35th CMD Workshop Beginner Course

Friday, 6th September 2019

First step in first-principles calculation- Precision Criterion for DFT simulation -Hiroki Funashima(Department of Physics, Kobe University)
Masaaki Geshi(INSD, Osaka University)

Translational Symmetry

Bloch State, Bloch function
1-electron Schrödinger Equationh̄22m

2+ V(r)( r ) = E (r)Born - von Karman condition

Periodicity of Potential

V r

tn = V(r)t n = n1 t 1 + n2 t 2 + n3 t 3

eigenstateeffect?

1-electron Schrödinger Equationh̄22m

Eigenvalue
2+ V(r)Hamiltonian Operator

O ( |t n )( r ) = E (r)Eigenfunction

Born - von Karman condition

Periodicity of PotentialO(r)(1V rr)tn = V(r)t n = n1 t 1 + n2 t 2 + n3 t 3h̄22m

Eigenvalue
2+ V(r)Hamiltonian OperatorO( |tn ) =(r(rtn ) = E (rtn )same eigenvalue

Eigenfunctiontn ) also is Eigenfunction whose eigenvalue is E

Bloch Theorem

O ( |t n )phase factork (r) = eik·tnk (r)Bloch condition(r) = eik·r uk (r) Bloch function
periodic function u kr

t n = ukr

Bloch state is characterized by kKey Point

Born-Karmann Condition
same periodicity

Periodic Potential

tn = V(r)V r

Wave function must be written as Bloch Function<latexit sha1_base64="3IwMFi7oAuZDgJDqwZyfCWNErGs=">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</latexit>r) = e~k (~i~k·~rplane wave

u~k (~r)periodic function
u~ (~r ~tn ) = u~ (~r)<latexit sha1_base64="I6WO7+fACc11dg0R2HHf7k9UaTY=">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</latexit>kkr) = (plane wave) x (periodic function)~k (~complex mathematical functional form…

<latexit sha1_base64="U88D8ucuLWWTFvX9w5ztM/BJtAA=">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</latexit>summation of simple mathematical function

Bloch function and basis set<latexit sha1_base64="4Co7YdPgwLE0qbKllCeMG3EFKFA=">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</latexit>r) = e~k (~i~k·~ru~k (~r)

How do we represent the periodic function?

r) = e~k (~i~k·~rX

Cn n (~r)

n<latexit sha1_base64="GdKQx54vuGinZy02vet70xKvJUc=">AAADnHicjVJNaxRBEK3N+JHEj2z0EhBlcbOSXJYeERRBWIgHUYR8uJtAJg4zvb1JszM9w0zvYjLMH8gf8OBJwYN48T948Q8o5KRX8RjBiwff9A4RDVntZaeqXr/XVdVdfhzIVDN2UJmwTp0+c3Zyavrc+QsXZ6qzlzppNEi4aPMoiJIN30tFIJVoa6kDsREnwgv9QKz7/aVif30oklRG6onejcVW6G0r2ZPc04Dc6kMnTqWbOUPBs36eLxgnyRfviaeZLFGHdyNdbuROOgjdTOVLrnLiHemqI4lbrbMmM6t23LFLp07lWo5mK1fJoS5FxGlAIQlSpOEH5FGK3ybZxCgGtkUZsASeNPuCcpqGdgCWAMMD2sd3G1FGPpgeYgH+ybzNElWIi9ypycJRTYB/AmWNGuwje8MO2Qf2ln1lP8fkLM4oat6F9UdaEbsz+3NrP/6pCmE17fxWja1ZU4/umFolao8NUnTBR/rh3vPDtburjewGe8W+of6X7IC9Rwdq+J2/XhGrL8zpXWh6sB1YDl0dd50hLqqP6JnxfXMX3aP6ajRvePPIVNxr48S+BPDilZL/4PXK1xr3osLgHExp5gFcTJv992wddzo3mzZr2iu36q1WOXeTdIWu0wL6uE0tekDL1MbJ7+gTfaYv1jXrvvXIejyiTlRKzWX6Y1mdXw+61Tc=</latexit>

basis sete.g., planewave basis set~r) = eik·~r~k (~kyX~iG·~r
CG(~r)~e~G<latexit sha1_base64="7rBPQaG0xUG0ek2F8BUMh0pdqm4=">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</latexit>kx

cut-oﬀ energy is defined as

Ecut-o↵ =~2 G2max
2mer) = e~k (~i~k·~rX

Cn n (~r)

n<latexit sha1_base64="GdKQx54vuGinZy02vet70xKvJUc=">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</latexit>

basis set

In generally speaking, Mathematics requires an infinite number of bases!

