Adaptively Compressed Polarizability Operator For Accelerating Large Scale ab initio Phonon Calculations Ze Xu 1 Lin Lin 2 Lexing Ying 3 1 , 2 UC Berkeley 3 ICME, Stanford University BASCD, December 3rd 2016 Ze Xu (UC Berkeley) ACP BASCD 2016 1 / 40
Table of Contents 1 What is Phonon? 2 How to Compute? 3 Adaptively Compressed Polarizability Operator 4 Numerical Examples 5 What’s More? 6 Conclusion Ze Xu (UC Berkeley) ACP BASCD 2016 2 / 40
Phonon Table of Contents 1 What is Phonon? 2 How to Compute? 3 Adaptively Compressed Polarizability Operator 4 Numerical Examples 5 What’s More? 6 Conclusion Ze Xu (UC Berkeley) ACP BASCD 2016 3 / 40
Phonon What is Phonon? “In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter... it represents an excited state in the quantum mechanical quantization of the modes of vibrations of elastic structures of interacting particles.” —wikipedia Ze Xu (UC Berkeley) ACP BASCD 2016 4 / 40
Phonon Lattice Wave Figure 1: Lattice wave example. Figure from wikipedia. Ze Xu (UC Berkeley) ACP BASCD 2016 5 / 40
Phonon Phonon Dispersion Relations Figure 2: Phonon dispersion and densities of states of gallium arsenide (GaAs). Figure from S. Baroni et al. , 2001 . Ze Xu (UC Berkeley) ACP BASCD 2016 6 / 40
Phonon Why Care? Ze Xu (UC Berkeley) ACP BASCD 2016 7 / 40
Phonon Why Care? • Thermal conductivity; specific heat; Ze Xu (UC Berkeley) ACP BASCD 2016 7 / 40
Phonon Why Care? • Thermal conductivity; specific heat; • Elastic neutron scattering; Ze Xu (UC Berkeley) ACP BASCD 2016 7 / 40
Phonon Why Care? • Thermal conductivity; specific heat; • Elastic neutron scattering; • Electrical conductivity; Ze Xu (UC Berkeley) ACP BASCD 2016 7 / 40
Phonon Why Care? • Thermal conductivity; specific heat; • Elastic neutron scattering; • Electrical conductivity; • Electron-phonon interaction related topics; Ze Xu (UC Berkeley) ACP BASCD 2016 7 / 40
Phonon Why Care? • Thermal conductivity; specific heat; • Elastic neutron scattering; • Electrical conductivity; • Electron-phonon interaction related topics; Ze Xu (UC Berkeley) ACP BASCD 2016 7 / 40
Phonon Elastic neutron scattering • Thermal conductivity; specific heat; • Elastic neutron scattering; • Electrical conductivity; • Electron-phonon interaction related topics; Figure 3: In elastic neutron-scattering, the neutron bounces off the bombarded nucleus without exciting or destabilizing it. Figure from The Schlumberger Oilfield Glossary. Ze Xu (UC Berkeley) ACP BASCD 2016 8 / 40
Phonon Electron-phonon interaction • Thermal conductivity; specific heat; • Elastic neutron scattering; • Electrical conductivity; • Electron-phonon interaction related topics; ◦ major contributor to electrical resistance in most inorganic metals and semiconductors above zero (very low) temperature; ◦ Overheating problem; Ze Xu (UC Berkeley) ACP BASCD 2016 9 / 40
