3212
J. Phys. Chem. 199498, 3212-3219
Wigner Method Dynamics in the Interaction Picture Klaus B. Merller, Jens Peder Dahl, and Niels E. Henriksen' Technical University of Denmark, Chemistry Department B, Chemical Physics, D T H 301, DK-2800, Denmark Received: August 24, 1993; In Final Form: October 25, 1993"
The possibility of introducing a n interaction picture in the semiclassical Wigner method is investigated. This is done with an interaction picture description of the density operator dynamics as starting point. W e show that the dynamics of the density operator dynamics as starting point. W e show that the dynamics of the interaction picture Wigner function is solved by running a swarm of trajectories in the classical interaction picture introduced previously in the literature. Solving the Wigner method dynamics of collision processes in the interaction picture ensures that the calculated transition probabilities are unambiguous even when the asymptotic potentials are anharmonic. An application of the interaction picture Wigner method to a Morse oscillator interacting with a laser field is presented. The calculated transition probabilities are in good agreement with results obtained by a numerical integration of the Schrodinger equation.
Introduction In the treatment of time-dependent molecular phenomena (collisions, photodissociation, etc.) the time-dependent Schrijdinger equation is an obvious starting point, and several numerical methodsof solving thisequationexist.' But, most ofthesemethods are very time consuming for systems of more than a few dimensions, and one often has to treat thedynamicsof a molecular process approximately. In this paper we focus on thesemiclassical Wigner method introduced by Heller.z This is a method based on the Weyl-Wigner phase space representation of quantum mechanics.34 In this representation the states are represented by distribution functions called Wigner functions and operators are represented by classical-like phase space functions. In the Wigner method thecorrect equation ofmotion is replaced by the classical Liouville equation, which ensures that the dynamics of the system can be solved by running a swarm of classical trajectories. The method is exact when the potential under consideration is at most quadratic. Otherwise, severe errors can occur. Previously, the ability of the Wigner method to reproduce the dynamics of a free one dimensional Morseoscillator have been These studies showed a fatal degradation of the distribution function for both stationary and nonstationary states even in the lowest quantum states of an oscillator with many bound states. In the study of state-to-state transition probabilities in collision and photodissociation processes, where the asymptotic potentials are harmonic, the Wigner method has been used with success due to the fact that the potential is only anharmonic under the (shorttime) interaction between subsystems (see, for example, the references in ref 9). But in systems where the asymptotic potentials are anharmonic it is impossible to obtain unique values for the transition probabilities, since the distribution function continues to degrade in the asymptotic regi~n.~JOSince real systems usually contain anharmonic parts in their asymptotic potentials this puts a strong limit to the usefulness of the Wigner method. Similar errors also show up in the semiclassical Gaussian wavepacket method,lIJz when the asymptotic potential is anharmonic.l3 With this method several authors have tried to overcome the problem by solving the dynamics in an interaction picture, where the asymptotic motion is, in a way, subtracted This is done either by a direct implementation of the usual quantum mechanical interaction picture or by the use of a classical interaction picture. The first approach involves
successive propagations of the state vector within the Gaussian wavepacket method using the full Hamiltonian or a zeroth-order channel Hamiltonian, respectively. In the second approach introduced by Skodje'4 a canonical transformation of the classical coordinates into a set of classical interaction coordinates is performed, and the interaction Hamiltonian to be used in the Gaussian wavepacket method is found by applying the quantization procedure for generalized canonical coordinates proposed by Heller.lz In this paper we examine the possibilities of improving the Wigner method by propagating the Wigner function in an interaction picture. It is organized in the following way: We begin by recalling the main features of the Weyl-Wigner representation of quantum mechanics. Then we give a short description of the quantum and the classical interaction pictures. We discuss subsequently the Wigner method in the interaction picture and show that the classical interaction picture, in a natural way, comes into play. Finally we present a numerical application and discuss the results. For simplicity all derivations are kept in one dimension, the extension to several dimensions being straightforward. To avoid confusion, quantum mechanical operators are written in capital letters (with the exception of the position and momentum operators) supplied with a hat, the corresponding phase space functions denoted by the same letter without the hat and classical functions are written in script.
