On rotational coherent states in molecular quantum dynamics

On rotational coherent states in molecular quantum dynamicsJorge A. Morales, Erik Deumens, and Yngve Öhrn Citation: J. Math. Phys. 40, 766 (1999); doi: 10.1063/1.532684 View online: http://dx.doi.org/10.1063/1.532684 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v40/i2 Published by the American Institute of Physics. Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors

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JOURNAL OF MATHEMATICAL PHYSICS VOLUME 40, NUMBER 2 FEBRUARY 1999

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On rotational coherent states in molecular quantumdynamics

Jorge A. Morales, Erik Deumens, and Yngve OhrnQuantum Theory Project, Departments of Chemistry and Physics, University of Florida,Gainesville, Florida 32611-8435

~Received 5 June 1998; accepted for publication 16 November 1998!

Coherent states suitable for the description of molecular rotations are developedand their connection to similar coherent states in the literature are explored. Inparticular their quasiclassical properties are developed. The use of such coherentstates in time-dependent electron nuclear dynamics studies of molecular collisionprocesses is discussed. ©1999 American Institute of Physics.@S0022-2488~99!04602-2#

I. INTRODUCTION

Coherent states~CS! are a set of elements$um&% of a Hilbert spaceH. All CS share twoproperties in common:1

~1! continuity, i.e., the statesum& are continuous functions1 of a parameter setm,

limm→m0

um&5um0&, ~1!

~2! resolution of the identity, i.e., there exists a positive measuredm1>0 for which

15E dm1um&^mu. ~2!

There exists a weaker formulation of the second property which will allow a larger class of1

(28) The closed linear span of$um&% is the Hilbert spaceH. This means that any state vector in thHilbert space may be represented as a~possibly infinite! linear sum of CS.1 Such CS may satisfya resolution of the identity with an indefinite measuredm6 ,

15E dm6um&^mu. ~3!

Both in the stronger and the weaker definitions, the CS form a nonorthogonal overcompleThere are a great variety of CS known and used in various areas of physics. For probl

molecular physics and in chemistry the canonical CS2,1 also referred to as Glauber states3 arecommonly used.4 These states$ua&% are associated with the harmonic oscillator HamiltonHvib5\v(a†a1 1

2), wherev is the angular frequency. The harmonic oscillator creation operacan be expressed as

a†51

&SAmv

\x2

i

Am\vpD ~4!

in terms of the self-adjoint operators of positionx and momentump, wherem is the oscillatormass. The complex parametera can be expressed in terms of the real parameters of avepositionxa5âuxua& and average momentumpa5âu pua& as

7660022-2488/99/40(2)/766/21/$15.00 © 1999 American Institute of Physics

2013 to 160.36.192.221. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

Sntum

s

d

767J. Math. Phys., Vol. 40, No. 2, February 1999 Morales, Deumens, and Ohrn

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a51

&SAmv

\xa2

i

Am\vpaD . ~5!

An expansion in terms of harmonic oscillator stationary states$un&, n50,1,...% exists,

ua&5expS 21

2uau2D (

n50

àn

An!un&5expS 2

1

2uau2Dexp~aa†!u0&, ~6!

from which the resolution of the identity is readily proven with the positive measure

dm1~a!51

pd Read Im a. ~7!

The spin coherent states$ub&%, with a complex parameterb, constitute another example of Cused in molecular physics.1 These states are associated with the total spin angular momeSW 5(Sx Sy Sz) and an expansion in terms of spin eigenstates$uSM&, S50,1/2,1,...;M5S,S21,...2S% exists,

ub&5 (M52S

S A ~2S!!

~S2M !! ~S1M !! F bS1M

~11ubu2!sG uSM&51

~11ubu2!s exp~bS1!uS 2S&, ~8!

whereS65Sx6 iSy . The resolution of the identity exists with the positive measure

dm1~b!52S12

p~11ubu2!2 d Rebd Im b. ~9!

It suffices here to mention as a third example the fermion CS,1 also known as the Thoulesdeterminant.5,6 These CS are used, e.g., in the description of many-electron systems.4 For Nelectrons in a basis of rankK>N the normalized Thouless CS$uz&% can be expressed as

uz&5det~ I 1z†z!21/2expF (h51

N

(p5N11

K

zphbp†bhG uC0&, ~10!

wherez denotes the set of complex parameters$zph%, thebi† andbi are the fermion creation an

annihilation operators, respectively, and where

uC0&5)i 51

N

bi†uvac&. ~11!

The resolution of the identity exists with the positive measure

dm1~z!5h det~ I 1z†z!2Kd2z, ~12!

where

d2z51

p )ph

d Rezphd Im zph , ~13!

and

h51!2!¯K!

1!2!¯~K2J!!1!2!¯J!. ~14!


of theUot

tion.

ime asn of a

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.

imum

768 J. Math. Phys., Vol. 40, No. 2, February 1999 Morales, Deumens, and Ohrn

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A set of coherent states may be related to a particular Lie group. Rasetti7 and Solomon8 havemade seminal contributions to the theory of group-related CS. Perelomov9 introduced a systematicprocedure for the construction of such group-related CS. For instance, the canonical CSharmonic oscillator are related to the Weyl group, the spin CS to the special unitary group S~2!,and the Thouless CS to the unitary group U~K!. There are, however, important CS that are ngroup related. The construction of coherent states requires a portion of mathematical intui

II. QUASICLASSICAL COHERENT STATES

A prominent property of many CS is their quasiclassical dynamics. A stateuc& is said to bequasiclassical when the evolution of average position, momenta, and energy,

xqc5^cuxuc&, pqc5^cu puc&, Hqc5^cuHuc&, ~15!

satisfy classical Hamilton equations, i.e.,

xqc5]Hqc

]xqc, pqc52

]Hqc

]pqc. ~16!

