Analytic dynamics of the Morse oscillator derived by semiclassicalclosures

Eric M. Heatwole and Oleg V. Prezhdoa�

Department of Chemistry, University of Washington, Seattle, Washington 98195-1700, USA

�Received 24 October 2008; accepted 25 May 2009; published online 29 June 2009�

The quantized Hamilton dynamics methodology �O. V. Prezhdo and Y. V. Pereverzev, J. Chem.Phys. 113, 6557 �2000�� is applied to the dynamics of the Morse potential using the SU�2� ladderoperators. A number of closed analytic approximations are derived in the Heisenberg representationby performing semiclassical closures and using both exact and approximate correspondencebetween the ladder and position-momentum variables. In particular, analytic solutions are given forthe exact classical dynamics of the Morse potential as well as a second-order semiclassicalapproximation to the quantum dynamics. The analytic approximations are illustrated with the O–Hstretch of water and a Xe–Xe dimer. The results are extended further to coupled Morse oscillatorsrepresenting a linear triatomic molecule. The reported analytic expressions can be used to accelerateclassical molecular dynamics simulations of systems containing Morse interactions and to capturequantum-mechanical effects. © 2009 American Institute of Physics. �DOI: 10.1063/1.3154143�

I. INTRODUCTION

The Morse interaction potential carries great importancein chemical systems and has been studied exhaustively. It hasbeen used to model water,1 alcohols,2 hydrocarbons,3,4 raregases,5,6 metals,7,8 surfaces,9,10 semiconductor particles,11,12

carbon nanotubes,13,14 biomolecules,7,15 and many othertypes of systems. The Morse oscillator belongs to the class ofsupersymmetric potentials, which also include the harmonicoscillator and the Coulumb interaction.16,17 A variety ofapproaches18–22 including factorization methods,21 can beused to find the wave functions and energy spectrum of theMorse oscillator. It is known that the SU�2� group is thedynamical group for the bounded region of the Morsepotential.23 Recently, the ladder operators associated withthis SU�2� group have been explicitly constructed,18–22 seealso Ref. 24. It is natural to consider the time evolution ofthese ladder operators, much like the time evolution of theraising and lowering operators of the harmonic oscillator25–27

or spin system.28,29 By considering equations of motion�EOM� for the ladder operators one can expect to extendstationary quantum mechanics to quantum dynamics and toobtain useful analytic expressions in time domain. Comparedto the alternative representations of the Morsedynamics,24,30,31 the SU�2� operators provide a concise rep-resentation that limits the number of EOM.

The Heisenberg EOM for the Morse ladder operators21

form an infinite hierarchy of coupled linear differential equa-tions. Valuable approximations to the exact hierarchy can beobtained using the recently developed quantized Hamiltondynamics �QHD� methodology.28,32–46 QHD truncates the hi-erarchy by closures47,48 and represents the higher-order ex-pectations values of quantum-mechanical operators in termsof classical-like products of the lower-order expectation val-

ues. A number of semiclassical approximations can be ob-tained this way, depending on the decomposition level. Al-ready at low orders, QHD can provide dramaticimprovements over classical mechanics at very little cost. Aclosed analytic solution for the classical evolution of theMorse system can be helpful as well. It can replace the cor-responding numerical solution used in molecular dynamics�MD� codes, substantially accelerate MD simulations in thegreat variety of systems1–15,49–52 and capture essentialquantum-mechanical effects, such as zero-point energy andtunneling. For example, vibrational motions of chemicalbonds involving hydrogen atoms are often described byMorse potentials. These motions are usually the fastest onesin molecular and solid state systems and, therefore, limit thelength of the time step during numerical integration of MDEOM. Replacing the numerical solutions with analytic dy-namics for the fast modes can speed up classical MD simu-lations. Further, dynamics of the hydrogen modes exhibitpronounced zero-point energy and tunneling features. Thesequantum-mechanical effects can be efficiently captured bythe semiclassical QHD approximations,28,32–46 both numeri-cally and especially if the solutions to the QHD EOM canalso be given in an analytic form.

The current paper presents the first examples of QHDclosures performed with the Morse raising and lowering op-erators, thereby extending the earlier studies carried out us-ing the ordinary harmonic operators. It derives the second-order QHD EOM for the Morse oscillator and finds ananalytic solution to these equations. The first-order QHD clo-sure naturally leads to the classical dynamics of the Morseoscillator as well as to the exact and approximate analyticsolutions to the classical dynamics. These formal results areillustrated with one and two-dimensional examples.

This paper is constructed as follows. First, we give abrief overview of the SU�2� ladder operators and examinesome of their properties. In particular, we discuss how to

a�Author to whom correspondence should be addressed. Electronic mail:prezhdo@u.washington.edu.

THE JOURNAL OF CHEMICAL PHYSICS 130, 244111 �2009�

0021-9606/2009/130�24�/244111/12/$25.00 © 2009 American Institute of Physics130, 244111-1

express physical observables and the Morse Hamiltonian interms of the ladder operators. Next, we derive approximateEOM for the ladder operators at the second-order in QHDand consider the first-order limit. The first-order of QHDallows us to derive a closed analytic form for the exact clas-sical EOM of the Morse system. For some applications itmay be desirable to have a simpler approximate form of thesolution. The simpler solution is obtained in a closed analyticform by considering a harmonic-like correspondence be-tween the ladder and position-momentum operators. Themethod is then expanded to a coupled linear chain of Morseoscillators. In this example the closed analytic EOM are notexact in the classical limit, however, they do provide an ac-curate representation of the classical dynamics. Once again,the harmonic-like operator correspondence is used to derivea simpler set of equations for the coupled Morse oscillators.In both one and two-dimensional cases, the harmonic-likeequations provide a great improvement over the true har-monic approximation of the Morse potential. Finally, theanalytic results are exemplified with three chemical systems:the O–H stretch of water, the Xe–Xe dimer, and a lineartriatomic. A number of initial conditions are considered foreach example. Different amounts of kinetic energy are givento the one-dimensional systems. The initial conditions for thetwo-dimensional example are chosen to represent the sym-metric stretch, the antisymmetric stretch and a mixed stretch-ing motion. This paper concludes with a summary of ourresults and a discussion of possible extensions and applica-tions.

