Defect Formation Energy Calculation

Tutorial Created with Software Release: 2026-1
Topics: Energy Capture & Storage
Methodology: Periodic Quantum Mechanics
Products Used: MS Maestro, Quantum Espresso

Tutorial files

450 MB
This tutorial is written for use with a 3-button mouse with a scroll wheel.
Words found in the Glossary of Terms are shown like this: Workspacethe 3D display area in the center of the main window, where molecular structures are displayed

 

Tip: You can hover over a glossary term to display its definition. You can click on an image to expand it in the page.
Abstract:

 

In this tutorial, we will learn how to generate point defects and how to calculate their defect formation energy, which includes a correction term for charged defects. Additionally, we will have a brief look at the density of states and the projected density of states.

 

Tutorial Content

  1. Introduction to Defect Energy Calculations

  1. Creating Projects and Importing Structures

  1. Generating Defect Structures

  2. Analyzing Density of States and Projected Density of States

  1. Calculating Defect Formation Energies

  1. Conclusion and References

  1. Glossary of Terms

1. Introduction to Defect Energy Calculations

Defects are a discontinuity of the crystal lattice that allow for a high degree of chemical, structural, and functional flexibility. Point defects, such as vacancies, substitutions, interstitials, and antisites, are ubiquitous in any crystal, whether unintentional or by design (e.g., doping). Even at low concentrations they significantly impact the electronic, optical, structural, and mechanical properties of crystals. Therefore, predicting defect concentrations is crucial in solid state materials science.

Defect concentration is directly related to the defect formation energy. Periodic density functional theory (DFT) provides valuable insight into structures with point defects and their energetics. In atomistic theory, the defect formation energy is calculated as follows:

Edef(q, EF) = E(defect) - E(bulk) - ∑niμi + q(EF + EVBM) + Ecorr

E(defect) is the total energy of the optimized system containing the defect. E(bulk) is the total energy of the optimized, perfect reference system (bulk or slab). ni depends on the defect type, where ni = 1 when the defect involves the addition of an atom to the system (e.g. an interstitial defect) and ni = -1 when the defect involves the removal of an atom from the system (e.g. a vacancy defect). μi is the defect chemical potential. q is the total charge of the defect system. EF is the Fermi energy relative to the valence band maximum, which, in this context, takes values between zero and the band gap. EVBM is the valence band maximum energy. Ecorr is the defect correction energy for charged systems.

Here, we will model the defects using plane wave DFT with periodic boundary conditions (PBC), as implemented in the Quantum ESPRESSO package. In systems with PBC, special care is needed when studying local charged defects, as their energetic properties slowly converge with the periodic cell size. Accurate and converged defect formation energies require correction for the artificial long-range Coulomb interactions between charged defects. Various correction methods have been published. The Freysoldt, Neugebauer, and Van de Walle (FNV) correction is particularly notable as it only requires two supercell calculations and avoids non-interpretable fitting constants (see the References for more information). The FNV correction is implemented in the Ecorr term in the above equation.

In this tutorial, we will demonstrate how to set up and optimize several vacancy defects (neutral and charged) and how to calculate and analyze their defect formation energies. This involves calculating all the terms in the above equation for a neutral, one- , and two-fold positively charged oxygen vacancy in a 2x2x2 crystal supercell of MgO. We will use the Defect Setup Calculations and Defect Formation Energy panels for periodic DFT calculations in Materials Science (MS) Maestro. Additionally, we will briefly look at the density of states (DOS) and projected density of states (PDOS) from the DFT calculations.

If you are unfamiliar with handling periodic structures or performing solid-state calculations with MS Maestro, two foundational tutorials are available and recommended: Building and Manipulating Crystal Structures and Electronic Structure Calculations of Bulk Crystals Using Quantum ESPRESSO.

2. Creating Projects and Importing Structures

At the start of the session, change the file path to your chosen Working Directorythe location where files are saved in MS Maestro to make file navigation easier. Each session in MS Maestro begins with a default Scratch Projecta temporary project in which work is not saved, closing a scratch project removes all current work and begins a new scratch project, which is not saved. A MS Maestro project stores all your data and has a .prj extension. A project may contain numerous entries corresponding to imported structures, as well as the output of modeling-related tasks. Once a project is saved, it is automatically saved each time a change is made.

