Point of Study: 

The purpose of this work is to investigate the transferability of pseudopotentials in copper systems. When periodic atomic systems are modeled in Density Functional Theory (DFT), plane waves are often used as a basis set for wave functions. Though convenient for the periodicity of the system, large plane wave basis sets are needed to model the core of atoms where the wave function oscillates considerably.

Figure 1 ) To demonstrate the oscillations in the all electron wave function of an atom, a model radial wave function is plotted as the solid blue line. In the region closer to the nucleus (denoted as the core region in red), the wave function oscillates significantly over a short spatial extent in this region.

Approximations are often made to smooth out this portion of the wave function to reduce the need for massive PW basis sets. The pseudopotential approximation does just this; being constructed such that they reproduce the all electron wave function beyond a designated cutoff radius r_c. (See figure 2) Pseudopotentials are defined by the potential of the core plus the screening from the core electrons. Though there are a number of formalisms to construct pseudopotentials,^1–4 not all of these pseudopotentials are able to accurately reproduce the all electron wave-function in all environments. Most chemistry happens with the valence electrons, comprised mostly from the outer portion of the radial wave function, but this varies from system to system. The amount of electrons participating in bonding is system dependent,⁴ and pseudopotentials that work well for one system may not for one with different chemistry. This work investigates this ‘transferability’ in a Cu and Cu/O system.

Background:

Consider the following figure illustrating the comparison between pseudo-wave functions and the true all electron wave function reproduced from Methodology of Quantum Mechanics/Atomic Simulations.⁴

Figure 2 (left) Is a plot of the pseudo and all-electron wave functions. The pseudo wave function as the dashed line, and the all-electron is solid. Notice that the pseudo-wave function matches the all electron wave function at values at or above the core radius r_c. (right) The screened pseudopotential is plotted (dashed) in comparison with the true unscreened, all-electron potential. Here the core radius is marked at the outermost node. The r_c used for pseudization is not necessarily the true demarcation for core electrons, and its position can vary per pseudopotential formalism.

Pseudo wave functions (figure 2r) are constructed to obey the above principle constraint and others that depend on the method of generation (e.g. norm conservation). Simple pseudopotentials are constructed by inversion of the radial Kohn Sham equation after the pseudo wave function has been defined, with the screened all-electron potential. (See figure 3 below after introduction of the key equations) Usually this is done to match the single-atom all electron wave function in a certain electronic configuration from DFT. The radial Kohn-Sham equation is given by,

(1)

where, l is the angular quantum number, V is the Kohn-Sham potential, epsilon is the eigen energy, and R (r) is the radial wave function. Note that this is a function of principal quantum number as well as the angular quantum number but is left out for simplicity. Good pseudopotentials are expected to yield the same eigenvalues as the all-electron results, among other requirements the valence charge, and smoothness conditions outlined in reference 2. These vary per formalism, but it is typically required that:

(2)

The radial portion of the pseudo-wave function, R ^PP should match the all electron wave function R ^{all e-} outside the cutoff radius r_c.

Figure 3 outlines the key steps in the generation of a pseudopotential. Note that this can be done with different electronic configurations. This is done for each angular momentum channel. See Kleinman and Bylander for further information on how this is done in practice.⁶ In the step of determining the cutoff radius, the core and valence electrons are partitioned; this should be done such that there is minimal core charge leakage. The Pseudo-Wave function yields a certain screened potential. The screening contributions from the Hartree and Exchange-Correlation potentials are removed to obtain the pseudopotential.

Note that the radial Kohn-Sham (eqn. 1) is a second order differential equation. When the energy and the potential are constant, the solution at a given position r_o is uniquely determined by the wave function and its derivative. Or even more simply by the logarithmic derivative since it contains both:

(3)

The effect of evaluating this for different energies gives different (fixed) Kohn-Sham potentials. It offers a crude (but very fast) test of the transferability. If the Pseudo and all-electron wave function produce the same logarithmic derivative for more energies (and potentials implicitly) the pseudopotential used is more transferable. This is succinctly written:

(4)

This should be true when evaluated at r_o greater than or equal to r_c. High-quality pseudopotentials should reproduce the all-electron logarithmic derivative evaluated at energies near an eigenvalue of eqn 1. In this work, different Cu pseudopotentials commonly used in Quantum Espresso are tested with equation 4, with more rigorous and applied testing to follow in future posts.

