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Hindawi Publishing CorporationAdvances in Physical ChemistryVolume 2011, Article ID 593872, 38 pagesdoi:10.1155/2011/593872

Review Article

Potential Energy Surfaces Using Algebraic MethodsBased on Unitary Groups

Renato Lemus

Instituto de Ciencias Nucleares, Universidad Nacional Autonoma de Mexico, Apartado Postal 70-543, 04510 Mexico, DF, Mexico

Correspondence should be addressed to Renato Lemus, [email protected]

Received 14 July 2011; Revised 13 October 2011; Accepted 21 October 2011

Academic Editor: Sylvio Canuto

Copyright © 2011 Renato Lemus. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This contribution reviews the recent advances to estimate the potential energy surfaces through algebraic methods based on theunitary groups used to describe the molecular vibrational degrees of freedom. The basic idea is to introduce the unitary groupapproach in the context of the traditional approach, where the Hamiltonian is expanded in terms of coordinates and momenta.In the presentation of this paper, several representative molecular systems that permit to illustrate both the different algebraicapproaches as well as the usual problems encountered in the vibrational description in terms of internal coordinates are presented.Methods based on coherent states are also discussed.

1. Introduction

The description of molecular systems involves the solutionof the corresponding Schrodinger equation. This task is sodifficult that an approach involving just numerical methodsneeds powerful computers even for three-or four-particlesystems. An alternative approach is based on choosingthe basis functions in such a way that they resemble theexact eigenfunctions as much as possible. The suitable basisare obtained by making approximations that simplify theHamiltonian of the molecule. The advantage of this methodis that the functions reproduce correctly the gross featuresof the spectrum, and consequently they provide a betterphysical insight in understanding the solutions. The firststep in simplifying the molecular problem consists in takingadvantage of the large difference between the nucleus andelectron masses, a fact that leads to the Born-Oppenheimerapproximation [1, 2]. As a result of this approximationthe original Schrodinger equation is split into two coupledequations, one corresponding to the electronic degrees offreedom which is solved for many nuclear geometries and theother one associated with the rotation-vibration Schrodingerequation for the nuclei whose potential is basically providedby the electronic energy [3, 4]. On the other hand, therotation-vibration Schrodinger equation is usually solvedmaking the rigid-rotor approximation together with theharmonic oscillator approximation. The total wave function

is then approximated as the direct product of three contribu-tions: electronic, rotational, and vibrational wave functions.Corrections to this description are allowed by introducingthe braking of the Born-Oppenheimer approximation, dis-tortion effects, anharmonicity, centrifugal distortion, andCoriolis coupling [3–5].

Within the Born-Oppenheimer approximation the po-tential energy surface (PES) is provided by the solution of theelectronic Schrodinger equation. Following this approach thecalculation of a PES represents a major problem because oftwo main reasons. On one hand an accurate calculation ofthe electronic wave functions and energies is a quite difficultproblem due to the correlation and exchange interactions, aswell as the relativistic effects [6]. Hence, this problem is byitself a challenge of current interest. On the other hand theestimation of a PES involves the solution of the electronicSchrodinger equation for different nuclear configurations,a formidable computational task feasible only for smallmolecules. Even, when this situation is possible, the predictedvibrational spectrum generated from the obtained PES is farfrom the standards of high-resolution molecular vibrationalspectroscopy, being necessary to refine the force constantsin order to obtained a good vibrational description [7]. Insummary, for medium and large molecules, the harmonicapproximation is usually considered, with the proviso of apoor quality concerning the prediction of vibrational spectra.

2 Advances in Physical Chemistry

An alternative approach to obtain an estimation of thePES is provided by the vibrational spectrum of molecularsystems [3, 4, 8, 9]. An expansion of the kinetic energy andthe potential in terms of normal coordinates, for instance,involves a set of force constants which can be estimated byfitting the vibrational spectrum. In this case the estimation ofthe PES is computationally cheap, with force constants fittedto provide a description of high quality. These force constantsmay in turn be used to predict spectra of isotopic species.The advantage of this approach is that it is relatively easyto implement, although the identification of the resonancesand anharmonic interactions to establish the appropriateHamiltonian to produce a high-quality fitting is in generalnot a simple task. The most common approach to carryout the vibrational description is through the use of theharmonic basis associated either with the normal modes orthe symmetry-adapted coordinates. The inconvenience oftaking a harmonic basis is that as the energy increases theadmixture of states becomes evident due to the intrinsicproperties of the basis. An alternative approach consists inconsidering a change to a more realistic basis. In particulara local basis turns out to be appropriate for two reasons.On one hand it allows the results provided by the normalbasis to be recovered when local harmonic oscillators areconsidered, but on the other hand this selection opens thepossibility of considering alternative bases, like Morse and/orPoschl-Teller functions, which reflect more accurately themain physical properties of a pure local bond. Local basesemerged as a natural way to explain the spectra of moleculesinvolving bonds with large differences in the masses of thebegin and end atoms [10–13]. Although both alternatives,local or normal basis, may be worked out in configurationspace, their corresponding algebraic representations providean effective and elegant route to deal with the descriptionof the vibrational degrees of freedom. When a harmonicbasis is considered, the algebraic approach appears in naturalform by introducing the bosonic operators associated withthe creation and annihilation operators of the harmonicfunctions [14]. This is a relatively easy task that allows usto exploit the concept of polyad, a pseudo-quantum numberthat defines a subspace of states connected through themain interactions of the system [15]. In contrast, whenconsidering anharmonic potentials, bosonic operators donot appear anymore, making the description a nontrivialtask. Stimulated by this problem algebraic methods based onthe unitary approach emerged as an alternative to describethe rovibrational degrees of freedom.

Unitary groups are proved to be relevant in the descrip-tion of many-body systems. In fact bilinear products ofcreation and annihilation fermionic or bosonic operatorsform sets of generators of unitary groups, which constitutethe dynamical group of a great variety of the systems [16]. Inparticular great attention has been paid to the description ofelectronic degrees of freedom in atoms and molecules [17–20], nuclear physics [16, 21] and subnuclear physics [22],and also to the rovibrational degrees of freedom [23, 24].When the vibrational excitations are described in terms ofa basis of harmonic oscillators, unitary groups appear innatural form [14, 19, 25]. Bilinear products of creation

and annihilation operators of a set of ν oscillators in ndimensions constitute the generators of the symmetry groupof the system U(νn), while sp(2νn,R) is the correspondingdynamical group [14, 26]. A set of harmonic oscillatorspresents an infinite number of states, which explains theappearance of a noncompact group as a dynamical group.It is possible to work with compact dynamical groups ifthe space of harmonic oscillators is cut off. To achievethis goal an extra boson is added in such a way that thetotally symmetric representation (total number of bosons)of the unitary group is fixed. This approach was for thefirst time proposed in the context of the description ofcollectives states of nuclei [27, 28]. Later on the same ideawas applied by Iachello et al. in the field of molecular physicsto describe the rotation-vibration excitations of molecularsystems, establishing what is known as the vibron model[23, 29, 30]. In the framework of this approach a U(4)group is associated with each bond providing a dynamicalgroup in terms of the direct product of U(4) groups. Thevibron model was successful in describing linear molecules[31, 32], but, because of its increasing complexity, it does notbecome suitable for nonlinear molecules beyond triatomicsystems. An alternative model to overcome this difficultywas proposed by considering the rovibrational degrees offreedom in unified form, proposing a unique unitary groupU(ν + 1) for a system of ν degrees of freedom includingboth vibrations and rotations [33]. In any case these modelsare phenomenological in the sense that the Hamiltonianis expanded in terms of the generators of the dynamicalalgebra (usually in terms of Casimir operators), providingeigenvectors and eigenvalues, but in purely algebraic form.In the framework of this formalism the PES may beextracted through the use of coherent states, an approachassociated with the classical limit [23]. A relevant featureof the unitary approach is that the addition of the extraboson besides providing a compact group as a dynamicalgroup, permits to take into account anharmonicities fromthe outset, enriching the model through the appearanceof orthogonal subgroups, which play a preponderant rolein the description of nonrigid molecules. In this type ofsystems, however, the polyad concept stops being useful. Thefull space has to be considered, allowing phase transitionsto appear and giving rise to the possibility of describingmolecular systems with several structural minima [34–38].

Although most of the applications of the unitary groupapproach in the field of molecular spectroscopy have beendeveloped to describe the rovibrational degrees of freedom,there exists a model for diatomic molecules including thefull set degrees of freedom. In this model the sp shellis considered for the electrons, while the rovibrationaldegrees of freedom are introduced through the U(4) vibronmodel. The full Hamiltonian is then expanded in termsof the generators of the dynamical group Ue(8) × Urv(4)[39–41]. A remarkable feature of this model is that theBorn-Oppenheimer approximation is not assumed. Theexpectation values with respect to the coherent states allowthe PES to be extracted for each electronic state. However,because the electronic degrees of freedom are taken in the

Advances in Physical Chemistry 3

united atom limit, the PESs do not reproduce the expectedshape in the separated atoms limit [42, 43].

A unitary approach restricted to describe only vibrationalexcitations was developed by Michelot and Moret-bailly [44],which later on was further analyzed to include an additionalsubgroup in order to introduce the most important localinteractions as a part of an expansion of the Hamiltonianin terms of Casimir operators [45]. This approach has beenapplied to several molecular systems, for instance, tetrahe-dral [46] and pyramidal molecules [47, 48], and it is based onthe methodology of algebraic techniques where each chainof groups provides a dynamical symmetry that establishesa basis to diagonalize a general Hamiltonian, whose maincontributions correspond to the Casimir operators associ-ated with different chains [19]. In the framework of thisapproach for a set of ν equivalent oscillators, the dynamicalgroup becomes U(ν + 1). The relevance of this approach isnot just the restriction of the space by itself but the fact thatthis treatment allows us to introduce anharmonicities fromthe outset. The one-dimensional version of this approachdeserves special attention due to the fact that it is connectedwith the Morse and Poshl-Teller potential, widely appliedin molecular problems [19]. In this case the vibron modeland the U(ν + 1) approach coincide. This model was firstidentified to describe the stretching degrees of freedom, butlater on it was extended to include the bending degrees offreedom [49, 50]. As previously mentioned the extractionof the PES is a non-trivial task, although it is possibleto estimate it through the coherent state formalism [23,51, 52]. However, in order to be in position to calculateforce constants useful to predict spectra of isotopic species,it is necessary to establish the appropriate correspondencebetween the coordinates and momenta of the system and thegenerators of the dynamical group [53].

In the last ten years successful efforts to connect thealgebraic approaches to describe molecular systems basedon unitary groups with their corresponding description inconfiguration space have been made. First the exact con-nection of the U(2) model with the Morse and Poschl-Teller potentials was established [54–56]. This connectionallowed force constants to be obtained, and consequently thePES became available to predict spectra of isotopic species[57–62]. Later on a connection was established betweenthe U(ν + 1) model of ν equivalent oscillators proposed byMichelot and Moret-bailly [44] and the space of coordinatesand momenta [63, 64]. This connection allowed for the firsttime the calculation of PES in the framework of this algebraicmodel [65, 66]. More recently, an explicit connectionbetween the U(3) algebraic model, used to describe thebending degrees of freedom of linear triatomic molecules[67], and the configuration space was established [68, 69].

The new approach developed to extract the PES maysubstitute the traditional approach in which the Casimiroperators play a preponderant role, since it has the remark-able feature that every Hamiltonian written in terms ofbosonic creation and annihilation operators associated withharmonic oscillators can be translated into the U(ν + 1)approach in such a way that in the harmonic limit bothtreatments coincide. As a consequence, in this scheme it is

not necessary to know the complex machinery of Lie algebrasto apply the models. An additional feature of the algebraicapproach is that it possesses particular features withoutanalog in the configuration space regarding the descriptionswhere the polyad plays a fundamental role. It is alwayspossible to obtain a better description of the vibrational exci-tations, compared with the analog models in configurationspace at the same level of approximation. In addition analgebraic approach contrasts with the models based on abinitio methods to extract the PES. The algebraic approachesare relatively simple to apply because the resonances canbe established in a straightforward way and the matrixelements are calculated in a simple way. This feature allowsa vibrational description to be done in a relatively short timecompared with ab initio methods or even with variationalapproaches where the kinetic energy is calculated in exactform and the potential is expanded to be fitted. However,it should be clear that the algebraic approaches cannotsubstitute the ab initio calculations; they represent just analternative to estimate the PES in a relatively simple andeconomic way when a molecular structure can be identifiedeither for semirigid or nonrigid molecules, although in thispaper we regard solely semirigid molecules. For instance,variational approaches are more reliable to make predictionout of the range of energies considered in the fits, andab initio calculations allow a potential energy surface ofreactive molecular systems to be calculated. An algebraicapproach is capable of providing PESs of two separatemolecular systems, but it is not possible to obtain theeffect of the molecular interaction over the PESs. It shouldbe also mentioned that current computational proceduresare capable of describing the rovibrational spectrum on ahigh level of approximation for molecules with four atomslike NH3, for instance [70]. Algebraic approaches, however,cannot contend with such calculations, they are proposedto establish approximated methods that may be applied tomore complex systems where ab initio calculations are tooexpensive to be applied.

In addition, in the context of molecular physics apotential energy surface may be obtained using an algebraicapproach either by means of the connection of the spectro-scopic parameters with the structure and force constants orby means of the introduction of the coherent states. However,only through the connection with configuration space it ispossible to predict the spectra of isotopic species.

In this paper we review the recent advances in establish-ing the connection between the algebraic approaches basedon the unitary groups and the physical space of coordinatesand momenta. Our goal is to show the method to extractthe PES for the different models and situations found inthe description of vibrations in terms of internal coordinate.We start with the basic concepts involved in the algebraicapproaches by analyzing the case of one oscillator. Lateron the case of two oscillators is analyzed in detail to showthe intrinsic advantages of an algebraic method. Here theapproach using coherent states to extract a PES is alsodiscussed. Thereafter several molecular systems are analyzed:nonlinear and linear triatomic molecules, pyramidal andplanar molecules. This selection of molecules permits to


illustrate the application of the different algebraic approachto estimate a PES.

This contribution is organized as follows. The basic con-cepts involved in the Born-Oppenheimer approximationis presented in Section 2. In Section 3 some fundamentalconcepts on symmetry are discussed. Section 4 is devoted tointroduce the main ingredients of the algebraic approaches,while in Section 5 the connection of theU(2) model with theMorse Potential is presented. In Section 6 we discuss in detailthe case of two interacting Morse oscillators. An approachto extract the PES using coherent states is presented inSection 7. In Section 8 an approach to obtained the PES isillustrated presenting the study of the water molecule. InSection 9 a more elaborated system, the BF3, is presented.Section 10 is devoted to establish an approach to obtainthe PES when the U(ν + 1) algebraic models for nonlinearmolecules are used. In Section 11 the case of linear moleculesusing the U(3) model is analyzed. Finally in Section 12 thesummary and conclusion are presented.

2. Molecular Hamiltonian

A molecule is a collection of N nuclei and λ − N electronsheld together by forces and obeying the laws of quantummechanics through the Schrodinger equation HΨ = EΨ,where the Hamiltonian in the axis system at the molecularcenter of mass (parallel to the laboratory system) takes thegeneral form [3, 4, 71]

H = T0 + T′ + V , (1)

where T0 is the sum of the kinetic energy of λ − 1 particles(center of mass excluded), T′ is a kinetic energy term thatinvolves crossed terms among the λ − 1 particles, whileV is the electrostatic potential energy. In our discussioninteractions between electron-spin magnetic moments andbetween nuclear magnetic and electric moments are notconsidered. Although the molecular center of mass systemallows the center of mass contribution to be eliminated,the problem of the cross-terms T′ arises. To eliminate suchcross-terms between nuclei and electrons, a reference systemparallel to the laboratory at the nuclear center of mass isintroduced. This new reference system induces in the kineticenergy the transformation T0 + T′ → Te + TN , where Teand TN are purely electronic and nuclear contributions. Theelectronic kinetic term Te contains diagonal T0

e and crossedcontributions T′e . If the latter is neglected, the Schrodingerequation takes the form

{

T0e + TN + V(RN , re)

}

Ψrve(RN , re) = ErveΨrve(RN , re),

(2)

where RN and re stand for the nuclear and electronic coordi-nates, respectively. Equations (2) is the starting point to carryout the Born-Oppenheimer (BO) approximation, whichassumes that the motion of the electrons are unaffected bythe motion of the nuclei. There are two ways of makingthe BO approximation: the perturbation theory approach

[1] and the variation theory approach [2]. In the formerapproach, the fundamental idea consists in expanding therovibronic Hamiltonian in powers of the parameter κ =(me/M0)1/4, whereme is the electron mass andM0 is the meannuclear mass. Identifying the different terms in powers of κ,the solutions are obtained in successive form. On the otherhand, in the latter approach the rovibronic wave equation iswritten in the form

Ψrve,m(re, UN ) =∑

n′Φmrv,n′(UN )Φelec,n′(re, UN ), (3)

where UN are the vibrational displacement coordinates,Φelec,n′(re, UN ) are a complete set of solutions to theelectronic problem, and the coefficients Φm

rv,n′(UN ) are tobe determined. Both treatments can be consulted in theliterature [1, 2]. Here we are going to provide only physicalarguments for the zeroth-order solution.

The wave function in (2) may be factorized as afirst approximation as the direct product of an electronicand a nuclear wave functions in the following form:Ψrve(RN , re) � Φe(RN ; re)Φrv(RN ), where the electronicfunction Φe(RN ; re) is parametric in the nuclear positionsand TNΦe(RN ; re)Φrv(RN ) � Φe(RN ; re) TNΦrv(RN ). Hence,freezing the nuclei in (2) and subtracting the nuclear repul-sion term VNN , the following equation for the electronicwave function is obtained:

{

T0e + V(RN , re)− VNN

}

Φe(re) = Ve(RN )Φe(re). (4)

If we now go back to (2) taking into account (4) and theprevious considerations, we have for the nuclear equation

{

TN + VN

}

Φrv(RN ) = ErvΦrv(RN ), (5)

where VN = Ve + VNN − Eelec and Erv = Erve − Eelec,and the rotation-vibration energy for a bound electronicstate is chosen so that the zero of energy is the minimumvalue of Ve. As a result of making the Born-Oppenheimerapproximation, the original problem simplifies to the twodifferential equations (4) and (5), where the electronicequation has to be solved before (5) since the correspondingenergy Ve provides the potential in the nuclear equation.