Structures can be built in MS Maestro or can be imported using File > Import Structures (or drag-and-dropped), and are added to the Entry Lista simplified view of the Project Table that allows you to perform basic operations such as selection and inclusion and Project Tabledisplays the contents of a project and is also an interface for performing operations on selected entries, viewing properties, and organizing structures and data. The Entry Lista simplified view of the Project Table that allows you to perform basic operations such as selection and inclusion is located to the left of the Workspacethe 3D display area in the center of the main window, where molecular structures are displayed. The Project Tabledisplays the contents of a project and is also an interface for performing operations on selected entries, viewing properties, and organizing structures and data can be accessed by Ctrl+T (Cmd+T) or Window > Project Table if you would like to see an expanded view of your project data.

  1. Double-click the Materials Science icon

Figure 2-1. Changing the Working Directory.

  1. Go to File > Change Working Directory
  2. Find your directory, and click Choose
  3. Pre-generated files are included for running jobs or examining output. Download the zip file here: https://www.schrodinger.com/sites/default/files/s3/release/current/Tutorials/zip/defect_formation_energy.zip
  4. After downloading the zip file, unzip the contents in your Working Directorythe location where files are saved for ease of access throughout the tutorial

Figure 2-2. Saving the project.

  1. Go to File > Save Project As
  2. Change the File name to defect_formation_energy_tutorial, click Save
    • The project is now named defect_formation_energy_tutorial.prj

Figure 2-3. The imported MgO structure file.

  1. Go to File > Import Structures
  2. Navigate to where you downloaded the provided tutorial files, choose Chapter_02 > input_structure.maegz and click Open
  3. A new entry titled MgO_64_bulk has been imported that is both selected(1) the atoms are chosen in the Workspace. These atoms are referred to as "the selection" or "the atom selection". Workspace operations are performed on the selected atoms. (2) The entry is chosen in the Entry List (and Project Table) and the row for the entry is highlighted. Project operations are performed on all selected entries and includedthe entry is represented in the Workspace, the circle in the In column is blue in the Workspacethe 3D display area in the center of the main window, where molecular structures are displayed
    • To remove the bonds for a cleaner image, open the workspace configuration panel ()and unclick bonds

Note: Please refer to the Glossary of Terms for the difference between includedthe entry is represented in the Workspace, the circle in the In column is blue and selected(1) the atoms are chosen in the Workspace. These atoms are referred to as "the selection" or "the atom selection". Workspace operations are performed on the selected atoms. (2) The entry is chosen in the Entry List (and Project Table) and the row for the entry is highlighted. Project operations are performed on all selected entries.

The imported entry is an optimized 2x2x2 supercell. The structure of the crystal unit cell  is available from the Materials Project Database and can be imported using the Query Materials Project Database panel (mp-1265). The 2x2x2 supercell can be generated using the Workspace Tools for Periodic Structures ( > Build Cell > Extents). See the Electronic Structure Calculations of Bulk Crystals Using Quantum ESPRESSO tutorial for more details.

 

 

3. Generating Defect Structures

In this section, we will generate charge defects for oxygen vacancies in MgO with charges of 0, +1 and +2 using the Defect Setup Calculations panel. This already includes most calculations necessary for the calculation of the defect formation energy.

Figure 3-1. Importing the bulk system into the Defect Setup Calculations panel.

  1. With MgO_64_bulk includedthe entry is represented in the Workspace, the circle in the In column is blue in the workspacethe 3D display area in the center of the main window, where molecular structures are displayed and selected(1) the atoms are chosen in the Workspace. These atoms are referred to as "the selection" or "the atom selection". Workspace operations are performed on the selected atoms. (2) The entry is chosen in the Entry List (and Project Table) and the row for the entry is highlighted. Project operations are performed on all selected entries in the entry lista simplified view of the Project Table that allows you to perform basic operations such as selection and inclusion, go to Tasks > Materials > Quantum Mechanics > Quantum ESPRESSO > Defect Setup Calculations
  2. Click Import to load the MgO_64_bulk structure

Figure 3-2. Selecting the vacancy atom for the defect and creating the first defect.

