Exact wave packet propagation using timedependent basis setsJ. Kucar and H.D. Meyer Citation: The Journal of Chemical Physics 90, 5566 (1989); doi: 10.1063/1.456410 View online: http://dx.doi.org/10.1063/1.456410 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/90/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Construction of basis sets for time-dependent studies J. Chem. Phys. 131, 064104 (2009); 10.1063/1.3202442 Accurate wave packet propagation for large molecular systems: The multiconfiguration time-dependentHartree (MCTDH) method with selected configurations J. Chem. Phys. 112, 8322 (2000); 10.1063/1.481438 Timedependent discrete variable representations for quantum wave packet propagation J. Chem. Phys. 102, 5616 (1995); 10.1063/1.469293 Reactive scattering using efficient timedependent quantum mechanical wave packet methods on an Lshapedgrid J. Chem. Phys. 94, 7098 (1991); 10.1063/1.460243 Exact timedependent wave packet propagation: Application to the photodissociation of methyl iodide J. Chem. Phys. 76, 3035 (1982); 10.1063/1.443342

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.113.111.210 On: Tue, 23 Dec 2014 01:09:30

Exact wave packet propagation using time-dependent basis sets J. Kucars> and H.-D. Meyer Theoretische Chemie, Physikalisch-Chemisches 1nstitut, 1m Neuenheimer Feld 253, D-6900 Heidelberg, West Germany

(Received 7 December 1988; accepted 27 January 1989)

The use of a time-dependent basis set for solving the time-dependent Schr6dinger equation is proposed. The basis set is a generalization of the harmonic oscillator functions. The proposed method is similar to the approaches formulated by Lee and Heller and by Coalson and Karplus. However, in the present method the time dependence of the basis functions is determined differently. Numerical studies on the quartic oscillator and on the Morse potential are carried out to investigate the performance of the various methods.

I. INTRODUCTION

The time-dependent approach to problems of quantum dynamics has attracted growing interest over the last de­cade. In many cases it first offers more physical insight and second, results in technical, i.e., computational advantages, compared to the time-independent formulation. Adopting a time-dependent formalism one is faced with a great variety of computational methods ranging from semiclassical I-I I and approximate quantum mechanical '2- '8 to (numerical) exact quantum mechanical procedures. 19-29 For recent re­views on time-dependent methods see Refs. 30 and 31. In the following we shall concentrate on the discussion of the nu­merically exact propagation schemes. There are two main streams of approach. In the first one the wave function is given pointwise on a numerical grid in coordinate space22

-25

or on a grid in both coordinate space as well as momentum space. 26-29 In the second route the wave function is expanded in a basis set. The time evolution of the wave function is then reduced to the time evolution of the expansion coefficients. We shall use this second route.

If the wave packet is expanded in a basis then the speed of convergence within this basis will certainly improve if the basis is allowed to follow the motion' of the wave packet. Such time dependence of a basis is of particular importance for the computation of scattering states. A fixed (and neces­sarily incomplete) basis can describe the (almost free) mo­tion of a scattering wave packet only over a very short time period. Lee and Heller '9•

2o and Coalson and Karplus21

(called LHCK in the following) independently introduced a basis of time-dependent generalized harmonic functions. The time dependence of these oscillator functions (see also Ref. 32) is described by time-dependent parameters which are identical to those introduced by Heller in his work on Gaussian wave packet propagation. I

Other authors 15.16 have expanded the wave function in a set of generalized Gaussians. The Gaussians are propagated according to Heller '·

2 and the time dependence of the expan­sion coefficients is determined variationally. The time-de­pendent basis of Gaussians is not orthonormal in contrast to the time-dependent harmonic basis used by LHCK. Hence

aJ Permanent address: R. Boskovic Institute, Zagreb, Yugoslavia.

the former basis may suffer from linear dependence during the course of the propagation.5

•15

A detailed comparative analysis of several methods for solving time-dependent problems can be found in the litera­ture. 15.16 In this article we shall introduce a time-dependent basis of generalized harmonic oscillator functions similar to the formulation ofLHCK. However, the time dependence of the parameters is determined in a different way. In Sec. II we shall derive and discuss our approach by concentrating on the one-dimensional case. All the advantages as well as diffi­culties of the present method can be discussed by considering one degree of freedom only. The multidimensional case merely increases the complexity. In Sec. III we briefly dis­cuss the multidimensional case and derive the working equa­tions for the case of two degrees of freedom. Section IV is devoted to the discussion of the numerical studies and Sec. V summarizes our findings.

II. THEORY

Assume one wants to expand a wave function 'I'(x) in an orthonormal set offunctions {9?n (x)}. Since one has the freedom of choosing the origin and the scale of the coordi­nate system one rather expands 'I' in the set {9?n [(x - X 1)/

x2 ]}. The determination of the parameters XI andx2 is in fact the only adjustment necessary if one expands real eigenfunc­tions of a time-independent formalism. Turning to time-de­pendent problems one may let x I and X 2 become time depen­dent. However, in order to be consistent one should also allow for translations in momentum space if one allows for translations in coordinate space. More generally, in time­dependent theory one should consider transformations in phase space. These transformations can be conveniently de­scribed by unitary operators acting on the basis functions. Hence we define the time-dependent basis {¢n (x,t)} as

¢n (x,t) = U(t)9?n (x)

m

= II exp [iadt)A d9?n (x) , (2.1 ) k~1

where ak denotes a real parameter and the generator Ak is a Hermitian operator. Note that we order products from right to left, i.e., U = expUamAm)' . 'expUa,A I)' The wave func­tion 'I'(x,t) is now expanded in the basis (2.1),

5566 J. Chern. Phys. 90 (10), 15 May 1989 0021-9606/89/105566-12$02.10 © 1989 American Institute of Physics

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.113.111.210 On: Tue, 23 Dec 2014 01:09:30

J. Kucar and H. -D. Meyer: Exact wave packet propagation 5567

'I'(X,t) = L an (t)"'n (X,/) n

= U(t) Lan (t)rpn (x) n

= U(t)¢(x,t) , (2.2)

where the definition of ¢ is obvious. We assume for a mo­ment that the ak (t) are given functions of time. Since 'I' satisfies the time-dependent Schr6dinger equation

iq,=fl\l! (2.3)

(we use ji = 1 throughout) it follows that ¢ satisfies

i(p = H eff ¢ (2.4a)

with

(2.4b)

The rotated Hamiltonian 1I and the partially rotated genera­tors Ak are defined as 17,18

1I = ut HU, (2.5a)

(2.5b)

The coefficients an hence satisfy the equations of motion

(2.6)

The solution of this equation obviously provides the exact solution to the Schr6dinger equation (2.3) regardless of the choice of the functions a k (I). In practice, however, the infi­nite sum appearing in Eq. (2.6) has to be truncated. It is therefore the task to determine ak (I) in such a way that the convergence of'l'(x,/) in the time-dependent basis is opti­mal, i.e., that the sum in Eq. (2.6) can be truncated to the smallest possible size.