Both the electronic and rotation-vibration equations aretoo complicated to be solved in exact form. The electronicequation belongs to the field of quantum chemistry [18,71, 72] and will not be discussed in this paper. We willconcentrate our presentation on the calculation of the PESthrough the analysis of the vibrational degrees of freedom.We thus move to the analysis of the rotation-vibrationequation, which is referred to as (ξ,η, ζ) axis system parallelto the laboratory with its origin at the nuclear center ofmass. This equation can be separated into a rotational anda vibrational part by introducing a rotated (x, y, z) systemwith its origin at the nuclear center of mass, also known asthe molecule-fixed axis system. In matrix form we have

rmol = R(

χ, θ,φ)

rlab, (6)

where R(χ, θ,φ) is a rotation specified by the Euler angles[73]. In order to optimize the separation of the rotational


and the vibrational parts, the Euler angles are determinedthrough the Eckart equations [74]

∑

i

mirei × ri = 0, (7)

where rei are the coordinates of the nuclear equilibriumconfiguration in the molecular-fixed system. For this config-uration the orientation of the axis is chosen to correspondto the principal axis of inertia. When the nuclear coordinatesare not far from the equilibrium, it turns out to be suitable tointroduce the coordinates {Δαi = αi − αei , α = x, y, z} whichare known as the vibrational displacement coordinates. Inparticular a linear combination of them corresponding tothe normal coordinates Qj is appropriate to write downthe molecular Hamiltonian. The rotation-vibration Hamil-tonian involved in (5), when rewritten in terms of themolecular coordinates {χ, θ,φ,Qi, i = 1, . . . , 3N − 6}, fornonlinear molecules for instance, takes the form [75–77]

Hrv = H0rot + H0

vib +Vanh + Vcen + Vcor, (8)

where H0rot corresponds to the rotational contribution corre-

sponding to the rigid-rotor Hamiltonian

H0rot =

12

∑

α

μeαα J2α , (9)

the second contribution H0vib is a sum of independent har-

monic oscillators associated with the normal modes

H0vib =

12

∑

r

(

P2r + λrQ

2r

)

, (10)

while Vanh contains the anharmonic contributions

Vanh = 16

∑

r,s,t

crst QrQsQt +1

24

∑

r,s,t,u

crstu QrQsQtQu + · · · ,

(11)

where the coefficients crst... are the force constants whichdetermine the PES. The fourth term in (8) corresponds tothe centrifugal distortion

Vcen = 12

∑

α,β

(

μαβ − μeαβ)(

Jα − pα)(

Jβ − pβ)

, (12)

and the last term to the vibrational Coriolis coupling

Vcor = 12

∑

α

μeαα p2α −

∑

α

μeαα Jα pα. (13)

In these expressions Pα is the conjugate momentum to thenormal coordinate Qα, Jα and pα are the components ofthe rotational and vibrational angular momenta, and μαβis the inverse of the matrix I′ which involves the momentsof inertia and the normal coordinates [75]. μeαβ is the samematrix evaluated at equilibrium. Hence, the zeroth-orderapproximation corresponds to the rigid-rotor harmonicHamiltonian H0

rv = H0rot+ H

0vib, whose eigenfunctions provide

suitable basis to diagonalize the complete Hamiltonian (8)for semirigid molecules.

Let us now focus on the vibrational degrees of freedom.Up to the seventies, the standard approach was based on theuse of normal bases as stated by (10). The success of sucha description at that time is explained because the analysiswas restricted to the low lying region of the spectrum [8].In the last decades, however, due to the development of newexperimental techniques, it has been possible to obtain high-resolution spectroscopic data for highly excited vibrationalstates [78, 79]. In this region, however, it is rarely possible tofind a dominant component to characterize the eigenstates,a feature due to the strong mixture of the harmonic basis.On the other hand, for high energies, the density of statesincreases rapidly, and, although the spectrum is expected tobe more complex, in some cases some regularities appear,which can be explained in terms of a local oscillators scheme[11–13]. The pattern of the spectrum can be understoodwhen one takes into account that in the chemical reactivitylimit the energy tends to accumulate in some particularbonds where the reaction evolves. This behavior leads to con-sider the vibrational problem in terms of a set of interactinglocal oscillators.

The vibrational Hamiltonian Hvib in terms of curvilinearinternal displacement coordinates qi acquires the generalform [80, 81]

H = 12

pG(

q)

p +V ′(

q)

+V(

q)

, (14)

where q and p are column vectors corresponding to the inter-nal displacement coordinates and their conjugate momentapk = −i�∂/∂qk, respectively, while the G(q) matrix estab-lishes the connection between the internal and Cartesiancoordinates. V(q) is the Born-Oppenheimer potential, whileV ′(q) is a kinetic energy term not involving momentumoperators which is usually neglected.

In a variational approach, the solution of the Schrodingerequation associated with the Hamiltonian (14) is obtainedwithout any additional approximation. Since this approachis not viable for large- and even medium- sized molecules,approximate methods are welcome. The usual approachto obtain a suitable Hamiltonian to deal with consists inexpanding both the G(q) matrix and the potential V(q)as a Taylor series around the equilibrium configuration,truncating the expansion where an adequate convergenceis achieved. In this scheme the zeroth-order Hamiltoniancorresponds to a sum of harmonic local oscillators

H0vib =

12

∑

α

Ge(

q)

p2α +

12

∑

α

λαq2α, (15)

where λα are the force constants. This Hamiltonian isdiagonal in the basis of direct product of harmonic oscillatorfunctions providing the basis to diagonalize a more generalHamiltonian. To obtain the zeroth-order Hamiltonian (15)from (14), it has been assumed an expansion in terms of thelocal coordinates qi, a fact that leads to identify harmonicoscillators. A more suitable coordinates to carry out theexpansion are Morse or Poschl-Teller coordinates [60, 82],


since in this way the zeroth-order Hamiltonian is identifiedwith a set of Morse and/or PT oscillators. This provides fromthe outset a better description because of their extra degreeof freedom of anharmonicity. An additional advantage is thatboth potentials can be treated in a unified form in terms ofsu(2) algebras, a crucial advantage that allows the vibrationaldescription to be improved without analog in configurationspace when the polyad is preserved. This point will bediscussed later, but before going into details we briefly sketchsome symmetry aspects of the molecular Hamiltonian.

3. Symmetry Considerations

The set of transformations that leaves the Hamiltonianinvariant satisfies the properties of a group G. Technicallyspeaking, if Ri ∈ G, then the associated operator ORi actingon the physical space commutes with the Hamiltonian

[

ORi , H]

= 0, i = 1, 2, . . . , |G|, (16)

where |G| stands for the number of elements of the group.When the set of operations corresponds to the maximumnumber of transformation, G is called the symmetry groupof the system, and according to Wigner’s theorem theeigenfunctions of the Hamiltonian span irreducible repre-sentations (irreps) of the group G [83]. We thus have thatthe symmetry group depends on the Hamiltonian, and theaction of the operators ORi over the physical space should bespecified in accordance with the approximation involved.

An exact Hamiltonian for any molecular system free ofexternal fields is invariant under the following operations:(a) any translation along a space fixed direction, Euclideangroup E(3), (b) any rotation about a space fixed axis passingthrough the center of mass of the molecule, rotation groupSO(3), (c) any permutation of the space and spin coordinates

of the electrons, permutation group S(e)n , (d) any permutation

of the space and spin coordinates of identical nuclei,complete nuclear permutation groupGCNP, and (e) inversionof the coordinates of all particles, nuclei and electrons, in thecenter of mass of the molecule, inversion group E . Theseoperations follow from the Hamiltonian and the natureof the space, uniformity, isotropy, indistinguishability ofidentical particles, and nature of the electromagnetic force.The full group is then given by

E(3)⊗ SO(3)⊗ S(e)n ⊗GCNP ⊗ E , (17)

where GCNP is given in terms of the direct product of sym-metric groups associated with identical nuclei. The group(17) provides irreps to label the states. For the Hamiltonian(1), where the kinetic energy of the center of mass has beeneliminated, the Euclidean group is not included in (17). Asnoted this group does not contain the point group of themolecule. The reason is that the point group is associatedwith a specific structure of the molecule, which in fact isintrinsic to the Born-Oppenheimer approximation. Hence,there should be a connection between the true symmetriesand point symmetries.

The group GCNP depends only on the chemical formula,and it should be noted that its order can be very large. Inpractice systematic accidental degeneracies appear accordingto the label scheme provided by (17). These degeneraciesare caused by the presence of more than one version ofthe equilibrium structure, in a given electronic state [3].Different versions are connected through potential barrierswhich, when they are too high to be experimentally detected,the labeling scheme (17) provides extra labels manifestedthrough degeneracy. From the point of view of symmetry,the operations of the complete permutation inversion groupGCNPI ≡ GCNP ⊗ E associated with the insuperable penetra-tion of the barrier are said to be unfeasible, and a subgroupis more suitable to label the states. Hence, the group GCNPI

may contain a subgroup composed of all feasible operationsconnecting several structural versions. This subgroup isknown as molecular symmetry group (MS group). The MSgroup turns out to be isomorphic to the molecular pointgroup when only one structural version is present. It shouldbe stressed that the MS group depends on the resolution ofthe experiment, since the splitting of the inversion tunnelingmay or may not be detected, depending of the quality of theexperimental devices.

From the technical point of view the distinction betweendifferent versions of the equilibrium structure is carried outby labeling the nuclei of the molecule in its equilibriumstructure. By permuting the labels on identical nuclei withand without inverting the molecule, the number of versionscan be identified, but it is only through the experimentthat the MS group can be established. Permutation-inversionoperations of the elements of MS group affect the coordinatesof the nuclei and electrons in the molecule. It is through thiseffect that we are able to establish its connection with themolecular point groups. Each operation p of the MS groupcan then be written in the following form [84–86]:

p = ϕ(

p)

v(

p)

s(

p)

, (18)

where v(p) is an operation that produces the change inthe vibronic coordinates (they translate into changes inthe vibrational displacement coordinates as well as in theelectronic coordinates) caused by p, ϕ(p) is an operationchanging the rotational coordinates (Euler angles), whiles(p) is an operation that generates a nuclear spin permuta-tion. Since the operations v(p), ϕ(p), and s(p) affect differentsubspaces, they commute with each other. In semirigidmolecules (where only one structural version is present)the vibronic operations v(p) constitute the elements ofthe molecular point group isomorphic to the point groupwidely used when in the context of the Born-Oppenheimerapproximation.

Let us now turn our attention to the identification of theirreps as a set of quantum numbers when discrete groupsare involved. In essence, symmetry plays a central role inthe necessity of establishing a complete set of commutingoperators (CSCO) to label the eigenstates |Ψ〉 of the time-independent Schrodinger equation. The Hamiltonian itselfcan be considered in the set of the CSCO, since the energyE provides a label for the state. If α stands for an index


introduced to distinguish different energies, the Schrodingerequation can be written as

H∣

∣Ψαi

⟩ = Eα∣

∣Ψαi

⟩

, i = 1, . . . , gα, (19)

where gα is the degeneracy. Because of the property (16),we may thus think that the set of operators { H , ORi , i =1, . . . , |G|} is useful to define a CSCO, but in general[ORi , ORj ] /= 0, unless the group is Abelian. This problem issolved by identifying subsets of G, which turn out to be theconjugate classes Ki of the group, since [ Kj , Ki] = 0, forall i, j. Hence, the Hamiltonian together with the classes ofthe group constitute a set of commuting operators, whoserepresentations can be diagonalized simultaneously in anyspace of independent functions.

We may now construct the representation matrix of theclass Kp in the basis of eigenvectors of the Hamiltonian:

‖〈Ψαj | Kp|Ψα

i 〉‖. The diagonalization of this matrix provides

eigenvectors of type {|Ψα,λpk 〉, k = 1, . . . , gα,λp}, with the

property H|Ψα,λpl 〉 = Eα|Ψα,λp

l 〉, Kp|Ψα,λpl 〉 = λp|Ψα,λp

l 〉,where λp is the label that distinguishes the different eigenval-

ues of the class operator Kp and l accounts for the degeneracy.We may now proceed to obtain the matrix representation

of the next class Kq in the new basis |Ψα,λpl 〉, to obtain

eigenvectors labeled also with the eigenvalues λq. Followingthis approach for the rest of the classes leads to the states

of the form {Ψα,λ1,...,λ|K|l } characterized by the eigenvalues

{α, λ1, . . . , λ|K|}. However, this set of labels is not complete,since the number of irreps is equal to the number ofclasses, and consequently the set of labels {λν

1, λν2, . . . , λν

|K|}specifies just the possible irreps ν. The question whicharises is concerned with the identification of the new set ofoperators capable to distinguish the states associated with thedegeneracy of the irreps. The answer is given by the classes ofa subgroup H , being H ⊂ G a canonical chain. Suppose thatH has |k| classes {k1, . . . , k|k|}, which clearly satisfy [kp, kq] =0. But the classes Ki of the group G commute with anyelement of the group and consequently commute also withthe classes of the subgroup [Ki, kp] = 0, for all i, p. This factsuggests to diagonalize the representation of the operatorskp in the basis {| Ψα,λ1,...,λ|K|

l 〉} to obtain a complete labelingfor the components of the irreps. After this procedure ofdiagonalization of the classes of H , we arrive to the completelabeling scheme

∣

∣

∣

∣Ψα,λν

1,...,λν|K|

λμ1 ,...,λ

μ|k|

�

, (20)

where the subindices λμp are defined by

kp

∣

∣

∣

∣Ψα,λν

1,...,λν|K|

λμ1 ,...,λ

μ|k|

�

= λμp

∣

∣

∣

∣Ψα,λν

1,...,λν|K|

λμ1 ,...,λ

μ|k|

�

. (21)

Let us now turn our attention to the identification ofthe labels involved in (20) as quantum numbers. The timeevolution of the expected value of an operator A is given by[87]

d

dt

⟨

Ψ∣

∣

∣A∣

∣

∣Ψ⟩

=⟨

Ψ∣

∣

∣

[

H , A]∣

∣

∣Ψ⟩

+

⟨

Ψ

∣

∣

∣

∣

∣

∂ A

∂t

∣

∣

∣

∣

∣

Ψ

⟩

,

(22)

where [ H , A] is the commutator of the Hamiltonian withthe operator A. Hence, if the operator A does not dependexplicitly on time and commute with the Hamiltonian, thenthe expected value is constant in time.

Suppose now that the states are chosen to be eigenstatesof the Hamiltonian together with the classes of the group Gand subgroup H :

|Ψ〉 −→∣

∣

∣

∣Ψα,λν

1,...,λν|K|

λμ1 ,...,λ

μ|k|

�

, (23)

then (22) translates into

d

dtEα = 0,

d

dtλνi = 0,

d

dtλμp = 0,

i = 1, . . . , |K|, p = 1, . . . , |k|,(24)

when A is substituted by H , Ki, and kp. Hence, the eigenval-

ues of the set of operators { H ; K1, . . . ,K|K|; k1, . . . , k|k|} areindependent of time and consequently are quantum numbers.

For a given α there are |K| sets of λνi values. As mentioned

before, this fact suggests a connection between the λνi values

and the characters χ(ν)i of the group. Indeed it has been

proved the following connection [88]:

λνi =

|Ki|nν

χ(ν)∗i , i = 1, . . . , |K|, (25)

where ν stands for the irreps of the group, while nν refers toits dimension. A similar relation is also valid for λ

μp and the

characters of the subgroup H .The expression (25) provides a projection method. This

assertion may be appreciated because of the following result:

any symmetry-adapted function |ψ(ν)i 〉 spanning the νth

irreps of dimension nν satisfies [88]

Ki∣

∣

∣ψ(ν)i

⟩

= λνi

∣

∣

∣ψ(ν)i

⟩

, i = 1, . . . ,nν, (26)

which means that the functions |ψ(ν)i 〉 are eigenvectors of

the class operators with eigenvalue λνi . This remarkable

result suggests to follow the inverse procedure to obtain(23). We can start diagonalizing the class operators toend up with the Hamiltonian. Consider this idea startingwith an arbitrary set of functions {|φi〉, i = 1, . . . ,n}.First we chose a subset of classes that allows the irreps tobe distinguished. The simultaneous diagonalization of theselected classes provides eigenvectors carrying the νth irreps.The eigenvectors spanning the same irreps are then usedto diagonalize the set of classes of the subgroup H . Theresulting eigenvectors span irreps of the group G as wellas the subgroup H , allowing the components of degenerateirreps to be distinguished. This approach, proposed by Chen[88], turns out to be very powerful and has been usedto generate general codes to project vibrational [89] androtation-vibration functions [90].

4. Basic Concepts of an Algebraic Approach

The simplest model to describe the vibrational excitationsof a molecule consists in modeling the system as set of


interacting harmonic oscillators, in accordance with (10) and(11)

Hvib = 12

∑

r

(

P2r + λrQ

2r

)

+16

∑

r,s,t

crst QrQsQt

+1

24

∑

r,s,t,u

crstu QrQsQtQu + · · · .(27)

An algebraic realization of this Hamiltonian can be obtainedby introducing the operators [3]

a†r =1√2

(

Pr + iλ1/2r Qr

)

, ar =(

1 | √2) (

Pr − iλ1/2r Qr

)

,

[

ar , a†r]

= λ1/2r �,

(28)

which have the effect of ladder operators over the harmonicwave functions. This approach is known as the traditionalalgebraic approach for vibrational excitations [91]. Theadvantages of this description are that all matrix elementsare calculated in algebraic form and that the resonances canbe expressed in precise mode. As a consequence, it is easyto identify the interactions that preserve the polyad [15].The harmonic basis intrinsic to this description, however,presents the disadvantage of an infinite dimension of thebasis, a consequence that the potential does not reflect theappropriate behavior in the high energy region of the spectra.To overcome this problem, new algebraic approaches basedon unitary groups emerged to take into account anharmon-icities from the outset.

A fundamental concept intrinsic to the algebraic modelsis the dynamical group. The generators of this group forma Lie algebra, and any dynamical variable, including theHamiltonian, can be expanded in terms of them. In thisway the generators of the dynamical group possess thefundamental property that any pair of eigenstates of theHamiltonian can be connected by them. Bilinear products ofcreation and annihilation operators of a set of ν harmonicoscillators in n dimensions constitute the generators of thedynamical group SP(2νn,R) [14, 26, 92], a noncompactgroup presenting an infinite number of representations, inaccordance with the infinite number of levels that present theharmonic oscillators. It is possible to work out with a com-pact dynamical group by adding an extra boson in such a waythat the totally symmetric representation (total number ofbosons) of the unitary group is fixed, as previously explainedin the introduction. This ingredient simplifies enormouslythe description since the technical machinery involved inthe compact groups is much less complicated than theone involved for the noncompact groups. To illustrate thisapproach, we will start considering the case of one oscillator.

Let us consider a one-dimensional harmonic oscillator,whose Hamiltonian and eigenvectors |n〉 in the algebraicrepresentation take the form

H = �ω

2

(

t†t + tt†)

, |n〉 = 1√n!

(

t†)n|0〉. (29)

The symmetry group of this system isU(1) since its generatorn = t†t commutes with (29) [19]. On the other hand, the

dynamical group is the noncompact group SP(2,R). Theunitary group approach consists in adding an extra boson s

in such a way that the total number of bosons N = t†t+ s†s ≡n + ns is constant. The bilinear products {t†t, s†s, t† s, s†t}are now generators of the SU(2) group. The states associatedwith this group according to the chain

U(2) ⊃ U(1) (30)

are given by [19]

|[N],n〉= 1√

(N − n)!n!

(

s†)N−n(

t†)n|0〉, n = 0, 1, . . . ,N ,

(31)

where the generator of U(1) is n = t†t. Because of thebosonic nature of the vibrations, these states span thetotally symmetric representation [N] of the U(2) group.A connection of the generators of U(2) with the knownangular momentum generators in the Cartesian basis is giventhrough [19]

Jx = 12

(

t† s + s†t)

, Jy = 12i

(

t† s− s†t)

,

Jz = 12

(

t†t − s†s)

(32)

and in the spherical basis

J+ = t†s, J− = s†t,

J0 = 12

(

t†t − s†s)

.(33)

Let us now consider the realization (33) with the followingnormalization:

c† ≡ J+√k= t

† s√k

, c ≡ J−√k= s†t√

k,

c0 ≡ 2J0k= 1k

(

t†t − s†s)

,

(34)

where k = N + 1. The action of these operators over the U(1)kets (31) is

c†|[N],n〉 =√

(n + 1)(1− (n + 1)/k)|[N],n + 1〉,

c|[N],n〉 =√

n(1− n/k)|[N],n− 1〉.(35)

Note that the operators c†(c) connect the whole space of thefunctions |[N],n〉, and consequently U(2) is the dynamicalgroup of the system; any dynamical variable can be expressedin terms of the generators of the unitary group SU(2).However, it is clear from (35) that it is only in the harmoniclimitN → ∞ for finite n that the matrix elements of the one-dimensional harmonic oscillator is recovered, for example,limN→∞c† = t†. Because of the constraint over N , a moregeneral Hamiltonian is expected to be expanded in termsof the generators of the dynamical group U(2) in such away that [ H , N] = 0. The total number of bosons is related


to j, the angular momentum label, through j = N/2 [19].In general it is convenient to establish the different chainsof groups and the invariant operators associated with themsince they are used to expand the Hamiltonian. In our casewe have three more chains in addition to (30) that providealternative bases

U(2) ⊃ SOx(2),

U(2) ⊃ SOy(2),

U(2) ⊃ SOz(2),

(36)

where the subindex in the SO(2) groups refers to the operatorin (32) that generates the group. Hence, a possible simpleexpansion of the Hamiltonian is

Hx = AJ2x . (37)

This Hamiltonian is diagonal in the Ox(2) basis. If thebasis associated with this chain is denoted by |[N],μ〉; Jx|[N],μ〉 = μ|[N],μ〉 and if in addition the followingquantum number is defined v = j+μ, with v = 0, 1, . . . , j−1when only the branch with negative values of μ is considered,the eigenvalues of Hx take the form

Ex(v) = Aμ2 = Aj2 − 2Aj

(

v − 12 jv2

)

. (38)

This is a Morse-like spectrum, which explains the factthat the SOx(2) chain is associated with the Morse func-tions [19]. Analytical solutions obtained from Hamiltoniansinvolving invariant operators associated with a chain arecalled dynamical symmetries. In general, the Hamiltonianthat best suits the description involves invariant operatorsof several chains, and any basis can be used to diagonalizeit. Hence, the introduction of the s-boson not only deformsthe potential, but also enriches the description providingdynamical symmetries and alternative bases to carry out thecalculations.