  1. In the workspacethe 3D display area in the center of the main window, where molecular structures are displayed, select(1) the atoms are chosen in the Workspace. These atoms are referred to as "the selection" or "the atom selection". Workspace operations are performed on the selected atoms. (2) The entry is chosen in the Entry List (and Project Table) and the row for the entry is highlighted. Project operations are performed on all selected entries an oxygen atom in the middle of the cell
    • Here we use oxygen atom number 28, but we could use any other symmetry equivalent oxygen atom
  2. In Vacancy atoms click Load selected atoms
    • 28 should appear as the vacancy atom
  3. Set the Charge to 0
  4. Click Create defect
    • The defect position is recorded

Figure 3-3. Creating the second defect and opening the Initial Parameters panel.

  1. Click Add Defect to create a second defect
    • Ensure the following steps are performed for Defect 2
  2. In case it is no longer selected, select the same oxygen atom (28) in the workspace
  3. In Vacancy atoms click Load selected atoms
  4. Set the Charge to 1
  5. Click Create defect
  6. Go to Initial Parameters

Figure 3-4. Setting the total magnetization in the Initial Parameters panel.

  1. Check Total magnetization and set it to 1
  2. Click Save
    • A new entry is added to the entry lista simplified view of the Project Table that allows you to perform basic operations such as selection and inclusion and is automatically includedthe entry is represented in the Workspace, the circle in the In column is blue in the workspacethe 3D display area in the center of the main window, where molecular structures are displayed
    • The vacancy atom site is shown with a large red sphere

Specifying a total magnetization of +1 is necessary for the oxygen vacancy with a charge of +1, because of the uneven number of electrons (doublet state)

Figure 3-5. Creating the third defect and opening the Pseudopotentials dialog box.

  1. Uncheck Show in Workspace for Defect 2
  2. Includethe entry is represented in the Workspace, the circle in the In column is blue the first entry again and select oxygen atom 28 in the workspace
  3. Click Add Defect
    • Ensure the following steps are performed for Defect 3
  4. In Vacancy atoms click Load selected atoms
  5. Set the Charge to 2
  6. Click Create defect
  7. Go to Pseudopotentials
    • The Defect Setup Calculations - Pseudopotentials dialog box opens  

Figure 3-6. Specifying the pseudopotentials for the periodic DFT runs.

Starting with release 25-3, the set of recommended PBE ultrasoft pseudopotentials is distributed with the suite and is automatically available and selected. In case you want to utilize a different set of pseudopotentials, some sites are referenced here. If you want to select non-default pseudopotentials, choose Browse and navigate to your files.

  1. Click OK to close the Pseudopotentials panel

Figure 3-7. Loading predefined options for the DFT calculations and running the job.

  1. Click Load Options
  2. Navigate to where you saved the tutorial files and select Chapter_03 > MgO_V_O_q012.cfg, click Open
    • A message appears informing you about the successful import of the required settings for the Quantum Espresso calculations. You can inspect the settings and adjust them by clicking Advanced Options.

For more information about parameter settings and performing convergence testing, see the Electronic Structure Calculations of Bulk Crystals Using Quantum ESPRESSO tutorial and the Quantum ESPRESSO Calculations documentation.

  1. Ensure the option to Optimize atomic positions is checked

You can uncheck this option in case you only want to recalculate the defect formation energy with a different functional on already optimized structures.

  1. Change the Job Name to V_O_q012
  2. Adjust the job settings () as needed
    • This job requires a CPU host and can be completed in about 12 hours with 32 CPU cores
    • This job runs a series of 10 subjobs for the bulk system and every defect, totaling to 40 in this case
  3. If you would like to run the job yourself, click Run. Otherwise, import the provided results from the zip file via File > Import Structures: Section_03 > V_O_q012 > V_O_q012.maegz
  4. Close the Defect Setup Calculations panel

4. Analyzing Density of States and Projected Density of States

The defect setup job also calculates the density of states (DOS) and the projected density of states (PDOS). The DOS plot shows the energies at which the electronic states are located. States at energies below the Fermi level are occupied (or bound) states, while states above the Fermi level are unoccupied. The highest-energy band of occupied states is commonly referred to as the valence band and its maximum energy value as Valence Band Maximum (VBM) , while the lowest-energy band of unoccupied states is referred to as the conduction band, and its minimum energy value as Conduction Band Minimum (CBM). The DOS is calculated for the bulk and defect structures. While the DOS plot provides information about the total density of states, the projected density of states (PDOS) reveals individual contributions of atoms and atomic orbitals to the total DOS.

