Development of a full-potential self-consistent N MTO method and code
26
Development of a full- potential self-consistent NMTO method and code Yoshiro Nohara and Ole Krogh Andersen
description
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 - PowerPoint PPT Presentation
Transcript of Development of a full-potential self-consistent N MTO method and code
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
NEW
NEW
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
NEW
NEW
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, 165113 (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
NMTOPotentialHamiltonian 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
NEW
NEW
NEW
as
charge sphere(hard sphere for spherical-harmonicsprojection and charge-density fitting)
potentialsphere
V1(r)
s1
s2
V2(r)
R1 R2
Superposition of potentials
Spheres and potentials defining the NMTO basis
An NMTO is an EMTO made energy-independent by N-ization
Kink
0
rsa
KPW: )()ˆ()()ˆ()()( 0 rrYrrYrr LLLL
Kinked partial wave (KPW)This enables the treatment of potential overlap to leading order
0))(( rvwhere
0)(
Finally, we need to define
the set of screened spherical waves (SSW):
0)( 0 and
LLRRRLRLR aP ''''' )( Projection onto an arbitrary radius r ≥ aR’ :
RLLRlRRLLRlRRLLR SrjrnrP '''''''''' )()()(
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 functions satisfying the following boundary conditions:
2/1)('0)(0)(',1)( aajajanan 1
000 0 0ψ
Kink
0
rsa
RLLR rP )(''YR’L’ projection:
Kink matrix: RLLRRLLR
Rla
RLLR Sr
aK ''''
,
0
'' ln
ln
(KKR matrix)
Logarithmic derivative Structure matrix
Log.der.S
Kinked partial wave (KPW)
An NMTO is a
NMTOs with N≥1 are smooth: Kink cancellation
1]...0[]...0)[( NGNG
where
NMTO:
1KG
: divided energy difference
nm
nmXnmXnmX
]...1[]1...[
]...[
]...0[ N
: 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
NEW
NEW
NEW
Charge from NMTOs)()()()()()()( 00 rrrrrrr
where is the occupation matrixRLLR ,'''
as
charge sphere(hard sphere)
potential sphere
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
PW x PW = PW
Gauss x Gauss = Gauss
YL x YL = YL
But, our problem is that SSW x SSW ≠ SSW
Charge from PW, Gaussian, or YL 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 YlmYl’m’=ΣYl’’m’’
these projections are simple to square:
SjjSnjSSjnnnP ~
• We use Methfessel’s method (Phys. Rev. B 38, 1537 (1988)) of interpolating across the hard-sphere interstitial using sums of SSWs:
RL
RLRLRLLR
RLRLLRLR c
'''
'''''' RL
RLRLbD
• 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.
RLD
The structural value and derivative (v&d) functions
Example: L=0 functions (for the diamond structure):
value
LLRRaRLLR DrrPdrd '''''' )(
1. deriv 3. 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:
)(rDRL
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
NEW
NEW
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: 0]1[
0]...0[]...0,1[
Connect smoothly to Laplace solutions inside the hard spheres
Add multipole potentials to cancel the ones added inside the hard spheres
Charge Hartree potential
VPoisson’s eq is simple to solve for SSWs:
Poisson’s eq.
Wave eq.
Potential 1 Potential 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
NEW
NEW
NEW
Getting the valence charge densityDiamond-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.
as
charge sphere(hard sphere)
potential sphere
full potential
Hartree + xc
Potentials and the OMTADiamond-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-onlyOMTA
Si+EOMTA
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+EOMTA + 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
NEW
NEW
NEW
Lattice parameter and elastic constants of Sifor each method
a(a.u.) C440 C11 C12(Mbar)
LMTO-ASA 10.18 0.54 2.60 0.25
LMTO-FP 10.25 1.14 1.64 0.62
NMTO-FP 10.18 1.09 1.78 0.59
Other LDA 10.17 1.10 1.64 0.64
Expt. 10.27 1.68 0.65
FP LMTO with v&d function technique was also implemented.
NEW
NEW
Timing for Si2E2
Setup timeTime persc-iteration
LMTO-ASA 500 1
LMTO-FP 3000 10
NMTO-FP 4000 13
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.
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
NEW
NEW
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
