Real-Space Electronic Structure Calculations using Meshfree Particle Method

(1)

Real-Space Electronic Structure Calculations using Meshfree Particle Method

Soichiro Sugimoto

Abstract—^This paper presents a meshfree particle method for real- space electronic structure calculations using symmetric smoothed particle hydrodynamics (SSPH). In the presented method, the real-space is repre sented by a finite set of particles without using any mesh. The Schrodinger and Poisson equations are self-consistently solved in real-space electronic structure calculations based on the density functional theory. These equations are discretized using particles, which are distributed on the real-space as computation points. As numerical tests, SSPH is applied to the Poisson and Schrodinger equations which can be solved analytically.

The results using SSPH are compared to those of the finite-difference method. As an application of the presented method, we have calculated the electronic structure of atoms such as H, He, and Li. The orbital energy eigenvalues are compared to those of the finite-difference method and the literature. The results calculated using SSPH are in good agreement with those of the finite-difference method. This means that SSPH has the application capability to practical electronic structure calculations.

I.

[1], [2].

[3], [4].

LT, m. [s], [6].

tfz, [6], n .

L-cv^^fc46, i c L < m t L .

XUnir^. Lfzr^^X,

tcft. :: t

6. yys^cDmm^^mxh^, mm

.#,) ^mznb,

nmn^(Dmmz'^ib^txm^ (§+«) ^

4a V 'b i^v 1 ^ tu^.

v^75^

m^itni:>tix\^^?>. Jun <Dm^ [8] XiX Galerkin

Reproducing kernel particle method (RKPM) ^ Kronig-Penney <DX- r/w. Si,

^ lc4oit a ^. Nicomedes

[9] "Cit, Meshless local petrov-galerkin (MLPG) method

^ 3 ^7Ca^?nlSffilJ^<7) Schrodinger 3 ^kixi^F'M^y' y y^^/uo:) Schrodinger Kronig-Penney if(OM^

^zmcz y=LyV~-m

oid Smoothed particle hydrodynamics (SPH)

[10], [II] SPH

'^mxMt^(r>:}3m^^m< tzi^{zmm^tix\^^^^mxh^ [12].

SPH SPH

Poisson Maxwell ir'

[i3]-[i6]. L^-uir^h,

ilia t h V\

SPH T-i± Kernel

Kernel

iCj:o-ri£fa$ix6. LtS^L. SPH cD Kernel

[13], [17].

SPH iCtoit'S Kernel i: LT, Reproducing kernel particle method (RKPM) [18], Corrective smoothed particle method (CSPM) [19], Modified smoothed particle hydrodynamics (MSPH) [20], Symmetric smoothed particle hydrodynamics (SSPH)

[21H23] /cC $ tL-CV^-5. (Moving Least

Square Approximation: MLS ififlil) MLSPH [24]

^nri/^a. MSPH t SSPH T-lt, iiJ!SjC</)TaylorJlFj|^ffll^5C i: ict: oT, iiM(Dmm(D^m'i^^^im(r>}k^(Dmmxn^ :i RKPM ^ MLS i£[a^fflV^fc:#^feic4av^-C

^Moyd:.^(Dmxn^xbt^^mxh^. lts^l, nd^(Dm.t.

T-#lt-r5 <b, MSPH t RKPM Tit Kernel Mic, MLS ififfilTit SSPHcO:^^(t, Kernel!]!^

(D'^W'iL^Wl^^fz^, Taylor ®P^iii2fe(7)3ltT'#l®-t6 C i:

fi^^m^Xh^. ttz, SSPH it MSPH (CJt'<TlhS[;0"^®^Tfe«7

MSPH ^ [22].

Chelikowski h [5] ^ Iwata h [2]

Ltz^^^^x,

v^tc^?gP.^®™iiH-mTit, SSPH

z t ^mmxh^

«^co@6^it, «/^fzn'f^mvcm<Di7m^(om^

^Tit, \^xn

—5T'#)^ SSPH i: L

T5^i)tar^igc&ics<5< SSPH

Z t ictoTtSaf^ffl ^

<9 }i,/ufz^myf^yyy^

yucpco—Schrodinger (Kohn-Sham

^) Tfe'5. Hartree I''^ i: Kohn-Sham

Hipkir) = £ki>k{r) (1) // = -F V;xt(r) -P VH(r) -h VMir) (2) : : t t , tpkir)

/u:3r—[g^fi,

, £k

(2)

K-xt kh t Kx.- Vh t Vc.:

P(t-) ^ A |T/'fc(T-)

C-T', A^„ri,u, Afifc#@<7)|liiI(ciolt5'fl-7-C)t^^

/N- h y - • .-Kr Vh ir{k(D Poisson ^ t

V^VH(r) = —47rp(r) (4)

y. ctifimi'-i®® p om

—doi^TIi, Perdew-Zunger W [25].

Kohn-Sham ')j^^\t,

-7-'^;® p{r) /j\Wf=T'fc-5 i: I ^9 ^f1^C0TT'ft?/j>4ntn(±Vj^ 1^4^

i.\ 55K^c/)iTi-^T-[i.

Self-Consistent ig-t (Fig.l).

III.

•5 Schrodinger Kohn-Sham Poisson

:^$-r'i±, KiiAfec^-o-r'^6 SSPH Schrodinger Poisson

ir;cioT(i}»'/>AfS5Ci75Bi^fi:^n9#r^T-^b6.

T, (Fig.2).

^.a^^iii p(r)

Poisson^?iiC^ftS<

V^FhW = -47rpCr)

1

^3ci^°f'>v-\';ucolfK Fcii(r) = FextCr) + + K„,|p]{r)

1

Kohn-Shain^liS$^S<

-\^'^ +FeffCr) ipkir) =gfci/'kCr)

" o r P i e

Pir)= ^ fk\^k(.r)l'

Self-Consistent ?

Convergence clieek im.«w-l'oUlll'<'-?

Fig. I. •^myaf^lSi:t/:iCj:6®^4ft®IF»£/)#iil^. Kohn-Sham

K.fr(r) 75>®M«-C-J).5i:l'9-^l4^c/)T -Cft?75>^citnrt4C5/.cv^. %^.omWX-\i, Vff t P 5rli®?)it'S:(b$-d:r

Selt-Consisteni

A. Smoothed particle hydrodynamics <Tyl^.M.

4>{r) (t Dirac (D Delta t 9

= f ip{r')6{r —r')dr'

Jii

it(5)}C#^it^Delta^l^^ Kernel Mi: WtlSr^mW(r-r'./i)

•C'S#^^-5i:, ^coJ;9!cM-C§^.

i)[r) fn 1 •jp{r')W{r —r', h)dr' (6)

Jii

ZXX, /i fbSEBt (smoothing length) Xh^. Kernel

li Delta Um^X'h^^f)^h, Gauss

(Fig.3) . Gauss 9(c4-5.

14^(7- —7"', h) = VV(|r - T*'!, /j) —U''(;-. h) —aexp

r-T'. Q \ a ^ l/{h^), 2

^TctO^-n a = l/ih'^n), 3 lk7cd:>^^ a = l/(/i^7r2) Xh^.

^W9E-C'(±, —^Xh^^ Wendland [26], [27] ^

Kernel m ^ t LX^mX^. h ^W\ Ltim

"n, Wendland kernel Gaussian kernel X '0 [32].

Wir,h) =

0 (^>2)

1 lk7€o:>m'^ Q = 3/(4/1), 2 t^CTccT)^^ Q = 9/(57r/i2). 3 a = 45/(32x/i^) i:4-5.

iiiiw

^Hri)

p(ri) l^eff(n) Fig.2. 3k^mm[~iS\iroU-T'ryt>(Dd Z--^.

Fig.3. 2 Kernel f^li!&. Kernel

tl^XJkii'^^m^X-h^^. K/t (± Kernel 0 . « {4^^^

(DmzX.-oX%t£^o. Wendland kernel C)^^(4 k = 2 Xh^.

(3)

J:o-Cg|icfli$:fT9. Lfz-/)'^^X, j

mdr' ^^m(Oi^mAVj T-®#^;t-5Ci:(Cj;oT, ^(6) (7)^

^x(DW^(Dmmm\: $ n-s.

N

^{'r) = Yl^^'^~'^i) (9)

3 = 1

ZZX, 2\-m.^(ny{y^y^7., N itn^^Xh^. Kernel (i

N

CO)

3 = \

m.^3 AVj \m.^mm^^m\^^x'^k(ox. o ic

N

fp{r)p^^3i)(rj)W{r-rj,h)AVj (12)

j = i

B. Symmetric smoothed particle hydrodynamics

SSPH li T a y l o r - 5 riT'SPH Kernel

Sfa^l^jE-rS. Kernel ififajt (6) <7);&i21 V hdiil-

L-C^ic ^(r') « Taylor

1. ^xmmti-^:: <t (CJ; o T^ (13) . ii:(13)cOjiSr:^mii:^)^?<::<tlcJ;oT, ^T-icoMri {c4o(t tpi = ^(xi,yi,2i) t :: t ii^X^ -5.

^To^a^{cov^-c^(l3)^l^?< ^i-e, #^T-(DM{cjQ{t5

m^(Dig. t ^(DW^^imXi^hti^.

KQ = T

tT^lj K h/\^T

(13)

Kij = ^Pi{x3,yj,Zj)Pj{xj,yj,Z3)W{ri -rj,h)AVj (14)

i = i

T/ = Pi{xj,yj,Zj)i){rj)W{ri - rj, h)AVj

j = \

(15) ZZX, NnW.'f-^, i,j {tn^(D4y^-y^:^

Xh^. Sfc, h3i^ P t Q

P{x, y, z) = [l,(a: - Xi), (y - y<), (^ - Zi),

{x - Xif, {y - yif, {z- Zi)^,

(x - Xi){y - yi), {y - yi){z - Zi),

(z - Zi){x - Xi), {x - Xi)^, (y - yi)^,

^ r, drpi dipi dipi

w '

1 d^tpi 1 d^tpi 1 dîpi dîpi d'îpi d'^tpi

2 5x2 ' 2

(16)

(17) ' 2 5^2 ' dxdy' dydz' dzdx'

K M daE:^ffn, T,Q,P\±Mk^-<i^ VJ^Xh^, I,J = ,MXh^. M ( i T a y l o r Z Z X W

9 Taylor Tm m

«icov^-Climl5g^-C•<7)^i'^5)•;i^5^^"ST|gr'fe6J

3 «KicV(r) = i3{x,y,z)

^ m T a y l o r i , M= g(m+l)(m+2)(m+3)

T'fea. 2 Taylor ®li M = 10

(18)

Laplacian (Ogp^(c^(13) (0)^?^(-liAi-'5<Jr. ^(18)(0

Poisson l3U^\-m<Dk 9

-Lu = / (20)

v= [<f>{ri), (i){r2), ••• , 0(ryv)]^ (21) /= [f{ri), f{r2), ••• , /(riv)]^ (22)

Laplacian I> (Df^jr^{-i}k<D X 9 Ibtl-S.

Lij = [0, 0, 0, 0, 2, 2, 2, 0, ••• 0] K'^T' (23) fr^i Ar(Ofij£5)-liit(14)-r'-^;te)ti, h/i^T'cDf^'M

T'l = Pi(xj,yj,Zj)W{n - rj, h)AVj (24)

D. SSPH 1/ Schrodinger 1jn^(Dmmt llk(P> Schrodinger 9.

H3j}{r) = ei}}{r) (25)

// = -iv^ + V;ff(r) (26)

coR®ri lC4olt5Ti:(25)

5x2 Q^p. + V4ff(ri)V'(ri) = e^(ri) (27) Laplacian t i){ri) (OgB5)"lC^ (13) (DM^iXAX^b,

Schrodinger l3U^mk(r>X 9 fi-m\M^{mm(DW^[mmtX

Hu = eSu (28)

u= ['ip{ri),ip{r2),--- ,^(r/v)]

| T

Sij = [1, O,--- 0] K-^T'

(29)

(30)

C. 55P// Poisson

k(Z> Poisson 0.

