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Faraday Discuss. Chem. Soc., 1983, 75, 141153
The Correspondence Principle and Intramolecular Dynamics
BY ERIC J. HELLER
Los Alamos National Laboratory, University of California, Los Alamos, New Mexico 87544, U.S.A.
Received 10th Jaiiuary, 1983
It is the purpose of this paper to describe some recent developments in the semiclassical approximations to quantum vibrationaI dynamics. In section 1 we describe a new and very simple method for finding uniformized semicIassica1 wavefunctions. The wavefunctions are given expIicitly as contractions over Gaussian functions, with the parameters of the Gaussians chosen according to an arbitrary classical trajectory. The semiclassical functions moreover are highly suitable for use as basis functions in ab initio work. In section 2 we present new resuIts for the quantum dynamics of a highly anharmonic classically chaotic system with an infinite number of quantum bound states. The results show the utility of the spectral criterion we have been advocating as a measure of phasespace flow in molecular systems, and they show some interesting quantum effects. Also, we show that the classical and quantum dynamics of a localmode CH stretch agree extremely well as to the fraction of available phase space covered in the course of the subsequent dynamics. In quantum mechanical systems the optical spectrum gives a direct measure of this fraction.
1. A SIMPLE QUANTIZATION SCHEME FOR WAVEFUNCTIONS
We have two previous publications on finding multidimensional, uniformized semiclassical wavefunctions.'*2 The first made use of methods developed by Sorbie and Handy for finding semiclassical eigenvalues. Our idea was to.run the trajectory that semiclassically quantized a dynamical system, and to allow this trajectory to " guide " a Gaussian wavepacket (or, equivalently, a coherent state) as in our previous
The wavepacket was taken to be " frozen " about the trajectory, i.e. it did not spread, but followed the trajectory (both in position and momentum). Then, using the idea that
we replaced the exact quantum dynamics of v/,(x) by the moving frozen Gaussian (FG), used the classical action plus Maslov corrections for the phase of the FG, and used the semiclassical energy E of Sorbie and Handy in the Fourier transform. This technique worked extremely well (fig. l), but required that the quantizing trajectories be found. This is a tedious procedure at best, and made calculations in three and more dimensions seem forboding.
led us to conclude that the success of the SorbieHandy method for eigenvalues was surprising in view of the fact that it was not a topologically " proper " quantization in many cases. Instead of holding the iV topologically invariant action integrals to integerplusMaslov corrections, the Sorbie and Handy (S.H.) method held only the sum of the actions to the correct value,
Discussions with Marcus and Noid
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http://dx.doi.org/10.1039/dc9837500141http://pubs.rsc.org/en/journals/journal/DChttp://pubs.rsc.org/en/journals/journal/DC?issueid=DC1983_75_0
142 CORRESPONDENCE PRINCIPLE AND INTRAMOLECULAR KINETICS
while making approximations to the individual actions. Yet we found the S.H. method worked quite well. This led us to investigate its success further, and we came upon a general technique for using arbitrary trajectories to find energies and semi classical wavefunctions of comparable quality.2 It turned out that the S.H. method is one of a class of extrapoZation methods that work away from the true E.B.K. quantizing trajectories, but extrapolate to the E.B.K. values. This realization freed us from having to make the search for the quantized actions in the first place, but did
Fig. 1. (left) Semiclassical wavefunctions generated as a superposition of Gaussian wave packets along a classical trajectory. (right) Numerically converged ab initio wavefunction
corresponding to the semiclassical state on the left.
require us to find the value of the independent actions for the (arbitrary) trajectories used to quantize the system. We found a new way of getting these true actions by running trajectories nearby the arbitrary trajectory. This arbitrary trajectory necessarily falls somewhere in action space near E.B.K. quantized actions, and a simple linear extrapolation to the corresponding energies and wavefunctions of those states was satisfactory. As the extrapolation distance in action space increases, the accuracy naturally decreases, but several energies and wavefunctions can be obtained from a single trajectory. The method for finding the actions is, quite importantly, independent of the topology of the trajectory. This means that classical resonances of any order present no difficulties. (Chaotic trajectories are not quantizable, how ever, by this method.)