Theory Weyl-Wiper Representation. In the Weyl-Wigner representation a quantum mechanical system characterized by the Schrijdinger density operator p ( t ) is represented by the Wigner distribution function3
r(q,p,t) = (l/rwJdy(q-Yl;(olq
+ ~ ) e ~ ' p y / (1) ~
This function is not a phase space probability density in the usual senses, but acts as a phase space probability density in the sense that the expectation value of a physical observable, denoted by (A), is found as
(-4 Tr(&$(t))
= JJdq
dpA(q,p,O r(q,p,t)
(2)
where the quantum mechanical operator 2 and the phase space function A are related by the Weyl correspondence.l* This correspondence gives the following relation between A and A:5
Abstract published in Advance A C S Abstracts, March 15, 1994.
0022-3654/94/2098-3212~04.50~0 0 1994 American Chemical Society
Wigner Method Dynamics
The Journal of Physical Chemistry, Vol. 98, No. 13, 1994 3273
where &.79P,$,B)
whereas in classical mechanics a canonical transformation of the phase space coordinates is performed. Quantum mechanics: In this section we implement the interaction picture in the density operator formulation of quantum mechanics, which is straightforward to do from the interaction picture in the state vector formalism.*g We recall some of the main features in a notation that mimics the description of the classical interaction picture by MarcusZOJ in order to express the differences and similarities between the two pictures as clearly as possible. The system Hamiltonian is divided into two terms
=
2h or put the other way around6
Equation 5 can be used to show that the phase space function corrEqnding to a product of two quantum mechanical operators, = A-B, is6
e
The subscripts 1 or 2 on a differential operator indicate that the operator acts on the first or the second function, respectively, in the product A(q,p,t).B(q,p,t). Using this equation, it is seen that the phase space function, called a Moyal bracket, corresponding to a quantum mechanical commutator is given by the correspondence:
It is worth noting that in certain cases the Moyal bracket reduces to a Poisson bracket if one of the phase space functions in the product only depends on either q or p (or it is a sum of functions that only depend on either q or p ) . These cases are (i) both A and B are at most harmonic in either q or p or (ii) either A or B is at most harmonic in both q and p . For later use we also note that a comparison of eq 5 and eq 1 shows that the phase space function corresponding to the projection operator P, = IJ/n)(rl.nl, which is a pure state density operator, is exactly the Wigner function for the state divided by 2uh. Thus, quantum mechanical transition probabilities, defined as the expectation value of pn,can be calculated from
(P,)= 2 r h J J d q dPr,(q,P) ~ ( 4 , P , t )
(8)
The time evolution of the Schriidinger density operator is controlled by the quantum Liouville equation:19
(9) where fI is the Hamiltonian. The equation of motion for the Wigner function can now be found by applying eq I and the definition, eq 1. Thus
where H(q,p,t) is found from the quantum mechanical Hamiltonian by the use of eq 5. InteractionPicture. When parts of the Hamiltonian governing the dynamics of the system under consideration describe a trivial and uninteresting motion it may be useful to handle the dynamics of the system in an interaction picture. In this picture the influence on the dynamics from these parts of the Hamiltonian is subtracted out in the sense that the quantities describing the system are constants of the motion for these parts of the Hamiltonian. The basic physical properties of the system should, of course, remain unaffected. In quantum mechanics the interaction picture is obtained by a unitary transformation of the Schriidinger picture,
