CP2K: the gaussian plane wave (GPW) method

Transcription

1 CP2K: the gaussian plane wave (GPW) method Basis sets and Kohn-Sham energy calculation R. Vuilleumier Département de chimie Ecole normale supérieure Paris Tutorial CPMD-CP2K

2 CPMD and CP2K CPMD CP2K Licensed by IBM and MPI Stuttgart Fortran 77 with extension Plane wave basis set heavy use of FFT possibility of k-point sampling GNU license high level fortran 90 Mixed gaussians and plane waves order N construction of the KS Hamiltonian

3 The problem We want to solve numerically the self-consistent Kohn-Sham equations: v KS ( r ) = v( r ) + d 3 r n 0( r ) r r + v xc[n 0 ]( r ) with N n 0 ( r ) = ϕ 0 i ( r ) 2 i= ϕ 0 i ( r ) + v KS ( r )ϕ 0 i ( r ) = ɛ i ϕ 0 i ( r ) (1) We need 1. a representation of φ i ( r ) and n( r ), 2. to be able to compute the total Kohn-Sham energy and its derivatives 3. a method to solve the non-linear the Kohn-Sham equation.

4 Plane wave vs gaussian basis sets: plane waves pros and cons Advantages independent of the nuclei position (good for forces) no BSSE one parameter controls the basis set size orthogonal numerical efficiency through use of FFT Disadvantages large number of basis set elements needed Necessary use of pseudo-potentials loss of chemical insight Border line fill the whole simulation box naturally periodic

5 Plane wave vs gaussian basis sets: gaussians pros and cons Advantages Good already for small basis set sizes correspond to chemical insight Computationally efficient (multi-centre integrals) Possibility to perform all-electrons calculations Disadvantages non-orthogonal atomic position dependent (Pulay forces) Basis set superposition error (BSSE) Systematic improvment not straightforward linear dependencies, over-completeness wrong asymptotic behaviour Border line no implicit periodicity can be tuned for each application

6 Plane wave basis set: Definition Basis set {ϕ α } of plane waves: (Ω is the volume of the box) ϕ α ( r ) = 1 Ω e i G α r The wavefunctions are expanded as φ i ( r ) = 1 c i ( G )e i G r Ω G (2) (3) For an orthorhombic box with lengths L x, L y and L z, the wavevectors G are G = i 2π L x x + j 2π L y y + k 2π L z z ; with i, j, k Z real wavefunctions only half space is needed

7 Representation of the electronic density n( r ) n( r ) = φ i ( r ) 2 = 1 Ω i n( r ) = 1 ñ( G Ω )e i G r G <2G max i G G c i ( G )c i ( G )e i( G G ) r (4) E n cut = 4E φ cut

8 Auxiliary real space grid Sampling interval = L N Nyquist critical frequency f c = 2π 2 G max related to E cut X i = (i 1) on the real space grid We can go from the real-space grid to the reciprocal-space grid and back using Fourier transform techniques If N is a product of small prime numbers one can use Fast Fourier Transform (FFT) techniques which are very efficient (N log N instead of N 2 )

9 Density from wavefunction coefficients Real grid spacing adapted to Ecut n ñ( G ) = Ω n( R N k )e i G Rk = Ω φ i ( R N k ) 2 e i G R k (5) R k i To calculate ñ( G ) we heavily use the FFT: R k c i ( G) (S)INV FFT φ i ( R k ) n( R k ) = i φ i ( R k ) 2 FFT ñ( G )

10 Density derivatives An important quantity will be n( R k ) c i ( G) This is n( R k ) c i ( G) = Rk n( R k ) φ i ( R k ) φ i ( R k ) ci ( G) = 1 Ω φ i ( R k )e i G R k

11 Gaussian basis set Primitive cartesian gaussian functions: g( r, η, l, A) = N c (x A x ) lx (y A y ) ly (z A z ) lz e η( r A) 2 (6) Problems: r: Electron coordinate A: Atomic coordinate l: Angular momentum No cusp at the origin wrong asymptotic behaviour Contracted cartesian gaussian functions: ϕ µ ( r) = i d µi g µi ( r) (7)

12 CP2K: the GPW method Joost VandeVondele et al. Quickstep: Fast and accurate density functional calculations using a mixed gaussian and plane waves approach. Computer Physics Communications, 167: , Two representations of the density: local basis set: n( r) = µν P µν ϕ µ ( r)ϕ ν ( r), (8) P µν : density matrix and plane waves: n( r) = 1 ñ( G) Ω exp(ig r) (9) G <G max

13 P µν is the central object to be determined E KS P µν = K µν is the Kohn-Sham matrix We need to solve the generalized eigenvalue problem K µν ci ν = ɛ i S µν ci ν ν where S µν = ϕ µ ϕ ν is the overlap matrix, with the constraint n occ and P µν = c µ i ci ν i=1 c µ j S µνci ν µ,ν ν = δ ij

14 Going from local basis set to plane waves: auxiliary grid Knowledge of ñ( G) is determined by the knowledge of n( R k ) on a regular mesh R k : n( R k ) = µν P µν ϕ µ ( R k )ϕ ν ( R k ), (10) Here too, heavy use of FFT

15 Order N construction Product of two gaussians is a gaussian centered at: P = η a A + η b B η a + η b (11) with prefactor exp η a η b η a + η b A B 2 mesh points far from P are put to zero (µ, ν) pairs far apart are not considered order N achieved using linked-cell method