Phonon Dynamical Matrix ∂ 2 E tot ( { R I } ) 1 D I,J = √ M I M J , ∂ R I ∂ R J • M I mass of the I -th atom, I = 1 , . . . , N A . • R I ∈ R d position of the I -th atom, I = 1 , . . . , N A . • D ∈ R dN A × dN A Ze Xu (UC Berkeley) ACP BASCD 2016 10 / 40
DFT Table of Contents 1 What is Phonon? 2 How to Compute? 3 Adaptively Compressed Polarizability Operator 4 Numerical Examples 5 What’s More? 6 Conclusion Ze Xu (UC Berkeley) ACP BASCD 2016 11 / 40
DFT Density Functional Theory Ab initio calculation – DFT. • most widely used; • exact description of ground state properties: • electron density, energy, and atomic forces. Ze Xu (UC Berkeley) ACP BASCD 2016 12 / 40
DFT Kohn-Sham Density Functional Theory E KS ( { ψ i } ; { R I } ) � � N e � =1 |∇ ψ i ( r ) | 2 d r + V ion ( r ; { R I } ) ρ ( r ) d r 2 (1) i =1 �� + 1 v c ( r , r ′ ) ρ ( r ) ρ ( r ′ ) d r d r ′ + E xc [ ρ ] 2 + E II ( { R I } ) . Ze Xu (UC Berkeley) ACP BASCD 2016 13 / 40
DFT Derivative � ∂V I ( r − R I ) ρ ( r ) d r − ∂E II ( { R I } ) F I = − . ∂ R I ∂ R I ∂ 2 E tot ( { R I } ) ∂ R I ∂ R J � � ρ ( r ) ∂ 2 V I ∂V I ( r − R I ) δρ ( r ) = d r + δ I,J ( r − R I ) d r ∂ R 2 ∂ R I δ R J I + ∂ 2 E II ( { R I } ) . ∂ R I ∂ R J Ze Xu (UC Berkeley) ACP BASCD 2016 14 / 40
DFT Derivative � ∂V I ( r − R I ) ρ ( r ) d r − ∂E II ( { R I } ) F I = − (2) . ∂ R I ∂ R I ∂ 2 E tot ( { R I } ) ∂ R I ∂ R J � � ρ ( r ) ∂ 2 V I ∂V I ( r − R I ) δρ ( r ) = d r + δ I,J ( r − R I ) d r (3) ∂ R 2 ∂ R I δ R J I + ∂ 2 E II ( { R I } ) . ∂ R I ∂ R J Ze Xu (UC Berkeley) ACP BASCD 2016 15 / 40
DFT Polarizability Operator � δρ ( r ) δρ ( r ) ∂V J ( r ′ − R J ) d r ′ . = δV ion ( r ′ ) δ R J ∂ R J δρ ( r ) echet derivative χ ( r , r ′ ) := Fr´ δV ion ( r ′ ) is the reducible polarizability operator. Ze Xu (UC Berkeley) ACP BASCD 2016 16 / 40
DFT KSDFT Cont. Kohn-Sham equations (W. Kohn and L. Sham 1965) � � − 1 H [ ρ ] ψ i = 2∆ + V [ ρ ] ψ i = ε i ψ i , (4) � � N e | ψ i ( r ) | 2 . ψ i ( r ) ψ j ( r ) d r = δ ij , ρ ( r ) = (5) i =1 • ε i energy level; ψ i orbitals; • index i = 1 , . . . , N e called occupied states; i = N e + 1 , . . . unoccupied states; • ε g = ε N e +1 − ε N e band gap; � v c ( r , r ′ ) ρ ( r ′ ) d r ′ + V xc [ ρ ]( r ) • V [ ρ ]( r ) = V ion ( r ; { R I } ) + • V ion = � g J = ∂V J ∂ R J . I V I , Ze Xu (UC Berkeley) ACP BASCD 2016 17 / 40
DFT Sternheimer Equations Q ( ε i − H ) Qζ ij = Q ( ψ i ⊙ g j ) . (6) i = 1 , . . . , N e , j = 1 , . . . , dN a • Q projection onto unoccupied states; • ζ ij defined as the solution to the equation. This is the core part of density-functional perturbation theory (DFPT) (S. Baroni et al. 2001). Ze Xu (UC Berkeley) ACP BASCD 2016 18 / 40
DFT How to solve that? Q ( ε i − H ) Qζ ij = Q ( ψ i ⊙ g j ) . i = 1 , . . . , N e , j = 1 , . . . , dN a • SVD? • Frozen phonon approach? (Or finite difference in math) • Compression? Ze Xu (UC Berkeley) ACP BASCD 2016 19 / 40
