunctions that need to be computed.

When fixing the position of the particle, ensure that it is not fixed exactly at the center of an atom or coincides with a node of the wavefunction. To avoid that, 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 position is fixed at:

or

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 CHG.XX files, which can be visualized using standard tools like VESTA, ASE, etc. Here, 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 ${\rm NKX}\times{\rm NKX}\times{\rm NKX}$ . The size of the supercell is written in the OUTCAR file

FFT grid for supercell:

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

(\mathrm{NGX*NKX})\times(\mathrm{NGX*NKX})\times(\mathrm{NGX*NKX})*12

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

Warning: The exciton charge density only accounts for the plane-wave part of the wavefunction, 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.

Mind: Calculation of the exciton wavefunction is only supported with IBSE=0 and ANTIRES=0.

Related tags and sections

CHG, NBSEEIG, BSEHOLE, BSEELECTRON, BSE

References

@@ Line 1: / Line 1: @@
-Plotting the wavefunction of an exciton can be instrumental for analyzing the symmetry, position, and localization of the excitonic state.
+[[File:HBN exciton.png|400px|thumb|Charge density of the first bright exciton in hBN.]]
+Plotting the wavefunction or charge density corresponding to an exciton can be instrumental in analyzing the excitonic state's symmetry, position, and localization.
-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>.
-# 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 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.}}
-{{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.
-# Example:
-In the figure, we provide an example, of the exciton wavefunction of the first bight excitonic state in hexagonal BN.
-[[File:HBN exciton.png|500px|thumb|Charge density of the first bright exciton in hBN.]]
+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 \phi_v^*(\mathbf{r}_h)\phi_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 ===
+The position of the fixed particle is provided in direct (fractional) coordinates by the tags {{TAG|BSEHOLE}} or {{TAG|BSEELECTRON}} for a hole or an electron, respectively. The tag {{TAG|NBSEEIG}} sets the number of exciton wavefunctions that need to be computed.
+When fixing the position of the particle, ensure that it is not fixed exactly at the center of an atom or coincides with a node of the wavefunction. To avoid that, 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 position is fixed at:
+or
+ electron position is fixed at:
+=== 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|CHG}}.XX files, which can be visualized using standard tools like VESTA, ASE, etc. Here, 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>.
+The size of the supercell is written in the {{FILE|OUTCAR}} file
+ FFT grid for supercell:
+{{NB|mind| The size of {{FILE|CHG}}.XX files can get very large. Estimate the {{FILE|CHG}}.XX file size as follows <math>(\mathrm{NGX*NKX})\times(\mathrm{NGX*NKX})\times(\mathrm{NGX*NKX})*12</math> bytes. Here, NG{X,Y,Z} is the number of grid points and NK{X,Y,Z} is the number of k-points along the axis.}}
+{{NB|warning|The exciton charge density only accounts for the plane-wave part of the wavefunction, 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.
+{{NB|mind|Calculation of the exciton wavefunction is only supported with {{TAG|IBSE}}{{=}}0 and {{TAG|ANTIRES}}{{=}}0.}}
 == Related tags and sections ==
-{{FILE|CHGCAR}}, {{TAG|NBSEEIG}}, {{TAG|BSEHOLE}}, {{TAG|BSEELECTRON}}, [[Bethe-Salpeter-equations calculations|BSE]]
+{{FILE|CHG}}, {{TAG|NBSEEIG}}, {{TAG|BSEHOLE}}, {{TAG|BSEELECTRON}}, [[Bethe-Salpeter-equations calculations|BSE]]
 == References ==
 [[Category:VASP]][[Category:Many-body perturbation theory]][[Category:Bethe-Salpeter equations]]