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Development of a full-potential self-consistent N MTO method and code

Development of a full-potential self-consistent N MTO method and code. Yoshiro Nohara and Ole Krogh Andersen. Contents. Introduction (motivation) Defining the N th-order muffin tin orbitals Output charge density Solving Poisson’s equation

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Development of a full-potential self-consistent N MTO method and code

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  1. Development of a full-potential self-consistent NMTO method and code Yoshiro Nohara and Ole Krogh Andersen

  2. Contents • Introduction (motivation) • Defining the Nth-order muffin tin orbitals • Output charge density • Solving Poisson’s equation • Input for Schrödinger’s equation:FP for Hamiltonian and OMTA for NMTOs • Total-energy examples • Summary NEW NEW NEW NEW

  3. Contents Introduction (motivation) Defining the Nth-order muffin tin orbitals Output charge density Solving Poisson’s equation Input for Schrödinger’s equation:FP for Hamiltonian and OMTA for NMTOs Total-energy examples Summary NEW NEW NEW NEW

  4. N-th order Muffin-Tin Orbitals are Basis sets Advantages of NMTO over LMTO: Accurate, minimal and flexible Accurate because the NMTO basis solves the Schr.Eq. exactly for overlapping MT potentials (to leading order in the overlap) Example: NiO and flexible because the size of the set and (the heads of) its orbitals can be chosen freely Minimal but if the chosen orbitals do not describe the eigenfunctions well for the energies (en) chosen, the tails dominate Example: Orthonormalized NMTOs are localized atom-centered Wannier functions, generated in real space with Green-function techniques, without projection from band states. Future: Order-N metod M.W.Haverkort, M. Zwierzycki, and O.K. Andersen, PRB 85, 165113 (2012)

  5. So it was only possible to get reliable band dispersions and model Hamiltonians using good potential input from e.g., FP LAPW } Overlap matrix eigen energies Potential NMTO eigen states Hamiltonian matrix But sofar no self-consistent loop and no full-potential treatment This talk concerns Work in progress on a FP-SC method and code

  6. Contents Introduction (motivation) Defining the Nth-order muffin tin orbitals Output charge density Solving Poisson’s equation Input for Schrödinger’s equation:FP for Hamiltonian and OMTA for NMTOs Total-energy examples Summary NEW NEW NEW NEW

  7. Spheres and potentials defining the NMTO basis potentialsphere R1 R2 V2(r) s2 s a s1 V1(r) charge sphere(hard sphere for spherical-harmonics projection and charge-density fitting) Superposition of potentials An NMTO is an EMTO made energy-independent by N-ization

  8. where KPW: and Finally, we need to define the set of screened spherical waves (SSW): Kinked partial wave (KPW) This enables the treatment of potential overlap to leading order Kink

  9. Projection onto an arbitrary radius r ≥aR’: But before that, define the operator, PR’L’(r), which projects onto spherical Harmonics, YL’, on the sphere centered at R’ with radius r. The SSW, ψRL(r), is the solution of the wave equation with energy ε which satisfies the following boundary conditions at the hard spheres of radii aR’ : where S is the structure matrix and n and j are generalized (i.e. linear combinations of) spherical Neumann (Hankel) and Bessel functionssatisfying the following boundary conditions: ψ 1 0 0 0 0 0

  10. Kinked partial wave (KPW) Kink S Log.der. YR’L’projection: Kink matrix: (KKR matrix) Logarithmic derivative Structure matrix

  11. : divided energy difference An NMTO is a superposition of KPWs with N+1 different energies,en , which solves Schrödinger’s equation exactly at those energies and interpolates smoothly in between NMTOs with N≥1 are smooth: Kink cancellation NMTO: where : Green matrix = inverted kink matrix

  12. Contents Introduction (motivation) Defining the Nth-order muffin tin orbitals Output charge density Solving Poisson’s equation Input for Schrödinger’s equation:FP for Hamiltonian and OMTA for NMTOs Total-energy examples Summary NEW NEW NEW NEW

  13. where is the occupation matrix The first two terms are single-center Ylm-functions going smoothly to zero at the potential sphere. The last, SSW*SSW term is multi-center and lives only in the hard-sphere interstitial potential sphere s a charge sphere (hard sphere) Charge from NMTOs Charge from PW, Gaussian, or YL basis sets is: PW x PW = PW Gauss x Gauss = Gauss YL x YL = YL But, our problem is that SSW x SSW ≠ SSW