II. THEORY

A. Ladder operators for the Morse potential

Setting the zero of energy at infinite separation, we de-fine the following Morse potential:

V�x� = V0�e−2�x − 2e−�x� . �1�

Here, V0 is the depth of the potential well, � is the width ofthe well, and x is the distance from the equilibrium position,which is set at x=0. This Morse potential gives rise to thewave function of the following form:18–22

�n��y� = Nn

�e−y/2ysLn2s�y� , �2�

where variable y is related to the coordinate x by

y = �e−�x, �3�

Ln2s�y� is the Laguerre function, and Nn

� is the normalizationconstant

Nn� =���� − 2n − 1���n + 1�

��� − n�. �4�

The constants � and s depend on the well depth V0 and width�, the total energy E, and the particle mass �

� =�8�V0

�2�2 ,

�5�

s =�− 2�E

�2�2 .

They are subject to the constraint 2s=�−2n−1, which givesrise to the energy spectrum of the Morse oscillator.

Frank and co-workers18–22 define the ladder operators K−

and K+

K− = − � d

dy�2s + 1� −

1

ys�2s + 1� +

�

2��s + 1

s,

�6�

K+ = � d

dy�2s − 1� +

1

ys�2s + 1� −

�

2��s − 1

s.

These ladder operators act on the wave function of the Morsepotential in the following way:

K−�n��y� = k−�n−1

� �y� ,

�7�K+�n

��y� = k+�n+1� �y� ,

where the eigenvalues k− and k+ are

k− = �n�� − n� ,

�8�k+ = ��n + 1��� − n − 1� .

The step-down operator, K−, destroys the ground state, whilethe step-up operator destroys the last bound state of theMorse potential, since s=1 for this state. It is now possible to

define the algebra associated with the operators K− and K+.By using Eq. �7� one can determine the commutator

�K+,K−��n��y� = 2k0�n

��y� , �9�

where the eigenvalue k0 is defined as

k0 = n −� − 1

2. �10�

From this one can define the operator K0

K0 = n −� − 1

2, �11�

and establish the identity

K0 = �yd2

dy2 +d

dy−

s2

y−

y

4+ n +

1

2 . �12�

Now the following commutation relations can be found for

the K−, K+, and K0 operators

�K+,K−� = 2K0, �13�

�K0,K−� = − K−, �14�

244111-2 E. M. Heatwole and O. V. Prezhdo J. Chem. Phys. 130, 244111 �2009�

�K0,K+� = K+. �15�

This set of equations defines the SU�2� group for the Morsepotential and allows the Hamiltonian to be written in thefollowing simple form:

H = −��

�K0

2, �16�

where

� =��2�

2�. �17�

B. Dynamics of the Morse oscillator

The set of operators defined in Sec. II A is useful fordetermining the static properties of the Morse oscillator.18–22

Can one apply these operators in order to obtain the evolu-tion of a particle in the Morse potential? It turns out that the

Heisenberg EOM for the ladder operators K−, K+, and K0

generate an infinite hierarchy of equations rather than aclosed set as for the harmonic oscillator. Approximations arerequired in order to solve the hierarchy in a closed form. Thiscan be done by utilizing the QHD approach.28,32–40,44–46

QHD is a simple approximation to quantum dynamics inthe Heisenberg representation. Except for special cases,where the dynamic variables and the Hamiltonian form aclosed Lie algebra, such as the harmonic oscillator or spinsystems, the Heisenberg EOM result in an infinite hierarchyof linear differential equations. QHD approximates higher-order variables by products of lower-order variables, gener-ating a closed set of nonlinear differential equations, muchlike in classical Hamilton dynamics. For example, a third-order variable can be decomposed in terms of the second andfirst-order variables as follows:32

ABC� � AB�C� + AC�B� + BC�A� − 2A�B�C� ,

�18�

preserving quantum effects to the second order. The higher isthe order of the decomposition, the closer the approximationlies to the exact quantum result. Decomposition to the first-order gives classical mechanics.

Using Eq. �16� for the Hamiltonian allows us to write thegeneral form of the Heisenberg EOM as

dA

dt= −

i

��A,H� =

i�

��A,K0

2� . �19�

Utilizing the commutator relations for the Morse ladder op-erators, Eq. �13�, gives the following equations to the secondorder:

dK+�t��dt

= −2i�

�K+K0�t��s,

dK−�t��dt

=2i�

�K−K0�t��s,

dK+K0�t��s

dt= −

i�

2��K+K0

2�t�� + 2K0K+K0�t��

+ K02K+�t��� ,

dK−K0�t��s

dt=

i�

2��K−K0

2�t�� + 2K0K−K0�t��

+ K02K−�t��� , �20�

where the K�K0�s terms refer to the symmetrized variables,

�K�K0�+ K0K��� /2. The hierarchy continues indefinitely,

since dK+K0i �s /dt K+K0

i+1�s. Applying the QHD approachand using Eq. �18� to decompose the third-order variables tothe second-order generates the following set of approximateequations:

dK+�t��dt

= −2i�

�K+K0�t��s,

dK−�t��dt

=2i�

�K−K0�t��s,

�21�dK+K0�t��s

dt= −

2i�

��2K+K0�t��sK0�t��

+ �K02�t�� − 2K0�t��2�K+�t��� ,

dK−K0�t��s

dt=

2i�

��2K−K0�t��sK0�t��

+ �K02�t�� − 2K0�t��2�K−�t��� .

Because K0 is a constant of motion, the resultant set of dif-ferential equations is linear and can be solved analytically.Specifically, we obtain the following expressions for the timeevolution of the expectation values of the raising and lower-ing operators:

K+�t�� =e−2i�/�K0�t�e−t − et��K+K0�0��s − K0�K+�0���

2�K02� − K0�2

+ K+�0��e−2i�/�K0�t �e−t + et�

2,

�22�

K−�t�� =e2i�/�K0�t�e−t − et��K−K0�0��s − K0�K−�0���

2�K02� − K0�2

+ K−�0��e2i�/�K0�t �e−t + et�

2,

where = �2i� /���K02�− K0�2. This provides an analytic

solution to the dynamics of the Morse oscillator that includesapproximate quantum-mechanical effects. Further analysis of

244111-3 Analytic dynamics of the Morse oscillator J. Chem. Phys. 130, 244111 �2009�

Eq. �22� will continue in future work. In particular, determin-

ing the initial conditions for K+K0�0�� and K02�− K0�2 may

be quite cumbersome in general. We will concentrate on thefirst-order limit for the remainder of this work.

