Higher order exponential split operator method for ...

Higher order exponential split operator method for solving time-dependent Schrodinger equations ANDRED.

BANDRAUK'

DPpartement de chimie, FacultP des sciences, Universitk de Sherbrooke, Sherbrooke (QuCbec), Canada J l K 2RI AND

HAISHEN Dkpartement de mathPmatique-infortnatique,FacultP des sciences, UniversitP de Sherbrooke, Sherbrooke (QuPbec), Canada J l K 2Rl

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Received July 2, 1991 This paper is dedicated to Professor Sigeru Huzinaga on the occasion of his 65th birthday ANDRED. BANDRAUK and HAISHEN.Can J. Chem. 70, 555 (1992). A new method of splitting exponential operators is proposed for the exponential form of the operator solution to the time-dependent Schrodinger equation. The method is shown to hold for any desired accuracy in the time increment. A comparison of different algorithms is made as a function of accuracy and computation time. Key words: splitting operator, Fast Fourier Transform (FFT), Schrodinger equations. ANDRED. BANDRAUK et HA^ SHEN.Can. J. Chem. 70, 555 (1992). On propose une nouvelle mCthode pour diviser les operateurs exponentiels de la forme exponentielle de la solution basCe sur I'opCrateur de I'Cquation de Schrodinger. On demontre que la mCthode possbde la prkcision dCsirCe lors de l'augmentation du temps. On prCsente une comparaison des divers algorithmes en fonction de leur prkcision et de leur temps de calcul. Mots clks : mkthode pour diviser les opkrateurs, FFT, equations de Schrodinger. [Traduit par la rCdaction]

I

I

I. Introduction Recent developments in laser physics and laser chemistry have pointed out the need for efficient algorithms to solve the time-dependent Schrijdinger equation (1-4). In general, in describing molecular multiphoton transition, we have recourse to a linear parabolic partial differential equation, which can be written as 111

acpl,(R,t) ih-=-at

h2 a2cpn(R,t) + V,,,,(R)cp,,(R,t) 2m a~~

extend here our previous work in refs. 5 and 6. The new can give any desired order of accuracy. A the accuracy verwill be made of the sus time (CPU),

2.

Symmetric formulas for splitting exponential operators exp[A(A B)] The most general method of splitting exponential operators is based on the Trotter product form for the operator exp[A(A + B)] (10-15), [3]

+

=

eh(A+B)

lim(eu/n: eXB/~n)" ))l--tc=

In the above, V,,,(R) is the time-independent field free molecular (electronic) potential and Vnn.(R,t)is the time-dependent electromagnetic-molecule interaction,

From this follows the most currently used standard second order accurate formula (7-9), [41

=

eh(A+B)

s2( ) + c3k3+ 0(k4)

where R is a molecular coordinate whereas r is the electromagnetic pulse coordinate, and pnn.(R)is a transition moment representing transitions between different electronic states of quantum number n. In refs. 5 and 6, we developed a fourth order accurate formula for splitting the exponential operator exp[A(A + B)] and used it to solve the time-dependent Schrodinger equation. The comparison of this new method and the standard second order accurate method (7-9) showed the new method to be more accurate and more efficient. In the present note we wish to present an improved scheme for solving numerically the linear parabolic equation [I] to high order by the splitting exponential operator method. We ' ~ u t h o to r whom correspondence may be addressed.

[5]

&(A)

=

eh4/2 e MI eh 4 / 2

and C, is some remnant combination of commutators of A and B (5, 6). We showed previously that eq. [4] can be expressed as [6]

eh(A+B)

=

eA.7(A+B)

eh ( l

- 2s)(A+B)

e hs(A+B)

so that higher order accuracy is obtained, i.e., by substituting eq. [5] in eq. [6], [7]

eA'A+B' = S2(sA)S2((1- 2s)A)S2(sA)

+ c 3 x 3 ( h 3+ (1 - 2 ~ ) +) 0(x4) To get third order accuracy, the leading error term 0(A3) must vanish, i.e.,

556

CAN. J . CHEM. VOL. 70. 1992

Thus, if s is a root of eq. [8], we can obtain [9]

eX'A+B' = &(A) + C4h4+ 0(h5)

then the formal solution of eq. 1161 is

where we have now defined [ 101

S3(h) = S,(sh)S2((1

[I81

-

2s)h)Sz(~h)

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Equation [ l o ] gives therefore a third order accurate formula for the exponential operator [3]. By iterating [7] further, we can obtain the general iteration:

11 11 eX'"'"' = S,,-,(sX) + C,,hn + o(A"+" from which follows, by one more iteration, that 1121 e " A + B ' = S,,(sh) + O(hn+')

We can therefore apply eq. [15] to solve 1161 with [17] numerically with any desired accuracy in the time increment At. On the other hand if one is dealing with a time-dependent problem, [19]

[20]

S,l(h) = S,,-,(sh)S,,-,((1 - 2s)X)S,,-,(sh)

[21] In ref. 5, we pointed out that if eq. [14] does not have real roots, the formula S,,(h) will not preserve unity since h = iAt, a complex number. This will create divergent exponentials in any iteration with concomitant loss of accuracy. One can shown that eq. [14] always has a real root when n is odd, i.e., for such cases, the symmetric expansion [13] for S,, is always unitary. In summary, we have developed the iterations, eA(A+B' = Sr,,-l(h) + ~(h'""),

n = 2,3, . . .

Formal solution of TDSE and higher order propagation We consider the Schrodinger equation

3.

where

U(t

+ sAt,t) = exp

cp(x,t + At)

=

[ i l"' + --

V(x,t)

U(t

+ At,t)cp(x,t)

+ At,t) = Texp

U(t

[22]

+ At,t) = exp

U(t

Since U(t

[

-iiAIHdt]

[

;[AIHdt]

- -

+ At,t) is unitary, we have at all time

U(t

+ At,t) = exp

H dl]

[

;[AIHdt] --

Clearly, by using S2,eq. [4], to approximate U(t + At,t), we will obtain only a second order accurate propagation scheme. We will now develop a propagation technique based on the group property of the operator U(t At,t) and eq. [ 151 to solve the Schrodinger equation for time-dependent potentials, in order to obtain high order accurate algorithms. We divide [t,t At] into three subintervals [t,t + sAt], [t + sAt, t + (1 - s)At], and [t (1 - s)At, t At]. On each subinterval, we use eqs. [22] and [4]. Thus for the first interval we obtain

+

+

dcp(x,t) = Hcp(x,t) at

If we choose a time-independent Hamiltonian,

[25]

2m ax-

and T is the time ordering operator. One can truncate eq. [20] and obtain

[24]

The symmetric decomposition of the general exponential operator [13] by the above algorithm furnishes, by construct, unitary approximations to propagator e-fl'l to any order of accuracy, 0(h2"+'),in the time increment At when we choose h = iAt, as shown in the next section.

ih-

d2

Since the operator exp[- i/h J ~H ~ dt] + is also ~ unitary, ~ then by comparing the two unitarity conditions, one finds .that any approximation to U(t + At,t) has always odd order in the leading error, i.e.,

and s is the real root of equation 2~2"-1 + (1 - 2s)2"- 1 = 0

[16]

h'

where

and s is a root of the more general equation

[15]

H =+ ,-

the formal solution of eq. [16] become more complicated. The general time-dependent solution can now be written (16),

where now 1131

cp(x,t + At) = e-iA'H/fi cp(x,t)

~ ~ ( t ) ( s A+t ).(At)~

+

+

BANDRAUKANDSHEN

For the second interval, one has [27]

U(t

+ (1

-

s)At, t

+ D,(t)((l - 2s)At13 + 0(At)'

+ sAt) = exp

where

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Finally for the last interval one obtains the equation, [28]

U(t

+ (1 - s)At, t + At) = exp

where

+ At,t) we have U(t + At,t) = U(t + sAt,t)U(t + (1 s)At, t + sAt)U(t + At, t + (1 - s)At)

By the group property of U(t [29]

-

I

and substituting eqs. [25], [27], and [28] into eq. 1291, we obtain

1

[30]

U,(t

+ At,t) = S2(sAt,t)Sz((l- 2s)At, t +(

A t + (1 + o(A~') where s is the real root of eq. [8]. The above procedure gives a fourth order accurate algorithm to calculate the evolution operator U for the time-dependent Schrijdinger equation, using products of these second order accurate propagators. The results holds for both timedependent and time-independent Harniltonians. Iterating eq. [30] according to the scheme explicited in eq. [15] will generate schemes of any desired accuracy in the time increment At.

