公募研究：ワニエ関数を軸とする準粒子自己無撞着法の新しい展開
代表：
鳥取大学 小谷岳生
連携研究者：ＮＩＭＳ
木野日織（ 阪大（赤井研）Æ アリゾナ州立大学Æ 鳥取大学 ）第一原理計算の基礎的な方法論
一体問題Æ低エネルギー励起、準粒子の寿命など
Æ不純物や分子、表面などへの応用＊一体問題解法を整備する（オーダーＮではない範囲）。局在基底と非局在基底。
＊「準粒子self-consistent GW法」の整備（異方性、取り扱いやすさ、収束性、計算速度）。
＊ワニエ関数の構成。低エネルギー励起（フォノンやマグノンなど）。
それらによる自己エネルギーなど。

The PMT method: a newlinearized method
PMT =APW + MTO
Muffin-tin

Takao Kotani (tottori‐u, Japan)

Outline1. Introduction
Motivation, linearized method, APW, MTO
2. Formalism A few key points
3. Resultsecalj google
from H2 through Kr2 in PBE
Not organized well yet …
(MostofAll are still in Japanese…

Motivation:
Quasiparticle Self-consistent GW (QSGW) methodHˆ = Hˆ 0 + ( Hˆ − Hˆ 0 )と分ける方法
をFP-LMTOをもとに開発してきた。
Æ方法論的には成功した。現状では数値的に問題点は色々ある。
Empty spheres,Not systematic convergence,計算のパラメータ設定が面倒。
Æ Look for a better linearized method.

Look for a better linearized method!

Requirements:＊速度、精度、ロバストネス:
Reliable in any atomic configurations.
Stable convergence (intrinsic stability).
Systematic convergence check.
＊更なる発展:GW, Wannier functions, magnon, phonons…
excitations, their coupling, and total energy.
My conclusion:
To overcome shortcomings in FP-LAPW and FP-LMTO, weshould use both of APW and MTO together.

The linearized method•Basis set
Æ eigenfunctions are given as:ψ p (r ) = ∑ α p F j (r )jj

{F j }:basis set{ }.

•Total energy in DF as a functional of ψ p
Kohn-Sham equation{ }{ }•For fixed Fj , minimization with respect to α pj .

{ }•For force, we take into account the changes of Fj ; the Puley term.

ＡＰＷ

Envelope function = Plane Wave(PW)

PW in MTMT region

MT region

3-components
PW ＋ Atomic-like parts － Counter parts
within MTs
within MTs
0th1st2nd•Good for extended state such as Na(3s).
•Systematic•Inefficient for Cu(3d), O(2p)Linearized APW

fcc Cuplane wave is not
so efficient for 3d
15Ry or more for~1mRy convergence

D.Singh et al PRB49,17424

“Atom-centered Hankel function” ×Ylm ,e.g, exp(−κ r )ＭＴＯrHead part

MT region

for S -channel

Tail part

MT region

Smooth Hakel function(not divergent at r=0)
given by M.Methfessel.

3-components
Hankel ＋ Atomic-like parts － Counter parts of Hankel
within MTwithin MT
0th1st2nd•Good for localized orbitals such as Cu(3d), O(2p)•Ｎot so good for extended states.
•Not systematic

PMT =APW + MTO
This is physically reasonable. Complementary.
T.K and M.van Schilfgaarde
arXiv:0808.1604
Phys. Rev. B 81, 125117 (2010)ÆHighly accurate for solids.
No empty sphere.
Small number of APWs.
(Parameters of MTO controlled by hand).
--- Recent development --T.K and Hiori Kino
•Formulation based on a general theory of augmentation.
•Results for dimers from H2 through Kr2 in PBE.
Simple setting of MTOs with APWs (Energy cutoff 2~4Ry) .

2.Formalism
A general formalism of augmentation wave.
Key points:• 3-component Hilbert space• Augmentation for product of envelope functions.

A basis function

MT centers at R a . Radis Ra .
•Spin index is not shown explicitly.

a basis F j (r ) consists of 3-kinds of components

F0 j (r ),{F1 j,a (r)}{F2 j,a (r)}Envelope function
Atomic-like parts
Counter parts
PW orfor r < R
for r < RaaHankel(Bloch sum)
Cutoff: l ≤ lmax ~ 4, Radial-part : ~5th-order Laguere polynomials
Augmentation procedure depend on Envelop functions
Conventional representation of augmented wave (Soler-Williams type)F j (r ) = F0 j (r ) + ∑ F1 j ,a (r − R a ) − ∑ F2 j ,a (r − R a )aaConventional Soler-Williams formalism

