Lecture on First-Principles Computation (11): The LAPW Method

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1 Lecture on First-Principles Computation (11): The LAPW Method 任新国 (Xinguo Ren) 中国科学技术大学量子信息实验室 (Key Laboratory of Quantum Information, USTC)

2 Recall: OPW and Pseudopotential Methods Problems: rapid oscillation of wave functions near the nuclei. Orthogonalized plane waves χ OPW k+g (r)= 1 u j k+g u j (r) ] Ω [ ei( k+ G) r j The pseudopotential (PP) method V nul (r)+v Hxc (r) V PP v ion (r)+v,pp Hxc (r)

3 Different pseudopotentials exist... Ionic pseudopotentials for Cu s p d Troullier-Martins Kerker HSC Vanderbilt

4 Separable Pseudopotential Operators V SL =V local (r)+ Y lm >δv l (r)<y lm lm Computationally inefficient! Nonlocal (NL) separable potential Reference pseudo atomic functions V NL =V local (r)+ lm ψ lm PS δv l >< δv l ψ lm PS ψ lm PS δv l ψ lm PS Projector functions ψ i δ V NL ψ j = lm ψ i ψ lm PS δv l δ V l ψ lm PS ψ j ψ lm PS δv l ψ lm PS Kleinman & Bylander (1982) The coupling between ψ i, ψ j are removed, Three-center integrals become two-center integrals!

5 Calculations with plane-wave basis sets Plane waves are the most natural basis for periodic systems G cut ψ n k (r)= G=0 c n,g(k)e i (k+g) r Kinetic energy of plane waves: E kin = ħ2 2 E kin cut = ħ2 k+g cut 2 2m 2m ei (k+g) r = ħ2 k+g 2 2m The energy cutoff determines the resolution of the wave functions, and hence the numerical accuracy of the calculation.

6 Softwares Based on Pseudopotential Plane-Wave Technique Ab-initio Quantum Espresso

7 Augmented plane waves (APW) J. Slater (1937) Muffin-tin(MT) sphere χ APW k +G (r)= Interstitial(I) region e i(k +G) r, r I a Lα (k+g)u lα (r, E)Y L ( r α ), r MT α L L=(lm) [ 1 2m d 2 dr + l(l+1) ] +V 2 r 2 s (r) E ru lα(r, E)=0 Multi-atom unit cell I α α R MT β

8 The APW method Requiring the APW to be continuous at the muffin-tin boundary e i(k +G) (R α+r α ) MT = a Lα (k+g)u lα (R α MT L, E)Y L ( R α MT ) Ce Basic formula: e i p r =4 π L i l j l ( pr)y L * ( p)y L ( r) a Lα (k+g)=4 πi l e (k +G) R α Bessel function α ) j l( k+g R MT u l α (R α MT, E) Y * L( k+g) Sjoestedt, Nordstroem, Singh, Solid State Comm, 2000

9 APW as basis functions χ APW k +G (r)= ei(k +G) r, r I a Lα (k+g)u lα (r, E)Y L ( r α ), r MT α L ψ n k (r)= G = G L A Lα (k)= G c n G (k)χ APW k +G (r) c n G (k)e i (k +G) r, r I A Lα (k)u lα (r, E)Y L ( r α ), r MT α a Lα (k+g)c n G (k) A Lα (k) are not independent variational parameters; rather, they are fixed by matching conditions.

10 Eigenvalue problem within the APW basis set G=G j = j 1 b 1 + j 2 b 2 + j 3 b 3, j>= χ APW k+g j > ψ n k (r)= j c jn (k)χ APW k+g j (r) Nonlinear eigenvalue problem: j [ i, E Ĥ E j, E + i, E Ĥ S j, E ] c jn (k)=0 Surface term: i Ĥ S APW j = S d S χ * k+gi (r) ( r χ APW k+g j (r - ) r χ APW k+g j (r + ) ) The secular equation: det {H (E) EO(E)}=0

11 A real example: band structure of copper Calculated by the APW method, Burdick (1963)

12 APW is an all-electron method The u l (r)y lm ( r) are orthogonal to core states. This basis can be used to obtain true valence states in the real potential. (1) Calculate core states separately in each SCF cycle. (2) Use the same potential for core and valence and calculate the charge density from the sum of these.

13 Pros and Cons of the APW method Exact for muffin-tin potentials, but be extended to full potential in principle, but very involved in practice. Truly all-electron method, applicable equally well for both light and heavy elements. Energy-dependent basis set => nonlinear eigenvalue problem => solution cannot be found from a single diagonalization, costly!

14 Linearlization: the LAPW method O.K. Andersen, Phys. Rev. B 12, 3060 (1975) χ LAPW k +G (r)= e i(k+g) r, r I L u l α (r, E l )=[ d u l α(r, E) d E ( A Lα (k+g)u l α (r, E l )+B Lα (k+g) u l α (r, E l )) Y L ( r α ), χ LAPW k +K (r) is required to be continuously differentiable at the MT boundary A Lα (k+g), B Lα (k+g) ]E= E l Fixed parameter r MT α

15 Pros and Cons of the LAPW method Energy-independent basis functions => linear eigenvalue problem! The slope is continuous => No surface term More plane waves are needed to achieve the same accuracy as APW Not flexible enough to describe the semicore states

16 The semicore states The standard LAPW method is not flexible enough to describe the semi-core states delocalized Localized, but not confined in the MT sphere Confined in the MT sphere

17 The local orbital (LO) correction to LAPW D. J. Singh (1991) In addition to the LAPWs, one adds LOs Φ LO Lα (r)= 0, r I ( A LO LO Lα u lα (r, E l )+B Lα u l α (r, E l )+C LO Lα u l α (r, E LO )) Y L ( r α ), L r MT α Both the value and slope of zero at the boundary, and LO (r) LO (r) Φ Lα Φ Lα are required to be is normalized The semicore states can now be adequately described with mininum extra cost!

18 What determines the basis size Large MT sphere => few plane waves Small MT sphere => many plane waves R MT G max is a good measure of a converged basis

19 Can one do better? Can one combined the advantages of the APW and LAPW method, i.e., to find an energy-independent basis that does not demand a noticeable increase of the plane-wave cutoff than the original APW method? YES: APW+lo

20 χ APW k +G (r)= The APW+lo method Sjoestedt, Nordstroem, Singh (2000) Φ lo l α (r)= Fixed parameter e i(k +G) r, r I a Lα (k+g)u lα (r, E l )Y L ( r α ), r MT α L 0, r I ( A lo Lα L + lo u lα (r, E l )+B L α u l α (r, E l ))Y L ( r α ), r MT α Determined by zero value at the boundary and the normalization condition Much less plane waves are needed, but the surface term is back!

21 Test example Cu One order of magnitude speed up!

22 The fast approach: (L)APW+lo Madsen et al. (2001) Mixed approach: APW+lo for physically important l channels, and LAPW for the higher ones Force acting on one O atom in sodium electro sodalite(40 atoms/unit cell) K. Schwartz, P. Blaha, G. Madsen (2002)

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24 Comparison of Different Solid-State DFT Codes, Basis Sets and Potential Website: