Fast and stable method for simulating quantum electron dynamics

Fast and stable m ethod for sim ulating quantum electron dynam ics N aokiW atanabe,M asaru T sukada

arXiv:physics/0011068v1 [physics.comp-ph] 29 Nov 2000

D epartm ent ofPhysics,G raduate SchoolofScience,U niversity ofTokyo 7-3-1 H ongo,113-0033 B unkyo-ku, Tokyo,Japan (Published from PhysicalR eview E.62,2914,(2000).) A fast and stable m ethod is form ulated to com pute the tim e evolution of a w avefunction by num erically solving the tim e-dependentSchrodingerequation. T hism ethod isa realspace/realtim e evolution m ethod im plem ented by several com putational techniques such as Suzuki’s exponential product, C ayley’s form , the nite di erential m ethod and an operator nam ed adhesive operator. T his m ethod conserves the norm of the w avefunction, m anages periodic conditions and adaptive m esh re nem ent technique,and is suitable for vector-and parallel-type supercom puters. A pplying this m ethod to som e sim ple electron dynam ics, w e con rm ed the e ciency and accuracy of the m ethod for sim ulating fast tim e-dependent quantum phenom ena. 02.70.-c,03.67.Lx,73.23,42.65.-k

so thatthe norm ofthe wavefunction isconserved during the tim e evolution. Stability and accuracy are im proved by C ayley’s form so we can use a longer tim e slice than those ofthe other m ethods. C ayley’s form is a kind of im plicit m ethods,this is the key to the stability,but im plicit m ethods are not suitable for periodic conditions and parallelization. W e have avoided these problem s by introducing an operator nam ed adhesive operator. T his adhesive operatoris also usefulforadaptive m esh re nem ent technique. O urm ethod inheritsm any advantagesfrom m any ordinary m ethods,and yet m ore im proved in m any aspects. W ith these advantages, this m ethod w ill be useful for sim ulating large-scale and long-term quantum electron dynam ics from rst principles. In section II, we form ulate the new m ethod step by step. In section III,we apply it to som e sim ulations of electron dynam icsand dem onstrate its e ciency. In section IV ,we draw som e conclusions.

I. IN T R O D U C T IO N

T here are m any com putationalm ethod ofsolving the T D -Schrodinger equation num erically. C onventionally, a wavefunction has been represented as a linear com bination of plane waves or atom ic orbitals. H owever, these representations entailhigh com putationalcost to calculate the m atrix elem ents for these bases.T he plane wave bases set is not suitable for localized orbitals,and the atom ic orbitalbases set is not suitable for spreading waves. M oreover, they are not suitable for parallelization, since the calculation of m atrix elem ents requires m assive data transm ission am ong processors. To overcom e those problem s,som e num ericalm ethods adopted realspace representation [1{4]. In those m ethods,a wavefunction is descritized by grid points in real space,and w ith them som e dynam ic electron phenom ena were sim ulated successfully [6{8]. A m ong these real space m ethods, a m ethod called C ayley’s form or C rank-N icholson schem e is know n to be especially useful for one-dim ensional closed system s because this m ethod conserves the norm of the wavefunction exactly and the sim ulation is rather stable and accurate even in a long tim e slice. T hese characteristics are very attractive for sim ulations over a long tim e span.U nfortunately,thism ethod isnotsuitable fortwoor three-dim ensionalsystem s. T his problem is fatalfor physically m eaningfulsystem s. T hough there are m any other com putationalm ethods that can m anage two- or three-dim ensionalsystem s,these m ethods also have disadvantages. In the present work,we have overcom e the problem s associated w ith C ayley’s form and have form ulated a new com putationalm ethod w hich ism ore e cient,m ore adaptable and m ore attractive than any other ordinary m ethods. In ourm ethod,allcom putationsare perform ed in real space so there isno need ofusing Fouriertransform .T he tim e evolution operatorin ourm ethod isexactly unitary by using C ayley’sform and Suzuki’sexponentialproduct

II. F O R M U L A T IO N

In this section, we form ulate the new m ethod step by step from the sim plest case to com plicated cases. T hroughout this paper, we use the atom ic units h = 1;m = 1;e = 1.

A . O ne-dim ensional closed free system

Forthe rststep,we considera one-dim ensionalclosed system w here an electron m oves freely but never leaks out ofthe system . T he T D -Schrodinger equation ofthis system is sim ply given as @ (x;t) i = @t 1

@x2 (x;t) : 2

(1)

T he solution ofEq.(1) is analytically given by an exponentialoperator as h @2 i (x;t) ; (x;t+ t)= exp i t x 2

decom posed into the LU form as 32 2 1 0 0 0 u1 1 u1 76 0 1 6 1 u1 0 0 2 74 6 4 0 1 u31 0 5 0 0 0 0 0 0 1 u41 2 1 (t+ t) 6 2(t+ t) 4 (t+ t) 3 4 (t+ t)