V'<?i(r) = /(r)

i (OR® n lC4o{t^^(l8)

a^+8i;3-+a;r = /(••')

nm K (14) x^:khtl, h T ' (24)

X^^hti?). h=^ry^m-rrrmH <Dfi^'j^itk(DXo{:i

Hij = "2^0 + leff(7'«)'S'tj (31)

CCT'. Lij (23), Sij (30) x^khix6.

E. mmt(p>fzdy(Dm^m

SSPH Schrodinger Poisson

6^iiiMtcov>T5feig-r^.

^fT9Hin-. rn^mtLxsirtD^m^'^-z^mxh^.

1) RT-(0^feR!c

2) ^n^(D^mimm h (dwm

kdA (fc-dimensional tree) ici

Kernel

3)

kd ni- i

l x v ^

•rs. ^-C(oRT^{cov^ri£(§R^(oy 7;

4)

^T'li Kernel mWiO:>m> 0 t f£^(r>X, i £ ( f 9 7. h(C (Cov^Tco;^ Kernel

LX't<Dmi^^d7^.

5) ff?ii(Ofig^(Og+W

^(23), (30), (31) {^J;oTfT?iJ(Ofig^^S:§fi[-rS.

^Ti± Kernel mW:<Dmr> 0 t tl^(DX, 9 X h(c (19)

(4)

IV. POISSON:^^^^ SCHRODINGER

SSPH 3 Poisson Schrodinger

fffi-re. ttz,

SSPH .tit|5?-r6.

0 <x,y,z <7 (DiL:fji^mi^t L,

SSPHd^a

«, «^AitT(7/A + iy^-C'^5.

SSPH Taylor MMli 2 <!J;(7)JSS-C'%ltLfc.

SSPH Poisson :^m.^^mmit-t^t^mwnm^wM

ttz, Kemel|liclt^ii=-(7)ifi 0 tt£^(Dx% %mr

LT^lbtLT 1/^-5 BiCGSTAB [28] \Z.X-^X%m

Lfc. Poisson

LT^btl"CV'*6 Conjugate Gradient (CG) jfelCtoTfi?

mLfz.

SSPH Schrodinger

% m i n t t t ^ ^ t z f t .

LX^htlX^^^ conjugate Residual &

(Orthomin(l)?i<t'bWtl6) [29], [30] I-1 oT^tlf Lfc. WPS

Schrodinger Lfzm^\±nwmm

Itfihfzih, CG}fe{cJ:o-C^¥^L/c.

A. Poisson

^(D Poisson

V"0(r) = /(r) It/(r) = /(r) = (4r^—6)exp(—r^) t't^t,

0

<f>{r) = 0(r) = exp(—r^)

(32)

^0(r) = (33)

(x,y,2) = (3.5,3.5,3.5) t Ltz,

^ (32) (7) Poisson

Fig.4 |:i/T^-r. maxi \<l>{ri) - ^{ri)\ Ltz.

zzx, <f>{tmmm, ^itmm^xh^. Fig.4

(m^, ^iiAD$-tirTPd]|iSA^/h$<i-SI5t'^

(t^<-|ScL-C4y'9, SSPH Poisson

I^tz, SSPH

B. Schrodinger

<7) Schrodinger Hartree mXt^(DXol:it£^.

-ts7^ + V{r)

ip{r) = Eip{r) (34)

V(r) = \{u}W + +(^32^)

⁽³⁵⁾

0 t«^2 +^na +^ cua (36)

bti^. ZZX, n = ni+712+na "Cfe^. n = 0

S(±^'C^ilL'C:]o <9, n = ni +712 + na tti:^ ^ o t£ ni,n2,n3 (nf^^^^^(Dmz\mm'- h ^.

a;i = 0^2 = Wa = 1 <t LTl^fiSft^tff ^ff ofc.