C. The classical limit

The classical limit of Eq. �21� is obtained by decompos-ing all variables to the first-order,32

dK+�t��dt

= −i�

��K0�K+�t�� + K0�K+�t��� ,

�23�dK−�t��

dt=

i�

��K−�t��K0� + K0�K−�t��� .

Since K0� is a constant of motion, Eq. �23� are a set of lineardifferential equations and can be solved easily

K+�t�� = K+�0��exp�−2i�

�K0�t ,

�24�

K−�t�� = K−�0��exp�2i�

�K0�t .

These first-order approximate solutions can also be obtainedfrom the second-order solutions �22� by applying the com-

mutator relations �13� to the K�K0� terms and using theQHD decomposition, Eq. �18�. The second-order solutions,Eq. �22�, reduce to the classical solutions in the first order, asthey should.

The expectation value of K0 can be identified with thevalue of s from Eq. �5� and related to the initial conditionsusing the form of the Hamiltonian from Eq. �16�

K0� = − s = −�−2�E0

�2�2 . �25�

The initial total energy of the system E0 is negative andshould be handled with care, such that all contributions to theenergy term are decomposed to the first order. It is nowpossible to rewrite the EOM �24� in the following way:

K+�t�� = K+�0��exp�i�t� ,

�26�K−�t�� = K−�0��exp�− i�t� ,

where � is given by

� =2s�

��27�

and can be considered an effective frequency.The effective frequency, Eq. �27�, approaches the correct

harmonic limit of �=� as 2s→� and goes to the correctclassical frequency away from the harmonic limit. This canbe seen by using the Bohr semiclassical condition and relat-ing the classical action J=n� to the energy,53 allowing theuse of the classical relationship

dEn

dJ= �cl. �28�

The energy can be given in terms of the action as

E = ��2V0

�J −

�2

2�J2, �29�

and J=J /2�. With these relationships, the classical fre-quency becomes

�cl = � − ��2n , �30�

which is the same as Eq. �27� with 2s=2n−�.At this point it necessary to determine the initial condi-

tions for K+� and K−�. For this purpose we represent d /dyas21

d

dy= −

1

�y

d

dx= −

i

��

1

ypx �31�

and rewrite Eq. �6� in the following form:

K− = � i

��

1

ypx�2s + 1� +

1

ys�2s + 1� −

�

2��s + 1

s,

�32�

K+ = �−i

��

1

ypx�2s − 1� +

1

ys�2s − 1� −

�

2��s − 1

s.

Here, the y operator precedes px, according to the conventionoriginating from the chain rule used in Eq. �31�. As a result,the initial conditions for the ladder operator are expressed interms of the initial conditions for the classical momentum px

and Morse coordinate y, Eq. �3�. Inversely, rearranging Eq.�32�

px

y= i��� K+

2�2s − 1�� s

s − 1−

K−

2�2s + 1�� s

s + 1

+�

2�2s + 1��2s − 1� ,

�33�1

y=

K+

2s�2s − 1�� s

s − 1+

K−

2s�2s + 1�� s

s + 1

+�

�2s + 1��2s − 1�

one can express the original x coordinate in terms of theMorse ladder operators

x =1

�ln� �K+

2s�2s − 1�� s

s − 1+

�K−

2s�2s + 1�� s

s + 1

+�2

�2s + 1��2s − 1� . �34�

It is now possible to write down a closed analytic ex-pression for the dynamics of the Morse oscillator in the px

and y coordinates

244111-4 E. M. Heatwole and O. V. Prezhdo J. Chem. Phys. 130, 244111 �2009�

px�t�y�t�

=px�0�y�0�

cos �t −��s

y�0�sin �t

+���s

�2s − 1��2s + 1�sin �t ,

�35�1

y�t�=

px�0�y�0���s

sin �t +cos �t

y�0�+

�

�2s − 1��2s + 1��1

− cos �t� .

In going from Eqs. �33�–�35� we neglected two imaginaryterms. These terms are inversely proportional to s and aresmall in the classical limit, in which the total energy farexceeds the zero-point energy, s 1 /2, see Eq. �25�. Equa-tion �35� can be rearranged to give the following closed formfor the usual x and px coordinates

px�t� =

px�0�cos �t − ��s sin �t +y�0����s

�2s − 1��2s + 1�sin �t

px�0���s

sin �t + cos �t +y�0��

�2s − 1��2s + 1��1 − cos �t�

,

�36�

x�t� =1

�ln� e�x�0�px�0�

��ssin �t + e�x�0� cos �t

+�2

�2s − 1��2s + 1��1 − cos �t� .

It is easy to show using Eq. �35� that these are the exactsolutions for the classical Morse oscillator with the constantvalue of s, which depends on the initial conditions.

The steady-state solution of Eq. �35� is

px,eq = 0,

�37�

yeq =�

2+

��2 − 4

2,

which in the classical limit of � 1, Eq. �5�, goes to

px,eq = 0,

�38�yeq = � .

This is the expected answer for the classical Morse oscillator.

D. The harmonic-like limit

It is useful to consider simpler forms for the above equa-tions that will be valid near the harmonic limit. In the har-monic limit �→�, and the following approximations can bemade

2s � 2s + 1 � 2s − 1 � � ,

�39�

� s

s + 1�� s

s − 1� 1.

This allows us to approximate Eq. �33� as

px �i

2�2����K+

��−

K−

�� ,

�40�1 − e−�x

��� �

2���K+

��+

K−

�� ,

where Eq. �17� was used. The above result was already ob-tained by Lemus.22 It converges to the correct harmonic

limit, since21 K� /��→a† ,a.The above approximations allow us to write the follow-

ing simple analytic solution for the Morse oscillator

p�t� = px�0�cos �t −z�0���

�sin �t ,

�41�z�t��

=z�0�

�cos �t +

px�0���

sin �t ,

where z=1−exp�−�x�. Rewriting the equations in terms of xand p we obtain

p�t� = p�0�cos �t −��

�z�0�sin �t ,

�42�

x�t� = −1

�ln�1 − z�0�cos �t −

�

��p�0�sin �t .

