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

Contents 1.Introduction (motivation) 2.Defining the Nth-order muffin tin orbitals 3.Output charge density 4.Solving Poisson’s equation 5.Input for Schrödinger’s equation: FP for Hamiltonian and OMTA for NMTOs 6.Total-energy examples 7.Summary NEW

Contents 1.Introduction (motivation) 2.Defining the Nth-order muffin tin orbitals 3.Output charge density 4.Solving Poisson’s equation 5.Input for Schrödinger’s equation: FP for Hamiltonian and OMTA for NMTOs 6.Total-energy examples 7.Summary NEW

Advantages of NMTO over LMTO: N-th order Muffin-Tin Orbitals are Basis sets 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: 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 and flexible because the size of the set and (the heads of) its orbitals can be chosen freely but if the chosen orbitals do not describe the eigenfunctions well for the energies (  ) chosen, the tails dominate M.W.Haverkort, M. Zwierzycki, and O.K. Andersen, PRB 85, (2012) Example: NiO Minimal

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

Contents 1.Introduction (motivation) 2.Defining the Nth-order muffin tin orbitals 3.Output charge density 4.Solving Poisson’s equation 5.Input for Schrödinger’s equation: FP for Hamiltonian and OMTA for NMTOs 6.Total-energy examples 7.Summary NEW

a s charge sphere (hard sphere for spherical-harmonics projection and charge-density fitting) potential sphere V 1 (r) s1s1 s2s2 V 2 (r) R1R1 R2R2 Superposition of potentials Spheres and potentials defining the NMTO basis An NMTO is an EMTO made energy-independent by N-ization

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

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

Kink Y R’L’ projection : Kink matrix: (KKR matrix) Logarithmic derivativeStructure matrix Log.der. S Kinked partial wave (KPW)

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

Contents 1.Introduction (motivation) 2.Defining the Nth-order muffin tin orbitals 3.Output charge density 4.Solving Poisson’s equation 5.Input for Schrödinger’s equation: FP for Hamiltonian and OMTA for NMTOs 6.Total-energy examples 7.Summary NEW

Charge from NMTOs where is the occupation matrix a s charge sphere (hard sphere) potential sphere The first two terms are single-center Y lm -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 PW x PW = PW Gauss x Gauss = Gauss Y L x Y L = Y L But, our problem is that SSW x SSW ≠ SSW Charge from PW, Gaussian, or Y L basis sets is:

How do we represent the  charge 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 Y lm Y l’m’ =ΣY l’’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: 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.

The structural value and derivative (v&d) functions Example: L=0 functions (for the diamond structure): value1. deriv3. deriv2. deriv 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:

Contents 1.Introduction (motivation) 2.Defining the Nth-order muffin tin orbitals 3.Output charge density 4.Solving Poisson’s equation 5.Input for Schrödinger’s equation: FP for Hamiltonian and OMTA for NMTOs 6.Total-energy examples 7.Summary NEW

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

Contents 1.Introduction (motivation) 2.Defining the Nth-order muffin tin orbitals 3.Output charge density 4.Solving Poisson’s equation 5.Input for Schrödinger’s equation: FP for Hamiltonian and OMTA for NMTOs 6.Total-energy examples 7.Summary NEW

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

full potential Hartree + xc 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 Least squares fit to the OMTA = potential defining the NMTO basis for the next iteration

Sphere packing Si-only OMTA Si+E OMTA 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 Matrix elements NMTO Potential Hamiltonian matrix Charge Overlap matrix } eigen energies eigen states SCF loop was closed

Contents 1.Introduction (motivation) 2.Defining the Nth-order muffin tin orbitals 3.Output charge density 4.Solving Poisson’s equation 5.Input for Schrödinger’s equation: FP for Hamiltonian and OMTA for NMTOs 6.Total-energy examples 7.Summary NEW

Lattice parameter and elastic constants of Si for each method a(a.u.)C 44 0 C 11 C 12 (Mbar) LMTO-ASA LMTO-FP NMTO-FP Other LDA Expt FP LMTO with v&d function technique was also implemented. NEW

Timing for Si 2 E 2 Setup time Time per sc-iteration LMTO-ASA5001 LMTO-FP NMTO-FP 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.

Contents 1.Introduction (motivation) 2.Defining the Nth-order muffin tin orbitals 3.Output charge density 4.Solving Poisson’s equation 5.Input for Schrödinger’s equation: FP for Hamiltonian and OMTA for NMTOs 6.Total-energy examples 7.Summary NEW

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