Here we report a similar but greatly simplified procedure to obtain semiclassical energies and uniformized semiclassical wavefunctions.6 The method works even in the quasiintegrable (partly chaotic) domain. To derive it we return to eqn (l), and project both sides of the equation into some state, Ip), which for the moment is arbitrary
The amplitude (plyE) will be nonzero only if a correct quantum eigenvalue E is used in eqn (2). We again use FGs to approximate ly,, call this ly;". This gives
This equation is none other than that used previously to obtain molecular spectra, using FG dynamic^.^.^ The idea is to examine the FG spectrum and use the energies
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E. J. HELLER 143
E obtained as spectral peaks as Fouriertransform energies in eqn (l), with yo and yt replaced by y;", I,$". The resulting YE", obtained as a finite superposition of Gaussians by making the time integration discrete, are equivalent to those obtained by the earlier method of DeLeon and Heller.2 The present approach is greatly simplified, however. The FG spectrum is trivial to obtain: only a single classical trajectory, overlaps of Gaussians and a Fourier transform are required. Then the same trajec tory and the same Gaussians are used with the E values obtained from the spectrum in eqn (1) to " build up " the wavefunctions. A typical spectrum and four of the wavefunctions obtained are shown in fig. 2. The arrows show the peak energy used
3 1 C B A D
10  6  2 2 6 10 El
Fig. 2. Frozen Gaussian spectrum, with arrows showing four spectral peaks whose energies were used in the Fourier transform to generate the four FG semiclassical wave
functions AD.
in the Fourier transform. All four arise from the same trajectory, but have a different nodal structure. All the larger peaks in the spectrum give a reasonable wavefunction, each of which is given in terms of the same trajectory.
In the original paper on the FG a p p r o ~ i m a t i o n , ~ ~ it was noted that the spectral peaks fell near the true energies. There are several choices for ways to refine the
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1 44 CORRESPONDENCE PRINCIPLE AND 1NTRAMOLECULAR KINETICS
phase of ly,"", and we see from eqn (1) and (3) that we can multiply yr" by el(dEE'), and shift the Fourier transform energy E and dE, without changing the wavefunction t,uEc. Thus the semiclassical energies will vary according to the convention used for the phase. One convention, however, gives energies identical to those of ref, (2) without having to know the actions. The classical frequencies of motion are needed, however. These are usually easy to obtain by Fourier transform of a classical variable. Details may be found in ref. (6).
The eigenvalues, and indeed the contractedGaussiansum wavefunctions are of poorer quality than if the true E.B.K. quantizing trajectories were used. However, the important advantages of this method are (1) its extreme ease of implementation and (2) the ability to use the semiclassical, Gaussiancontracted wavefunction as basis functions for solving the timeindependent Schrodinger equation. These functions start out very close to true eigcnfunctions, even for very anharmonic potentials. Then a very small set of these can be used to converge on the true eigenfunctions. We have just started to explore the possibilities of this " contractedclassical Gaussian basis set " idea.
Returning to the spectrum seen in fig. 2, we note it is best to pick the large peaks in the spectrum as Fourier energies in eqn (1). The physical reason for this i s easily understood, If the initial Gaussian yrc has a large component in a particular eigen state, that means the action variables of the Gaussian match those of the eigenstate, i.e. the guiding trajectory of the Gaussian looks like the eigenstate itself. In such a case the Gaussian superpositions can readily represent the eigenstates. Tho further away in action space that the eigenstate is from the guiding trajectory, the smaller the intensity of the spectral peak. To get all the eigenstates we need to move the guiding trajectory to several new spots in action space.
Finding the eigenvalues by this method is not really the point, however.
The basis functions are roughly of the quality shown in fig. 1 !