F j (r ) = F0 j (r ) + ∑ F1 j ,a (r − R a ) − ∑ F2 j ,a (r − R a )aa•Eigenfunctions ψ p = ∑ α pj F jjzProduct⎛ **Fi (r)Fj (r) = ⎜ F0i (r) +⎞F1i,a (r − Ra ) − ∑ F2i,a (r − Ra ) ⎟∑aa⎝⎠⎛⎞× ⎜ F0 j (r) + ∑ F1 j,a (r − Ra ) − ∑ F2 j,a (r − Ra ) ⎟aa⎝⎠**contains cross terms.Æ Approximaiton: we neglect these terms
ÆDF total energy, Kohn-Sham equation.
A problematic logic…
It is better to say no cross terms by definition.
How to formulate?(next page)

we start over from the definition of the Hilbert space.

3-component Hilbert space
A basis itself is a set of functions :{{} {}} {...} means a set.
The set of F j , { F j } is a Hilbert space
F j = F0 j (r ), F1 j ,a (r ) , F2 j ,a (r )(as a direct sum of linear spaces).

Æ 3-component Hilbert space
We define a quantum mechanical system in thisspace. Then we apply DFT.

In the 3-component Hilbert space{{}}} {F j = F0 j (r ), F1 j ,a (r ) , F2 j ,a (r ) ,jψ=α•Eigenfunction:∑ pFjpj•Definition of product for bilinear quantities：{{}{}}

Fi*(r)Fj (r ') = F0*i (r)F0 j (r '), F1*i,a (r)F1 j,a (r ') , F2*i,a (r)F2 j,a (r ')ψ *p ' (r )ψ p (r ') = ∑ α ip*'α pj Fi* (r ) F j (r ')i, jProduct is definedÆ How to connect to real world?

Mapping from the 3-component Hilbert space to real world:{{}}}{Fi*(r)Fj (r ') = F0*i (r)F0 j (r '), F1*i,a (r)F1j,a (r ') , F2*i,a(r)F2 j,a(r ')R-mapping, R […]R ⎡⎣Fi*(r)Fj (r ')⎤⎦ = F0*i (r)F0 j (r ') + ∑F1*i,a (r − Ra )F1 j,a (r '− Ra ) − ∑F2*i,a (r − Ra )F2 j,a (r '− Ra )aMeaningful when r ≈ r ' :a(1)Both are in the same MT;(2)Both are in interstitial region.
augmentation for the product of Envelope functions(the essence of Soler-Williams formulation)c.f.

R ⎡Fi*(r)⎤R ⎡⎣Fj (r ')⎤⎦⎣⎦

R-mapping is not to basis functions, but to the products.

“density-matrix-like product”

ψ p = ∑α p F jjj

ρij =OCCUPIED∑

α ip*α pjP⎧∑ρ F* (r)F (r '),⎫3-component
ij 0i0j⎪⎪⎪,ij⎪*ρij Fi (r)Fj (r ') = ⎨⎬∑**i, j⎪ ∑ρij F1i (r)F1 j (r '), ∑ρij F2i (r)F2 j (r ')⎪i, j⎩⎪ i, j⎭⎪R-mapping⎡⎤*R ⎢ ∑ ρij Fi (r ) F j (r ') ⎥ = ∑ ρij F0*i (r ) F0 j (r ')⎣ i, j⎦ i, j+ ∑ ρij F1*i (r − R ) F1 j (r '− R ) − ∑ ρij F2*i (r − R ) F2 j (r '− R )i, ji, jÆ density, kinetic energy density, and so on in the Hilbert space.
(Another approximation to the Hartree term (Weinert method)Æ as a functional of the 3-component density n(r ) = ∑ ρij Fi* (r ) F j (r )i, jFinally, we can give total energy in DFT
as a functional of ψ p = ∑ α pj F jj

Minimization of the total energy gives the Kohn-Sham equation.
Our formalism results in the equivalent equations with previous articles, but :•Consistency is transparent.
•Definition of diatomic quantities in the cutoff model is clear.
•Convenient to derive atomic forces (clear about surface term).

Hilbert space in the Real‐world

3‐component Hilbert space (artificial)

A basis function

A basis function

Fi (r) = F0i (r) + F1i (r −R) − F2i (r −R)
product*Fi (r ) F j (r ')Fi (r ) = { F0i (r ), F1i (r ), F2i (r )}Define productFi* (r ) F j (r ')By definition, No cross terms
Bilinear quantities:
norm, density, and so on.
Coulomb interaction also.

Define bilinear quantities,
Coulomb interaction andso on.