One can derive Eq. �42� starting from Eq. �35� and employ-ing the same set of approximations. It should be emphasizedthat solution �42� is not equivalent to the harmonic approxi-mation to the Morse potential. It provides a significantly bet-ter description of the dynamics of the Morse oscillator, asillustrated with examples below. Equation �42� does con-verge to the harmonic solution in the appropriate limit.

E. Linear triatomic

For a more complicated example than the simple one-dimensional Morse oscillator, consider a linear triatomic, inwhich the atoms are coupled via Morse potentials. Ignoringthe translational and rotational components, the kinetic en-ergy of the triatomic can be written as

T =m

2�x1

2 + x32� +

M

2x2

2. �43�

Here, the end atoms x1 and x3 have mass m, and the middleatom x2 has mass M. The potential energy of the linear tri-atomic, in which the neighboring atoms interact by theMorse potential, is given by

V�x� = V0�e−2��x2−x1� − 2e−��x2−x1��

+ V0�e−2��x3−x2� − 2e−��x3−x2�� . �44�

Following the standard procedure,54 the center of mass of thetriatomic is set stationary, such that

x2 = −m

M�x1 + x3� . �45�

Eliminating x2 in Eqs. �43� and �44� gives the following formfor the kinetic:

244111-5 Analytic dynamics of the Morse oscillator J. Chem. Phys. 130, 244111 �2009�

T =m

2��1 +

m

M�x1

2 + x33� +

2m

Mx1x3 �46�

and potential

V�x� = V0�exp�− 2��− �1 +m

Mx1 −

m

Mx3

− 2 exp�− ��− �1 +m

Mx1 −

m

Mx3

+ exp�− 2���1 +m

Mx3 +

m

Mx1

− 2 exp�− ���1 +m

Mx3 +

m

Mx1

energies.Defining the new variables u, v

u = − �1 +m

Mx1 −

m

Mx3,

�47�

v =m

Mx1 + �1 +

m

Mx3

eliminates the cross terms in the potential energy andchanges the kinetic energy to

T =m

2�2m + M���m + M��u2 + v2� + 2muv� . �48�

Since the momentum can be defined as

pi =�T

� x, �49�

the kinetic energy can be rewritten in terms of the momen-tum, and the total Hamiltonian for the triatomic system canbe expressed as

Htot =1

2�pu

2 +1

2�pv

2 −pupv

M+ V�u� + V�v� , �50�

where V�u� and V�v� are identical one-dimensional Morsepotentials. The reduced mass � is defined as

� =mM

m + M. �51�

Using Morse ladder operators, the Hamiltonians Hu andHv for the noninteracting degrees of freedom u and v, can bewritten as18

Hu =pu

2

2�+ �2�1 − exp�− �u��2 = −

��

��K0

u�2,

�52�

Hv =pv

2

2�+ �2�1 − exp�− �v��2 = −

��

��K0

v�2.

The total Hamiltonian Htot including the interaction term is

Htot = Hu + Hv −pupv

M. �53�

The u and v modes are coupled through the kinetic energyterm. If the system is close to the harmonic limit, it is pos-sible to use Eq. �40� and to rewrite the coupling term usingthe K+, K− operators. This produced the following form ofthe total Hamiltonian:

Htot = −��

���K0

u�2 + �K0v�2 −

�

2M�K+

uK+v − K+

uK−v − K−

uK+v

+ K−uK−

v� . �54�

The expression for the coupling term with the exact corre-spondence between the momentum and ladder operators, Eq.�33�, is significantly more complex. Now one can apply thecommutator relationships, Eq. �13�, and to derive HeisenbergEOM for the linear triatomic.

Following a procedure that is similar to the steps takenin deriving the EOM for the single Morse oscillator, we ob-tain the following set of coupled differential equations forthe linear triatomic

dK0u�t��

dt=

i��

2�M��K+

u�t�� + K−u�t����K+

v�t�� − K−v�t���� ,

dK+u�t��

dt= −

i�

��2K0

u�t��K+u�t�� +

�

MK0

u�t���K+v�t��

− K−v�t��� ,

dK−u�t��

dt=

i�

��2K0

u�t��K−u�t�� −

�

MK0

u�t���K+v�t��

− K−v�t��� ,

dK0u�t��

dt=

i��

2�M��K+

v�t�� + K−v�t����K+

u�t�� − K−u�t���� ,

dK+v�t��

dt= −

i�

��2K0

v�t��K+v�t�� +

�

MK0

v�t���K+u�t��

− K−u�t��� ,

dK+v�t��

dt=

i�

��2K0

v�t��K+v�t�� −

�

MK0

v�t���K+u�t��

− K−u�t��� . �55�

It should be noted that in the above set of equations the

expectation values of K0u,v are no longer constants of motion,

in contrast to the single Morse oscillator. They evolve asenergy flows from one mode to the other. Therefore, the

244111-6 E. M. Heatwole and O. V. Prezhdo J. Chem. Phys. 130, 244111 �2009�

above differential equations are nonlinear. They cannot bereadily solved by analytic methods and must be propagatednumerically. Nevertheless, further progress can be made byanalytic techniques.

The EOM for K0u,v� can be rewritten in terms of mo-

menta and coordinates as

dK0u�t��

dt=

�2�

2Mpvu ,

�56�dK0

u�t��dt

=�2�

2Mpuv .

When the system is near the harmonic limit, and in particular

��2� /2M��1, the K0u,v terms can be approximated as con-

stants. This leads to the following simplified set of EOM

dK+u�t��

dt= −

i�

��2K0

u�t��K+u�t�� +

�

MK0

u�t���K+v�t��

− K−v�t��� ,

dK−u�t��

dt=

i�

��2K0

u�t��K−u�t�� −

�

MK0

u�t���K+v�t��

− K−v�t��� ,

dK+v�t��

dt= −

i�

��2K0

v�t��K+v�t�� +

�

MK0

v�t���K+u�t��

− K−u�t��� ,

dK+v�t��

dt=

i�

��2K0

v�t��K+v�t�� −

�

MK0

v�t���K+u�t��

− K−u�t��� . �57�

The K0u,v terms are set by the initial conditions. This new set

of equations is now linear with respect to the dynamic vari-ables, and it is possible to find a general analytic solution forthis system. However, the general form is quite complex andis not presented here. It is simpler to determine the solutionson a case by case basis.