DFT How to solve that? cont. Q ( ε i − H ) Qζ ij = Q ( ψ i ⊙ g j ) . i = 1 , . . . , N e , j = 1 , . . . , dN a • SVD? • Frozen phonon approach? (Or finite difference in math) • Compression! ACP formulation reduces the computational complexity of phonon calculations from O ( N 4 e ) to O ( N 3 e ) for the first time ( X . et al. 2016). Ze Xu (UC Berkeley) ACP BASCD 2016 20 / 40
DFT The Dyson Equation Irreducible polarizability operator χ 0 = δρ δ V χ = χ 0 + χ 0 v hxc χ (7) or U = χ 0 G + χ 0 v hxc U, U = χG. So compression of χ 0 can only be done adaptively . Ze Xu (UC Berkeley) ACP BASCD 2016 21 / 40
ACP Table of Contents 1 What is Phonon? 2 How to Compute? 3 Adaptively Compressed Polarizability Operator 4 Numerical Examples 5 What’s More? 6 Conclusion Ze Xu (UC Berkeley) ACP BASCD 2016 22 / 40
ACP Adaptively Compressed Polarizability Operator Q ( ε i − H ) Qζ ij = Q ( ψ i ⊙ g j ) . i = 1 , . . . , N e , j = 1 , . . . , dN a • Chebyshev interpolation – disentangle the energy dependence on the left • Interpolative separable density fitting method – compress the rhs vectors. • Adaptively compressed – cost stays the same across iterations. Ze Xu (UC Berkeley) ACP BASCD 2016 23 / 40
ACP Chebyshev Interpolation Pick { ε c } ∈ I ≡ [ ε 1 , ε N e ] . Lagrange interpolation N c � � ε − � ε c ′ � ζ = ζ c ε c ′ . � ε c − � c =1 c ′ � = c Ze Xu (UC Berkeley) ACP BASCD 2016 24 / 40
ACP Interpolative Decomposition M ij = ψ i ⊙ g j , or M ij ( r ) = ψ i ( r ) g j ( r ) . N µ N µ � � M ij ( r ) ≈ ξ µ ( r ) M ij ( r µ ) ≡ ξ µ ( r ) ψ i ( r µ ) g j ( r µ ) . (8) µ =1 µ =1 (H. Cheng, Z. Gimbutas, P. G. Martinsson, and V. Rokhlin, 2005) Ze Xu (UC Berkeley) ACP BASCD 2016 25 / 40
ACP Interpolative Separable Density Fitting Two-step procedure: subsampled random Fourier Transform and QR decomposition. (J. Lu and L. Ying, 2015) Ze Xu (UC Berkeley) ACP BASCD 2016 26 / 40
ACP Reconstruction To avoid O ( N 4 e ) complexity, construct N e N c � � � ε i − � ε c ′ � ψ i ( r µ ) . W µ = 2 ψ i ⊙ (9) ζ cµ � ε c − � ε c ′ i =1 c =1 c ′ � = c N µ � χ 0 g j ≈ W µ g j ( r µ ) , (10) µ =1 χ 0 := W Π T . or formally, we have χ 0 ≈ � Ze Xu (UC Berkeley) ACP BASCD 2016 27 / 40
ACP Iterative scheme Introduce the following change of variable U = � B = v − 1 U − B, hxc G, (11) � χ 0 [ � U ] v hxc � U = � U + B. (12) Iterative scheme: 0 [ � χ k U k ] = W k (Π k ) T ( a ) Construct � � − 1 U k +1 = I − W k (Π k ) T v hxc ( b ) � B (13) = B + W k � I − (Π k ) T v hxc W k � − 1 (Π k ) T v hxc B. Ze Xu (UC Berkeley) ACP BASCD 2016 28 / 40
Numerical Examples Table of Contents 1 What is Phonon? 2 How to Compute? 3 Adaptively Compressed Polarizability Operator 4 Numerical Examples 5 What’s More? 6 Conclusion Ze Xu (UC Berkeley) ACP BASCD 2016 29 / 40
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