In other words, the average position and momentum of the quasiclassical state evolve in tthe position and momentum of their classical analogs. One should note that the definitioquasiclassical state does not demand the semiclassical limit\→0 to be invoked. Neither is therea priori any guarantee that a quasiclassical state even exists for a given Hamiltonian. Ehretheorem10 offers a means to investigate the quasiclassical property, i.e., the equations

i\d

dt^cuxuc&5^cu@ x,H#uc&,

~17!

i\d

dt^cu puc&5^cu@ p,H#uc&

should reduce to the classical ones of Eq.~16! for the stateuc& to be quasiclassical.In this manner it is straightforward to show that the canonical CS of Eq.~6! are quasiclassical

In particular,

âuxua&5xa~ t !5A2\

mvRe@a exp~2 ivt !#,

~18!âu pua&5pa~ t !5A2m\v Im@a exp~2 ivt !#

and the total energy using the harmonic oscillator Hamiltonian is

Ea[Hqc5âuHvibua&5\vuau21\v

25Ec

a1\v

2, ~19!

where

Eca5

1

2mpa

211

2mv2xa

2 ~20!

is the classical energy of the harmonic oscillator. This particular set of CS satisfies the minuncertainty relation

Dxa~ t !Dpa~ t !5\/2, ~21!


il-

calssical-

nciple

namics

yingle-

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oduct


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where the widthsDxa(t) andDpa(t) are

Dxa~ t !5Dxa5A \

2mv, Dpa~ t !5Dpa5Am\v

2. ~22!

The coordinate representation of the canonical CS is

ca~x,t !5^xua~ t !&5exp~ iua~ t !!S mv

p\ D 1/4

exp~2 ivt !expS 2S x2xa

2DxaD 2DexpS ipa~ t !x

\ D ,

~23!

whereua(t) is a global phase. The spin CS, Eq.~8!, are quasiclassical with respect to a Hamtonian describing the spin dynamics under a time-dependent magnetic field.11,1 Minimum uncer-tainty conditions are also known for this CS.11,1 There are CS that do not exhibit the quasiclassiproperty. The Thouless CS is not a quasiclassical state. However, it is possible to obtain clalike equations for the Thouless parameters via the time-dependent variational pri~TDVP!.4,12

III. ELECTRON NUCLEAR DYNAMICS AND COHERENT STATES

In this section, we make the connection between the CS and the electron nuclear dy~END! theory.13,4,12The END wave function4 is

CEND~ t !51

NnuclFnucl@R~ t !,P~ t !# f el@z~ t !,R~ t !#expF i

\gEND~ t !G , ~24!

wheregEND(t) is the total phase. At the simplest level of approximation the nuclear partFnucl isthe product of Gaussian wave packets of positionsR(t) and momentaP(t),

Fnucl~X;R,P!5)k51

nucl

exp~2ak@Xk2Rk~ t !#21 iPk~ t !•@Xk2Rk~ t !# !, ~25!

and the electronic partf el@zph(t),R(t),#5uz& is the fermion~Thouless! CS shown above. The verrole of the fermion CS is to provide a nonredundant and continuous parametrization of the sdeterminant electronic wave function. It should be noted that the total system END wave fuis given in the laboratory frame and includes translational and overall rotational motion. UsinapproximateCEND(t) and the TDVP a set of classical Hamilton-like equations are obtained foThouless parametersz(t) andz* (t).13,14,4,12In order to obtain the proper END equations for thlevel of approximation the quantum mechanical Lagrangian is first obtained in the limit ofwidth nuclear Gaussian wave packets. This approach leads to a classical treatment of thethat retains the nonadiabatic electron–nuclear coupling terms. This approximation mayscribed as full, nonlinear time-dependent Hartree–Fock~TDHF! for electrons and narrow wavpacket nuclei.

The time propagation of a molecular system undergoing a reaction may produce aproduct fragments. One important aim of molecular reaction dynamics is to predict the distribof products over rovibrational states. The treatment of such a reactive process at the simplelevel of approximation leads to a fragment in some electronic state with its system of nvibrating and rotating as a classical object. It has been demonstrated how the END represeof the nuclear part of such a fragment under very general conditions factors as

Fnucl@X;R~ t !,P~ t !#'F0FvibF rot . ~26!

Viewing this wave function, even in the narrow wave packet limit, as an evolving staterepresenting this state in terms of suitable CS makes possible ana posteriori quantum stateresolution of the nuclear dynamics. In this way we adopt the approximate labeling of pr


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rotortionalm in

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states in terms of vibrational~harmonic oscillator! quantum numbers and rotational~rigid rotor!quantum numbers. Obviously, more ambitious CS could be attempted, but this level of socation seems reasonable for low energy reactive collisions.

Thea posteriorivibrational analysis in terms of harmonic oscillator CS has been outlined~seeRef. 12! and applied to obtain vibrationally resolved differential cross sections for proton csions with hydrogen molecules at 30 eV,15 in excellent agreement with experiment. A corresponing analysis for a quasiclassical treatment of the rotational dynamics is an interesting prospethe necessary development is discussed in this paper.

IV. ROTATIONAL COHERENT STATES

The term rotational CS denotes those CS which are quasiclassical with a field-freeHamiltonian. It is important to emphasize that the previously discussed spin CS is not a rotaCS by virtue of the preceding definition. Most of the rotational CS known in the literature stesome way from the spin CS. Except for one case discussed below,16 the majority of the rotationalCS concerns the description of the linear rotor.

The first known rotational CS were derived by Atkins and Dobson.17 The Atkins–Dobson CSare group generated by the Schwinger boson operators of the angular momentum,18 have a posi-tive measure, and can in principle be applied to linear rotors. An interesting and morecontribution to the theory of rotational CS was made by Janssen,16,1 who constructed rotational CSfor the general asymmetric rotor. Janssen CS$uxyz&% can be expressed as

uxyz&5 (IMK

JIMK~x,y,z!uIMK &, ~27!

where uIMK&; I 50,12,1,...; M, K50,6 1

2,...6I are the integer~boson! and half-integer~fermion!rotational states associated with the asymmetric rotor Hamiltonian,x, y, andz the CS parametersandJIMK(x,y,z) a set of coefficients. Janssen CS are not group generated and have an indmeasure. These CS satisfy quasiclassical dynamical equations when evolved by the asymrotor Hamiltonian in the Hilbert space spanned by the statesuIMK&. An interesting feature is thathe Janssen CS are identical to the Atkins–Dobson CS in the limit of the linear rotor. The prthat identity involves a proper reparametrization of Janssen CS. Both sets of CS have the pof mixing half-integer and integer quantum numbers. Therefore, they are not directly useful fdiscussion of molecular rotational spectra for which a corresponding development in terms ointeger quantum numbers is necessary.

