Making Electronic Structure Theory work

Transcription

1 Making Electronic Structure Theory work Ralf Gehrke Fritz-Haber-Institut der Max-Planck-Gesellschaft Berlin, 23 th June 2009 Hands-on Tutorial on Ab Initio Molecular Simulations: Toward a First-Principles Understanding of Materials Properties and Functions Outline Density Mixing Linear Mixing, Pulay Mixer, Broyden Mixer Electronic Smearing Fermi-Dirac, Gaussian, Methfessel-Paxton Preconditioning Direct Minimization Spin Polarization Energy Derivatives Local Atomic Structure Optimization Outlook on Global Structure Optimization

2 Outline Spin Polarization Fixed Spin vs. Unconstrained Spin Calculation Energy Derivatives Local Atomic Structure Optimization Outlook on Global Structure Optimization Outline Spin Polarization Energy Derivatives Atomic Forces Numeric vs. Analytic Second Derivative Vibrational Frequencies Local Atomic Structure Optimization Outlook on Global Structure Optimization

3 Outline Spin Polarization Energy Derivatives Local Atomic Structure Optimization Steepest Descent Conjugate Gradient Broyden-Fletcher-Goldfarb-Shannon (BFGS) Cartesian vs. Internal Coordinates Outlook on Global Structure Optimization Outline Spin Polarization Energy Derivatives Local Atomic Structure Optimization Outlook on Global Structure Optimization

4 0 Density Mixing h KS 2 2 V eff n in, r h KS l l l n out r 2 l r l R n in n out n in, rn in r n r nn 2 d 3 r? no yes finished! Density Mixing 1 n in n out charge sloshing! N 2 = 1.1 Å PBE

5 0 Density Mixing h KS 2 2 V eff n in, r h KS l l l 1 n in n out 1 n out r 2 l r l Let's try some damping... R n in n out n in, rn in r n r nn 2 d 3 r? no yes finished! Density Mixing n 1 in n out 1 =0.5 N 2 = 1.1 Å PBE seems to work!

6 0 h KS 2 2 V eff n in, r Density Mixing but we can still do better... h KS l l l n 1 in f n in,n 1 2 in, n in,, n out,n 1 out, n j2 out, n out r 2 l r l R n in n out n in, rn in r n r nn 2 d 3 r? no yes finished! 0 h KS 2 2 V eff n in, r Density Mixing but we can still do better... h KS l l l n 1 in f n in,n 1 2 in, n in,, n out,n 1 out, n j2 out, n out r 2 l r l? R n in n out n in, rn in r n r nn 2 d 3 r? no yes finished!

7 Density Mixing N 2 = 1.1 Å PBE The Pulay Mixer take more previous densities into account under the constraint of norm conservation n opt in 1 P. Pulay, Chem. Phys. Lett. 73 (1980), 393

8 The Pulay Mixer take more previous densities into account under the constraint of norm conservation n opt in 1 main assumption: the charge density residual is linear w.r.t. to the density R n opt in R R P. Pulay, Chem. Phys. Lett. 73 (1980), 393 The Pulay Mixer take more previous densities into account under the constraint of norm conservation n opt in 1 main assumption: the charge density residual is linear w.r.t. to the density R n opt in R R coefficients are determined by minimizing the residual l R n opt in R n opt in 0 system of equations for P. Pulay, Chem. Phys. Lett. 73 (1980), 393

9 The Pulay Mixer illustratio one dimension with two densities R 1 R 1 RR R 2 R 2 R n opt in R opt R 1 1R opt 2 n P. Pulay, Chem. Phys. Lett. 73 (1980), 393 The Pulay Mixer illustratio one dimension with two densities R 1 R 1 RR R 2 R 2 R n opt in R opt R 1 1R opt 2 n to prevent trapping in subspace, add fraction of residual 1 opt n true x 3 1 n in n opt opt in R 2 P. Pulay, Chem. Phys. Lett. 73 (1980), 393

10 The Pulay Mixer additional preconditioner might accelerate convergence 1 n in n opt in GR n opt preconditioning matrix: G A k2 in k 2 2 k 0 long-range components of density changes are stronger damped than short-range G. Kerker, Phys. Rev. B 23 (1981), 3082 P. Pulay, Chem. Phys. Lett. 73 (1980), 393 The Pulay Mixer additional preconditioner might accelerate convergence 1 n in n opt in GR n opt preconditioning matrix: G A k2 in k 2 2 k 0 long-range components of density changes are stronger damped than short-range G. Kerker, Phys. Rev. B 23 (1981), 3082 two spin channels are to be mixed in the case of spin polarization charge density: n rn r spin density: n rn r combine both channels to one matrix! R n, n R n, n d r n 2 r n rn out, r n in r stable convergence P. Pulay, Chem. Phys. Lett. 73 (1980), 393