Before we continue we mention that the choice of the generators Ak as well as of the basis functions rpn is quite general. The only restrictions are that the Ak 's are Hermi­tian and form a basis of a Lie algebra.33 The set {rp n } must be a complete set of eigenfunctions of some Hamiltonian Ho =!p2 + Vo(x), To be specific we choose rpn to be the oscillator functions

rpn (x) = (2nn!) -1/21T-1/4e - x'12H n (x) , (2.7)

where Hn denotes the nth Hermite polynomial. These func­tions are the eigenfunctions of the Hamiltonian

Ho = !(p2 + X2) . (2.8)

The requirement that 'I' converges optimal with respect to the basis {rpn} can be expressed by the variational princi­ple

(2.9) n

because optimal convergence implies that only the first few coefficients carry nonnegligible weights. Because of the pa­rameter j we have a whole family of variational principles. The choice, of, say, j = 3 or 4 seems to be more appropriate than the choice j = 1 because the variational principle tends to suppress higher coefficients more strongly for larger j.

However, the choice j = 1 considerably simplifies the alge­bra to follow. We therefore first apply Eq. (2.9) with j = 1 and define

(2.10)

According to the variational principle (2.9) the parameters are implicitly given by the equation

JG =0 Ja k

which results in

(2.11a)

(¢I [Ak,Ho] I¢) =0. (2.11b)

The time evolution of the parameters follows from

!!..- JG = 0 dt Ja k

which yields

(¢I [[Ak,Ho] ,Heff ] I¢) = O.

Adopting a matrix notation one finds

Fc:l=f,

where

Fij = (¢I [[A;.Ho], Aj] I¢) ,

/; = - (¢I [[A;.Ho],lI] I¢) .

(2.12a)

(2.12b)

(2.13 )

(2.14a)

(2.14b)

Equation (2.11) establishes the initial conditions and Eqs. (2.13) and (2.14) yield the desired equations of motion for the parameters. The evaluation of Eqs. (2.13) and (2.14) can be simplified if one makes use of the fact that the genera­tors A k form a basis of a Lie algebra.33

-35 The partially rotat­

ed generatorsAk are hence members of the Lie algebra and can be expressed as linear combinations of the generators;

(2.15 )

The matrix D is called "structure matrix". 18 The matrix ele­ments D kk , can be computed from the commutation-rela­tions of the generators. 34

,35 It is now obvious that the set of equations (2.11b) is equivalent to

(2.11c)

provided the structure matrix is nonsingular. Equation (2.11c) is simpler than Eq. (2.11b) becauseAk is usually a simpler operator than Ak • Next we introduce

FOij = (¢I [[A;.Ho], Aj] I¢) ,

fOi = - (¢I [[Ai,Ho],lI] I¢) ,

and arrive at the equations of motion

Foc:lo = fo

and

or

(2.16a)

(2.16b)

(2.17a)

(2.17b)

c:l = DT-IFo-lfo, (2.17c)

where DT denotes the transpose ofD. Equation (2.17c) shows that both D and Fo must be nonsingular matrices in order to obtain meaningful equations of motion. The regu-

J. Chern. Phys., Vol. 90, No. 10, 15 May 1989

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.113.111.210 On: Tue, 23 Dec 2014 01:09:30

5568 J. Kucar and H. -D. Meyer: Exact wave packet propagation

larity of D can usually be guaranteed by choosing an appro­priate ordering of the generators. 18.34.35

It is now time to introduce a specific choice of the gener­ators. We shall adopt the harmonic Lie algebra. This algebra is spanned by the operators {1,x,p,x2,xp,p2}. Since we must use a basis of Hermitian operators we replace xp by

XP= !(xp + px) . (2.18)

We found it further convenient to replace p by - P and p2 by Ho; i.e., we use {l,x, - p,x2,XP,Ho} as generators. This set is obviously a basis of the harmonic Lie algebra. By choosing a specific ordering of the generators we define the transforma­tion Uas

U( ) - ix,p ip,x ib,x' ia,xp iy,H" i8,

t =e e e e e e, (2.19)

where the time-dependent parameters are characterized by the sUbscript t. The time-dependent basis functions (2.1) hence become

tPn (x,t) = exp [aJ2 + ib, (x - x, )2

+ ip, (x - x,) + i(n + 1I2)y, + iO,]

(2.20)

It is obvious that the unitary operator exp(iy,Ho)exp(io,) changes the phase of the time-dependent basis function tPn (x,t) but does not influence its form. Hence the inclusion of this operator does not improve the convergence in the basis. However, the inclusion of this operator may improve the numerical performance of the coupled differential equa­tions (2.4). This is discussed in Appendix A. For sake of brevity we drop the operator exp(iy,Ho)exp(io,) in the fol­lowing.

Defining aCt) = (a"b"p"x,) T it is now easy to solve Eq. (2.11). We obtain

(¢lx2 _ p21¢) = 0,

(¢IXP I¢) = 0,

(¢Ixl¢) = 0,

(¢Ipl) = o.

(2.21a)

(2.21b)

(2.21c)

(2.21d)

Using ¢ = ut 'I' one can derive the explicit equations for the parameters:

a = J.. lo [('I'I (p - p, )21'1') _ 4b 2] (2.22a) , 4 g ('1'1 (x _ x, )21'1') "

('1'1 (x - x,)(p - p,) + (p - p,)(x - x,) 1'1') b, = 4('1'1 (x _ x, )21'1') ,

x, = ('I'lxl'l') ,

p, = ('I'lpl'l') .

Assuming that the Hamiltonian is of the form

1 2 H=-p + V(x), 2m

(2.22b)

(2.22c)

(2.22d)

(2.23 )

we can derive the equation of motion (for details see Appen­dix B)

iI, = - 2bJm - (¢li[p2,V] 1¢)/(¢14Hol¢) , (2.24a)

b, = 4a'/2m - 2b ;2m - e2a'(¢li[XP,V] 1¢)/(¢12Hol¢) , (2.24b)

X, =pJm.

(2.24c)

(2.24d)

Replacing ¢ by Ut'l' in Eq. (2.24c) one finds that p, = ('1'1 - av lax I '1') holds. Hence the time evolution of x, and p, is given by the Ehrenfesttheorem. Ifthe potential V is harmonic then the equations of motion simplify to

iI, = - 2bJm , (2.25a)

b, = e4a'/2m - 2b ;/m - !V" (x,) , (2.25b)

p, = - V'(x,) , (2.25c)

X, = pJm , (2.25d) where the prime denotes differentiation with respect to x. These equations are equivalent to those derived by Heller l

and used by LHCK. (To make the comparison note that

Heller's complex parameter a, is a, = i!ea, + b, in our nota­

tion.) Equations (2.25) may, of course, be used for nonhar­monic potentials as well, but then they are different from our equations of motion (2.24). In order to discuss some of the differences we first note that the parameters as defined by Eqs. (2.25) depend only on the initial values ofthe param­eters and on the potential. Their time evolution is indepen­dent of the wave function. Our equations of motion (2.24), on the other hand, do explicitly depend on the wave func­tion. The most important difference lies perhaps in the per­formance of the parameter a,. Equation (2.24a) may be re­written as

iI, = - 2bJm + G /2G, (2.26)

where G is defined by Eq. (2.10). Equation (2.25a) just misses the last term of the above equation. Since one expects G to be positive most ofthe time (the wave function becomes more complicated as time evolves) we find that our a, is larger than the one computed according to Eq. (2.25a). Since the width of the basis functions scales like e - a, we find that our basis functions are less diffuse than those ofLHCK. These arguments are supported by our numerical studies (see Sec. IV).