Similar developments were published by Bhaumiket al., but again for the case of the linearotor statesuIMK 50&. A review of some rotational CS for linear rotors was published by Foet al.19 This study discusses the Atkins–Dobson CS among many others but misses thoJanssen and Bhaumiket al. More importantly, new CS generated by the SO(3)^ R5 group aredeveloped there to study diatomic molecules in the presence of an electromagnetic field.

In this section we introduce a new set of rotational CS following closely the definitioJanssen. Only integer quantum numbers are used for these CS and their quasiclassical beanalyzed.

A. Rotor Hamiltonian

The pure rotor Hamiltonian for a molecular system can be written as20

H rot5Lx

2

2Ax1

Ly2

2Ay1

Lz2

2Az, ~28!

whereAi ( i 5x,y,z) are the moments of inertia andL i the body-fixed components of the orbitangular momentum. Please note that from here on\51. The analogous space-fixed componeof orbital angular momentum areJi and the following relations hold (J25L2):


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@ Ji ,J j #5 i e i j l Jl , @ L i ,L j #52 i e i j l L l ~29!

and

@ Ji ,Lk#5@ L2,Lz#5@ J2,Jz#50, ~30!

wheree ikl are the components of the Levi–Civita tensor. Note the so-called anomalous cotation relationship20 of the L i components. As a result of these commutation relationships, texists a complete set of rotor eigenstatesuIMK& so that

L2uIMK &5I ~ I 11!uIMK &, I 50,1,2,...,

LzuIMK &5KuIMK &, K50,61,...,6I , ~31!

JzuIMK &5M uIMK &, M50,61,...,6I .

These rotor eigenstates in angular representation are

^f,u,xuIMK &5F2I 11

8p2 G1/2

DMKI* ~f,u,x!, ~32!

whereDMKI (f,u,x) are elements of a rotation matrix~Wigner D functions!.20

It follows from the above commutator relations that the rotational Hamiltonian satisfies

@H rot ,Ji #50, @H rot ,J2#50. ~33!

The Hamiltonian eigenfunctionsC IMa must satisfy

H rotC IMa 5Erot

a C IMa ,

J2C IMa 5I ~ I 11!C IM

a , I 50,1,2,..., ~34!

JzC IMa 5MC IM

a , M50,61,...,6I ,

where the superscripta is an additional label of a particular rotational eigenstate. Another serelations implied by the above commutation relations is

@H rot ,L i #5 i(j

e i jk

2Ak~ L j Lk1LkL j !. ~35!

The C IMa eigenfunctions are expressed in the symmetric rotor basis as

C IMa 5(

KcK

IM auIMK &, ~36!

where the coefficientscKIM a are to be determined.

In the special case of a spherical rotorA5Ax5Ay5Az ~e.g., CH4 and SF6), the eigenvalueproblem simplifies to

H rot5L2

2A5

J2

2A, C IM

a 5C IM 5uIMK &, ErotIM a5Erot

I 5I ~ I 11!

2A. ~37!


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r,

, it is,

sheroup

ese

ith the


Downloaded 11 Apr

In the case of a prolate symmetric rotor, the moments of inertia satisfyAx<Ay5Az ~e.g., CH3Cland PCl5). The eigenvalue problem then becomes

H rot5L2

2Az1S 1

2Ax2

1

2AzD Lz

2,

C IMa 5C IM 5uIMK &, ~38!

ErotIM a5Erot

IM 5I ~ I 11!

2Az1S 1

2Ax2

1

2AzDK2.

The equivalent expressions for the case of an oblate symmetric rotorAx5Ay<Az ~e.g., CHCl3 andC6H6) can be obtained by interchanging theAx with the Az in the last equations. The case of thlinear rotor ~e.g., all diatomics, CO2, and C2H2) is obtained as theAx50 limit of the prolatesymmetric case. Then

H rot5J2

2A, C IM

a 5C IM 5uIM 0&,

~39!

ErotIM a5Erot

I 5I ~ I 11!

2A, û,fuIM 0&5YIM ~u,f!,

where theYIM (u,f) are the spherical harmonics.20 Finally, in the case of an asymmetric rotowith the moments of inertia satisfyingAx<Ay<Az ~e.g., CH2H2), the eigenfunctionsC IM

a keeptheir linear combination form, and thecK

IM a coefficients must be specifically calculated.

B. Groups

Although the set of CS under construction is not group related in the Perelomov senseof course, connected to the rotation groups. Specifically, the statesuIMK& span the irreduciblerepresentations of the semidirect product of SO~3!^SO~3! with an Abelian group. The generatorof the first SO~3! group are theL i , referring to the molecule-fixed frame, while those of tsecond one are theJi referring to the space-fixed frame. The generators of the Abelian g

R(2l11)2 belong to a family of tensor operatorsTmnl (l50,1

2,1,32,...; m,n50,6 12,...,6l). We

select the tensors withl51 in order to limit the CS to integer rotational quantum numbers. Thtensor operators commute among themselves, i.e.,

@ Tmnl ,Tm8n8

l#50 ~40!

and satisfy the relations

Tmnl†5~21!n2mT2m2n

l , ~41!

and

(m

Tmnl† Tmn8

l5dnn8 , (

nTmn

l† Tm8nl

5dmm8 . ~42!

In addition to the commutation relations among these tensor operators we need the ones wSO~3! generators in Eq.~29! and the relations~30!,

@ Lz ,Tmnl #5nTmn

l , @ Jz ,Tmnl #5mTmn

l , ~43!


l

theher


Downloaded 11 Apr

as well as

@ L6 ,Tmnl #5@l~l11!2n~n71!#1/2Tmn7l

l ,~44!

@ J6 ,Tmnl #5@l~l11!2m~m61!#1/2Tm6ln

l ,

where

L65Lx7 i L y , J65 Jx6 i Jy . ~45!

The rotation matrix elementsDmnl (a,b,g), (l50,1

2,1,32,...; m,n50,6 12,...,6l) are a real-

ization of theTmnl operators and define their action on the statesuIMK&. Here it suffices to know

that

T21211 uI 2I 2I &5S 2I 11

2I 13D 1/2

uI 11 2I 21 2I 21&. ~46!

C. Construction of coherent states

The straightforward application of Perelemov’s prescription9 would make the set of rotationacoherent states be

X~x,c!Z~z,v!Y~ymn1 !u000&, ~47!

where theX(x,c) andZ(z,v) operators are two parametrizations of the SO~3! group

X~x,c!5NXexJ1e2x* J2e2 icJz,~48!