If the initial energy of the system is split equally be-tween the two degrees of freedom, such that

K0u� = K0

v� = − s = −� �E0

�2�2 , �58�

the following EOM can be obtained

dK+u�t��

dt= i��K+

u�t�� +C

2�K+

v�t�� − K−v�t��� ,

dK−u�t��

dt= i��− K−

u�t�� +C

2�K+

v�t�� − K−v�t��� ,

�59�dK+

v�t��dt

= i��K+v�t�� +

C

2�K+

u�t�� − K−u�t��� ,

dK−v�t��

dt= i��− K−

v�t�� +C

2�K+

u�t�� − K−u�t��� .

Here, C is the mass ratio

C =�

M, �60�

and � is defined in Eq. �27�. The solutions of these equationsare derived in Appendix. The exact correspondence betweenthe position-momentum and ladder operators, Eq. �33�, gives

1

yu�t�= sin �1t

�1 + C

2s��� pu�0�

yu�0�−

pv�0�yv�0�

+ sin �2t�1 − C

2s��� pu�0�

yu�0�+

pv�0�yv�0�

+1

2cos �1t� 1

yu�0�−

1

yv�0�+

1

2cos �2t� 1

yu�0�+

1

yv�0� +��1 − cos �2t�

�2s + 1��2s − 1�,

1

yv�t�= sin �1t

�1 + C

2s��� pv�0�

yv�0�−

pu�0�yu�0�

+ sin �2t�1 − C

2s��� pu�0�

yu�0�+

pv�0�yv�0�

+1

2cos �1t� 1

yv�0�−

1

yu�0� +1

2cos �2t� 1

yu�0�

+1

yv�0� +��1 − cos �2t�

�2s + 1��2s − 1�. �61�

The harmonic-like correspondence, Eq. �40�, gives

zu�t��

= sin �1t�1 + C

2���pu�0� − pv�0��

+ sin �2t�1 − C

2���pu�0� + pv�0�� +

1

2cos �1t�zu�0�

− zv�0�� +1

2cos �2t�zu�0� + zv�0�� ,

zv�t��

= sin �1t�1 + C

2���pv�0� − pu�0��

+ sin �2t�1 − C

2���pu�0� + pv�0�� +

1

2cos �1t�zv�0�

− zu�0�� +1

2cos �2t�zu�0� + zv�0�� , �62�

244111-7 Analytic dynamics of the Morse oscillator J. Chem. Phys. 130, 244111 �2009�

where zu=1−e�u and similarly for zv. The frequency terms�1 and �2 are

�1 = �C + 1� ,

�63��2 = �1 − C� .

Both solutions, Eqs. �61� and �62�, reduce in the appropriatelimit to the EOM for the linear triatomic with the atomsinteracting by harmonic potentials.

III. RESULTS AND DISCUSSION

The Morse oscillator is used extensively in literature andexamples to choose from abound.1–15,49–52 Here, we considertwo cases for the single Morse oscillator and one example ofcoupled Morse oscillators.

A. O–H stretch

The first example is the O–H stretch of water, as repre-sented in the SPC-F potential,1 which is widely used inphysical-chemical modeling of bulk water. The potential in-cludes both harmonic and anharmonic descriptions of theO–H stretching motion. We consider the more accurate an-harmonic version represented by a Morse potential of theform given in Eq. �1�. The two parameters of the Morsepotential are V0=0.708 mdyn Å and �=2.567 Å−1. Know-ing the reduced mass of the O–H stretch allows us to calcu-late the parameter �, Eq. �5�, as �=34.7547. It should benoted that there exist only 16 bound states for this Morsepotential. Therefore, even at high energies the system is farfrom the classical limit, where a large number of boundstates is required. In spite of this, the classical dynamicsgenerated by the SPC-F potential is regarded as a faithfulrepresentation of the O–H motion of water.1

The exact classical-mechanical evolution was generatednumerically using the fifth-order Runge–Kutta algorithm.The numerical results were compared to both exact and ap-proximate analytic formulas, Eqs. �36� and �42�, respectively,as well as to the harmonic oscillator approximation with thefrequency determined by the curvature of the Morse poten-tial at the bottom of the well. Three separate initial condi-

tions were considered. The initial potential energy was set tothe minimum, and the initial kinetic energy was varied fromnear the harmonic limit to far from the harmonic limit. Fig-ures 1�a�–1�c� shows the results for the initial kinetic ener-gies of ��, 3��, and 7��, respectively. As expected, Eq.�36� gives the exact result for all initial conditions chosen,while the harmonic-like approximation, Eq. �42�, shows asubstantial improvement over the harmonic oscillator de-scription.

B. Xe–Xe dimer

Next, we investigate an example with a larger value of �,bringing the system closer to the classical limit. For thisexample we choose the Morse potential used in Ref. 5 tomodel clusters of xenon atoms. In present, we consider twoxenon atoms interacting through the Morse potential. Ref. 5gives the following parameters for the potential, Eq. �1�:V0=3.05�10−21 J and �=2.439 nm−1. Given the reducedmass of Xenon, these parameters correspond to �=283.5729. There are 141 bound states for this system, and itis expected to be much closer to the classical limit than thefirst example.

As before, three cases are considered with the initialpotential energy set to the minimum for all three. Figures1�d�–1�f� shows the results for the initial kinetic energies of��, 20��, and 50��, respectively. These initial kinetic en-ergies correspond to the average thermal energies at 3, 60,and 150 K. Equation �36� gives the exact results for thisexample. Both the harmonic oscillator approximation and theharmonic-like solution, Eq. �42�, fail for the high energy casedue to the large anharmonicity accessible near the dissocia-tion limit. Nevertheless, the harmonic-like approximationprovided by Eq. �42� shows a substantial improvement overthe true harmonic description.

