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Advances in Physical Chemistry
Volume 2011 (2011), Article ID 593872, 38 pages
Potential Energy Surfaces Using Algebraic Methods Based on Unitary Groups
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal 70-543, 04510 México, DF, Mexico
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 the unitary groups used to describe the molecular vibrational degrees of freedom. The basic idea is to introduce the unitary group approach 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 algebraic approaches 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.
The description of molecular systems involves the solution of the corresponding Schrödinger equation. This task is so difficult that an approach involving just numerical methods needs powerful computers even for three-or four-particle systems. An alternative approach is based on choosing the basis functions in such a way that they resemble the exact eigenfunctions as much as possible. The suitable basis are obtained by making approximations that simplify the Hamiltonian of the molecule. The advantage of this method is that the functions reproduce correctly the gross features of the spectrum, and consequently they provide a better physical insight in understanding the solutions. The first step in simplifying the molecular problem consists in taking advantage of the large difference between the nucleus and electron masses, a fact that leads to the Born-Oppenheimer approximation [1, 2]. As a result of this approximation the original Schrödinger equation is split into two coupled equations, one corresponding to the electronic degrees of freedom which is solved for many nuclear geometries and the other one associated with the rotation-vibration Schrödinger equation for the nuclei whose potential is basically provided by the electronic energy [3, 4]. On the other hand, the rotation-vibration Schrödinger equation is usually solved making the rigid-rotor approximation together with the harmonic oscillator approximation. The total wave function is then approximated as the direct product of three contributions: electronic, rotational, and vibrational wave functions. Corrections to this description are allowed by introducing the braking of the Born-Oppenheimer approximation, distortion effects, anharmonicity, centrifugal distortion, and Coriolis coupling [3–5].
Within the Born-Oppenheimer approximation the potential energy surface (PES) is provided by the solution of the electronic Schrödinger equation. Following this approach the calculation of a PES represents a major problem because of two main reasons. On one hand an accurate calculation of the electronic wave functions and energies is a quite difficult problem due to the correlation and exchange interactions, as well as the relativistic effects . Hence, this problem is by itself a challenge of current interest. On the other hand the estimation of a PES involves the solution of the electronic Schrödinger equation for different nuclear configurations, a formidable computational task feasible only for small molecules. Even, when this situation is possible, the predicted vibrational spectrum generated from the obtained PES is far from the standards of high-resolution molecular vibrational spectroscopy, being necessary to refine the force constants in order to obtained a good vibrational description . In summary, for medium and large molecules, the harmonic approximation is usually considered, with the proviso of a poor quality concerning the prediction of vibrational spectra.
An alternative approach to obtain an estimation of the PES is provided by the vibrational spectrum of molecular systems [3, 4, 8, 9]. An expansion of the kinetic energy and the potential in terms of normal coordinates, for instance, involves a set of force constants which can be estimated by fitting the vibrational spectrum. In this case the estimation of the PES is computationally cheap, with force constants fitted to provide a description of high quality. These force constants may in turn be used to predict spectra of isotopic species. The advantage of this approach is that it is relatively easy to implement, although the identification of the resonances and anharmonic interactions to establish the appropriate Hamiltonian to produce a high-quality fitting is in general not a simple task. The most common approach to carry out the vibrational description is through the use of the harmonic basis associated either with the normal modes or the symmetry-adapted coordinates. The inconvenience of taking a harmonic basis is that as the energy increases the admixture of states becomes evident due to the intrinsic properties of the basis. An alternative approach consists in considering a change to a more realistic basis. In particular a local basis turns out to be appropriate for two reasons. On one hand it allows the results provided by the normal basis to be recovered when local harmonic oscillators are considered, but on the other hand this selection opens the possibility of considering alternative bases, like Morse and/or Pöschl-Teller functions, which reflect more accurately the main physical properties of a pure local bond. Local bases emerged as a natural way to explain the spectra of molecules involving bonds with large differences in the masses of the begin and end atoms [10–13]. Although both alternatives, local or normal basis, may be worked out in configuration space, their corresponding algebraic representations provide an effective and elegant route to deal with the description of the vibrational degrees of freedom. When a harmonic basis is considered, the algebraic approach appears in natural form by introducing the bosonic operators associated with the creation and annihilation operators of the harmonic functions . This is a relatively easy task that allows us to exploit the concept of polyad, a pseudo-quantum number that defines a subspace of states connected through the main interactions of the system . In contrast, when considering anharmonic potentials, bosonic operators do not appear anymore, making the description a nontrivial task. Stimulated by this problem algebraic methods based on the unitary approach emerged as an alternative to describe the rovibrational degrees of freedom.
Unitary groups are proved to be relevant in the description of many-body systems. In fact bilinear products of creation and annihilation fermionic or bosonic operators form sets of generators of unitary groups, which constitute the dynamical group of a great variety of the systems . In particular great attention has been paid to the description of electronic degrees of freedom in atoms and molecules [17–20], nuclear physics [16, 21] and subnuclear physics , and also to the rovibrational degrees of freedom [23, 24]. When the vibrational excitations are described in terms of a basis of harmonic oscillators, unitary groups appear in natural form [14, 19, 25]. Bilinear products of creation and annihilation operators of a set of oscillators in dimensions constitute the generators of the symmetry group of the system , while is the corresponding dynamical group [14, 26]. A set of harmonic oscillators presents an infinite number of states, which explains the appearance of a noncompact group as a dynamical group. It is possible to work with compact dynamical groups if the space of harmonic oscillators is cut off. To achieve this goal an extra boson is added in such a way that the totally symmetric representation (total number of bosons) of the unitary group is fixed. This approach was for the first time proposed in the context of the description of collectives states of nuclei [27, 28]. Later on the same idea was applied by Iachello et al. in the field of molecular physics to describe the rotation-vibration excitations of molecular systems, establishing what is known as the vibron model [23, 29, 30]. In the framework of this approach a group is associated with each bond providing a dynamical group in terms of the direct product of groups. The vibron model was successful in describing linear molecules [31, 32], but, because of its increasing complexity, it does not become suitable for nonlinear molecules beyond triatomic systems. An alternative model to overcome this difficulty was proposed by considering the rovibrational degrees of freedom in unified form, proposing a unique unitary group for a system of degrees of freedom including both vibrations and rotations . In any case these models are phenomenological in the sense that the Hamiltonian is expanded in terms of the generators of the dynamical algebra (usually in terms of Casimir operators), providing eigenvectors and eigenvalues, but in purely algebraic form. In the framework of this formalism the PES may be extracted through the use of coherent states, an approach associated with the classical limit . A relevant feature of the unitary approach is that the addition of the extra boson besides providing a compact group as a dynamical group, permits to take into account anharmonicities from the outset, enriching the model through the appearance of orthogonal subgroups, which play a preponderant role in the description of nonrigid molecules. In this type of systems, however, the polyad concept stops being useful. The full space has to be considered, allowing phase transitions to appear and giving rise to the possibility of describing molecular systems with several structural minima [34–38].