16 density derivatives We will need derivatives fo the density: n( R k ) P µ,ν n( R k ) P µ,ν = ϕ µ ( R k )ϕ ν ( R k ) (12) at the points R µ,ν where ϕ µ ( R k )ϕ ν ( R k ) is calculated

17 Multi-grids For some products, a lower cutoff the G m ax can be used. In principle every pair could have a different cutoff; in practice 3 to 5. n( r) = 1 ñ 1 ( G) Ω exp(i G r) + ñ 2 ( G) exp(ig r) G <G 1 G <G 2 Products with a lower cutoff are colocated on a coarser grid adapted to the lower cutoff. The coarser grid is map on to the finer grid by Fourier transforms (FFT s) (13)

18 derivatives: chain rule n( R k ) P µ,ν = n( R k ) n( R k) n( R R k) P µ,ν k = 1 exp ig N ( R k R k) ϕ µ ( R k)ϕ ν ( R k) R k G <G 2 = 1 exp ig Ω R k Ω exp ig N R k ϕ µ ( R k)ϕ ν ( R k) G <G 2 R k Use of FFT

19 Calculation of the Kohn-Sham energy and its derivatives E[n] = T s [n] + Kinetic energy d 3 r n( r )v( r ) + J[n] + E xc [n] I J Z I Z J R I R J ϕ µ ϕ ν is analytic

20 Goedecker-Tetter-Hutter (GTH) pseudopotentials Local part v PP loc LR SR (r) =vloc (r) + vloc (r) = Z r erf(αr) + 4 i=1 C PP i ( 2αr) 2i 2 exp [ (αr) 2] Non-local part with the projectors v SR nl = lm ij r p lm i r p i lm = Ni l Y lm (ˆr)r l+2i 2 exp hij p l j lm r [ 1 2 ( ) ] r 2 r l

21 all short range terms are analytic in the local basis set representation

22 Electrostatic contribution E ES = 1 2 Ω d 3 rd 3 r n( r )n( r ) r r + I Ω d 3 r vi c ( r )n( r ) I J Z I Z J R I R J Using the Fourier transformed representation and with PBC: just like previously 1 2 Ω d 3 rd 3 r n tot( r )n tot ( r ) r r = 2πΩ G 0 ñ tot ( G ) 2 G 2 ñ tot ( G ) = ñ( G ) + I ñ c I ( G ) exp( i G R I )

23 Exchange-correlation energy E xc [n] = is, just like in CPMD, approximated by E xc [n] = Ω N Differences with cpmd Ω d 3 r ɛ xc (n( r ))n( r ) R k ɛ xc (n( R k ))n( R k ) Derivatives of the energy by Fourier transformed or smoothed finite difference Density is first smoothed using nearest-neighbours (reducing the ripples )

24 Smoothing operator q 3 (S q f ) i,j,k = q 3 + 6q q q l m n f i+l,j+m,k+n q = 10 to 50 l= 1 m= 1 n= 1 Chain rule used again for derivatives smoothing of the xc potential

25 Derivatives of the electrostatic and xc contributions (E ES + E xc ) n( R k ) = Ω N (V H( R k ) + V xc ( R k )) V H ( R k ) obtained by FFT from ṼH( G) From chain rule we obtain (E ES + E xc ) P µν = Ω N R k ( V H ( R k ) + V xc ( R ) k ) ϕ µ ( R k )ϕ ν ( R k )

26 Optimization Two ways of solving the Kohn-Sham equation: 1. Reach for self-consistency (traditional scheme) Obtain lowest occupied orbitals from diagonalization of the KS matrix for a test density Density from occupied orbitals Update test density until self-consistency 2. Minimization of the energy functional Use a non-linear minimization scheme to minimize the KS energy

27 CP2K (1): traditional scheme Diagonalization Given the Kohn-Sham Hamiltonian K µ,ν constructed previously, we need to solve Kc = Scɛ (14) where, c are the wavefunction coefficient in the local basis set and S the overlap matrix. ɛ is the diagonal matrix of eigenvalues. S in the local basis set can be calculated analytically.

28 Direct diagonalization Since the number of local basis set elements is not prohibitively large (up to a few thousands) the diagonalization can be done directly using linear algebra packages (lapack or scalapack for parallelized versions) The generalized eigenvalue problem is first transformed to a normal eigenvalue problem introducing U T U = S: K c = c ɛ (15) with ( ) K = U T 1 K (U) 1 c = Uc (16) Cholesky decomposition or symmetric decomposition U = S 1 2

29 Update of the density matrix P A new density matrix is obtained: (P out ) µ,ν = N occ n=1 c n,µc n,ν The resuting density is used to update the test density matrix. Mixing scheme P i+1 in = αp i out + (1 α)p i in (17)

30 Orbital transformation: direct minimization in CP2K Use of an orbital transformation: c(x) = c 0 cos U + xu 1 sin U (18) with c O Sc 0 = I and the linear constraint c O SX = 0 and U = x T Sx

31 Forces E[n] = T s [n] + d 3 r n( r )v( r ) + J[n] + E xc [n] I J Z I Z J R I R J We need to calculate I E[n] at the optimum density n. Three types of terms: 1. v depends explicitly on R I : term d 3 r n( r ) I v( r ) 2. the basis set elements ϕ µ depend on R I : term 2 µ,ν Pµ,ν I ϕ µ E n ϕ ν 3. P µ,ν itself depends on R I : term µ,ν K µ,ν I P µ,ν If the overlap matrix is independent of R I (e.g. plane waves), the last term vanishes. Otherwise, extra force: Pulay forces