In closing this section we briefly discuss the case of mini­mizing (¢IH61¢) rather than (¢IHol¢). As before, our ap­proach is defined by Eqs. (2.10)-(2.16), but wherever Ho appear in these equations one now has to replace it by H 6. Instead ofEq. (2.21) we now obtain

(¢lx41¢) = (¢lp4 1¢) ,

(¢lx3p + px3 1¢) = _ (¢I Xp3 + p3x l¢) ,

(¢lp31¢) = _ (¢Ixpxl¢) ,

(¢lx3 1¢) = - (¢Ipxpl¢) ,

(2.27a)

(2.27b)

(2.27c)

(2.27d)

as conditions which minimize G. Unfortunately these equa­tions cannot be inverted analytically to yield the parameters as we could do previously [compare with Eq. (2.22) ] . How­ever, the initial wave function is usually of such a simple form that the initial values of the parameters can be deter­mined without numerically searching for the minimum of G. The equations of motion follow in the usual way. Some de­tails are provided in Appendix B. Here we only mention that

J. Chern. Phys., Vol. 90, No. 10,15 May 1989

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.113.111.210 On: Tue, 23 Dec 2014 01:09:30

J. Kucar and H. -D. Meyer: Exact wave packet propagation 5569

first the numerical evaluation of the elements of fo is more elaborate than previously and that second the matrix Fo is now a full matrix which must be inverted numerically at each step of the integration. Hence minimizing H ~ is com­putationally much more costly than minimizing Ho. This additional computational labor largely compensates the gain which is achieved by the now better convergence properties of the time-dependent basis (compare with Sec. IV).

III. THE MULTIDIMENSIONAL CASE

Our approach is easily generalized to n dimensions. The wave function ifJ is now expanded in a product of harmonic oscillator functions and the generators are now derived from the n-dimensional harmonic Lie algebra. A basis of this har­monic algebra is given by all products of the operators {1,xi,Pi Ii = l, ... ,n} up to second order. As in the one-di­mensional case it is useful to first identify all linear combina­tions of these operators which commute with Ho = !:2:(P7 + x7)· A generator which commutes with Ho cannot improve the convergence of the basis functions and its time-dependent parameter cannot be determined by the variational principle (2.9). To identify the above mentioned linear combinations we introduce the operators

Jij = xiP; + PiX;, 1 <J<J<n ,

Kij = XiX; + PiP;, 1 <i<J<n ,

(3.1a)

(3.1b)

L i; = xiP; - PiX;, l<i <j<n, (3.1c)

and use {1,xi,Pi,xix;,Jij,Kij,Lij} as a basis of the harmonic Lie algebra. The operators Kij and Lij commute with Ho and should thus b~.removed. However, only the operators Kij can be removed. If we remove L ij from the above set then the remaining operators do not form a Lie algebra and a struc­ture matrix does not exist. Hence one has to keep the L i; and determine their time-dependent parameters by other means rather than the variational principle (2.9).

To proceed we determine via Eq. (2.11c) the conditions which minimize G:

Ak=Xi: (ifJIPilifJ) =0,

Ak=Pi: (ifJlxilifJ) =0,

( 3.2a)

(3.2b)

Ak=XiX/ (ifJIJijlifJ) =0, (3.2c)

Ak =Jij: (ifJlxix; -PiP;lifJ) =0. (3.2d)

Next we have to state conditions for the parameters of the operators Lij because-as discussed above-the variational principle (2.9) does not determine these parameters. The angular momentum operator Lij generates rotations in the Xi-X; plane U=l-j). It is thus possible to determine the pa­rameter of Lij such that (ifJlxix; lifJ) vanishes. This choice of additional constraints turns out to be very convenient and we hence replace Eq. (3.2d) by the conditions

(ifJlx; - p;lifJ) = 0,

(ifJlxix;lifJ) =0, kj,

(ifJIPiP; lifJ) = 0, kj.

(3.2e)

( 3.2f)

(3.2g)

Equations (3.2a)-(3.2c) and (3.2e)-(3.2g) can now be used to determine the parameters and their time evolution.

In the following we will concentrate on the discussion of

the two-dimensional case. In order to simplify the notation we use x,Y and p,q rather than X1,X2 and PI,P2' respectively. The operators J 12 and L 12 are abbreviated to J and L, respec­tively, and we use the definitions

! J II = XP,

! J22 = YQ.

Next we define the unitary operator U as

(3.3a)

(3.3b)

U(t) = e - iy,qe - ix,p eiq,y /p,xeih,y'eig,xy eif,x'eit3,Leia,Jeib,YQeia,xp.

(3.4 )

We have chosen this particular ordering of the generators because it does not only lead to a nonsingular structure ma­trix but also to a splitting of D into three blocks (see Appen­dix C). The time-dependent basis functions are now given by

tPnm (x,y,t) = exp[!(a, + b, ) + ip, (x - XI)

+iq,(y-y,) +ift(X-XI)2

+ ig, (x - xI)(Y - YI)

+ ih , (y - YI )2]q;>n (x')q;>m (Y') , (3.5)

where the primed variables are related to the coordinates through the transformation matrix T,

[X:]=[TI T2]rX-XI]. Y T3 T4 Lv - YI

The matrix elements of Tare

TI =ea'(cos,Bcosha+sin,Bsinha),

T2 = ea'(cos,Bsinh a - sin,B cosh a) ,

T3 = /'( cos,B sinh a + sin,B cosh a) ,

T4 = /'(cos,B cosh a - sin,B sinh a) .

(3.6)

(3.7a)

(3.7b)

(3.7c)

(3.7d)

The equations of motion for the 11 parameters are de­rived in Appendix C. Here we only note that the matrix Fo splits into three blocks and that F 0- I can be obtained explicit­ly. For the case of a harmonic Hamiltonian

H = p2/2ml + q2/2m2 + V(x,y) , (3.8)

where Vis a polynomial up to second degree, we could derive the equations of motion (see Appendix C). Rather than to give the explicit form of the equations of motion for all of the parameters we found it more convenient to replace the pa­rameters a"b"a" and,B, by the maxtrix elements T 1, ... ,T4 of the transformation matrix T. After some algebra one arrives at the following set of equations:

XI =pJm l ,

XI =qJm2 ,

PI = - Vx (X"YI) ,

iII = - Vy(x"YI),

C = ITTTM-1TTT - 2CM- 1C - IV 22'

T= -2TM- 1C,

where we have used the abbreviations

and

( 3.9a)

(3.9b)

(3.9c)

(3.9d)

(3.ge)

(3.9f)

(3.10a)

J. Chern. Phys., Vol. 90, No.1 0, 15 May 1989

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.113.111.210 On: Tue, 23 Dec 2014 01:09:30

5570 J. Kucar and H. -D. Meyer: Exact wave packet propagation

(3.lOb)

and where Vx = av lax, etc. The matrix M is a diagonal matrix with the masses m I and m2 on its diagonal. The con­nection to the LHCK formalism is established by recogniz­ing that Heller's complex matrix At (Ref. 19) is in our nota­tion At = C + (i12)TTT. As in the one-dimensional case our approach [Eq. (3.9)] is found to be equivalent to LHCK if the Hamiltonian is harmonic. For anharmonic po­tentials our approach [Eqs. (3.2), and (Cl)-(C4)] is dif­ferent from LHCK.