4. Numerical comparison We previously compared the second and fourth order accurate splitting operators [26] and [30]. The latter is always more accurate (5, 6). In the preceding two sections we obtained higher order accurate formulas. Now we try to find the optimum algorithm with respect to maximum accuracy and minimum computation time (CPU). First, we note that the second order formula, S2 in eq. [5], has three exponential operators; the fourth order one, S, in eq. [lo], has seven exponential operators; the sixth order one, S,, has 19 exponential operators, and the eighth order one, S,, has 55 exponential operators. Thus from the eighth order formula, one does not seem to have practical algorithms due to the growing number of exponential operators. We consider eq. [18] (we set m = fi = 1) with V(x) = x' as a test. This time-independent harmonic oscillator model has an exact ground state solution,

[-I

fi

[ill

v(x.0

=

'I4

exp[

+ it) fi

-(x'

]

We solved this problem numerically using the various exponential splitting schemes presented in Sect. 2, using Fast Fourier Transform techniques (FFT) for the exponential of the Laplacian. All reported calculations were performed in an IBM RISC 6000/530 workstation (10 megaflop performance). In Fig. 1 a detailed comparison is presented of the L' errors (defined as: L' error = ( A x C e.'.)"',. . e., = ,Isn - s'l where s n is a numerical solution and s' is an exact solution) of the amplitudes of wavefunctions as a function of CPU time (seconds). w e see directly that the second order accurate formula S2 (3rd order error) takes much CPU time for high accuracy. The eighth order accurate expansion S, is also not efficient because of too many exponential operators. So we focus on the fourth (S,) and sixth (S,) order accurate formulas. In the high accurate range (about lo-,), the sixth order formula S, is better than the fourth one S,. In view of the numerical results of Fig. 1, in general, the sixth order formula S, seems to be the optimum method, i.e., with minimum CPU seconds one obtains the highest accuracy. In practice, if a problem does not require high accuracy, the fourth order accuracy formula S, seems to be preferred to the sixth order one (seventh order error), S,. We next solve eq. [20] (m = fi = 1) with V(x,t) = x 2 + x * sin(t), i.e., the forced harmonic oscillator, using 2nd, 4th, 6th order accurate formulas S1, S,, S,. We compare with these numerical results, the exact solution to eqs. [16] and [19] for an initial vibrational ground state (17),

where e(t) = f i [sin(fit) and

v? sin(t)]/2

CAN. J. CHEM. VOL. 70. 1992

2nd 4th - 6th - 8th -

A

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*

-

order order order order

accurate accurate accurate accurate

S2

53 55 57

FIG. 1. Log of L' error of the amplitude of the time-dependent wavefunction (eq. [31]) as a function of CPU computation time on an IBM RISC 6000-530 workstation for different splitting operators, S2, S,, S,, and S,.

TABLE1. The second order accurate splitting operator Sz for time-dependent potential V(x,r) = x2 + x sin(t). -3.57~ 5 x 5 3.57~, Ax = 77~1128, t = 600 (- 100 cycles) Time

Amplitude error

Phase error

CPU(s)

L is the Lagrangian of the classical forced harmonic oscillator. This 'time-dependent problem was solved numerically in Sect. 3 using the exponential splitting scheme for the propagator U . In Tables 1-3 we present, for various orders of accuracy in the time increments, the L~ errors for the amplitude and the phase of the numerical time-dependent wavefunctions as a function of the computation time in CPU seconds. These tables show that the sixth order accurate formula S , requires fewer CPU seconds and gives the best results. For instance, when the errors of the amplitude and phase are about and l o - , respectively, the second order formula S , requires 739 CPU seconds (Table I ) , the fourth order one S , re-

TABLE2. The fourth order accurate splitting operator S, for time-dependent potential V(x,t) = .r2 + x sin@). -3.57~ 5 x 5 3.5.rr, Ax = 77~1128, t = 600 (-100 cycles) Time

Amplitude error

Phase error

CPU(s)

TABLE3. The sixth order accurate splitting operator S, for time-dependent potential V(x,r) = x2 + x sin(t). -3.57~ 5 x 5 3.5.rr, Ax = 77~1128, t = 600 (- 100 cycles) Time

Amplitude error

Phase error

CPU(s)

BANDRAUK AND SHEN

TABLE 4. The second order accurate splitting operator Sz for time-dependent potential V(x,t) = x' + x sin(t). - 3 . 5 ~5 x 5 3 . 5 ~ fLr , = 7 ~ / 1 2 8 t, = 1200 (-200 cycles)