C. Linear triatomic

Next, consider two coupled Morse oscillators represent-ing a linear triatomic molecule. For this example, the systemis taken to be far into the classical limit with the followingMorse parameters V0=1250 and �=�0.0003. The reducedmasses are �=0.75 and M =3, Eq. �50�. The units are such

-0.15-0.1-0.05

00.050.10.150.2

0 0.5 1 1.5 2 2.5 3 3.5 4

X(a)

-0.3-0.2-0.10

0.10.20.30.4

0 1 2 3 4 5

(b)

-0.4-0.20

0.20.40.60.81

0 1 2 3 4 5 6 7 8

(c)

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0 0.02 0.04 0.06

X

Time

(d)

-0.3-0.2-0.10

0.10.20.30.4

0 0.02 0.04 0.06Time

(e)

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 0.02 0.04 0.06 0.08 0.1Time

(f)

FIG. 1. �Color online� Evolution of the x-coordinate ofthe Morse potential, Eq. �1�, representing the O–Hstretch of water ��a�–�c�� and the vibration in the Xe–Xedimer ��d�–�f��. The exact numerical result �solid line�is compared to the exact analytic solution, Eq. �36�,which coincided with the numerical result. Also shownare the harmonic-like approximation �dashes�, Eq. �42�,and the harmonic solution �dashed-dotted�. For the O–Hstretch, �=34.7547, and the initial conditions are �a�px�0�=����, �b� px�0�=���3�, and �c� px�0�=���7�.For the Xe system, �=283.5789, and the initial condi-tions are �d� px�0�=����, �e� px�0�=���20�, and �f�px�0�=���50�. All examples started at the equilibriumposition x�0�=0. The harmonic-like solution, Eq. �42�,significantly improves over the harmonic solution whenmore energy is introduced into the system.

244111-8 E. M. Heatwole and O. V. Prezhdo J. Chem. Phys. 130, 244111 �2009�

that �=1. The preceding parameters lead to values of �=5000 and �=1. Three specific cases for the initial condi-tions are considered. These are the antisymmetric stretch, thesymmetric stretch and a mixed stretching mode, where all ofthe initial kinetic energy is put into one of the Morse poten-tials. The approximate analytic solutions shown in Figs. 2–4are obtained using Eq. �61� and harmonic-like Eq. �62�.

Figure 2 gives the evolution of one of the coordinates forthe antisymmetric stretching motion of the triatomic. Theinitial conditions are pu�0�= pv�0�, and not pu�0�=−pv�0� asmight be intuitively expected. This arises from the relation-ship expressed in Eq. �47�. The initial conditions for the co-ordinates u�0� and v�0� are chosen to be at their minima forall examples. Four values for the initial momenta are chosen,pu�0�= pv�0�=−1,−10,−20,−30 in parts �a�–�d� of Fig. 2,respectively. Since the time evolution of u�t� and v�t� willgive identical results, only u�t� is shown. The exact numeri-cal data for the classical dynamics of the triatomic �solidline� are compared to the results from Eq. �61� �long dash�and harmonic-like Eq. �62� �short dash�, as well as to theevolution of two coupled harmonic oscillators with the fre-quency given by the curvature at the bottom of the Morsepotential �dashed-dotted�. Equation �61� is exact for the an-

tisymmetric stretch. The harmonic-like approximation, Eq.�62�, shows significant improvement over the coupled har-monic oscillators. All examples converge to the appropriateharmonic limit, as illustrated in Fig. 2�a�.

Figure 3 illustrates the symmetric stretch. The approxi-mation made in Eq. �59� no longer holds away from theharmonic limit for the symmetric stretch, as it did for the

antisymmetric stretch, and the K0� terms are not constants ofmotion. Therefore, the defined effective frequencies, Eqs.�27� and �63�, and the values of s will change over time, andEq. �61� will not be exact away from the harmonic limit.This is indeed the case, as seen in parts �b� and �d�–�f� of Fig.3. However, both Eqs. �61� and �62� show a significant im-provement over the straight harmonic approximation. Theresults are good in part because the fluctuations of the effec-tive time-dependent frequency have a tendency to averageout over time.

Figure 4 represents a situation where one degree of free-dom is given an initial kinetic energy, while the other degreeof freedom starts out with no kinetic energy. In this case, thetime evolution of both coordinates u�t�, parts �a�–�c�, andv�t�, parts �d�–�f�, are shown. As in the case for the symmet-

-1.5

-1

-0.5

0

0.5

1

1.5

0 5 10 15 20 25

U(a)

-15

-10

-5

0

5

10

15

0 5 10 15 20 25

(b)

-30

-20

-10

0

10

20

30

0 5 10 15 20 25Time

(c)

-40-30-20-100102030405060

0 5 10 15 20 25Time

(d)

U

FIG. 2. �Color online� Evolution of the u coordinate ofthe coupled Morse oscillators representing a linear tri-atomic molecule, Eq. �44�, with parameters �=1, �=1, and �=5000. The following initial conditionspu�0�= pv�0�=−1,−10,−20,−30 are used in parts �a�–�d�, respectively. They represent antisymmetric stretch,see Eq. �47�. The exact numerical results �solid line�,the approximate analytic solutions, including Eq. �61��long dashed� and harmonic-like Eq. �62� �shortdashed�, and the harmonic oscillator approximation�dashed-dotted� are shown. Note that the solution of Eq.�61� coincides with the exact result.

-1.5-1

-0.50

0.51

1.52

0 5 10 15 20 25

U

(a)

-20-15-10-50510152025

0 10 20 30 40 50

(b)

-30-20-10010203040506070

0 10 20 30 40 50

(c)

-1.5-1

-0.50

0.51

1.52

0 5 10 15 20 25

V

Time

(d)

-15-10-50510152025

0 10 20 30 40 50Time

(e)

-30-20-10010203040506070

0 10 20 30 40 50Time

(f)

FIG. 3. �Color online� Evolution of the u and v coordi-nates of the coupled Morse oscillators, Eq. �44�, parts�a�–�c� and �d�–�f�, respectively, for the initial condi-tions pu�0�=−pv�0�=−1,−10,−20, representing sym-metric stretch, see Eq. �47�. The lines are the same as inFig. 2.