In this article we will not report on numerical calcula­tions treating more than one degree offreedom. We want to mention, however, that using the 2D formalism developed in Appendix C we have recently successfully redone a model problem which we have solved previously17 using a fixed (time-independent) basis.

IV. ILLUSTRATIVE EXAMPLES

In this section we discuss two one-dimensional model problems in order to examine the numerical performance of the methods proposed in Sec. II. We have used three differ­ent ways of determining the time-dependent parameters, namely, minimizing (¢IHol¢) [cf. Eqs. (2.24)], minimizing <¢IH~I¢) (see Appendix B), and adopting the LHCK method [cf. Eqs. (2.25)]. For brevity we will call the first two approaches the Ho and H ~ method, respectively.

Our first model problem consist of a particle of unit mass moving in the quartic potential Vex) = 0.01x4. We choose the initial wave function to be Gaussian of width ~5/18 localized at 6.0 and moving with a momentum of - 6.5. For all three methods we hence obtain the initial pa-

rameters at = 0.2939, h, = 0, p, = 6.5, and x, = 6.0. The expansion coefficients are an = Dno initially. The time evolution of the expectation values ('I' Ixl'l') and ('I'lpl '1') as well as of the variances ('I'lx21'1') - ('I'lxl'l')2 and ('I'lp21'1') - ('I'lpl'l')2 is shown in Fig. 1. All three methods give, of course, identical results for these expectation values as for all expectation values with respect to '1'. Expectation values with respect to 1,6, however, are different. In Fig. 2 we show the time evolution of (¢IHol¢) and (¢IH6I¢) 1/2 for all the three methods. Figure 2 does not only indicate the con­vergence properties of the time-dependent basis, it also gives us some information on the structure of the wave function. For short times (1<; 1) the wave function remains to be a generalized Gaussian since (¢IHol¢)::::: 1/2 holds. When the wave packet hits the wall for the first time (t::::: 2) it acquires a more complex structure and some higher coefficients are populated. When the wave packet is in the middle of the well (1:::::3.3) it recovers almost a simple Gaussian shape. But as the wave packet hits the other wall (1:::::4) it acquires such a rich structure that it has to be represented by several time­dependent basis functions. It never again acquires a Gaus­sian shape.

The LHCK method shows a very sharp rise of < 1,6 I Ho 11,6) and (¢IH6I¢) close to t 3.5. This indicates that the wave function is very poorly represented by the time-dependent basis of the LHCK method for times larger than t = 3.5. To

10

5

10~ ____________________________ ~

30

20

10

,. I \ I \ I \ I \ I \ I I

FIG. I. Upper part: Time evolution of the expectation values ('I'lxl '1') (full line) and ('I'lpl'l') (dashed line). The quartic potential is used. Lower part: Time evolution of the variances ('I'lx'I'I') - ('I'lxl'l')' (full line) and ('I'lp'I'I') - ('I'lpl'l')' (dashed line).

check the convergence as well as to increase the computa­tional speed we have set up the computer code in such a way that it allows for a dynamic increase ofthe basis size. At each step of the integration the program compares the sum of the moduli square of the last two coefficients with some thresh-

lSr-----------r---------------,

FIG. 2. Upper part: Time evolution of (¢'I Hoi¢,) for all three methods inves­tigated. Dotted line: LHCK method, full line: Ho method, dashed line: H t method. The quartic potential is used. The Ho method yields the smallest values for (¢,IHol¢,) by construction. Lower part: Similar to upper part but shown is (</1IH~ 1</1) 112.

J. Chern. Phys., Vol. 90, No. 10,15 May 1989

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.113.111.210 On: Tue, 23 Dec 2014 01:09:30

J. Kucar and H. -D. Meyer: Exact wave packet propagation 5571

old E. If the sum is larger than E then the program adds two new coefficients to the Schrodinger equation (2.6). In this way it automatically determines the appropriate size of the basis set. (An automatic decrease of the basis set size, how­ever, was not implemented.) Figure 3 shows the required number of basis functions. As expected, minimizing (¢IH61¢) yields the best and using the LHCKmethod gives the worst convergence. In fact the LHCK method requires so many basis functions for times larger than t = 4 that it becomes a useless method for solving our problem. Compar­ing the other two methods one finds that for shorter times where the number of basis functions used in both methods is not too different it is more efficient to minimize (¢!Hol¢) because minimizing (¢IH61¢) leads to equations of motion which are more elaborate to evaluate. At longer times this labor is overcompensated by the gain obtained through the smaller size of the basis.

We now turn to the discussion of the parameters. The parameters X t and Pt are closely related to the expectation values ("'lxl"') and ("'11'1"'), however, these parameters and expectation values become identical only if one mini­mizes (¢!Hol¢). We found that all three methods yield a similar time dependence for X t • The differences are most pronounced in the vicinity of the turning points. At the first turning point (t = 1.96) we found X t = - 7.64, - 7.51, - 7.36 for the LHCK, Ho. and H6 method, respectively. For the other turning points we obtained: t = 4.36, X t = 7.64, 7.13, 6.59; t = 6.76, X t = - 7.64, - 6.57, - 5.71; t = 9.16, X t = 7.64, 5.88, 4.78. Hence the differ-

ences between the methods increase for longer times where the wave function becomes more complicated. In the LHCK method the center of the basis functions approaches the re­pulsive walls most closely. The Ho method-and even more so the H 6 method-tend to pull the center of the basis func­tions away from the repulsive walls, particularly so if the wave function becomes diffuse.

200.-----------------------------------,

100

r-~

r I

I 1

. r .'~r--'-..r

~.J 1-

r

~_I-,....-

" I .-1 __ -1

FIG. 3. Number of basis functions required by the program. The threshold parameter E was set to E = 10-' (see the text) and the initial number of basis functions was set to N = 8. Note that the LHCK method requires an extreme number of basis function when the wave function acquires a more complicated structure, i.e., when (t,6IH"It,6) > 2.