where fP is the time independent zeroth-order Hamiltonian and kl describes an interaction with another system (or, in two or more dimensions, between subsystems). This term may be time dependent. A state vector in the interaction picture is defined through a unitary transformation of the Schriidinger statevector, ($(t))r = exp(itH0/h)(J/(t)).22 This unitary transformation can be given a simple inierpretation recalling that IJ/(t)) = U(H, t,to)l+(to)), where U(H,t,to)is the unitary Schradinger evolution operator. In the case of a time indep:ndent Hamiltonian the evolution operator equals exp(-i(t - to)H/ h ) ,and we see that the interaction state vector is obtained from the Schriidinger state vector ,by a propagation of the latter from the time t to time zero using Ho. Using the unitary transformation of the state vector we get the following definition of the interaction density operator:
where we have used the Baker-Hausdorff identityzzin the second line. This transformation can be given the same interpretation as the transformation between states if we introduce a time evolution operator in the density operator formalism. This evolution operator, S(H,t,to), is called the quantum Liouville propagator, and from the time evolution of the state vector we get the following definition:
&t) = v(fi;i;t,t,) ;(to)
v+(fi;t,t,)
S(ii;t,to);(to)
(13)
In the simple case, where k is time indepecdent, the definition shows that S(&;;t,t,)equals exp(-(t - to)[ ,Hl/ih). In general, S(k;t,to)can be found from the equation
zs(fi;r,to) a = x[fi, 1 ]s(fi;t,t,)
(14)
which is obtained by insertion of eq 13 in the quantum Liouville equation, eq 9, andomitting thedensityoperatorp(t0). A solution to an equation of this type is given by Magnus.23 With these considerations eq 12 can be written &(t) = ~ @ ; O , t )
&t)
(15)
which shows that the interaction density operator can be obtained from the Schradinger density operator by pefforming a propagation from the time t to time zero using Ho. Furthermore, insertion of eq 13 in eq 12 gives &t) = S ( p ; O , t )s(fi;t,t,) S(p;to,O);,(to)
(16)
which means that the time evolution of the interaction density
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The Journal of Physical Chemistry, Vol. 98, No. 13, 1994
operator can be interpreted as successive time propagations using the quantum Liouville propagator in the following way. Fi_rst one performs a propagagon from time zero to time to using HO, then from to to t using H and finally from t back to zero using #. We see that if to = 0, these propagations can be interpreted as a forward propagation from zero tot using the full Hamiltonian followed by a backward propagation using i?P. The equation of motion for the interaction density operator is easily obtained from the equation of motion for the interaction picture state vector.19 This is
where
is given by the unitary transformation
= S(P;O,t)8'
Mprller et al. old coordinates by a propagation of the latter from the time t to time zero using 7 f O :
Inverting these relations we see that (q1,pI) play the role as initial coordinates for (q,p)when these are propagated using go.Thus, we can get an analytical expression for the relation between the coordinates in the new and the old phase space if the zeroth-order problem can be solved analytically. Comparison of eqs 21 and 23 shows that the dynamical variables qdt) and p d t ) are related to their initial values through
(18)
We have used the definition of the quantum Liouville propagator, eq 13, in the last line. For later reference we expand in t using the Baker-Hausdorff identity:
Equation 17 can be solved exact only if this transformation can be performed analytically, which is impossible in most cases. Classical mechanics: The classical interaction picture has been considered thoroughly by Marcus using Hamilton-Jacobi theory,*oJl and it was reintroduced by Skodje17in a more operational way based on the idea of successive time propagations after different Hamiltonians. In this section we connect the two points of view, and to avoid misunderstandings we adopt the notation used by Marcus which is expounded in ref 21, Appendix B. We consider a system in the phase space with the coordinates q, p , whose motion is generated by the Hamiltonian, %. The classical interaction picture is introduced as a canonical transformation of this phase space into a new phase space with the coordinates 41, PI. Again, the Hamiltonian is divided in a zerothand first-order term, % = go+ %I, where %O is considered time independent as in the quantum mechanical case. The transformation between the coordinates in the two phase spaces is given by
