Transcript スライド 1

Development of a full-potential selfconsistent NMTO method and code
Yoshiro Nohara and Ole Krogh Andersen
Contents
NEW
NEW
NEW
NEW
1.
2.
3.
4.
5.
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
6. Total-energy examples
7. Summary
Contents
NEW
NEW
NEW
NEW
1.
2.
3.
4.
5.
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
6. Total-energy examples
7. Summary
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
Minimal 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 (en) chosen,
the tails dominate
Example:
Orthonormalized NMTOs are localized atomcentered 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)
But sofar no self-consistent loop
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
Overlap matrix
Potential
NMTO
Hamiltonian matrix
}
eigen energies
eigen states
This talk concerns
Work in progress on a FP-SC method and code
Contents
NEW
NEW
NEW
NEW
1.
2.
3.
4.
5.
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
6. Total-energy examples
7. Summary
An NMTO is an EMTO made energy-independent by N-ization
Spheres and potentials defining the NMTO basis
potential
sphere
R1
R2
V2(r)
s2
s a
s1
charge sphere
V1(r)
(hard sphere for spherical-harmonics
projection and charge-density fitting)
Superposition
of potentials
Kinked partial wave (KPW)
This enables the treatment of potential overlap to leading order
0
KPW: L
L
L
L


 (r )   (r )Y (rˆ)   (r )Y (rˆ)  (r )
Kink


0
where
(  e  v(r ))  0
and
(  e )  0

a s
0
r
Finally, we need to define
the set of screened spherical waves (SSW):
(  e )  0
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’ :
PR'L' (aR' ) RL   R'R L'L
Projection onto an arbitrary radius r ≥ aR’ :
PR'L' (r ) RL  nR'l ' (r ) R'L'RL  jR'l ' (r )SR'L'RL
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:
n(a)  1, n' (a)  0
j(a)  0
j' (a)  1/ a 2
ψ
1
0
0
0
0
0
Kinked partial wave (KPW)
Kink



0
S
Log.der.
r
a s
YR’L’ projection:
PR'L' (r ) RL
 ln 
a
 ln r
0
Kink matrix: K R 'L 'RL
(KKR matrix)
 R 'L 'RL  S R 'L 'RL
a , Rl
Logarithmic derivative Structure matrix
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
NMTO:
where
1
(G)[0...N ]G[0...N ]
1
G  K : Green matrix = inverted kink matrix
[0...N ] : divided energy difference
X [m...n  1]  X [m  1...n]
X [m...n] 
em en
NMTOs with N≥1 are smooth: Kink cancellation
Contents
NEW
NEW
NEW
NEW
1.
2.
3.
4.
5.
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
6. Total-energy examples
7. Summary
Charge from NMTOs





0 
0 
 (r )   (r )  (r )   (r )  (r )  (r ) (r )
where
 n 'R'L',nRL 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
Charge from PW, Gaussian, or YL basis sets is:
potential sphere
s
a
charge sphere
(hard sphere)
PW x PW = PW
Gauss x Gauss = Gauss
YL x YL
= YL
But, our problem is that
SSW x SSW ≠ SSW
How do we represent the  charge so
that also Poisson’s equation can be solved?
•
We use Methfessel’s method (Phys. Rev. B 38, 1537 (1988)) of
interpolating across the hard-sphere interstitial using sums of SSWs:
n n

D





c
 n 'R'L' n 'R'L'nRL nRL  nRL nRL  RLbRL
n ' R ' L 'nRL
nRL
nRL
• 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:
P ~ n n  n Sj  jS n  jS Sj
•
For this, we construct, once for a given structure, a set of so-called
value-and-derivative functions DnRL each of which is 1 in its own Rlmνchannel and zero in all other.
The structural
value and derivative (v&d) functions

DRL (r ) is given by
The n -th derivative function (ν=0,1,2,3) for the RL channel:
a superposition of SSWs with 4 different energies and boundary conditions:
n
d dr rP
n'
n
(
r
)
D
R 'L '
RL

a
 n 'n  R 'R L 'L
Example: L=0 functions (for the diamond structure):
value
1. deriv
2. deriv
3. deriv
Contents
NEW
NEW
NEW
NEW
1.
2.
3.
4.
5.
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
6. Total-energy examples
7. Summary
Solving Poisson’s equation for v&d functions
Poisson’s eq is simple to solve for SSWs:
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:
Poisson’s eq.
 V  

Wave eq.


e
 [1,0...n ]  [0...n ]  0
 [1]  0
Diamond structure
Charge
s value function
Potential 1
Convert to the divided
energy difference one
order higher. This
potential is localized.
Potential 2
Connect smoothly
to Laplace
solutions inside
the hard spheres
Hartree potential
Add multipole
potentials to cancel
the ones added inside
the hard spheres
Contents
NEW
NEW
NEW
NEW
1.
2.
3.
4.
5.
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
6. Total-energy examples
7. Summary
Getting the valence charge density
Diamond-structured Si
On-site,
sphericalharmonics 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 hardsphere interstitial
using the v&d
functions.
The valence potential sphere
charge density
s
is the sum of
a
the right and
left-hand parts.
charge sphere
(hard sphere)
Potentials and the OMTA
Diamond-structured Si
Hartree
potential
Values below
-2 Ry deleted
xc potential
full potential
Calculated on radial
and angular
meshes and
interpolated across
the interstitial using
the v&d functions
Hartree + xc
Least squares fit
to the OMTA
= potential defining
the NMTO basis for
the next iteration
Sphere packing Matrix elements
Si-only
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
Si+E
OMTA
Overlap matrix
NMTO
Hamiltonian matrix
}
eigen energies
eigen states
SCF loop was closed
Potential
Charge
Contents
NEW
NEW
NEW
NEW
1.
2.
3.
4.
5.
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
6. Total-energy examples
7. Summary
Lattice parameter and elastic constants of Si
for each method
FP LMTO with v&d function technique was also implemented.
a(a.u.)
C440
C11
C12(Mbar)
LMTO-ASA
10.18
0.54
2.60
0.25
NEW
LMTO-FP
10.25
1.14
1.64
0.62
NEW
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
Timing for Si2E2
Setup time
Time per
sc-iteration
LMTO-ASA
500
1
NEW
LMTO-FP
3000
10
NEW
NMTO-FP
4000
13
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
NEW
NEW
NEW
NEW
1.
2.
3.
4.
5.
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
6. Total-energy examples
7. Summary
Summary
Goal
Accurate total energy with small accurate basis sets
Implementation
Examples
Future work
v&d functions / full potential / self-consistency
Si (total energy / elastic constant)
Improve the implementation and
computational speed, general functionals,
forces, order-N method, etc