In materials design, an understanding of the properties of the electrons in a material is essential. Electrons are what hold materials together, and they determine properties important for engineering, such as photovoltaic or catalytic properties. Computer models of electron behavior are widely used in materials design research, but modeling exact electron behavior is very computationally expensive. To increase computational efficiency, approximations are used which decrease computational cost while still providing quite accurate descriptions of electron behavior. These approximations rely on pseudopotentials, which are a modified form of the electrostatic potential between the electrons and nucleus.
The configuration of electrons in an atom changes as the atom’s chemical environment changes, so pseudopotentials must give accurate approximations for a range of electron configurations. To create a pseudopotential, a reference configuration is chosen, and the pseudopotential is designed so that electron behavior in this configuration is nearly exact. For configurations other than the reference configuration, however, the pseudopotential is less accurate. An arbitrary function of one variable, called the augmentation function, can be added to the pseudopotential for these non-reference configurations in order to increase their accuracy. Finding the augmentation function that provides the greatest accuracy is the goal of the project.
The augmentation function can, in principle, be any function whatsoever. We used augmentation functions in the form of a sum of cosine terms, each multiplied by a coefficient that can be adjusted to fit the sum to any smooth function. Thus, this form provides much flexibility for the augmentation function. Additionally, the number of terms can be adjusted. More terms provides more flexibility in fine-tuning the augmentation function, but more terms also complicates the problem of determining the optimal coefficients. We pursued the problem with 4 terms, but we believe the technique we developed can be extended to greater numbers of terms.
The optimal coefficients can be defined mathematically as the coefficients which minimize the error of the pseudopotential (where error is the difference in energy plus electron density between the exact electron calculation and pseudopotential calculation). While pursuing the problem of minimizing error, we discovered a surprising relation between the coefficients and error: graphs of the error ‘surface’ showed valley-like regions which complicate the optimization process. To address this, we developed an algorithm which parametrizes these ‘valleys’ and thereby reduces the dimensionality of the problem, simplifying the procedure. From preliminary data, it appears this algorithm was successful at finding the coefficients which minimize error.
In future work, we hope to gain an understanding of the physical reason for the valleys in the error surface. An understanding of this might lead to more efficient algorithms for optimizing the augmentation function. In addition, pseudopotentials produced with this algorithm must be tested in a range of materials models to ensure that they are accurate and reliable. Further into the future, we hope that these improved pseudopotentials will contribute to the development of new photovoltaic materials and catalysts, which can address environmental and energy concerns and improve our quality of life.