The expression on the right-hand side of eqs 20 should be understood as time-dependent phase space functions (the first parenthesis) evaluated a t the point (q,p). This transformation corresponds to a time evolution as we shall see below. The time evolution of the dynamical variables q ( t ) andp(t) in the original phase space is governed by*'
A general expression for T*(%;t,to) is given in refs 20 and 21. If % is time independent, T*(%;t,to) equals exp((t - to)( ,%)), and we see that the interaction coordinates are obtained from the
This equation can be read in the following way: qdt) and p d t ) are the coordinates of the point which is obtained by propagating the point (ql(to),p,(to))from time zero to time to using So,then from to to t using %, and finally from t to zero using Po.Hence, we see that the evolution of the dynamical variables in the classical interaction picture, as in the quantum mechanical case, can be interpreted as successive propagations after different Hamiltonians. The dynamical variables in the classical interaction picture satisfy a set of Hamilton's eq~ations:'~*~O
with an interaction Hamiltonian %;(qppr) that equals 7f1(q,p), whereq,pareexpressedin termsofqI,prthrougheq 23. In practice, this means that an analytical expression for %;(qr,p,), which is necessary for the direct use of eq 25, can only be found provided that the classical zeroth-order problem can be solved analytically. Inversion of eq 23 and insertion in %1 gives a transformation between 7fl and %; analogous to the quantum mechanical, eqs 18 and 19:
This relation illustrates that since the Poisson bracket and the commutator divided by ih both are realizations of a Lie bracket, the quantum mechanical and the classical interaction Hamiltonian, apart from ordering, share the same functional dependence on the coordinate and momentum. Wigner Method. In this section we suggest an implementation of the Wigner method based on the quantum mechanical interaction picture. We begin by recalling the Wigner method in the Schradinger picture and show how the dynamics of the Wigner method distribution function can be described using a time evolution operator. Schrbdinger picture: The scheme suggested by Heller2 is to set up the initial density operator for the problem of interest, transform it to the corresponding Wigner distribution function and then propagate this function according to the equation
The Journal of Physical Chemistry, Vol. 98, No. 13, 1994 3275
Wigner Method Dynamics
with the initial condition f(q,p,to) = I'(q,p,to). This equation arises from eq 10 if the expansion of the Moyal bracket is cut off after the first term. We have supplied the distribution function in thisequationwitha tilde toindicatethat,ingeneral,this function differs from the correct Wigner distribution function at times different from the initial time. Only when the phase space Hamiltonian is separated in a sum of functions of q or p, which are at most harmonic, eq 27 gives the exact Wigner function at any time. It is assumed that in the Schradinger picture the quantum *mechanical Hamiltonians have the usual form, fZ = jP/2m V(Q,r),which implies that thephasespaceHamiltonians are simply the classical ones for the corresponding classical systems. Thus, eq 27 is approximative whenever the potential is anharmonic. Equation 27 is recognized as the classical Liouville equation, and Heller2 states the following procedure to obtain f(q,p,t). Insertion shows that the delta function
