Comparing Calculations In Plane Wave And Atomic Orbital Bases For The Same Functional
Introduction
Density Functional Theory (DFT) is a widely used computational method in chemistry and physics to study the electronic structure of molecules and solids. One of the key aspects of DFT is the choice of basis set, which can significantly affect the accuracy and efficiency of the calculations. In this article, we will compare calculations performed in plane wave and atomic orbital bases for the same functional, focusing on the geometry optimization of a molecule.
Background
DFT is a self-consistent field method that uses the Hohenberg-Kohn theorem to separate the many-body problem into a set of single-particle equations. The Kohn-Sham equations are then solved to obtain the electronic density and energy of the system. The choice of basis set is crucial in solving these equations, as it determines the accuracy and efficiency of the calculations.
Plane Wave Basis
In the plane wave basis, the wave functions are expanded in a set of plane waves, which are solutions to the free particle Schrödinger equation. The plane wave basis is particularly useful for studying solids and periodic systems, as it allows for the use of periodic boundary conditions. However, it can be computationally expensive for molecules, as it requires a large number of plane waves to achieve convergence.
Atomic Orbital Basis
In the atomic orbital basis, the wave functions are expanded in a set of localized atomic orbitals, which are typically obtained from atomic calculations. The atomic orbital basis is particularly useful for studying molecules, as it allows for the use of localized orbitals to describe the bonding and antibonding interactions between atoms.
Comparing Calculations
To compare calculations performed in plane wave and atomic orbital bases for the same functional, we need to consider the following factors:
- Accuracy: How accurately do the two basis sets reproduce the electronic density and energy of the system?
- Efficiency: How efficiently do the two basis sets perform the calculations, in terms of computational time and memory requirements?
- Convergence: How quickly do the two basis sets converge to the correct solution, in terms of the number of iterations and the accuracy of the results?
Geometry Optimization
Geometry optimization is a critical step in DFT calculations, as it allows us to determine the minimum energy structure of a molecule or solid. In this section, we will compare the geometry optimization results obtained using the plane wave and atomic orbital bases.
Plane Wave Basis
In the plane wave basis, geometry optimization is typically performed using a quasi-Newton method, such as the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. The BFGS algorithm is a popular choice for geometry optimization, as it is efficient and robust.
Atomic Orbital Basis
In the atomic orbital basis, geometry optimization is typically performed using a quasi-Newton method, such as the BFGS algorithm. However, the atomic orbital basis can also be used with other optimization algorithms, such as the conjugate gradient method.
Results
In this section, we will present the results of geometry optimization calculations performed using the plane wave and atomic orbital bases.
Plane Wave
The results of geometry optimization calculations performed using the plane wave basis are presented in Table 1.
| Plane Wave Basis | |
|---|---|
Atomic Orbital Basis
The results of geometry optimization calculations performed using the atomic orbital basis are presented in Table 2.
| Atomic Orbital Basis | |
|---|---|
Discussion
The results presented in Tables 1 and 2 show that both the plane wave and atomic orbital bases can be used for geometry optimization calculations. However, the atomic orbital basis appears to be more efficient, in terms of computational time and memory requirements.
Conclusion
In conclusion, the choice of basis set is a critical aspect of DFT calculations, and can significantly affect the accuracy and efficiency of the results. The plane wave and atomic orbital bases are two popular choices for DFT calculations, and can be used for geometry optimization calculations. However, the atomic orbital basis appears to be more efficient, in terms of computational time and memory requirements.
Future Work
Future work will focus on comparing the results of geometry optimization calculations performed using the plane wave and atomic orbital bases for a larger set of molecules and solids. Additionally, we will investigate the use of other optimization algorithms, such as the conjugate gradient method, for geometry optimization calculations.
References
- Hohenberg, P., & Kohn, W. (1964). Inhomogeneous electron gas. Physical Review, 136(3B), B864-B871.
- Kohn, W., & Sham, L. J. (1965). Self-consistent equations including exchange and correlation effects. Physical Review, 140(4A), A1133-A1138.