5. Connection of the U(2) Algebraic Model withthe Morse Potential

The Hamiltonian for one-dimensional Morse potential hasthe form [93]

H = 12μp2 +D

(

1− e−βq)2

, (39)

where μ is the reduced mass, q = x − x0 is the displacementcoordinate, D is the depth of the potential, and β standsfor its range. The solution of the Schrodinger equationassociated with this Hamiltonian is given by

Ψjv(z) = N

jv e−z/2zsL2s

v (z), (40)

where L2sv (z) are the associated Laguerre functions, the

argument z is related to the physical displacement coordinate

q by z = (2 j + 1)e−βq, Njv is the normalization constant and

the variables j and s are related to the depth of the potentialand the energy, respectively through [19]

κ = 2 j + 1 =√

8μDβ2�2

, s =√

−2μEβ2�2

, (41)

with the constraint condition s = j − v. Using the factoriza-

tion method, it is possible to obtain creation b† and annihi-

lation b operators of the Morse functions, which turn out tohave the following effect:

b†Ψ jv(z) =

√

(v + 1)(

1− v + 1κ

)

Ψjv+1(z),

bΨjv(z) =

√

v(

1− v

κ

)

Ψjv−1(z),

(42)

with

vΨjv(z) = vΨ

jv(z), (43)

and whose explicit form in terms of the coordinate andmomentum are given in [94]. The operators {b†, b}, togetherwith the number operator v, satisfy the commutation rela-tions[

b, b†]

= 1− 2v + 1κ

,[

v, b†]

= b†,[

v, b]

= −b,

(44)

which can be identified with the usual su(2) commutationrelations through {b† = J−/

√κ, b = J+/

√κ, v = j − J0},

where Jμ satisfy the usual “angular momentum” commuta-tion relations [73]. Hence, the SU(2) group is the dynamicalgroup for the bound states of the Morse potential [19].From the group theoretical point of view, the parameter jlabels the irreducible representations of the SU(2) group.The projection of the angular momentum m is related to vby m = j − v. From this relation we see that the groundstate (v = 0) corresponds to m = j, while from thedissociation condition the maximum number of quanta isv = j(m = 0). The state corresponding to v = j, however,is not normalizable, and consequently the allowed valuesfor v are v = 0, 1, . . . , j − 1. The Morse functions are thenassociated with one branch (in this case to m ≥ 1) of theSU(2) representations, although a recent work associates anoncompact group to the bound Morse space [95]. Thebound solutions (40), however, do not form a completeset of states in the Hilbert space [96]. However, when thevibrational excitations are far from the dissociation limit, itis a reasonable approximation to consider the bound states asa complete space [97, 98]. A similar situation appears in thecase of the Poschl-Teller potential [96, 99].

The realization of the Morse Hamiltonian in terms of thesu(2) algebra is given by [58]

HM = �ω

2

(

b†b + bb† +1

2κ

)

. (45)

From (42) we obtain the corresponding eigenvalues

EM(v) = �ω

[

(

v +12

)

− 1κ

(

v +12

)2]

. (46)


The harmonic limit is obtained by taking κ → ∞:

limκ→∞

b = a, limκ→∞

b† = a†, (47)

where the operators {a†, a} satisfy the usual bosonic com-

mutation relations. Since the set of operators {b†, b, v}constitutes a dynamical algebra for the Morse potential, anydynamical variable can be expanded in terms of them. Inparticular we are interested in the expansion of the momentaand Morse coordinates. For the momentum the followingsecond-order expansion is obtained [55]:

p = i

2

√

2�ωμ{

fvb† − b fv +

1√κ

(

gvb†b† − bbgv

)

+O(

1κ

)}

,

(48)

while for the Morse coordinate [100, 101]

y

β=√

�

2ωμ

×{

fvb† + b fv +

1√κ

(

f dv + gvb†b† + bbgv

)

+O(

1κ

)}

,

(49)

where fv, gv, and f dv are functions of the number operator vgiven by

fv =√

(κ− 2v − 1)(κ− 2v + 1)

(κ− v)2 ,

gv = −√

√

√

√

κ2(κ− 2v − 1)(κ− 2v + 3)

(κ− v)2(κ− v + 1)2 ,

f dv = 1 + 2v.

(50)

Since v is diagonal, we have substituted v by v. For highnumber of quanta v or κ small, the terms of order 1/

√κ

must be included. In contrast, if we are interested in the lowlying region of the spectrum, a reasonable approximationconsists in neglecting the terms of order 1/

√κ and taking

the harmonic limit of the diagonal functions limκ→∞ fv = 1,limκ→∞gv = −1. We may thus propose the linear approxima-tion

p � i

2

√

2�ωμ(

b† − b)

,

y

β= q �

√

�

2ωμ

(

b† + b)

.

(51)

We immediately note the similarity of these expressions withthe harmonic oscillator case. In fact, taking the harmoniclimit we recover the usual expressions for the harmonic case.In this approximation it is clear that effective interactionspreserving the polyad can be established in a straightforwardway.

Let us now come back to the previous analysis of thealgebraic SU(2) model. If we compare the matrix elements(35) with (42), we immediately identify the isomorphism

c† ≈ b†, c ≈ b, |[N],n〉 ≈ ∣∣ j, v⟩. (52)

However, we still have to identify the dynamical symmetryassociated with the Morse Hamiltonian. The isomorphism(52) suggests the SOz(2) symmetry since the eigenfunctionsof Jz are also eigenfunctions of n. In fact, if σ is the eigenvalueof Jz, we have the identity σ = n − j. Taking the negativebranch of σ , we have n = 0, 1, . . . , j−1. The SOz(2) dynamicalsymmetry takes the form

H = �ω

2

(

c†c + cc† +1

2k

)

, (53)

which up to a constant is basically J2z , as expected. We

have thus established the exact connection between thesu(2) algebraic model and the eigenfunctions of the one-dimensional Morse Hamiltonian.

The connection we have presented deserves an additionalcomment. First we note that, in the linear approximation(51), the generator Jx is proportional to the Morse coordinatey/β, while Jy corresponds to the momentum. The dynamicalsymmetries SOx(2) and SOy(2) are then associated withthe coordinate and momentum representation in the linearapproximation. On the other hand, the SOz(2) dynamicalsymmetry is associated with the energy representation, whilethe U(1) corresponds to the energy representation in theharmonic limit.

The treatment presented here may be followed for thePoschl-Teller potential arriving to a similar result; SU(2) isthe dynamical group of the system [56]. The matrix elements(35) are isomorphic to the corresponding matrix element ofthe creation and annihilation operator of the PT functions.Of course, the explicit form of the wave function is differentas well as the expansion of the coordinate and momenta,but the energy spectrum is the same, as established by thesu(2) algebra. There is a fundamental difference betweenthese potentials, Morse and PT. While the Morse potentialis asymmetric, the PT potential is symmetric, allowing bothsymmetric and asymmetric local modes to be modeled in theunified framework of the su(2) algebra.

6. Two Interacting Morse Oscillators

When the masses of the atoms involved in a set of equivalentbonds are very different, a scheme of interacting localoscillators represents a suitable approach to carry out thevibrational description. For example, the stretching modesin H2O present a strong local behavior because of the largeratio 16 : 1 of the masses. In contrast, when the massesare similar, the behavior is normal and starting from aset of local oscillators does not represent a good zeroth-order approximation. It is possible, however, to describemolecules presenting a strong normal behavior in terms oflocal oscillators, even taking into account the preservationof the polyad, a fact that provides unique advantage ofthe algebraic models over the traditional description inconfiguration space, as we next explain.

We start our discussion with a treatment in terms oflocal harmonic oscillators. The Hamiltonian (14) for two


equivalent oscillators up to quadratic order and neglectingV ′(q) takes the form

H = 12gorr

2∑

i=1

pi +12frr

2∑

i=1

q2i + gorr′ p1 p2 + frr′q1q2. (54)

Introducing the bosonic operators

pi = i

2

√

2�ωμ(

a†i − ai)

qi =√

�

2ωμ

(

a†i + ai)

i = 1, 2,

(55)

with μ = 1/gorr and ω =√

frrgorr , the Hamiltonian (54) takesthe form

limN→∞

H = �

2

√

frrgorr

2∑

i=1

(

a†i ai + aia†i

)

+�

2

√

frrgorr

(

frr′

frr+gorr′

gorr

)

(

a†1a2 + a1a†2

)

+�

2

√

frrgorr

(

frr′

frr− gorr′

gorr

)

(

a†1a†2 + a1a2

)

.

(56)

This Hamiltonian does not preserve the total numberof quanta (polyad), and consequently the dimension ofits matrix representation is infinite. In practice, a polyadpreserving Hamiltonian is considered

H = ωloc

2∑

i=1

(

a†i ai + aia†i

)

+ λloc

(

a†1a2 + a1a†2

)

, (57)

where we have introduced the definitions

ωloc = �

2

√

frrgorr ,

λloc = �

2

√

frrgorr

(

frr′

frr+gorr′

gorr

)

.

(58)

It should be clear that the Hamiltonian (57) is expected toprovide a good description as long as the polyad breakingterm in (56) is neglected. In this section we will see that it ispossible to work with a Hamiltonian of type (57) to describesystems where the full Hamiltonian (56) is expected to beconsidered.

Let us now introduce the symmetry-adapted coordinatesin the Hamiltonian (54)

Sg = 12

(

q1 + q2)

, Su = 12

(

q1 − q2)

, (59)

with the corresponding induced transformation in themomenta. In the new coordinates the Hamiltonian trans-forms into two independent harmonic oscillators (normalmodes)

H = 12

(

GggP2g +GuuP

2u

)

+12

(

FggS2g + FuuS

2u

)

, (60)

where the following definitions are introduced:

Ggg = gorr + gorr′ , Guu = gorr − gorr′ ,Fgg = frr + frr′ , Fuu = frr − frr′ .

(61)

An algebraic representation is obtained through the intro-duction of the bosonic realization

a†Γ = αΓSΓ − i

2�αΓPΓ, aΓ = αΓSΓ +

i

2�αΓPΓ, (62)

where Γ = g,u with parameters

α2g =

12�

√

√

√frr + frr′

g orr + g o

rr′, α2

u =1

2�

√

√

√frr − frr′

g orr − g o

rr′. (63)

The Hamiltonian (60) takes thus the form

H = ωg(

a†g ag + aga†g

)

+ ωu(

a†uau + aua†u

)

, (64)

with the definitions

ωg = �

2

√

(

frr + frr′)(

gorr + gorr′)

,

ωu = �

2

√

(

frr − frr′)(

gorr − gorr′)

.

(65)

The exact connection between the Hamiltonians (56) and(64) is given by the relation between the bosons involved,which is given by

a†g =1

2ααg√

2

{(

α2g + α2

)(

a†1 + a†2)

+(

α2g − α2

)

(a1 + a2)}

,

a†u =1

2ααu√

2

{

(

α2u + α2)

(

a†1 − a†2)

+(

α2u − α2)(a1 − a2)

}

,

(66)

where α2 = (2�)−1√

frr /grr . However, we can return to aHamiltonian of the form (57) introducing the canonicaltransformation

a†g =1√2

(

c†1 + c†2)

, a†u =1√2

(

c†1 − c†2)

, (67)

where c†i (ci) are also bosonic operators associated with theith oscillator. The substitution of (67) into (64) yields

H = ωnor

2∑

i=1

(

c†i ci + cic†i

)

+ λnor

(

c†1 c2 + c1c†2

)

, (68)


ωnor

= �

4

(√

(

frr+ frr′)(

gorr+gorr′)

+√

(

frr− frr′)(

gorr−gorr′)

)

,

λnor

= �

2

(√

(

frr+ frr′)(

gorr+gorr′)−√

(

frr− frr′)(

gorr−4gorr′)

)

.

(69)


The local operators c†i (ci) do not correspond to the physicallocal operators a†i (ai), but their action on an isomorphiclocal basis may be chosen to be the same. In fact we establishthe isomorphism

c†i (ci)←→ a†i (ai), (70)

allowing the Hamiltonian (68) to be expressed in the form

H = ωnor

2∑

i=1

(

a†i ai + aia†i

)

+ λnor

(

a†1a2 + a1a†2

)

. (71)

We thus have that the same algebraic Hamiltonian maydescribe molecules with both local and normal mode behav-iors, but the interpretation of the spectroscopic parametersmust be considered appropriately in order to obtain thecorrect physical results for the force constants. This is aremarkable feature, because it implies that both Hamiltoni-ans provide the same fit to experimental energies, but theforce constants derived from the optimized spectroscopicparameters are different. We should also note that the correctforce constants can be obtained from a local interpretationof the Hamiltonian as long as the polyad breaking term isincluded. Hence, the use of the Hamiltonian (71) avoidsbreaking the polyad keeping the correct physical informationthrough the parameters (69). However, this is possible onlyin the quantum mechanical treatment. The use of coherentstates to extract the potential energy surface does not providethe correct results, as we will shortly discuss [53].

The quantitative criterion to choose between the localand the symmetrized description in order to evaluate appro-priately the force constants is provided by the connectionbetween (57) and (71), which is obtained by rewriting thespectroscopic parameters in (69) in the form

√1 + x to carry

out the Taylor series expansion with the identifications x =frr′ / frr and x = gorr′ /g

orr . This analysis leads to the conditions

∣

∣

∣

∣

∣

frr′

frr

∣

∣

∣

∣

∣

� 1,

∣

∣

∣

∣

∣

gorr′

gorr

∣

∣

∣

∣

∣

� 1, (72)

to be able to apply the local mode Hamiltonian (57). In otherwords,

lim|x|�1

ωnor = ωloc, lim|x|�1

λnor = λloc. (73)

We should also stress that in this limit the correspondencec†i (ci)↔ a†i (ai) is satisfied

lim|x|�1

a†g =1√2

(

a†1 + a†2)

,

lim|x|�1

a†u =1√2

(

a†1 − a†2)

.

(74)

When the conditions (72) are not satisfied the normal modeHamiltonian (71) should be taken. For the water moleculefor instance, we have(

gorr′

gorr

)2

� 2.16 × 10−4,

(

frr′

frr

)2

� 3.76 × 10−4

(75)

while for the CO2 molecule

(

gorr′

gorr

)2

� 0.326,

(

frr′

frr

)2

� 0.01, (76)

which are illustrative of local and normal behaviors, respec-tively.

The Hamiltonians (64) and (71) are completely equiv-alent, since they provide the same energy spectra with thesame force constants. We may now take a further step byapplying the anharmonization procedure [58, 59]

a†i −→ b†i , ai −→ bi, (77)

in such a way that the Hamiltonian (71) transforms into

H = ωnor

2∑

i=1

(

b†i bi + bib†i

)

+ λ nor

(

b†1b2 + b1b†2

)

, (78)

which is intended to be diagonalized in the direct product oflocal Morse oscillators. On the other hand, the Hamiltonianof two interacting Morse oscillators in configuration spacetakes the form

H = 12gorr

2∑

i=1

pi +12frrβ2

2∑

i=1

y2i + gorr′ p1 p2 +

frr′

β2y1y2, (79)

whose algebraic representation up to a constant term takesthe form

H = ωloc

2∑

i=1

(

b†i bi + bib†i

)

+ λloc

(

b†1b2 + b1b†2

)

, (80)

with spectroscopic parameters given by (58). In practicethe Hamiltonians (78) and (80) are equivalent, since bothprovide the same energy spectrum. The difference is givenin the relation between the spectroscopic parameters and thestructure and force constants. While the anharmonizationprocedure provides the correct results for the force constantsthrough (78), the treatment in configuration space does notgive equivalent results, unless the polyad is broken. Hence,this approach is a consequence of a purely algebraic analysiswithout analog in the treatments in configuration space. Inthis paper we will present several applications where thisanalysis is crucial to obtain the correct force constants.

7. PES Using Coherent States

With the advent of algebraic techniques based on unitarygroups to describe nuclei [19, 27, 28], the use of coherentstates became a valuable tool to understand the physicalcontent of the models [51, 102]. The same kind of alge-braic techniques were also applied to describe rovibrationalspectra of molecules, and, because of the phenomenologicalfeature of models, the analysis of the classical limit throughthe use of coherent sates was crucial in providing a physicalinsight into the models [51, 52, 102–105]. According to thismethod, the diagonal matrix elements of the Hamiltonianin the basis of coherent states provide through a suitable


transformation a potential energy surface, whose deriva-tives evaluated at equilibrium give the force constants. Inmolecular physics however a potential energy surface can beobtained by means of the connection of the spectroscopicparameters with the structure and force constants. A naturalquestion which arises is concerned with the comparison ofthe force constants provided by both methods. To answer thisquestions a system of two interacting Morse oscillators willbe considered [53].

The quantum mechanical Hamiltonian of two equivalentinteracting Morse oscillators in configuration space up toquadratic terms is given by (79). This Hamiltonian may betranslated into an algebraic representation by means of theapproximated expressions (51), where there the frequency

of the oscillators ω is given by ω =√

frr /μ =√

frrgorr . In

the expansions (51) all terms of order 1/√k and higher are

neglected. The substitution of (51) into the Hamiltonian(79) gives rise to the algebraic realization

H = �ω

2

2∑

i=1

(

b†i bi + bib†i +

12κ

)

+ λ(

b†1 b2 + b1b†2)

+ λ′(

b†1 b†2 + b1

b2

)

,

(81)

where

λ = �ω

2

(

frr′

frr+gorr′

gorr

)

,

λ′ = �ω

2

(

frr′

frr− gorr′

gorr

)

.

(82)

The expressions ω =√

frrgorr . and (82) constitute explicitdependence of general functions

ω = ω(

frr)

, λ = λ(

frr , frr′)

, λ′ = λ′(

frr , frr′)

,(83)

which will be used in the following discussion. We mayinterpret the above Hamiltonian as a phenomenologicalexpansion with parameters ω, λ, and λ′, and the questionwhich arises is concerned with the possibility to recover theoriginal Hamiltonian from the classical expression obtainedthrough the coherent states. The PES extracted in this way(imposing the condition p1 = p2 = 0) and denoted by V(r)is given as a function of the parameters. In symbolic form

V = V(r1, r2;ω, λ, λ′). (84)

From this expression, the force constants are given by

frr(ω, λ, λ′) ≡(

∂2V(r)∂r2

1

)

0

,

frr′(ω, λ, λ′) ≡(

∂2V(

y1, y2)

∂r1∂r2

)

0

.