Z~z,v!5NZezL1e2z* L2e2 ivLz,

with the real parameters 0<c<2p, 0<v<2p, and the complex parametersx (2`<Rex<`,2`<Im x<`) andz (2`<Rez<`, 2`<Im z<`), respectively.NX andNZ denote normaliza-tion constants. The unitary operatorY(ymn

1 ) is a general element of the Abelian groupR9 gener-ated by the tensor operatorsTmn

1 , i.e.,

Y~ymn1 !5expS (

m,nymv

1 Tmn1 D ~49!

with the complex parametersymn1 (2`<Reymn

1 <`, 2`<Im ymn1 <`) satisfying ymn

1 5(21)m2v11y2m2n

1* . This mode of construction combines two spin CS with CS belonging toabelian groupR9. This produces a set of CS of some complexity, which will not be furtanalyzed.

Instead, in analogy with the Janssen’s approach,16 we propose the simpler construction

uxyz&5exp@2 12yy* ~11xx* !2~11zz* !2#exJ1ezL1 exp@y f~ I !T2121

1 #u000&, ~50!

where the parametery relates to the above discussion such thaty5y21211 . The functionf ( I ) is

f ~ I !5A~2I 21 I !/~2I 21!, ~51!

where the operatorI is defined by~compare Ref. 16!

I uIMK &5I uIMK &. ~52!


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fol-


Downloaded 11 Apr

The effect of thef ( I ) function is to generate a desired Poisson distribution by canceling sfactors occurring in Eq.~46!. The operatorI can be expressed in terms of the Schwinger booperators, but in the present context it can be seen as a purely formal construct that sesimplify some expressions. Note the subtle differences in the normalization factor and in theexponential operator in comparison to those in Ref. 16. It follows straightforwardly that

exp~y f~ I !T21211 !u000&5 (

I 50,1,...

`~y f~ I !T2121

1 ! I

I !u000&5 (

I 50,1,...

`yI

AI !uI 2I 2I & ~53!

and that

ezL1uI 2I 2I &5 (n50

`zn

n!L1

n uI 2I 2I &

5 (n50

2Izn

n!$2I ~2I 21!¯@2I 2~n21!#%1/2~n! !1/2uI 2I 2I 1n&

5 (K52I

Iz~ I 1K !

@~ I 1K !! #1/2F ~2I !!

~ I 2K !! G1/2

uI 2IK &, ~54!

wheren5I 1K has been used from the second to the third line. By changingM to K, andL1 to

J1 , the analogous expansion ofexJ1uI 2I 2I & is obtained. These results make possible thelowing expression for the CS:

uxyz&5exp@2 12 yy* ~11xx* !2~11zz* !2#

3 (IMK

H @~2I !! #2

~ I 1M !! ~ I 2M !! ~ I 1K !! ~ I 2K !! J 1/2x~ I 1M !yIz~ I 1K !

~ I ! !1/2 uIMK &. ~55!

Each member in this set of CS is normalized to unity since

^xyzuxyz&5exp@2yy* ~11xx* !2~11zz* !2#

3 (IMK

@~2I !! #2

~ I 1M !! ~ I 2M !! ~ I 1K !! ~ I 2K !!

~xx* !~ I 1M !~yy* ! I~zz* !~ I 1K !

I !

5exp@2yy* ~11xx* !2~11zz* !2#(I 50

`~yy* ! I

I !

3 (M52I

I~2I !! ~xx* !~ I 1M !

~ I 1M !! ~ I 2M !! (K52I

I~2I !! ~zz* !~ I 1K !

~ I 1K !! ~ I 2K !!

5exp@2yy* ~11xx* !2~11zz* !2#(I 50

`@~yy* !~11xx* !2~11zz* !2# I

I !51, ~56!

where the power expansions, such as

~11zz* !2I5 (K52I

I~2I !!

~ I 1K !! ~ I 2K !!~zz* ! I 1K ~57!


e

se CS.

n, we

second

eed toe bino-


Downloaded 11 Apr

have been used. It is noteworthy that we now have a Poisson distribution in the variableyy* (11xx* )2(11zz* )2. In addition, the overlapx8y8z8uxyz& between two different members of thset of CS is

^x8y8z8uxyz&5exp@2 12yy* ~11xx* !2~11zz* !2#exp@2 1

2y8y8* ~11x8x8* !2~11z8z8* !2#

3exp@yy8* ~11xx8* !2~11zz8* !2#, ~58!

which can be obtained in an analogous manner showing the general nonorthogonality of theBecause of their construction, the$uxyz&% states satisfy the condition Eq.~1! for CS. In order to

verify whether these CS satisfy the stronger or the weaker formulation of the second conditioneed to construct a proper measuredm. Using the measure of Ref. 16 as a guide we obtain

dm6~x,y,z!51

p3 $4@~11xx* !~11zz* !#4~yy* !228@~11xx* !~11zz* !#2yy* 11%dx dy dz,

~59!

where

dx5d Rexd Im x, dy5d Reyd Im y, dz5d Rezd Im z ~60!

so that

E dm6~x,y,z!uxyz&^xyzu5 (IMK

uIMK &ÎMK u51. ~61!

Then, it follows immediately@see Eq.~55!# that

uIMK &5E dm6~x,y,z!expF21

2yy* ~11xx* !2~11zz* !2Gx* I 1My* Iz* I 1K

3H @~2I !! #2

I ! ~ I 1M !! ~ I 2M !! ~ I 1K !! ~ I 2K !! J 1/2

uxyz&. ~62!

Note that both the CS from Ref. 16 and the present ones satisfy the weaker version of thecondition for CS, because the measure of neither is positive.

D. Coherent state operator averages

In order to develop the dynamics related to the rotational CS certain operator averages nbe determined. Evaluation of the necessary integrals involves using some properties of thmial power expansion and the Poisson distribution. The final results are as follows:

^ I &5yy* ~11xx* !2~11zz* !25z, ~63!

^Lx&5z1z*

11zz*z, ^Ly&5

i ~z* 2z!

11zz*z, ^Lz&5

~zz* 21!

11zz*z, ~64!

^L i2&5H z

21^L i&

2F111

2zG if z.0,

0 if z50,

~65!