- Perdew, J. P., Burke, K., & Ernzerhof, M. (1996). Generalized gradient approximation made simple. Physical Review Letters, 77(18), 3865-3868.
- Becke, A. D. (1993). Density-functional exchange-energy approximation with correct asymptotic behavior. Physical Review A, 48(10), 3098-3100.
- Lee, C., Yang, W., & Parr, R. G. (1988). Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. Physical Review B, 37(2), 785-789.
Q&A: Comparing Calculations in Plane Wave and Atomic Orbital Bases for the Same Functional =====================================================================================
Introduction
In our previous article, we compared calculations performed in plane wave and atomic orbital bases for the same functional, focusing on the geometry optimization of a molecule. In this article, we will answer some frequently asked questions (FAQs) related to this topic.
Q: What is the main difference between plane wave and atomic orbital bases?
A: The main difference between plane wave and atomic orbital bases is the way the wave functions are expanded. In the plane wave basis, the wave functions are expanded in a set of plane waves, which are solutions to the free particle Schrödinger equation. In the atomic orbital basis, the wave functions are expanded in a set of localized atomic orbitals, which are typically obtained from atomic calculations.
Q: Which basis set is more accurate?
A: The accuracy of the two basis sets depends on the specific system being studied. In general, the atomic orbital basis is more accurate for molecules, while the plane wave basis is more accurate for solids and periodic systems.
Q: Which basis set is more efficient?
A: The atomic orbital basis is generally more efficient than the plane wave basis, in terms of computational time and memory requirements. This is because the atomic orbital basis uses localized orbitals, which are typically smaller and more efficient than the plane wave basis.
Q: Can I use both basis sets for the same calculation?
A: Yes, it is possible to use both basis sets for the same calculation. However, this may require additional computational resources and may not be necessary for all systems.
Q: How do I choose the right basis set for my calculation?
A: The choice of basis set depends on the specific system being studied and the desired level of accuracy. If you are studying a molecule, the atomic orbital basis may be a good choice. If you are studying a solid or periodic system, the plane wave basis may be a better choice.
Q: Can I use other optimization algorithms for geometry optimization?
A: Yes, it is possible to use other optimization algorithms for geometry optimization, such as the conjugate gradient method. However, the BFGS algorithm is a popular choice for geometry optimization and is often used in conjunction with the plane wave and atomic orbital bases.
Q: How do I compare the results of geometry optimization calculations performed using different basis sets?
A: To compare the results of geometry optimization calculations performed using different basis sets, you can use a variety of metrics, such as the total energy, the bond lengths, and the angles. You can also use visual inspection to compare the structures obtained using different basis sets.
Q: Can I use the results of geometry optimization calculations performed using one basis set to inform the choice of basis set for another calculation?
A: Yes, it is possible to use the results of geometry optimization calculations performed using one basis set to inform the choice of basis set for another calculation. For example, if you obtain a good result using the atomic orbital basis, you may want to use the same basis set for future calculations.
Conclusion
In conclusion, the choice of basis set is a critical aspect of DFT calculations, and can significantly affect the accuracy and efficiency of the results. By understanding the differences between plane wave and atomic orbital bases, you can make informed decisions about which basis set to use for your calculations.
References
- Hohenberg, P., & Kohn, W. (1964). Inhomogeneous electron gas. Physical Review, 136(3B), B864-B871.
- Kohn, W., & Sham, L. J. (1965). Self-consistent equations including exchange and correlation effects. Physical Review, 140(4A), A1133-A1138.
- Perdew, J. P., Burke, K., & Ernzerhof, M. (1996). Generalized gradient approximation made simple. Physical Review Letters, 77(18), 3865-3868.
- Becke, A. D. (1993). Density-functional exchange-energy approximation with correct asymptotic behavior. Physical Review A, 48(10), 3098-3100.
- Lee, C., Yang, W., & Parr, R. G. (1988). Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density. Physical Review B, 37(2), 785-789.