11 The Broyden Mixer secant method f x n f x n1 f' x n x n x n 1 f x n 1 f x n 1! 0 x n 1 x n 1 f'x n f x n f x n x n x n 1 x n 1 The Broyden Mixer secant method f x n f x n1 f' x n x n x n 1 f x n 1 f x n 1! 0 x n 1 x n 1 f'x n f x n f x n x n x n 1 x n 1 n 1 in n in G Rn in approximate Jacobian: n in G Rn in 0 D.D. Johnson, Phys. Rev. B 38 (1988), 12807

12 The Broyden Mixer secant method f x n f x n1 f' x n x n x n 1 f x n 1 f x n 1! 0 x n 1 x n 1 f'x n f x n f x n x n x n 1 x n 1 n 1 in n in G Rn in under-determined approximate Jacobian: n in G Rn in 0 G 1 G n G R R G ij G 1 G 0 as little change as possible D.D. Johnson, Phys. Rev. B 38 (1988), The Broyden Mixer secant method f x n f x n1 f' x n x n x n 1 f x n 1 f x n 1! 0 x n 1 x n 1 f'x n f x n f x n x n x n 1 x n 1 n 1 in n in G Rn in under-determined approximate Jacobian: n in G Rn in 0 G 1 G n G R R G ij G 1 G 0 as little change as possible modified Broyden: minimize Ew 0 G 1 G 2 i D.D. Johnson, Phys. Rev. B 38 (1988), w i n i G 1 R i 2

13 level crossing may lead to charge sloshing Electronic Smearing i i i E F N occ n r 2 l r l level crossing may lead to charge sloshing Electronic Smearing i i i E F N occ n r 2 l r l electronic smearing softens the change of electronic density i i i E F n r l f l l 2 r 1 f 1 l f 1 l f l

14 Electronic Smearing Fermi-Dirac 1 : f E F 1 exp E F 1 Gaussian 2 : f E F 1 1erf E F 2 Methfessel-Paxton 3 : f 0 x 1 1erf x 2 x E F N f N x f 0 x A m H 2m 1 xe x2 m1 N x 1 N. Mermin, Phys. Rev. 137 (1965), A C.-L. Fu, K.-H. Ho, Phys. Rev. B 28 (1983), M. Methfessel, A. Paxton, Phys. Rev. B 40 (1989), 3616 Electronic Smearing Kohn-Sham functional not a variational quantity anymore E { c} E { c},{ f } E f in 0 0

15 Electronic Smearing Kohn-Sham functional not a variational quantity anymore E { c} electronic free energy F is variational FE S,{ f n } entropy term E { c},{ f } F f in 0 0 E f in 0 0 Electronic Smearing Kohn-Sham functional not a variational quantity anymore E { c} electronic free energy F is variational FE S,{ f n } entropy term E { c},{ f } F f in 0 0 E f in 0 0 systems with continuous density of states at the Fermi edge: F E O2 0 O 2 N E 0 N F E 2 N 1 F E Fermi-Dirac / Gaussian Methfessel-Paxton desired total energy E for zero smearing can be backextrapolated M. Gillan, J. Phys.: Condens. Matter 1 (1989), 689 M. Methfessel, A. Paxton, Phys. Rev. B 40 (1989), 3616 F. Wagner, T. Laloyaux, M. Scheffler, Phys. Rev. B 57 (1998), 2102

16 Alternative: Direct Minimization E KS g l E tot l constrained minimum of Kohn-Sham functional: gradient: steepest-descent: EE KS l l l l 1 E F l E l g l f l h KS l l 1 l l g l f l N Spin Polarization Fixed Spin vs. Unconstrained Spin 12 2 V eff r i i i V XC r E XC n r E F E F E F N i f E i F i N w i MN N f E i F i f i i w M i N E i F w f f i i total multiplicity kept fixed total spin is relaxed

17 Spin Polarization Fixed Spin vs. Unconstrained Spin fixed spin Co 3 PBE obtained with unconstrained spin obtained multiplicity the lowest one Energy Derivatives Atomic Forces total energy of the system is the total minimum of the Kohn-Sham-functional under the normalization constraint: E tot min c E KS i i i 1 i E c 0 R, R, R E ccc 0 0 E cc 0 0