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Time

Amplitude error

Phase error

CPU(s)

TABLE 5. The fourth order accurate splitting operator S , for tlme-dependent potential V(x,t) = x2 + x sin(t). - 3 . 5 ~5 x 3 . 5 ~ ,hx = 7 ~ / 1 2 8 ,t = 1200 (=200 cycles) Time

Amplitude error

Phase error

CPU(s)

559

In conclusion, we have shown that the sixth order accurate formula S5 as given by eq. [14] allows explicit exponential numerical integration of parabolic partial differential equations, such as the time-dependent Schrodinger equation, with sixth order accuracy, i.e., errors depend on (At)'. This method is shown to be superior in accuracy over the second and fourth order accurate methods. The second order scheme is the most frequently used method in current timedependent problems (7-9). For high order accuracy (= the sixth order operator, S5, is optimum. For low order accuracy (- lo-'), the fourth order accurate operator S3, previously proposed by us ( 5 , 6), seems to be a more useful algorithm (see Fig. 1). Both methods are always superior to the standard second order accurate methods based on S2 (79). We are currently examining the efficiency of these algorithms for coupled equations, as occur in multiphoton electronic transitions (1). Of note is that the present algorithms are always unitary as compared to other efficient propagation schemes that are not unitary, such as Chebyshev polynomial expansions (1 8).

Acknowledgment We thank the Natural Sciences and Engineering Research Council of Canada for grants supporting this research. We also thank IBM Canada for partial support of a RISC machine. TABLE 6. The sixth order accurate splitting operator Ss for time-dependent potential V(x,t) = x2 + x sin(t). - 3 . 5 ~ 5 x 5 3 . 5 ~ ,hx = 7 ~ / 1 2 8 ,t = 1200 (=200 cycles) Time

Amplitude error

Phase error

CPU(s)

quires 322 s (Table 2), and the sixth, S5, gives 0.85* in amplitude and 0.5*10-~in phase using only 235 s (Table 3). When we double the time interval from [0,600] to [0,1200], the results for 2nd, 4th, and 6th order accurate formulas in Tables 4-6 show that accumulations of the error in the sixth order formula are fewer than those of the other two lower order schemes. Therefore in long time propagation the sixth order formula offers the best possible algorithm in terms of maximum accuracy in amplitude and phase for the least CPU time.

1. A . D. Bandrauk. Atomic and molecular processes with intense laser pulses. Vol. 171B. NATO Advanced Studies Institute. Plenum Press, New York. 1988. 2. A. D. Bandrauk. In Advances in multiphoton spectroscopy. Vol. 6. Edited by S. H. Lin. Academic Press, New York. 1990. 3. S. Chelkowski and A. D. Bandrauk. J. Chem. Phys. 98, 3618 (1988). 4. S. Chelkowski, A. D. Bandrauk, and P. B. Corkum. Phys. Rev. Lett. 65, 2355 (1990). 5. A. B. Bandrauk and H. Shen. Chem. Phys. Lett. 176, 428 (1991). 6. H. Shen. M. Sc. Thesis, UniversitC de Sherbrooke. 1991. 7. M. D. Feit and J. A. Fleck. J . Chem. Phys. 78, 2578 (1984). 8. H. Kono and S. H. Lin. J . Chem. Phys. 84, 107 1 (1986). 9. J . N. Barddey, A. Szoke, and J . M. Comella. J. Phys. B: At. Mol. Opt. Phys. 21, 3899 (1989). 10. H. F. Trotter. Proc. Am. Math. Soc. 10, 545 (1959). 11. R. M. Willcot. J . Math. Phys. 8, 962 (1967). 12. S. I. Berstein and A. A. Mirin. J. Comput. Phys. 5 , 547 (1970). 13. H. D. Raedt and B. D. Readt. Phys. Rev. A, 26, 3575 (1983). 14. M. Suzuki. J. Math. Phys. 26, 601 (1985). 15. Q. Sheng. IMA J . Numer. Anal. 9, 199 (1988). 16. P. Pechukas and J. C. Light. J. Chem. Phys. 44, 3897 (1966). 17. H. Husimi. Prog. Theor. Phys. 9, 381 (1953). 18. C. Leforestier e t a l . J. Comput. Phys. 94, 59 (1991).