Lecture on First-principles Computations (11): The Linearized Augmented Plane-wave Method

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1 Lecture on First-principles Computations (11): The Linearized Augmented Plane-wave Method 任新国 (Xinguo Ren) 中国科学技术大学量子信息重点实验室 Key Laboratory of Quantum Information, USTC Hefei,

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

3 Augmented plane waves (APW) J. Slater (1937) Muffin-tin (MT) sphere Interstitial(I) region χ APW k+g r, E = e i k+g r, r I a Lα k + G u lα r, E Y L r α, r MT α L Multi-atom unit cell L = l, m 1 2m d2 l l dr2 r 2 + V s r E ru lα r, E = 0 I α α R MT β Spherically averaged crystal potential

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

5 Spherical Bessel functions Solution to the Heimholtz equation: 2 k 2 ψ r = 0 The regular solution (finite at r = 0): ψ r ~ j l kr Y lm ( r) Ƹ Spherical Bessel function, satisfying, 1 r 2 d dr r2 dj l kr dr + l l + 1 r 2 j l kr = k 2 j l (kr) j 0 x = sin(x) x j 1 x = sin(x) x 2 cos(x) x

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

7 Eigenvalue problem within the APW basis set G = G j = j 1 b 1 + j 2 b 2 + j 3 b 3, ψ nk r = j c jn k χ APW k+gj r ȁj = APW ฬχ k+gj Nonlinear eigenvalue problem: j i, E H E j, E + i, E H S j, E c jn k = 0 The surface term: i H S j = නdS χ APW k+gi r S APW r r χ k+g j The secular equation: Det H E EO E = 0 r χ k+g APW j r +

8 A real example: band structure of copper Calculated by the APW method, empirical potential, 2048 k points. G. A. Burdick, Phys. Rev. 129, 138 (1963).

9 Ƹ 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 states and calculate the charge density from the sum of these.

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

11 Linearization: the LAPW method O.K. Andersen, Phys. Rev. B 12, 3060 (1975) Fixed parameter χ LAPW k+g r = e i k+g r, r I A Lα k + G u lα r, E l + B Lα k + G uሶ lα r, E l Y L r α, r MT α L uሶ lα r, E l = du lα r, E de E=E l χ LAPW k+g r is required to be continuously differentiable at the MT boundary. This can be achieved by choosing proper A Lα k + G, B Lα k + G

12 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

13 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

14 The local orbital (LO) correction to LAPW D. J. Singh, Phys. Rev. B 43, 6388 (1991) In addition to the LAPWs, one adds LOs Φ LO Lα r = A LO Lα u lα r, E l + B LO Lα uሶ lα r, E l 0, r I + C LO Lα u lα r, E LO Y L r α, r MT α L Both the value and slope of Φ LO Lα r are required to be zero at the boundary, and Φ LO Lα r is normalized. The semicore states can now be adequately described with minimum extra cost!

15 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 R MT

16 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 planewave cutoff than the original APW method? YES: APW+lo

17 ሶ The APW+lo method Sjöstedt, Nordström, Singh, Solid State Commun. 114, 15 (2000) χ APW k+g r = e i k+g r, r I a Lα k + G u lα r, E l Y L r α, r MT α L + Fixed energy parameter lo r = Φ lα L A lo Lα u lα r, E l + B lo Lα 0, r I u lα 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!

18 Test example Cu One order of magnitude speed up!

19 Madsen et al. (2001) The fast approach: (L)APW+lo 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, Comput. Phys. Commun. 147, 71 (2002)

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