(2)

w here t is a sm alltim e slice. B y using Eq.(2) repeatedly,the tim e evolution ofthe wavefunction is obtained. A n approxim ation is utilized to m ake a concrete form ofthe exponentialoperator.W e haveto be carefulnotto destroy the unitarity ofthe tim e evolution operator,otherw ise the wavefunction rapidly diverges. W e adopted C ayley’s form because it is unconditionally stable and accurate enough. C ayley’s form is a fractionalapproxim ation ofthe exponentialoperator given by h @2 i 1 + i t@ 2=4 x exp i t x ’ : 2 1 i t@ x2=4

i

3 0 0 7 u3 5 1

3 b1(t) 7 6 b2(t) 7 5 = 4 b (t) 5 3 b4(t)

(8)

H ere bi and ui are auxiliary vectors de ned as below bi(t) ui

i1

1=(A

(t)+ B i(t)+ ui 1 ); u0

i+ 1 (t);

0

(9) (10)

T he auxiliary vector ui is determ ined in advance,and it is treated as a constant vector in Eq.(10). 26N oating operations are heeded to solve Eq. (10); here N is the num berofthe grid pointsin the system ,abouttw ice thatoftheEulerm ethod.U nliketheEulerm ethod,itexactly conservesthe norm because the m atricesin Eq.(6) are unitary. M oreover,the expected energy is conserved because the tim e evolution operator com m utes w ith the H am iltonian in this case.

(3)

It is second-order accurate in tim e. B y substituting Eq.(3)forEq.(2)and m oving the denom inatoronto the left-hand side,the follow ing basic equation is obtained: h 1

0 u2 1 0 3 2

h t @x2 i t @x2 i (x;t+ t)= 1 + i (x;t) : (4) 2 2 2 2

T his is identical w ith the well-know n C rank-N icholson schem e. T he wavefunction is descritized by grid points in realspace as

B . T hree-dim ensional closed free system

It is easy to extend this technique to a threedi m ensional system . T he form al solution of the T D (xi;t) ; xi = i x; i= 0; ;N 1 (5) i(t)= Schrodinger equation in a three-dim ensional system is given by an exponentialofthe sum ofthree second difw here x isthe span ofthe grid points.W e approxim ate ferentialoperatorsas the spatial di erential operator by the nite di erence h m ethod (FD M ). T hen Eq. (4) becom es a sim ultaneous @2 @y2 @z2 i (r;t+ t)= exp i t x + + (r;t) : (11) linear equation for the vector quantity i(t+ t). For 2 2 2 exam ple,in a system w ith six grid points,Eq.(4) is apT hese di erentialoperators in Eq.(11) are com m utable proxim ated in the follow ing way: am ong each other,so the exponentialoperatorisexactly 3 32 2 decom posed into a product of three exponentialoperaA 1 0 0 1 (t+ t) tors: 1 0 7 6 2 (t+ t) 7 6 1 A 5 4 5 4 0 1 A 1 h @2 i h @2 i 3 (t+ t) y 0 0 1 A (r;t+ t)= exp i t x exp i t 4 (t+ t) 2 2 3 32 2 h @2 i B 1 0 0 1 (t) exp i t z (r;t) : (12) 6 1 B 1 0 7 6 2 (t) 7 (6) = 4 2 0 1 B 1 5 4 3 (t) 5 Each exponentialoperatorisapproxim ated by C ayley’s 0 0 1 B 4 (t) form as In the above, 1 + i t@ x2=4 1 + i t@ y2=4 ( r;t+ t )= 1 i t@ x2=4 1 i t@ y2=4 x2 x2 + 2;B 4i 2 (7) A 4i t t 1 + i t@ z2=4 (r;t) : (13) 1 i t@ z2=4 and 0 and 5 are xed at zero due to the boundary condition. 78N oating operations are required to com pute Itiseasy to solve thissim ultaneouslinearequation beEq.(13);w here N is the totalnum ber ofgrid points in cause the m atrix appearing on the left-hand side iseasily the system . T he norm and energy are conserved exactly. 2

T he decom position (16)isa second-orderone.H igherorder decom positions are derived using Suzuki’s fractal decom position [9{11,13]. For instance, a fourth-order fractaldecom position S 4( t) is given by

B y the way, a conventional m ethod, Peacem anR achfold m ethod [1,8],utilizessim ilarapproxim ation appearing on Eq.(13), w hich is a kind of the alternating direction im plicit m ethod (A D I m ethod). H owever,by using exponentialproduct,we have found that there is no need ofA D I.T his fact m akes the program m ing code sim pler and it runs faster.