  14. For this, we construct, once for a given structure, a set of so-called value-and-derivative functions each of which is 1 in its own Rlmν-channel and zero in all other. How do we represent theyGycharge so that also Poisson’s equation can be solved? • SSWs are complicated functions and products of them even more so. What we have easy access to, are their spherical-harmonics projections at and outside the hard spheres, and using YlmYl’m’=ΣYl’’m’’these projections are simple to square: • We use Methfessel’s method (Phys. Rev. B 38, 1537 (1988)) of interpolating across the hard-sphere interstitial using sums of SSWs:

  15. The structural value and derivative (v&d) functions The -th derivative function (ν=0,1,2,3) for the RL channel: is given by a superposition of SSWs with 4 different energies and boundary conditions: Example: L=0 functions (for the diamond structure): value 1. deriv 2. deriv 3. deriv

  16. Contents Introduction (motivation) Defining the Nth-order muffin tin orbitals Output charge density Solving Poisson’s equation Input for Schrödinger’s equation:FP for Hamiltonian and OMTA for NMTOs Total-energy examples Summary NEW NEW NEW NEW

  17. For a divided energy difference of SSWs the solution of Poisson’s eq is the divided energy difference one order higher with the energy = zero added as # -1: Solving Poisson’s equation for v&d functions Poisson’s eq. Poisson’s eq is simple to solve for SSWs: Wave eq. Diamond structure Charge Hartree potential Potential 1 Potential 2 s value function Convert to the divided energy difference one order higher. This potential is localized. Connect smoothly to Laplace solutions inside the hard spheres Add multipole potentials to cancel the ones added inside the hard spheres

  18. Contents Introduction (motivation) Defining the Nth-order muffin tin orbitals Output charge density Solving Poisson’s equation Input for Schrödinger’s equation:FP for Hamiltonian and OMTA for NMTOs Total-energy examples Summary NEW NEW NEW NEW

  19. potential sphere s a charge sphere (hard sphere) Getting the valence charge density Diamond-structured Si On-site, spherical-harmonics part. This part is discontinuous at the hard sphere and vanishes smoothly outside the OMT. SSW*SSW part of the valence charge density interpolated across the hard-sphere interstitial using the v&d functions. The valence charge density is the sum of the right and left-hand parts.

  20. Potentials and the OMTA Diamond-structured Si Hartree potential Values below -2 Ry deleted xc potential Calculated on radial and angular meshes and interpolated across the interstitial using the v&d functions full potential Hartree + xc Least squares fit to the OMTA = potential defining the NMTO basis for the next iteration

  21. } Overlap matrix eigen energies NMTO eigen states Hamiltonian matrix SCF loop was closed Charge Potential Matrix elements Sphere packing Since in the interstitial, both the potential perturbation and products of NMTOs are superpositions of SSWs, integrals of their products (= matrix elements) are given by the structure matrix and its energy derivatives. Si+E OMTA + on-site non-spherical + interstitial perturbations Si-only OMTA Si+E OMTA

  22. Contents Introduction (motivation) Defining the Nth-order muffin tin orbitals Output charge density Solving Poisson’s equation Input for Schrödinger’s equation:FP for Hamiltonian and OMTA for NMTOs Total-energy examples Summary NEW NEW NEW NEW

  23. Lattice parameter and elastic constants of Sifor each method FP LMTO with v&d function technique was also implemented. NEW NEW

  24. Timing for Si2E2 NEW NEW Setup time is mainly for the constructing structure matrix. Huge and not usual cluster size including 169 sites with lmax=4 is used for the special purpose of the elastic constants. This cost is controllable for purpose, and reducible with parallelization.

  25. Contents Introduction (motivation) Defining the Nth-order muffin tin orbitals Output charge density Solving Poisson’s equation Input for Schrödinger’s equation:FP for Hamiltonian and OMTA for NMTOs Total-energy examples Summary NEW NEW NEW NEW

  26. Summary Accurate total energy with small accurate basis sets Goal Implementation v&d functions / full potential / self-consistency Examples Si (total energy / elastic constant) Future work Improve the implementation and computational speed, general functionals, forces, order-N method, etc

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