244111-9 Analytic dynamics of the Morse oscillator J. Chem. Phys. 130, 244111 �2009�

ric stretch, we expect the approximation made in Eq. �59� to

fail away from the harmonic limit. K0�, s and the effectivefrequencies are no longer constants with time. Parts �a� and�c� of Fig. 4 show that the appropriate behavior in the har-monic limit is obtained. Parts �b,c,d,f� indicate that the ap-proximations break down when the energy is increased, asexpected. The analytic results reported for the symmetric andmixed cases, Figs. 3 and 4, can be improved further by goingbeyond the approximations made in Eqs. �61� and �62� andsolving Eq. �57� directly.

IV. CONCLUSIONS

The current paper presented the first application of theQHD approach to chemical systems described by the Morseraising and lowering operators, thereby extending the earlierstudies performed with the ordinary harmonic operators. Thesecond order QHD EOM for the Morse oscillator were de-rived and an analytic solution to these equations was found.The first-order QHD closure gave classical dynamics of theMorse oscillator and the analytic solutions to the classicaldynamics.

The Morse potential was represented in terms of the lad-der operators, which satisfy the commutator relations for theSU�2� group. While other types of raising and lowering op-erators can also be used to represent the Morsedynamics24,30,31 the SU�2� operators provide a concise repre-sentation that is particularly useful within the QHD frame-work, since it limits both the number and the order of theQHD variables that are required in order to capture quantum-mechanical effects. The SU�2� operators lead to a faster con-verging QHD series compared to other operators for tworeasons. First, the SU�2� operators form a closed algebra.Even though this algebra does not include the Morse Hamil-tonian, Eq. �16�, and as a result the QHD EOM form aninfinite hierarchy, the terms of the SU�2� hierarchy tend toclose partially and to branch less than terms based on otheroperators. Second, the Morse Hamiltonian itself has a verysimple form in the SU�2� representation, Eq. �16�. Therefore,the Heisenberg commutators that give rise to EOM for theexpectation values are also simple and contain few terms. Incomparison, the operators defined in Refs. 24, 30, and 31

were introduced with molecular and spectroscopic applica-tions in mind. For instance, the operators of Ref. 24 giveexact matrix elements for the optical transition dipole mo-ments involving the Morse momentum operator. Therefore, aQHD hierarchy involving these alternative Morse operatorswould be particularly useful for a semiclassical spectroscopicanalysis.

The evolution of the ladder operators was considered inthe Heisenberg representation. The exact quantum dynamicsof the system can be found with an infinite set of coupleddifferential equations. In order to allow for useful analyticsolutions they must be decoupled. We applied QHD closuresto decouple the EOM for the lower-order operators fromthose for the higher-order variables. The resulting approxi-mate EOM could be solved analytically in a closed form.The accuracy of the approximate solutions compared to thefull quantum result depended on the level at which the QHDapproximation is taken. This work considered the first andsecond-order QHD approximations. The first-order approachis equivalent to classical dynamics. Therefore, our semiclas-sical approach allowed us to derive closed analytic solutionsfor the classical dynamics of the Morse oscillator. It shouldbe noted that the semiclassical results obtained in this workare applicable to the bound region of the Morse potentialand, in general, cannot be used in the dissociative region.

The reported analyses were performed using the Morseladder operators, which can be related exactly to the positionand momentum operators. The exact relationship betweenthe ladder and position-momentum operators is fairly com-plex. A simpler, approximate relationship, which becomesexact near the harmonic limit, can be used instead. The EOMobtained using this approximate harmonic-like relationshipare simple and significantly outperform the straightforwardharmonic approximation. Both the accurate and simpler re-lationships between the ladder and position-momentum op-erators lead to EOM that conserve the total energy and con-tain other constants of motion originating from the

conservation of K0� in Eq. �23�.The semiclassical analytic expressions for the dynamics

of the Morse oscillator were illustrated with two realisticsystems, including the O–H stretch of water and the Xe–Xe

-1.5

-1

-0.5

0

0.5

1

1.5

0 10 20 30 40 50

U(a)

-30-20-10010203040

0 10 20 30 40 50

(b)

-60-40-20020406080

0 10 20 30 40 50

(c)

-1.5

-1

-0.5

0

0.5

1

1.5

0 10 20 30 40 50

V

Time

(d)

-30-20-10010203040

0 10 20 30 40 50Time

(e)

-60-40-20020406080

0 10 20 30 40 50Time

(f)

FIG. 4. �Color online� Same as Fig. 3, but for the asym-metric initial conditions pu�0�=−1,−20,−30, pv�0�=0.Note that the harmonic-like approximation, Eq. �62�,�short dashed� gives better results than the more generalapproximation, Eq. �61�, �long dashed� due to changing

values of K0� and s, which were assumed constant inderiving Eq. �57�.

244111-10 E. M. Heatwole and O. V. Prezhdo J. Chem. Phys. 130, 244111 �2009�

dimer. The simple, harmonic-like approximation providedsubstantial improvements over the true harmonic description,while the full solution of the first-order equations gave theexact closed form of the classical dynamics.

Next, the semiclassical approach was applied to coupledMorse oscillators. The system consisted of three particles.The two identical outside particles formed a linear chain withthe middle particle and interacted with the middle particlethrough Morse potentials. Expressing the Hamiltonian usingthe Morse ladder operators, we derived analytic first-order

EOM. Because the K0 ladder operator was no longer a con-stant of motion for the coupled system, in contrast to thesingle Morse oscillator, the first-order QHD closure gavenonlinear EOM. Close to the harmonic limit it was possible

to treat the expectation value of K0 as a constant, reducingthe number of differential equations from six to four andmaking them linear. This set of differential equations wassolved analytically, generating a closed form solution for theapproximate classical dynamics of two coupled Morse oscil-lators.

Using the exact and approximate correspondence be-tween the Morse ladder operators and the position-momentum variables, we compared the semiclassical expres-sions for the coupled Morse oscillators with the exactnumerical results and the harmonic limit. Three sets of initialconditions were considered, including the symmetric stretch,the antisymmetric stretch and a mixed mode. The exact op-erator correspondence produced an exact analytic expressionfor the classical dynamics of the antisymmetric stretch. Theharmonic-like correspondence showed a significant improve-ment over the harmonic description. The symmetric and

mixed modes were described approximately. Since K0 is timedependent for the symmetric stretch and the mixed mode, thesemiclassical solutions deviated from the exact numerical an-swer away from harmonic limit. In this case, the results ob-tained using the harmonic-like correspondence between theladder and position-momentum operators were of the samequality as the results obtained using the exact correspon-dence.