Figure 4-1. Opening the Projected Density of States Viewer.

The Defect Setup Calculations panel provides the results in a single group with 40 entries. Here, we will analyze the PDOS for two specific orbitals of the oxygen vacancy defect with charge +1.

  1. With the V_O_q012 (40) entry group selected(1) the atoms are chosen in the Workspace. These atoms are referred to as "the selection" or "the atom selection". Workspace operations are performed on the selected atoms. (2) The entry is chosen in the Entry List (and Project Table) and the row for the entry is highlighted. Project operations are performed on all selected entries, and the V_O_q012_defect_2_2_pdos_2 entry included, click on the WAM button and select Projected Density of States Viewer > Row 25: V_O_q012_defect_2_2_pdos_2
    • The Projected Density of States Viewer panel opens
    • Alternatively, go to Tasks > Materials > Quantum Mechanics > Quantum ESPRESSO > Projected Density of States Viewer and load the results via the Import button at the top

Figure 4-2. Adding a set for the O atoms in the Projected Density of States Viewer.

We will now analyze the contributions of the O-2p orbital.

  1. Change Selected atoms to Atom set O
  2. In the workspacethe 3D display area in the center of the main window, where molecular structures are displayed, in the atom selection menu, click on Define and choose Element > O > Add > OK.
    • All 31 O atoms in the includedthe entry is represented in the Workspace, the circle in the In column is blue structure will be highlighted
    • You can also manually select all O atoms by Ctrl-clicking (Cmd-clicking) the Mg atoms in the workspacethe 3D display area in the center of the main window, where molecular structures are displayed
  3. Back in the Projected Density of States Viewer, you will see all the O atoms listed as the Selected atoms. Click Add to List.

Figure 4-3. Settings for the plot in the Projected Density of States Viewer.

  1. Set the Gaussian broadening to 0.2 eV and click Recompute
  2. Check Zero to Fermi energy
    • This will set the zero of energy on the horizontal axis to the Fermi energy. The actual value of the Fermi energy is reported in the status bar at the foot of the panel.
    • Please note: Do not use this option when comparing the DOS of differently charged defects. The Fermi energy is different for differently charged systems!
  3. Set the Spin plot to Mirror spins

 

Figure 4-4. Selecting the orbitals for the PDOS plot and adjusting the axes.

  1. In the Atomic PDOS section, select Total DOS, and in the Atomic Orbital PDOS section select Atom set O (n = 2, l = 1)
    • You have now plotted the total DOS and the contributions of the O-2p orbital
  2. Open the Edit axis, curve and image parameters panel
  3. For X-Axis, set the Min to -10 and the Max to 10
  4. For the Y-Axis, set the Min to -50 and the Max to 50
  5. Click Apply
  6. Click Ok

The plot has been adjusted to see the areas of interest. Feel free to explore the other DOS/PDOS plots before moving onto the next section.

The PDOS indicates that the defect electronic state has a strong oxygen p-orbital character, suggesting that it is localized on the oxygen atoms surrounding the vacancy. However, this is a rare case when such a conclusion can be misleading. Further analysis can be conducted by examining the spin density of the system.

Figure 4-5. Visualizing the spin density information.

  1. Close the Projected Density of States Viewer panel
  2. Select(1) the atoms are chosen in the Workspace. These atoms are referred to as "the selection" or "the atom selection". Workspace operations are performed on the selected atoms. (2) The entry is chosen in the Entry List (and Project Table) and the row for the entry is highlighted. Project operations are performed on all selected entries and includethe entry is represented in the Workspace, the circle in the In column is blue the V_O_q012_defect_2_2_polarization_surface_8 entry
  3. Optional: Choose a visualization state of your liking, here we have used the ball-and-stick representation and undisplayed bonds
  4. Click the S next to the entry and select the only available surface to visualize it in the workspacethe 3D display area in the center of the main window, where molecular structures are displayed

 

Most of the spin density is evident as a spheroid localized at the vacancy site, with only minor contributions from the p-orbitals of the surrounding oxygen atoms. However, this characteristic of the defect is not reflected in the PDOS because the projectors are only constructed on atomic sites. There are no vacancy-centered atomic orbitals that could reflect the so-called F-center character of the oxygen vacancy in MgO.

5. Calculating Defect Formation Energies

In this section, we will calculate the defect formation energy for an oxygen vacancy in three charge states using the Defect Formation Energy panel and the output of Section 3.

Figure 5-1. Opening the Defect Formation Energy panel.

  1. With the group V_O_q012 (40) selected(1) the atoms are chosen in the Workspace. These atoms are referred to as "the selection" or "the atom selection". Workspace operations are performed on the selected atoms. (2) The entry is chosen in the Entry List (and Project Table) and the row for the entry is highlighted. Project operations are performed on all selected entries in the entry lista simplified view of the Project Table that allows you to perform basic operations such as selection and inclusion, use the WAM (workflow action menu) button () to open the Defect Formation Energy panel for the first defect (Row 22: V_O_q012_defect_1_1_potential_surface_9)
    • Alternatively, go to Tasks > Materials > Quantum Mechanics > Quantum ESPRESSO > Defect Formation Energy
    • The Defect Formation Energy panel opens

 

Figure 5-2. Setting the dielectric constant in the Defect Formation Energy panel.

Opening the panel via the WAM button will automatically import the calculation results including defect structure, its charge state, and other important settings. When opening the panel via the Tasks menu, the results have to be imported manually.

  1. Set the Dielectric constant type to Scalar and the value to 10.66. This value is used for the FNV defect correction energy calculation.
    • This value corresponds to the static dielectric constant for MgO taken from literature (see References)

If the atoms are allowed to relax after introducing the defect, as it is the case in this example, the electronic and the ionic contribution to the dielectric constant are needed. If the atoms are not allowed to relax during the calculation, only the electronic contribution to the screening of the defect charge needs to be considered.

Additionally, this example uses the dielectric constant for a cubic MgO and thus assumes isotropic screening. For anisotropic systems the dielectric constant setting would need to be switched from Scalar to Tensor, and the principal components of the dielectric tensor should be set.

Figure 5-3. The defect formation energy for the neutral defect.

  1. In the Defect formation energy section at the bottom, set n*μ to 437.475 eV

The expected value here is the product of ni and the chemical potential μi for the defect. ni depends on the defect type, where ni = 1 when the defect involves the addition of an atom to the system (e.g. an interstitial defect) and ni = -1 when the defect involves the removal of an atom from the system (e.g. a vacancy defect). Thus, for vacancy defects, n equals -1.

μi is the defect chemical potential. It is calculated with respect to the defects reservoir. The choice of chemical potential depends on presumed experimental conditions. In this tutorial, we assume that the vacancy is formed by thermal reduction of MgO, whereby the O2 molecule escapes the crystal into a gas phase leaving two O vacancies behind. The chemical potential, in this case, is defined as ½ of the ground state energy of the O2 molecule in the gas phase.  This energy can be calculated separately as a triplet state of the O2 molecule in the box using the same settings that were used for the defect calculation. The chemical potential is -437.475 eV, and the value to be set in the box is 437.475 eV.

The total defect formation energy at EF=0 above EVBM is 6.7 eV. For the neutral vacancy, the defect energy correction is zero. Therefore, we do not perform any calculations to determine the defect correction energy. The term ΔEtot describes the difference in total energy between the optimized defect and bulk structures. VBMBulk is the valence band maximum as calculated for the bulk system, and Ef is the Fermi energy.  By changing the EF energy above EVBM we observe that the defect formation energy remains unchanged, which is expected for the neutral defect.