No cutoff limit

No cutoff limit

Exact Solution

Exact Solution

Both should be the same
Basis is in real worldÆApproximation

Numerically solvable model.
Well-defined for finite cutoff.

c.f. PAW treated the Hilbert space of “pseudo wave functions”.
( F0i (r ) determins F1i (r ) and F2i (r ) uniquely. )zPrepare 3-comonents Hilbert space
zDefine products and so on in the spacejψ=αzFor given p ∑ p F j ,jwe can evaluate (or define) bilinear quantities
as a functional in the space through R-mapping.

Problems:
Positive definiteness of overlap matrix
Null vector problems
Orthogonality to core (since we use frozen core approximation).

RESULTS•PBE, Colinear Spin, Scalar Relativistic DFT•Large supercell 13.5Å× 15Å×16.5Å.
•APW and Atom-centered localized basis (MTO and lo)•Ferro-magnetic. Only inversion symmetry. Γ point only.
MTO set(part)×２

Number of local basis (1+3+5+7)×２+ (1+3)=36
Atom-independent setteing of MTO parameters(not shown here)

Atom in supercell
22

34(Ry)1973,3025
NAPW =1081,
NBASIS=1081+36, 1973+36, 3025+36

Diatomic molecules(From H2 through Kr2)

2Sz=42Sz=22Sz=0
APW( EAPW means EMAX)EAPW=2

EAPW=3

EAPW=4

NAPW

1081

1973

3025

NBASIS

1081+36x2

1973+36x2

3025+36x2

Cr2 at equilibrium atomic distanceVibrational frequency
Equilibrium Atomic distanceAtomization energy

2Sz=4

2Sz=2

Nbasis=1081+36×2, 1973+36×２, 3025+36×２

Nbasis=1081+36×2, 1973+36×２, 3025+36×２

Comparison with Gaussian

ＮＲ：non relativistic

ＧＴＯ：6-311+G(d,p)

Sumary• Formalism based on “3-component Hilbert space”.

Theoretical consistency is transparent for cutoffed model.
Atomic force is derived systematically (not shown).
• With APW (cutoff energy is 2～4Ry), we can perform accurate
calculations for homo nuclear diatomic molecules.
•Two kinds of augmented wave Ævery efficient calculations.
For example, 300 atoms in 15Åcell Æ roughly speaking,basis ~ 30×300 + 2000 (~ 10000 basis)*Construction of the Wannier functions. (superposition of MTO).
* Better control for Positive definiteness of Norm.
Null space problem:
vectors with zero eigenvalues for Hamiltonian and Overlap matrix.

Total energy minimization in the Linearized method
previous iteration

One-particle potential V(r )Basis set {Fi (r )}augmented wave

−ΔHamiltonian H ij = Fi+ V Fj ,2m
Overlap matrix Oij = Fi Fj
Diagonalization (H − ε p O )α p = 0

electron density n(r )Next iteration•inner product:

Fi F j =3*
3*
3*rrrrdrF()
F()+drF()
F()−drF∫ 0i 0 j∫∫1i1j2i (r) F2 j (r)r <RCellr <R

3*3*
drF(r)F(r)−drF∫ 0i 0 j∫
2i (r) F2 j (r)

Cell

r <R

should be positive definite. Also for the Coulomb matrix.

Hilbert space in the Real‐world

3‐component Hilbert space(model space)

A basis function

A basis function

Fi (r) = F0i (r) + F1i (r −R) − F2i (r −R)
Basis for*Fi (r ) F j (r ')product
Bilinear Operators

Fi (r ) = { F0i (r ), F1i (r ), F2i (r )}Basis for *NoF(r)F(r')
jproduct icross terms

R-mapping

Define bilinear operatorsin this space

No cutoff limit
Exact Solution

No cutoff limit
Exact Solution

Both should be the same
Approximations to Æ Solve it

Numerically solvable model

1. Numerically solvable model Æ then apply DFT
2. The exact solution at the infinite cutoff limit.
c.f. PAW gives “quantum mechanism” in space of “pseudo wave functions”.

Hilbert space in the Real‐world

3‐component Hilbert space(model space)

A basis function

A basis function

Fi (r) = F0i (r) + F1i (r −R) − F2i (r −R)
Basis for*Fi (r ) F j (r ')product
Bilinear Operators

Fi (r ) = { F0i (r ), F1i (r ), F2i (r )}Basis for *NoF(r)F(r')
jproduct icross terms

R-mapping

Define bilinear operatorsin this space

No cutoff limit
Exact Solution

No cutoff limit
Exact Solution

Both should be the same
Approximations to Æ Solve it

Numerically solvable model

1. Numerically solvable model Æ then apply DFT
2. The exact solution at the infinite cutoff limit.
c.f. PAW gives “quantum mechanism” in space of “pseudo wave functions”.