^L i L j1L j L i&5H 2^L i&^L j&F111

2z G if z.0,

0 if z50,

~66!


rderrences

own

r

. Thejec-


Downloaded 11 Apr

whereiÞ j , and the notation¯&5^xyzu¯uxyz& is used. By changingL i to Ji , andz to x* in theabove expressions, the averages of the components ofJ are obtained. The integral^xyzu I uxyz&turns out to be slightly different from that of Ref. 16. However, the functionality of the first-oaverages with respect to that basic integral remains essentially the same and real diffeappear in the second-order averages.

Uncertainty relationships for the CS can be derived by combining the well-knrelationship10

~DL i !2~DL j !

2> 14u^Lk&u2 ~ iÞ j Þk!, ~67!

where

~DL i !25^L i

2&2^L i&2, ~68!

with Eqs.~64! and ~65! to obtain

~DLi !2~DL j !

25H 1

4F11

^L i&2

z2 GF11^L j&

2

z2 Gz2 if z,0

0 if z50.

~69!

Note that in the special case of^L i&5^L j&50, z.0 the uncertainty relationship is minimized fothat pair of components, i.e., (DL i)

2(DL j )25 1

4u^Lk&u2, (iÞ j Þk).

E. Reparametrization

For the purpose of physical interpretation a new parametrization of the CS is introducedparameters related to the spin CS, i.e.,x andz, are changed by adopting the stereographic protion onto a plane18

x5e2 ia cotb

2, z5e1 ig cot

d

2, ~70!

where 0<a,g<2p, and 0<b,d<p. The remaining parametery is expressed as

y5z1/2sin2b

2sin2

d

2ei ~a2g2e!, ~71!

where 0<z<` and 0<e<2p. In terms of these new parameters one finds

xx* 5cot2b

2, zz* 5cot2

d

2, 11xx* 5cosec2

b

2,

~72!

11zz* 5cosec2d

2, yy* 5z sin4

b

2sin4

d

2, yy* ~11xx* !2~11zz* !25z,

and


e

e


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uxyz&5uabgdez&

5expS 2z

2D (IMK

H @~2I !! #2

~ I 1K !! ~ I 2K !! ~ I 1M !! ~ I 2M !! J 1/2Fe2 iM a cosI 1MS b

2 D sinI 2MS b

2 D G3Fe1 iKg cosI 1MS d

2D sinI 2KS d

2De2 i I eG z1/2

AI !uIMK &

5expS 2z

2D (IMK

DMII ~a,b,0!DKI

I ~2g,d,e!z1/2

AI !uIMK &, ~73!

where the definition of the rotational matricesDMKI 18,20 has been used.

We write

âbgdezu I uabgdez&[^ I &5z ~74!

and similarly

^Lx&5z cosg sind, ^Jx&5z cosa sinb,

^Ly&5z sing sind, ^Jy&5z sina sinb, ~75!

^Lz&5z cosd, ^Jz&5z cosb,

and

^L2&5^ J2&5z~z12!. ~76!

From these expressions, it follows that the parameterz is the angular momentum modulus, thpairs of anglesg,d, anda,b are the azimuthal and the polar angle of the^L & and^J& vectors in thebody-fixed and the space-fixed frame, respectively. The anglee is associated with the relativorientation of the body-fixed and the space-fixed frames. Finally, the probabilityPIMK(z,b,d) tofind the rotational stateuIMK& in the CS is

PIMK~z,b,d!5H @~2I !! #2

~ I 1K !! ~ I 2K !! ~ I 1M !! ~ I 2M !! J Fcos2~ I 1M !b

2sin2~ I 2M !

b

2 G3Fcos2~ I 1K !

d

2sin2~ I 2K !

d

2 Gexp~2z!z I

I !

5H @~2I !! #2

~ I 1K !! ~ I 2K !! ~ I 1M !! ~ I 2M !! J p~ I 1M !~12p!~ I 2M !

3q~ I 1K !~12q!~ I 2M ! exp~2z!z I

I !, ~77!

wherep5cos2 d andq5cos2 b. Note thatPIMK(z,b,d) combines binomial distributions inp andq, and a Poisson distribution inz. The result

(I 50,1,...,

`

(M52I

I

(K52I

I

PIMK~z,b,d!51 ~78!

is readily obtained. Alternatively, one can write


tion of

ent


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PIMK~z,b,d!5uDMII ~a,b,0!u2uDKI

I ~2g,d,e!u2z I

I !exp~2z!

5@dMII ~b!#2@dKI

I ~d!#2z I

I !exp~2z!. ~79!

This probability exhibits more detail than is commonly needed and when used in the calculasuitableS-matrix elements averaging procedures will wash out the excessive details.

F. Coherent state dynamics

Applying Ehrenfest’s theorem to the operatorsL i for i 5x,y,z, i.e.,

d

dt^L i&52 i ^@H rot ,L i #&5K (

j« i jk

1

2Ak~ L j Lk1LkL j !L , ~80!

for z.0 results in@see Eq.~66!#

d

dt^Lx&5^Ly&^Lz&S 11

1

2z D S 1

Az2

1

AyD ,

d

dt^Ly&5^Lz&^Lx&S 11

1

2z D S 1

Az2

1

AxD , ~81!

d

dt^Lz&5^Lx&^Ly&S 11

1

2z D S 1

Ax2

1

AyD ,

and forz50,

d

dt^L i&50, i 5x,y,z. ~82!

The quasiclassical rotation vector or angular velocityv is then defined with components

v i5H 1

Ai

z1 12

z^L i&, z.0

0, z50.

~83!

The quasiclassical nature of this rotation vector is evident from

vx5vyvz

Ax~Ay2Az!,

vy5vxvz

Ay~Ax2Az!, ~84!

vz5vxvy

Az~Ax2Ay!,

which are the classical Euler equations for the motion of a rigid body without torque.21 It followsthat the rotation vectorvW behaves exactly as that of a classical rigid body with the same mom


.

ngularl CS.

to be

tiontotal

same

ith and by

s ofher can

CS,valueelocitye by anundds tocopic

oundhave


Downloaded 11 Apr

of inertia when the CS are propagated by the Schro¨dinger equation with the Hamiltonian of Eq~28!. This is analogous to the classical motion Eq.~18! of the harmonic oscillator CS Eq.~6!.