S 4( t)= S 2(s t) S 2(s t) S 2((1 4s) t) S 2(s t) S 2(s t) (17) w here h h 4 i t i t i exp i V V exp i t 2 2 2 p3 4) : (18) s 1=(4

h

C . Static potential

S 2( t)

exp

i

N ext we considera system subjected to a static externalscalar eld V (r). T he T D -Schrodinger equation and its form alsolution in this system are as follow s: i @ (r;t) h 4 i = + V (r) (r;t) : @t 2 i h 4 i tV (r) (r;t) : (r;t+ t)= exp i t 2

D . D ynam ic potential

(14) To discuss high-speed electron dynam ics caused by a tim e-dependent external eld V (r;t),we should take account ofthe evolution ofthe potentialitselfin the T D Schrodinger equation given as

(15)

To cooperate w ith the potentialin the fram ework ofthe form ula described in the previoussubsections,wehaveto separate the potentialoperatorfrom the kinetic operator using Suzuki’s exponentialproduct theory [9,10]as h (r;t+ t)= exp

@ (r;t) = H (t) (r;t) ; H (t)= i @t

T he analytic solution ofEq.(19) is given by a D yson’s tim e ordering operator P as " Z # t+ t n o 0 4 0 (r;t+ t)= P exp i dt V (r;t ) (r;t) : 2 t (20)

h 4 i t i i V exp i t 2 2 h t i (r;t) : (16) exp i V 2

T his decom position is correct up to the second-order of t. T he exponentialofthe potentialis com puted by just changing the phase ofthe wavefunction ateach grid point. T he exponentialofthe Laplacian is com puted in the way described in the previous subsections. Each operatorisexactly unitary,so thenorm isconserved exactly. B utdueto theseparation oftheincom m utableoperators, the energy isnotconserved exactly.Yetitoscillatesnear around itsinitialvaluesand itneverdriftsm onotonously. T his algorithm is quite suitable for vector-type supercom putersbecausealloperationsareindependentby grid points,by row s,or by colum ns. T he outline ofthis procedure for a two-dim ensionalsystem is schem atically described by Fig.1.

V

Kx

Ky

4 + V (r;t) : (19) 2

T hetheory ofthedecom position ofan exponentialw ith tim e ordering was derived by Suzuki[12]. T he result is rather sim ple. For instance,the second-order decom position is sim ply given by h

h 4 i t t i V (r;t+ ) exp i t 2 2 2 h t i t ) (r;t) (21) exp i V (r;t+ 2 2

(r;t+ t)’ exp

i

and the fourth-order fractaldecom position is given by (r;t+ t)= S 2(s t;t+ (1 s) t) S 2(s t;t+ (1 2s) t) S 2((1 4s) t;t+ 2s t) S 2(s t;t+ s t) S 2(s t;t) (r;t) ;

V

h S 2( t;t)

FIG .1. T he procedure for a tw o-dim ensional closed system w ith a static potential. H ere V show s the operation of the exponentialofthe potential,w hich changes the phase of the w avefunction at each grid point. K x and K y show the operation ofC ayley’s form along the x-axisand the y-axis respectively. T hey are com puted independently by grid points, by row s,or by colum ns.

exp

i

(22)

h 4 i t t i V (r;t+ ) exp i t 2 2 2 h t t i exp i V (r;t+ ) : (23) 2 2

T heseoperatorsarealso unitary.T heseproceduresare quite sim ilar to those ofthe static potentialexcept that we take the dynam ic potentialat the speci ed tim e. 3

E . P eriodic system

In a crystalorperiodicsystem ,thewavefunctionsm ust obey a periodic condition: (r + R ;t)=

(r;t)exp [i ];

k

R ;

(24)

w here k is the B loch wave num ber and R is the unit vector of the lattice. T he m atrix form equation corresponding to Eq. (6) in this system takes the follow ing form : 2 32 3 A 1 0 e+ i 1 (t+ t) 6 1 A 1 0 7 6 7 6 2(t+ t) 7 4 0 1 A 1 5 4 3(t+ t) 5 ei 0 1 A 4 (t+ t) 32 2 3 B 1 0 ei 1 (t) 6 1 B 1 0 7 6 2(t) 7 7 = 6 4 0 1 B 1 5 4 3(t) 5 (25) e+ i 0 1 B 4 (t)

(26)

= I+

1

e 2iC 2

1 ei e 1 +i

Y-adhesive

Ky

Y-adhesive

V

3 2 1 0 0 2 1 0 7 6 1 4 0 1 2 1 5 0 0 1 2 3 3 2 2 0 0 0 0 2 1 0 0 1 0 0 7 6 0 1 1 07 6 1 : (30) + = 4 1 05 0 0 1 1 5 40 1 0 0 0 0 0 0 1 2 T he interior of the rst m atrix on the right-hand side is separated into two blocks, w hich m eans this system is separated into two physically independent areas. T he second m atrix,w hich is the adhesive operator,connects the two areas. A large system is separated into m any sm allareas,and each area is m anaged by a single processor. Since the exponential of a block diagonal m atrix is also a block diagonalm atrix,each block is com puted by a single processor independently. D ata transm ission is needed only to com pute the adhesive operator. T he am ount of data transm ission is quite sm all, nearly negligible.T he outline ofthe procedure fora twodim ensionalclosed system on two processorsis schem atically described by Fig.3.

