ResearchOne Page Summary: Density Functional Theory

Dingyi SunUpdated: December 15, 2016

One Page Summary: Density Functional Theory

Dingyi Sun

Updated: December 15, 2016

Figure 0.1: Cross-sectional high resolutiontransmission electron microscopy image of anasymmetric three-component superlattice con-taining (Ca, Sr, Ba)TiO3 from [1].

In the computational design of materials, accuracy of results isparamount. When examining a complex physical problem for whichthere is no precedent and all of the minute effects and interactionsneed to be known (e.g. a situation such as that in Figure 0.1), onemust use these so-called ab-initio methods , i.e. methods which usefirst principles. These techniques are essentially approximations ofquantum mechanics, sacrificing some fidelity in favor of computabil-ity, a sacrifice which is necessary when the physical problem beingcomputed is, for instance, dealing with material defects, which occurat concentrations on the order of parts per million1.

One such technique that falls under this umbrella of ab-initiomethods is density functional theory. While density functionaltheory in itself has a number of variants, one of the most popularformulations of it was introduced by Kohn and Sham [2] (consultParr and Yang [3] for additional background). The core componentof Kohn-Sham density functional theory was the construction of an

energy functional to represent a many-body system, in terms of kinetic energy Ts , Hartree energy (classical electrostaticinteraction energy between electrons) EH , energy of interaction between electrons and external potentials Eext, inter-nuclei repulsive energyEzz , and the so-called "exchange-correlation" energyExc2, all of which are functions of spin-upor spin-down electron density �f˛;ˇg and the positions of the atomic nuclei R. This energy functional had the form

EŒ ;R� D TsŒ�˛; �ˇ �C Exc Œ�˛; �ˇ �CEH Œ�˛; �ˇ �CEextŒ�˛ C �ˇ ;R�C Ezz ŒR�: (0.1)

Figure 0.2: Illustration of electrostatic field deviationfrom a perfect crystal behavior due to existence of atwin plane.

Given that this formulation was in terms of energy, the variationalnature of this problem would lead to the objective being tominimize the energy of Equation 0.1 in order to obtain a ground-state energy, E0ŒR� D inf EŒ ;R�, subject to the constraint oforthonormality of the quantum mechanical wavefunctions .

Herein lies an issue; the minimization problem evolves intoan eigenvalue problem which, without further caution, comesat an unacceptable OŒN 3� problem, where N is the number ofelectrons. To circumvent this, earlier work in our group (seeSuryanarayana et al. [4, 5]) employed a technique of spectralGauss quadratures in order to reel in the cost to a much more-tractable OŒN �. Ponga et al. [6] then introduced coarse grainingto the mesh in order to further reduce the cost and improvescalability, showing the viability of this implementation with ahost of calculations examining defects in magnesium. This workhas enabled the use of density functional theory to examine additional problems of interest - such as the energiesassociated to a twin boundary in magnesium (Figure 0.2), and paves the way for upcoming studies on alloys, where,once again, accuracy will be paramount.

1A consequence of this is that, in order to accurately simulate defects, an enormous computational domain consisting of millions (or more) ofatoms must be used.

2The approximations made to this term govern the accuracy of the solution, and the inner-workings of this term are sometimes controversial.

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ResearchOne Page Summary: Density Functional Theory

Dingyi SunUpdated: December 15, 2016

References[1] F.M. Granozio, G. Koster, and G. Rijnders. Functional oxide interfaces. MRS Bulletin, 38:1017–1023, 2013.

[2] W. Kohn and L.J. Sham. Self-Consistent Equations Including Exchange and Correlation Effects. Physical Review,140(4A):A1133–A1138, 1965.

[3] R.G. Parr and W. Yang. Density-Functional Theory fo Atoms and Molecules. Oxford University Press, 1989.

[4] P. Suryanarayana, V. Gavini, T. Blesgen, K. Bhattacharya, and M. Ortiz. Non-periodic finite-element formulationof Kohn-Sham density functional theory. Journal of the Mechanics and Physics of Solids, 58:256–280, 2010.

[5] P. Suryanarayana, K.Bhattacharya, and M. Ortiz. Coarse-graining Kohn-Sham Density Functional Theory. arXiv,pages 1–34, 2012.

[6] M. Ponga, K. Bhattacharya, and M. Ortiz. A Sublinear-Scaling Approach to Density-Functional-Theory Analysisof Crystal Defects. Journal of the Mechanics and Physics of Solids (Accepted), 2016.

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