Although most of the applications of the unitary group approach in the field of molecular spectroscopy have been developed to describe the rovibrational degrees of freedom, there exists a model for diatomic molecules including the full set degrees of freedom. In this model the shell is considered for the electrons, while the rovibrational degrees of freedom are introduced through the vibron model. The full Hamiltonian is then expanded in terms of the generators of the dynamical group [39–41]. A remarkable feature of this model is that the Born-Oppenheimer approximation is not assumed. The expectation values with respect to the coherent states allow the PES to be extracted for each electronic state. However, because the electronic degrees of freedom are taken in the united atom limit, the PESs do not reproduce the expected shape in the separated atoms limit [42, 43].
A unitary approach restricted to describe only vibrational excitations was developed by Michelot and Moret-bailly , which later on was further analyzed to include an additional subgroup in order to introduce the most important local interactions as a part of an expansion of the Hamiltonian in terms of Casimir operators . This approach has been applied to several molecular systems, for instance, tetrahedral  and pyramidal molecules [47, 48], and it is based on the methodology of algebraic techniques where each chain of groups provides a dynamical symmetry that establishes a basis to diagonalize a general Hamiltonian, whose main contributions correspond to the Casimir operators associated with different chains . In the framework of this approach for a set of equivalent oscillators, the dynamical group becomes . The relevance of this approach is not just the restriction of the space by itself but the fact that this treatment allows us to introduce anharmonicities from the outset. The one-dimensional version of this approach deserves special attention due to the fact that it is connected with the Morse and Pöshl-Teller potential, widely applied in molecular problems . In this case the vibron model and the approach coincide. This model was first identified to describe the stretching degrees of freedom, but later on it was extended to include the bending degrees of freedom [49, 50]. As previously mentioned the extraction of the PES is a non-trivial task, although it is possible to estimate it through the coherent state formalism [23, 51, 52]. However, in order to be in position to calculate force constants useful to predict spectra of isotopic species, it is necessary to establish the appropriate correspondence between the coordinates and momenta of the system and the generators of the dynamical group .
In the last ten years successful efforts to connect the algebraic approaches to describe molecular systems based on unitary groups with their corresponding description in configuration space have been made. First the exact connection of the model with the Morse and Pöschl-Teller potentials was established [54–56]. This connection allowed force constants to be obtained, and consequently the PES became available to predict spectra of isotopic species [57–62]. Later on a connection was established between the model of equivalent oscillators proposed by Michelot and Moret-bailly  and the space of coordinates and momenta [63, 64]. This connection allowed for the first time the calculation of PES in the framework of this algebraic model [65, 66]. More recently, an explicit connection between the algebraic model, used to describe the bending degrees of freedom of linear triatomic molecules , and the configuration space was established [68, 69].
The new approach developed to extract the PES may substitute the traditional approach in which the Casimir operators play a preponderant role, since it has the remarkable feature that every Hamiltonian written in terms of bosonic creation and annihilation operators associated with harmonic oscillators can be translated into the approach in such a way that in the harmonic limit both treatments coincide. As a consequence, in this scheme it is not necessary to know the complex machinery of Lie algebras to apply the models. An additional feature of the algebraic approach is that it possesses particular features without analog in the configuration space regarding the descriptions where the polyad plays a fundamental role. It is always possible to obtain a better description of the vibrational excitations, compared with the analog models in configuration space at the same level of approximation. In addition an algebraic approach contrasts with the models based on ab initio methods to extract the PES. The algebraic approaches are relatively simple to apply because the resonances can be established in a straightforward way and the matrix elements are calculated in a simple way. This feature allows a vibrational description to be done in a relatively short time compared with ab initio methods or even with variational approaches where the kinetic energy is calculated in exact form and the potential is expanded to be fitted. However, it should be clear that the algebraic approaches cannot substitute the ab initio calculations; they represent just an alternative to estimate the PES in a relatively simple and economic way when a molecular structure can be identified either for semirigid or nonrigid molecules, although in this paper we regard solely semirigid molecules. For instance, variational approaches are more reliable to make prediction out of the range of energies considered in the fits, and ab initio calculations allow a potential energy surface of reactive molecular systems to be calculated. An algebraic approach is capable of providing PESs of two separate molecular systems, but it is not possible to obtain the effect of the molecular interaction over the PESs. It should be also mentioned that current computational procedures are capable of describing the rovibrational spectrum on a high level of approximation for molecules with four atoms like NH3, for instance . Algebraic approaches, however, cannot contend with such calculations, they are proposed to establish approximated methods that may be applied to more complex systems where ab initio calculations are too expensive to be applied.
In addition, in the context of molecular physics a potential energy surface may be obtained using an algebraic approach either by means of the connection of the spectroscopic parameters with the structure and force constants or by means of the introduction of the coherent states. However, only through the connection with configuration space it is possible to predict the spectra of isotopic species.
In this paper we review the recent advances in establishing the connection between the algebraic approaches based on the unitary groups and the physical space of coordinates and momenta. Our goal is to show the method to extract the PES for the different models and situations found in the description of vibrations in terms of internal coordinate. We start with the basic concepts involved in the algebraic approaches by analyzing the case of one oscillator. Later on the case of two oscillators is analyzed in detail to show the intrinsic advantages of an algebraic method. Here the approach using coherent states to extract a PES is also discussed. Thereafter several molecular systems are analyzed: nonlinear and linear triatomic molecules, pyramidal and planar molecules. This selection of molecules permits to illustrate the application of the different algebraic approach to estimate a PES.
This contribution is organized as follows. The basic concepts involved in the Born-Oppenheimer approximation is presented in Section 2. In Section 3 some fundamental concepts on symmetry are discussed. Section 4 is devoted to introduce the main ingredients of the algebraic approaches, while in Section 5 the connection of the model with the Morse Potential is presented. In Section 6 we discuss in detail the case of two interacting Morse oscillators. An approach to extract the PES using coherent states is presented in Section 7. In Section 8 an approach to obtained the PES is illustrated presenting the study of the water molecule. In Section 9 a more elaborated system, the BF3, is presented. Section 10 is devoted to establish an approach to obtain the PES when the algebraic models for nonlinear molecules are used. In Section 11 the case of linear molecules using the model is analyzed. Finally in Section 12 the summary and conclusion are presented.
2. Molecular Hamiltonian
A molecule is a collection of nuclei and electrons held together by forces and obeying the laws of quantum mechanics through the Schrödinger equation , where the Hamiltonian in the axis system at the molecular center of mass (parallel to the laboratory system) takes the general form [3, 4, 71] where is the sum of the kinetic energy of particles (center of mass excluded), is a kinetic energy term that involves crossed terms among the particles, while is the electrostatic potential energy. In our discussion interactions between electron-spin magnetic moments and between nuclear magnetic and electric moments are not considered. Although the molecular center of mass system allows the center of mass contribution to be eliminated, the problem of the cross-terms arises. To eliminate such cross-terms between nuclei and electrons, a reference system parallel to the laboratory at the nuclear center of mass is introduced. This new reference system induces in the kinetic energy the transformation , where and are purely electronic and nuclear contributions. The electronic kinetic term contains diagonal and crossed contributions . If the latter is neglected, the Schrödinger equation takes the form where and stand for the nuclear and electronic coordinates, respectively. Equations (2) is the starting point to carry out the Born-Oppenheimer (BO) approximation, which assumes that the motion of the electrons are unaffected by the motion of the nuclei. There are two ways of making the BO approximation: the perturbation theory approach  and the variation theory approach . In the former approach, the fundamental idea consists in expanding the rovibronic Hamiltonian in powers of the parameter , where is the electron mass and is the mean nuclear mass. Identifying the different terms in powers of , the solutions are obtained in successive form. On the other hand, in the latter approach the rovibronic wave equation is written in the form where are the vibrational displacement coordinates, are a complete set of solutions to the electronic problem, and the coefficients are to be determined. Both treatments can be consulted in the literature [1, 2]. Here we are going to provide only physical arguments for the zeroth-order solution.
The wave function in (2) may be factorized as a first approximation as the direct product of an electronic and a nuclear wave functions in the following form: , where the electronic function is parametric in the nuclear positions and . Hence, freezing the nuclei in (2) and subtracting the nuclear repulsion term , the following equation for the electronic wave function is obtained: If we now go back to (2) taking into account (4) and the previous considerations, we have for the nuclear equation where and , and the rotation-vibration energy for a bound electronic state is chosen so that the zero of energy is the minimum value of . As a result of making the Born-Oppenheimer approximation, the original problem simplifies to the two differential equations (4) and (5), where the electronic equation has to be solved before (5) since the corresponding energy provides the potential in the nuclear equation.
Both the electronic and rotation-vibration equations are too complicated to be solved in exact form. The electronic equation belongs to the field of quantum chemistry [18, 71, 72] and will not be discussed in this paper. We will concentrate our presentation on the calculation of the PES through the analysis of the vibrational degrees of freedom. We thus move to the analysis of the rotation-vibration equation, which is referred to as axis system parallel to the laboratory with its origin at the nuclear center of mass. This equation can be separated into a rotational and a vibrational part by introducing a rotated system with its origin at the nuclear center of mass, also known as the molecule-fixed axis system. In matrix form we have where is a rotation specified by the Euler angles . In order to optimize the separation of the rotational and the vibrational parts, the Euler angles are determined through the Eckart equations  where are the coordinates of the nuclear equilibrium configuration in the molecular-fixed system. For this configuration the orientation of the axis is chosen to correspond to the principal axis of inertia. When the nuclear coordinates are not far from the equilibrium, it turns out to be suitable to introduce the coordinates which are known as the vibrational displacement coordinates. In particular a linear combination of them corresponding to the normal coordinates is appropriate to write down the molecular Hamiltonian. The rotation-vibration Hamiltonian involved in (5), when rewritten in terms of the molecular coordinates , for nonlinear molecules for instance, takes the form [75–77] where corresponds to the rotational contribution corresponding to the rigid-rotor Hamiltonian the second contribution is a sum of independent harmonic oscillators associated with the normal modes while contains the anharmonic contributions where the coefficients are the force constants which determine the PES. The fourth term in (8) corresponds to the centrifugal distortion and the last term to the vibrational Coriolis coupling In these expressions is the conjugate momentum to the normal coordinate , and are the components of the rotational and vibrational angular momenta, and is the inverse of the matrix which involves the moments of inertia and the normal coordinates . is the same matrix evaluated at equilibrium. Hence, the zeroth-order approximation corresponds to the rigid-rotor harmonic Hamiltonian , 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 the use of normal bases as stated by (10). The success of such a description at that time is explained because the analysis was restricted to the low lying region of the spectrum . In the last decades, however, due to the development of new experimental techniques, it has been possible to obtain high-resolution spectroscopic data for highly excited vibrational states [78, 79]. In this region, however, it is rarely possible to find 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 states increases rapidly, and, although the spectrum is expected to be 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 understood when one takes into account that in the chemical reactivity limit the energy tends to accumulate in some particular bonds where the reaction evolves. This behavior leads to consider the vibrational problem in terms of a set of interacting local oscillators.
The vibrational Hamiltonian in terms of curvilinear internal displacement coordinates acquires the general form [80, 81] where and are column vectors corresponding to the internal displacement coordinates and their conjugate momenta , respectively, while the matrix establishes the connection between the internal and Cartesian coordinates. is the Born-Oppenheimer potential, while is a kinetic energy term not involving momentum operators which is usually neglected.
