Poisson Solver, Pseudopotentials, Atomic Forces in the BigDFT code

Transcription

1 CECAM Tutorial on Wavelets in DFT, CECAM - LYON,, in the BigDFT code Kernel Luigi Genovese L_Sim - CEA Grenoble 28 November 2007

2 Outline, Kernel 1 The with Interpolating Scaling Functions in DFT for Interpolating scaling functions for electrostatic problems Caclulation of the Kernel Benchmarks and performances A solver for surfaces BC 2 GTH-HGH pseudopoentials in BigDFT Convenience of wavelet basis representation 3 Calculation of Treatment of the different terms

3 The Hartree potential In the DFT calculation in the Kohn-Sham formalism we have:, Find a set of orthonormal orbitals Ψ i (r) that minimizes: E = 1 2 N/2 i=1 Ψ i (r) 2 Ψ i (r)dr + 1 ρ(r)v H (r)dr 2 + E xc [ρ(r)] + V ext (r)ρ(r)dr Kernel where N/2 ρ(r) = 2 i=1 2 V H (r) = 4πρ(r) Ψ i (r)ψ i (r)

4 The in electronic structure calculation During the minimization procedure we need to perform, Kernel s equation Calculation of the self-consistent potential: 2 V H (r) = 4πρ(r) Such equation should be solved at each minimisation iteration. Need of having an efficient and accurate formalism. Plane waves approach The most common approach. Uses the fourier components f(x,y,z) = p x,p y,p z e 2πi ( px Lx ) py pz x+ y+ Ly Lz z f px,p y,p z The equation is algebraic in the Fourier coefficients

5 with plane waves treatment The Laplacian is diagonal in Plane waves representation, Kernel Immediate solution V px,p y,p z = 1 1 ( ) π 2 ( ) 2 ( ) 2 ρ px,p y,p z, p x py L x + L y + p z L z Characteristics Simple and fast, easy to parallelize (FFT) Automatically implement Periodic BC on a finite volume Do not fix the value of V px,p y,p z May result in problems for systems with other BC How to solve this equation for other BC?

6 Isolated BC: the Green function treatment Consider the equation for isolated BC. In this case the solution is given by, Kernel Green s function for (kernel) 2 1 r = 4πδ(r) = V(r) = An unambiguous solution dr ρ(r ) r r Such prescription is unique and is compatible with. How to implement it? Plane-wave based approaches: Truncated kernel, numerically or analytically Screening Functions Approximate treatment, large box required. Need of an accurate and efficient algorithm

7 with Interpolating Scaling Functions A convenient basis for an electrostatic problem, Kernel Interpolating Scaling Functions A set of localised functions centered on the nodes of a uniform mesh 1.5 ϕ j (x) = ϕ 0 (x j) Undergo multiscale relation ϕ(x) = m h j j= m}{{} filters ϕ(2x j) LEAST ASYMMETRIC DAUBECHIES-16 scaling function wavelet The expansion coefficients are the real space values ρ jx,j y,j z = ρ(h x j x,h y j y,h z j z ) Represents exactly an order m polynomial The first m discrete and continuous moments coincide

8 A finite three-dimensional convolution, Kernel The expression of the potential in this basis is thus intuitive: V(i) = K ij ρ j j Where the central object is the Kernel in the K ij = K i j, K i = K ( r )ϕ i (r)dr, K (r) = 1 r Values of the potential are obtained via a convolution V(i) = K i j ρ j j It can be treated via a zero-padded FFT algorithm Exact, easy to parallelize (FFT) For a box of N 3 points, it reduces the scaling from O(N 6 ) to O(N logn)

9 Characteristics of the approach This approach is: Explicit, guarantees the good BC, Kernel Real space-based, immediate interpretation of the expansion coefficients Can be combined with other real-space treatments of the density (e.g. XC) combined with ABINIT XC routines Can be used independently from the DFT code Preserves the first m (multipole) moments of the electrostatic potential Requires only the evaluation of the kernel We need to evaluate N 3 integrals K i = K ( r )ϕ i (r)dr