5 o(7)^î{cov^TIaWît<tll^lll^^^}?:^6it. 7i = 0(O 71 = 1 CO^-^ (îfi 3) , n = 2 O^-a 7)*5}^46 b>tbfc

Flg.5 l:i^i^-r.

Error = \E —-fi/cxactl

IEcxactI ⁽³⁷⁾

t-^mLfz. zzx, Eimm^, Eoxactinmnwm^mxh^.

Fig.5 {tihi^Rxj^ii^^(Dr^m A

-CFpI]!^ a < -r^lSi'^H/ivh^ < /cCoTi/^-S

SSPH ssph

SSPH Laplacian (7)JlfdltT'3^.t< tp{r) g

Kernel m^^m^^X'j&m-^fz^, 0

<t£^fzt^^hi^^. tfz, SSPH

tz CR ?£{i CG ifect >9 P CM:kfxW.M'^xn

t+lSS7)'^+5^(cilX^L/cj:7)^ofc - <t

10-

10--

s - x

'•e- I

> w 'C

•6-

X ra...

E 10-

10"

! SSPH-e-

1 1

i i

FD i

i 1

i i

1 i

l""""""^1

;

i 1 1 1

1

u ra X

UJ

1

10-

tu

10-

EqSSPH-^

1\ i

EiSSPH-"- _j

E2 SSPH -f- j j

EoFD-«- _i

Ej FD -B— — E2FD-5-

1 i I

L / ^

y\f^\

\

₁

t'"""

1 1

i i

«

0.05 0.1 0.15 0.2 0.25 0.3

Particle and Grid spacing A

0.35

Fig.4. Poisson SSPH

?rii)JP$-itrF.gRS A j^CoT 1/^2) 54^5)-

0.1 0.15 0.2 0.25 0.3

Particle and Grid spacing A

0.35

Fig.5. Schrodinger SSPH i:

A < -rsiiir'ia^;4vh$ < /j:o-ci/^-5 c i ;4'>^

/4^-5.

(5)

V.

tK^.

V^J^,i=^(Cov^T^i^t>C^$:tf-^-f 5. SSPH Poisson loU Kohn-sham iju^^mmtLx 3 d:yz(Dm&'\Xm^'i^m^

-8 < x,y,2 < 8 L-C/^?^lr^tfofc.

ssph (c^o{t^5Ki^c»f4®?5^i3ic(c

.^M) II, Pp1Pg^AiL-C(16/A + lf T'fe6. ^

lUli A = 0.2 <t Ltztzib, 531441 Xh^. SSPH (c 46It6 Taylor ®|llli 2 Lfc.

SSPH V^T^tScfb Lfzm^lX Poisson BICGSTAB

&. Kohn-Sham CRifelcJ;o-Cfi?tffLfc.

L/cii'a II, Poisson Kohn-Sham CG &lcj;oTft?tlfLfc.

ffl^/^fc:^-'a•|-oV^TI^ TABLE 11 {C;^^-. :Scl^fiI [31] ^ TABLE

III ic/T^-r. xm, [31] ssph

^^l±, tK^ : 1.5 X 10-2, ^y A : 4.7x10-2, yf-i^/A (Is ig/lil) : 8.3 X 10-2, (2s llljl) : 1.1 x 10-^-Cfe6. ^

tK^ : 1.2 X 10-2, ^ijryj^, 4.2x10-2, y^-^A (IsUlil) :8.0x 10-2, yf-[>A(2s|ll

it) : 8.5 X 10-2 X'h^. SSPH t A/if-

i:L-rv^5);()\ SSPH

^y h y — • Flg.6 (v:;^^-. Jg

i=-]^7J^P>cDl®gi r lc^-r6/^- h y - • ^J^>T-

r ~ 2 J^±Tii 2/r (7)ft^<t-icLT46<9, ji;^-C"b^Pg^£fiSS:

tK^, y^f>i»mT-toi:4^sg®^Fig.7ic.T^i-. Fig.?