(85)

However, we have now the explicit forms (83), and conse-quently the following consistency relations are to be expected

frr(

ω(

frr)

, λ(

frr , frr′)

, λ′(

frr , frr′)) = frr ,

frr′(

ω(

frr)

, λ(

frr , frr′)

, λ′(

frr , frr′)) = frr′ .

(86)

Let us now proceed to consider the coherent states ap-proach. A possible way to express the SU(2) coherent statesis the following [53]:

|N ;α〉 = 1√N !

1√

(

1 +∣

∣ζ∣

∣2)N

(

s† + ζt†)N |0〉, (87)

which may be recast in the form

∣

∣N ; ζ⟩ = 1

√

(

1 +∣

∣ζ∣

∣2)N

eζ J+∣

∣ j,− j⟩, (88)

where, the operator J+ is given by (33). Using the directproduct of two coherent states of the form (88), we obtainthe expectation value of the Hamiltonian (81):

⟨

N ; ζ1ζ2∣

∣H∣

∣N ; ζ1ζ2⟩ = �ω

κ

(

N +14

)

+�ω

κN(N − 1)

×{∣

∣γ1∣

∣2 +∣

∣γ2∣

∣2}

+ λN2

κ

{

γ∗1 γ2 + γ1γ∗2

}

+ λ′N2

κ

{

γ∗1 γ∗2 + γ1γ2

}

,

(89)

where for the sake of simplification, we have introduced thedefinition

γi ≡ ζi(

1 +∣

∣ζi∣

∣2) . (90)

The following transformation to the classical variables y andp is now proposed [53]:

γ = ζ(

1 +∣

∣ζ∣

∣2) = 1√

2κ

⎧

⎪

⎨

⎪

⎩

y

βλ0− i p√

�ωμ

⎫

⎪

⎬

⎪

⎭

,

γ∗ = ζ∗(

1 +∣

∣ζ∣

∣2) = 1√

2κ

⎧

⎪

⎨

⎪

⎩

y

βλ0+ i

p√

�ωμ

⎫

⎪

⎬

⎪

⎭

,

(91)

where

λ0 =√

�

ωμ. (92)

It can be proved that (91) reduces to the expected results inthe harmonic limit [53]. The substitution of (91) into (89)yields the energy surface

E(

y1, y2, p1, p2) = �ω

κ

(

N +14

)

+�ω

2

(

N(N − 1)k2

)

1β2λ2

0

× {y21 + y2

2

}

+�ω

2

(

N(N − 1)k2

)

1�ωμ

{

p21 + p2

2

}

+N2

κ2

(λ + λ′)β2λ2

0

{

y1y2}

+N2

κ2

(λ− λ′)�ωμ

× {p1p2}

,(93)


which is a function of the spectroscopic parameters. Theexpression (93) resembles the original Hamiltonian in con-figuration space, but we have to substitute the functions (83)in order to prove that the full Hamiltonian is recovered. Infact, taking into account the explicit expressions (82), theenergy function (93) transforms into

E(

y1, y2, p1, p2) = h0 +

N(N − 1)k2

2∑

i=1

H(i)M +

N2

k2gorr′ p1p2,

+N2

k2

frr′

β2y1y2,

(94)

where

h0 = �ω

k

(

N +14

)

. (95)

The original Hamiltonian is recovered in the limit ofN large,except for a constant term:

limN→∞

E(

y1, y2, p1, p2) = 5

4�ω +

2∑

i=1

H(i)M + gorr′ p1p2 +

frr′

β2y1y2.

(96)

From this expression, the PES is obtained at once by takingthe condition p1 = p2 = 0,

V(

y1, y2) = lim

N→∞E(

y1, y2, p1 = 0, p2 = 0)

= frr2β2

y2i +

frr′

β2y1y2,

(97)

with the consistency result(

∂2V(

y1, y2)

∂r21

)

0

= frr ,

(

∂2V(

y1, y2)

∂r1∂r2

)

0

= frr′ .

(98)

In this way the PES has been reproduced [53].From this analysis one point should be remarked: the

reason why it was possible to fully recover the PES, besideshaving used the appropriate transformation (91), is that thefull Hamiltonian (up to quadratic order) was involved. Ingeneral this situation is not satisfied, as we next discuss.

7.1. Polyad Approximation. In molecular spectroscopy mostof the times an approximation involving polyad conservationis considered as zeroth-order calculation. If necessary a VanVleck perturbation theory is applied to take into accountpolyad mixing [101, 106, 107]. In our case of two interactingoscillators the polyad P = v1 + v2 is considered as a goodquantum number, and therefore the corresponding algebraicHamiltonian to be taken is given by

H = �ω

2

2∑

i=1

(

b†i bi + bib†i +

12κ

)

+ λ(

b†1 b2 + b1b†2)

.

(99)

Let us now proceed to analyze the coherent state method toextract the PES. For this Hamiltonian we have

⟨

N ; ζ1ζ2∣

∣H∣

∣N ; ζ1ζ2⟩ = �ω

κ

(

N +14

)

+�ω

κN(N − 1)

×{∣

∣γ1∣

∣2 +∣

∣γ2∣

∣2}

+ λN2

κ

{

γ∗1 γ2 + γ1γ∗2

}

,

(100)

and applying the transformation (91) we obtain

E(

y1, y2, p1, p2) = �ω

κ

(

N +14

)

+�ω

2

(

N(N − 1)k2

)

1β2λ2

0

× {y21 + y2

2

}

+�ω

2

(

N(N − 1)k2

)

1�ωμ

{

p21 + p2

2

}

+N2

κ2

λ

β2λ20

{

y1y2}

+N2

κ2

λ

�ωμ

{

p1p2}

.

(101)

This expression reproduces the independent Morse oscil-lators, but it involves only λ. In this case the consistencyrelations (86) are not strictly satisfied [53].

If we now impose in the Hamiltonian (101) the con-straint p1 = p2 = 0, we obtain for the PES

V(

y1, y2;ω, λ) = �ω

κ

(

N +14

)

+�ω

2

(

N(N − 1)k2

)

1β2λ2

0

× {y21 + y2

2

}

+N2

κ2

λ

β2λ20

{

y1y2}

.

(102)

From this expression we finally obtain for the force constants

frr(k,ω) =(

∂2V(

y1, y2;ω, λ)

∂r21

)

0

=(

N(N − 1)k2

)

�ω

λ20

,

frr′(k, λ) =(

∂2V(

y1, y2;ω, λ)

∂r1/∂r2

)

0

= N2

κ2

λ

λ20.

(103)

If we now take into account the relations for ω and (82), wearrive at

limk→∞

frr(k,ω) ≡ limk→∞

(

∂2V(

y1, y2;ω, λ)

∂r21

)

0

= frr ,

limk→∞

frr′(k, λ) ≡ limk→∞

(

∂2V(

y1, y2;ω, λ)

∂r1∂r2

)

0

= frr′ ,

(104)

as long as the condition λ′ = 0 is taken into account.As an example, we next consider the water molecule. A

fitting of the vibrational levels of water is a difficult task.


However, if we consider only the 19 stretching levels (up topolyad 5) reported by Halonen in his contribution in [2],a pretty good fit with an rms = 0.50 cm−1 is obtained, aslong as up to quartic interactions are included (6 parametersinvolved). Keeping the same energies, but considering thesimple Hamiltonian (99), it is not possible to describe thespectrum with a reasonable quality. Reducing the numberof energies by considering levels up to polyad 3, a fittingwith rms = 24.23 cm−1 is obtained with the following setof spectroscopic parameters:

(k = 60 −→ N = 59), ω = 3808 cm−1, ˜λ = −54.64 cm−1,(105)

where the tilde is introduced to emphasize that the param-eters are given in wave numbers. Taking into account thedefinition (92) and the explicit expressions for the Wilsonmatrix elements, we obtain for the force constants [53]

frr(k) =(

N(N − 1)k2

)

�ω

λ20=(

N(N − 1)k2

)

(�ω)2

�2gorr

=(

N(N − 1)k2

)

8.0409aJ

A2 ,

frr′(k) = N2

κ2

λ

λ20= N2

κ2

�ω

�2gorrλ = −N

2

κ20.1154

aJ

A2 .

(106)

The force constants can be also calculated from thequantum mechanical Hamiltonian using the connection ofthe spectroscopic parameters with the force constants givenin (82), whose results are displayed in Table 1 with the labelQM. In the same table the results (106) are labeled (CS),and (CS(N → ∞)) when the limit N → ∞ is considered.According to the previous analysis the difference between thefirst column (QM) and the third one (CS(N → ∞)) lieson the approximation λ′ = 0 in the Hamiltonian (81). ForH2O this parameter is approximately given by λ′ ≈ 7.7810−4,which when compared with λ ≈ −0.0286918 justify the useof the Hamiltonian (99).

It should be clear that the good agreement shown inTable 1 is due to the fact that λ′ is approximately null, whichin turn is satisfied for molecules with local behavior. In otherwords, the vibrational excitations in water molecule can beapproximated as interacting local oscillators. The naturalquestion which arises is concerned with the results in systemswith a normal mode behavior.

7.2. Normal Mode Behavior. In this subsection we addressthe problem of obtaining the PES when a Hamiltonian oftype (99) preserving the polyad is considered in the fittingof a spectrum associated with a normal mode behavior. Westart with the Hamiltonian (81) in the harmonic limit, whichtakes the form (56). The corresponding polyad preserving

Table 1: Force constants for H2O obtained from the fit (118)using (a) the quantum mechanical connection (QM) given by (82),(b) the coherent states method (CS), and (c) the coherent stateapproach taking the harmonic limit N → ∞.

Force constant QM CS CS (N → ∞)

frr aJ A−2

8.0409 7.6389 8.0409

frr′ aJ A−2 −0.1122 −0.1115 −0.1154

Hamiltonian is given by (57). As we previously discussed, theHamiltonian (56) in configuration space is given by

H = 12gorr(

p21 + p2

2

)

+12frr(

q21 + q2

2

)

+ gorr′ p1p2 + frr′q1q2,

(107)

which can be transformed into the algebraic representation(64), when symmetry-adapted coordinates are introduced.The exact connection between the Hamiltonians (56) and(64) is given by the relation between the bosons involved(66), but we can return to the Hamiltonian of the form(56) introducing the canonical transformation (67). Thesubstitution of (67) into (64) yields (68). As we stressed, thelocal operators c†i (ci) do not correspond to the physical localoperators a†i (ai), but their action on an isomorphic localbasis may be chosen to be the same. In fact we will establishthe isomorphism c†i (ci) ↔ a†i (ai), allowing the Hamiltonian(68) to be expressed in the form

H = ωnor

3∑

i=1

(

a†i ai + aia†i

)

+ λnor

∑

i> j

(

a†i a j + aia†j

)

. (108)

We may now consider the anharmonization procedure [58,59]

a†i −→ b†i , ai −→ bi, (109)

to obtain the Hamiltonian (78)

H = ωnor

2∑

i=1

(

b†i bi + bib†i

)

+ λnor

(

b†1b2 + b1b†2

)

. (110)

This approach of identification of the spectroscopic param-eters in terms of the structure and force constants does nothave analog in the treatments in configuration space.

As an example of a normal mode behavior, we will con-sider the molecule of carbon dioxide. Although the strongFermi interaction of this molecule makes it inappropriateto use the Hamiltonian (99) to obtain good values of forceconstants, we will proceed to estimate them by taking therelevant stretching parameters from a fit using the U(2) ×U(3)×U(2) model given in [68], where the purely stretchingHamiltonian up to quadratic terms coincides with (99). Theparameters are given by

(k = 160, N= 159), ω=916.15 cm−1, ˜λ=−474.22 cm−1.(111)


Table 2: Force constants for CO2 by means of (a) [68] (QM∗), (b)using the quantum mechanical connection (QM) given by (110),(c) the coherent states method (CS) using (103), (d) the coherentstate approach taking the harmonic limit N → ∞, and (e) thecoherent states approach using the Hamiltonian (81), but estimatedthrough the connection with the Hamiltonian (110) (CS-PB).

Force constant QM∗ QM CS CS (N → ∞) CS-PB

frr aJ A(−2)

15.55 17.48 13.32 13.58 17.48

frr′ aJ A(−2)

1.86 6.48 −3.46 −3.51 6.48

If we intend to calculate the force constants by means of(103) according to our coherent states method, we obtain theresults displayed in the columns CS and CS (N → ∞) ofTable 2. In contrast, when the force constants are calculatedquantum mechanically using (69) the results of columnlabeled QM is obtained. As a reference, the force constantsobtained from [68] by means of the U(2) × U(3) × U(2)model are displayed in the column labeledQM∗. In the lattercase the bending interactions are involved in the calculationswhich explains the difference. Since in this work we areinterested in evaluating the coherent states method, we willbe interested in comparing the predicted force constants withthe column QM.

While for frr the values are not too different, in the caseof the force constant frr′ the results are fairly different evenqualitatively. We should stress the wrong sign of the forceconstant frr′ , which is determined by the sign of λ accordingto (103). This result is a consequence of having used a polyadpreserving Hamiltonian to obtain the PES. This assertion canbe proved by means of the criterion (76), which reflects thenormal behavior that characterizes this molecule. Hence, it isclear that for carbon dioxide the coherent state method failswhen a polyad preserving Hamiltonian is used. In moleculeswith normal mode behavior, the appropriate Hamiltonian torecover the PES must be the full Hamiltonian (81), whichmeans that in principle the polyad has to be broken. In otherwords, in this case the parameter λ′ in (81) is not negligibleif one intends to calculate the force constants. In fact for CO2

we have

λ′

λ= frr′ / frr − gorr′ /gorr

frr′ / frr + gorr′ /gorr≈ −1.42. (112)

From the previous analysis we know that the coherent stateapproach applied to the full Hamiltonian (81) taking thelimit k → ∞ provides the same force constants that thequantum mechanical result labeled QM in Table 2. This isa remarkable result that must be taken into account whenusing this treatment. To emphasize this point in Table 2, acolumn labeled as coherent state-polyad breaking (CS-PB) hasbeen added, whose results are expected to be obtained froma fit using (81).

We have thus presented the method of coherent statesto extract the PES of molecular systems in the contextof the Born-Oppenheimer approximation. The analysis hasbeen presented for the one-dimensional version of thealgebraic models based on unitary groups. In particular thealgebraic representation of two interacting Morse oscillators

was considered. Special attention was paid to the case whenthe polyad is considered as a good quantum number. Itwas shown that in this approximation the coherent statesapproach reproduce the PES only for systems with localmode behavior. For molecules with a normal mode behaviorthe coherent state approach is still valid, but a Hamiltonianbreaking the polyad must be considered, as a consequenceof the strong coupling. The two independent oscillators stopbeing a good zeroth-order Hamiltonian, and the parameterω in (81) is strongly influenced by the interaction associatedwith the parameter λ′. Since the polyad concept is crucialto be able to carry out the calculations in the descriptionvibrational excitations in molecules, we conclude that inpractice the suitability of the coherent state approach isrestricted to molecules with local mode behavior. In the nextsections we proceed to show how to obtain the PES quantummechanically for different representative systems of semirigidmolecules.

8. Application to Water Molecule

In an algebraic description, where the polyad is consideredas a pseudo-quantum number, the possible force constantsto be estimated from a fit are restricted. The reason is thatmany terms should be taken away from the Hamiltonian,eliminating the possibility of calculating the correspondingforce constants. When the linear approximation (51) isconsidered, a similar situation is present. In this section wewill show that, when the quadratic terms are included inthe expansions of the coordinates and momenta (see (48)and (49)), the whole set of force constants can be estimatedup to the order considered in the Hamiltonian. To this endwe will present the vibrational description of H2

16O in theframework of the SU(2) model, the one-dimensional case ofthe unitary group approach.

To estimate the PES of H216O beyond the linear approx-

imation (51), a tensorial formalism is developed to expandthe Hamiltonian in powers of 1/

√k in terms of symmetry-

adapted operators, which in turn are given in terms of localcreation and destruction Morse operators in the spirit toestablish the connection between the effective Hamiltonianapproach and the standard local mode models in configu-ration space [58]. Hence, in the framework of the algebraicmodel, independent Morse oscillators are considered as azeroth-order approximation. In a Morse oscillator basis,however, the interaction terms couple the whole space in theHamiltonian matrix, not allowing to take advantage of thesimplifications brought about by approximately conservedpolyad numbers. This problem, however, can be avoided bykeeping only the terms which preserve the polyad. Here wewill first proceed to obtain the Hamiltonian in terms of theMorse variables yi and their momenta, to thereafter carry outtheir expansions in terms of the symmetry-adapted opera-tors, keeping the terms preserving the polyad. A remarkableconsequence of keeping the terms of order (1/

√κ) in the

expansion in the Morse coordinates and momenta is that allthe force constants up to quartic terms in the potential can bedetermined, even with the constraint of the polyad as a goodquantum number, as previously mentioned.


The equilibrium structure of the water molecule isnonlinear with structure parameters rOH = 0.9575 A and∠HOH = 104.51◦ [108]. The point symmetry is C2v, but itis enough to consider the subgroup C2 since the vibrationstake place on a plane. This molecule has three degrees offreedom, two of them associated with the stretching modes(A ⊕ B) and the other to the bending mode (A). Theharmonic approximation provides a complete basis in termsof normal modes which can be used to diagonalize a generalHamiltonian. In the standard notation this basis is labeled by|ν1ν2ν3〉, where ν1 and ν3 are the number of quanta in thestretching A and B modes, respectively, while ν2 is associatedwith the bending A mode [8].

If we use internal displacement coordinates, the quantummechanical Hamiltonian that describes the vibrational exci-tations of the H2O molecule takes the form (14)

H = 12˜pG(

q)

p +V(

q)

. (113)

Here we have omitted the purely quantum mechanical termderived from the kinetic energy not involving momentumoperators. The components qr and qr′ will be assignedto the O-H stretching displacements coordinates from theequilibrium, while qφ corresponds to the displacement fromequilibrium of the ∠HOH-bending coordinate. In this waywe have

qr = Δr, qφ = re Δφ, qr′ = Δr′, (114)

where re has been added in the definition of the bendingcoordinate in order to have the same distance units. For semi-rigid molecules a reasonable approach consists in expandingboth the G(q) matrix as well as the potential V(q) in aTaylor series about the equilibrium configuration. It has beenpointed out, however, that, in order to obtain convergencefor large oscillations, the Morse variables yi = 1 − e−βiqi areappropriate [109–114]. Hence, here we consider an expan-sion of the variables {yi, i = r, r′,φ} for both stretchingand bending coordinates. In this spirit the elements gi j ofthe G(q) matrix are expanded up to quadratic terms, sincewe intend to consider an expansion in the Hamiltonian upto quartic terms. Consequently an expansion up to quarticterms is carried out for the potential. It is worth notingthe relevance of the ratio yi/βi. In the rigid limit this ratiotends to the variable qi. The zeroth-order Hamiltonian isthen given by

H0 = HMr + HM

r′ + HMφ , (115)

where

HMi =

12goii p

2i +

12fii y

2i , i = r, r′,φ, (116)

while the complete Hamiltonian up to quartic terms takes theform [58]

H = H0 +(

grr′)

0 pr pr′ +12

(

∂gφφ∂qr

)

0

β−1r

(

yr + yr′)

p2φ

+

(

∂grφ∂qφ

)

0

β−1φ

(

pr + pr′)

(

yφ pφ + pφ yφ)

2

+14

[(

∂2gφφ∂q2

r

)

0

+

(

∂gφφ∂qr

)

0

βr

]

β−2r

(

y2r + y2

r′)

p2φ

− 12

(

∂2gφφ∂q2

φ

)

0

β−2r yr yr′ p

2φ

+14

⎡

⎣

(

∂2gφφ∂q2

φ

)

0

+

(

∂gφφ∂qφ

)

0

βφ

⎤

⎦β−2φ pφ y

2φ pφ

+12

⎡

⎣

(

∂2grr′

∂q2φ

)

0

+

(

∂qrr′

∂qφ

)

0

βφ

⎤

⎦β−2φ pr pr′ y

2φ

+(

grφ)

0

(

pr + pr′)

pφ

+12

(

∂gφφ∂qφ

)

0

β−1φ pφ yφ pφ

+

(

∂grr′

∂qφ

)

0

β−1φ pr yφ pr′

+

(

∂grφ∂qr′

)

0

β−1r

(

pr yr′ + pr′ yr)

pφ

+14

(

∂2gφφ∂q2

φ

)

0

β−1r β−1

φ

(

pr yr′ + pr′ yr)

(

yφ pφ + pφ yφ)

+12

(

∂2gφφ∂qr∂qφ

)

0

β−1r

(

yr + yr′)

pφ yφ pφ

+12

[(

∂2qrφ∂q2

r

)

0

+

(

∂2grφ∂qr

)

0

βr

]

β−2r

(

pr y2r′ + pr′ y

2r

)

pφ

+12

⎡

⎣

(

∂2grφ∂q2

φ

)

0

+

(

∂grφ∂qφ

)

0

βφ

⎤

⎦β−2φ

(

pr + pr′)

×(

y2φ pφ + pφ y

2φ

)

2+ frr′β

−2r yr yr′

+12

(

βφ frφ + frφφ)

β−1r β−2

φ

(

yr + yr′)

y2φ

+14!