The classical-like analytic solutions for the dynamics ofthe one and two-dimensional Morse oscillators can be usedto simplify numerical simulations of multidimensional sys-tems. For instance, the fast motions of the OH and CH bondsare often described by Morse potentials. In a complex mo-lecular system involving multiple time scales, the evolutionof these bonds create numerical bottlenecks and limit the sizeof the integration time-step. The analytic solutions of theMorse problem can be used to eliminate the need to propa-gate the fast motions numerically and to allow one to in-crease the time step and extend the simulation to longertimes. In such cases, the exact and approximate analytic so-lution of the classical-mechanical dynamics of the Morse os-cillator, Eqs. �36� and �42� will depend parametrically on thecoordinates of the slower modes. Additionally, zero-point en-ergy and other quantum-mechanical effects associated withthe OH and CH motions can be incorporated into numericalsimulation using the analytic solutions �22� of the second-order QHD equations.

Quantum effects in the Morse oscillator dynamics werecaptured with the second-order approximation. The numberof the second-order QHD Eq. �21� is only twice that of theclassical mechanics, Eq. �23�. It is straightforward to extendthe technique to higher order. The general approach intro-duced and advocated in this work allows one to studyquantum-mechanical effects in the Morse oscillator dynam-ics at little computational cost.

ACKNOWLEDGMENTS

Multiple and fruitful discussions with Yuriy Pereverzevare greatly appreciated. The research was supported bygrants from the NSF under Grant No. CHE-0701517 andACS-PRF 46772-AC6.

APPENDIX: SOLUTIONS FOR THE LINEARTRIATOMIC

Here, we obtain solutions to Eq. �59�, which was derivedin Sec. II E for the linear triatomic molecule represented bycoupled Morse oscillators. The system of linear differentialEq. �59� for the expectation values of the ladder operatorscan be solved by standard techniques, for instance by findingthe appropriate eigenvalues and eigenvectors. For simplicity,we drop the expectation value notation ¯ �, since all termshave been decoupled to the first-order. The expectation val-ues evolve according to

K+u�t� =

1

8� ei�1t − e−i�1t

�C + 1�K+

u�0��C + 2� − K−u�0�C − K+

v�0�

��C + 2� + K−v�0�C� +

e�2t − e−�2t

�1 − C�K+

u�0��2 − C�

+ K−u�0�C + K+

v�0��2 − C� + K−v�0�C� + 2�K+

u�0�

− K+v�0���ei�1t + e−i�1t� + 2�K+

u�0� + K+v�0���ei�2t

+ e−i�2t� ,

K−u�t� =

1

8� ei�1t − e−i�1t

�C + 1�K+

u�0�C − K−u�0��C + 2�

− K+v�0�C + K−

v�0��C + 2�� −ei�2t − e−i�2t

�1 − C�K+

u�0�C

+ K−u�0��2 − C� + K+

v�0�C + K−v�0��2 − C��

+ 2�K−u�0� − K−

v�0���ei�1t + e−i�1t� + 2�K−u�0�

+ K−v�0���ei�2t + e−i�2t� ,

K+v�t� =

1

8� ei�1t − e−i�1t

�C + 1�− K+

u�0��C + 2� + K−u�0�C + K+

v�0�

��C + 2� − K−v�0�C� +

ei�2t − e−i�2t

�1 − C�K+

u�0��2 − C�

+ K−u�0�C + K+

v�0��2 − C� + K−v�0�C� + 2�K+

v�0�

244111-11 Analytic dynamics of the Morse oscillator J. Chem. Phys. 130, 244111 �2009�

− K+u�0���ei�1t + e−i�1t� + 2�K+

v�0� + K+u�0���ei�2t

+ e−i�2t� ,

K−v�t� =

1

8� ei�1t − e−i�1t

�C + 1�− K+

u�0�C + K−u�0��C + 2�

+ K+v�0�C − K−

v�0��C + 2�� −ei�2t − e−i�2t

�1 − C�K+

u�0�C

+ K−u�0��2 − C� + K+

v�0�C + K−v�0��2 − C��

+ 2�K−v�0� − K−

u�0���ei�1t + e−i�1t� + 2�K−u�0�

+ K−v�0���ei�2t + e−i�2t� , �A1�

where C is given by Eq. �60�, and �1, �2 are given by Eq.�63�.

In order to obtain the time dependence in the position-momentum representation, we can relate the coordinate vari-ables to the Morse raising and lowering operators using ei-ther the exact expression, Eq. �33�, or the much simplerharmonic-like approximation, Eq. �40�. The latter is moreappropriate in the limits used for deriving the analytic formof Eq. �A1� and gives rise to fairly simple expressions com-pared to the much more complex form of the expressionsobtained using Eq. �33�.

Starting with Eq. �A1� and using Eq. �33�, we find thetime dependence of yu�t�

1

yu�t�=

1

4s��

sin �1t�1 + C

�� pu�0�yu�0�

−pv�0�yv�0�

��C��2s + 1��s + 1 + �2s − 1��s − 1�2

2�2s + 1��2s − 1���s + 1��s − 1�+ 2�

+1

4s��

sin �2t�1 − C

�� pu�0�yu�0�

+pv�0�yv�0�

��2 − C��2s + 1��s + 1 + �2s − 1��s − 1�2

2�2s + 1��2s − 1���s + 1��s − 1��

+1

2cos �1t� 1

yu�0�−

1

yv�0� +1

2cos �2t� 1

yu�0�

+1

yv�0� +��1 − cos �2t�

�2s + 1��2s − 1�. �A2�

This expression can be further simplified by noticing that thefollowing term rapidly approaches its limit when s 1

��2s + 1��s + 1 + �2s − 1��s − 1�2

2�2s + 1��2s − 1���s + 1��s − 1�� 2. �A3�

This allows us to rewrite Eq. �A2� in its the final form, Eq.�61�, as seen in Sec. II E. The EOM for the harmonic-likelimit are found in a similar fashion by starting with Eq. �A1�and using Eq. �40� to relate the position-momentum variablesto the raising and lowering operators. After simplification weobtain Eq. �62� for the harmonic-like limit.

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