The time dependence of the parameter a t is depicted in Fig. 4. As discussed in Sec. II the LHCK method leads in the average to a smaller at-and hence to broader basis func­tions-than the other two methods. Only close to the turn­ing points where the basis functions are narrow does the LHCK method yields even less diffuse basis functions than the Ho and H 6 method. In general, we found that the LHCK method yields the strongest and the H 6 method the weakest oscillations of the time-dependent parameters.

The bounded motion in a fairly narrow potential well which we have discussed so far is certainly not the prototype of a problem which one would like to attack with the present methods. In such a case it is simpler to use a fixed time­independent basis. A time-dependent basis must be used, however, if one considers the unbounded motion of a scat­tered particle. We therefore investigated the motion of a wave packet scattered by the Morse potential V(x) = D(1 - e - x/x" ), as a second example. (The particu­lar numerical implementation of the method required that the potential was given as a polynomial. The Morse potential was expanded up to sixth power in x - X t and this time­dependent potential was actually used in the computation). We have set the potential parameters to D = 10.25 and Xo = 4.5269 because such a potential has been discussed pre­viously.3 The initial wave function was chosen to be a Gaus­sian and the initial values of the parameters were set to X t = 20, Pt = - 6.5, at = - 0.346 (i.e., ("'lx2 1"') - ("'lxl",)2 = 1), and ht = O. In the following we concen­

trate on the discussion of the parameters at and ht because the present example clearly illuminates the strange behavior of these parameters in the vicinity of a turning point.

Figure 5 shows the time evolution of the parameters at and ht . In the beginning at slowly decreases, i.e., the basis functions become more diffuse. When the wave packet starts to "feel" the repulsive wall then the parameter at rapidly rises. Close to the turning point «"'lpl"') = 0 occurs at

2r----------------------.-------;-.

-1

" " ,.

2

:

II 6

.. .

' ..

8 10

FIG. 4. Time evolution of the parameter a, for all three methods investigat­ed. The quartic potential was used. Note that the LHCK method (dotted line) yields a parameter a, which is more strongly oscillating than the pa­rameter of the Ho method (solid line) and of the H ~ method (dashed line). Remember that the width of the time-dependent basis functions scales like exp( - a,).

J. Chern. Phys., Vol. 90, No. 10, 15 May 1989

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.113.111.210 On: Tue, 23 Dec 2014 01:09:30

5572 J. Kucar and H. -D. Meyer: Exact wave packet propagation

2 6

, , " ,: l\ , '" ~

I'l: \ '----__ _ ,t \ __

I' , -_ " " " " " -2 , ' , ' , ' 2 , ' , ' , ' , ' '.

FIG. 5. Upper part (left-hand scale): Similar to Fig. 4 but using the Morse potential. See the text. Lower part (right -hand scale) : Time evolution of the parameter b, when using the Morse potential. See the text. For sake of clar­ity of the figure we have suppressed to draw b, when b, ;;.3.5.

t = 3.92) the parameter at goes through a maximum. After the maximum at rapidly decreases but when the wave packet leaves the interaction region(t;;;.6) then the decrease of at is slowed down and accounts for the usual spreading of a free wave packet. Whereas the Ho and the H6 methods yield similar functions at it is eye catching that the LHCK method shows a different result. The parameter of this method goes to a deeper minimum and through a higher maximum than the parameters of the other two methods. Hence the width of the basis functions varies more strongly when the LHCK method is used. Moreover, the maximum of at is shifted to later times and definitely occurs after the turning point when the LHCK method is used.

Turning now to the parameter bt [Fig. 5(b)] one ob­serves that the time evolution of this parameter computed with the LHCK method differs even more strongly from the parameter computed by the other two methods. The LHCK parameter bt shows an almost discontinuous behavior. We have found that such a rapid variation of bt generally occurs at turning points. The variation of bt is more rapid the "harder" the potential wall, i.e., in our example the smaller is the potential parameter xo or the higher is the scattering energy. It is clear that the numerical integration of such rap­idly varying quantities requires small time steps.

To summarize the findings of our numerical studies we recall that the Ho and H 6 methods perform better than the LHCK method because they require fewer basis functions. However, it is not really the number of basis functions which matters but computational speed. It was noted previously by Heather and Metiu, 16 who studied the time-dependent mul­tiple Gaussian approach,15 that their equations of motion may become stiff36 and force the integrator to use very small

time steps. This increases dramatically the required comput­er time. The present methods suffer from similar problems. Concentrating on the Ho and H 6 method we have observed that our differential equations become stiff closely before and after but not at a turning point. To be specific in our second example the integrator takes the smallest step sizes around t;:::;2.7 and around t;:::;5. The wave function is still close to the repulsive wall at these times but the basis func­tions have now become quite diffuse. As a consequence, the basis functions penetrate deeply into the classical forbidden region glvmg rise to very large matrix elements (rpn IHeft" Irpm), in particular for large nand m. These large matrix elements turn the differential equation (2.6) into a stiff equation. Hence the equations of motion become more stiff the larger the basis set used and the smaller is at while the wave packet is close to a repulsive wall. The LHCK method suffers particularly from the stiffness of the equa­tions. In comparison to the other two methods it first re­quires the largest basis set size, second, it assumes the small­est values for the parameter at and third, its basis functions approach the repulsive wall most closely.

v. CONCLUDING REMARKS

We have studied time-dependent quantal dynamics by expanding the wave function in a time-dependent basis. This has the obvious advantage that the wave function can be represented more efficiently in a time-dependent basis com­pared to a time-independent one. Propagating the wave function in a time-dependent basis, however, there appear two additional pieces of work. First, one has to solve the equations of motion for the parameters which determine the time-dependent basis functions. Second, one has to evaluate the matrix elements of the effective Hamiltonian at each step of the integration [cf. Eq. (2.6)]. In our numerical studies we therefore have restricted ourselves to the investigation of model systems the Hamiltonians of which are polynomials in coordinates and momenta. The matrix elments of such a Hamiltonian can be evaluated very efficiently, because we are using a basis of harmonic oscillator functions.

We have determined the equations of motion for the parameters by adopting a variational principle which, in some sense, guarantees for an optimal convergence of the wave function in the time-dependent basis. Lee and Hell­er 19.20 and Coalson and Karplus21 have introduced a time­dependent basis set similar to ours but they have determined the time-dependent parameters by truncating the potential to second degree. For harmonic potentials all the various methods discussed in this paper are equivalent. For anhar­monic potentials we found the LHCK approach to be less efficient than our approach which is based on a variational principle for determining the parameters. When the wave function acquires a more complicated structure then the LHCK approach may even cease to be a feasible method.

A numerical problem which seems to be common to all methods using time-dependent basis sets 16 arises because the differential equations to be solved may become stiff36 in the course of the propagation. The numerical solution of stiff differential equations requires very small integration steps. This significantly increases the computation time. Although

J. Chem. Phys., Vol. 90, No. 10, 15 May 1989

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.113.111.210 On: Tue, 23 Dec 2014 01:09:30

J. Kucar and H. -D. Meyer: Exact wave packet propagation 5573

our method certainly suffers from the stiffness problem it seems to us that here this problem is not as severe as in the other approaches (at least it is much less severe than in the LHCK approach). One may think of improving the present approach by deriving the equations of motion for the param­eters from a variational principle which not only tries to achieve an optimal convergence but also tries to avoid stiff equations, i.e., tries to avoid large matrix elements of Heff .