(

11β2φ fφφ + 6βφ fφφφ + fφφφφ

)

β−4φ y4

φ

+14!

(

11β2r frr + 6βr frrr + frrrr

)

β−4r

(

y4r + y4

r′)

+14!

6(

βrβφ frφ + βφ frrφ + βr frφφ + frrφφ)

× β−2r β−2

φ

(

y2r + y2

r′)

y2φ

+14!

12(

βφ frr′φ + frr′φφ)

β−2r β−2

φ yr yr′ y2φ

+14!

6(

β2r frr′ + 2βr frr′r′ + frrr′r′

)

β−4r y2

r y2r′

+14!

4(

2β2r frr′ + 3βr frrr′ + frrrr′

)

β−4r

(

yr′ y3r + yr y

3r′)


+ frφβ−1r β−1

φ

(

yr + yr′)

yφ

+16

(

3βr frr + frrr)

β−3r

(

y3r + y3

r′)

+16

(

3βφ fφφ + fφφφ)

β−3φ y3

φ

+12

(

βr frr′ + frrr′)

β−3r

(

yr′ y2r + yr y

2r′)

+12

(

βr frφ + frrφ)

β−1φ β−2

r

(

y2r + y2

r′)

yφ

+ frr′φβ−2r βφ yr yr′ yφ

+14!

4(

2β2r frφ + 3βr frrφ + frrrφ

)

β−3r β−1

φ

(

y3r + y3

r′)

yφ

+14!

4(

2β2φ frφ + 3βφ frφφ + frφφφ

)

β−3φ β−1

r

(

yr + yr′)

y3φ

+14!

12(

βr frr′φ + frr′r′φ)

β−3r β−1

φ

(

yr y2r′ + yr′ y

2r

)

yφ.

(117)

Since we consider noninteracting Morse oscillators as thezeroth-order Hamiltonian, the natural basis to diagonalizethe Hamiltonian (117) consists in the direct product ofMorse functions

∣

∣

∣ jr , jφ; vr , vr′ , vφ⟩

= ∣∣ jrvr⟩⊗

∣

∣

∣ jφvφ⟩

⊗ ∣∣ jrvr′⟩

, (118)

where vr and vr′ stand for the number of quanta associatedto qr and qr′ , respectively, vφ corresponds to the bending qφcoordinate, and we have taken into account the equivalenceof the stretching bonds through the equality jr = jr′ . Tosimplify the matrix representation of the Hamiltonian, thepolyad should be introduced. The polyad is established fromthe resonances of the first overtones with the fundamentals.For H2O the energy of the symmetric mode is approximatelytwice the bending energy [115]. Hence, the appropriatepolyad number is defined by

P = 2(vr + vr′) + vφ = 2(ν1 + ν3) + ν2. (119)

We should note that P can be written in terms of bothlocal and normal quantum numbers. This situation is dueto the fact that this molecule presents a strong local behavior.In cases where a normal behavior appears, the polyad canbe given only in terms of normal quantum numbers. Anexample of this case is the BF3, a molecule that will bediscussed in the next section.

Although we may use the basis (118) to diagonalize theHamiltonian, it is more convenient to propose a symmetry-adapted basis. The new basis should carry quantum numbersisomorphic to the normal quanta, as well as irreduciblerepresentations of the symmetry group. A basis with thesecharacteristics can be constructed in several ways, but themost efficient approach consists in a combination of theeigenfunction method developed by Chen [88] and thediagonalization of the symmetry-adapted number operatorsin a harmonic basis. This approach is explained in detail

in [89, 90]. The Hamiltonian, on the other hand, beingtotally symmetric, can be expressed in terms of symmetry-adapted tensors coupled to the totally symmetric irreduciblerepresentation. Hence, we introduce the following tensors[58]:

YΓ0,x =

∑

i

αΓx,i fdvi ,

YΓ†1,x =

∑

i

αΓx,i fvib†i ,

YΓ†2,x =

∑

i

αΓx,i gvib†ib†i ,

(120)

where the subindex x = s, b was included in order todistinguish tensors arising from the stretching and bending

degrees of freedom, respectively, and the coefficients αΓ,γx,i

correspond to the coefficients of the symmetry-adaptedfunctions of one quantum.

When the expressions (48) and (49) are substituted into(117), each contribution of the Hamiltonian will be givenin terms of an expansion in powers of the parameter 1/

√κ,

which, according to the approximation, ranges from (1/√κ)0

to 1/κ2. If we restrict ourselves to keep only the polyadpreserving terms, several contributions disappear. Giventhis substitution, we now invoke the symmetry-adaptedtensors (120) in order to make clear the invariance of theHamiltonian as well as the order of the interactions. Hence,in terms of symmetry-adapted tensors, the Hamiltonian(117) takes an expansion of the form

H = H0 − x1

[(

YA†1,s Y

A1,s −YB†

1,sYB1,s

)]

− x2

√2[

YA†1,s Y

A1,bY

A1,b +

1√κYA

0,sYA†1,bY

A1,b + hc

]

. . . ,

(121)

whose complete expression is given in [58]. In [58] theHamiltonian is expressed in terms of two kinds of tensorsassociated with the coordinates and momentum. The reasonis that the coefficients involved in the expansions of (48)and (49) were written in different ways. The coefficientsassociated with the kinetic energy are named xi, while theones associated with the potential expansion are zi. They aregiven in Tables 1 and 2 of [58]. In spite of the apparent com-plexity of the Hamiltonian (121), the form of the tensorialexpansion is very enlightening. On one hand it makes clearthe order of the interaction involved in each term, and onthe other hand provides the type of additional interactionswhich are taken into account when the polyad is preserved.If in (121) we carry out the limit κ → ∞, all the tensors YΓ

0,x

and YΓ†2,x vanish and consequently some of the coefficients

disappear. As a consequence only the subset of force con-stants { frr , fφφ, frr′ , frrrr , frrrr′ , frrr′r′ , fφφφφ, frrφφ, frφφ, frr′φφ}are able to be determined. These force constants are theones that are determined in a traditional calculation usingthe linear expansions (51) preserving the polyad [57]. Incontrast, in this approach all the force constants up to quarticorder are determined. In the same spirit we may break thepolyad taking terms up to order 1/

√κ [101].


Table 3: Force constants for H2O. In the first column is indicated the order that should be considered in order to obtain the correspondingforce constant.

Order Force constant This worka Jensenb Halonen and Carrington Jr.c Lemus et al.d Ab initioe

1 βs/A−1

2.1573 2.053 2.1542

1 Ds/aJ 0.8694 0.999 0.9050

1 βb/A−1

0.8279 0.7296

1 Db/aJ 0.5190 0.7055

1 frr /aJA−2

8.093 8.4393 8.428 8.401 8.443

1 r2e fφφ/aJ 0.652 0.7070 0.699 0.688 0.7921

1 frr′ /aJA−2 −0.1571 −0.1051 −0.101 −0.11 −0.100

1/√κ re frφ/aJA

−1 −1.0578 0.3064 0.219 0.2743

1/√κ frrr /aJA

−3 −35.880 −55.40 −51.91 −56.400

1/√κ re frr′ /aJA

−2 −3.098 −0.447 0.414 −0.505

1/√κ frrr′φ/aJA

−3 −0.358 −0.318 0.645 −0.076

1 r2a frφφ/aJA

−1 −2.6243 −0.3383 −0.314 −0.51 −0.3210

1/√κ re frrφ/aJA

−221.442 −0.252 1.341 −0.084

1/√κ r3

e fφφφ/aJ −8.4312 −0.7332 −0.918 −0.7482

1 frrrr /aJA−4

130.87 306.0 248.7 275.39 338

1 r2e frrφφ/aJA

−2 −15.89 −0.950 −2.0 −15.699 −0.28

1 frrrr′ /aJA−4

7.861 2.57 3.728 −0.30

1 r2e frr′φφ/aJA

−22.3910 0.1150 −0.632 0.62

1 frrr′r′ /aJA−4 −9.843 1.93 −12.68 0.52

1 r4e fφφφφ/aJ 23.349 −0.238 −0.1 2.32 −0.74

1/√κ re frrrφ/aJA

−3 −100.98 −6.14 −1.2

1/√κ re frrr′φ /aJA

−33.5584 −3.22 0.2

1/√κ r3

e frφφφ/aJA−1

8.955 0.87 0.648aFrom [58]. bFrom [138]. cFrom [109]. dFrom [57]. eFrom [139].

Following this approach, in [58] is described the vibra-tional spectrum of the most abundant isotopic speciesof water, H16

2 O. An energy fit was carried out for 72experimental energies up to 23, 000 cm−1, obtaining an rmsdeviation of 5.00 cm−1. The Hamiltonian (121) includes 17force constants, the Morse parameters βr and βφ, plus theChild’s parameters κs and κb. Since the latter parameters areconnected with the anharmonicity of the oscillators, theyare estimated by considering the energy levels for one andtwo quanta. This procedure gives rise to the parametersκs = 47, κb = 86. The energy fit was carried out varyingthe linear parameters zi and the Morse frequencies ωi,i = r,φ, providing the set of force constants displayed inTable 3. In the first column of the table, the order neededin the expansions (48) and (49) in order to determine hecorresponding force constants is indicated. All the forceconstants of order 1/κ are not determined in the linearapproximation (51).

A comparison with several description is presented. Ingeneral the force constants of second order follow the generaltrend of the previous calculations. For third- and quartic-order constants, however, large differences appear. This maybe explained because of the difference in the calculations,in particular compared with the calculations of Jensen andthe ab initio results. This approach is quite simple, and theresults may be compared with similar results, like the ones

by Halonen and Carrington jr. [109] and Lemus et al. [57].Indeed the force constants are similar.

Because of the differences in the more elaborated calcu-lations, the reliability of the force constants was tested bypredicting the energies for the isotopes H17

2 O, H182 O, D16

2 O,and T16

2 O, which can be found in Tables x, x, and xxx of [58].Finally, we should mention that the Hamiltonian (121),

which is impossible to obtain in more complex situations,is fundamental to interpret the different interactions derivedfrom the use of the expansions (48) and (49). In practice,however, the symmetry-adapted expansion is not necessary.

9. Application to Trifluoride of Boron

In the description of vibrational excitations of moleculeswith high symmetry, the appearance of spurious statesemerges in natural form when internal coordinates are used.The elimination of these extra degrees of freedom must bedone from both the space and the Hamiltonian. Trifluorideof boron is a molecule that presents this feature, but it isalso interesting because of its strong normal behavior. Thedescription of this molecule in terms of normal modes isnot a problem, the interesting situation appears when it isdescribed in terms of a local mode model. As we know,the latter description cannot be carried out preserving thepolyad. We show, however, that our algebraic approach


allows this analysis to be done with the advantage of improv-ing the description considerably.

Recently in a series of works the high-resolution infraredand Raman spectra of 11B F3 from 400 to 4600 cm−1 wereanalyzed [116–120]. In these studies the analysis of thecombination and overtone states led to the assignment ofover 25,000 transitions. The number of pure vibrationalenergies, however, is reduced to around 25 band origins.The equilibrium configuration of the 11B F3 molecule is D3h

planar with structure parameter re = 1.3070 A [120]. Itpresents six vibrational degrees of freedom, three of themassociated with the stretching modes (A′1 ⊕ E′), two of themcorresponding to the bending modes (E′), and one out-of-plane mode (A′′2 ). The harmonic approximation providesa complete basis in terms of normal coordinates. In thestandard notation, this basis is labeled by |ν1ν2νl33 νl44 〉, whereν1 and ν3 correspond to the symmetrical and degenerateB-F stretching modes A′1 and E′, respectively, while ν2 andν4 correspond to the out-of-plane and degenerate F-B-Fbending modes A′′2 and E′, respectively. Since the dipoleoperator spans the A′′2 and E′ representations, only theν1 vibrational mode is infrared inactive, the other threefundamentals are infrared active. Many of the combinationand overtones states turn out to be infrared inactive as directtransitions from the ground states. However these states maybe accessed as infrared-active hot bands transitions fromthermally populated levels [118]. The fundamentals of 11BF3

are (in cm−1)

ν1 = 885.84, ν2 = 691.21,

ν3 = 1453.96, ν4 = 479.35,(122)

from which we identify the following allowed symmetry res-onances:

ν1 ≈ 2ν4, ν3 ≈ 3ν4, 2ν2 ≈ 3ν4. (123)

The identification of these resonances permits to establishthe polyad P, which in terms of integer numbers may bedefined as

P = 4ν1 + 3ν2 + 6ν3 + 2ν4. (124)

The structure of this polyad (209) reflects its strong normalbehavior, which in turn is manifested in the fact that theinteractions up to quadratic order will be dominated bydiagonal contributions in a normal basis.

In Figure 1 of [66] the assignment of local displacementcoordinates is displayed. The first set (ri, i = 1, 2, 3)corresponds to the space of stretching oscillators (s), thesecond set (θi, i = 4, 5, 6) stands for the space of bendingoscillators (b), while the set (δi, i = 7, 8, 9) is associated withthe out-of-plane angles (γ):

qi = Δri, i = 1, 2, 3,

qj = reΔθj , j = 4, 5, 6,

qk = reΔγk, k = 7, 8, 9,

(125)

where re is the equilibrium distance B-F. We now proceed toobtain the symmetry-adapted coordinates Sα, given by

Sαx =∑

i

miαqi, α = 1, . . . , 9, (126)

where α ≡ {Γ,μ} stand for the irreps Γ and μ of the D3h

group and the subgroup Ca2 , respectively, and x refers to the

subspace (s, b, γ). Explicitly, we have for the stretching modes

SA′1s = 1√

3

(

q1 + q2 + q3)

,

SE′,As = 1√

6

(

2q1 − q2 − q3)

,

SE′,Bs = 1√

2

(

q2 − q3)

(127)

and for the bending coordinates

SA′1b =

1√3

(

q4 + q5 + q6)

,

SE′,Ab = 1√

6

(

q4 − 2q5 + q6)

,

SE′,Bb = 1√

2

(

q4 − q6)

,

(128)

while for the out-of-plane subspace

SA′′2γ = 1√

3

(

q7 + q8 + q9)

,

SE′′,Aγ = 1√

6

(−q8 + q9)

,

SE′′,Bγ = 1√

2

(

2q7 − q8 − q9)

.

(129)

The coordinates associated with the bending and out-of-plane coordinates in (125) are not independent. For thebending coordinates, the following redundant equation isgiven [62]:

6∑

i=4

qi +1

re√

3

⎡

⎣

12

6∑

i=4

q2i +

6∑

i> j=4

(

qiq j)

⎤

⎦ + · · · = 0, (130)

while for the out-of-plane coordinates

(

q9 − q8)− 1

re√

3

(

q4q9 − q6q8)

+ · · · = 0,

(

2q7 − q8 − q9)− 1

re√

3

(

2q5q7 − q6q8 − q4q9)

+ · · · = 0.

(131)

These expressions, obtained through vectorial relationsbetween the stretching vectors, allow the coordinates

{SA′1b , SE′′,Aγ , SE

′′,Bγ } to be identified as redundant in the linear

approximation. The transformation (127) is linear, butthe transformations (128) and (129) are not linear in thesense that their inverse are nonlinear since the redundant


equations (130) and (131) are to be satisfied. Indeed, in goingup to the second order, the nonlinear transformations fromthe symmetrized coordinates to internal valence coordinatesare found to be

q4 = 1√6SE

′,Ab +

1√2SE

′,Bb ,

q5 = − 2√6SE

′,Ab ,

q6 = 1√6SE

′,Ab − 1√

2SE

′,Bb ,

q7 = 1√3SA′′2γ − 1

3re

√

23SA′′2γ SE

′,Ab ,

q8 = 1√3SA′′2γ +

13√

6reSA′′2γ SE

′,Ab − 1

3√

2reSA′′2γ SE

′,Bb ,

q9 = 1√3SA′′2γ +

13√

6reSA′′2γ SE

′,Ab +

13√

2reSA′′2γ SE

′,Bb .

(132)

Once the spurious coordinates are identified, the usualapproach consist in rewriting the Hamiltonian (14) interms of symmetry-adapted coordinates. According to theLagrange’s definition of classical momentum and usingthe chain rule, the local momenta are obtained from thesymmetrized ones by the linear transformation

pi =∑

α

miαPα, (133)

a relation that allows to express the kinetic energy in the form

T = 12˜PG

(

q)

P , (134)

where P stands for the column vector of conjugate momentaassociated with the symmetry coordinates. The transformedWilson’s matrix takes the form

G(

q) =M†G

(

q)

M, (135)

where M = ‖mi,α‖. The Hamiltonian is obtained by addingto the kinetic energy the potential as a function of the newcoordinates

H = 12˜PGP +V(S). (136)

A suitable Hamiltonian of practical interest is obtained byexpanding the Wilson matrix as well as the potential in termsof physical symmetry-adapted coordinates up to quarticorder. The resonances of higher order are later taken intoaccount. Hence, the Hamiltonian is given in terms of a sumof contributions according to the different subspaces (s-stretching, b-bending, and γ-out-of-plane) plus additionalinteractions preserving the polyad

H = Hs + Hb + Hγ + Hsb + Hsγ + Hbγ. (137)

For the sake of simplification, we now introduce the variables

SA′1s −→ S1, SE

′,Ab −→ S4a,

SE′,As −→ S3a, SE

′,Bb −→ S4b,

SE′,Bs −→ S3b, S

A′′2γ −→ S2,

(138)

with similar notation for the momenta. The stretching Ham-iltonian Hs, for instance, takes the form

Hs = 12G0

11P2

1 +12G0

3a3a

(

P23a + P2

3b

)

+12F11S

21

+12F3a3a

(

S23a + S2

3b

)

+14!F1111S

41 +

64!F113a3aS

21

(

S23a + S2

3b

)

+14!F3a3a3a3a

(

S23a + S2

3b

)2.