Figure 5-4. Running the defect correction calculation for the defect with +1 charge.

  1. Via the WAM button (), open the Defect Formation Energy panel for the second defect (Row 32: V_O_q012_defect_2_2_potential_surface_9)
  2. Make sure the settings for the dielectric constant have not changed

Now we want to calculate the defect correction energy for this charged defect.

  1. Change the Job Name to defect_correction_V_O_q1
  2. Adjust the job settings () as needed
    • This job requires a Linux CPU host and can be completed in about 1 minute on a single CPU core
  3. If you would like to run the job yourself, click Run. Otherwise, import the provided results from the zip file via File > Import Structures: Section_05 > defect_correction_V_O_q1 > defect_correction_V_O_q1.maegz

Figure 5-5. The results for the defect with +1 charge.

  1. With the new defect_correction entry group (defect_correction_V_O_q1 (2)) selected(1) the atoms are chosen in the Workspace. These atoms are referred to as "the selection" or "the atom selection". Workspace operations are performed on the selected atoms. (2) The entry is chosen in the Entry List (and Project Table) and the row for the entry is highlighted. Project operations are performed on all selected entries in the entry lista simplified view of the Project Table that allows you to perform basic operations such as selection and inclusion, use the WAM (workflow action menu) button () to open the Defect Formation Energy panel again
    • Alternatively, you can load the job directly in the Defect Formation Energy panel via the Load Existing Job option (select the .spxgz file)
    • The graph in the panel is now populated
  2. The boundaries and legend in the graph are movable. Move the fitting region boundaries such that they encompass the region at large distances from the defect after the minimum. Move the legend below the data points to have a better view on the data. Click Fit.
    • The defect correction energy and the defect formation energy values are updated

The zero defect correction and the alignment value are used to calculate the defect correction energy. The zero defect correction value describes the contribution of the macroscopically screened lattice energy of the defect charge with compensating background. Fitting the potential far away from the defect position generates the alignment value. For more detailed information see the original publications by Freysoldt et al. (see References).

The total defect formation energy for the V+ is 3.9 eV. In this case, the defect correction is not zero because we are calculating a charged defect. Additionally, the term containing VBMBulk and Ef is no longer zero, since the charge is now +1.

Figure 5-6. Running the defect correction calculation for the defect with +2 charge.

  1. Via the WAM button (), open the Defect Formation Energy panel for the second defect (Row 42: V_O_q012_defect_3_3_potential_surface_9)
  2. Make sure the settings for the dielectric constant have not changed
  3. Change the Job Name to defect_correction_V_O_q2
  4. Adjust the job settings () as needed
    • This job requires a Linux CPU host and can be completed in about 1 minute on a single CPU core
  5. If you would like to run the job yourself, click Run. Otherwise, import the provided results from the zip file via File > Import Structures: Section_05 > defect_correction_V_O_q2 > defect_correction_V_O_q2.maegz

Figure 5-7. The results for the defect with +2 charge.

  1. With the new defect_correction entry group (defect_correction_V_O_q2 (2)) selected(1) the atoms are chosen in the Workspace. These atoms are referred to as "the selection" or "the atom selection". Workspace operations are performed on the selected atoms. (2) The entry is chosen in the Entry List (and Project Table) and the row for the entry is highlighted. Project operations are performed on all selected entries in the entry lista simplified view of the Project Table that allows you to perform basic operations such as selection and inclusion, use the WAM (workflow action menu) button () to open the Defect Formation Energy panel again
    • Alternatively, you can load the job directly in the Defect Formation Energy panel via the Load Existing Job option (select the .spxgz file)
    • The graph in the panel is now populated
  2. Move the fitting region boundaries such that they encompass the region at large distances from the defect after the minimum. Move the legend below the data points to have a better view on the data. Click Fit.
    • The defect correction energy and the defect formation energy values are updated

The calculated defect formation energy of the oxygen vacancy with charge +2 is 1.4 eV.

