HybriD3 Theory Training Workshop -- organized by Y. Kanai (UNC) and V. Blum (Duke) ***Practical Density Functional Theory with Plane Waves)*** Natalie A. W. Holzwarth, Wake Forest University, Department of Physics, Winston-Salem, NC, USA Acknowledgements: • Cameron Kates, Jamie Drewery, Hannah Zhang, Zachary Pipkorn (former WFU undergrads) • Zachary Hood (WFU chemistry alum, Ga Tech Ph. D) • Jason Howard, Ahmad Al-Qawasmeh, Yan Li, Larry E. Rush, Nicholas Lepley (current and former WFU grad students) • NSF grant DMR-1507942 9/29/2018 HybriD3 2018 1 Perspectives on Materials Simulations Fundamental Physical Numerical Software Theory approximations approximations packages and tricks quantum-espresso input (http://www.quantum-espresso.org/) parameters abinit (https://www.abinit.org/) useful results Qbox (http://qboxcode.org/) FHI-aims (https://aimsclub.fhi-berlin.mpg.de/) It is important to know what is inside the box! 9/29/2018 HybriD3 2018 2 Fundamental Physical Numerical Software Theory approximations approximations packages and tricks Quantum mechanics and electrodynamics as it applies to many particle systems 9/29/2018 HybriD3 2018 3 Fundamental Physical Numerical Software Theory approximations approximations packages and tricks Born-Oppenheimer approximation [Born & Huang, Dynamical Theory of Crystal Lattices , Oxford (1954)]: Nuclear motions treated classically while electronic motions treated quantum mechanically because MN>>me Density functional theory [Kohn, Hohenberg, Sham, PR 136, B864 (1964), PR 140, A1133 (1965)]: Many electron system approximated by single particle approximation using a self- consistent mean field. Frozen core approximation[von Barth, Gelatt, PRB 21, 2222 (1980)]: Core electrons assumed to be “frozen” at their atomic values; valence electrons evaluated variationally. 9/29/2018 HybriD3 2018 4 Fundamental Physical Numerical Software Theory approximations approximations packages and tricks Plane wave representations of electronic wavefunctions and densities Pseudopotential formulations 9/29/2018 HybriD3 2018 5 Numerical methods more generally -- Plane waves or Grids Muffin Muffin tins Numerical methods Numerical Density Functional Theory 9/29/2018 HybriD3 2018 6 ***Practical Density Functional Theory with Plane Waves)*** Outline • Treatment of core and valence electrons; frozen core approximation • Use of plane wave expansions in materials simulations • Pseudopotentials • Norm conserving pseudopotentials • Projector augmented wave formalism • Assessment of the calculations 9/29/2018 HybriD3 2018 7 Summary and conclusions: • Materials simulations is a mature field; there are many great ideas to use, but there still is plenty of room for innovation. • Maintain a skeptical attitude to the literature and to your own results. • Introduce checks into your work. For example, perform at least two independent calculations for a representative sample. • On balance, static lattice results seem to be under good control. The next frontier is more accurate treatment of thermal effects and other aspects of representing macroscopic systems. • Developing first-principles models of real materials to understand and predict their properties continues to challenge computational scientists. 9/29/2018 HybriD3 2018 8 ***Practical Density Functional Theory with Plane Waves)*** Outline • Treatment of core and valence electrons; frozen core approximation • Use of plane wave expansions in materials simulations • Pseudopotentials • Norm conserving pseudopotentials • Projector augmented wave formalism • Assessment of the calculations 9/29/2018 HybriD3 2018 9 9/29/2018 HybriD3 2018 10 Two examples -- Partitioning electrons into core and valence contributions nr() ncore () r n vale () r 9/29/2018 HybriD3 2018 11 For spherically symmetric atom: (r) ()r Y () rˆ nlmiii nl ii lm ii P( r ) (r ) nli i nli i r Example for carbon 2 nr() w () r niil n ii l i 2 =4 2()r2 2 () r 2 2 ( r ) 1s 2 s 2 p 4 2 2 2 = 2 Pr() 2 Pr ( ) 2 P () r 2 1s 2 s 2 p 9/29/2018r HybriD3 2018 12 Radial wavefunctions for carbon 1s (r) 2p nl P 2s r (Bohr) 9/29/2018 HybriD3 2018 13 Electron density of C atom 2 ncore(r) [1s ] /4 2 nr 2 2 nval(r) [2s 2p ] r (Bohr) 9/29/2018 HybriD3 2018 14 Electron density of copper Example for Cu 1s22s22p63s23p63d104s1 ncore(r) /4 2 nr nvale(r) r (Bohr) 9/29/2018 HybriD3 2018 15 Radial wavefunctions for Cu 3s 3d 4p (r) nl P 4s 3p Note that in this case, the “semi-core” states 3s23p63d10 are included as “valence” r (Bohr) 9/29/2018 HybriD3 2018 16 Frozen core approximation n( r) ncore()r n vale () r Example for Cu Variationally optimize energy wrt nvale ( r) 1s22s22p63s23p63d104s1 ncore(r) /4 2 nr nvale(r) r (Bohr) 9/29/2018 HybriD3 2018 17 Systematic study of frozen core approximation in DFT