which shows that the value of the distribution function in a given phase space point ( q p ) at the time t is found by taking the value of the initial distribution in the phase space point that results from a propagation of the point ( q p ) from the time t to time to according to Hamilton's Yuations. This is, of course, in accordance with the relation I'(q,p,t) = I'(qo,po,to), where (q,p) and (qo,po)are related through Hamilton's equations as stated above. Interactionpicture: In the Wigner equivalent of theinteraction picture, the equation of motion, obtained from eq 17 by applying the correspondence, eq 7 , and the definition, eq 1 , should be
+
f(Q,P,t) =
- q(t))S(P - P ( 0 )
(28)
where q(t) and p ( t ) are solutions to Hamilton's equations, is a solution to the classical Liouville equation. A linear combination of these delta functions with constant coefficients will also be a solution to the classical Liouville equation. Therefore,the solution to eq 27 can be given in the form (which was already given by Moya14):
where q(t) and p ( t ) are solutions to Hamilton's equations with the initial conditions q(t0) = qo and p(to) = PO:
Thus, a solution to eq 27 can be obtained from the initial Wigner function by running a swarm of classical trajecrories from to to t weighted with the initial weight such that I'(q(t),p(f),t)= f(qo,pO,to). This procedure gives the value of the distribution function at points unknown from the beginning. Alternatively, we could find a solution to eq 27 in terms of an evolution operator, S(%;t,to), that connects the value of the distribution function at different times in the same phase space point:
f(q,p,t) = w;t,to)f(q,P,to)
= (s(%;t,ro)f(t,))(q,p)
(31)
This operator is called the classical Liouville propagator. Inserting eq 31 in the classical Liouville equation, eq 27, we get
a
--S(%;t,to)
at
= {%, )S(%;t,to)
Using this and the automophism of
T*(%;t,,t)
Foll_owingthe program given by Heller? the equation of motion for r1(q,p,t)should read
with the initial condition f,(q,p,to) = I'Aq,p,to). In the Schradinger picture the phase space Hamiltonians are trivial to construct since they simply equal the classical ones. The interaction picture phase space Hamiltonians are more complicated or even impossible to construct. For a construction based on the direct application of the Weyl correspondence, eq 5 , we need an analytical expression for the quantum mechanical interaction Hamiltonian. But, as mentioned after eq 19, such an expression is not obtainable in general, and we must seek other ways to find the phase space interaction Hamiltonian. To do that, we use,the formal exnansion of eq 19, to point out some general aspects in the construction of the correct phase space interaction Hamiltonian. In principle, the phase space interaction Hamiltonian can be found by successive applications of the correspondence,eq 7 , to the terms in the expansion of fi:. If we still assume that the full quantum mechanical Hamgtonian in the Schriidinger picture has the form fZ = j 2 / 2 m V(Q,t),the phase space Hamiltonians corresponding to fio and fP are the classical Hamiltonians 7;fO and 7f1. These, of course, still separate in terms that depend on either q or p and are at most harmonic in the latter. Then, the phase space correspondence of the first term in fZ: is simply H1.The phase space correspondence of the second term, the commutator divided by i h between fio and HI, becomes a Poisson bracket between 7foand 7f1,because only the first term in the expansion of the Moyal bracket is nonvanishing:
fZi,
+
(32)
Thisequation is identical to theequation satisfiedby T*-l(%;t,tO), which is seen by a differentiation of T* P-1 = 1 and introduction of eq 22. Thus S(%;t,to)= rc-'(%;t,to) =
where H: is the phase space function corresponding to hi,and the interaction picture Wigner function is given by
(33)
we can write eq 3 1 as
Consequently, we find the phase space correspondence of the third term in eq 19 by applying the Moyal bracket operator to a product of a Poisson bracket between W a n d 7f1and the function 7 f O . In general, the Poisson bracket between two Hamiltonians having the usual form may contain mixed terms, but it is at most linear in the momentum and, since 7 f O is harmonic in the momentum, the phase space correspondence to the third term in eq 19, too, is a Poisson bracket:
Merller et al.