The formal development of the present method bears some similarities to the recently developed time-dependent rotated Hartree (TDRH) approach. 17

,18 The combination of the present method with TDRH, i.e., the solution of the one-dimensional SchrOdinger equations appearing in the· TDRH formalism by introducing a time-dependent basis, seems to be very promising. In TDRH one has, similar to the present method, to evaluate the matrix elements of some effective Hamiltonian at each step of the integration. Hence this computational drawback does not count when combin­ing both methods. Work on implementing the two methods in a combined fashion is in progress.

In the course of this paper we have expanded the wave function tP in a harmonic oscillator basis. It is obvious how to proceed if one wants to choose another set of basis functions. Here we want to emphasize that the introduction of the time­dependent unitary operator UU) may also be useful when adopting a spectral grid method. 27.31 In a spectral grid meth­od the wave function (its Fourier transform) is assumed to have nonvanishing values only on some interval in coordi­nate space (momentum space). With the aid of the operator U( t) one can center the wave function on these intervals and optimize its width. An obvious choice for the operator the expectation value of which is to be minimized reads

1 In 2' 2' H,=- p/+x/, j 2 il' I

where j is a similar parameter as in Eq. (2.9). Minimizing ('I'IU~utl'l') leads to an improved representation of tP = Ut'l' on the spectral grid. For j = 1 the operator Hj

reduces to the harmonic oscillator Hamiltonian for which we have derived the equations of motion of the parameters ak in this article.

As a final remark we want to mention that Lie algre­braic methods for solving time-dependent problems have been extensively used by Levine and co-workers.37

•38 They,

however, sought for approximate solutions and determined the parameters in front of the generators by the principle of maximum entropy.

ACKNOWLEDGMENTS

One of us (J. K.) thanks the Deutschen Akademischen Austauschdienst (DAAD) for financial support. The other one (H.-D. M.) gratefully acknowledges the financial sup­port by the Deutsche Forschungsgemeinschaft (DFG) through the Sonderforschungsbereich 91.

APPENDIX A

In deriving equations of motion for the parameters ak [Eqs. (2.21 )-(2.24)] we found that operators exp (iY,Ho) exp (iD, ) influence just the phase of the harmonic

oscillator basis functions and thus did not include these op­erators explicitly. For solving the mean field equations, i.e., integrating tP. it could be numerically more convenient to extract the phase factor before the integration. We write

'H '[j </>(x,t) = e'Y' He' X(x,t) . (Al)

The function X is, similar to tP, expanded in the basis

(A2) n

with lP n defined in Eq. (2.7). Instead of solving mean-field equations (2.6) for the coefficients an one arrives now to somewhat modified equations

ibn = I (lPn IHeff + 8, + 1',HollPm )exp [iy, (m - n)] bm , m

(A3)

where Heff is the effective Hamiltonian propagating tP and with the obvious relation

_ i[j, iy,(n + 112lb an - e en' (A4)

In order to improve the numerics concerning integration of the expansion coefficients we determine 1'1 and 8, such as to minimize IIx1l2. This leads to the following equations:

. (Ho)(Heff ) - !(HoHeff + HeffHO) y, = (H~) - (HO)2 •

(ASa)

8, = - (Heff ) 1', (Ho> , (ASb)

where the expectation values are in terms of tP. It is obvious from the Eq. (AS) that problems arise if (H~) (HO)2 becomes equal or close to zero. The denominator in Eq. (ASa) becomes exactly zero, if tP is represented by a single harmonic oscillator function. However, for a expansion con­sisting of several terms but still dominated by a single func­tion, the denominator is numerically close to zero. If the equation is close to the singularity then the set of coupled differential equations to be solved becomes stiff and the nu­merical integration requires small times steps. Then the ad­vantage of integrating X rather than tP is lost. To avoid this we found it convenient to determine 1', as follows:

1',= (lPn IHeff IlPn) - (lPoIHeff IlPo)

n-l (A6)

where n is chosen by convenience. We keep 8, as defined in Eq. (ASb).

In all our numerical tests we found that such choice of 1', and 8" although somewhat arbitrary, improves the numeri­cal performance, i.e., allows for larger step sizes.

APPENDIXB

In this Appendix we consider more explicitly the one­dimensional case. The operator U(t) which acts on the basis function is of the form

(BI)

The partially rotated generators A k as defined in Eq. (2.Sb) can be obtained using the Baker-Hausdorff formula,34,35 and they read

J. Chern. Phys., Vol. 90. No. 10, 15 May 1989

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.113.111.210 On: Tue, 23 Dec 2014 01:09:30

5574 J. Kucar and H. -D. Meyer: Exact wave packet propagation

-XP=XP, .... 2 -20, 2 X =e x,

x = e-a,x,

(B2a)

(B2b)

(B2c)

(B2d)

The operators Ak are linear combinations of the generators A k • The last term in Eq. (B2d) indicates that apart from the operators appearing in U(t) [Eq. (Bl)] the unit operator is necessary to build a basis of a Lie algebra. However, as al­ready discussed [Sec. II and Appendix A) the unit operator changes only the phase of the wave function and can be dropped. According to Eq. (B2) we write the structure ma­trix D [Eq. (2.15)] as follows:

o e- 2a,

o o

o o

2b,e- a,

0] o o .

eO,

(B3)

To obtain the equations of motion for the parameters (1 = (a,b,p,x,) T one has to evaluate the matrix Fo, and the vector fo as defined by Eq. (2.16).

Making use of Eqs. (2.21a)-(2.21d) the matrix Fo be­comes diagonal:

4(Xp3 + p3X )

4(x4) + 3 (xpxp + pxpx)

- 4(x3 )

8(p3)

The matrix elements appearing in the above expression are more complicated to evaluate than the ones obtained by min­imizing (t,6IHolt,6). The complexity increases even more if one considers the computation of fo [see Eq. (2.16b) but Ho replaced by H 6 ] . In evaluating F 0 and fo we made use of the fact that

[Ak,H6] = Ho[Ak,Ho] + [Ak,Ho]Ho (B8)

holds. The use ofEq. (B8) simplifies the calculation because the operation of Ho on a wave function is trivial.

APPENDIXC

In the following we discuss the two-dimensional case in more detail. The conditions which minimize G are given by Eqs. (3.2a)-(3.2c) and (3.2e)-(3.2g) for the general multi­dimensional case, and we rewrite them here for the particu­lar case under consideration

o o

o

(B4)

Assuming the Hamiltonian of Eq. (2.23), the rotated Hamiltonian as defined by Eq. (2.5a) takes the appearance

1I = _1_ (/ap2 + 4b;e - 2a'x2 + 4b,XP + 2eap,p 2m

+ 4b,e - ap,x + p;) + Vee - a,x + x,) (B5)

and it follows for the vector fo,

8b,(x2)/m + (i[p2,V)) 20, 8b 2e - 20, _

- ~ (x2) + ' (x2) + 2(i[XP, V]) fo= - m m

2b,e - ap,lm + (i[p, V))

(B6)

It is obvious that both DT and Fo can be inverted analytically. Solving Eq. (2.17c) is straightforward and yields the equa­tions of motion for the parameters a"b"x"p, as given by Eqs. (2.24a)-(2.24d).