18 Energy Derivatives Atomic Forces total energy of the system is the total minimum of the Kohn-Sham-functional under the normalization constraint: E tot min c E KS i i i 1 i E c 0 R, R, R E ccc 0 0 E cc 0 0 since the Kohn-Sham-functional is variational, only the partial derivative remains: F d E tot d R = E R c 0 E c 0 =0 c 0 R = E E KS R R R i E R =0 i i i 1 Energy Derivatives The Individual Force Contributions classical Coulomb interaction: F HF, Z Z R R R R Z R r d r nr 3 R r 3 Pulay-correction due to incomplete or nuclear-dependent basis set: F Pulay, 2 l l R h KS l l moving basis functions Multipole-correction due to incomplete Hartree-potential: F MP, d r n rn MP r V H n F GGA, d r n r R XC n 2 n 2 R MP moving multipole components gradient-dependency of exchange-correlation energy in case of a GGA functional:

19 Energy Derivatives The Convergence of the Forces total energy is variational error in the density enters total energy to second order E 0 ccc 0 atomic forces are not variational error in the density enters total energy to first order F 2 E 0 ccc 0 R ccc 0 N-N (d=1.0 Å) total energy forces convergence of atomic forces more expensive than total energy Energy Derivatives Forces and Electronic Smearing Kohn-Sham functional not variational w.r.t. the occupation numbers E { c},{ },{R },{ f },E F E f in 0 0 atomic forces are not consistent with the total energy of the system! F d E tot d R = E R f E f 0 f E R E F =0 E F R E R atomic forces are consistent with the electronic free energy F f in 0 0 F F R M. Weinert and J.W. Davenport, Phys. Rev. B 45 (1992), R.M. Wentzcovitch, J.L. Martin and P.B. Allen, Phys. Rev. B 45 (1992), 11372

20 Energy Derivatives The Second Derivative and Vibrations first derivative was simple analytical F E R Energy Derivatives The Second Derivative and Vibrations first derivative was simple analytical second derivative more nasty F E R d2 E tot H d F d R d R d R 2 E R R c 2 E c 0 R c R e.g. A. Komornicki and G. Fitzgerald, J. Chem. Phys. 98 (2), 1398 (1993)

21 Energy Derivatives The Second Derivative and Vibrations first derivative was simple analytical second derivative more nasty F E R d2 E tot H d F d R d R d R 2 E R R c 2 E R c c 0 R set of coupled equations needs to be solved c 2 E R R c 2 E c 0 c 2 e.g. A. Komornicki and G. Fitzgerald, J. Chem. Phys. 98 (2), 1398 (1993) Energy Derivatives The Second Derivative and Vibrations numerical d F F R F, 0 R, 0 d R 2 R,0 vibrations M R d E R d R E R E R d 2 E d R d R R 0 u u u R R, 0 u tu e i t det H 2 M! 0 talk by Gert von Helden, Friday, 26 th June, 11:30, Fingerprinting molecules: Vibrational Spectroscopy, Experiment and Theory

22 Local Atomic Structure Optimization Potential Energy Surface (PES) E tot R Local Atomic Structure Optimization Potential Energy Surface (PES) E tot R

23 Local Atomic Structure Optimization Steepest Descent R,i1 R,i i F R,i Local Atomic Structure Optimization Steepest Descent too small too large R,i1 R,i i F R,i R

24 Local Atomic Structure Optimization too large too small Steepest Descent R,i1 R,i i F R,i R ill-conditioned PES might lead to zig-zag behaviour Local Atomic Structure Optimization too large too small Steepest Descent R,i1 R,i i F R,i ill-conditioned PES might lead to zig-zag behaviour R Conjugate Gradient take information of previous search directions into account R,i1 R,i i G,i R,i G,i F,i i G,i1 i Fletcher-Reeves F,i F,i line minimization constructions of search directions based upon the assumption that PES is perfectly harmonic F,i1 F,i1 F Polak-Ribiere,i F,i F,i1 i F,i1 F,i1