In a variational approach, the solution of the Schrödinger equation associated with the Hamiltonian (14) is obtained without any additional approximation. Since this approach is not viable for large- and even medium- sized molecules, approximate methods are welcome. The usual approach to obtain a suitable Hamiltonian to deal with consists in expanding both the matrix and the potential as a Taylor series around the equilibrium configuration, truncating the expansion where an adequate convergence is achieved. In this scheme the zeroth-order Hamiltonian corresponds to a sum of harmonic local oscillators where are the force constants. This Hamiltonian is diagonal in the basis of direct product of harmonic oscillator functions providing the basis to diagonalize a more general Hamiltonian. To obtain the zeroth-order Hamiltonian (15) from (14), it has been assumed an expansion in terms of the local coordinates , a fact that leads to identify harmonic oscillators. A more suitable coordinates to carry out the expansion are Morse or Pöschl-Teller coordinates [60, 82], since in this way the zeroth-order Hamiltonian is identified with a set of Morse and/or PT oscillators. This provides from the outset a better description because of their extra degree of freedom of anharmonicity. An additional advantage is that both potentials can be treated in a unified form in terms of algebras, a crucial advantage that allows the vibrational description to be improved without analog in configuration space when the polyad is preserved. This point will be discussed later, but before going into details we briefly sketch some symmetry aspects of the molecular Hamiltonian.
3. Symmetry Considerations
The set of transformations that leaves the Hamiltonian invariant satisfies the properties of a group . Technically speaking, if , then the associated operator acting on the physical space commutes with the Hamiltonian where stands for the number of elements of the group. When the set of operations corresponds to the maximum number of transformation, is called the symmetry group of the system, and according to Wigner’s theorem the eigenfunctions of the Hamiltonian span irreducible representations (irreps) of the group . We thus have that the symmetry group depends on the Hamiltonian, and the action of the operators over the physical space should be specified in accordance with the approximation involved.
An exact Hamiltonian for any molecular system free of external fields is invariant under the following operations: (a) any translation along a space fixed direction, Euclidean group , (b) any rotation about a space fixed axis passing through the center of mass of the molecule, rotation group , (c) any permutation of the space and spin coordinates of the electrons, permutation group , (d) any permutation of the space and spin coordinates of identical nuclei, complete nuclear permutation group , and (e) inversion of the coordinates of all particles, nuclei and electrons, in the center of mass of the molecule, inversion group . These operations follow from the Hamiltonian and the nature of the space, uniformity, isotropy, indistinguishability of identical particles, and nature of the electromagnetic force. The full group is then given by where is given in terms of the direct product of symmetric 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 been eliminated, the Euclidean group is not included in (17). As noted this group does not contain the point group of the molecule. The reason is that the point group is associated with a specific structure of the molecule, which in fact is intrinsic to the Born-Oppenheimer approximation. Hence, there should be a connection between the true symmetries and point symmetries.
The group depends only on the chemical formula, and it should be noted that its order can be very large. In practice systematic accidental degeneracies appear according to the label scheme provided by (17). These degeneracies are caused by the presence of more than one version of the equilibrium structure, in a given electronic state . Different versions are connected through potential barriers which, when they are too high to be experimentally detected, the labeling scheme (17) provides extra labels manifested through degeneracy. From the point of view of symmetry, the operations of the complete permutation inversion group associated with the insuperable penetration of the barrier are said to be unfeasible, and a subgroup is more suitable to label the states. Hence, the group may contain a subgroup composed of all feasible operations connecting several structural versions. This subgroup is known as molecular symmetry group (MS group). The MS group turns out to be isomorphic to the molecular point group when only one structural version is present. It should be stressed that the MS group depends on the resolution of the experiment, since the splitting of the inversion tunneling may or may not be detected, depending of the quality of the experimental devices.
From the technical point of view the distinction between different versions of the equilibrium structure is carried out by labeling the nuclei of the molecule in its equilibrium structure. By permuting the labels on identical nuclei with and without inverting the molecule, the number of versions can be identified, but it is only through the experiment that the MS group can be established. Permutation-inversion operations of the elements of MS group affect the coordinates of the nuclei and electrons in the molecule. It is through this effect that we are able to establish its connection with the molecular point groups. Each operation of the MS group can then be written in the following form [84–86]: where is an operation that produces the change in the vibronic coordinates (they translate into changes in the vibrational displacement coordinates as well as in the electronic coordinates) caused by , is an operation changing the rotational coordinates (Euler angles), while is an operation that generates a nuclear spin permutation. Since the operations , , and affect different subspaces, they commute with each other. In semirigid molecules (where only one structural version is present) the vibronic operations constitute the elements of the molecular point group isomorphic to the point group widely used when in the context of the Born-Oppenheimer approximation.
Let us now turn our attention to the identification of the irreps as a set of quantum numbers when discrete groups are involved. In essence, symmetry plays a central role in the necessity of establishing a complete set of commuting operators (CSCO) to label the eigenstates of the time-independent Schrödinger equation. The Hamiltonian itself can be considered in the set of the CSCO, since the energy provides a label for the state. If stands for an index introduced to distinguish different energies, the Schrödinger equation can be written as where is the degeneracy. Because of the property (16), we may thus think that the set of operators is useful to define a CSCO, but in general , unless the group is Abelian. This problem is solved by identifying subsets of , which turn out to be the conjugate classes of the group, since , for all , . Hence, the Hamiltonian together with the classes of the group constitute a set of commuting operators, whose representations can be diagonalized simultaneously in any space of independent functions.
We may now construct the representation matrix of the class in the basis of eigenvectors of the Hamiltonian: . The diagonalization of this matrix provides eigenvectors of type , with the property , where is the label that distinguishes the different eigenvalues of the class operator and accounts for the degeneracy. We may now proceed to obtain the matrix representation of the next class in the new basis , to obtain eigenvectors labeled also with the eigenvalues . Following this approach for the rest of the classes leads to the states of the form characterized by the eigenvalues . However, this set of labels is not complete, since the number of irreps is equal to the number of classes, and consequently the set of labels specifies just the possible irreps . The question which arises is concerned with the identification of the new set of operators capable to distinguish the states associated with the degeneracy of the irreps. The answer is given by the classes of a subgroup , being a canonical chain. Suppose that has classes , which clearly satisfy . But the classes of the group commute with any element of the group and consequently commute also with the classes of the subgroup for all. This fact suggests to diagonalize the representation of the operators in the basis to obtain a complete labeling for the components of the irreps. After this procedure of diagonalization of the classes of , we arrive to the complete labeling scheme where the subindices are defined by
Let us now turn our attention to the identification of the labels involved in (20) as quantum numbers. The time evolution of the expected value of an operator is given by  where is the commutator of the Hamiltonian with the operator . Hence, if the operator does not depend explicitly on time and commute with the Hamiltonian, then the expected value is constant in time.
Suppose now that the states are chosen to be eigenstates of the Hamiltonian together with the classes of the group and subgroup : then (22) translates into when is substituted by , , and . Hence, the eigenvalues of the set of operators are independent of time and consequently are quantum numbers.
For a given there are sets of values. As mentioned before, this fact suggests a connection between the values and the characters of the group. Indeed it has been proved the following connection : where stands for the irreps of the group, while refers to its dimension. A similar relation is also valid for and the characters of the subgroup .
The expression (25) provides a projection method. This assertion may be appreciated because of the following result: any symmetry-adapted function spanning the th irreps of dimension satisfies  which means that the functions are eigenvectors of the class operators with eigenvalue . This remarkable result suggests to follow the inverse procedure to obtain (23). We can start diagonalizing the class operators to end up with the Hamiltonian. Consider this idea starting with an arbitrary set of functions . First we chose a subset of classes that allows the irreps to be distinguished. The simultaneous diagonalization of the selected classes provides eigenvectors carrying the th irreps. The eigenvectors spanning the same irreps are then used to diagonalize the set of classes of the subgroup . The resulting eigenvectors span irreps of the group as well as the subgroup , allowing the components of degenerate irreps to be distinguished. This approach, proposed by Chen , turns out to be very powerful and has been used to generate general codes to project vibrational  and rotation-vibration functions .
4. Basic Concepts of an Algebraic Approach
The simplest model to describe the vibrational excitations of a molecule consists in modeling the system as set of interacting harmonic oscillators, in accordance with (10) and (11) An algebraic realization of this Hamiltonian can be obtained by introducing the operators  which have the effect of ladder operators over the harmonic wave functions. This approach is known as the traditional algebraic approach for vibrational excitations . The advantages of this description are that all matrix elements are calculated in algebraic form and that the resonances can be expressed in precise mode. As a consequence, it is easy to identify the interactions that preserve the polyad . The harmonic basis intrinsic to this description, however, presents the disadvantage of an infinite dimension of the basis, a consequence that the potential does not reflect the appropriate behavior in the high energy region of the spectra. To overcome this problem, new algebraic approaches based on unitary groups emerged to take into account anharmonicities from the outset.
A fundamental concept intrinsic to the algebraic models is the dynamical group. The generators of this group form a Lie algebra, and any dynamical variable, including the Hamiltonian, can be expanded in terms of them. In this way the generators of the dynamical group possess the fundamental property that any pair of eigenstates of the Hamiltonian can be connected by them. Bilinear products of creation and annihilation operators of a set of harmonic oscillators in dimensions constitute the generators of the dynamical group [14, 26, 92], a noncompact group presenting an infinite number of representations, in accordance with the infinite number of levels that present the harmonic oscillators. It is possible to work out with a compact dynamical group by adding an extra boson in such a way that the totally symmetric representation (total number of bosons) of the unitary group is fixed, as previously explained in the introduction. This ingredient simplifies enormously the description since the technical machinery involved in the compact groups is much less complicated than the one involved for the noncompact groups. To illustrate this approach, we will start considering the case of one oscillator.
Let us consider a one-dimensional harmonic oscillator, whose Hamiltonian and eigenvectors in the algebraic representation take the form The symmetry group of this system is since its generator commutes with (29) . On the other hand, the dynamical group is the noncompact group . The unitary group approach consists in adding an extra boson in such a way that the total number of bosons is constant. The bilinear products are now generators of the group. The states associated with this group according to the chain are given by  where the generator of is . Because of the bosonic nature of the vibrations, these states span the totally symmetric representation of the group. A connection of the generators of with the known angular momentum generators in the Cartesian basis is given through  and in the spherical basis Let us now consider the realization (33) with the following normalization: where . The action of these operators over the kets (31) is Note that the operators connect the whole space of the functions , and consequently is the dynamical group of the system; any dynamical variable can be expressed in terms of the generators of the unitary group . However, it is clear from (35) that it is only in the harmonic limit for finite that the matrix elements of the one-dimensional harmonic oscillator is recovered, for example, . Because of the constraint over , a more general Hamiltonian is expected to be expanded in terms of the generators of the dynamical group in such a way that . The total number of bosons is related to , the angular momentum label, through . In general it is convenient to establish the different chains of groups and the invariant operators associated with them since they are used to expand the Hamiltonian. In our case we have three more chains in addition to (30) that provide alternative bases where the subindex in the groups refers to the operator in (32) that generates the group. Hence, a possible simple expansion of the Hamiltonian is This Hamiltonian is diagonal in the basis. If the basis associated with this chain is denoted by and if in addition the following quantum number is defined , with when only the branch with negative values of is considered, the eigenvalues of take the form This is a Morse-like spectrum, which explains the fact that the chain is associated with the Morse functions . Analytical solutions obtained from Hamiltonians involving invariant operators associated with a chain are called dynamical symmetries. In general, the Hamiltonian that best suits the description involves invariant operators of several chains, and any basis can be used to diagonalize it. Hence, the introduction of the -boson not only deforms the potential, but also enriches the description providing dynamical symmetries and alternative bases to carry out the calculations.
5. Connection of the Algebraic Model with the Morse Potential
The Hamiltonian for one-dimensional Morse potential has the form  where is the reduced mass, is the displacement coordinate, is the depth of the potential, and stands for its range. The solution of the Schrödinger equation associated with this Hamiltonian is given by where are the associated Laguerre functions, the argument is related to the physical displacement coordinate by , is the normalization constant and the variables and are related to the depth of the potential and the energy, respectively through  with the constraint condition . Using the factorization method, it is possible to obtain creation and annihilation operators of the Morse functions, which turn out to have the following effect: with and whose explicit form in terms of the coordinate and momentum are given in . The operators , together with the number operator , satisfy the commutation relations which can be identified with the usual commutation relations through , where satisfy the usual “angular momentum’’ commutation relations . Hence, the group is the dynamical group for the bound states of the Morse potential . From the group theoretical point of view, the parameter labels the irreducible representations of the group. The projection of the angular momentum is related to by . From this relation we see that the ground state corresponds to , while from the dissociation condition the maximum number of quanta is . The state corresponding to , however, is not normalizable, and consequently the allowed values for are . The Morse functions are then associated with one branch (in this case to ) of the representations, although a recent work associates a noncompact group to the bound Morse space . The bound solutions (40), however, do not form a complete set of states in the Hilbert space . However, when the vibrational excitations are far from the dissociation limit, it is a reasonable approximation to consider the bound states as a complete space [97, 98]. A similar situation appears in the case of the Pöschl-Teller potential [96, 99].