For the case of minimizing (t,6IH61t,6) the matrix Fo be­comes

4(P3)

- 4(x3)

3(P2) + (x2)

- (xp +px)

I (t,6lplt,6) = (t,6lqlt,6) = 0,

(t,6lxlt,6) = (t,6lylt,6) = 0,

(t,6IXP 1t,6) = (t,61 YQ 1t,6) = (t,6lxq + pylt,6) = 0,

(t,6lx2 - p21t,6) = (t,6ly2 - q21t,6) = 0 ,

(t,6lxYIt,6) = (t,6lpqlt,6) = o.

(B7)

(CIa)

(Clb)

(Clc)

(Cld)

(Cle)

Next one has to calculate the partially rotated operators Ak and determine the structure matrix D [Eq. (2.15)]. For the specified ordering of the generators [see Eq. (3.4)] the structure matrix splits into three blocks: the 4 X 4 matrix (D I ) for the operators XP, YQ, J, and L, the 3 X 3 matrix (D2 ) for the operators x 2

, xy, and y2, and the 4 X 4 matrix (D3 ) for the linear operators x,y,P, and q. In evaluating the equations of motion for the parameters the inverse of the transpose of the D matrix is required. Since D has quite a simple structure, it is possible to invert the three blocks ana­lytically, and we prefer to give (Dj) -I directly:

[

1 0

o 1 (DT)-I = 100

o 0

- sinh(a, - b, )tanh 2a,

sinh(a, - b, )tanh a,

- cosh(a, - b,)

sinh(a, - b, )/cosh 2a,

cosh(a, - b, )tanh 2a, ] cosh(a, - b, ) tanh 2a,

- sinh(a, - b,) ,

cosh(a, - b, )/cosh 2a,

(C2a)

J. Chern. Phys., Vol. 90, No.1 0, 15 May 1989

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.113.111.210 On: Tue, 23 Dec 2014 01:09:30

J. Kucar and H. -D. Meyer: Exact wave packet propagation 5575

(C2b)

[

TIT3 - (2/,T4 -g,T3)ITo (2/,Tz -g,T,)ITo ]

(D )-1 = TzT4 (2h,T3 -g,T4)ITo - (2h,T, -g,Tz)ITo 3 0 0 T41To - TzlTo '

o 0 - T31To T,1To

(C2c)

where T I,Tz,T3,T4 are defined by Eq. (3.7), and To is the determinant of the Tmatrix [Eq. (3.6)]:

T. T T T T (a,+b,)

0= 14- Z 3=e •

The matrix Fo, to be calculated next, is very sparse due to the fact that certain expectation values vanish [Eq. (Cl)]. Some manipulating with the ordering of its rows and columns brings Fo into a block form. The equations of motion for the parameters aOk can then be written

[-1 -1 qo (i[q,H» ] m [(I~~])] -1

= -- , Xo (i[x,H])

- 1 '0 (i[y,H»

[4(~') 0 - (L) ][0°] [(I[X' - P,.H])] 4(yZ) (L ) ho = (i[yZ - qZ,H]) ,

(L) - (L) - «XZ) + (yZ» go (i[xq + py,H])

[-t 0 0 (L) ][ ~] [(I[ YQ.H])]

-2(rL 0 - (L ) io (i[XP,H ] )

0 (XZ) _ (yZ) (XZ) + (yZ) Po = (i[xy,H ]) ,

(L) - (L) (XZ) - (T) - «XZ) + (yZ» ao (i[pq,H ])

where (A ) = (cpiA icp), etc. The matrices in Eq. (C3) can be inverted analytically:

(L )Z _ (XZ) (y2) _ (yZ)Z (L )2 (L) (y2)

4 4 (Fab»-' = (L )2 (L )2 _ (x2) (y2) _ (X2)2 - (L ) (x2)

4 4 4(x2) (y2) - (L )(T) (L) (XZ)

_ (L)2 _ (L )2 + 4(x2) (yZ) + 4(X2)2 _ (L)2 _ (L )2 + 4(x2) (y2) + 4(T)Z

(FaC» -I = _ 2(L ) (x2) (x2) + (y2)

(x2) _ (y2)

2(L) (x2)

- 2(L ) (x2)

2(L )(T)

2(L )Z(y2) (x2) + (T)

(x2) - (T)

_ 2(L )(y2)

2(L) (x2)

_ 2(L )(y2)

2( (L )2 _ 2(x2) (y2» (XZ) + (y2) (x2) _ (y2)

_ 4(x2) (y2) (x2) + (T)

(x2) _ (y2)

A

2

_ 4(XZ)(y2) 4(X2)(y2)

I

'A,

with (FTF + El)-IFT, where F stands for Fab) or FaC

).

(C3a)

(C3b)

(C3c)

(C4a)

(C4b)

A = [( (L )2 _ 4(x2) (y2» «x2) + (yZ»)) -I .

One can show that A averages only if cp is an eigenfunction of L. More important, the row which determines Po becomes singular if (XZ) = (T). To avoid numerical difficulties one should replace 1/( (x2) - (y2» by «x2) - (T»I [( (x2 ) - (T»2 + E) with some convenient chosen value for E. More general, one should replace's F- ' by

The vector fo [right-hand side in Eqs. (C3a)-(C3c)) depends upon the Hamiltonian under consideration. In the following we evaluate it explicitly for the harmonic Hamilto­nian [Eq. (3.8)], and finally derive the equations of motion for the parameters. Since the harmonic Hamiltonian H is a member of the harmonic Lie algebra one finds that the rotat­ed Hamiltonian H is also in the Lie algebra and hence can be expanded as

J. Chem. Phys., Vol. 90, No. 10, 15 May 1989

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.113.111.210 On: Tue, 23 Dec 2014 01:09:30

5576 J. Kucar and H. -D. Meyer: Exact wave packet propagation

II = Co + Cxx + CyY + CpP + Cqq + Cx,x2 + Cy'y2

+ CxyXY + Cp'p2 + Cq' q2 + Cpqpq + CxpXP + Cyq YQ + Cxqxq + CpyPY,

where the following abbreviations are used:

I (P; q;) Co=- -+- + V(x"y,) ,

2 m l m2

Cx = T 0- I [~(2J, T4 - g, T3) - ~(2h, T3 - g, T4) + Vx T4 - Vy T3] m l m 2

Cy = TO-I[ -~(2J,T2 -g,TI) +~(2h,TI -g,T2) - Vx T2 + VyTI] , m l m2

Cp =~TI +~T2' m l m2

Cq =~T3+~T4' m l m2

Cx' = J.. T 0- 2 (_1_(21, T4 - g, T3)2 + _I_(2h, T3 - g, T4)2 + Vxx T~ + Vyy T~ - 2 Vxy T3T4) , 2 m l m2