However, the definition of the rotation vectorv in Eq. ~83! differs from the definition

v i51

Ai^L i& ~85!

by Janssen,16 which corresponds exactly to the classical definition21

v i51

AiLi . ~86!

This means that for the Janssen all-spin rotational CS the expectation values of the amomentaL i are quasiclassical variables, whereas they are not for our integer-only rotationaInspection of Eq.~83! reveals that the variables

L i5H z1 12

z^L i&, z.0

0, z50,

~87!

are quasiclassical. This difference requires some explanation, which, as will be shown, isfound in the Heisenberg uncertainty principle.

The proportionality factor (z1 12)/z between the quasiclassical variables and the expecta

values ofL i is a constant of the motion related to the total angular momentum. The limit ofangular momentum tending to zero is equivalent toz→0 @Eq. ~76!#. This means that for theJanssen all-spin CS

v25vx21vy

21vz2<

1

Ax~^Lx&

21^Ly&21^Lz&

2!<z2

Ax, ~88!

where we have used the conventionAz>Ay>Ax . From this it follows thatv2→0 when theangular momentum is decreased to zero and the CS are ‘‘spinning down.’’ In contrast, thecalculation for the integer-only CS shows that

1

AxS z1

1

2D 2

>v2>1

AzS z1

1

2D 2

~89!

and, thus, as the angular momentum decreases toward zero these CS keep ‘‘spinning’’ wangular momentum of12 in units of\, i.e., the zero angular momentum state cannot be reachecontinuously decreasing the value of the angular momentum.

This result is related to the uncertainty principle. The orientation of the rigid body in termthe Euler angles with decreasing angular momentum cannot be determined precisely, neitthe rotation vector being the time derivative of the orientation. As a result the integer-onlywhich are the appropriate ones to describe rotating bodies, yield a finite lower bound to theof the angular velocity as the angular momentum decreases. Thus, since the angular vcannot decrease below a certain value, one cannot assume that the orientation can changarbitrarily small amount in a given time interval. It is noteworthy that apart from the lower boEq. ~89! in the rotation vector length the motion is completely classical. The lower bound tenzero for increasing moments of inertia indicating that this effect is not present for macrosbodies.

The remaining question is why the Janssen all-spin CS do not exhibit this finite lower bon the angular velocity vector. The answer lies in the fact that half-integer spin systems


lwaysits

ntationn

eculardict the

iveults inemical

lysis islishedry candy a

in thisrotor.

llor,

atof the

am-


Downloaded 11 Apr

angular momenta that do not correspond to any position coordinate. As a result they apossess an uncertainty of1

2\ in the orientation part of the total angular momentum even whenvalue tends to zero. For such CS the attempt to precisely define the angular momentum oriedoes not introduce additional constraints allowing the angular velocityv to decrease to zero whethe total angular momentum does.

V. APPLICATIONS

The rotational CS can with advantage be applied to spectroscopic analysis of molprocesses. For instance, the harmonic oscillator CS have been employed to successfully predistribution of reaction products over vibrational quantum states.22 The role of the CS in thisanalysis was to allow a detaileda posterioriquantum level analysis of calculations on a reactmolecular system that only employed essentially classical trajectories. This procedure resgreat savings of computer time and thus increases the range of studies to quite complex chreactions.

Because rotations and vibrations are intimately coupled in molecular systems, such anamost successful when both kinds of motion are treated on an equal footing. This is accompby combining the canonical CS and the integer-only rotational CS developed here. The theobe applied without difficulties to general molecular fragments and it is straightforward to stugeneral asymmetric rotor. However, in order to avoid inessential technicalities we describesection how the theory applies to a diatomic molecule, which can be thought of as a linear

A. Linear rotor CS

By selecting the body-fixedz axis of a diatomic molecule (Ax5Ay , Az50) as that of themolecular bond, the component of the angular momentum vectorL in that direction vanishes at atimes (K50). In that case, the rotational statesuIMK& become the eigenstates of the linear roti.e., the spherical harmonicsYIM (u,w), see Eq.~39!. Furthermore, insertingvz50 into Eq.~84!makes the other two components ofv in the body-fixed frame constant as well. This implies ththez parameter does not vary in time, its specific value being dependent on the orientationx and they axes in the body-fixed frame. The CS of Eq.~55! then become

ûwuxy&diat5exp@2 12yy* ~11xx* !2#(

IMH ~2I !!

~ I 1M !! ~ I 2M !! J 1/2x~ I 1M !yI

AI !YIM ~u,w!, ~90!

where the superfluous terms inzI 1K have been omitted. Alternatively, these CS can be reparetrized so that

ûwuxy&diat5ûwuabz&diat

5expS 2z

2D(IM

H ~2I !!

~ I 1M !! ~ I 2M !! J 1/2Fe2 iM a cosI 1Mb

2sinI 2M

b

2 G z I /2

AI !YIM ~u,w!

5expS 2z

2D(IM

DMII ~a,b,0!

z I /2

AI !YIM ~u,w!. ~91!

When the CS are prepared with the vectorâbzuJuabz& normal to thexy plane ~i.e., withâbzuJxuabz&5âbzuJyuabz&50) it holds that the initial value ofb is 0, or p. Consider thecaseb50. Then,

ûwu00z&diat5expS 2z

2D(I

z I /2

AI !YII ~u,w!, ~92!


allye

dion

th thee

vector

.


Downloaded 11 Apr

where the anglea has been arbitrarily set to zero. The CS time evolution ofu0b(t)z&diat from thatinitial condition, is

ûwu0b~ t !z&diat5ûwuexp~2 iH t !u00z&diat

5ûwuexpS 2z

2D(I

z I /2

AI !expF2 i

I ~ I 11!

2At GYII ~u,w!,

5expS 2z

2D(I

z I /2

AI !YII ~u,w2v I t !, ~93!

whereH is the diatomic Hamiltonian,A the moment of inertia, and

v I5~ I 11!

2A. ~94!

In the last line of Eq.~93!, the explicit form23

YI 6I~u,w!5~21! I1

2I I ! S ~2I 11!!

4p D 1/2

sinI u exp~ i I w! ~95!

has been used. From Eqs.~93! and~95!, it is easy to see that the rotational CS peak symmetricaroundu5p/2 ~i.e., the maximum lies in thexy plane! throughout the evolution. The shape of throtational CS with respect to the anglew is more difficult to describe analytically. For largezvalues, the superposition of the spherical harmonics given by Eq.~93! is sharply peaked arounI 5I max;z. When this holds, the sum overI can be approximated by an integration. The evaluatof that integral by the stationary phase method24 reveals that the CS peak around the valuef5fmax where

fmax~ t !;~ I max1

12!

At;

~z1 12!

At;vzt. ~96!

This implies that when the total angular momentum is high the peak’s center moves wiconstant angular velocityvz along the equator of theu,w sphere. The general properties of thdiatomic rotational CS can be easily derived form this example prepared withb50. If the rotationR(a,b,0) is applied tou00z&diat then the general diatomic rotational CS, Eq.~91!, are recovered

ûwuabz&diat5ûwuR~a,b,0!u00z&diat

5expS 2z

2D(IM

DMII ~a,b,0!

z I /2

AI !YIM ~u,w!

5expS 2z

2D(I

z I /2

AI !YII ~u8,w8!, ~97!

where the properties of the spherical harmonics have been applied and where the anglesu8 andw8are given with respect to the rotated space-fixed frame. The rotation transforms the

âbzuJuabz& from thez direction of the space-fixed frame to the~a,b! direction, in accordancewith Eq. ~75!. The final expression in Eq.~97! is formally identical to the nonrotated CS in Eq~92!. The properties ofuabz&diat are the same as those ofu00z&diat but now referred to a planenormal to the vectorabzuJuabz&.


akes

thexed

ent in

se of akets,

ase

lD

pen-


Downloaded 11 Apr

No closed form expression is known for the rotational CS Eq.~55! or Eq. ~90!. However, inSec. V B a wave function for a diatomic molecule is constructed that clarifies the CS and mtheir structure more explicit.

B. The diatom END wave function

For purposes of interpretation in terms of CS we rewrite the END wave function incenter-of-mass~c.m.! frame. The actual END propagation is always done in the space-filaboratory frame, but the analysis of reagents~at the initial time! and products~at the final time!is better done in the c.m. frame of each fragment. The END wave function for each fragmthe narrow wave packet limit for the nuclei and with a Thouless determinant5 for the electronstakes the form

CEND~X,x,t !5Fnucl@X;R~ t !,P~ t !# f el@x;z~ t !,R~ t !#exp@ igEND~ t !#, ~98!

when there no longer exist any overlaps or exchange terms between fragments. For the cadiatomic fragment the nuclear part consists of two generalized frozen Gaussian wave pac

Fnucl~X;R,P!51

Nnucl)k51

2

expH 2ak@Xk2Rk~ t !#21i

\Pk~ t !•@Xk2Rk~ t !#J . ~99!

The electronic partf el@z(t),R(t)# is a Thouless single determinant wave function. The total phgEND(t) is the quantum mechanical action

gEND~ t !5E0

t

ds L@R~s!,P~s!,z~s!#, ~100!

in terms of the END quantum LagrangianL(R,P,z).The Gaussian wave packets have finite width, 1/ak , explicitly defined as those of canonica

CS @compare Eqs.~22! and ~23!#. The SC limit of zero width then yields the simplest ENapproximation.

The transformation to the c.m. coordinates is using Jacobi coordinates, i.e.,

X051

M~m1X11m2X2!, x5X22X1, ~101!

with a similar transformation of the average nuclear positions

R0~ t !51

M@m1R1~ t !1m2R2~ t !#,

~102!r ~ t !5R2~ t !2R1~ t !,

and withM5m11m2 the total mass. This transformation results in the product of two indedent Gaussian wave packets:

Fnucl~X;R,P!5F0~X0;R0,P0!F int~x;r ,p! ~103!

only if the condition:v15v25v is imposed. Then it follows that

F0~X0;R0,P0!51

N0exp$2a0@X02R0~ t !#21 iP0~ t !•@X02R0~ t !#%, ~104!

and


learhase

s

hase

f thefineon

eterstion


Downloaded 11 Apr

F int~x;r ,p!51

Nintexp$2am@x2r ~ t !#21 ip~ t !•@x2r ~ t !#%. ~105!

Here, the new momentum parameters are

P0~ t !5P1~ t !1P2~ t ! ~106!

and

p~ t !5m~R22R1!5m r , ~107!

respectively, wherem is the reduced mass. The transformed exponents turn out to be

a05Mv

2~108!

and

am5mv

2, ~109!

in units of 1/\. The electronic part of the wave function, being a function of relative nucpositions only, is not be affected by this transformation. A similar partition of the total pmeans that

gEND~ t !5g0~ t !1g int~ t !, ~110!

where g0(t) contains only the nuclear c.m. variables andg int(t) the internal nuclear variablealong with the electronic parameters.

The c.m. part is a Gaussian with a trivial time evolution. This part together with its pg0(t) can be totally separated from the rest leaving the internal END wave function

C intEND~ t !5F int@r ~ t !,p~ t !# f el@z~ t !,r ~ t !#exp@ ig int~ t !#. ~111!

The internal wave function is now split into a vibrational and a rotational part. Because ocoupling of rotations and vibrations in molecules this separation must be approximate. Dex5xn andr5rm, wheren andm are unit vectors. For a vibrating molecule rotating with rotativectorv we write the momentum

p5m r5pvm1mbv3m, ~112!

with pv the vibrational part of the momentum parallel to the axis of the molecule andb theequilibrium bond length. The END evolution of the molecule is described through the paramr (t), p(t), andz(t) only. When the coupling of rotation and vibration is neglected the rotavector is constant yielding the angular momentum

L5Av5mb2v. ~113!

The exponent ofF int(x;r ,p) in Eq. ~105! becomes

2am@x2r #21 ip•@x2r #52am~x2r !222amr ~x2r !n•~n2m!

2amr 2~n2m!21 i ~x2r !pvm•n1 imb~x2r !~v3m!•n

1 irp vm~n2m!1 imbr~v3m!•~n2m!. ~114!

Introducing the anglesh betweenn andm andj betweenv3m andn yields


rof

s

ion


Downloaded 11 Apr

2am~x2r !222amr ~x2r !~12cosh!22amr 2~12cosh!,~115!

i ~x2r !pv cosh1 imbv~x2r !cosj2 irp v~12cosh!1 imbvr cosj.