25 Local Atomic Structure Optimization The Broyden-Fletcher-Goldfarb-Shanno Method Newton-Method: additional to the atomic forces, the Hessian is taketo account: f x f x i xx i f x i 1 2 xx Axx i i f x f x i Axx i R f x! 0 x x i A 1 f x i W. H. Press et al., Numerical Recipes, 3 rd Edition, Cambridge University Press Local Atomic Structure Optimization The Broyden-Fletcher-Goldfarb-Shanno Method Newton-Method: additional to the atomic forces, the Hessian is taketo account: f x f x i xx i f x i 1 2 xx Axx i i f x f x i Axx i R f x! 0 x x i A 1 f x i Quasi -Newton-Scheme: inverse Hessian is approximated iteratively R,i1 R,i i G,i R,i G,i H i F,i H 0 1 R,i1 R,i H i 1 F,i F,i1 H i 1 H i 1 perfectly harmonic PES is assumed line minimization W. H. Press et al., Numerical Recipes, 3 rd Edition, Cambridge University Press

26 Local Atomic Structure Optimization Cartesian vs. Internal Coordinates d 2 R 1,R 2, R 3 q 1,q 2,q 3 q 1 d 1 q 2 d 2 q 3 qb RR 0 RR q difficult! d 1 (3N-6) x (3N) not invertable! P. Pulay, G. Fogarasi, F. Pang, J.E. Boggs, J. Am. Chem. Soc. 101, 2550 (1979) Local Atomic Structure Optimization Cartesian vs. Internal Coordinates d 2 R 1,R 2, R 3 q 1,q 2,q 3 q 1 d 1 q 2 d 2 q 3 qb RR 0 RR q difficult! d 1 (3N-6) x (3N) not invertable! EE 0 i i q i 1 2 ij F ij q i q j 1 6 ijk F ijk q i q j q k i E q i as simple as possible F ijk 3 E q i q j q k faster convergence of the local geometry optimizer P. Pulay, G. Fogarasi, F. Pang, J.E. Boggs, J. Am. Chem. Soc. 101, 2550 (1979)

27 Local Atomic Structure Optimization Inverse-Distance Coordinates appropriate for weekly bound systems R P.E. Maslen, J. Chem. Phys. 122, (2005) J. Baker, P. Pulay, J. Chem. Phys. 105 (24), (1996) Local Atomic Structure Optimization Inverse-Distance Coordinates appropriate for weekly bound systems R E q E R R 2 q 1 R 2 E 2 E 2 q R 2 R4 2 E R R3 P.E. Maslen, J. Chem. Phys. 122, (2005) J. Baker, P. Pulay, J. Chem. Phys. 105 (24), (1996)

28 Local Atomic Structure Optimization Inverse-Distance Coordinates appropriate for weekly bound systems R q 1 R E q E R R 2 2 E 2 E 2 q R 2 R4 2 E R R3 R,i1 R,i i G,i R,i displacements are accelerated for large interatomic distances P.E. Maslen, J. Chem. Phys. 122, (2005) J. Baker, P. Pulay, J. Chem. Phys. 105 (24), (1996) Global Structure Optimization Potential Energy Surface (PES) E tot R

29 Global Structure Optimization Potential Energy Surface (PES) E tot R Global Structure Optimization Basin-Hopping random variations of atomic coordinates followed by local structural relaxations many different flavours of trial moves possible D. J. Wales, J. P. K. Doye, M. A. Miller, P. N. Mortenson, and T. R. Walsh, Adv. Chem. Phys. 115, (2000) Z. Li, and H. A. Scheraga, Proc. Natl. Acad. Sci. U.S.A. 84, 6611 (1987)

30 Global Structure Optimization further methods... Genetic Algorithms 1 : two parents mate and create a child the fittest (energetically lowest) children survive talk by Matt Probert, Friday, 26 th June, 9:00, Structure predictions in materials science Minima-Hopping, Basin-Paving, Conformational Space Annealing,... 1 D. M. Deaven, K. M. Ho, Phys. Rev. Lett. 75, 288 (1995) Summary How to get E tot in DFT density-mixing schemes (Pulay, Broyden) direct minimization Taking the derivative of E tot atomic forces molecular vibrations What to do with E tot and its derivative local structural relaxation (steepestdescent, conjugate gradient, BFGS) global structure optimization (Basin- Hopping, Genetic Algorithms,...)

31 Summary How to get E tot in DFT density-mixing schemes (Pulay, Broyden) direct minimization Let's do it!! Taking the derivative of E tot atomic forces molecular vibrations What to do with E tot and its derivative today, 14:00 18:00 first practical session lecture hall local structural relaxation (steepestdescent, conjugate gradient, BFGS) global structure optimization (Basin- Hopping, Genetic Algorithms,...)