Cy' = J.. T 0- 2(_I_(2J,T2 - g,TI)2 + _I_(2h,TI - g,T2)2 + Vxxn + Vyyn - 2Vxy T IT2) , 2 m l m2

Cxy = - To-2(_I_(2J,T4 -g,T3)(2J,T2 -g,TI) m l

+ _I_(2h, TI - g, T2)(2h, T3 - g, T4) + Vxx T2T4 + Vyy TI T3 - VXY (TI T4 + T2T3») , m2

Cp ' =J..(Ti + n), 2 m l m2

Cq' =J..(n + n), 2 m l m2

C - TI T3 T2T4 pq---+--' m l m2

Cxp = T 0- I (.IJ..(2J, T4 - g, T3) - .!i.(2h, T3 - g, T4») , m l m2

Cyq = - T O-I(I1..(2J,T2 - g,TI) - ~(2h,TI - g,T2») '

m l m2

Cxq = T O-I(I1..(2J,T4 -g,T3) -~(2h,T3-g,T4»)'

m l m 2

Cpy = - T <> I (.IJ..(2J, T2 - g, T I) - .!i.(2h, TI - g, T2») . m l m 2

The evaluation of the vector fo is straightforward and yields

(C6a)

(C5)

MUltiplying Fo- I which fo gives the equations for the aOk

parameters:

Po = - Cx , qo = - Cy , Xo = CP ' Yo = Cq ,

flo = - CXP ' bo = - Cyq ,

J. Chem. Phys., Vol. 90, No.1 0, 15 May 1989

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.113.111.210 On: Tue, 23 Dec 2014 01:09:30

J. Kucar and H. -D. Meyer: Exact wave packet propagation 5577

ho = Cq, - Cy', io = Cp' - Cx' ,

go = Cpq - Cxy , ao = -! (Cxq + Cpy ) ,

. (L) Po =!( Cpy - Cxq ) + (Cp' - Cq') (x2) _ (y2)

(C7)

These parameters are to be transformed according to Eq. (2.17b) thus completing the evaluation of equations of mo­tion for the parameters. The algebra involved in multiplying the matrix (DT) -1 with the vector containing aok param­eters is straightforward.

It is convenient to drop the term containing the expecta­tion value (L) in Eq. (C7). We recall that the equation of motion for Po was just chosen for convenience. A different choice for the value of Po does not increase (tPIHoltP). It is easy to show that a redefinition of Po does not influence the other equations (C7). Dropping the term containing (L) one arrives at equations of motion which first are simpler than before and second which depend only on the Hamilto­nian and not on the wave function 1,6. When we made the comparison with the LHCK method (see Sec. III) we have dropped all terms containing (L) in the final equations. Note, however, that this simplification is only possible for harmonic Hamiltonians.

IE. J. Heller, J. Chern. Phys. 62, 1544 (1975). 2E. J. Heller, J. Chern. Phys. 64, 63 (1976). 3D. Huber, and E. J. Heller, J. Chern. Phys. 87, 5302 (1987). 4J. Grad, Y. J. Yan, and S. Mukamel, Chern. Phys. Lett. 134, 291 (1987). 'R. T. Skodje and D. G. Truhlar, J. Chern. Phys. 80, 3123 (1984). 6D. Kumamoto and R. Silbey, J. Chern. Phys. 75, 5164 (1981). 7R. B. Gerber, M. A. Ratner, and V. Buch, Chern. Phys. Lett. 91, 173 ( 1982).

8K. C. Kulander, and E. J. Heller, J. Chern. Phys. 69, 2439 (1978). 9J. Recarnier, D. A. Micha, and B. Gazdy, Chern. Phys. Lett. 119, 383 (1985).

101. Benjamin, J. Chern. Phys. 85, 5611 (1986). 11R. Gilmore and J.-M. Yuan, J. Chern. Phys. 86, 130 (1987). 12A. B. Weglein, Phys. Rev. A 17,1810 (1978). 13R. B. Gerber, V. Buch, and M. A. Ratner, J. Chern. Phys. 77, 3022

(1982). 14G. Jolicard and G. D. Billing, Chern. Phys. 104, 357 (1986). I'S._I. Sawada, R. Heather, B. Jackson, and H. Metiu, J. Chern. Phys. 83,

3009 (1985). 16R. Heather and H. Metiu, J. Chern. Phys. 34, 3250 (1986). 17J. Kucar, H.-D. Meyer, and L.-S. Cederbaurn, Chern. Phys. Lett. 140, 525

(1987). 18H._D. Meyer, J. Kucar, and L. S. Cederbaurn, J. Math. Phys. 29, 1417

( 1988). 19S. Y. Lee and E. J. Heller, J. Chern. Phys. 76, 3035 (1982). 20S. Y. Lee, Chern. Phys. 108,451 (1986). 21R. D. Coalson and M. Karplus, Chern. Phys. Lett. 90, 301 (1982). 22E. A. McCullough and R. E. Wyatt, J. Chern. Phys. 54, 3578 (1971). 23A. Askar and A. S. Cakrnak, J. Chern. Phys. 68, 2794 (1978). 24R. C. Mowrey and D. J. Kouri, J. Chern. Phys. 84, 6466 (1986). "Y. Sun, R. C. Mowrey, and D. J. Kouri, J. Chern. Phys. 87, 339 (1987). 26M. D. Feit, J. A. Fleck, and A. Steiger, J. Cornput. Phys. 47, 412 (1982). 27D. Kosloffand R. Kosloff, J. Cornput. Phys. 52, 35 (1983). 28H. Tal-Ezer and R. Kosloff, J. Chern. Phys. 81, 3967 (1984). 29R. Kosloffand H. Tal-Ezer, Chern. Phys. Lett. 127, 223 (1986). 30R. B. Gerber, R. Kosloff, and M. Berman, Cornput. Phys. Rep. 5, 59

(1986). 31R. Kosloff, J. Phys. Chern. 92, 2087 (1988). 32H._D. Meyer, Chern. Phys. 61, 365 (1981). 33R. Gilmore, Lie Groups, Lie Algebras and some a/their Applications (Wi-

ley, New York, 1974). 34J. Wei and E. Norman, J. Math. Phys. 4,575 (1963). 35J. Wei and E. Norman, Proc. Am. Math. Soc. 15, 327 (1964). 36C. W. Gear, Numerical Initial Value Problems in Ordinary Differential

Equations (Prentice-Hall, Englewood Cliffs, 1971). 37y. Alhassid and R. D. Levine, Phys. Rev. A 18, 89 (1978). 38R. D. Levine, in New Horizons a/Quantum Chemistry, edited by P. O.

Lowdin and B. Pullman (Reidel, Dordrecht, 1982), p. 135.

J. Chern. Phys., Vol. 90, No. 10,15 May 1989

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.113.111.210 On: Tue, 23 Dec 2014 01:09:30