3276 The Journal of Physical Chemistry, Vol. 98, No. 13, 1994
Unfortunately, this simple pattern does not continue in general. The Poisson brackets analogous to the term on the right-hand side of eq 39 become more and more complicated as we consider the higher terms, which means that unless the potential in 7fois at most harmonic, the expansion of the Moyal bracket does not terminate after the first term. These considerations show that the correct phase space interaction Hamiltonian equals the classical interaction Hamiltonian, eq 26, plus terms of order h2, h4, ...:
We began the analysis by saying that in most cases we cannot construct the phase space interaction Hamiltonian from the quantum mechanicalone simply because the latter is inaccessible. Equation 40 shows that we may try, as an approximation, to use the classical interaction Hamiltonian (which can be found as described in the previous section) in the equation of motion instead of the correct phase space interaction Hamiltonian. This procedure becomes exa:t in the classical limit, h 0, and whenever the potential JP is harmonic. Besides solving the problem of construction the correct phase space interaction Hamiltonian this approximation has a close resemblanceto the approximation used in the first place to obtain the equation of motion in the Wigner method. If we compare eq 31:
-
h , p , t ) = S(7f;t,to)f(q,p,to)
(41)
to the time evolution of the Schrbdinger picture density operator, eq 13: i4t)
= S(fi;t,t,);(to)
(42)
we see that in the Wigner method the phase space operator corresponding to the quantum Liouville propagator is approximated with the classical Liouville propagator. Earlier we saw that both the classical and the quantum mechanical interaction pictures are defined as a transformation brought about by a Liouville propagator. A comparison of the quantum mechanical and the classical interaction Hamiltonian, eqs 18 and 26, and the use of eq 33 shows that the approximate “translation” of the quantum Liouville propagator into the classical Liouville propagator causesa translation of the quantum mechanicalinteraction Hamiltonian into the classical interaction Hamiltonian:
= s(r;P;o,t)R’
-, S(%0;0,t)7f1 = 7f:
With eqs 24 and 34 in mind we see that the right-hand side of eq 45 equals r,(qo,po,to),where (q0,pO) is obtained from (q,p) by propagating the latter point from time t to time to in the classical interaction picture, Thus, we find that introducingthe interaction picture in the Wigner method by the useof successive propagations supports the idea of using the classical interaction picture in the interaction picture implementation of the Wigner method. The classical interaction picture can be implemented either by constructing the classical interaction Hamiltonian and integrate the interaction picture equations of motion or by performing successive propagations of the initial distribution function using different Hamiltonians. The explicit method based on theconstruction of the interaction Hamiltonian may be the more elegant one and it can give more time efficient propagations. But, it suffers from the fact that the classical zeroth-order problem must be solved analytically, which restricts the class of problems where it can be used. Furthermore, it may be a little cumbersome to construct the derivatives of the interaction Hamiltonian to be used in the equations of motion. The other method can be used for any 7 f 0 , and it requests no analytical work aside from differentiation of the terms in the original Hamiltonian.
A Numerical Application In this section we consider a numerical application of the interaction picture implementation of the Wigner method. For simplicity we choose a one-dimensional system, namely, a nonrotating Morse oscillator that interacts with a time-dependent (pulse-shaped) laser field, where the interaction is described semiclassically in the electric dipole appr~ximation.~~ To investigate the accuracy of propagating in the interaction picture, we use this method to calculate the transition probabilities for various excitations of a Morse oscillator induced by lasers having different field strengths and pulse lengths. Performing the propagations in the interaction picture ensures that we obtain unique results. The obtained results are compared to results obtained by exact numerical propagation of the state vector according to the time dependent Schrbdinger equation using the FFT scheme by Kosloff.1 The Hamiltonian is
(43)
Thus, the use of the classical interaction Hamiltonian in an interaction picture formulation of the Wigner method is a natural way to overcome the problems in constructing the phase space interaction Hamiltonian. Up to now we have focused on the construction of a phase space interactionHamiltonian which makes it possible to integrate the equation of motion in the interaction picture formulation of the Wigner method. Instead we could introduce the interaction picture in the Wigner method by translating eq 16 according to the considerations above. Thus, the Wigner method analog of eq 16 should read
where m is the reduced mass and ~ ( 4 is) the dipole operator of the oscillator and EOis the electric field strength, w is the (central) frequency,and a(t)is the shape of the laser field. In the treatment of the dynamics of a Morse oscillator it is convenient to introduce the variables25
where A = */ah,
wo = (2Da2/m)’/’
(48)
A specifiesthe number of bound states in the Morse oscillator and w0 is the
Using eq 33, this can be written
frequency of the corresponding harmonic oscillator. In these variables the Hamiltonian becomes
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The Journal of Physical Chemistry, Vol. 98, No. 13, 1994 3277
TABLE 1: Parameters in the Morse Potential and Dipole Operator Corres nding to HF, Based on the Parameters in Ref 27 ( 7 = lCrpl0s)
= wop + w 0 (A,h- 2 6 / A - Z e 4 / A ) -
"7
2h
2
L
