The dynamical basis setMario Blanco and Eric J. Heller Citation: The Journal of Chemical Physics 83, 1149 (1985); doi: 10.1063/1.449478 View online: http://dx.doi.org/10.1063/1.449478 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/83/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in An “optimal” spawning algorithm for adaptive basis set expansion in nonadiabatic dynamics J. Chem. Phys. 130, 134113 (2009); 10.1063/1.3103930 Basis set study of classical rotor lattice dynamics J. Chem. Phys. 120, 5695 (2004); 10.1063/1.1649735 On the optimization of Gaussian basis sets J. Chem. Phys. 118, 1101 (2003); 10.1063/1.1516801 Quantum dynamics via mobile basis sets: The Dirac variational principle J. Chem. Phys. 96, 4266 (1992); 10.1063/1.462820 Multidimensional quantum eigenstates from the semiclassical dynamical basis set J. Chem. Phys. 87, 6592 (1987); 10.1063/1.453444

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:

155.33.120.209 On: Fri, 21 Nov 2014 20:13:08

The dynamical basis seta) Mario Blancob)

Department 0/ Chemistry and Biochemistry, University 0/ California, Los Angeles, California 90024

Eric J. HelierC)

Los Alamos National Laboratories, Los Alamos, New Mexico 87545

(Received 26 September 1984; accepted 24 January 1985)

A new Cartesian basis set is defined that is suitable for the representation of molecular vibration­rotation bound states. The Cartesian basis functions are superpositions of semiclassical states generated through the use of classical trajectories that conform to the intrinsic dynamics of the molecule. Although semiclassical input is employed, the method becomes ab initio through the standard matrix diagonalization variational method. Special attention is given to classical­quantum correspondences for angular momentum. In particular, it is shown that the use of semiclassical information preferentially leads to angular momentum eigenstates with magnetic quantum number 1M I equal to the total angular momentum J. The present method offers a reliable technique for representing highly excited vibrational-rotational states where perturbation techniques are no longer applicable.

I. INTRODUCTION

The Born-Oppenheimer approximation assumes that fast electrons adjust adiabatically to the sluggish nuclei. Sim­ilarly, the standard quantum theory of molecular motion assumes that vibrations adjust adiabatically to the rotations of the molecule. For this reason trial functions for the matrix representation of the Hamiltonian are commonly chosen as products of vibrational wave functions and symmetric-top eigenfunctions I

(1.1)

This choice of basis functions makes use of, and is con­sistent with, the rearrangement of the nuclear Hamiltonian into vibrational and rotational contributions. The vibration­al-rotational (VR) rearrangement facilitates the use of per­turbation techniques by allowing direct inspection of the rel­ative magnitude of individual terms in the vibration-rotation Hamiltonian.

Cast within the Eckart-frame formalism, the most suc­cinct form of the VR-Hamiltonian was obtained by Watson.2

Watson simplified the Darling-Dennison VR-Hamiltonian for a nonlinear N-atom molecule, through the use of com­mutation relations and sum rules of an effective reciprocal inertia tensor J.lap, to the following form:

1 3N-6 1 K=2" L p~ +2" L (lla - 1Ta)J.lap(lIp -1Tp)

k a,p

fi2 -8 ~J.laa + V(QI,Q2,···,q3N-6)· (1.2)

(Reference 2 gives a full explanation of each term in Wat­son's Hamiltonian.) Briefly, the first term is a sum over the diagonal kinetic energy contributions of the vibrating nuclei, Pk being the conjugate momenta of the Qk normal coordi-

al supported by the National Science Foundation and The Los Alamos Na­tional Laboratories.

bj Present address: Rohm and Haas Research Laboratories, 727 Norristown Road, Spring House, PA 19477.

cj Present address: Chemistry Department, University of Washington, Seat­tle, WA 98195.

nate. The second sum (a, {3 = x, y,z) takes into account the angular momentum and vibrational angular momentum contributions as well as the Coriolis coupling. The third sum, known as the Watson term has no classical analog3 and acts as a mass-dependent contribution to the potential ener­gy V(q). V(q) contains an isotopically invariant part, Veff

which is given by the eigenvalue of the electronic Hamilton­ian calculated as a function of the nuclear coordinates q, as well as a comparatively small term dependent on the nuclear masses that arises as a correction to the adiabatic approxi­mation. Pioneering work to calculate solutions of Schroe­dinger's equation with Watson's form of the VR Hamilton­ian can be found in the work of Whitehead and Handy4(a)

and also Carney and Kern.4(b)

The main advantage of a rotationally adapted Hamil­tonian results when we use eigenfunctions of the square of the angular momentum operator J 2 (e.g., spherical harmon­ics or symmetric top eigenfunctions) as trial functions for the angular coordinates, as in expression (1.1). Because J 2 com­mutes with the Hamiltonian this choice leads to independent matrices of smaller size so that the diagonalization effort is significantly reduced; there is one matrix for each value of J. Progress has been made, however, to transform Cartesian trial functions into eigenfunctions of J 2 and Jz through the use of angular momentum projection operators,5 closing the advantage gap between Cartesian and rotationally adapted Hamiltonians.

The method developed in this communication is useful because the computation of Hamiltonian matrix elements, required in the variational approach to vibration-rotation spectroscopy of polyatomic molecules, I is usually cumber­some due to the complexity of the differential operators in curvilinear coordinates implicit in expression (1.2). More­over, the basis functions used in these computations must (i) resemble the true molecular eigenstates, such that errors in the calculated energies and spectral intensities that arise from using a truncated basis set are within the bounds of experimental resolution (usually a few wave numbers) and (ii) avoid the singularities that always appear in the Hamil­tonian as a result of the embedding of a body-fixed frame in a

J. Chern. Phys. 83 (3), 1 August 1985 0021-9606/85/151149-13$02.10 ® 1985 American Institute of Physics 1149

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:

155.33.120.209 On: Fri, 21 Nov 2014 20:13:08

1150 M. Blanco and E. J. Heller: The dynamical basis set

nonrigid system (Eckart frame transformation). These sin­gularities are attributed to the impossibility of parametrizing the proper rotation group in such a way that the parametri­zation function is analytic everywhere.6

As pointed out by Sutcliffe,6 an embedded Hamiltonian has a restricted domain defined by the set of functions over which integrals in the form of Eq. (1.3) converge.

Hij = Iv tJI/7t"'tJlj dV. (1.3)

In any particular application, however, there is no a priori reason to suppose that the domain of the transformed Ha­miltonian [defined by (ii) above] will contain the states of physical interest [as required by (i)]. To take the Watson Hamiltonian as an example, its domain cannot include func­tions of qk such that expectation values of f.1a/3 diverge. The chosen functions must either strongly vanish in the regions where the effective reciprocal inertia tensor is singular or give divergent contributions of opposite signs that exactly cancel each other upon taking the expectation value of the Hamiltonian (a specific triatomic example is given in Ref. 6).

Although these singularities are quite unavoidable as consequences of the embedding, they can be chosen to occur as and where desired by different choices of embedding be­cause the singularities are not essentially physical in origin but are artifacts of the transformation process. Notice that the primitive form of the Hamiltonian in normal Cartesian coordinates has no domain of restriction:

If 3N- 3 J2 Yt'= -- L -+ V(ql,q2' ···,Q3N-6)· (1.4)

2 k JQi

All "smooth" square integrable functions of the qk normal coordinates yield finite matrix elements. In addition notice that in Cartesian normal coordinates the computation of the matrix elements Hij is greatly simplified by the diagonal form of the kinetic energy operator and the simplicity of the differential operators. If one could generate the required so­lutions effectively and economically then the Cartesian form (1.4) is less troublesome to use than any body-fixed form of the Hamiltonian, including (1.2).

The problem of constructing multidimensional wave functions using a Cartesian basis set has been the subject of recent research in quantum dynamics.7

-1O Semiclassical

wave functions are given explicitly as sums of Gaussian func­tions, with the parameters of the Gaussians chosen accord­ing to a quantizing classical trajectory7 or according to an arbitrary classical trajectory for which only quasi periodicity is required.8 Although the method proved to be simple and, under most circumstances, a very accurate technique to ap­proximate quantum eigenfunctions, it was applied only to multidimensional model vibrational Hamiltonians.

A large number of chemically interesting processes, in­cluding photodissociation, photoisomerization, one photon, and multiphoton ionization, occur predominantly in the nonseparable VR dynamical regime. To understand these processes it becomes important to study new ways to repre­sent molecular states with basis functions that can handle an arbitrary degree of vibration-rotation coupling. Semiclassi­cal wave functions offer some hope in this respect because

they are defined with the help of classical trajectories that can follow the whims of the Hamiltonian in the presence of any degree of VR ·coupling, including centrifugal distor­tions, effects of anharmonicities, and vibrational angular momenta.

We define here a new basis set for the molecular VR problem which we term the dynamical basis set. It is based on the semiclassical wave functions presented in Sec. II C. Unlike traditional basis sets, such as products of harmonic or Morse oscillator eigenfunctions times spherical harmon­ics or symmetric top eigenfunctions, the accuracy of the dyn­amical basis set in representing the true vibrational-rota­tional eigenstates of the molecule is expected to be independent of the size of the VR coupling terms because the basis set is not coupled to any particular zeroth order Hamil­tonian. The main ideas behind this new basis set are summar­ized below.

In principle, molecular eigenstates can be obtained through the Fourier decomposition of an arbitrary time-de­pendent state tJI (q,t ).

tJln (q) = cn- I J: 00 ei(E. - JY)t IlltJI (q,t = O)dt,

= cn- 1 J: 00 eiE

•t III tJI (q,t )dt,

Cn = (tJln(q)ltJI(q,t=O)¥O. (1.5)

In Eq. (1.5) Yt' is the Hamiltonian of the system, tJln (q) is the eigenstate of interest with discrete energy En (here we con­sider only the bound region of the molecular BO potential energy surface; Ref. 11 gives applications to scattering prob­lems) tJI(q,t = 0) is in general an arbitrary initial state that evolves in time under the influence of the propagator

e-iKtllltJI(q,t=O) = tJI(q,t). (1.6)

In general, the exact time evolution of an arbitrary ini­tial state, tJI (q,t = 0) is unknown. For a large variety of sys­tems, however, the action of the propagator on a state that is initially well localized can be approximated by a time-depen­dent wave packet [4'> (q,t n, with averaged position and mo­mentum that coincide with the solutions of the classical equations of motion. The integral (1.5) is in practice discre­tized to a sum and each point in the sum represents the wave packet at different points in time during its classical motion

(1.7)

Llt is a constant time interval and E~c is a semiclassical ap­proximation to En. Thus, the exact eigenstate tJln (q) is ap­proximated by a sum, tJI~C(q) of semiclassical wave packets 4'> (q,tk ). The real or imaginary part of tJI~C(q) is then used as a dynamical basis function for a variational calculation of more exact solutions through standard matrix diagonaliza­tion procedures described in Sec. II C.

Figure 1 shows a typical dynamical basis function that is being prepared by a wave packet that moves on the xy plane of a two dimensional central Morse field potential (req = 2ao)' The x andy axes are labeled in units of the Bohr

J. Chem. Phys., Vol. 83, No.3, 1 August 1985 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:

155.33.120.209 On: Fri, 21 Nov 2014 20:13:08

M. Blanco and E. J. Heller: The dynamical basis set 1151

100 300 400

FIG. 1. A time-dependent frozen Gaussian wave packet generates a dyna­mical basis set function. The wave packet is guided by a classical trajectory (connecting solid line). See the text for further details.

radius ao and for clarity they are omitted from all but the first snapshot at t = O. At t = 0 the center of the wavepacket was displaced in the y direction (yo = 2.2ao) and was given some linear momentum in the x direction (Px = 1.36 Ii/ ao)' These displacements were chosen such that the classical angular momentum (L = rXp) was approximately equal to 31i. The winding solid line gives the path of the classical trajectory that defines the center of the wave packet at t = tk • After t = 25 fs (roughly one vibrational period) the wave packet has begun to move and changed sign twice. At t = 100 fs it begins to interfere with its past history, i.e., those wave pack­ets that were laid down along the trajectory at equal time intervals according to Eq. (1.7). After five rotational periods (400 fs) the trajectory has nearly closed upon itself, rendering as redundant the rest of the infinite time dynamics. At this point the sum of wave packets yields the characteristic nodal structure of an eigenstate, here the (v = 0, J = M = 3) state of this 2D model system.

As we will show, by changing the value of E~c in Eq. (1.7) one classical trajectory can generate a large number of dynamical basis functions. Also, several independent classi­cal trajectories can be taken to cover a larger portion of the relevant phase-space of the molecule.9

Similarly, we used classical trajectories to generate a dynamical basis set for the representation of the eigenstates of the three internal degrees of freedom of a rotating-vibrat­ing diatomic molecule H 2+ . We intentionally did not incor­porate the separability of the Schroedinger equation into ro­tational (angular) and vibrational (radial) degrees offreedom, but dealt directly with the three internal Cartesian degrees of freedom [r = (x, y, z)] in the center of mass coordinate frame [see Fig. (2)]. Because the vibration-rotation separation is more difficult to achieve in polyatomics, the demonstration that we can avoid the introduction of complicated curvilin­ear coordinates and remain in Cartesian space is important.

Previously/,8 in vibration-only systems, we had stopped at the level of the semiclassical states, using them directly to compute molecular properties, energies, spectra and so forth. The semiclassical states are surprisingly accu­rate approximations to the true eigenfunctions, but certainly can be improved. Here, for the first time, we use their poten-

x

x

z

C,M .

...+--+----~-y

\ I "

Lab.

FIG. 2. Standard center of mass frame transformation for a diatomic molecule. The two position vectors r 1

and r 2 in the space fixed ref­erence frame are reduced to one r = r, - r2 in the cen­ter of mass coordinate frame.

tial as basis functions in a variational determination of more accurate eigenfunctions. The coupled vibration-rotation problem for a diatomic molecule is presented, however, the method is in principle applicable to rotating-vibrating poly­atomics as well.

In Sec. II we present the method. Section III contains results for the rotation-vibration problem of H 2+ as the first numerical test of the method. In the course of examining the dynamical basis functions, we found that the expectation values of~, J2, and'Jz , were good approximations to the three quantum eigenvalues, namely, E, J, and M. We do not separate out rotations explicitly and, therefore, all three eigenvalUes are tests of the quality of the basis set. Because the M quantum number is associated with a 2J + 1 degener­acy in the absence of a magnetic field, it is somewhat surpris­ing that our approximate dynamical projection [Eq. (1.7)] could have singled out one Mvalue {1M I = J)from the others (1M 1= J - 1, J - 2, ... ,0). Section III and the Appendix show why this should be the case when using semiclassical input information. Section IV briefly outlines the applica­bility of the method to the rotating-vibrating polyatomic molecule.

II. METHOD

A. Projection operator equations

Expression (2.1)

!/in(q) = cn-I f: 00 ei(En-$')tlli!/i(q,t = O)dt (2.1)

can be looked upon as a projection operator equation for the unitary Hermitian propagator defined in Eq. (2.2)

U(t,to)!/i (q,to) = e - iJY(t- to)/li!/i (q,to),

= !/i (q,t - to). (2.2)

A stationary state of the system !/in (q) is projected out of the dynamical state !/i (q,t ) through the Fourier transform inte­gral (2.1). This transformation can easily be checked by sub­stituting !/i (q,t ) in Eq. (2.1) with its expansion in the complete set { !/in J of eigenstates

J. Chern. Phys., Vol. 83, No.3, 1 August 1985 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:

155.33.120.209 On: Fri, 21 Nov 2014 20:13:08

1152 M. Blanco and E. J. Heller: The dynamical basis set

(2.3) n

Projection operator equations of this sort are important because practical algorithms to compute quantum mechani­cal observables can be developed from them. For example, the analogous expression for the projection of eigenfunctions of the angular momentum operators J 2 and Jz is:

I/IJM(q) = cJl,;p~M' 1/1 (q),

= cJ!.J, (2J + 1) Jd OD"J,;M,(O)Rn I/I(q) , (2.4a) 8r where

(2.4b)

D ~M' (0) are the irreducible representations of the rotation group R 3, also known as the Wigner D functions, and o = (a, /3, y) are Euler angles that specify the orientation of the body-fixed frame with respect to a space-fixed reference frame. The effect of the rotation operator on the state 1/1 (q) is given by

Rn I/I(q) = I/I(q'),

(2.5)

q= An 'q.

An is a 3 X 3 matrix of cosine directions between the unit vectors of the fixed and rotated frames (Rij = i)).

It was recently shown5 that Eq. (2.4) could be taken in its literal sense to adapt a Cartesian basis function to the symmetries of the rotation group R 3• The projected states I/IJM(q) were used in numerical calculations as basis set func­tions to compute the matrix representation of a molecular Hamiltonian. Only a small matrix diagonalization was re­quired because P~M' and,:;>r commute, causing the,:;>r ma­trix to be block diagonal in J.

The projection power of the propagator Eq. (2.1) is po­tentially greater than that of the angular momentum projec­tion operator (2.4), in the sense that the propagator's equa­tion projects out as many invariants (quantum numbers) as the Hamiltonian operator contains. Among these we have the energy eigenvalues (En), the total angular momentum (J label), and all nondegenerate group symmetry invariants.

In practice, however, Eq. (2.1) cannot be used in its li­teral sense, as was done with the projection equation for the rotation group, because of two inherent limitations:

(i) The exact evolution of an arbitrary initial state, the integrand in Eq. (2.1) is not known. Expression (2.3) shows that an exact knowledge of 1/1 (q,t ) would amount to a com­plete description of all the stationary states of the system.

(ii) The coefficients exp (iEn t 1-17), unlike the coefficients D ~M' (0) in the angular momentum projection operator, are system dependent and generally unknown. In fact, the eigen­values En are generally the quantities that one wishes to compute.

B. The frozen Gaussian approximation

In view of the above limitations, the following semiclas­sical approximations to the exact eigenvalues and to the inte­grand in Eq. (2.1) have been proposed 10:

(i) The evolution of a state 1/1 (q) that is well localized at t = 0 can be approximated by a complex Gaussian wave packet [tP (q,t)] that moves according to the classical equa­tions of motion.

(ii) The eigenvalues (En) required in the Fourier-trans­form integral can be approximated by the semiclassical val­ues E ~c. These are obtained through the time-dependent expression of the Franck-Condon (FC) spectrum lO

•12

,13 of tP (q,t ) according to Eq. (2.6).

E(W) = J: 00 eiw' (tP (q,O)ltP (q,t )dt, (2.6)

where

w=EI-I7, (2.6a)

E ~c 1-17 are the values of the frequencies at which E(W) peaks [see Fig. 4]. We call E(W) theJrozen Gaussian (FG) spectrum for reasons that will become clear in the following para­graph.

The Gaussian wave packet tP (q,t ) can be given at differ­ent levels of approximation, depending on the number of independent varying parameters that are included in it. The simplest form is the Jrozen Gaussian approximation 10

(FGA):

tP(q,t)=7](t)[ exp -(q-q,).A.(q-q,)

(2.7)

The parameters qt ,p" and S, are obtained by integra­tion of the following first order differential equations:

. aHI q, = -:;:- , v.., p=p(

. aHI Pt= -- , aq q=q,

St = p, . qt - (Ec1ass + Eo)·

These equations are subject to the initial conditions:

qo = (tP (t = O)lqltP (t = 0),

Po= (tP(t=O)I-i-ItVltP(t=O).

(2.7a)

(2.7b)

(2.7c)

(2.7d)

(2.7e)

In these expressions H represents the classical form of the Hamiltonian of the molecule. Eqs. (2.7a) and (2. 7b) are readi-

z f---~---------------1 , , , , , , , , , , , , , , , , , , , , , , , , , , : .. :::::::: ~ ~ ~ ~ -(~---~----~ --~ --~

a FIG. 3. Typical classical trajectories for a diatomic molecule in the c.m. frame (Fig. 2). (a) Total angular momentum with nonzero Lx, L y , and L z components. (b) Trajectory with angular momentum vector oriented along the z axis only. Both trajectories are restricted to move on a plane, perpen­dicular to L, regardless of their initial conditions.

J. Chem. Phys., Vol. 83, No.3, 1 August 1985 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:

155.33.120.209 On: Fri, 21 Nov 2014 20:13:08

M. Blanco and E. J. Heller: The dynamical basis set 1153

ly recognized as Hamilton's equations of motion. Eclass + Eo is the classical plus zero point energies of the frozen Gaus­sian (FG) wave packet [Eclass = H (qo, Po)].

The termJrozen refers to the fact that the wave packet does not change size during its evolution, i.e., the matrix A is taken as constant in time

OJ· A .. =_' /j ..

IJ 21i IJ' (2.7f)

In Eq. (2.7f) OJ; is the ith-normal mode frequency of the mole­cule. Finite frequencies, based on Gaussian fits to symmetric top eigenfunctions, should be used instead of the zero rota­tional frequencies, as suggested in Sec. III. Finite frequencies are needed to have a normalizable wave packet. It is easy to show that after normalization <1> (q,t) obeys the minimal un­certainty principle:

ilq;ilp; = [(q:) - (qi)2] 112 [ (P:) - (Pi)2 r 12,

= (4A .. )-1/2(1i2A .. )1/2 =!!:.. U U 2 (2.7g)

The parameter 1] (t ) is given by the expression

(

3n - 3 )114

1](t) = i~1 OJ;Pi,i + pi,; (2.7h)

This prefactor represents a normalization correction which depends on the instantaneous position and momentum of the FG wave packet. 7.8

Solutions to Eqs. (2.7a)-(2.7c) are obtained through nu­merical integration routines, such as Runge-Kutta and Ad­ams-Moulton algorithms. We used both methods with no significant difference in calculated energies and eigenvalues. For polyatomic systems, however, the Adams-Moulton al­gorithm will be more efficient since it requires fewer evalua­tions of the potential energy function.

As the reader can easily check, at any time t:

(<1> (t )lql<1> (t) = ql'

(<1>(t)l- iliVl<1>(t) = p,.

(2.8)

(2.9)

Therefore, the center of the FG wave packet follows the clas­sical trajectory, just as the "center" of the exact quantum amplitude I/' (q,t ) in Eq. (2.1) satisfies Erhenfest's theorem:

:t (I/' Iqll/') z( I/' I :11/') , (2. lOa)

:t (I/' I - iliVII/') z - (I/' I: 11/')· (2. lOb)

<1> (q,t), however, yields the exact quantum dynamics and is not just an approximation for a multidimensional har­monic oscillator. 17 Furthermore, when applied to "smooth" anharmonic systems, the frozen Gaussian approximation worked surprisingly well for a large variety of purposes, in­cluding determination of semiclassical eigenvalues and ei­genfunctions for model vibrational Hamiltonians,7.8 Franck-Condon (absorption) spectra,1O three-dimensional gas-surface scattering, II direct photodissociation,14 and Ra­man scattering intensities. 15 Smooth here means that the an­harmonicities are locally small, i.e., within the width ilq of the FG wave packet, but they can be globally large over the entire potential surface.

Semiclassical wave functions are defined through the discretized form of Eq. (2.1) with the FG wave packet as argument of the integral (sum):

M 'c I/'n SC(q) = L eiE

• 'J"<1> (q,tk ). (2.11) k

The discretization is required because the integral gen­erally cannot be evaluated analytically. Each point of this sum represents a particular time during the classical motion of the wave packet. However, the constant time difference ilt = tk + I - tk needs not to be equal to the integration time step used in Hamilton's Eqs. (2.7a) and (2.7b). For higher accuracy the time step used in the integration of Hamilton's equations is usually made smaller than ilt, say ilt 110.

The M th term in Eq. (2.11) must satisfy the quasiperio­dic condition

l(q'M,P'M) - (qo,Po)I :som. I (qo,Po) I

(2.1Ia)

Quasiperiodic trajectories are preferred because the sum (2.11) can be stopped once the dynamics become redundant.

Because the semiclassical wave functions are dynami­cally adapted to the Hamiltonian, they are a first good ap­proximation to the exact eigenstates of the system. It is natu­ral to suggest the use of these functions as a basis set for quantum mechanical molecular bound state calculations. Unlike zeroth order basis sets, e.g., harmonic and Morse oscillator eigenfunctions for vibrations or spherical harmon­ics and rigid symmetric top eigenfunctions for rotations, this basis set can incorporate the combined rotation-vibration dynamics of individual molecules.

C. The dynamical basis set We define dynamical basis set functions as the real or

the imaginary part of semiclassical wave functions I/'~B(q) = [I/'~C(q) + 5' 21/'~SC(q)]/25',

(2.12)

5'= {I,il· These basis functions are convenient to use because they give real Hamiltonian matrices that are easier to diagonalize than complex matrices. The procedure is similar to the construc­tion of Px and Py atomic orbitals out of Y1m and Y1 _ m

spherical harmonics. Either choice of 5 should give the same eigenvalues because both linear combinations are degener­ate. This definition also has the advantage of giving real wave functions with amplitudes that can be plotted to observe their nodal structure.

The complex conjugate state I/'~SC(q) differs from I/':,c(q) only by a change in sign of the linear momentum and action of the frozen Gaussians:

<1> *(qlt,p, ,S,) = <1> (ql - t, - p, - S,). (2.13)

Thus, the linear combinations (2.12) can be associated with the adaptation of the semiclassical state to time-inversion symmetry.

The dynamical basis set is not orthonormal. The vari­ational eigenvalue problem in a nonorthonormal set is dis­cussed in depth elsewhere. 16 Here it suffices to say that the variational eigenvalue problem reduces to the solution of the matrix equation:

J. Chem. Phys., Vol. 83, No.3, 1 August 1985

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:

155.33.120.209 On: Fri, 21 Nov 2014 20:13:08

1154 M. Blanco and E. J. Heller: The dynamical basis set

HU = SUE, (2.14)

where H is the Hamiltonian matrix, U is the matrix of eigen­vector coefficients, S is the overlap matrix (Sij = (lJIfB(q)1 IJIfB(q)), and E is the diagonal energy-eigenvalue matrix. The nth column of U contains the coefficients of an "exact" eigenstate

(2.15) m

The overlap and Hamiltonian matrix elements are sums of integrals over FG wave packets. For example, the overlap of the nth and mth dynamical basis set is written as follows:

( IJIDBIIJIDB) = _1_ [ (lJIsc + f;- 21J1*scllJI sC + f;- 21J1*SC)] n m 4151 2 n ~ n m ~ m

= ! Re [ (IJI~CIIJI~) + 5 2( IJI ~SCIIJI~)] , (2.16)

where

(IJI~CIIJI~) = I exp - i(E~Ct,_ E~tkl/Ii( <1>11<1>k)' (2.16a) I,k

FG multidimensional integrals are easy to compute due to the orthogonality of the q coordinates. The overlap of two frozen Gaussians is given by Eq. (2.17).

(<1>II<1>k) =1/i1h[det(;A-I)f12xexp[ - ~ q_.A.q_

i 1 A-I i C"] (2 17) + 2i P+ . q- - 81r P- . . P- -?-' . where

q- =ql-qk'

P± =PI ±Pk'

S_ =SI -Sk'

(2.17a)

The kinetic energy matrix elements are simply given by:

fz2 2 fz2 2 (<1>11- - V l<1>k) = - - (<1>II<1>k)[ l..:il - TrA],

2 2 (2.18)

where

TrA= IA jj • (2.18a) i

For the evaluation of the potential energy matrix ele­ments it should be noted that the product of two FG's gives a Gaussian kernel also:

<1> r(q)<1>dq) = <1>ldq) = 1llk exp [ - 2(q - qlk) . A . (q - qlk)

i i ] +-,;Plk ·(q-qlk)+-,;Slk , (2.19)

where

1llk = 1l11lk exp { - 2qlk • A . qlk

Plk =Pk -PI'

Sik =Sk -SI' (2.19a)

<1>lk is highly localized around the midpoint qlk between the I th and k th Gaussians; more so than the individual Gaus­sians are localized around qk and ql [notice the factor of2 in the exponential argument of Eq. (2.19)]. Therefore, a low order polynomial expansion of the potential energy around qlk should be sufficient to give a good approximation of the exact integral in Eq. (2.20)

(<1>d V(q)l<1>k) = f: 00 <1>rV(q)<1>k dq

= f: 00 <1>lk(q)[ V(qlk) + V'(qld(q - qlk)

+ ~ V"(qlk)(q - qlk)2 + o (q3)] dq. (2.20)

Convenient formulas for the computation of powers in q have been developed.5 For the first powers:

(<1>llqil<1>k) =5i(<1>d<1>k)' (2.20a)

where

!: i A-I ':I = qlk +- 'Plk'

4fz (2.20b)

For second and higher powers in one component we use the recursion formula:

(<1>llq;"l<1>k) =5i(q;,,-I) + :;1 (q;,,-2). (2.20c) /I

For cross products of powers in more than one component we can use the simple expression below:

(2.20d)

An alternative method to calculate the expectation val­ue of V (q) is to use a quadrature. A ten-point Gauss-Hermite quadrature in each of the components of q has worked well for systems with a maximum dimensionality of three [a total of 1000 evaluations of V (q) perintegral]. The quadrature and the low order polynomial expansion methods can be applied when using ab initio electronic surfaces or analytic fits of the potential energy surface.

We now proceed to show that the method is numerical­ly feasible for a three-dimensional quantum system defined by one vibrational and two rotational degrees offreedom of a diatomic molecule. This case is intended only as an illustra­tion, as the system is obviously separable in spherical polar coordinates. The insight gained into the use of the dynamical basis set, however, can be applied to the polyatomic case (see Discussion).

III. NUMERICAL RESULTS

We focus on the H/ molecular Hamiltonian which takes the following form in the center of mass coordinate system [see Fig. 2]

JY'= -~(~+~+~)+ V(x,y,z). (3.1) 2/l Jx2 Jy2 JZ2

In Eq. (3.1), x, y, and z are the usual relative coordinates

J. Chem. Phys., Vol. 83, No.3, 1 August 1985 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:

155.33.120.209 On: Fri, 21 Nov 2014 20:13:08

M. Blanco and E. J. Heller: The dynamical basis set 1155

x = x2 - Xl' Y = Y2 - Yl' Z = Z2 - Zl • Setting ,:z = (X2 + y2 + r) the interatomic potential energy V(r) for the ground electronic lsug state can be approximated by a Morse poten­tial l8

(3.2)

with Do = 0.10262 H, P = 0.72 ao- 1, and re = 2ao. (In the present case we used an analytic potential to avoid numerical integration errors. We were more interested in the differ­ences between the separable VR Hamiltonian and the Carte­sian coordinate method advocated here, rather than the ab­solute energy eigenvalues from an ab initio potential energy surface.)

The rotation-vibration dynamics of Ht, represents a difficult, perhaps even limiting, case for the present method due to the long de Broglie wavelength associated with the small reduced mass (0.5 amu) and binding energy (3.79 eV) of

H2+ (A = h /~2JlDo). For comparison, the exact (converged) eigenfunctions and energy eigenvalues were first calculated in the decoupled spherical polar coordinate system:

1/1 vJM(r,8,l/> ) = YJM (8,l/> )~ vJ(r). (3.3)

The product form [Eq. (3.3)) of the eigenstates is exact in the diatomic case [however, notice that the radial solutions ~ vJ(r) still depend parametrically on the total angular mo­mentum J]. The energy eigenvalues can be determined by solving the one dimensional radial equation

[If a2 .,,2J(J+1) ] - - - + ,:z + V(r) ~ vJ(r) = EvJ~ vJ(r). 2Ji a,:z 2Ji

(3.4) Because the vibration-rotation separation is more difficult to achieve in polyatomics, the demonstration that we can avoid the introduction of complicated curvilinear coordinates and remain in Cartesian space is important as was pointed out in the Introduction.

A. The frozen Gaussian spectrum

The FG spectrum can be thought of as a Franck-Con­don emission spectrum of the molecule from an excited elec­tronic state whose equilibrium configuration is displaced with respect to the ground electronic eqUilibrium configura­tion by the coordinates of the center of the wave packet at t = O. The spectrum provides approximate energy values (E:.c) to project out the eigenstates of the system [Eq. (2.11)]. The FG spectrum is somewhat more general than an FC spectrum in that wave packets with (p) #0 at t = 0 are also used when needed (see below).

According to Eqs. (2.6) and (2.7), a central guiding clas­sical trajectory is required to generate the FG spectrum. To ensure that a proper measure of phase space has been cov­ered by the wave packet in a finite amount of time, the trajec­tory is run until it comes back near the original starting point (quasiperiodicity). Figure 3 shows two quasiperiodic classi­cal trajectories for H2+ in the c.m. frame. Both trajectories move inside equipotential energy surfaces (spheres) oscillat­ing back and forth around r e while executing a full rotational period. Furthermore, the trajectories are restricted to move on a plane (perpendicular to the total angular momentum

vector L) because the four constants of the motion (E, Lx, Ly , L z ) act as constraints over the six degrees of freedom (x, y, Z, P x' P y' P z) leaving only two independently varying parameters. The trajectory shown in Fig. 3(b) has its total angular momentum vector L directed along the Z axis and, therefore, moves entirely on the xy plane. Without loosing generality, we selected only trajectories on the xy plane for the present study.

Figure 4 shows typical spectra for two different starting conditions of the wave packet. As the energy of the wave packet increases, the spectrum acquires more structure be­cause the wave packet begins to overlap with the high-lying quantum states. As an example, Fig. 4(a) shows one well­defined vibrational envelope, and its associated rotational substructure, whereas Fig. 4(b) shows three vibrational bands for which the rotational structure is also resolved. The complexity of each spectrum is consistent with the energies of the corresponding wave packets (0.37 and 0.50 eV, respec­tively).

Each feature in the FG spectrum can be labeled and each band can be assigned to a specific vibrational (v) and rotational (J) quantum number by plotting the wave func­tions or by computing (J 2

) and (H). For example, a small portion of the rich spectrum in Fig. 5(a) was completely re­solved and analyzed in Fig. 5(b). It was found that the states represented by the largest amplitudes at the center of each vibrational band had the rotational quantum number J closely matching the classical angular momentum of the wave packet. Therefore, as a general rule, the initial condi­tions of the wave packet (ro, Po,) should be adjusted to give values for the classical observables nearly matching the quantum expectation values of the state that one wishes to approximate semiclassically.

B. The dynamical basis functions

Figure 6(a) shows xy-counter plots of a dynamical basis set function for Ht. Positive and negative amplitudes are

3,-----,-------------,

a

iJ 2 6 8 ~~-----------------.

€(w) b

E

w FIG. 4. Frozen Gaussian spectra. Typical frozen Gaussian spectra for two different starting conditions of the wave packet. (a) Xo = 0.0, PxO = - 5.0, Yo = 2.0, P)IJ = 0.0, Eel = 0.37 eV. (b) Xo = 0.0, PxO = - 3.987, Yo = 2.508, P)IJ = 0.0, and Eel = 0.50 eV. The arrows indicate the semi­classical energies used to generate the first and second dynamical basis func­tions of Fig. 7.

J. Chern. Phys., Vol. 83, No.3, 1 August 1985 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:

155.33.120.209 On: Fri, 21 Nov 2014 20:13:08

1156 M. Blanco and E. J. Heller: The dynamical basis set

3.,-----------------,

3.4 3.5

W

3.6

FIG. 5. Highly structured fro­zen Gaussian spectrum. (a) A frozen Gaussian spectrum de­picting six vibrational bands (b) a small resolved portion of the spectrum in (a) shows the v and J labels of the corresponding dynamical basis set functions.

represented by solid and broken line types, respectively. Fig­ure 6(b) depicts the converged quantum state If/ 1,5,5 (r,e,¢ ) computed through the usual separation of radial and angular variables. The area between the inner and outer circles give the classically allowed values ofx andy. Notice how closely the dynamical basis function resembles the exact quantum eigenstate. Both the radial (r) and the angular (¢ ) nodal struc­ture are correct and also the small but finite amplitude into the classically forbidden regions (quantum tunneling) is qualitatively correct.

Figure 7 shows xy-contour plots for another six dyna­mical basis set functions. This set was used to compute the first six states ofHt (v = 0, 1,2,3,4,5,M = J = 10) shown for comparison in Fig. 8. The number of radial nodes gives the vibrational quantum number v, and the number of angular nodal planes gives the m quantum number that was found to be equal to the total angular momentum J for all dynamical basis functions. (To the best of our knowledge this is the first time the vibrational as well as rotational combined structure of a molecular eigenfunction has been displayed.)

The same comments about the goodness of the nodal structure of the dynamical basis functions, prior to the vari­ational calculation, apply to this set of functions of higher total angular momentum. In fact, the approximations made here will get better as we increase the total angular momen­tum of the system. The irregularities in the last two basis functions in Fig. 7 were corrected by taking a larger number of points in the sum (2.11) which effectively increased the resolution of these states.

FIG. 6. Comparison between a dynamical basis function and an exact quan­tum state. (a) (v::::; I, J::::;5) dynamically projected state ofH,+. (b) The exact quantum converged state 'l'= !('l''''M + 'l''''_M)' (v= 1,/= 5, M= 5).

FIG. 7. Dynamical basis set functions. xy-contour plots of6 dynamical ba­sis set functions used to compute the first 6 (v = 0,1, ... ,5, J = 10) states of H,+ . Compare with the exact states shown in Fig. 8. See the text for details on the labeling of the dynamical basis functions.

The energy eigenvalues for the first six vibrational lev­els were obtained by diagonalizing a small (6 X 6) Hamilton­ian matrix. This reduction in matrix size resulted from the goodness oftheJ label of the projected states. States of differ­ent J values are used in separate matrices reducing the com­putational effort considerably in the same manner that rota­tional-vibrational product states factor out the Hamiltonian matrix in blocks labeled by J. In Table I, these values are compared with those obtained by using the method of sepa­ration of vibrational and rotational degrees of freedom for J = 10. Also shown are the purely semiclassical results. The semiclassical values refer to the diagonal elements in the Ha­miltonian matrix:

( If/fBJJ¥'Jlf/fB) H .. = .

II ( If/ fBJlf/ fB) (3.5)

The semiclassical values are good first approximations to the fully converged calculations (a maximum error of 2%). A clear improvement occurs, however, when the semi­classical states are used as basis set functions for the vari­ational calculation shown in the last column in Table I. All of the dynamical-basis-set eigenvalues are within 0.4% of the converged values.

FIG. 8. Exact (quantum converged) vibrational-rotational eigenstates of H2+ . The labels of the states are as in Fig. 7 with v varying between ° and 5 from the upper left-hand comer to the lower right-hand comer. M and J are 10 for all depicted states.

J. Chem. Phys., Vol. 83, No.3, 1 August 1985 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:

155.33.120.209 On: Fri, 21 Nov 2014 20:13:08

M. Blanco and E. J. Heller: The dynamical basis set 1157

TABLE I. Comparison of eigenvalues forJ = 10 (energy in em-I).

Vibrational Semiclassical Dynamical quantum number Exact error basis set

(v) (a) (b) (c)

0 4184.55 39.51 16.46 1 6301.90 -18.66 2.19 2 8290.32 - 50.48 1.32 3 10 149.88 - 132.13 7.90 4 11 880.70 - 211.36 -14.05 5 13482.85 - 222.99 6.15

a Quantum converged calculations Ref. 5 using the method of separation of angular and radial variables

bEnergy difference between the semiclassical and the exact values in col­umn(a).

C Energy difference obtained with a (6 X 6) dynamical basis set.

The initial positions and momenta for the quasiperiodic classical trajectories used to generate the dynamical basis functions shown in Fig. 7 are summarized in Table II. Notice that L = roXPo = 10, i.e., the classical angular momentum for all trajectories was set equal to the quantum number J = 10 of the basis set.

Because the classical trajectory does not introduce any amplitude in the z direction we resorted to the quantum an­gular solutions to account for the finite width of the states out of the xy plane. This amplitude has a purely quantum mechanical origin and it is better described as quantum dyn­amical tunneling and is due to the noncommutative algebra ofthe angular momentum component operators that causes the total angular momentum vector to be delocalized by an amount8J = J l/2forM = Jstates. Consequently, theampli­tude of the states out of the xy plane (z direction) was ap­proximated by fitting the value of the JJ-spherical harmonic YJJ in the z direction with the following Gaussian:

(2J ,)1/2 ¢lIz) =--' _e-a"i'

J!2J '

(3.6)

where

az = {rv'2tg- 1 [1r/2 - sin -lIe - 1/2J)] )-2 . (3.7) Harter and co-workersl9 have used a similar procedure in their discussion of semiclassical states of symmetric polya­tomic molecules based on D~ symmetric top functions.

Figure 9(b) shows that approximation (3.6) gets better at higher values of J, as one would expect from its semiclassical origin. For low values of J the exact quantum state tunnels

TABLE II. Initial conditions for the dynamical basis set classical trajector­ies.a

PxO Yo

- 5.000 2.000 - 3.987 2.508 - 3.549 2.817 - 3.277 3.052 - 3.035 3.295 - 2.786 3.589

·Yo in units of Bohr radius (ao)'pxO in units ofl;/ao' All other initial condi­tions were set equal to zero Xo = Zo = 0.0 and PyO = P.o = 0.0.

0.6~--------·------~

1 J 10 (\j

O. 3 -;t-

'. 0.0

0 2 4 60.00 1. 25 2.50

Z (0 ) Z (0 ) 0 0

FIG. 9. The exact amplitude of the states out of the xy-plane, I Yjj

[8 = tr l (z/re ),!6 = oW (solid line type), is approximated by the Gaussain shape of the wave packet (broken line type). Not surprisingly the semiclassi­cal approximation is more accurate at larger values of the total angular mo­mentum J. For J~ 10 the shape of the exact amplitude along the z direction is for all practical purposes Gaussian.

deeper to higher z values, whereas the semiclassical state drops to zero too quickly. For J = 10, however, the quantum state amplitUde is essentially Gaussian.

In Sec. II we noted that because the projection equation for the propagator excludes invariants that are (accidentally or otherwise) degenerate in energy, we do not expect the propagator's projection to quantize the angular momentum along an axis fixed in space. We have observed, however, that the projected dynamical states are not admixtures of 2J + 1 M-degenerate states, but instead they compared well only with 1M I = J states as shown in Figs. 6-8. We can ex­plain the apparent projection of this degenerate invariant by pointing out that classical mechanical information was used in the basis set. In the Appendix, it is shown that for 1M I = J the expectation value of the angular momentum operator (JM IJ IJM) shows the vector-like transformation properties typical of classical systems. Thus, projection of states with low values of M is greatly hindered by the small dispersion in the expectation value of the angular momentum components 8Jx , 8Jy , and 8Jz of the semiclassical wave packet.

c. General comments

The most important technical experience gained with the dynamical basis set is summarized below.

(1) A large number of peaks may be present in a FG spectrum depending upon the initial conditions of the wave packet. The features that are clearly resolved generate states that best resemble the eigenstates of the system. Among these states, those with large amplitudes should be preferred because their Franck-Condon factors I (!{In 1(1) 12 measure the extent of the overlap between the wave packet and the desired eigenstate.

(2) If quasiperiodic orbits are taken as the guiding tra­jectory of the wave packet, there is no need to include more terms in the sum (2.11) because the ensuing dynamics are redundant (or nearly so) after the trajectory closes (or nearly closes) upon itself.

(3) Sometimes a single classical trajectory will not give all the semiclassical states that one would like to include in the basis set. This happens when the wave packet overlap integral with a subset of eigenstates vanishes for symmetry

J. Chem. Phys., Vol. 83, No.3, 1 August 1985 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:

155.33.120.209 On: Fri, 21 Nov 2014 20:13:08

1158 M. Blanco and E. J. Heller: The dynamical basis set

reasons. To generate the missing states it is required to run additional trajectories. In most cases the initial conditions for these additional trajectories are simply given by the ap­plication of the symmetry elements of the point group ofthe molecule over the initial conditions of the first trajectory.

(4) A discrete Fourier-transform numerical routine is used to generate the FG spectrum. Because the trajectory is run fora finite amount of time the integrand (<P (q,O)I<P (q,t) in Eq. (2.6) should be multiplied by a cut-off window func­tion to make the discrete numerical Fourier transform con­verge more rapidly. Several cut-off window functions are available in the literature but we recommend the four-point Blackman-Harris window20 because it gives a highly con­centrated lobe with very low sidelobe structure in the fre­quency domain. This feature helps to resolve two or more nearby "states" that have significantly different amplitudes.

(5) The slow step in the calculation of matrix elements is the computer evaluation of exponential functions. It helps to sum the exponents in the product Gaussian kernel (2.19) be­fore computing the exponential. A large savings in computer time is realized if the overlap is set to zero and exponentials are not computed whenever the real part of the exponential argument is less than, say, - 8.

(6) Finally, the use of matrix-vector multiplications to compute the overlap, kinetic and potential energy matrix elements should allow the present method to take full advan­tage of the rapidly developing technology of parallel process­ing. Furthermore, it is even conceivable that in the near fu­ture a large array of microprocessors to store and manage the information of individual dynamical basis functions can be developed.

IV. DISCUSSION

This paper has presented several new developments in our program of "projecting out" eigenfunctions from semi­classical wave packet dynamics. First, we have learned that the projection works well for rotations, at least for diato­mics. Previous dynamical projection work has been applied to vibrations only.7.8 Second, we have used for the first time the approximate eigenstates obtained via dynamical projec­tion as a basis set off unctions to be used in the time indepen­dent Schroedinger equation. The need to embed the Hamil­tonian in a body-fixed frame to account for angular momentum invariance and the appearance of singularities as a result of the embedding were altogether avoided by re­maining in Cartesian coordinates and using the propagator's projection equation.

The method shares similarities with the generator coor­dinate formalism. 21 Both are based on an integral represen­tation of Schroedinger's equation [the projection operator Eq. (1.5), on the one hand, and the Hill-Wheeler21 equation, on the other] and both lead to the the nonorthonormal vari­ational problem. Our underlying philosophy, however, is that classical mechanics is an excellent starting place for the determination of quantum wave functions (for both numeri­cal procedures and intuitive understanding). Consequently, our "intrinsic states," are not purely quantum mechanical states but contain classical information.

In effect, most purely quantum mechanical techniques

solve all the dynamics in one large step. The method intro­duced here will still require significant investment in com­puter time, but for the completely coupled vibration-rota­tion problem the resulting basis-set size will be very much smaller and more reliable than that obtained by convention­al basis functions, because the dynamical wave functions re­semble more closely the exact eigenstates of the molecule at the intermediate and high energy regimes. The techniques applied here share many of the simplifying features of classi­cal trajectories. Each trajectory "solves" only one piece of the dynamics, such pieces, however, can be computed even for complicated dynamical systems. The full dynamics, the accumulation of information over several dynamical func­tions, is then obtained in the form of eigenvectors through the diagonalization of the Hamiltonian matrix.

In a previous paper,5 we used angular momentum pro­jection operators, Euler angles, and irreducible representa­tions of the rotation group (Wigner D functions) to build the angular nodal structure of IJ,M ) states into a Cartesian basis set. Rigorously speaking however, no dynamical variables were employed in that approach. By employing the unitary operator U(t,to) however, time appears as the natural vari­able for the dynamical projection of a time-dependent wave packet into approximate eigenstates of the system. The pro­jection builds all nondegenerate invariants of the system into the basis set functions (a discussion of the approximate pro­jection of the M-quantum label is given in the Appendix).

Angular momentum projection5.19.22 and the present method are fully quantal and converge to the exact quantum energies when the size of the basis set is increased. As com­pared with pure symmetry considerations of the former, however, the dynamical basis set method gives in addition a dynamical adaptation which, when applied to nonrigid po­lyatomic molecules, will prove to be crucial in the computa­tion of eigenstates and eigenvalues. Regardless of the sizes of the rotation-vibration interaction terms, they will be auto­matically taken into account through the dynamical infor­mation contained in the basis functions. The central classical guiding trajectory of the wave packet gives a built-in indica­tor that measures the vibration-rotation interactions, includ­ing angular momentum centrifugal effects, Coriolis effects, and vibrational angular momentum.

Additional work is needed to make the method more general. A method to select initial conditions for the trajec­tories that will uniformily sample the phase-space available to the molecule is needed. Semiclassical techniques that can deal with low values of J (J < 10) must be developed because low J transitions are of significant spectroscopic interest. Furthermore, the semiclassical representation of states with low values of 1M I <J is still an unsolved problem. As one possible solution we suggest to apply angular momentum shift operators to M = J states to obtain arbitrary values of M, such as in Eq. (4.1)

(4.1)

The numerical implementation ofEq. (4.1) will probably de­mand a discretization in Euler space of the P ~J operator in a manner consistent with the method in Ref. 5. Looking ahead we are optimistic that the dynamical basis set can be ex-

J. Chem. Phys., Vol. 83, No.3, 1 August 1985 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:

155.33.120.209 On: Fri, 21 Nov 2014 20:13:08

M. Blanco and E. J. Heller: The dynamical basis set 1159

tended to treat the six fully coupled vibrational-rotational degrees of freedom of a triatomic molecule. We expect that the use of Cartesian coordinates will greatly facilitate the computation of matrix elements and that the dynamical pro­jection will lead to the same degree of angular-momentum symmetry adaptation required to block-diagonalize the Ha­miltonian matrix in the same manner that symmetric top eigenfunctions do.

ACKNOWLEDGMENTS

We would like to thank W. G. Harter for valuable sug­gestions concerning classical-quantum correspondences for angular momentum. One of us (MB) would like to thank D. J. Tannor and R. C. Brown for their interest during the early preparation of this work. We want to thank the referee for his valuable comments. The kind assistance ofD. C. Hanson with the final manuscript is gratefully acknowledged.

APPENDIX: THE CLASSICAL LIMIT OF ANGULAR MOMENTUM EIGENSTATES

The quantum description of the angular momentum of a spinless molecule (within the Born-Oppenheimer approxi­mation) requires two constants of the motion, J2 and the projection of the angular momentum on an axis fixed in space, J. n. Furthermore, there is a finite uncertainty asso­ciated with the measurement of the total angular momentum of the molecule {jJ> ° even for pure angular momentum ei­genstates, ~,m) [see Eq. (A6)]. The noncommuting algebra of the angular momentum component operators is also respon­sible for a finite angular delocalization of the molecule. On the other hand, the classical mechanical description requires three constants of the motion instead of just two. The classi­cal vector description of angular momentum

Lx = ILl sin/3 cos a,

Ly = ILl sin/3 sin a,

L z = ILl cos/3,

(AI)

is deterministic with an uncertainty in the total angular mo­mentum and angular position of the system limited only by the precision of the measuring apparatus [here the Euler an­gles a and /3 correspond to the azimuth (¢ ) and polar (8) angles, respectively].

In view of this difference, we may ask: how can the delocalized angular momentum eigenstates ~,m) account for the localized L vector description in the macroscopic classical limit? To answer this question, the most common approach has been to investigate the superposition of angu­lar momentum eigenstates ~,m) that will give coherent states for which the uncertainty product of angular momen­tum and angular position is minimal23

-25

(A2)

with i = x, y, z (note: we will use capital J for angular mo­mentum component operators andj and m for the quantum numbers). In the same manner that one constructs Glauber coherent states by superposing eigenstates of the harmonic oscillator, one is lead to take linear combinations of angular

momentum states with coefficients that minimize the left­hand side ofEq. (A2). Thus, for example, we have the follow­ing ansatz for an angular momentum coherent state23

:

IS) =N L (k+a~ y+m(k_at_ y-mIO)

k j,m (j + m )!(j - m)! '

(A3)

where k+ and k_ are the complex eigenvalues of the boson annihilation and creation operators (a+,a_)

a"IS)=k"IS), ..1,= ±. (A4)

This superposition gives25

(AS)

There is room for at least one objection to this ap­proach. The reason to minimize the uncertainty product {j¢{jJ is the presumption that {j¢ = ° and {jJ = ° for all classi­cal systems. Although {jJ rigorously vanishes for all classical systems only for rigid bodies we have that {j¢ = 0, because there can be no ambiguities about the orientation of a body to which a reference frame can be rigidly attached. Nonrigid classical systems, however, can have arbitrarily large angu­lar position uncertainty while the uncertainty in J remains zero.

A simpler picture emerges if we choose to rely on the correspondence principle, which can be loosely stated as: the classical description of a system is reached by increasing the quantum numbers of a complete set of commuting observa­bles, including those which only add degeneracies (e.g., m), until the classical values of these observables are obtained. Accordingly, the uncertainty of the angular momentum op­erator as measured by the ratio {jJ / (J 2) 1/2 vanishes in the limit of large j for eigenstates with I m I = j,

{jJ _ [(j,mlJ2~,m) - (j,mIJ ~,m)2] 112

(J2)1/2 - (j,mIJ2~,m)1/2

= [1 - m 2/jIj + 1)] 112. (A6)

The vanishing of the uncertainty ratio suggests that classical states can be made to correspond with quantum states whose quantization axis (n) points in the direction of the total angu­lar momentum

J. n '/Ielass = fzj'/lc\ass • (A7)

Therefore, by setting I m I = j and allowing for large values of j, we should have a classical-like state even for a finite value of -Ii. Notice that merely lettingj ~ 00 is not enough, since the states with low values of m can give uncertainties as large as the total angular momentum of the system even in the limit of infinite j. We shall now prove that the expectation value of the angular momentum operator (j,m IJ I j,m) has the same transformation properties of the classical angular momentum vector L for I j, I m I = j ) states in the limit of largej.

The angular momentum eigenstates I j,m) z (where z in­dicates that the quantization axis is in the z direction) trans­form under the action of the rotation operator as follows26:

J. Chem. Phys., Vol. 83, No.3, 1 August 1985

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:

155.33.120.209 On: Fri, 21 Nov 2014 20:13:08

1160 M. Blanco and E. J. Heller: The dynamical basis set

91'(n,r)lj,m)z = I Djmm'(a, {3, rWm')z, m'

(A8)

n is a unit length vector in the direction of the rotated quanti­zation axis (a, {3 ) and r is the twist angle of rotation around n.

The transformation properties of Ij,j )z states under rotations can now be analyzed by setting m = j and by pro­ceeding to compute the expectation values of the angular momentum components of the rotated I j,j ) it state, The ex­pectation values of Jx and Jy are computed from those of J + andJ_

m= -j

X [sin ({3 /2)]2U- m)=F 1 (A9a)

with

N'm

= {U=t= m)[j U + 1) - m(m + I)]} 112

J j±m+1

2 '1 X J. (j + m)!(j - m)! '

(A9b)

where we have used the explicit form of the Wigner D func­tions. 27 Setting u = cos2

( {3 /2) we get

(J +) ii = -Ileia( U _ u2 ) 1/2 2j

i 1 (2j _ I) (2j) 1=0 I

(A 10)

but

~ l_k(/) k~O (-1) k =8/0' (A11)

After some simplifications

(J ± ) ii = 2fYe ± ia(u _ U2)1/2,

= fY sin {3e ± ia, (A12)

The value of (Jz) is similarly found, and the end result is

a b

FIG. 10. Three dimensional (c.m. frame) constant probability density (I If' 12) surface for two quantum states ofH2+' (a) Unlike classical trajectories, the

quantum states of the system can be highly delocalized the more so for low values of M [here, M = 0, see also Eq. (A6)]. (b) Localized angular momen­tum states (1M I = J) closely resemble the classical dynamics ofthe system [compare with Fig. 3(b), see the text for further details]. Here, M = J = 10.

(A13)

(JY)ii = ;/J+ -Jit = fY sin{3 sin a, (A13a)

(Jz)" =fljcos{3. (A13b)

For large values ofj, we have that [jU + 1)]1/2 ~ j, and we can state these results briefly as follows

(j,jIJ Ij,j)ii

= I (J) ii I (sin {3 cos a, sin {3 sin a, cos {3 ),

= L. (A14) In conclusion, the classical three-dimensional L vector

description is recovered in the limit oflargej from the expec­tation value of the angular momentum operator of a I j, j > it state. In addition, the I j,j) it states exhibit minimal uncer­tainty, which vanishes in the classical limit when expressed as the ratio of the total angular momentum. These two prop­erties provide strong evidence to support the observation that I j,j ) ii states represent the angular momentum dynam­ics of classical systems.

The classical-like character of j j-states becomes evi­dent by comparing the probability density plots of the Ij = 10, m = 0) state of Ht [Fig. lO(a)] with that of the I j = 10, m = 10) [Fig. lO(b)]. Notice that the probability distribution is largely restricted to the vicinity of the xy plane for the Ij = 10, m = 10) state depicted in Fig. lO(b). Al­though some dynamical tunneling is still present (the finite thickness of the doughnut-shaped state) this state shows a great resemblance to the classical trajectory depicted in Fig. 9(b). On the other hand, the m = 0 state in Fig, lO(a) is highly nonclassical, i.e., it has highly delocalized values for the oth­er two angular momentum components (Jx , J y)' In the case of a diatomic molecule, this angular momentum delocaliza­tion leads to states with significant probability density am­plitude out of the classical plane of motion as can be seen in Fig. lO(a). In particular the semiclassical behavior of D~M(O) (JJ-symmetric top eigenfunctions) has also been studied by Harter, Patterson, and da Paixao. 19

'G. D. Carney, L. L. Sprandel, and C. W. Kern, in Advances in Chemical Physics, edited by I. Prigogine and S. A. Rice (Wiley, New York, 1978), Vol. 37, p. 305-379. This reference also contains a general account of the variational method as applied to the vibration-rotation molecular prob­lem.

2J. K. G. Watson, Mol. Phys. 15,479 (1968). 3J. D. Louck, J. Mol. Spectrosc. 61,107 (1976). 4(a)R. J. Whitehead and N. C. Handy,J. Mol. Spectrosc. 55, 356(1975); 59, 459 (1976); (b) G. D. Carney and C. W. Kern, Int. J. Quant. Chem. Symp. 9,317 (1975).

'M. Blanco and E. J. Heller, J. Chem. Phys. 78, 2504 (1983). 6B. T. Sutcliffe, Mol. Phys. 48,561 (1983). 7M. J. Davis and E. J. Heller, J. Chem. Phys. 75, 3916 (1981). 8(a) N. DeLeon and E. J. Heller, J. Chem. Phys. 78,4005 (1983); (b) E. J. Heller, Faraday Diss. Chem. Soc. 75, 141 (1983); (c) N. DeLeon and E. J. Heller, in Semiclassical Spectral Quantization: Application to Two and Four Coupled Molecular Degrees of Freedom (to be published).

9M. J. Davis and E. J. Heller, J. Chem. Phys. 75, 246 (1981). See this refer­ence in connection with dynamically trapped trajectories and quantum dynamical tunneling.

lOE. J. Heller, J. Chem. Phys. 75, 2923 (1981). "G. Drolshagen and E. J. Heller, J. Chem. Phys. 79, 2072 (1983).

J. Chem. Phys., Vol. 83, No.3, 1 August 1985 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:

155.33.120.209 On: Fri, 21 Nov 2014 20:13:08

M. Blanco and E. J. Heller: The dynamical basis set 1161

12E. J. Heller, J. Chern. Phys. 68, 2066 (1978); 68, 3891 (1978). 13K. C. Kulander and E. J. Heller, J. Chern. Phys. 69, 2439 (1978). 14R. C. Brown, Ph. D. dissertation, University of California, Los Angeles,

1983. "(a) S.-Y. Lee, and E. J. HeUer, J. Chern. Phys. 71,4777 (1979); (b) D. J.

Tannor and E. J. HeUer, ibid. 77, 202 (1982); (c) E. J. Heller, R. L. Sund­berg, and D. J. Tannor, J. Phys. Chern. 86, 1822 (1982).

16J. H. Wilkinson, The Algebraic Eigenvalue Problem (Oxford University, London, 1965). See also M. J. Davis and E. J. HeUer, J. Chern. Phys. 71, 3383 (1979).

17E. J. Heller, J. Chern. Phys. 62,1544 (1975); ibid. 65, 4979 (1976). 18Shi_I Chu, J. Chern. Phys. 75, 2215 (1981). I"W. G. Harter, C. W. Patterson, and F. J. da Paixao, Rev. Mod. Phys. 50,

37 (1978). 2Op. J. Harris, Proc. IEEE 60,51 (1978). 21D. L. HiIl and J. A. Wheeler, Phys. Rev. 89,1102 (1953). See also L. Lath-

ouwers and P. Van Leuven, in Advances in Chemical Physics (Wiley, New York, 1982), Vol. 49, p. 115.

22L. Lathouwers, J. Phys. A: Math. Nucl. Gen. 13,2287 (1980). 23R. Delbourgo, J. Phys. A: Math. Nucl. Gen. 10, 1837 (1977). 24Regarding the properties of angular momentum coherent states see the

following: (a) J. M. Radcliffe, J. Phys. A: Gen. Phys. 4, 313 (1971); (b) H. A. Buchdahl, N. P. Buchdal, and P. J. Stiles, J. Phys. A: Math. Nucl. Gen. 10,1833 (1977); (c) A. J. Bracken and H. I. Leemon, J. Math. Phys. 22, 719 (1981); (d)R. Bonifacio, D. M. Kim and M. O. ScuUy, Phys. Rev. 187,441 (1969).

25p. W. Atkins and J. C. Dobson, Proc. Soc. London A 321,321 (1971). 26M. E. Rose, Elementary Theory of Angular Momentum (Wiley, New

York, 1967). 27E. Wigner, Gruppentheorie (Friedrich Vieweg und Sohn, Braunschweig,

1931).

J. Chem. Phys., Vol. 83, No.3, 1 August 1985

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:

155.33.120.209 On: Fri, 21 Nov 2014 20:13:08