0n1Ideal

0  n  nmax

realistic calculation<latexit sha1_base64="u2MHsUJTpmJqwWJazyL//KSNlWs=">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</latexit><latexit sha1_base64="z3IhIYL2TBLN7QGL3uENnuADDSc=">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</latexit>

Cutoﬀ

genealogy of DFT calculation
PP-PW(Pseudo Potential PlaneWave)

Planewave Basis Set
PW

USPP-PW
NCPP-PWClassical PPPW(Aschcroft,etc )(1991,D.Vanderbuilt)(D.R. Harman,
M. Schlüter and
C. Chiang(1979)

empirical pseudo potential

NC(norm-conserving)PAW

ab initio

projected core orbital

APW

FLAPW

LAPW

Linearized : O.K. Andersen(1979) Takeda and Kübler(1979)

Muffin-Tin Orbital
KKR(Korringa(1947), Kohn andRokstoker(1954)(P. E. Blöchl(1994))

PAW (Projector Augmented Wave Method )Augmented(using spherical wave)(J. C. Slater(1937) )US (ultrasoft)

reduce basis set

Full-potential M.Weinert(1980)

LMTO
O. K. Andersen and O. Jepsen(1984))

FP-LMTO

Linear Combination of Muffin-Tin Orbital

FP-KKR

Computational Experiment 1
To simplify, we try to visualize the plane wave expansion for Hydrogen Radial
wave function.(In this case, angular part is neglected)

Representation for Hydrogen Radial Wave function
as Summation of Plane wave Basis set

Reproducibility of Radial Wave function
Case: 3s orbital

Case: 4d orbital

important for physical properties

important for physical properties

center of Atomcenter of Atom

Practice 1
relation between number of plane wave bases and reproducibility of radial wave function(1) login to cmd2
ssh -Y stud??@cmd2.phys.sci.osaka-u.ac.jp
your id(2) launch App ``rPlaneWaves’’~teac14/bin/rPlaneWaves &(wait few second…)

Menu Window for rPlaneWaves(3) Select Orbital(4) Select num ofPlane wave basis set(5) Select num ofPlane wave basis set……

(X) quit application

Actually, wave function has angular part, we are required much number of bases…

Computational Experiment 2

k-dependency and cutoﬀ Energy dependency for real material
in DFT calculation using FLAPW method(ABCAP code)

GaNc-GaN(zincblende structure)

h-GaN(wurtzite structure)

Which is the most stable phase??

Etotal = 1.73910 2 (eV ) (calc. FLAPW; DFT-LDA)<latexit sha1_base64="Kh6vfBOC9C9ZlN/CP4LViPZT2jI=">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</latexit>

Does your calculation results have enough precision to compare total energy?

In these calculation, we vary 2 parameters
to determine the precision of DFT Calculation Results,••

number of k-point mesh(grid)
cutoﬀ energy to expand wave function

We perform to the 2-type of calculation.
(1) check k-point mesh dependency(cutoﬀ energy is static)(2) check cutoﬀ dependency(k-point mesh is static)Insulator(semiconductor) vs. Metal

Precision for total energy (k-point mesh dependency)
Si: semiconductor

Cu: Metal

Precision for total energy (cutoﬀ energy)
Si: semiconductor

Cu: Metal

Precision for magnetic moment (k-point mesh)

Precision for magnetic moment (cutoﬀ energy)small cell vs. large cellSi (2atoms / unit cell)

SrTiO3 (5atoms / unit cell)translational lattice vector in real spacet n = n1 t 1 + n2 t 2 + n3 t 3
t1 , t2 , t3 :primitive translational lattice vector

primitive reciprocal vector
g1 =

2 (t2t1 (t2t3 )
t3 )

g2 =

2 (t3t1 (t2t1 )
t3 )

g3 =

2 (t1t1 (t2t2 )t3 )g l = l1 g 1 + l2 g 2 + l3 g 3eigl ·tn = e2 ni = 1

orthogonality

gi · tj = 2

reciprocal lattice(n:integer)i,je i(k+gl )·tn = e ik·tn(periodicity in k-space)

Practice 2
relation between number of plane wave bases and reproducibility of radial wave function(1) login to cmd2
ssh -Y stud??@cmd2.phys.sci.osaka-u.ac.jp
your id(2) launch App ``PrecSimu’’~teac14/bin/PrecSimu &(wait few second…)

(3) target material(4:optional) check crystal structure(5) select calculation type(6) select fixed value

(7)select graph type(7) plot graph(if you don’t have DFT calculation results,
you submit the job to calculate
electronic structure)