The realization of the Morse Hamiltonian in terms of the algebra is given by  From (42) we obtain the corresponding eigenvalues The harmonic limit is obtained by taking : where the operators satisfy the usual bosonic commutation relations. Since the set of operators constitutes a dynamical algebra for the Morse potential, any dynamical variable can be expanded in terms of them. In particular we are interested in the expansion of the momenta and Morse coordinates. For the momentum the following second-order expansion is obtained : while for the Morse coordinate [100, 101] where , , and are functions of the number operator given by Since is diagonal, we have substituted by . For high number of quanta or small, the terms of order must be included. In contrast, if we are interested in the low lying region of the spectrum, a reasonable approximation consists in neglecting the terms of order and taking the harmonic limit of the diagonal functions , . We may thus propose the linear approximation We immediately note the similarity of these expressions with the harmonic oscillator case. In fact, taking the harmonic limit we recover the usual expressions for the harmonic case. In this approximation it is clear that effective interactions preserving the polyad can be established in a straightforward way.
Let us now come back to the previous analysis of the algebraic model. If we compare the matrix elements (35) with (42), we immediately identify the isomorphism However, we still have to identify the dynamical symmetry associated with the Morse Hamiltonian. The isomorphism (52) suggests the symmetry since the eigenfunctions of are also eigenfunctions of . In fact, if is the eigenvalue of , we have the identity . Taking the negative branch of , we have . The dynamical symmetry takes the form which up to a constant is basically , as expected. We have thus established the exact connection between the algebraic model and the eigenfunctions of the one-dimensional Morse Hamiltonian.
The connection we have presented deserves an additional comment. First we note that, in the linear approximation (51), the generator is proportional to the Morse coordinate , while corresponds to the momentum. The dynamical symmetries and are then associated with the coordinate and momentum representation in the linear approximation. On the other hand, the dynamical symmetry is associated with the energy representation, while the corresponds to the energy representation in the harmonic limit.
The treatment presented here may be followed for the Pöschl-Teller potential arriving to a similar result; is the dynamical group of the system . The matrix elements (35) are isomorphic to the corresponding matrix element of the creation and annihilation operator of the PT functions. Of course, the explicit form of the wave function is different as well as the expansion of the coordinate and momenta, but the energy spectrum is the same, as established by the algebra. There is a fundamental difference between these potentials, Morse and PT. While the Morse potential is asymmetric, the PT potential is symmetric, allowing both symmetric and asymmetric local modes to be modeled in the unified framework of the algebra.
6. Two Interacting Morse Oscillators
When the masses of the atoms involved in a set of equivalent bonds are very different, a scheme of interacting local oscillators represents a suitable approach to carry out the vibrational description. For example, the stretching modes in H2O present a strong local behavior because of the large ratio 16 : 1 of the masses. In contrast, when the masses are similar, the behavior is normal and starting from a set of local oscillators does not represent a good zeroth-order approximation. It is possible, however, to describe molecules presenting a strong normal behavior in terms of local oscillators, even taking into account the preservation of the polyad, a fact that provides unique advantage of the algebraic models over the traditional description in configuration space, as we next explain.
We start our discussion with a treatment in terms of local harmonic oscillators. The Hamiltonian (14) for two equivalent oscillators up to quadratic order and neglecting takes the form Introducing the bosonic operators with and , the Hamiltonian (54) takes the form This Hamiltonian does not preserve the total number of quanta (polyad), and consequently the dimension of its matrix representation is infinite. In practice, a polyad preserving Hamiltonian is considered where we have introduced the definitions It should be clear that the Hamiltonian (57) is expected to provide a good description as long as the polyad breaking term in (56) is neglected. In this section we will see that it is possible to work with a Hamiltonian of type (57) to describe systems where the full Hamiltonian (56) is expected to be considered.
Let us now introduce the symmetry-adapted coordinates in the Hamiltonian (54) with the corresponding induced transformation in the momenta. In the new coordinates the Hamiltonian transforms into two independent harmonic oscillators (normal modes) where the following definitions are introduced: An algebraic representation is obtained through the introduction of the bosonic realization where with parameters The Hamiltonian (60) takes thus the form with the definitions The exact connection between the Hamiltonians (56) and (64) is given by the relation between the bosons involved, which is given by where . However, we can return to a Hamiltonian of the form (57) introducing the canonical transformation where are also bosonic operators associated with the th oscillator. The substitution of (67) into (64) yields with the definitions The local operators do not correspond to the physical local operators , but their action on an isomorphic local basis may be chosen to be the same. In fact we establish the isomorphism allowing the Hamiltonian (68) to be expressed in the form We thus have that the same algebraic Hamiltonian may describe molecules with both local and normal mode behaviors, but the interpretation of the spectroscopic parameters must be considered appropriately in order to obtain the correct physical results for the force constants. This is a remarkable feature, because it implies that both Hamiltonians provide the same fit to experimental energies, but the force constants derived from the optimized spectroscopic parameters are different. We should also note that the correct force constants can be obtained from a local interpretation of the Hamiltonian as long as the polyad breaking term is included. Hence, the use of the Hamiltonian (71) avoids breaking the polyad keeping the correct physical information through the parameters (69). However, this is possible only in the quantum mechanical treatment. The use of coherent states to extract the potential energy surface does not provide the correct results, as we will shortly discuss .
The quantitative criterion to choose between the local and the symmetrized description in order to evaluate appropriately the force constants is provided by the connection between (57) and (71), which is obtained by rewriting the spectroscopic parameters in (69) in the form to carry out the Taylor series expansion with the identifications and . This analysis leads to the conditions to be able to apply the local mode Hamiltonian (57). In other words, We should also stress that in this limit the correspondence is satisfied When the conditions (72) are not satisfied the normal mode Hamiltonian (71) should be taken. For the water molecule for instance, we have while for the CO2 molecule which are illustrative of local and normal behaviors, respectively.