We established that at EF=0 above VBM the oxygen vacancy formation energy follows the trend:

Edef (VO2+) < Edef (VO1+) < Edef (VO0)

The defect formation energy is a function of the charge and the Fermi energy. The latter can range from zero to the band gap. With respect to the Fermi energy, the defect formation energy for the neutral defect (defect 1, green line) is a horizontal line as it does not depend on the Fermi energy. However, charged defects do show a linear dependency on the Fermi energy with the slope proportional to the charge (see formula 1). Thus, the defect energy as a function of EF for a one-fold positively charged oxygen vacancy (defect 2, orange line) shows a smaller incline than the two-fold positively charged vacancy (defect 3, blue line). The defect energy diagram indicates that in the EF energy range 0-2.2 eV above VBM the most stable oxygen vacancy charged state is +2. In the EF range 2.2-2.6 eV above VBM the most stable vacancy charge state is +1. At higher Fermi energies, the neutral vacancy is the most stable defect. While considering defect energy results it is important to keep in mind the following factors:

  1. The accuracy of the results significantly depends on the chosen DFT method. Specifically, the semilocal PBE functional used in this tutorial is known to underestimate the band gap energy and to lead to an artificial delocalisation of the wavefunction due to the self-interaction error. Therefore, it is advisable to refine the calculations using more accurate exact exchange hybrid functionals.
  2. The specifics of the experimental conditions are incorporated into the defect energy calculations through the chemical potential value, which must be calculated or estimated separately. Though the choice of chemical potential strongly influences the absolute defect formation energy, it does not affect the relative energies of the defects' different charged states. In this study, the chemical potential is calculated in a standard approach corresponding to weakly reducing experimental conditions. One can envisage strongly reducing conditions, e.g. in H2 atmosphere or in contact with a highly oxidizable metal or a semiconductor, or strongly oxidative conditions such as in high partial pressure of H2O, O2 or ozone. The chemical potentials for these cases must be calculated accordingly.
  3. It is important to consider different types of possible defects and their relative energies. Defects with higher energies are unlikely to exist in measurable concentrations and can be ignored.
  4. Low defect energies typically indicate an unstable crystal structure and a tendency toward phase transitions to more stable structures. Such cases warrant special investigation.
  5. The current workflow for calculating defect energies is suitable for both bulk and surface defects in the slab model. However, note that the FNW scheme currently implemented is designed for bulk systems only. Therefore, it must not be applied to surface defects. Recall that the correction energy is zero for neutral defects and is small for large systems.

6. Conclusion and References

In this tutorial, we used an example of oxygen vacancy in MgO to learn how to generate the defect geometry, how to set up and perform defect energy calculations for neutral and charged defects. We looked at the projected density of states (PDOS) and calculated the defect correction term for the charged defects as well as the total defect formation energy for all defects. We showed the dependency of the defect formation energy on the charge and the Fermi energy.

For further learning:

For introductory content, focused on navigating the Schrödinger Materials Science interface, an Introduction to Materials Science Maestro tutorial is available. Please visit the materials science training website for access to 100+ tutorials. For scientific inquiries or technical troubleshooting, submit a ticket to our Technical Support Scientists at help@schrodinger.com.

For self-paced, asynchronous, online courses in Materials Science modeling, including access to Schrödinger software, please visit the Schrödinger Online Learning portal on our website.

For some related practice, proceed to explore other relevant tutorials:

For further reading:

7. Glossary of Terms

Entry List - a simplified view of the Project Table that allows you to perform basic operations such as selection and inclusion

Included - the entry is represented in the Workspace, the circle in the In column is blue

Project Table - displays the contents of a project and is also an interface for performing operations on selected entries, viewing properties, and organizing structures and data

Recent actions - This is a list of your recent actions, which you can use to reopen a panel, displayed below the Browse row. (Right-click to delete.)

Scratch Project - a temporary project in which work is not saved, closing a scratch project removes all current work and begins a new scratch project

Selected - (1) the atoms are chosen in the Workspace. These atoms are referred to as "the selection" or "the atom selection". Workspace operations are performed on the selected atoms. (2) The entry is chosen in the Entry List (and Project Table) and the row for the entry is highlighted. Project operations are performed on all selected entries

Working Directory - the location where files are saved

Workspace - the 3D display area in the center of the main window, where molecular structures are displayed