10 Gaussian tensor product decomposition It can be shown that (Beylkin et al.), Kernel Approximation with gaussians 1 r ω k e p k r 2 k with k = 1,,89, p k, ω k suitably chosen Accuracy of 10 8 for r (10 9,1) We can rescale for R (0,L). The computational cost is reduced N 3 89 N K j = 89 k=1 ω k K jx (p k )K jy (p k )K jz (p k ) K j (p) = ϕ 0 (x)e p(x j)2 dx

11 Other properties of the scaling functions The computational cost is reduced N 3 89 N., Moreover The scaling property of the interpolets ϕ 0 (x/2) = h j ϕ j (x), j Implies similar condition of the one-dimensional function Kernel K j (4p) = 1 2 j h j K 2i j (p). Thus we can evaluate the integrals for low p (not too sharp gaussians), then rescaling. ISF properties allows us to gain in accuracy

12 Characteristics Very fast with moderate memory occupation: Elapsed Time for a grid on a Cray XT3, proc s More precise than other existing s e-04 Kernel 1e-05 1e-06 Max Error 1e-07 1e-08 Hockney Tuckerman 1e-09 8th 14th 20th 30th 1e-10 40th 50th 60th 100th 1e Grid step

13 Characteristics of the, Kernel In summary, we have developed a technique Free boundary conditions Very high accuracy Good computational performance, easy to parallelize Can be used also in other contexts and/or combined with other treatment (e.g. XC) Coupled with ABINIT XC routines In BigDFT code, for big systems it represents a small amount (typical values are less than 2-3% for big systems) of the overall computation L. Genovese, T. Deutsch, A. Neelov, S. Goedecker, G. Beylkin Efficient solution of s equation with free boundary conditions, [arxiv: cond-mat/ ], J. Chem. Phys. 125, (2006)

14 The Surfaces boundary conditions The same formalism can be applied to other BC, Kernel A domain isolated in one direction (say y) and periodic in x and z, with periods L x and L z. A function f which lives in such a domain can be expanded as px pz f(x,y,z) = e 2πi( x+ z) Lx Lz f px,p z (y) p x,p z without any loss of generality. Mixed representation For such functions the s equation become ( 2 y µ 2 p x,p z ) Vpx,p z (y) = ρ px,p z (y), where µ 2 p x,p z = 4π 2 (p x /L x ) 2 + (p z /L z ) 2.

15 A Green s function formalism, Kernel The Green s function for the one-dimensional Helmoltz equation can be used ( 2 y µ 2) G(µ;y) = δ(y) ; { 1 2µ G(µ;y) = e µ y µ > 0 1 y µ =, 2 0 the components of the potential can be carried out: A Green s function for each Fourier component V px,p z (y) = dy G(µ px p z ;y y )ρ px,p z (y ). We can use Interpolating Scaling Functions for the Isolated direction.

16 with Mixed representation, Kernel By defining the one-dimensional kernels K (µ px,p z ;j) = G(µ px p z ;y)ϕ j (y)dy We can have a treatment similar to the case 1-dim convolution for each reciprocal space component V px,p z (i) = K (µ px,p z ;i j)ρ px,p z (j) j From mixed representation to full real space The calculation can be performed with a (semi-)zero-padded FFT algorithm. The I/O are function of the real domain.

17 Speed-up the calculation The calculation of the kernel can be improved, Kernel Speed The analytic form of the Green functions allows for recursion relations improvement in speed of a factor of (N z + 1)/3 Accuracy The scaling relation will again help in the kernel K (2µ;j) = 1 2 j h j K (2µ,2i j) A Free-Lunch case We have the same advantages of the treatment with an explicit formalism conceived for surfaces BC

18 for surface boundary conditions Elapsed Time on a Cray XT3, grid, #proc sec More precise than other treatments e-04 Kernel Absolute relative error 1e-06 1e-08 1e-10 1e-12 Mortensen 8th 16th 1e-14 24th 40th h 8 curve 1e grid spacing L. Genovese, T. Deutsch, S. Goedecker Efficient and accurate three dimensinal solver for surface problems, [arxiv: cond-mat/ ], J. Chem. Phys. 127, (2007)

19 A solver for surface problems, Kernel Like the case We developed a technique Accurate and fast, easy to parallelize Can be applied both in real or reciprocal space codes Explicit treatment No supercell or screening functions More precise than other existing approaches Allows comparisons betweeen different backgrounds (charged systems) Example of the plane capacitor: Periodic Hockney Our approach V V V y y y x x x