Since x and r are close the angleh is small and so isp/22j. Using these facts in an ordeanalysis consideringx2r as small, omitting terms of third order or higher in the real part, andsecond order or higher in the imaginary part, and leaving out the coupling term result in

2am~x2r !222amr 2~12cosh!,~116!

1 i ~x2r !pv1 imvbr cosj.

Replacingr in the angular part by the equilibrium bond distanceb, using Eq.~109! for am , andexpressingh andj in terms of the polar coordinatesu,w of x in the laboratory frame using obviouright spherical triangles produces a vibrational and a rotational factor

F int~x,u,w;r ,0,0!'Fvib~x;r !F rot~u,w;0,0!, ~117!

with

Fvib~x;r !51

NvibexpF2

mv

2~x2r !21 ipv~x2r !G ~118!

and

F rot~u,w!51

Nrotexp@2mvb2~12sinu cosw!1 imvb2 sinu sinw#

51

Nrotexp@2mvb2~12sinueiw!#. ~119!

The general case is obtained by rotatingv to the orientationa,b in the laboratory frame.Identification of the CS parameterz can be accomplished by evaluation of the expectat

values ofJ and its square with respect to the END wave function Eq.~99!,

Fnucl5F0F int5F0FvibF rot . ~120!

One finds easily that

^Jx&5mvb2 sinb cosa,

^Jy&5mvb2 sinb sina, ~121!

^Jz&5mvb2 cosb,

and that

^Jx2&5^Jx&

21Axxv, ^Jy2&5^Jy&

21Ayyv, ^Jz2&5^Jz&

21Azzv. ~122!

Using the result

Tr~A!52mb2, ~123!

it follows that


S

states, ofpriatef fixedl factormatch

on iss andwave

lutionfor

of zeror wavetional

or

in the-

haveationalr-only

m de-tructing

factsherotors,from ateger-

endental CS


Downloaded 11 Apr

^J2&5^J&212mb2v5mvb2~mvb212!5z~z12!, ~124!

such that the CS parameterz may be identified with the total nuclear angular momentummvb2 ofthe diatomic molecule in units of\.

The expansion of the rotational factor of the END wave function Eq.~119!

F rot~u,w;0,0!51

Nrotexp~2z!(

I

~z sinueiw! I

I !~125!

should be compared to the CS Eq.~93! using Eq.~95!

ûwu00z&diat5exp~2z/2!(I

S ~2I 11!!!

4p~2I !!! D 1/2~2z1/2sinueiw! I

AI !. ~126!

The two series have the same structure for the argumentsu andw, but consecutive terms in the Cseries decrease less rapidly as functions ofI.

Even though the two functions are not identical the rotational CS are useful for finalanalysis. That the vibrational CS match the vibrational part of the END wave function icourse, not surprising as the END wave function is built from Gaussians with the approwidths. The CS, because of their nature, retain shape during evolution. Thus, the choice oa Gaussian shape for the END wave function ensures that also the shape of the rotationapersists during evolution. Our analysis has shown that the average orientation and the widththose of the rotational CS. Furthermore, the END wave function by the TDVP constructilimited to the dynamics of average values, which are the internal vibrational coordinateorientation of the body fixed frame. It then makes sense to replace the factors of the ENDfunction by their CS counterparts with the matching parameters, including the time evogenerated by the END wave function. This procedure is the recommended one to useaposterioriquantum state analysis even when the END evolution has been done for the casewidth Gaussians, i.e., classical nuclei. Thus, since it is now established that the END nucleafunction for low excitations can to a good approximation be represented as a product of vibraand rotational CS, the expressions in Eqs.~6! and ~79! can be used to compute probabilities fvibrational and rotational eigenstates, and, thus, providea posteriori quantum vibrational androtational resolution of cross sections obtained from classical trajectory calculations.

VI. CONCLUSION

The canonical CS have countless applications. First derived by Schro¨dinger2 and later ana-lyzed by many1 these CS display remarkable properties. One of the most useful propertiescontext of molecular processes is the quasiclassical evolution. Janssen16 constructed all-spin rotational CS that evolve quasiclassically. Involving both integer and half-integer spins these CSnot seen much application to physical systems. Our work establishes that integer-only rotCS can be constructed that exhibit quasiclassical dynamics. A notable property of the integerotational CS is the nonzero minimum angular velocity attained as the angular momentucreases to zero. The Janssen construction does not follow Perelomov’s prescription of consgroup-related CS. The metric of these CS for the resolution of the identity is nondefinite. Thethat the construction of Atkins and Dobson17 employs Perelomov’s prescription for the case of tsymmetric rotor CS, that it coincides with the Janssen construction restricted to symmetricand that these have a positive definite measure, make it plausible that there exists CSsuitable Perelomov construction with a positive measure for both the all-spin CS and the inonly CS.

It is known that the canonical CS remain quasiclassical in the presence of a time-deplinear external field. We have not investigated whether this holds true also for the rotationwhen subjected to some external torque.


els ofvalid

nona-

dged.

s


Downloaded 11 Apr

In this paper we have concentrated on extracting probabilities for rotational quantum levmolecules from nuclear trajectory calculations of molecular processes. This application isboth for dynamics involving predetermined potential energy surfaces as well as for directdiabatic approaches such as END.

ACKNOWLEDGMENTS

The support of this research by of the Office of Naval Research is gratefully acknowleThe authors thank Professor John Klauder for useful conversations about coherent states.

1J. R. Klauder and B.-S. Skagerstam,Coherent States, Applications in Physics and Mathematical Physics~World Scien-tific, Singapore, 1985!.

2E. Schrodinger, Naturwissenschaften14, 664 ~1926!.3R. J. Glauber, Phys. Rev. Lett.10, 84 ~1963!.4E. Deumens, A. Diz, R. Longo, and Y. O¨ hrn, Rev. Mod. Phys.66, 917 ~1994!.5D. J. Thouless, Nucl. Phys.21, 225 ~1960!.6D. J. Thouless,The Quantum Mechanics of Many-Body System~Academic, New York, 1961!.7M. Rasetti, J. Theor. Phys.13, 425 ~1973!.8A. I. Solomon, J. Math. Phys.12, 390 ~1971!.9A. M. Perelomov,Generalized Coherent States and Their Applications~Springer, New York, 1986!.

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On rotational coherent states in molecular quantum dynamics