2 T he exponentialof@xad is exactly calculated by the follow ing form ula:

i

X-adhesive

2

hi t i hi t i hi t i hi t i 2 2 2 exp exp exp : @x2 = exp @xad @xtd @xad 2 4 2 4 (28)

1 ei e 1

Kx

T he adhesive operator plays another im portant role. It m akes C ayley’s form suitable for parallelization. W e use the adhesive operator to represent the second nite di erence m atrix in the follow ing way:

T he rst m atrix on the right-hand side, w hich corre2 spondsto @xtd ,istri-diagonal,and the second one,w hich 2 ,is its rem ainder,and it has a quite corresponds to @xad sim ple form . T he exponentialofthe second di erential operator is decom posed by these term s:

+i

X-adhesive

F . P arallelization

M ultiplying by x 2, the above representation reads in the m atrix form : 2 3 2 1 0 ei 6 1 2 1 0 7 6 7 4 0 1 2 1 5 e+ i 0 1 2 3 3 2 2 1 0 0 ei 1 1 0 0 0 0 0 0 7 2 1 0 7 6 6 1 7 : (27) +6 = 4 5 4 0 0 0 0 5 0 1 2 1 0 0 1 1 e+ i 0 0 1

h exp iC

V

FIG .2. T he procedure fora tw o-dim ensionalperiodic system . H ere K x and K y show the operations ofC ayley’s form , and they operate as ifthis system is notperiodic. X -adhesive and Y -adhesive m ean the operations ofthe exponentialofthe adhesive operators along the x-axis and the y-axis, respectively. T he operation ofthe adhesive operator needs only the values at the edges ofthe system .

T hese m atrices have extra elem ents,so the equation can no longer be solve e ciently. W e propose a trick to avoid this problem . W e represent the second spatialdi erentialoperator @x2 as a sum oftwo operators: 2 2 @x2 = @xtd + @xad :

T his operation is exactly unitary and easy to com pute. 2 T he exponentialof @xtd is com puted in the ordinary 2 way. T hus the norm is conserved. W e nam ed @ad an \adhesive operator" because this operator plays the role ofan adhesion to connectboth edgesofthe system . T he outline of the procedure for a two-dim ensionalperiodic system is schem atically described by Fig.2.

: (29) 4

2 @xad =

V

Adhesive

Ky

Adhesive

Kx

T he exponentialofthe adhesive operatoris calculated using the follow ing form ula: 2

13 1=4 1=8 1=8 it @ 1=2 1=2 0 A 5 = exp 4 4 x2 1=2 0 1=2 0 1 2c1 c1 c1 I+ @ 4c1 2c1 + c2 2c1 c2 A ; (35) 4c1 2c1 c2 2c1 + c2

G . A daptive m esh re nem ent

It is necessary for real space com putation to be equipped w ith an adaptivem esh re nem entto reducethe com putationalcost or to im prove the accuracy in som e im portant regions. W e im proved the adhesive operator to m anage a connection of between two regions w hose m esh sizes are di erent,as illustrated in Fig.4.

2∆ x

1

1 x2

V

FIG .3. T he procedure for a tw o-dim ensionalclosed system on tw o processors. A dhesive show s the operation ofthe exponentialof the adhesive operator for parallelcom puting. T he operation ofthe adhesive operator needs only the values at the edges of the areas, so the data transm ission betw een the processors is quite sm all.

2∆ x

0

w here c1 c2

∆x 3

5

4

6

2

1 2 3 (34) 4 5 6

-1/4 1/8 1/8 1/2 -1/2 1/2 -1/2

∆x

h 3i t i 1 exp p 6 28 x 2 h 1 2i t i exp p 6 28 x 2

1 ; 6 1 : 6

(36) (37)

In this way,it is found that the adhesive operator is im portant to sim ulate a larger or a m ore com plicated system by the present m ethod.

x FIG .4. A n exam ple of adaptive m esh re nem ent. T he elem ent in the left area is tw ice as large as that in the right area. T he adhesive operator connects these areas.

III. A P P L IC A T IO N

In this section,we show som e applications ofour num ericalm ethod. T hough these applications treat sim ple physicalsystem s,they are su cient for verifying the reliability and e ciency ofthe m ethod. T hroughout this section,we use the atom ic units (a.u.).

T he second di erentialoperator@x2 should be H erm ite, butin thiscasethe condition required forthe m atrix representation (@x2 )ij is given by (@x2)ij x 2i = (@x2)ji x 2j ;

for alli;j:

(31)

C onsidering this condition, an approxim ation of the second di erentialoperator is given as

@x2

=

-1/2 1/4 1/4 -1/2 1/8 1/8 1 1/2 -3/2 1 1/2 -3/2 x2 1 -2 1

1 -2

A . C om parison w ith conventional m ethods

1 2 3 (32) 4 5 6

A s far as we know ,the conventionalm ethods ofsolving the T D -Schrodingerequation are classi ed into three categories: 1) the m ultistep m ethod [3],2) the m ethod developed by D e R aedt [2]and 3) the m ethod equipped w ith C ayley’s form [5]. In this section, we m ake brief com parisons between C ayley’s form and other conventionalm ethods by sim ply sim ulating a G aussian wave packetm oving in a onedim ensionalfree system as illustrated in Fig.5.

T he indices attached to this m atrix indicate the corresponding m esh indices described in Fig.4. T his m atrix isalso divided into a block-diagonalone and an adhesive operator as -1/2 1/4 1/4 -1/4 2 @xbd =

1 x2

-1

1 -1

1

1 -2

1

-2

1 2 3 (33) 4 5 6

po 2W

x xo

5

FIG . 5. T he m odel system for com parison w ith conventional m ethods. 256 com putational grid points are allocated in the physicallength 8:0a.u.A G aussian w ave packet is placed in the system ,w hose initialaverage location xo and m om entum po are set at xo = 2:0a.u.and po = 12:0a.u.,respectively.

M eanw hile, the second-order D e R aedt’s m ethod is given by hi t @2 i h @2 i hi t @2 i xa xa (t+ t)= exp exp i t xb exp (t) 2 2 2 2 2 (44) 2 2 w here @xa and @xb are the parts ofthe second di erential operator and are approxim ated by nite di erence m atrices as below : 3 2 1 1 0 0 0 0 6 1 1 0 0 0 0 7 7 1 6 1 1 0 0 7 6 0 0 2 (45) @xa ’ 7 ; 6 1 0 0 7 x26 0 0 1 5 4 0 0 0 0 1 1 0 0 0 0 1 1 3 2 1 0 0 0 0 0 6 0 1 1 0 0 0 7 7 1 6 1 0 0 0 7 6 0 1 2 @xb ’ (46) 7 : 6 1 1 0 7 x26 0 0 0 5 4 0 0 0 1 1 0 0 0 0 0 0 1

T he T D -Schrodingerequation ofthis system is sim ply given by @ (x;t) i = @t

@x2 2

(x;t) :

(38)

T he wavefunction at the initialstate is set as a G aussian: h jx x j2 i 1 o (x;t= 0)= p4 exp + i p x ; (39) o 4W 2 2 W 2 w here W = 0:25a.u.; xo = 2:0a.u.; po = 12:0a.u. T he evolution ofthis G aussian is analytically derived as (x;t)= p4

1

T he exponentials ofthose m atrices are exactly calculated using the follow ing form ula: i h 1 e 2iC 1 1 1 1 : (47) = I+ exp iC 1 1 1 1 2

+ ( =2)(t=W )2 i h (x x pot)2 o + i p x : (40) exp o 4W 2 + (t=W )2

2 W

2

T herefore,the average location ofthe G aussian hx(t)i is derived as ifit is a classicalparticle: hx(t)i= hx(t= 0)i+ pot:

T he tim e evolution ofthism ethod isexactly unitary,and the norm is exactly conserved unconditionally. H owever, it seem s that the accuracy tends to break dow n on the condition that t= x 2 > 1:0. T his m ethod needs 18N oating operations per tim e step, w hich is the fastest m ethod in unconditionally norm -conserving m ethods. C ayley’sform w ith the nitedi erencem ethod isgiven by

(41)

T his characteristic is usefulto check the accuracy ofthe sim ulation. W e use the second-order version of the m ultistep m ethod and the D e R aedt’s m ethod in orderto com pare w ith C ayley’s form since C ayley’s form is second-order accurate in space and tim e. T he second-order m ultistep m ethod we used in this system is given by (t+ t)=

(t

t)+ i2 t

w here @x2 isapproxim ated by a 2 2 1 0 6 1 2 1 1 6 2 6 0 1 2 @x ’ 6 2 0 0 1 x 6 4 0 0 0 0 0 0

@x2 2

(t) ;

(t+ t)=

1 + i t=4 @ x2 1 i t=4 @ x2

(t) ;

(48)

w here the spatial di erential operator is approxim ated by the ordinary way in Eq.(43). T he tim e evolution ofthis m ethod is exactly unitary, and thenorm isexactly conserved unconditionally.M oreover,this m ethod m aintains good accuracy even under the condition that t= x 2 > 1:0. T his m ethod needs 26N oating operationspertim estep,w hich isthefastest m ethod in unconditionally stable m ethods. W ehavesim ulated them otion oftheG aussian by those m ethods. First we show a com parison ofC ayley’s form w ith the conventionalm ethods in the fram ework ofthe FD M .Figure 6 show s the tim e evolution ofthe error in the energy, w hich is evaluated by the nite di erence m ethod as described below

(42)

nite di erence m atrix as 3 0 0 0 0 0 0 7 7 1 0 0 7 (43) 7 : 2 1 0 7 5 1 2 1 0 1 2

Extra m em oriesareneeded forthe wavefunction atthe previoustim e step (t t). T hough the tim e evolution ofthis m ethod is not unitary,the norm ofthe wavefunction is conserved w ith good accuracy on the condition that t= x 2 0:5. T his m ethod needs only 10N oating operationspertim e step,w hich isthe fastestm ethod in conditionally stable m ethods.

(t)= E (t) E (t)=

NX 1 1 Re 2 x i= 0

i (t)

E (t= 0) i1

(t)

2 i(t)+

(49) i+ 1 (t)

:

(50) 6

T he initialenergy is evaluated as 73:03a.u.,though it is theoretically expected to be 74a.u.T he ratio t= x 2 is set at 0:5 to m eet the stable condition required for the m ultistep m ethod.

Error of the mean velocity [a.u.]

0.9

T he energies violently oscillate in the results of the m ultistep m ethod and D e R aedt’s m ethod, as a result of the fact that these tim e evolution operators do not com m ute w ith the H am iltonian. T hese energies seem to convergeafterthe wavepacketisdelocalized in a uniform way overthe system .M eanw hile,the energy isconserved exactly in the result of C ayley’s form because C ayley’s form com m utesw ith the spatialsecond di erentialoperator w hich is the H am iltonian itselfin this system . Figure 7 show sthe relation ofthe tim e slice tto the error in the average m om entum ofthe G aussian,w hich is evaluated by the nite di erence m ethod as described below : hx(t= T )i hx(t= 0)i T

( t= x 2 )=

Multistep deRaedt Cayley

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 0.0625

0.125

0.25 0.5 ∆ t / ∆x 2

1

2

FIG .7. Errorsin theaverage m om entum com puted by the three m ethods in severaltim e slices. T he m ultistep m ethod cannot be perform ed w hen t= x 2 > 0:5. T he error of D e R aedt’s m ethod is too large w hen t= x 2 > 1. T he error ofC ayley’s form is rather sm all. T he spatialslice is set at x = 1=32a.u.

hp(t= 0)i ; (51)

NX 1

hx(t)i= x

xij i(t)j2

In the m ultistep m ethod, the com putation cannot be perform ed due to a oating exception, if the ratio t= x 2 exceeds0:5.In D eR aedt’sm ethod,theerrorbecom estoo large to plotin thisgraph ifthe ratio t= x 2 exceeds 1:0. M eanw hile, in C ayley’s form , the error is not so large even ifthe ratio t= x 2 exceeds 1:0. In this way, C ayley’s form is found rather stable. T herefore,wecan usea longertim eslicethan thoseofthe otherm ethods. A nd thisC ayley’sform becom essuitable forthree-dim ensionalsystem s,potentials,periodicconditions,adaptive m esh re nem ent,and parallelizations by our im provem ents in this paper.

(52)

i= 0

1 Im 2

hp(t)i=

NX 1 i(t)

i+ 1 (t)

i1

(t) ;

(53)

i= 0

w here T is a tim e span setat0:4a.u.T he initialm om entum hp(t= 0)iiscalculated as11:7a.u.,w hich isdi erent from the theoreticalvalue po = 12:0a.u.due to the nite di erence m ethod.

Error of the energy [a.u.]

0.25

Multistep De Raedt Cayley

0.2

B . T est of the adhesive operator 0.15 0.1

To verify the reliability and e ciency ofthe adhesive operator for periodic condition and parallelization, we have sim ulated the m otion ofa G aussian wave packetin a two-dim ensionalfree system . A s illustrated in Fig.8, thissystem hasperiodic conditionsalong both the x-axis and the y-axis,and it is divided into nine areas,each of them is m anaged by a single processing elem ent;the adhesive operator connects them . T he initialwavefunction is set as a G aussian given as

0.05 0 -0.05 0

0.02

0.04 0.06 Time [a.u.]

0.08

0.1

FIG .6. T im e variances in the energies com puted by the three m ethods. T he tim e slice is set at t= 1=2048a.u.and the spatial slice is set at x = 1=32a.u. so that the ratio t= x 2 is equal to 0:5. T he energies violently oscillates in the result of the m ultistep m ethod and D e R aedt’s m ethod. M eanw hile, the energy is conserved exactly in the result of C ayley’s form .

(r;t= 0)= p

1 2 W

2

exp

h jr r j2 o + ip o 4W 2

i r ; (54)

w here ro is set as the center of this system and p o = (1a.u.;1a.u.),W = 1a.u.T he energy ofthis G aussian is theoretically derived as 1:0625a.u. 7

PE9

PE7

PE8

PE9

PE7

PE3

PE1

PE2

PE3

PE1

PE6

PE4

PE5

PE6

PE4

PE9

PE7

PE8

PE9

PE7

PE3

PE1

PE2

PE3

PE1

FIG .10. T im e variance in the energy. T he initial energy is theoretically derived as 1:0625a.u., but it is evaluated as 1:0553a.u.by the FD M .T he energy oscillates near its initial value but never drifts m onotonously.

Second,we allocate 64 64 grid pointsonly in the centralarea asillustrated in Fig.11.W e utilize the adhesive operator for the adaptive m esh re nem ent. Figure 12 show s the snapshots, w ith the G aussian going through these areas sm oothly. Figure 13 show s the evolution of the energy,w hich is observed to oscillate near its initial value.In thisway,the reliability ofthe adhesiveoperator is proved.

FIG .8. T he m odel system for the test of the adhesive operatorfor periodic conditionsand parallelization. T hissystem is periodically connected and is divided into nine areas. Each area is m anaged by a single processing elem ent. 32 32 com putational grid points are allocated in each area w hose physicalsize is set at 8:0a.u. 8:0a.u.T he tim e slice is set at t= 1=16a.u.

Figure 9 show s snapshots ofthe tim e evolution ofthe G aussian, w hich is observed to go through these areas sm oothly. Figure 10 show s the evolution ofthe energy, w hich is observed to oscillate around its initialvalue.

PE9

PE7

PE8

PE9

PE7

PE3

PE1

PE2

PE3

PE1

PE6

PE4

PE5

PE6

PE4

PE9

PE7

PE8

PE9

PE7

PE3

PE1

PE2

PE3

PE1

FIG .11. T he m odelsystem forthe testoftheadhesive operator for the adaptive m esh re nem ent. T his system is also periodically connected and is divided into nine areas. Each area is m anaged by a single processing elem ent. T he size ofeach area is set at 8:0a.u. 8:0a.u. 32 32 com putational grid pointsare allocated in each areasexceptthe centralarea. T he centralarea has64 64 com putationalgrid points,w hich m akes it tw ice as ne as those of the other areas. T he tim e slice is set at t= 1=16a.u. FIG .9. Evolution of the density. T he G aussian is observed to go through these areas sm oothly.

1.0560

Energy [a.u.]

1.0555 1.0550 1.0545 1.0540 1.0535

FIG .12. Evolution of the density. T he G aussian is observed to go through these areas sm oothly.

1.0530 0

5

10

15 20 Time [a.u.]

25

30

8

and the electric force. W e follow the evolution for 32k iteration. Figure14 show sthetim evariancein thepolarization of theelectron.T heoscillation ofthe polarization generates anotherelectric eld,w hich correspondsto a non-linearly scattered light from the atom . B y Fourier-transform ing the polarization along the tim e axis, we obtained the spectrum ofthe scattered light show n in Fig.15.

1.063 1.062

Energy [a.u.]

1.061 1.060 1.059 1.058 1.057 1.056

0.2

1.055 1.054 5

10

15 20 Time [a.u.]

25

30

Polarization [angstrom]

0

FIG .13. T im e variance in the energy. T he initial energy is theoretically derived as 1:0625a.u., but it is evaluated as 1:0591a.u.by the FD M .T he energy oscillates near its initial value but it never drifts m onotonously.

C . E xcitation of a hydrogen

0

5

10

15

20

25

30

FIG .14. T im e variance in the polarization ofthe electron.

10

(55)

Intensity [arbitrary units]

eE zz :

-0.1

Time [fs]

T he spatialvariation ofthe electric eld ofthe lightis neglected, because the electron system is m uch sm aller than the order ofthe wave length. T hen the interaction term ofthe H am iltonian is approxim ated as H int =

0

-0.2

A s the last application of the present m ethod, we dem onstrate its validity and e ciency in describing the process of photon-induced electron excitation in a hydrogen atom in a strong laser eld. T he laser is treated as a classically oscillating electric force polarized in the z-direction: E z = E o sin !t:

0.1

(56)

In other words, we only take into account the electrodipole interaction ofthe electron w ith the light,and neglect the electro-quadrapole, the m agnetic-dipole, and other higher interactions. T he am plitude E o issetat1=64a.u.= 0:80V /A ,w hich isasstrong asa usualpulse laser.T he angularfrequency ! issetat0:3125a.u.= 8:5eV ,lessthan thetransition energy between 1S and 2P.O rdinarily,such low energetic electric force has no e ect on the electronic excitation. B ut w ith such a strong am plitude,various nonlinear opticale ects are caused by the electron dynam ics. W e allocate 1283 grid pointsin a 323a.u.3 cubic closed system . T he hydrogen nucleusislocated atthe centerof the system ,and the nucleus potentialis constructed by solving the Poisson equation in the discretized space to avoid the singularity of the nucleus potential. T he 1Sorbitalisassum ed astheinitialstateofthewavefunction. T hen we turn on the electric eld and start the sim ulation. T he tim e slice is set at 0:0785a.u.= 2:0 10 3 fs so as to follow the rapid variation of the wavefunction

8

6

4

2

0

0

5

10

15

20

25

30

Photon energy [eV]

FIG .15. Spectrum ofthe scattered lightgenerated by the oscillation ofthe electron.