20 Pseudopotential treatment Smooth the wavefunction, Kernel Eliminate the rapid variations of the valence wavefunction in the core region. Thanks to them, only two levels of resolutions are enough for good accuracy in BigDFT The pseudopotential used are of GTH-HGH type V loc (r) = ( ) [ ( ) ] eff erf(r) + e 1 r 2 2 r r l P r r r l V nonloc = l ij l m= l h (l) ij p l,m i r p l,m i = Y l,m (θ,ϕ)f (l) i (r l )e 1 2 p l,m j ( ) r r l

21 of the pseudopotential operators These terms act differently on the energy calculation, Kernel Different treatment E loc = R V loc (r)ρ(r)dr V loc (r) can be expressed in interpolating scaling functions basis (real space values) E nonloc = l ij l m= l h(l) ij ψ p l,m i p l,m j ψ The important quantities are scalar products The general term which appear in the application of the nonlocal operator is of the form drψ(r)p(r) where P(r) is a gaussian times a separable polynomial The projector is a tensor product P(r) = P x (x)p y (y)p z (z)

22 Agrees with the wavefunction representation, Kernel For a wavefunction expressed in Daubechies wavelets ψ(r) = i x,i y,i z c ix,i y,i z φ ix (x)φ iy (y)φ iz (z) the calculation of the scalar product can be decomposed From N 3 to 3N calculations dr ψ(r)p(r) = i x,i y,i z c ix,i y,i z φ ix (x)p x (x)dx φ iy (y)p y (y)dy φ iz (z)p z (z)dz The numbers R φ ix (x)p x (x) are the expansion coefficients in Daubechies basis of the projectors. Treatment simple and accurate The application of the nonlocal projectors is easy and exact in the basis representation (no intrinsic approx.)

23 Suppose we find the ground state energy of an electronic system. The forces on an atom in the position R are, Kernel Gradient of the energy F i = de dr i The DFT Hamiltonian is given in the Bohr-Oppenheimer approximation. For this approach it is valid the Feynman-Hellmann theorem de = ψ H dr i R i ψ for the normalised ground state eigenfunction Equivalently, for this case we can write F i = E ext R i

24 Which terms depend explicitly of the atomic positions? The external ionic energy:, E ion = 1 2 a b a b R a R b Kernel The local pseudopotential energy: E loc = drv loc (r,{r a })ρ(r) The nonlocal (separable) one: E nonloc = h i,j ψ p i (R) p j (R) ψ i,j Let us see the procedure used for each term separately

25 The calculation of each term Immediate ionic force calculation The contribution coming from the ionic energy is analytical, Kernel Local pseudopotentials The local pseudopotential term can be separated in two parts: V loc (r) = ( ) [ ( ) ] eff erf(r) + e 1 r 2 2 r r l P = V e + V g r r r l where V g is a gaussian times a polynomial V e is the solution of the isolated s equation where ρ e e V e = 4πρ e ( ) r r l is a gaussian

26 Local pseudopotential energy contribution The contribution from the local term is thus, Kernel Local pseudopotentials F a = a V g ρ + a = a V g ρ + a = a V g ρ + drv e (r)ρ(r) drdr ρ e (r ) r r ρ(r) dr a ρ e (r)v H (r) This approach Preserves locality Easy to parallelize Is of O(n) scaling

27 Contribution from the Nonlocal pseudopotentials, Kernel Same treatment of the energy calculation The projectors of the separable part are gaussian times polynomials (spherical harmonics). Their derivatives are still gaussian times polynomials. For each projector its derivative p(r) p R is a function of the same kind and can thus be expressed with the same technique. We calculate the forces contribution by applying the Leibnitz rule to the energy expression

28 In Summary We implemented the calculation of, where E ext = E I + E L + E S F i = E ext R i Ionic E I = 1 2 i j i j R ij Local E L = R dr V loc (r)ρ el (r) Separable E S = o n o ψ o H S ψ o Kernel Each term is calculated with a different method. The complete implementation Preserves the locality (easy to parallelize) Compatible with the adaptivity The calculation can be performed with arbitrary accuracy (no intrinsic approximations)