(139)

The factors G0αβ stand for the matrix elements of the Wilson

matrix evaluated at equilibrium. The next step in ourapproach consists in obtaining an algebraic representationof this Hamiltonian through the introduction of the bosonicoperators associated with the symmetry-adapted coordinates

S1 = 12β1

(

a†1 + a1

)

, P1 = i�β1

(

a†1 − a1

)

,

S3a = 12β3

(

a†3a + a3a

)

, P3a = i�β3

(

a†3a − a3a

)

,

S3b = 12β3

(

a†3b + a3b

)

, P3b = i�β3

(

a†3b − a3b

)

,

(140)

where βi = (1/2�)√

Fii/G0ii. The substitution of (140) into

(139) leads to the algebraic representation

Hs = �Ω1ν1 + �Ω3ν3 + x11ν21 + x33ν2

3 + x13ν1ν3 + gssl2s ,(141)

where νi are number operators defined by

ν1 = a†1 a1, ν3 = a†3aa3a + a†3ba3b, (142)

and ls is the vibrational angular momentum given by

ls = −i√

2[

a†3 × a3

]A′2, (143)

where × stands for the coupling E′ × E′ to the irreps A′2. Thespectroscopic parameters involved in (141) as well as in therest of the contributions are function of the structure andforce constants [62]. Following a similar approach for the restof the contributions in (137) and adding the resonance termsof higher order, we obtain [62]

H =4∑

i=1

�ωiνi +4∑

i≤ jxi j νiν j + K34/34 N34/34

+ f1 F1/44 + f2 F3/444 + f3 F22/444

+ gsl2s + gbl

2b + gsbls · lb,

(144)


with similar definitions for the number operators and the

vibrational angular momentum lb, and

F3/444 =[

a†3 ×[

[a4 × a4]A′1 × a4

]E′1]A′1

+H.c.,

N34/34 =[

a†3 × a†4]A′1 × [a3 × a4]A

′1 ,

F1/44 = a†1 × [a4 × a4]A′1 +H.c.,

F22/444 = a†2 × a†2 ×[

[a4 × a4]E′1 × a4

]A′1+H.c.

(145)

The Hamiltonian (144) may be diagonalized in a harmonicoscillator basis. Instead we will introduce, in similar form to(67), the following canonical transformation in terms of localbosons (after the substitution c†i → a†i ):

aΓμ†x =

∑

i

mi;Γ,μ,xa†i , (146)

where x stands for the subspace and the matrix M =‖mi;Γ,γ,x‖ corresponds to the coefficients in the projections(127), (128), and (129). We now introduce our proposal ofanharmonization

a†i −→ b†i , ai −→ bi, (147)

where b†i (bi) are identified with creation and annihilationoperators for the Morse functions for the stretching andbending spaces and Poschl-Teller functions for the out-of-plane subspace. This step is crucial in our method sincesignificant improvements are implied, without a straight-forward analog in configuration space. With the proposal(147), we interpret the new Hamiltonian as an algebraic localrepresentation in terms of operators defined in a space ofMorse functions for the stretches and bending oscillators andPoschl-Teller functions for the out-of-plane modes.

Hence, in the framework of the anharmonization proce-dure, the Hamiltonian that we propose takes the isomorphicform (144)

H =4∑

i=1

�ωiνi +∑

i≥ jxi j

12

(

νiν j + ν jνi)

+ K34/34N34/34

+ f1F1/44 + f2F3/444 + f3F22/444

+ gsl

2

s + gbl

2

b + gsbls ·lb,

(148)

but the new operators defined with bars are given by

ν1 =[

T†A′1s × TA′1

s

]A′1,

ν2 =[

T†A′′1γ × TA′′1

γ

]A′1,

ν3 =√

2[

T†E′

s × TE′s

]A′1,

ν4 =√

2[

T†E′

b × TE′b

]A′1,

(149)

while for the resonances

F1/44 =[

T†A′1s ×

[

TE′b × TE′

b

]A′1]A′1

+H.c.,

N34/34 =[

[

T†E′

s × T†E′b

]A′1 ×[

TE′s × TE′

b

]A′1]A′1

+H.c.,

F3/444 =[

T†E′

s ×[

[

TE′b × TE′

b

]A′1 × TE′b

]E′]A′1

+H.c.,

F22/444 = T†A′′2

γ × T†A′′2γ ×[

[

TE′b × TE′

b

]E′ × TE′b

]A′1+H.c.,

(150)

with tensors given by

T†Γ,μx =

∑

i

mi;xΓμb†i , (151)

where x stands for the space. The coefficients are theone obtained in the symmetry-adapted coordinates, inaccordance with (126), with the isomorphism

a†1 ←→ T†A′1s , a†2 ←→ T

†A′′1γ ,

a†3a ←→ T†E′,A

s , a†3b ←→ T†E′,B

s ,

a†4a ←→ T†E′,A

b , a†4b ←→ T†E′,B

b ,

(152)

and similar correspondence for the adjoint operators. TheHamiltonian (148) deserves the following comment. Oncethe anharmonization procedure is applied, the numberoperators involving the same space stop commuting, whichexplains the form of the symmetrized anharmonic contribu-tion. On the other hand, we may intend to diagonalize thisHamiltonian in the local basis given in terms of the directproduct (in configuration space)

Ψv(

y, u) =

6∏

i=1

ΨMvi

(

yi)

9∏

j=7

ΨPTvj

(

uj)

, (153)

where ΨMvi (yi) and ΨPT

vj (yj) are Morse and PT functions,respectively. However, this is not possible due to the existenceof spurious modes. These redundancies should be eliminatedfrom the basis. To achieve this task we follow an approachbased on the construction of a basis obtained from thediagonalization of the representation of a complete set ofcommuting operators in the local basis, which leads to asymmetry-adapted basis isomorphic to the normal basis[62, 89, 121].

The available vibrational experimental data is not asabundant as we could wish. We have considered the 25experimental energies available in [120]. The Hamiltonian(148) involves 21 interactions, but not all the parameters areable to be determined with the available experimental data.On the other hand, the diagonalization of the Hamiltonian(148) implies fixing the parameters κs, κb, and κγ associatedwith the depth of the local Morse and PT potentials. Theapproach we have followed to estimate the parameters


Table 4: Experimental and calculated energies, given as differencesof energy, for 11BF3. Fit 1 is obtained by diagonalizing the Hamilto-nian (148), while Fit 2 corresponds to the equivalent description inthe harmonic limit. In the last row the rms is displayed.

Polyad State Exp. [120] Fit 1 Fit 2

Symmetry A′14 11 885.84 0.4 2.4

4 42 959.31 0.9 0.2

6 22 1384.96 0. 0.3

8 31 41 1929.68 0 0.0

Symmetry E′2 41 479.35 −0.2 -3.

4 42 959.69 −0.7 0.6

6 11 41 1361.32 0.2 −1.

6 31 1453.96 0. −5.4

8 11 42 1837.73 −0.2 2.1

8 22 41 1866.12 0. 0.5

8 31 41 1931.92 0. 0

10 12 41 2240.94 0.1 0.3

10 11 31 2336.20 −0.7 −5.1

12 11 31 41 2810.70 −0.3 0.3

12 32 2905.36 0.5 7.8

14 13 41 3118.20 −0.1 1.

14 12 31 3216.32 −0.1 −5.3

16 12 31 41 3687.14 0.4 −0.1

16 11 32 3783.84 0.3 5.7

18 33 4310.26 −0.3 −4.5

Symmetry A′′23 21 691.21 0 0.4

7 11 21 1573.73 0 −0.2

9 23 2081.12 0. −0.3

Symmetry E′′5 21 41 1171.49 0. −1.

9 21 31 2139.68 0 0

rms 0.52 4.69

consists in taking the best values arisen from a series of fitsinvolving the 25 experimental energies considering only theharmonic ωi and anharmonic contributions xi j (14 param-eters). The best values were found to be κs = 50, κb = 170and κγ = 180. The spectroscopic parameters are thereafteroptimized by a least square method for fixed κs, κb, and κγ.In Table 4 we present the 25 experimental and the theoreticalenergies provided by two fits. Fit 1 refers to the case wherethe anharmonicities xi j as well as the resonance N34/34 aretaken into account. Adding additional interactions makesthe calculation unstable [62]. The deviation obtained wasrms = 0.52 cm−1. A criterion to evaluate our approachconsists in carrying out a comparison with the equivalentdescription in the harmonic limit (an equivalent descriptionin configuration space). This test is displayed in the columnlabeled as Fit 2, providing a square root deviation of rms =4.69 cm−1. A comparison of our description with this fit

Table 5: Force constants derived from the spectroscopic parametersprovided by Fit 3.

Parameter This work [62] Ab initio [140]

F11 (aJA−2

) 9.043 9.052

F22 (aJA−2

) 0.0979

F33 (aJA−2

) 7.018 7.018

F44 (aJA−2

) 0.2439 0.52

F1111 (aJ A−4

) 499.46 100.359

F2222 (aJ A−4

) 0.0428

F3a3a3a3a (aJ A−4

) 168.24 127.364

F4a4a4a4a (aJ A−4

) 0.2925 1.090

F1122 (aJ A−4

) −0.9345

F113a3a (aJ A−4

) 317.65 89.376

F114a4a (aJ A−4

) −0.0259 2.225

F223a3a (aJ A−4

) −0.8432

F224a4a (aJ A−4

) −0.0023

F3a3a4b4b (aJ A−4

) −0.748 −0.305

F3a4a4a4a (aJ A−4

) −1.335 1.820

allows us to conclude that the local approach based onanharmonic interacting oscillators is significatively better.The obtained force constants are displayed in Table 5.

To finish this section we will include some commentsconcerned with the approach to consider polyad breakingeffects. The problem of polyad breaking is an old subjectthat has attracted the attention since many years ago. Ithas been analyzed using different techniques, with theirassociated features and limitations. Full variational methodswith exact kinetic energy operators have been successful toobtain a good description of fairly high vibrational excita-tions, but unfortunately these methods are computationallyvery demanding and they cannot in practice be extendedto large or even medium-sized molecules [2]. Recursivetechniques based on Lanczos method have been developed,but it seems that they are not well suited to the study ofovertone line shapes [106]. Alternative approaches based onthe semiclassical description of molecular vibrations havebeen also developed, although difficulties arise at higherenergies, where chaotic behavior is dominant [106]. Apowerful approach to take into account the effect of polyadmixing is based on the canonical Van Vleck perturbationtheory [4, 9]. The basic idea consists in transforming theHamiltonian, expressed as an expansion in perturbativecontributions H(n), to a new representation via a series ofunitary transformations where the effect of the interactionsconnecting states of different polyads is taken into accountinside the polyad blocks. Because of the perturbative natureof this approach, a crucial ingredient consists in providingthe order of the different contributions to the Hamiltonian.Recently a general and systematic approach to deal with theproblem of polyad breaking when Morse potentials are usedto describe the internal degrees of freedom of molecularsystems has been established [101].


10. U(ν + 1) Algebraic Model for Vibrations

The SU(2) model obtained by adding an extra boson to theone-dimensional harmonic oscillator can be extended in twodirections. If we were interested in describing simultaneouslyboth vibration and rotations of diatomic molecules, theappropriate extension would lead to the U(4) model [23].Another possibility, however, consists in extending the modelby including several local vibrational degrees of freedomof the same subspace. This effort gives rise to the unitaryapproach U(ν + 1) proposed by Michelot and Moret-bailly[44]. In this section we will discuss this approach, presentinga proposal to establish the connection with configurationspace, and in this way being able to estimate the PES. Resultsfor pyramidal molecules are presented [63–66].

We start considering ν equivalent harmonic oscillators.Associated with the ith oscillator, we have bosonic creationa†i and annihilation ai operators. In the framework of theU(ν + 1) approach an additional s boson is added with theconstraint that the total number of bosons N is constant. Thegenerators of the group U(ν + 1) are given by

Cji = c†i c j , ci = ai, i = 1, . . . , ν, cν+1 = s (154)

with commutation relations[

Cji , C

qp

]

= Cqi δp, j − Cj

pδq,i, (155)

where the extra boson has been included. In this modelthe state vectors associated with a local mode descriptionof a molecular system consist of a set of ν + 1 independentharmonic oscillators. Explicitly this basis is given by

|[N];n1,n2, . . . ,nν,ns〉 = 1√

ns!∏ν

jn j !

(

s†)ns

ν∏

i

(

a†i)ni|0〉,

(156)

which are characterized by the total number of quanta N ,whose corresponding operator is given by

N = n + ns, (157)

with

n =ν∑

i=1

a†i ai, ns = s†s. (158)

The total number of bosons fixes the totally symmetricrepresentation [N] of the U(ν + 1) group, and because of therelation (157) we can rewrite the kets (156) in the form

|[N],n;n1,n2, . . . ,nν〉 ≡ |[N];n1,n2, . . . ,nν,ns〉. (159)

The addition of the s boson together with the fact that therepresentation [N] is fixed makes the unitary group U(ν + 1)a dynamical group for the set of ν oscillators.

From the generators (154) of the unitary group U(ν + 1)we identify ν-su(2) subalgebras with generators

Ji,+ = sa†i , Ji,− = s†ai, Ji,0 = −12

(

s†s− a†i ai)

,

(160)

with the usual angular momentum commutation relations[

Ji,+, Ji,−]

= 2Ji,0,[

Ji,0, Ji,±]

= ±Ji,±. (161)

We now introduce the normalized operators

b†i ≡Ji,+√N

, bi ≡Ji,−√N

, (162)

which satisfy the commutation relations

[

bi, b†j

]

= δi j − 1N

[

nδi j + a†j ai]

,[

b†i , b†j]

=[

bi, bj]

= 0.

(163)

In particular

[

bi, b†i

]

= 1− 1N

[n + ni] = − 2NJi,0. (164)

The action of these operators over the kets (159) is thefollowing:

b†i |[N],n;n1,n2, . . . ,nν〉 =√

(ni + 1)(

1− n

N

)

× |[N],n+1;n1,. . .,ni+1,. . .,nν〉,

bi|[N],n;n1,n2, . . . ,nν〉 =√

ni

(

1− n− 1N

)

× |[N],n−1;n1,. . .,ni−1,. . .,nν〉,(165)

while for the operators a†i (ai)

a†i |[N],n;n1,n2, . . . ,nν〉

=√

(ni + 1)|[N],n + 1;n1, . . . ,ni + 1, . . . ,nν〉,ai|[N],n;n1,n2, . . . ,nν〉

= √ni|[N],n− 1;n1, . . . ,ni − 1, . . . ,nν〉.(166)

From these results, it is clear that in the harmonic limit N →∞ the operators {b†i , bi} go to the bosonic operators {a†i , ai},as long as n remains finite. This is indeed the case when in thecalculations the polyad number is used. In fact, from (165)and (166) we obtain

limN→∞

b†i = a†i , limN→∞

bi = ai, (167)

and consequently

limN→∞

[

bi, b†j

]

= δi j . (168)

When α sets of equivalent oscillators are present in thevibrational description, a u(ν + 1) algebra is introduced foreach equivalent set, so that the dynamical group is given bythe direct product

U1(ν1 + 1)×U2(ν2 + 1)× · · · ×Uα(να + 1). (169)


In this case a total number of bosons Ni is associated witheach space, which in turn is related to the dissociation limitof the corresponding internal coordinates [44].

The model summarized above permits to obtain a spec-troscopic description based on energy and intensity fittings.However, extracting force constants is crucial to establishthe connection between the generators of the dynamicalalgebra and the local coordinates and momenta, at least inapproximate form. We propose the following approximationfor the local coordinates qi and momenta pi:

qi �√

�

2ωμ

(

d†i + di)

,

pi � i

2

√

2�ωμ(

d†i − di)

,

(170)

where d†( d) may be identified either with a†(a) or b†(b).The connection (170) represents just an approximation, notonly because we are proposing a linear expansion in termsof the operators (162) but also because the commutationrelations [qi, p j] = i�δi j are only nearly satisfied. We knowhowever that these relations are exact in the harmonic limit,a remarkable fact that suggests the approximate connection(170) with configuration space.

In order to decide the appropriate correspondence a†(a)

or b†(b), we will first analyze the case where all operatorsbelong to the same set of equivalent oscillators α. In this case

when the operators d†( d) are associated with a†(a), the usualcommutation relation for the coordinates and momenta aresatisfied. In contrast, using the operators b†(b) we obtain

[

qi, qj]

= �

2ωμ1N

[

a†j ai − a†i a†j]

, i, j ∈ α, (171)

while for the coordinates and momenta[

qi, p j]

= i�δi j − i�

2N

[

2nδi j + a j a†i + aia

†j

]

, i, j ∈ α.

(172)

We may thus conclude that the correspondence d†( d) →a†(a) must be considered for contributions of the Hamilto-nian that involve one set of equivalent coordinates, otherwisethe results of the fit would depend on the order of the localcoordinates and momenta in the expansion. However, we canbypass this problem by proposing a symmetrization when we

select the operators b†(b) as we will discuss.This approach allows the Born-Oppenheimer potential

surfaces to be obtained. As an example the spectroscopicdescription of Arsine will be presented, where the forceconstants are estimated as well as transition intensities. Theequilibrium structure of the arsine molecule is pyramidalwith structure parameters re = 1.51106 A, and ∠HAsH =92.069◦ [122]. The point symmetry group is C3v. Thismolecule has six degrees of freedom, three of them associatedwith the stretching modes (A1⊕E) and three bending modes(A1 ⊕ E). In the standard notation the harmonic basis islabeled by

∣

∣

∣ν1ν2νl33 νl44⟩

, (173)

where ν1 and ν3 correspond to the symmetrical and degen-erate As-H stretching modes A1 and E, respectively, while ν2

and ν4 correspond to the symmetrical and degenerate H-As-H bending modesA1 and E, respectively. l3 and l4 correspondto the projection of the angular momenta associated with thecorresponding degenerate modes. Since here we will restrictourselves to the study of the stretching degrees of freedom,the normal states will be labeled as |ν1νl33 〉 or |1ν1 , 3l3ν3〉. Wethus start by expanding both the Wilson matrix as well asthe potential in terms of the displacement local coordinates.Since the elements grr and grr′ of the G(q) matrix does notdepend on the stretching variables, the expansion reduces togi j(q) = g0

i j . The Hamiltonian takes thus the form

H = 12gorr

3∑

i=1

p2i + gorr′

3∑

i> j=1

pi p j +12frr

3∑

i=1

q2i + frr′

3∑

i> j=1

qiq j

+14!frrrr

3∑

i=1

q4i ,

+44!frrrr′

[

q31

(

q2 + q3)

+ q32

(

q1 + q3)

+ q33

(

q1 + q2)]

+34!frrr′r′

[

q21

(

q22 + q2

3

)

+ q22

(

q23 + q2

1

)

+ q23

(

q21 + q2

2

)]

+124!frrr′r′′

[

q21q2q3 + q2

2q1q3 + q23q1q3

]

,

(174)

where only terms leading to preservation of the polyad

P = n1 + n2 + n3 (175)

have been considered.The next step consists in obtaining the algebraic repre-

sentation of the Hamiltonian. In principle we could carry

out the substitution of (170) with the identification d†( d)with b†(b), but this realization would not be unique.Consequently we propose to substitute each term by itssymmetric form. For the quadratic terms we thus carry outthe substitutions

pi p j −→ 12

(

pi p j + p j pi)

, qiq j −→ 12

(

qiq j + qjqi)

(176)

while for the quartic terms appearing in the potential

q31

(

q2 + q3) −→ 1

2

[

q31

(

q2 + q3)

+(

q2 + q3)

q31

]

,

q21

(

q22 + q2

3

) −→ 12

[

q31

(

q22 + q2

3

)

+(

q22 + q2

3

)

q31

]

,

q21q2q3 −→ 1

6

[

q21

(

q2q3 + q3q2)

+(

q2q3 + q3q2)

q21

+q3q21q2 + q2q

21q3]

.(177)

We now reconsider the arguments to identify d†( d) either

with b†(b) or a†(a). We have found that the best selection


corresponds to the substitution d†( d) → a†(a) in the quad-ratic terms as well as in the higher-order terms involving

number operators, while d†( d) → b†(b) into the rest of theterms. The Hamiltonian takes thus the form in wave numbers(cm−1)

H

hc= ωen + λ

3∑

i /= j

a†i a j + α1I1 +4∑

i=2

αj

⎛

⎝

∑

p

OpIi

⎞

⎠, (178)

where the local interactions Ii are

I1 =3∑

i=1

(

a†i ai)2 =

3∑

i=1

n2i ,

I2 = b†21b1

(

b2 + b3

)

+ b21b†1(

b†2 + b†3)

+(

b†1 b1 + b1b†1)

×[

b†1(

b2 + b3

)

+ b1

(

b†2 + b†3)]

,

I3 = b†21

(

b22 + b2

3

)

+ b21

(

b†22 + b†2

3

)

+(

b†1 b1 + b1b†1)

×[

b†2 b2 + b2b†2 + b†3 b3 + b3

b†3]

,

I4 = 2(

b†21b2b3 + b2

1b†2 b

†3

)

+(

b†1 b1 + b1b†1)

×(

b†2 b2 + b2b†2 + b†3 b3 + b3

b†3)

+ b†3 b21b†2 + b3

b†21b2 + b†3

(

b†1 b1 + b1b†1)

b2

+ b3

(

b†1 b1 + b1b†1)

b†2 .(179)

The sum of the operators Op involved in the last term ofthe Hamiltonian (178) runs over the permutations p = e,[120, 123]. Following this approach the spectroscopic pa-rameters are given in terms of the structure and forceconstants. Note that with this identification the zeroth orderHamiltonian corresponds to a set of noninteracting harmon-ic oscillators.

The Hamiltonian (178) is diagonalized in a symmetry-adapted basis, which is obtained by projecting the functions(159). It has been carried out a fit of the available experimen-tal data [124, 125]. In Table 6 we present the 21 experimentalenergies as well as the theoretical energies provided by thefit, where an rms deviation of 1.59 cm−1 was obtained. Inthe calculation all parameters were freely optimized withequal weights keepingN = 28 fixed according to dissociationarguments [47, 63]. The rms provided by the fit in theharmonic limit is 1.62 cm−1, a slightly higher value. Fromthe spectroscopic parameters, the force constants displayedin Table 7 are obtained. In the same table we have includedfor comparison the parameters obtained by Lukka et al. [122]as well as ab initio results [126]. Although it is possible toobtain a fit of the same quality with less parameters, we havedecided to include all the interactions in order to estimateall the available force constants. As noted from Table 7, theresults are in good agreement with previous calculations.The discrepancy in the sign of the constant frrrr is due to

the approximation in the potential, where only terms up toquartic order were kept. In our approximation frrrr must beless than zero in order to reproduce the strong anharmonicbehavior in the spectrum, a behavior reproduced naturallywhen Morse oscillators are considered at zeroth order.

The analysis of transition intensities represents a test forthe wave functions provided by any model. In particular inthe framework of a local scheme the infrared absorptionintensities are calculated with a bond dipole model, wherethe dipole function is expressed as a sum of bond dipoles inthe following form [11, 12]:

�μ(r) =3∑

i=1

μi(ri)ei, (180)

where ri is the instantaneous bond length of the ith bond andei is a unit vector along the ith bond. The infrared transitionintensities from the initial state |ν〉 to the final state |ν′〉within a single electronic state is given by [123]

Iν→ ν′ = CΔEνν′∑

i, j

∣

∣

∣μ(ν′,i),(ν, j)

∣

∣

∣

2, (181)

where C is a constant, ΔEνν′ is the energy difference betweenthe states, the subindices {i, j} account for the degeneracy ofthe states, while the dipole matrix elements take the form

∣

∣μνν′∣

∣2 =

∑

ζ=x,y,z

∣

∣

⟨

ν′∣

∣�μ(r) · eζ∣

∣ν′⟩∣

∣

2

=∑

ζ=x,y,z

∣

∣

⟨

ν′∣

∣μζ(r)∣

∣ν′⟩∣

∣2,

(182)

where the degeneracies have not been considered explicitly.A crucial question that arises is concerned with the formof the bond dipole function. A Taylor series expansion asa function of the bond lengths is a possibility, but a moreattractive alternative consists in modeling the local dipolefunctions with an analytical function, such as the Meckedipole moment function [127]

ti = μoqmi e−γqi , (183)

where qi is the displacement local coordinate for the ithoscillator, γ and μo are parameters to be determined, and mis usually taken to be an integer ≥1. Here we will considerm = 1. The different components of the dipole function isthen expanded in terms of the local operators (183).

For pyramidal molecules the dipole operator spans theirreducible representations A1 and E in the following form:

μA1z ,

(

μEx ,μEy)

. (184)

To calculate the intensities, a representation of the dipoleoperators involved in (181) has to be attained in terms of thelocal dipole moment functions (183). Projecting accordingto the chain

C3v ⊃ Cas . Cas =

{

E, σav}

, (185)


Table 6: Energies (in cm−1) provided by the fit using the Hamiltonian (178). The rms deviation obtained is 1.59 cm−1. Experimental energieswere taken from [124, 125].

PolyadState Contribution State Contribution

Exp.Energies

ΔE(normal) (normal) (local) (local) theor.

Symmetry A1

1 11 1.00 100 1.00 2115.16 2115.17 0.0

2 32 0.52 200 0.98 4166.77 4165.65 1.1

2 12 0.52 110 0.98 4237.7 4238.95 −1.2

3 1132 0.66 300 0.99 6136.34 6136.01 0.3

3 13 0.71 210 0.97 6275.83 6277.62 −1.8

3 33 0.55 111 0.98 6365.95 6367.23 −1.3

4 1232 0.47 400 0.99 8028.98 8027.29 1.7

4 14 0.41 310 0.97 8249.51 8249.94 −0.4

4 34 0.44 220 0.96 — 8332.92 —

4 1232 0.35 211 0.98 — 8397.37 —

5 1134 0.35 500 0.99 9841.4 9841.06 0.3

5 1332 0.38 410 0.99 — 10143.2 —

5 1233 0.57 320 0.98 — 10289.1 —

5 15 0.31 311 0.95 — 10370.7 —

5 35 0.27 221 0.95 — 10443.9 —

6 1234 0.37 600 0.99 11577.5 11577.5 −1.2

6 1432 0.38 510 0.99 — 11957.0 —

6 36 0.32 420 0.95 — 12183.1 —

6 1234 0.50 411 0.90 — 12265.1 —

6 36 0.37 330 0.86 — 12270.1 —

6 16 0.47 321 0.94 — 12401.5 —

6 36 0.38 222 0.96 — 12500.3 —

Symmetry A2

3 33 1.00 210 1.00 — 6301.12 —

4 1133 1.00 310 1.00 — 8261.71 —

5 1233 0.71 410 0.99 — 10149.9 —

5 35 0.71 320 0.99 — 10311.7 —

6 1135 0.52 510 0.99 — 11962.9 —

6 1333 0.55 420 0.99 — 12193.7 —

6 36 0.76 321 0.99 — 12439.1 —

Symmetry E

1 31 1.00 100 1.00 2126.42 2126.72 −0.3

2 1131 0.73 200 0.99 4167.94 4168.74 −0.8

2 32 0.73 011 0.99 4247.52 4248.59 −1.1

3 1231 0.38 300 0.99 6136.33 6136.32 0.0

3 1231 0.55 012 0.99 6282.35 6283.46 −1.1

3 33 0.67 120 0.99 6294.71 6295.02 −0.3

4 1133 0.43 400 0.99 8028.97 8027.32 1.6

4 1331 0.58 310 0.98 8257.27 8253.0 4.3

4 1232 0.50 130 0.98 8258.37 8258.65 −0.3

4 34 0.31 022 0.97 — 8335.51 —

4 34 0.47 211 0.99 — 8417.15 —

5 1233 0.37 500 0.99 9841.4 9841.06 0.3

5 1431 0.35 410 0.99 — 10143.7 —

5 1134 0.32 140 0.99 — 10149.8 —


Table 6: Continued.

PolyadState Contribution State Contribution

Exp.Energies

ΔE(normal) (normal) (local) (local) theor.

5 1332 0.37 023 0.98 — 10290.4 —

5 35 0.65 230 0.99 — 10310.4 —

5 1233 0.31 311 0.98 — 10378.1 —

5 35 0.43 122 0.98 — 10460.9 —

6 1333 0.26 600 0.99 11576.3 11577.5 −1.2

6 1531 0.19 510 0.99 — 11957.1 —

6 1234 0.25 150 0.99 — 11962.8 —

6 1531 0.27 024 0.95 — 12184.1 —

6 1135 0.28 240 0.99 — 12192.7 —

6 1135 0.18 411 0.94 — 12266.3 —

6 36 0.24 033 0.91 — 12270.1 —

6 1234 0.24 123 0.99 — 12412.6 —

6 1135 0.36 231 0.99 — 12427.1 —

Table 7: Force constants derived from the spectrocopic parametersgiven in Table 2. The suface obtained by Lukka was taken from[122], while the ab initio surface from [126].

Parameter This work [65] Lukka et al. Ab initio

frr (aJA−2

) 2.836 2.841 2.829

frr′ (aJA−2

) −0.010 40 −0.009 95 −0.0097

frrrr (aJA−4

) −51.9074 46.323 54.4

frrrr′ (aJA−4

) 0.2066 — —

frr′r′ (aJA−4

) −0.9825 — —

frrr′r′′ (aJA−4

) 0.6596 — —

we obtain the following representation for the dipole com-ponents up to first order:

μE,A′x = δ1

1√6

(

2t1 − t2 − t3)

,

μE,A′′y = δ1

1√2

(

t2 − t3)

,

μA1,A′z = δ2

1√3

(

t1 + t2 + t3)

,

(186)

where A′ and A′′ label the irreps of the subgroup Cas . We

may add contributions of higher order by means of successivecouplings. Taking into account this analysis the transition,intensities (181) may be expressed as

Iνi→ ν f = ΔEν f νi

{∣

∣

∣

⟨

ν f Γ f∣

∣

∣μA1

∣

∣

∣νiΓi⟩∣

∣

∣

2

+∣

∣

∣

⟨

ν f Γ f A′∣

∣

∣μ(E1,A′)x

∣

∣

∣νiΓiA′⟩∣

∣

∣

2}

,

(187)

where we have considered the Wigner-Eckart theorem inorder to involve only the first component of the eigenvectors[26].

In order to calculate the matrix elements of the dipolefunction in the context of the U(ν + 1) model, it is necessary

to obtain an algebraic representation of the operators (183).To achieve this goal we simply substitute the algebraicrepresentation for the coordinates to obtain

ti = μo1√2

(

b†i + bi)

e(−γ/√2)(b†i +bi), (188)

taking dimensionless units. We thus will be interested in thecalculation of the matrix elements of this operators in thelocal basis (159)

Mm′,m ≡⟨

[N],n′;n′1,n′2,n′3∣

∣

∣ti∣

∣

∣[N],n;n1,n2,n3

⟩

,

(189)

where m denotes symbolically the set of local quantumnumbers {N ,n,n1n2,n3}. We can obtain closed analyticresults for the matrix elements (189) based on the fact thatthe operators (165) satisfy the su(2) commutation relations.Taking advantage of this fact, we can write

e(−γ/√2)(b†i +bi) = e−β(Ji++Ji−) = e−ζ1 Ji,+e−ζ2 Ji,−e−ζ3 Ji,0 ,(190)

where

β = γ√2N

,

ζ1 = tanh(

β)

,

ζ2 = sinh(

β)

cosh(

β)

,

ζ3 = 2 ln[

cosh(

β)]

,

(191)

which was obtained through a two-dimensional represen-tation of the su(2) algebra [128]. Hence, the transitionoperators (188) may be written in the following form:

ti = μo1√2

(

b†i + bi)

e−(ζ1√N)b†i e−(ζ2

√N)bi e−(ζ3

√N)Ji,0 ,

(192)

where we should take into account that

Ji,0 = −N2[

1− 1N

(n + ni)]

, (193)


in accordance with (164). The matrix elements (189) are nowable to be calculated in a straightforward way. The result is

Mm′,m = μ0

(

Π j /= iδn′j ,nj)

√2

eζ3(N/2)(

−ζ1

√

N)n′i−ni−1

×√

ni!(

n′i − 1)

!

√

n′i

(

1− n′ − 1N

)

×ni∑

q=0

(ζ1ζ2N)q

q!(

n′i − ni + q − 1)

!(

ni − q)

!

× f(

ζ3,N ;n′ − 1,n;n′i − 1,ni; q)

+ μ0

(

Π j /= iδn′j ,nj)

√2

eζ3N/2(

−ζ1

√

N)n′i−ni+1

×√

ni!(

n′i + 1)

!

√

(

n′i + 1)

(

1− n′

N

)

×ni∑

q=0

(ζ1ζ2N)q

q!(

n′i − ni + q + 1)

!(

ni − q)

!

× f(

ζ3,N ;n′ + 1,n;n′i + 1,ni; q)

,(194)

where for convenience we have defined

f(

ζ3,N ;n′,n;n′i ,ni; q) = e(ζ3/2)(n+ni)

Nq

×(

N − n + q)

!√

Nn′i−ni(N − n)!(N − n′)!.

(195)

This description contrasts with the analysis presented in [47],where the proposed dipole transition operator is based on thegeneral behavior of the dipole transition intensities and hasto be defined for each type of local states.

The dipole transition intensities (187) involve threeparameters in the linear approximation (186), namely{δ1, δ2, γ}. The parameters δ1 and δ2 fix the scale, while γ in(183) determines how the strengths decrease as a functionof the number of quanta. Note that since the stretchingcoordinates are equivalent the operators {ti, i = 1, 2, 3}contain the same parameter γ. These parameters should bedetermined through a fit to the experiment. The functiontaken to minimize is

rms =⎡

⎢

⎣

∑Nexp

i=1

[

Log(

Iiexp

)

− Log(

Iical

)]2

Nexp −Npar

⎤

⎥

⎦

1/2

, (196)

where Nexp and Npar stand for the number of experimentalstates and the number of parameters used in the fit, respec-tively. In Table 8 we present the calculated intensities for theexperimental data available [124]. The parameters obtainedin the fit were

δ1 = 0.0173, δ2 = 0.0122, γ = 0.193536. (197)

The δ parameters are concerned with the normalizationand consequently a small variation does not affect theresults considerably. In contrast, γ must be specified withhigh precision. In the same Table 8, results presented byHalonen et al. [124] and Pluchart et al. [47] are alsoincluded. On average our results are similar in accuracy tothe ones presented in [124]. This is expected since in essencewe are using the same expansion for the dipole operator.However, although at first sight the intensities provided by[47] are better, we should take into account that not all theexperimental data were considered and that the number ofparameters involved were 4, one more parameter that is usingthe linear approximation (186). This treatment to describedipole transition intensities is general and is readily extendedto include higher-order expansions in the dipole operators,which in the harmonic limit reduce to the usual expansionsconsidered in local mode models [11].

11. Algebraic Approach to Describe LinearTriatomic Molecules

The unitary model U(3) deserves special attention because itis suitable to be applied to the description of the degeneratebending degrees of freedom of linear molecules. In fact, theuse of the su(3) algebra for the bends and the su(2) algebrasfor the stretches turns out to be a successful scheme todescribe the vibrational excitations of linear molecules.

In this section we present the procedure to estimate thePES when the su(2) × su(3) × su(2) Lie algebra is usedas the dynamical group of linear triatomic molecules. Inthis scheme, each stretching degree of freedom is describedthrough an su(2) dynamical algebra, while the doublydegenerate bending degree of freedom is modeled withan su(3) dynamical algebra. Again to achieve this goal weestablish the connection between the traditional approachand the algebraic method. As an application, the analysis ofthe vibrational excitations of CO2 in its ground electronicstate is considered.

As we know in terms of internal displacement coor-dinates, qk, the quantum mechanical Hamiltonian thatdescribes the vibrational excitations of a molecule takes theform

H = 12

pG(

q)

p +V(

q)

. (198)

The explicit form of the Hamiltonian strongly dependson the molecular system. In the case of linear triatomicmolecules, two sets of internal coordinates are needed. Thefirst set, (q1, q2), spans the subspace of stretching vibrationswith

qi = Δri = ri − re, i = 1, 2, (199)

where ri corresponds to the ith bond length and re to the ithbond equilibrium length. The second set, (qa, qb), spans thesubspace of bending oscillators with the following definition[129]:

qa = reeY · r1 × r2

r1r2, qb = −reeX · r1 × r2

r1r2, (200)


Table 8: Observed and calculated relative intensities of the stretching vibrational bands in arsine from [65].

Local state νcalc/cm−1 Iobs Icalc Halonen et al. [124] Pluchart et al. [47]

Symmetry A1

(100) 2115.13 1.0a 1.0 1 1.0

(200) 4165.79 0.021 0.068 0.022 0.0209

(110) 4238.94 — 1.14 × 10−3 0.13× 10−3 —

(300) 6136.07 0.32× 10−3 1.32 × 10−3 0.49× 10−3 0.47× 10−3

(210) 6277.63 0.28× 10−4 0.07 × 10−4 0.0055× 10−4 —

(400) 8027.32 0.15 ×10−4 0.129× 10−4 0.071× 10−4 0.12 × 10−4

Symmetry E

(100) 2126.81 1.0a 1.0 1.0 1.0

(200) 4168.75 0.021 0.069 0.022 0.0209

(011) 4248.66 — 2.64× 10−4 0.58× 10−4 —

(300) 6136.36 0.32 ×10−3 1.32 × 10−3 0.49× 10−3 0.47× 10−3

(012) 6283.43 — 1.51× 10−6 0.51× 0−6 —

(120) 6295.01 0.14× 10−4 0.015× 10−4 0.0032 × 10−4 —

(400) 8027.35 0.15× 10−4 0.128 × 10−4 0.071 × 10−4 0.12× 10−4

aScaled value.

where r1 and r2 are vectors from the C-atom to each one ofthe O-atoms. The unit vectors eX and eY lie on the directionof the x and y-axes of the laboratory axis system, with itsorigin in the molecule’s center of mass.

With this choice of coordinates, the q and p vectors in(198) are given by

q = (q1, q2, qa, qb)

, p = (p1, p2, pa, pb)

. (201)

In general for linear molecules the matrix G(q) cannot beobtained in closed form. However, it is possible to evaluatethe G-matrix as well as its derivatives at equilibrium usingits explicit form in terms of the Cartesian coordinates. Thisprocedure is presented in [69].

We could expand the Wilson matrix as well as thepotential energy in terms of the local coordinates. Thisexpansion permits the identification of a set of localinteracting oscillators in (198). This approach, however, islargely improved when symmetry-adapted coordinates arefirst introduced.

Hence, we start introducing the symmetry coordinatessuitable for homonuclear molecules

QΣg =1√2

(

q1 + q2)

, QΣu =1√2

(

q1 − q2)

,

Q+ = − 1√2

(

qa + iqb)

, Q− = 1√2

(

qa − iqb)

,

(202)

where the coordinates associated with the stretches arelabeled with the subscripts Σg/u, and the bending degrees offreedom with +/−. The first case is a simplification of Σ+

g/u,while the bending labels are a shorthand notation for Π±

[8, 130]. Both labels refer to the irreducible representationsof the point symmetry group D∞h. In matrix form

Q = qS, (203)

where

Q =(

QΣu , QΣg , Q−, Q+

)

,

S = 1√2

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎝

1 1 0 0

−1 1 0 0

0 0 1 −1

0 0 −i −i

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎠

.(204)

The change to symmetry coordinates transforms the Hamil-tonian (198) into

H = 12PGP +V(Q), (205)

where

P =(

PΣu , PΣg , P−, P+

)

,

G = ˜SGS.(206)

The transformed Wilson matrix G is block diagonal, asexpected.