Severalsharp peaksarefound,w hich areinterpreted as follow s:T he peak at8:5eV com esfrom R ayleigh scattering,w hose frequency is identicalw ith the injected light: !. T he peak at 10:2eV com es from Lym an em ission, w hich isgenerated by theelectron transition from the2Porbitalto the 1S-orbital: !L . O n the other hand,the peak at 12:1eV com es from Lym an em ission,w hich is generated by the electron transition from the 3P-orbital to the1S-orbital:!L .T hepeak at6:8eV com esfrom hyper R am an scattering,w hose frequency is identicalw ith 2! !L . M oreoverthe peak at 25:5eV com es from the third harm onic generation,w hose frequency is identical w ith 3!. 9

T he sim ulation is also perform ed for a di erent laser frequency;the injecting photon energy ! issetat10:2eV , w hich isthesam easthetransition energy between 1S and 2P.In this case the electron starting from a 1S orbitalis expected to excite to a 2Pz orbital. Figure 16 show s the snapshotsofthedensity during the sim ulation tim e span.

O ne could obtain such behavior analytically by using perturbation theory;however,w ith the present m ethod, we could directly calculate them w ithout perturbation theory and w ithout inform ation on the excited states of the system .

IV . C O N C L U SIO N

W e have form ulated a new m ethod for solving the tim e-dependentSchrodingerequation num erically in real space. W e have found that by using C ayley’s form and Suzuki’s fractal decom position, the sim ulation can be fast, stable, accurate, and suitable for vector-type supercom puters. W e have proposed the adhesive operator to m ake C ayley’s form suitable for periodic system s and parallelization and adaptive m esh re nem ent. T hese techniques w ill also be useful for the tim edependent K ohn Sham equation, w hich is our future work.

FIG .16. Evolution ofthedensity oftheelectron in the hydrogen atom . T he density starting from a 1S orbitaloscillates w ith tim e and becom es a 2Pz orbital.

Figure 17 and Fig.18 show the polarization and the spectrum ,respectively.T hree peaksare found,at9:9eV , 10:2eV ,and 10:5eV . T hese peaks are derived from the theory of the D ressed atom or the A C stark e ect as below :

V .A C K N O W LE D G M E N T S

eE o h2P zjzj1Si; !; ! + eE o h2P zjzj1Si :

!

(57) W e are indebted to Takahiro K uga for his suggestions concerning non-linearoptics. C alculationswere done using the SR 8000 supercom puter system at the C om puter C entre,U niversity ofTokyo.

0.4

Polarization [angstrom]

0.3 0.2 0.1 0 -0.1

[1] R . Varga, M atrix Iterative A nalysis (Prentice-H all, Englew ood C li s,N J,1962),p.273. [2] H .D e R aedt and K .M ichielsen, C om puters in Physics, 8,600 (1994). [3] T .Iitaka,Phys.R ev.E 49,4684 (1994). [4] H .N atoriand T M unehisa,J.Phys.Soc.Japan 66,351 (1997). [5] N um erical R ecipes in C , chapter 19, section 2, W . H . Press,S.A .Teukolsky,W .T .Vetterling and B .P.Flannery,(C am bridge U niversity Press,1996). [6] H .D e R aedt and K .M ichielsen, Phys.R ev.B .50, 631 (1994) [7] T .Iitaka,S.N om ura,H .H irayam a,X .Zhao,Y .A oyagi and T .Sugano,Phys.R ev.E 56,1222 (1997). [8] H .K ono,A .K ita,Y .O htsukiand Y .Fujim ura,J.C om put.Phys.(U SA ),130,148 (1997). [9] M .Suzuki,Phys.Lett.A 146,319 (1990). [10] M .Suzuki,J.M ath.Phys.32,400 (1991). [11] K .U m eno and M .Suzuki,Phys.Lett.A 181,387 (1993). [12] M .Suzuki,Proc.Japan A cad.69 Ser.B ,161 (1993). [13] M .Suzukiand K .U m eno,Vol.76 ofSpringerProceedings in Physics,(C om puterSim ulation Studiesin C ondensedM atter Physics V I,editied by D .P.Landau,K .K .M on, H .B .Schuttler,Springer,B erlin,1993),p.74. [14] M .Suzuki,Phys.Lett.A 201,425 (1995).

-0.2 -0.3 -0.4

0

5

10

15

20

25

30

Time [fs]

FIG .17. T im e variance in the polarization ofthe electron.

Intensity [arbitrary units]

20

15

10

5

0

0

5

10

15

20

25

30

Photon energy [eV]

FIG .18. Spectrum ofthe scattered light generated by the oscillation ofthe electron.

10