http://journals.aps.org/prb/abstract/10.1103/PhysRevB.21.2222 9/29/2018 HybriD3 2018 18 ***Practical Density Functional Theory with Plane Waves)*** Outline • Treatment of core and valence electrons; frozen core approximation • Use of plane wave expansions in materials simulations • Pseudopotentials • Norm conserving pseudopotentials • Projector augmented wave formalism • Assessment of the calculations 9/29/2018 HybriD3 2018 19 Consider a periodic material with a unit cell described by primitive lattice vectors a1,a2, and a3: a 3 Multiple a2 unit cells a1 Any position r in the material is related to an infinite number of other points in the material r T r n1 a 1 n 2 a 2 n 3 a 3 because of its periodic symmetry. Because of Bloch's theorem, any wavefunction of the system having wavevector k, ikT i kr has the property: k(rT )e k ( r ). This means that kk( rr ) e u ( ), where uk (r ) is a periodic function. All directly measurable properties of the system also reflect the periodic nature of the system. For example, the electron density: n(rT ) n ( r ) 9/29/2018 HybriD3 2018 20 Mathematical relationships – The Fourier transform of a periodic function results in a discrete summation based on the reciprocal lattice. iGr For the electron density: n(r ) ne ( G ) , where Gbb mmm1 1 2 2 3b3 G The reciprocal lattice vectors are related to the primitive translation a j ak vectors according to bija =2 ij ; b i 2 ai (a j ak ) 1 3 iGr Here n(G ) d r e n (r ). unit cell Fourier transforms are a natural basis for periodic functions. For example, the periodic part of the Bloch wave function: iGr ik G r uuekk(r ) (G ) and kk( rG ) ue ( ) G G This works well, provided the summation converges in the sense thta uk (G ) fo r k+G Kmax . 9/29/2018 HybriD3 2018 21 ***Practical Density Functional Theory with Plane Waves)*** Outline • Treatment of core and valence electrons; frozen core approximation • Use of plane wave expansions in materials simulations • Pseudopotentials • Norm conserving pseudopotentials • Projector augmented wave formalism • Assessment of the calculations 9/29/2018 HybriD3 2018 22 The notion of pseudopotential has been attributed To Enrico Fermi in the 1930’s. “First principles” pseudopotentials were developed by Hamann, Schlüter, and Chiang, PRL 43, 1494 (1979) and J. Kerker, J. Phys. C 13, L189 (1980). Cs effective potential Cs 5s orbital r c rc Pseudo Pseudo V all-Electron VrVPseudo() all-Electron () r Potential (Ry) Potential r r c Pseudo()r all-Electron () r all-Electron V r rc r (bohr) r (bohr) 9/29/2018 HybriD3 2018 23 Justification for pseudopotential formalism Valence electron orthogonality to core electrons provide a repulsive effective potential, resulting in an effective smooth “pseudopotential” for valence electrons. Norm-conserving pseudopotential construction schemes 9/29/2018 HybriD3 2018 24 Norm-conserving pseudopotentials -- continued Constructed from all-electron treatments of spherical atoms or ions: 2 2 all-Electron all-Electron V( r ) nl nl (r ) 0 2m 2 2Pseudo Pseu d o Vr( ) nl nl ( r ) 0 2m Require: Pseudo all-Electron V( rV ) ( r ) for rr c Pseudo all-Electron nl(r ) nl (r ) for rr c Also require: 2 2 Norm conservation d3r Pseudo() rd 3r all-Electron () r nl nl condition; has several rrcr r c benefits Procedure can be carried out for one orbital nl at a time VPseudo()rV Pseudo () r V Pseudo () r P loc nl nl Non-local projector operator nl 9/29/2018 HybriD3 2018 25 Recent improvements to norm-conserving pseudopotentials Improves the accuracy of the norm conserving formulation and allows for accurate plane wave representations of the wavefunctions: ik Gr k()r u k () G e G for k+G Kmax 9/29/2018 HybriD3 2018 26 Other plane wave compatible schemes -- (1) (2) (3) Muffin Muffin tins References: 1. P. E. Blöchl, PRB 50, 17953 (1994) 2. D. Vanderbilt, PRB 41, 7892 (1990); K. Laasonen, A. Pasquarello, R. Car, C. Lee, D. Vanderbilt, PRB 47, 10142 (1993) 3. D. R. Hamann, M. Schlüter, C. Chiang, PRL 43, 1494 (1979); D. R. Hamann, PRB 88, 085117 (2013) Numerical methods Numerical Density Functional Theory 9/29/2018 HybriD3 2018 27 Basic ideas of the Projector Augmented Wave (PAW) method Peter Blöchl, Institute of Theoretical Physics TU Clausthal, Germany Blöch presented his ideas at ES93 --“PAW: an all- electron method for first-principles molecular dynamics” Reference: P. E. Blöchl, PRB 50, 17953 (1994) Features • Operationally similar to other pseudopotential methods, particularly to the ultra-soft pseudopotential method of D. Vanderbilt; often run within frozen core approximation • Can retrieve approximate ‘’all-electron” wavefunctions from the results of the calculation; useful for NMR analysis for example • May have additional accuracy controls particularly of the higher