We may now proceed to expand the Wilson Matrix aswell as the potential in terms of the symmetry-adaptedcoordinates, keeping the relevant terms associated with theresonances. It is thus necessary to establish the polyadcorresponding to the system of interest, in this case for theCO2 molecule. The fundamentals of CO2 are (in cm−1) [8]

ν1 = 1285.41, ν2 = 667.38, ν3 = 2349.14,(207)

from which we set up the following allowed symmetryresonances:

ν1 ≈ 2ν2. (208)


The identification of these resonances permits to establishthe polyad P, which in terms of integer numbers may bedefined as

P = 2(ν1 + ν3) + ν2. (209)

The expansion of the G matrix and the potential in termsof the symmetry coordinates leads to a Hamiltonian of theform [68]

H = Hs + Hb + Hsb, (210)

where the first term, Hs, is the pure stretching contribution

Hs = 12g0ΣgΣg

P 2Σg +

12g0ΣuΣu

P 2Σu +

12fΣgΣg Q

2Σg +

12fΣuΣu Q

2Σu

+14!fΣgΣgΣgΣg Q

4Σg +

14!fΣuΣuΣuΣu Q

4Σu +

64!fΣgΣgΣuΣu Q

2ΣgQ2Σu .

(211)

The second term, Hb, depends solely on bending coordinatesand momenta

Hb = g0+−P+P− + f+− Q+ Q− +

(

∂2g+−∂Q+∂Q−

)

0

P+ Q+ Q−P−

+64!f++−− Q2

+Q2−,

(212)

while, the third term, Hsb, embodies the stretch-bend inter-action terms

Hsb =(

∂g+−∂QΣg

)

0

QΣgP+P−+

(

∂gΣg+

∂Q+

)

0

QΣg

(

Q+P+ +P− Q−)

+63!fΣg+− QΣg

Q+ Q− +12

⎛

⎝

∂2g+−∂Q2

Σg

⎞

⎠

0

Q2ΣgP+P−

+12

(

∂2g+−∂Q2

Σu

)

0

Q2ΣuP+P−

+12

(

∂2gΣgΣg∂Q+∂Q−

)

0

P 2ΣgQ+ Q−+

12

(

∂2gΣuΣu∂Q+∂Q−

)

0

P 2ΣuQ+ Q−

+12

(

∂2gΣg+

∂QΣg ∂Q+

)

0

PΣgQΣg

(

Q+P+ + Q−P−)

+12

(

∂2gΣu+

∂QΣu∂Q+

)

0

PΣuQΣu

(

Q+P+ + Q−P−)

+124!fΣgΣg+− O2

ΣgQ+ Q−

+124!fΣuΣu+− Q2

ΣuQ+ Q−.

(213)

The Hamiltonian (210) can be translated into an algebraicrepresentation through the introduction of bosonic creation

and annihilation operators. For the stretching degrees offreedom, these operators are defined as

a†Γ = αΓ QΓ − i

2�αΓPΓ, aΓ = αΓ QΓ +

i

2�αΓPΓ,

(214)

where Γ = Σg ,Σu and the definitions

(

αΣg)2 = 1

2�

√

√

√

√

fΣgΣgg0ΣgΣg

= μΣg2�

√

fΣgΣg g0ΣgΣg =

ωΣg μΣg2�

,

(

αΣu)2 = 1

2�

√

√

√

√

fΣuΣug0ΣuΣu

= μΣu2�

√

fΣuΣug0ΣuΣu =

ωΣuμΣu2�

,

(215)

where μΓ = 1/g0ΓΓ. In the case of the bending mode, the

following relations apply:

a†± = α± Q± +i

2�α±P∓, a± = −α±Q∓ +

i

2�α±P±,

(216)

with

(α+)2 = (α−)2 = (α)2 = 12�

√

√

√

√

f+−g0

+−= 1

2�

√

√

√

√

fqaqag0qaqa

. (217)

We now carry out the additional canonical transfor-mation between symmetrized and local stretching bosonicoperators (67)

a†Σg =1√2

(

c†1 + c†2)

, a†Σu =1√2

(

c†1 − c†2)

. (218)

The operators {c†i }, with i = 1, 2, will be identified withlocal operators in accordance with (70). If we restrict theinteractions to preserve the polyad number

P = 2(

nΣg + nΣu)

+ (n+ + n−), (219)

where nΣg /u stands for the number of quanta in the Σg/uvibrational mode and n+ +n− = n is the number of quanta inthe degenerate bending mode, the Hamiltonian (210) in thelocal harmonic representation is

H = ωs

2∑

i=1

(

a†i ai + aia†i

)

+ λs

2∑

i /= j=1

a†i a j + αs1(

n21 + n2

2

)

+ αs2(

a†21 a2

2 + a†22 a2

1 + 4n1n2

)

+ αs3(

n1a†2a1 + n2a

†1a2 +H.c.

)

+ ωbn + αb1n2 + αb2 �

2 + αsb1{(

a†1 + a†2)

a+a− +H.c.}

+ αsb2 (n1 + n2)n + αsb3(

a†1a2 + a†2a1

)

n,

(220)



ni = a†i ai, i = 1, 2,

� = a†+a+ − a†−a− = n+ − n−,

n = a†+a+ + a†−a− = n+ + n−,

(221)

where ni is the number of quanta for the i-th stretchingoscillator, n is the total number of bending quanta, and � isthe vibrational angular momentum. The expressions for thespectroscopic parameters in terms of the structure and forceconstants are detailed in Appendix C of [68]. An appropriatebasis to diagonalize the Hamiltonian (220) is constructed bymeans of the symmetry projection of the direct product offunctions

∣

∣

∣n1n2,n�⟩

= N(

a†1)n1(

a†2)n2(

a†+)(n+�)/2(

a†−)(n−�)/2|0〉,

(222)

with normalization constant

N = 1√

n1!n2!((n + �)/2)!((n− �)/2)!. (223)

The present treatment is based on local harmonic oscillator.The next step consists in introducing the anharmonization asin the previous models.

Let us first proceed to introduce the algebraic approachbased on unitary groups. An algebraic description of a systemconsists in identifying an appropriate dynamical algebra. Afundamental feature of this approach is that the system’sHilbert space carries a specific irreducible representation ofthe dynamical algebra. In CO2 the stretching vibrationalmodes will be associated with su(2) algebras, while an su(3)dynamical algebra is proposed to describe the vibrationalbending modes. The result is a simple model able to copewith the rigidly linear and rigidly bent limits as well asthe situations in between these two limits [34–38]. Thedynamical group for the complete triatomic molecule is thengiven by SU1(2)× SU(3)× SU2(2) [131, 132].

The bosonic su(3) Lie algebra is not related to a knownpotential. It can be constructed with two Cartesian bosoncreation and annihilation operators {τ†a , τ†b , τa, τb}, togetherwith a scalar boson {σ†, σ} [23, 24]. To be consistent with(200), we label the Cartesian operators as a and b instead ofx and y. From the physical point of view it is convenient tointroduce spherical (circular) bosons

τ†± = ∓τ†a ± iτ†b√

2, τ± = ∓τa ∓ iτb√

2, (224)

since they carry irreducible representations of the symmetrygroup D∞h. Following [23, 24], the nine generators of U(3)

may be written as combinations of the bilinear products of acreation and annihilation operator:

n = τ†+τ+ + τ†−τ−, nσ = σ†σ ,

l = τ†+τ+ − τ†−τ−,

D+ =√

2(

τ†+σ − σ†τ−)

, D− =√

2(

−τ†−σ + σ†τ+

)

,

Q+ =√

2 τ†+τ−, Q− =√

2 τ†−τ+,

R+ =√

2(

τ†+σ + σ†τ−)

, R− =√

2(

τ†−σ + σ†τ+

)

.

(225)

The operator l is identified as the 2D angular momentum in(221). There are two possible chains starting from SU(3) andending in SO(2) (i.e., that conserve 2D angular momentum):

U(3) ⊃ U(2) ⊃ SO(2) −→ Chain (Ib),

U(3) ⊃ SO(3) ⊃ SO(2) −→ Chain (IIb),(226)

where the b label stands for bending. The correspondingsubalgebras are composed by the following elements:

U(2){

n,l, Q+, Q−}

,

SO(3){

l, D+, D−}

,

SO(2){

l}

.

(227)

Following the algebraic methodology, the Hamiltonian isexpanded in terms of powers and products of the Casimiroperators. However, in molecular systems, if the Hamilto-nian goes beyond one- and two-body operators, the numberof interactions is greater than the number of invariantoperators [133]. Consequently it is convenient to use anotherprescription to construct the Hamiltonian. One possibilityconsists in expanding the Hamiltonian in terms of thedynamical algebra [131, 132], but another alternative con-sists in establishing the connection between the coordinatesand momenta and the generators of the dynamical algebra insuch a way that all the algebraic interactions are extractedfrom the Hamiltonian in configuration space. In any casea suitable bending basis to diagonalize the Hamiltonian isgiven by

∣

∣

∣[N];nσ ,nl⟩

= Nb

(

σ†)nσ(

τ†+)(n+l)/2(

τ†−)(n−l)/2|0〉,

(228)

with normalization constant

Nb = 1√

nσ !((n + l)/2)!((n− l)/2)!. (229)

Note that the kets (228) may be denoted as∣

∣

∣[N];nσ ,nl⟩

=∣

∣

∣[N];nl⟩

, (230)

because of the constraint N = n + nσ .


Now we follow an approach starting from the Hamilto-nian (210). Following the approach given in [63, 64], fromthe generators (225) of the unitary group U(3) we identifytwo suk(2) subalgebras with generators

J+,k = στ†k ,

J−,k = σ†τk ,

J0,k = −12

(

σ†σ − nk)

, k = a, b,

(231)

where nk = τ†k τk . The operators {J±,k, J0,k} have the usualangular momentum commutation relations [73]. We nowintroduce the normalized operators

b†k ≡J+,k√N

, bk ≡J−,k√N

, k = a, b, (232)

and the circular operators

b†± = ∓b†a ± ib†b√

2, b± = ∓ba ∓ ibb√

2. (233)

The action of these operators over the kets (230) is thefollowing

b†±∣

∣

∣[N];nl⟩

=√

(

n± l2

+ 1)(

1− n

N

)∣

∣

∣[N]; (n + 1)l±1⟩

,

b±∣

∣

∣[N];nl⟩

=√

(

n± l2

)(

1− n− 1N

)∣

∣

∣[N]; (n− 1)l∓1⟩

.

(234)

The harmonic limit is obtained taking the N large limit [68]

limN→∞

b†± = τ†±, limN→∞

b± = τ±. (235)

Keeping in mind the estimation of the PES, it is crucialto establish the connection between the generators of thedynamical algebra and the local coordinates and momenta,at least in an approximate form. We propose the followingapproximation for the local coordinates, Q±, and momenta,P±, in the framework of the U(3) model [68]

Q±�√

�

2ωbμb

(

d†± − d∓)

, P±�− i2√

2�ωbμb(

d†∓ + d±)

,

(236)

where d†±(d±) may be identified either with b†±(b±) ora†±(a±), following the prescription discussed in [63–66].

Here the parameters ωb and μb are ωb =√

f+−g0+− =

√

fqaqag0qaqa and μb = 1/g0

+− = 1/g0qaqa .

The proposed relation (236) represents just an approx-imation, not only because we are truncating to a linearexpansion in terms of the operators (233), but also becausethe commutation relations,

[

Q j ,Pk

]

= i�δjk, (237)

have a correction of the order 1/N when the operators d†±(d±)are substituted by b†±(b±) [68]. We now suggest to use therelations (236) instead of (216), with the substitution ofthe operators d†±(d±) by a†±(a±) when number and angularmomentum operators are involved. Otherwise, we use thesubstitution rule d†±(d±) → b†±(b±), as it is the case for Fermiinteractions [63–66]. With regard to the stretching degrees offreedom, we may consider the anharmonization procedure

a†i −→ b†i , ai −→ bi, i = 1, 2, (238)

as discussed in Section 6.Following this approach, the following Hamiltonian is

obtained:

H = ωs

2∑

i=1

(

b†i bi + bib†i

)

+ λs

2∑

i /= j=1

b†i b j + αs1(

n2s,1 + n2

s,2

)

+ αs2(

b†21 b2

2 + b†22 b2

1 + 4ns,1ns,2)

+ αs3(

ns,1b†2b1 + ns,2b

†1b2 +H.c.

)

+ ωbn + αb1n2 + αb2 �

2 + αsb1{(

b†1 + b†2)

b+b− +H.c.}

+ αsb2(

ns,1 + ns,2)

n + αsb3(

b†1b2 + b†2b1

)

n,

(239)

with the definition

ns,i = b†i bi, i = 1, 2. (240)

The equations that relate the spectroscopic parameters andforce constants are given in Appendix C of [68]. In order tocarry out the necessary calculations, we select the basis

∣

∣

∣[Ns], [N]; v1v2;n�⟩

= |[Ns]; v1v2〉 ⊗∣

∣

∣[N];n�⟩

, (241)

where |[N];n�〉 is given by (228) and for the stretches

|[Ns]; v1v2〉 = |[Ns]; v1〉 ⊗ |[Ns]; v2〉, (242)

with

|[Ns]; vi〉 =√

κvi(Ns − vi)!vi!Ns!

(

b†i)vi|0〉. (243)

The matrix elements of the Hamiltonian (239) are obtainedby means of (234) and

b†i |[Ns]; vi〉 =√

(vi + 1)(

1− vi + 1κ

)

|[Ns]; vi + 1〉,

bi|[Ns]; vi〉 =√

vi

(

1− viκ

)

|[Ns]; vi − 1〉.(244)

We proceed to apply this approach to obtain the PESof carbon dioxide in its electronic ground state. In thiselectronic manifold, CO2 is a linear molecule with a bond


Table 9: Force constants for carbon dioxide from [68].

Force constant12CO2

12CO2

This work Reference [136]

fq1q1 (aJA−2

) 15.98 16.01

fq1q2 (aJA−2

) 1.5310 1.2526

fq1q1q1q1 (aJA−4

) 189.61 681.87

fq1q1q1q2 (aJA−4

) 152.73 12.27

fq1q1q2q2 (aJA−4

) 59.367 36.702

fqaqa (aJA−2

) 0.5721 0.5818

fqaqaqaqa (aJA−4

) 3.3714 0.75

−7.1862 —

fq1qaqa (aJA−3

) −0.9592 −0.8874

fq1q1qaqa (aJA−4

) −19.549 —

fq1q2qaqa (aJA−4

) −32.880 —

length re = 1.16 A [134]. Thus, its point symmetry groupis D∞h, with four vibrational degrees of freedom: twostretching modes (Σ+

g ⊕Σ+u) and a doubly degenerate bending

mode (Π±u ) [8]. The Hamiltonian (239) has 11 spectroscopicparameters. However, as shown in Appendix C of [68], weknow that the parameters αb1 and αb2 depend on the sameforce constant f++−−. Thus, we obtain two different estimatesfor the f++−− force constant because in the fit we consideredthese parameters as independent. This can be justified since,in principle, in a more complete description, additionalforce constants of higher order should affect differently bothvalues. The spectroscopic parameters are optimized with aniterative nonlinear least square method. The fit included allthe experimental energies up to polyad P = 9. The optimalvalues for the boson numbers Ns and N were found to beNs = 160 and N = 150. In Table C.1 of [68] the 101 fittedenergies are given, where an rms deviation of 0.53 cm−1 wasobtained. The set of optimized spectroscopic parameters thatproduced the fit in Table C.1 are presented in Table C.2 of[68].

From the spectroscopic parameters we calculate theforce constants that characterize the ground electronic stateof carbon dioxide. The results are shown in Table 9. Asmentioned above, since we are considering effective forceconstants, two values for the force constant fqaqaqaqa wereobtained due to the ambiguity in its determination. Theresults show that the general trend of the force constantsis satisfactorily reproduced. The quadratic and cubic forceconstants obtained by our approach are close to the valuesgiven by Chedin [135, 136], as can be noted. A remarkableresult regarding the Fermi parameter is that a change in signprovides the same fit. However, the predicted force constantsas well as the wave functions may be quite different. Hence,a fit with αsb1 = 36.005 cm−1 gives rise to the force constantfq1qaqa = −5.4 aJ A−3, while αsb1 = −36.005 cm−1 gives rise tothe force constant fq1qaqa = −0.9592 aJ A−3, close to the forceconstant fq1qaqa = −0.8874 aJ A−3obtained in [135, 136].However, the accordance between our results and Chedin’sforce constants in the quartic order case is sensitively worse.

This is an expected result, considering the difference betweenboth approaches.

This system is a representative case where a good fit doesnot guarantee good wave functions. The eigenvectors haveto be tested through the calculation of Raman transitionintensities, for instance [137].

12. Conclusions

In this paper we have presented a review of the recent devel-opments in establishing the connection between algebraicmethods based on unitary groups and the configurationspace, the basic ingredient to extract the potential energysurfaces. To achieve this goal we present the vibrationaldescription of representative molecules where the differentalgebraic methods as well as physical situations are present.

This contribution starts presenting a summary of thero-vibrational description of molecules in the framework ofspace of the molecular coordinates. The vibrational Hamil-tonian in terms of curvilinear coordinates is also discussed.Because of the importance of symmetry of the systems, thebasic concepts involved in the symmetry projection tech-nique based on the eigenvector method has been presented.A remarkable bonus of this approach is that the charactertables are naturally identified with sets of quantum numbers.This approach has been presented in the framework of thethe permutation inversion formalism, which is fundamentalin establishing the molecular symmetry group.

The basic concepts involved in an algebraic approach ispresented through the SU(2) model, the most simple modelassociated with the vibrational degrees of freedom. Conceptsof dynamical symmetry and chains of groups are illustrated.The connection of the SU(2) algebraic model with the Morsepotential is established in detail, providing the expansion ofthe coordinate and momentum in terms of the generators ofthe group. Once the analysis of one oscillator is established,we study the case of two interacting oscillators. This studyprovides a criterion to determine the local character of amolecule, but it also shows that the algebraic treatmentof anharmonization does not have analog in configurationspace when dealing with molecules with a normal modebehavior preserving the polyad. The extention of the modelto a set of interacting Morse or/and PT oscillators permits todescribe the vibrational excitations of polyatomic molecules.Hence, an approach for molecules with both local andnormal mode behaviors has been established in a unifiedform in the framework of local interacting Morse oscillators.

One of the traditional approaches to extract the PESin the framework of the U(ν + 1) algebraic models isbased on coherent states. We have presented the suitabletransformations to recover the PES for the case of twointeracting Morse oscillators. Although this approach can beextended to more than two oscillators, the transformation isrestricted to the case of interactions up to quadratic order.More work is needed in order to extend the treatment toHamiltonians of any order.

Water was the first molecule to be analyzed as an exampleto obtain the PES through the connection of the unitaryapproach with the coordinates and momenta going beyond


the linear approximation. Up to quadratic terms are takeninto account in the expansion, which allows the full PESto be obtained up to quartic order even preserving thepolyad. The more elaborated molecular system of trifluorideof boron was presented in order to show the power of thealgebraic model when applied to molecules with normalmode behavior. This system was interesting not only becauseof its normal mode behavior but also because the approachfollowed to deal with spurious states.

Regarding the U(ν + 1) model of ν-equivalent oscillators,we have presented a reformulation that allows the forceconstants to be obtained. In this scheme the differencebetween this approach and the models of interacting Morseoscillators is clearly established. This approach is applied tothe pyramidal molecule Arsine. It has been also shown howto model the dipole transition probabilities.

The equivalent approach to extract the PES is presentedfor the case of linear triatomic molecules. As an example thecarbon dioxide has been analyzed. This approach is generalfor any linear molecule as long as the polyad is considered asa good quantum number.

Summarizing, we have presented a general approach toestimate the PES when algebraic methods based on unitaryalgebras are used. This is based on the connection betweenthe algebraic approach and the configuration space. Onlyin the most simple case of the SU(2) model the connectioncan be established in exact form. For higher dimension onlyan approximation can be proposed. This approximation,however, is enough to obtain a reasonable PES for thecase of semirigid molecules. We believe that these efforts toconnect the algebraic approaches with configuration spacemay represent an important step for estimating the PESin a simple way. The algebraic approach contrasts withvariational methods, which are too expensive to be appliedto large- or even medium- sized molecules.

Acknowledgment

This work is partially supported by CONACyT, Mexico.

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