2. QUANTUM AND CLASSICAL ENERGY TRANSFER
For some time now, we have advocated a spectral intensity criterion 912 for measuring the flow of probability through the available quantum mechanical phase space. The basic ideas follow. An absorption or emission spectrum is a measurable. Any spectrum with a spread in energy corresponds to some nonstationary state, i.e. let the spectrum be
where p i are the spectral intensities, determined by
where pa is the spectral nonstationary state and II,Y,,) are the eigenstates of the system. In an electronic absorption, for example, qa would typically correspond to the ground vibrational state multiplied by the transition m~rnent .~ ' ~. Thus va is a displaced wave packet on the upper electronic potential surface. Other qa values may be produced by varying the experimental conditions, or may be produced at will by theorists trying to understand the dynamics of a particular potential surface. In any case, the distri bution ofpi values contains much information about the dynamics of va. Indeed, the P,* values may be used to calculate the fraction of available phase space covered in the course of the dynamics of q 3 , . I 2 By " available " phase space, we mean all those states or cells in phase space whose total energy E and energy dispersion AE match or are consistent with the mean energy E and spread AE of the state qa. This is the direct
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E. f. HELLER 145
analogue of classical phasespace flow. Because energy is conserved in both classical and quantum mechanics, the " available " phase is restricted by energy conservation.
A totally chaotic, or R.R.K.M. molecule, would be expected to cover 100% of its available phase space; a quasiperiodic molecule much less. However, the amount of phase space covered depends not only on the intrinsic dynamics but also on the initial state ya. Some states qa might already be distributed through much of phase space (like the metal filaments in an oldstyle Aash bulb) while other states pa might be more compact. Clearly, a filamentary pa is, in general, expected to cuver more phase space than the compact one, given the same dynamics. So two factors emerge as important: (1) the intrinsic dynamics and (2) the states qa used to test the dynamics (in the case of theorists) or the spectroscopic states va (avaiIable experimentally).
The key to understanding the importance of spectral intensities in elucidating energy transfer lies in the two formulae
and
where P(alb) = P(bla) is the timeaveraged probability of starting in the state qa and being found later in the state q b , i.e.
Thus by measuring a spectrum (to high resolution) we can say how much time qa spends in the vicinity of its own birthplace [i.e. P(ala), eqn (6)], or we can say where in phase space it goes [P(alb) for all b].
Note that for a = b in eqn (8), the integrand is the often studied autocorrelation function P ( t ) = ](~l,lq,(,)>l*.~~.'~,~~ Contrary to the impression which seems to exist. the fundamental quantity is not the functional form ofP(t), but rather its average area:
There is a very simple interpretation for P(a\a). Suppose qa is an eigenstate. Then P(a]a) = 1. Suppose pa is an equalamplitude sum of No eigenstates. Then
Suppose Pb is one of the eigenstates composing pa. Then also P(alb) = I/N,,.
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146 CORRESPONDENCE PRINCIPLE AND INTRAMOLECULAR KINETICS
Suppose qc is in a normalized state constructed out of the No vn, but otherwise arbitrary. Then
P(alc) = l /No. (12) Thus qa spends (I/N,)th of its time in the state qa, or (l/N,)th of its time in any state pb or pc, which is made up of one or more of the No states that have intensity in the spectrum of QI,. The spectrum is
where w,, = EJh. It is as if pa has No cells to visit in phase space, and it spends equal time in each cell. It does not matter how we divide up the cells, as long as each cell has unit volume and lies within the space of No states. The point of eqn (10)(13) is that
P(ala) 5 (no. of phase space cells visited by ya)I = 2 (p3' n
In general we have learned to deal with nonequal amplit~des,~ '~ but the interpret ation, eqn (14), remains the same. Eqn (14) is paramount to the realization that
" 7.0  3.5 0.0 3.5 7.0 U
Fig. 3. The anharmonic potential discussed in the text. Contours range from E = 80.0 to E = 25.0 in increments of 5.0,
measurable spectra contain information about energy flow in molecules. Note that assignments of the spectra are not necessary: We take a spectrum, normalize the line intensities pi so that
and then use eqn (14) to determine the number of phasespace cells visited by y,(t). This number becomes all the more significant if we know how many states N exist
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E. J. HELLER 147
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within a range AE of the mean energy E of the state qa (w...