The Hamiltonians (64) and (71) are completely equivalent, since they provide the same energy spectra with the same force constants. We may now take a further step by applying the anharmonization procedure [58, 59] in such a way that the Hamiltonian (71) transforms into which is intended to be diagonalized in the direct product of local Morse oscillators. On the other hand, the Hamiltonian of two interacting Morse oscillators in configuration space takes the form whose algebraic representation up to a constant term takes the form with spectroscopic parameters given by (58). In practice the Hamiltonians (78) and (80) are equivalent, since both provide the same energy spectrum. The difference is given in the relation between the spectroscopic parameters and the structure and force constants. While the anharmonization procedure provides the correct results for the force constants through (78), the treatment in configuration space does not give equivalent results, unless the polyad is broken. Hence, this approach is a consequence of a purely algebraic analysis without analog in the treatments in configuration space. In this paper we will present several applications where this analysis is crucial to obtain the correct force constants.
7. PES Using Coherent States
With the advent of algebraic techniques based on unitary groups to describe nuclei [19, 27, 28], the use of coherent states became a valuable tool to understand the physical content of the models [51, 102]. The same kind of algebraic techniques were also applied to describe rovibrational spectra of molecules, and, because of the phenomenological feature of models, the analysis of the classical limit through the use of coherent sates was crucial in providing a physical insight into the models [51, 52, 102–105]. According to this method, the diagonal matrix elements of the Hamiltonian in the basis of coherent states provide through a suitable transformation a potential energy surface, whose derivatives evaluated at equilibrium give the force constants. In molecular physics however a potential energy surface can be obtained by means of the connection of the spectroscopic parameters with the structure and force constants. A natural question which arises is concerned with the comparison of the force constants provided by both methods. To answer this questions a system of two interacting Morse oscillators will be considered .
The quantum mechanical Hamiltonian of two equivalent interacting Morse oscillators in configuration space up to quadratic terms is given by (79). This Hamiltonian may be translated into an algebraic representation by means of the approximated expressions (51), where there the frequency of the oscillators is given by . In the expansions (51) all terms of order and higher are neglected. The substitution of (51) into the Hamiltonian (79) gives rise to the algebraic realization where The expressions . and (82) constitute explicit dependence of general functions which will be used in the following discussion. We may interpret the above Hamiltonian as a phenomenological expansion with parameters , , and , and the question which arises is concerned with the possibility to recover the original Hamiltonian from the classical expression obtained through the coherent states. The PES extracted in this way (imposing the condition ) and denoted by is given as a function of the parameters. In symbolic form From this expression, the force constants are given by However, we have now the explicit forms (83), and consequently the following consistency relations are to be expected
Let us now proceed to consider the coherent states approach. A possible way to express the coherent states is the following : which may be recast in the form where, the operator is given by (33). Using the direct product of two coherent states of the form (88), we obtain the expectation value of the Hamiltonian (81): where for the sake of simplification, we have introduced the definition The following transformation to the classical variables and is now proposed : where It can be proved that (91) reduces to the expected results in the harmonic limit . The substitution of (91) into (89) yields the energy surface which is a function of the spectroscopic parameters. The expression (93) resembles the original Hamiltonian in configuration space, but we have to substitute the functions (83) in order to prove that the full Hamiltonian is recovered. In fact, taking into account the explicit expressions (82), the energy function (93) transforms into where The original Hamiltonian is recovered in the limit of large, except for a constant term: From this expression, the PES is obtained at once by taking the condition , with the consistency result In this way the PES has been reproduced .
From this analysis one point should be remarked: the reason why it was possible to fully recover the PES, besides having used the appropriate transformation (91), is that the full Hamiltonian (up to quadratic order) was involved. In general this situation is not satisfied, as we next discuss.
7.1. Polyad Approximation
In molecular spectroscopy most of the times an approximation involving polyad conservation is considered as zeroth-order calculation. If necessary a Van Vleck perturbation theory is applied to take into account polyad mixing [101, 106, 107]. In our case of two interacting oscillators the polyad is considered as a good quantum number, and therefore the corresponding algebraic Hamiltonian to be taken is given by Let us now proceed to analyze the coherent state method to extract the PES. For this Hamiltonian we have and applying the transformation (91) we obtain This expression reproduces the independent Morse oscillators, but it involves only . In this case the consistency relations (86) are not strictly satisfied .
If we now impose in the Hamiltonian (101) the constraint , we obtain for the PES From this expression we finally obtain for the force constants If we now take into account the relations for and (82), we arrive at as long as the condition 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 to polyad 5) reported by Halonen in his contribution in , a pretty good fit with an cm−1 is obtained, as long as up to quartic interactions are included (6 parameters involved). Keeping the same energies, but considering the simple Hamiltonian (99), it is not possible to describe the spectrum with a reasonable quality. Reducing the number of energies by considering levels up to polyad 3, a fitting with cm−1 is obtained with the following set of spectroscopic parameters: where the tilde is introduced to emphasize that the parameters are given in wave numbers. Taking into account the definition (92) and the explicit expressions for the Wilson matrix elements, we obtain for the force constants 
The force constants can be also calculated from the quantum mechanical Hamiltonian using the connection of the spectroscopic parameters with the force constants given in (82), whose results are displayed in Table 1 with the label QM. In the same table the results (106) are labeled (), and () when the limit is considered. According to the previous analysis the difference between the first column (QM) and the third one () lies on the approximation in the Hamiltonian (81). For H2O this parameter is approximately given by , which when compared with justify the use of the Hamiltonian (99).
It should be clear that the good agreement shown in Table 1 is due to the fact that is approximately null, which in turn is satisfied for molecules with local behavior. In other words, the vibrational excitations in water molecule can be approximated as interacting local oscillators. The natural question which arises is concerned with the results in systems with a normal mode behavior.
7.2. Normal Mode Behavior
In this subsection we address the problem of obtaining the PES when a Hamiltonian of type (99) preserving the polyad is considered in the fitting of a spectrum associated with a normal mode behavior. We start with the Hamiltonian (81) in the harmonic limit, which takes the form (56). The corresponding polyad preserving Hamiltonian is given by (57). As we previously discussed, the Hamiltonian (56) in configuration space is given by 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). The substitution of (67) into (64) yields (68). As we stressed, the local operators do not correspond to the physical local operators , but their action on an isomorphic local basis may be chosen to be the same. In fact we will establish the isomorphism , allowing the Hamiltonian (68) to be expressed in the form We may now consider the anharmonization procedure [58, 59] to obtain the Hamiltonian (78) This approach of identification of the spectroscopic parameters in terms of the structure and force constants does not have analog in the treatments in configuration space.
As an example of a normal mode behavior, we will consider the molecule of carbon dioxide. Although the strong Fermi interaction of this molecule makes it inappropriate to use the Hamiltonian (99) to obtain good values of force constants, we will proceed to estimate them by taking the relevant stretching parameters from a fit using the model given in , where the purely stretching Hamiltonian up to quadratic terms coincides with (99). The parameters are given by If we intend to calculate the force constants by means of (103) according to our coherent states method, we obtain the results displayed in the columns CS and CS () of Table 2. In contrast, when the force constants are calculated quantum mechanically using (69) the results of column labeled QM is obtained. As a reference, the force constants obtained from  by means of the model are displayed in the column labeled . In the latter case the bending interactions are involved in the calculations which explains the difference. Since in this work we are interested in evaluating the coherent states method, we will be interested in comparing the predicted force constants with the column .
While for the values are not too different, in the case of the force constant the results are fairly different even qualitatively. We should stress the wrong sign of the force constant , which is determined by the sign of according to (103). This result is a consequence of having used a polyad preserving Hamiltonian to obtain the PES. This assertion can be proved by means of the criterion (76), which reflects the normal behavior that characterizes this molecule. Hence, it is clear that for carbon dioxide the coherent state method fails when a polyad preserving Hamiltonian is used. In molecules with normal mode behavior, the appropriate Hamiltonian to recover the PES must be the full Hamiltonian (81), which means that in principle the polyad has to be broken. In other words, in this case the parameter in (81) is not negligible if one intends to calculate the force constants. In fact for CO2 we have From the previous analysis we know that the coherent state approach applied to the full Hamiltonian (81) taking the limit provides the same force constants that the quantum mechanical result labeled QM in Table 2. This is a remarkable result that must be taken into account when using this treatment. To emphasize this point in Table 2, a column labeled as coherent state-polyad breaking (CS-PB) has been added, whose results are expected to be obtained from a fit using (81).
We have thus presented the method of coherent states to extract the PES of molecular systems in the context of the Born-Oppenheimer approximation. The analysis has been presented for the one-dimensional version of the algebraic models based on unitary groups. In particular the algebraic representation of two interacting Morse oscillators was considered. Special attention was paid to the case when the polyad is considered as a good quantum number. It was shown that in this approximation the coherent states approach reproduce the PES only for systems with local mode behavior. For molecules with a normal mode behavior the coherent state approach is still valid, but a Hamiltonian breaking the polyad must be considered, as a consequence of the strong coupling. The two independent oscillators stop being a good zeroth-order Hamiltonian, and the parameter in (81) is strongly influenced by the interaction associated with the parameter . Since the polyad concept is crucial to be able to carry out the calculations in the description vibrational excitations in molecules, we conclude that in practice the suitability of the coherent state approach is restricted to molecules with local mode behavior. In the next sections we proceed to show how to obtain the PES quantum mechanically for different representative systems of semirigid molecules.
8. Application to Water Molecule
In an algebraic description, where the polyad is considered as a pseudo-quantum number, the possible force constants to be estimated from a fit are restricted. The reason is that many terms should be taken away from the Hamiltonian, eliminating the possibility of calculating the corresponding force constants. When the linear approximation (51) is considered, a similar situation is present. In this section we will show that, when the quadratic terms are included in the expansions of the coordinates and momenta (see (48) and (49)), the whole set of force constants can be estimated up to the order considered in the Hamiltonian. To this end we will present the vibrational description of in the framework of the model, the one-dimensional case of the unitary group approach.
To estimate the PES of beyond the linear approximation (51), a tensorial formalism is developed to expand the Hamiltonian in powers of in terms of symmetry-adapted operators, which in turn are given in terms of local creation and destruction Morse operators in the spirit to establish the connection between the effective Hamiltonian approach and the standard local mode models in configuration space . Hence, in the framework of the algebraic model, independent Morse oscillators are considered as a zeroth-order approximation. In a Morse oscillator basis, however, the interaction terms couple the whole space in the Hamiltonian matrix, not allowing to take advantage of the simplifications brought about by approximately conserved polyad numbers. This problem, however, can be avoided by keeping only the terms which preserve the polyad. Here we will first proceed to obtain the Hamiltonian in terms of the Morse variables and their momenta, to thereafter carry out their expansions in terms of the symmetry-adapted operators, keeping the terms preserving the polyad. A remarkable consequence of keeping the terms of order () in the expansion in the Morse coordinates and momenta is that all the force constants up to quartic terms in the potential can be determined, even with the constraint of the polyad as a good quantum number, as previously mentioned.
The equilibrium structure of the water molecule is nonlinear with structure parameters and . The point symmetry is , but it is enough to consider the subgroup since the vibrations take place on a plane. This molecule has three degrees of freedom, two of them associated with the stretching modes and the other to the bending mode (). The harmonic approximation provides a complete basis in terms of normal modes which can be used to diagonalize a general Hamiltonian. In the standard notation this basis is labeled by , where and are the number of quanta in the stretching and modes, respectively, while is associated with the bending mode .
If we use internal displacement coordinates, the quantum mechanical Hamiltonian that describes the vibrational excitations of the molecule takes the form (14) Here we have omitted the purely quantum mechanical term derived from the kinetic energy not involving momentum operators. The components and will be assigned to the O-H stretching displacements coordinates from the equilibrium, while corresponds to the displacement from equilibrium of the -bending coordinate. In this way we have where has been added in the definition of the bending coordinate in order to have the same distance units. For semirigid molecules a reasonable approach consists in expanding both the matrix as well as the potential in a Taylor series about the equilibrium configuration. It has been pointed out, however, that, in order to obtain convergence for large oscillations, the Morse variables are appropriate [109–114]. Hence, here we consider an expansion of the variables for both stretching and bending coordinates. In this spirit the elements of the matrix are expanded up to quadratic terms, since we intend to consider an expansion in the Hamiltonian up to quartic terms. Consequently an expansion up to quartic terms is carried out for the potential. It is worth noting the relevance of the ratio . In the rigid limit this ratio tends to the variable . The zeroth-order Hamiltonian is then given by where