2 = 0 1:146Its xy

av^I^.^^|c/^Slc^^eoT4''i:>^f|55)•<A)®^^^

< /£ 0, oT^^S ::

< ?tto-cv^S C t VI.

ofc;i\ 4^l9F5£T't^Lt:^?iimi^ (If^.^J (7)5^4]T t ^Hf^^llc^^^L?J^^t4^4^-CV^Sfc4(). P LT y^zf y XASrfflV^-r-]i-C/^VV^/•ct^•®

TABLE I

SSPH ^ffll/^rgf33ILfc:#UljI<^!aM. (^^FbIP^ a = 0.2)

Atom Is orbital 2s orbital

H -0.2300 -

He -0.5434 -

Li -1.7218 -0.0942

TABLE II

i^m^mi^xiYn L/i#«coiiwfiS. (t&T-Rnisa a = 0.2)

H -0.2308 ^-

He -0.5465 ^-

Li -1.7282 -0.0965

TABLE III

ATOMIC Reference Data for Electronic Structure Calculations [31]

H -0.2335 -

He -0.5704 -

Li -1.8786 -0.1055

T'fcSi:#;t?>nS. ^Pflco-gp

^l^^4^^#<«i()'C^PHl^^?|g^±lfS::i:;6-^T'tScoT', iiS i#x<b4xS.

46V^T®t^^a^/^- F y - • Tfs^yiy^fV(Dm\Lm^i^^^^(D / n - h y - •

V^t:«6^FHl^l^?^l4:S:4xai£i«S?^=cv ^<b %;t fbtiS.

VII. Xtib

LTJKE^&O—DXh^ SSPH

Poisson 3 d^7i.M^MW}^(D Schrodinger tlrlcJ;oTfP^M^«Lfc±T-.

ffl t LT, ^iaaPic&lcS<5< SSPH ^ iifflLtc. rtiicity,

^ Ltz.

om^,

t-c, ^FnllcmT-^i^^^-tirS:: tT-

0. < ®i^tiifiitm-ei4:-®T-(75

S Schrodinger Poisson LT^<

SSPH ^m\^^X^ixh<r)1]U^^mW\tX-h-)]

jfelcov^Tt^Ltc.

nmm \^x^x

SSPH ofc. Poisson

< —^Ltz^^^^^htltz. Schrodinger

^|5Sa5>t£{c46ltSt&4^,^<!:l^i:M(c^i^ ^ga®L

t z t ^ , SSPH >9

tK^, -^y-^A, yf-r^Ai^^cA)®

L, #i|yiiI(79liWfil^^Ci®(fit [31]

^iJrblSLfc:. WI5gll5>ti(-46ltSf&T-,^i:[^L(3:ffilc^-7- (If

^SaSL/ci#, SSPH tW[5gll5>?^(79|^l:l4:l5<k^^- S!ct/c:^^ SSPH (7):^7J^^^755X^v^^^<^/j;ofc.

SSPH ^fflV^tcim-C'li, Taylor®ia^i^i:fc(0^t-C'#lS-rS

::<^T'l|3^att^5|S]Jli-S^<^;65^^$4^Tv^S [32]. Ltzr^^X,

M'^<D Taylor il|«^#{ti-ix«, P LfiSlCim^-^.^ga® Ltc^

^xh<^xi,mm'7^i^x <9 < -rs i: ct s.

^tz, ff#coi|xai4^WPSII^?iJ: "9 [^±^-li:S:: tj-C# S.

3.5

2.5

> 1.5

0.5

•

He atom(SSPl

2 1) • /r

•

\ \

\

^J^L

6 8 10

r (a.u.)

12 14

Fig. 6. ^V <>2»ir,T-£0/N- hV~ • /u. "bcojgilr

(6)

-15 -I -OS ^ OS 1 15

-0.5 jp 0.5 1 1,5 2

-1 -OS I OS 1

Fig. 7. ( b ) - - y ( c ) V

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