Computational Methods

DFT calculations were performed using the Quantum Espresso 6.2 code. Though typically a plane wave code, the single atom calculations are done on a radial mesh using the ld1 module within Quantum Espresso. The single atom pseudopotential and all electron results were obtained on the default radial mesh designated in the pseudopotential files. The PSLib 1.0 PAW pseudopotential was used as the initial test pseudopotential due to its expected high transferability.⁷ This uses Cu in the configuration [Ar] 4s^1.5   3d^9.5 4p^0.0 with minimum soft cutoff radii (l– dependent from local potential constructed from Bessel functions) of bohr, and a core cutoff radius of 1.0 bohr. This pseudopotential implements a scalar relativistic correction developed by Koelling and Harmon.⁸

Transferability is often a point of discussion during the generation of pseudopotentials; atoms do not stay in the same electronic configurations in all chemical environments. The pseudopotential is tested in different electronic configurations.

Results

Figure 3 (Top) Ground State Configuration: The pseudopotential reproduces the logarithmic derivative of the wave function well over a large energy range. Noting the energy scale is in Ry (~13.6 eV). The slight dip at the end of the plot corresponds to a node in the wave function. This becomes more evident when using another, slightly larger r_o. Figure 3 (Bottom) Ground State at larger: The discontinuity in the derivative is due to a node in the pseudo and all electron wave functions.

In the results we see that the PAW pseudopotential provides an adequate description of the all electron wave function over a significant energy range. It is important to note that a deficiency of using logarithmic derivatives to test transferability is that the Kohn-Sham potential is considered at a fixed configuration. This changes as the chemical environment changes. In order to account for this to an extent, other electronic configurations are tested as well. (Including the configuration the pseudopotential was generated at) This was done for a range of r_o values larger than the soft cutoff radius, and the results are appended. The overall results show that the logarithmic derivatives agree fairly well when varying the electronic configuration as well. This is important for Cu considering that it tends to lose electrons in many chemical environments.

Conclusions

The PAW pseudopotential tested is transferrable over a considerable range of energies and selected evaluation radii. The results would likely be that the pseudopotential would still reproduce reasonable energies in a chemically varied environment. (E.g. in CuO). Rigorous testing can be done by comparing the system energies produced by all-electron DFT simulations in different crystals to those of the pseudized cores. Due to the expense of all electron calculations for extended solids, a comparison will not be made with an all electron configuration. Instead, a comparison will be made for another copper pseudopotential with worse transferability seen in logarithmic derivatives. The two pseudopotentials will be analyzed in bulk Cu, CuO and compared to other studies in the literature.

1. Hamann, D. R., Schlüter, M. & Chiang, C. Norm-Conserving Pseudopotentials. Phys. Rev. Lett. 43, 1494–1497 (1979).
2. Troullier, N. & Martins, J. L. Efficient pseudopotentials for plane-wave calculations. Phys. Rev. B 43, 1993–2006 (1991).
3. Kresse, G. & Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B 59, 1758–1775 (1999).
4. Umeno, Y., Shimada, T., Kinoshita, Y. & Kitamura, T. Methodology of Quantum Mechanics/Atomic Simulations. in Multiphysics in Nanostructures (eds. Umeno, Y., Shimada, T., Kinoshita, Y. & Kitamura, T.) 5–34 (Springer Japan, 2017). doi:10.1007/978-4-431-56573-4_2
5. Dal Corso, A. Pseudopotentials periodic table: From H to Pu. Comput. Mater. Sci. 95, 337–350 (2014).
6. Koelling, D. D. & Harmon, B. N. A technique for relativistic spin-polarised calculations. J. Phys. C Solid State Phys. 10, 3107–3114 (1977).

Appended Results: