S. A. Astashkevich. St. Petersburg State University, Peterhof, St. Petersburg, Russia. Abstract

Transcription

1 Algebraic Determination of Spectral Characteristics of Roibrational States of Diatomic Molecules. I. Diagram Technique for Determination of Vibrational Dependences of Matrix Elements S. A. Astasheich St. Petersburg State Uniersity, Peterhof, St. Petersburg, Russia Abstract Explicit algebraic expressions for the expansion of the ibrational matrix elements f() r ' in series of matrix elements on the wae functions of the ground ibrational state hae been obtained for arbitrary sufficiently differentiable functions of the internuclear distance f() r, arbitrary alues and ', and the potential cures whose ladder operators can be constructed. A diagram technique hae been deeloped for it that consists in: ) the numeration of the matrix elements M l l, m d l l f ( r( y)) m dy l (y - a dynamic ariable) by points of the D diagram with coordinates (l, ), ) the drawing arrows between points of this diagram corresponding to the action of the annihilation operators ˆK on the wae functions; 3) total taing into account of all possible path ectors formed by the continuous sequences of arrows from point (, ) towards points (0, ). The only requirement is that the action of the operator ˆK on the wae functions should gie the wae functions of the Schrödinger equation with the potential cure haing the same parameter alues. All necessary data for algebraic calculations of the ibrational dependence of matrix elements for the harmonic oscillator and the Morse potential hae been gien. Obtained expressions can be used to determine the absolute alues and ibrational dependences of arious spectroscopic characteristics of both ground and electronically excited states of diatomic molecules. Keywords: ibrational matrix elements, algebraic methods, ladder operators, diagram technique. PACS number(s): Fd, Ca, t address: astasheich@mail.ru

2 . Introduction Along with the using of methods based on the numerical solution of the ibrational Schrödinger equation [, ] for the calculation of matrix elements corresponding to different characteristics of molecules algebraic methods are actiely deeloped in last time [3 5]. These methods include the factorization [5, 6], group theory [7], the super-symmetric quantum mechanics (SUSY) [8] and other methods. The basic point of these methods is to find analytical solutions of the Schrödinger equation for the energy leels, the wae functions and matrix elements using the formalism of the quantum mechanics and mathematical physics (including the hyperirial theorems, the Hellmann-Feynman theorem, the sum rules formulas and etc.; the second quantization method, analytical properties of special (hypergeometric) functions, the theory of Lie algebra operators, etc.). It is important to emphasize that the algebraic methods, in contrast to numerical methods, proide a theoretical possibility of finding an explicit dependences of the matrix elements on ibrational and rotational quantum numbers. In particular this maes the algebraic methods are promising for the deelopment of semi-empirical methods of determination of ibrational and rotational dependences of the spectroscopic characteristics of molecules. It is also important to note that the algebraic methods, in principle, allow deeloping an error estimations mathematical deice for calculations of matrix elements using information about the inaccuracy of used potential energy cure parameters and some functions corresponding to matrix elements on the electronic wae functions of a certain quantum-mechanical operators (dependences on internuclear distance of the dipole moment, the transition dipole moment and others). It is important to note that algebraic methods can be effectiely applied to describe not only the energy of molecular leels but also non-energetic (radiatie, electrical, magnetic) characteristics of the ibrational and ibrational-rotational leels (see bibliography in monographs [3 5]). This is especially important taing into account that the non-energetic characteristics, in particular, the radiatie transitions probabilities and the g-factors of molecular leels may hae significantly higher information content about the internuclear dynamics than the energetic characteristics [9, 0]. Our cycle of papers deoted in the first place to the algebraic description namely non-energetic characteristics of the ibrational, ibrational-rotational and roibrational states of diatomic molecules. It should be noted that the published algebraic expressions for the ibrational dependence of matrix elements of an arbitrary function of hae been obtained only in the case of the harmonic potential [ 3]. For anharmonic potentials algebraic expressions for

3 dependences of the ibrational matrix elements on ibrational [4 ] and rotational [3 8] quantum numbers of the combining states were obtained only for some model functions (polynomial, exponential, and some others) for the following potentials: quartic anharmonic oscillator [4], Morse [5 0, 3 6], modified Pöschl Teller [, ]), and Kratzer [7, 8] (see also [5, 6]). These expressions were obtained by different methods using the properties of hypergeometric functions [5, 6, 9, ], the hyperirial theorem [7, 8, 4, 7, 8], the hyperirial theorems and sum rules formulas [4] and using the second quantization method [5, 6]. It should be emphasized that explicit (non-recurrence) algebraic expressions for the ibrational dependence of matrix elements for an arbitrary function of internuclear distance are absent in the literature. Receipt of such expressions is actual problem for the deelopment of algebraic methods of the study of ibrational dependences of arious characteristics of diatomic molecules. This article begins the cycle of papers deoted to the deelopment of an algebraic approach to the determination of ibrational and ibrational rotational dependences of different spectroscopic characteristics of the roibrational states of diatomic molecules by means of the ladder operators. The purpose of this cycle is to: ) deelop the algebraic model for the ibrational matrix elements of quantum mechanical operators of fairly general form using the apparatus of the second quantization for diatomic molecules; ) obtain missing in the literature algebraic expressions for the dependences of these matrix elements and different spectroscopic characteristics of diatomic molecules on ibrational and rotational quantum numbers of the combining states, and 3) mae an algebraic analysis of the influence of adiabatic effects (anharmonicity, ibrational-rotational interaction) as well as nonadiabatic effects (electron-ibrational and electronic-rotational interactions) on these dependences of spectroscopic characteristics. The present paper is deoted to the receiing algebraic expressions for matrix elements on arbitrary sufficiently differentiable function of internuclear distance r on the ibrational wae functions of the potential cure whose explicit expressions for the creation and annihilation operators are nown. The only condition whose fulfillment is essential for us is the fact that the action of creation and annihilation operators on the wae functions should gie the wae functions of the Schrodinger equation with the same parameter alues. Without going into details of the group theory description of these operators we still note that this situation occurs, in particularly, when these operators form the Weyl Heisenberg algebra (case of the harmonic oscillator [5,9]) and the su() algebra (the Morse potential 3

4 [0,30] and the modified Pöschl Teller potential []). The purpose of the present paper is to obtain analytical formulas for the matrix elements f ( r) ' for arbitrary alues of ibrational quantum numbers of the combining states and ' in terms of matrix elements on the wae functions of the ground ibrational state for these potentials as well as other possible potentials which satisfy to the conditions described aboe higher.. Notations Consider non-relatiistic Schrödinger equation for the ibrational wae function ( r ) of the diatomic molecule Hˆ ( r) E ( r), E is the energy of th ibrational leel. Disregarding the rotation of the molecule as well as the non adiabatic effects of intermolecular interactions the Hamiltonian of the molecule is gien by ˆ ( ) d H r V ( r dr ), where V () r is the potential energy cure of the electronic state under inestigation; is the reduced mass of the molecule. The factorization method is consists in to represent V () r in the form of an analytical potential of some parameters (,,..., n ) V( r) U (, r) () and reduce the Schrodinger equation with this potential to the two first order differential equations [5, 6]. This is achieed by the substituting r by a dynamic ariable y for which creation and annihilation operators can be constructed: For greater isibility here and further in the article the quantum numbers describing the electronic state of the molecule are not gien since the subject of this study is namely ibrational dependences of matrix elements. The influence of ibration-rotation interactions and also non-adiabatic effects on ibrational and rotational dependences of matrix elements will be analyzed in our next article. 4

5 K ˆ ( y) aˆ ( y), () K ˆ ( y) aˆ ( y). (3) The action of the annihilation operator on the wae function of the ground ibrational state is gien by: K ˆ 0 ( y ) 0. (4) In general the creation and annihilation operators hae the form (see [5]) 3 : ˆ d K b ˆ c ˆ ( y ). (5) dy Operator coefficients a ˆ, b ˆ and functions cˆ ( y) depend in general on the ibrational quantum number of the wae function on which they act, as well as the parameters of the potential of the molecule. For example in the case of the Morse potential such parameters are the dissociation energy De and the parameter α staying in the exponents of this potential (see Table). The result of the action these operators on the wae functions will be no longer denoted as operator but the numerical coefficients a, b and functions c ( y ). The explicit form of expressions for the coefficients a and b as well as the dependence of dynamic ariable on internuclear distance y=y(r) for the harmonic oscillator and the Morse potential necessary for further analysis are gien in Table. Now let us turn to an analysis of the matrix elements f ( r) '. In spectroscopic studies the function f() r can corresponds to dependences on the internuclear distance of the dipole moments (also higher order moments) of the electric (and magnetic) transitions, the dipole moments of molecule, the tensor coefficients of the electrical polarizability, the magnetic susceptibility and other physical characteristics of the molecule. 3 In the literature these operators are sometimes gien in the form ˆ d K g( x) bˆ ˆ c( x) (see [5]). dx The operators gien in this form can be represented in the form described by formula (5) by means of x the ariable substitution y ( g( z)) dz. x 0 5

6 Table. The form of the dependence y=y(r), the coefficients a, b and the wae function of the ground ibrational state 0 ( y ) for the harmonic oscillator and the Morse potential (according to data from [5, 9]); here q (8 D e ). Parameters Harmonic oscillator Potentials Morse potential U(, r ) [ ] 4 ( r r ) e De exp( ( r re)) exp( ( r re)) y=y(r) ( r r e) q exp( ( r re )) a ( q ) b ( q ) q + q 0 ( y ) 4exp( y ) ( q ) ( q ) y exp( y / ) 3. Diagram technique for determination of ibrational dependences of matrix elements Our goal is to express the matrix elements f ( y) ' on the wae functions of states with arbitrary alues of the ibrational quantum numbers and ' in terms of matrix elements on the wae functions of the ground ibrational state. Without loss of generality consider the matrix elements f ( y) m where m is an integer satisfying to the condition m ( max ); max is the maximum alue of the ibrational quantum number for bound ibrational states of the potentials haing the dissociation threshold, and max = for potentials are without such a threshold (for example, the harmonic oscillator). First, consider the case m 0. 6

7 According to the definition of the commutator the following relation taes place Kˆ f ( y) Kˆ f ( y) f ( y) K ˆ,. (6) where ˆ ˆ AB is the commutator of the operators  and ˆB. The expression for the commutator in the formula (6) has the form: ˆ df ( y) K f ( y) bˆ dy. Using this relation and Eqs. (), (3) and (6) and the mutually conjugation of the operators ˆK and ˆK we obtain () f ( y) m b m f ( y) m a m f ( y) m a. (7) Here and further it is used the following notation ( ) d f ( y ) f ( y ) dy. Note that Eq. (7) contains numerical coefficients corresponding operators. a, a m and b m instead It can be seen from Eq. (7) that the using creation and annihilation operators allows decreasing alue of the ibrational quantum number on for the wae function of one from the combining ibrational states. Applying formula (7) to the matrix elements to the right in this formula we decrease alue of the ibrational quantum number of the wae function of the ibrational state again on. Thus, the sequential application times of this formula leads decreasing from to 0 alue of the ibrational quantum number of the wae function of one from the combining ibrational states. This allows us to express the matrix elements f ( y) m as a linear sum of the matrix elements 0 ( ) f ( y ) (where =m, m+,..., (m+) and =0,,,..., ). Taing into account this we will be carried out the solution of our tas in two stages. At first, we will obtain a formula expressing the matrix elements f ( y) m in terms of the matrix elements 0 ( ) f ( y ). Then we will obtain 7

8 a formula expressing the matrix elements 0 ( ) f ( y ) in terms of the matrix elements ( ) ( 0 f y ) 0. To obtain the required expression and also illustrate the effect of creation and annihilation operators we use the following diagram technique. We construct a diagram with points (l, ) which conform to the matrix elements of the form ( l) l f ( y ) m (see Fig.). Value is plotted on the abscissa in descending order from to 0. Value l is plotted on the ordinate in ascending order from 0 up to. In accordance with Eq. (7) it is possible to go from each point (l, ) to two points (l, ) and (l, ) that corresponds to the first and second terms on the right in Eq. (7). The ertical arrow ("down") corresponds to the transition from point (l, ) to point (l, ) and the diagonal arrow ("down and right") corresponds to the transition from point (l, ) to point (l, ) (see the supplementary figure at the top right corner of Fig.). With regard to Eq. (7) these transitions corresponds to the multiplication on the coefficients b m a l and a m a l correspondingly. Let us introduce the definition of a path ector as a continuous line connecting the points of the diagram by a sequence of ertical and diagonal arrows. Then the result of repeated application of formula (7) is a set of path ectors. It is need to transit from points (, ) to the line l=0 (X-axis). This corresponds to a set of path ectors limited by the area enclosed by the triangle ABC whose ertices are points (, ), (0, ) and (0, 0) correspondingly (see Fig.). Thus, the required expression implies the summarizing of matrix elements through all these path ectors. This summarizing can be fulfilled as follows. Initially we fix point (0, j) (point D) on the X-axis (Fig.). This point corresponds to the matrix element ( j) 0 ( ) f y j m in the required expression. The set of all possible (in accordance with formula (7)) path ectors from point (, ) (point A) towards point (0, j) (point D) is limited by a parallelogram whose ertices are points A and D and points ( j, ) and (j, j) designated as points E and F accordingly. In Fig. this parallelogram AEDF is selected by the dot-dashed line. As an illustration it is shown one on Fig. a possible path ector AG G G D connecting point (, ) and point (0, j). It should be noted that as only the tip of the path ector falls on the line ED or FD then there is only one further route to get to the point D namely to go along one of these lines. 8

9 l (l, ) A G b m a l a m a l G l G l (l, ) (l, ) j G j F j E G j G G l=0 B D C = j =0 Fig. The diagram illustrating the sequential application of formula (7). Points (l, ) correspond to matrix elements ( l) l f ( y ) m. Value is plotted on the abscissa and alue l is plotted on the ordinate. The line AG G G l G j G j G G D corresponds to one of the possible path ectors appearing at the transformation the matrix element f ( y) m into the matrix elements ( j) 0 ( ) f y j m. A diagram illustrating a single action of formula (7) is in the upper right corner of the figure. The ertical arrow from point (l, ) to point (l, ) corresponds to the first term on the right in bracets in formula (7) and the diagonal arrow from point (l, ) to point (l, ) corresponds to the second term on the right in bracets in this formula. 9

10 Using this notation let us analyze the coefficients staying before the matrix elements in formula (7) into the required expression. (i) Any ector path that lies within parallelogram AEDF contains arrows in all that corresponds to the application times of formula (7). Therefore consideration the factors before bracets in formula (7) gies the factor ( A ( )) in the required expression where A ( ) a. (8) (ii) Since any path ector that lies within the AEDF area contains j diagonal arrows taing into account of the right term in bracets in formula (7) gies us the factor (for 0 j ): For j the coefficient A ( m, ) this case (see Fig.). Aj(, m) a. (9) m j since the path ector contains only ertical arrows in (iii) Tae into account the left term in bracets in formula (7) which is shown arrows "down" on Fig. Any path ector that lies within the AEDF area contains j ertical arrows. Note that the first arrow "down" on a ector path that lies in the AEDF area can hae any abscissa alue i from j to. The second arrow "down" on a path ector should not be to the left of the preious arrow "down" on this path ector. Therefore this second arrow "down" can hae any the abscissa alue i from j up to i. Continuing this reasoning further allows us to draw conclusion that the last j-th arrow "down" on the path ector can hae any abscissa alue i j from j up to i j. Thus, considering coefficients b times of the formula (7) gies the factor at the application i i i j B j(, m ) b i... m b i m b i 3 m b i j m. (0) i j i j i 3 j i j j The joint taing into account of the results (i) (iii) allows us to obtain the coefficient before the matrix element 0 ( j) f ( y ) j m in the required expression. This coefficient is equal to the factor A j (, m ) B j (, m ) A ( ). 0

11 To obtain an expression for the matrix element f ( y) m it is only need to sum oer j from 0 up to. At this summarizing point D in Fig. should run all the points lying on the X-axis from point C towards point B. Then we obtain the requared expression: f ( ) ( y ) m j (, ) (, ) 0 ( ) A( ) A j m B j m f y j m j 0. () At last we express the matrix elements to the right in Eq. () through the matrix elements of the wae functions of the ground ibrational leel. Using the Eqs. () (6) and mutually conjugation of the operators ˆK and ˆK we obtain j m ( j) ( b ( 0 ) j m 0 f y) j m 0 f ( y) 0 A( j m). () obtain Expressing the rightmost term in Eq. () with regard to formula () we finally ( ( )) (, ) 0 ( j m) f r y m j m f ( y ) 0, (3) j 0 here j m j (, m) ( b0 ) Aj(, m) B j(, m) A( j m) A( ). (4) So far we considered the case m 0. If m 0 it is need to replace by +m and m by m in the right-hand side of Eqs. (3) and (4). Denote m min (, m). Then for the general case of arbitrary alues m satisfying the condition 0 +m max Eq. (3) can be written as: m j m f ( r) m j( m, m ) 0 f ( y ) 0. (5) j 0 Eqs. (4) and (5) are the main result of the present paper. The expressions for particular cases of matrix elements can be deried as a consequence of these formulas. For example the formula for the diagonal matrix elements ( ' ) corresponding to the

12 expectation of a certain physical characteristics (described by the function f() r ) in the state with ibrational quantum number is obtained from Eqs. (4) and (5) by substituting m=0 and m. 4. Analysis It can be seen that the expression for the matrix elements f () r m (see Eq. (5)) is defined (see Eqs. (8) (0)) by: ) the coefficients a i (i=,,, m m ) and b i (i= m, m,, m m )) which, in turn, are determined by the explicit form of the annihilation operator (see Eqs. (3) and (4)) and the parameters of the potential cure (,,..., n ) (Eq. ()); ) the wae function of the ground ibrational state, 3) the p-th order deriaties of function f( y ) (p= ', ' +,, (+ )) (see Eq. (5)) that are determined by the analytic properties of this function. It is important to note that to calculate of matrix elements f ( r) ' for arbitrary alues and ' it is enough to hae information about analytic properties of the function f( y) only for those alue of r where the wae function of the ground ibrational state is not negligible. All necessary data for algebraic calculations of arious characteristics of the ibrational states of diatomic molecules by Eq. (5) for the harmonic oscillator and the Morse potential are gien in Table. These data were obtained using the data of [5, 9]. The coefficients j( m, m) can be easily calculated and tabulated for a range of the molecule parameters (,,..., n) of interest in spectroscopic studies if the explicit form of the operators ˆK and ˆK is nown. For further analysis expand function f( y ) in a Taylor series in the neighborhood of a point y (for example it can be the alue y corresponding to equilibrium internuclear 0 distance r e ): f ( y0) f ( y) ( y y0) 0!. (6)

13 Tae into account that j m j m f ( y ( 0) f y y y0 0! ) ( ). Substituting this expression into Eq. (5) we obtain ( ) (, ) 0 ( 0 ) f y m m m y y 0, (7) 0 here m ( j m ) ( m, m ) j( m, m ) f ( y0)! j 0. (8) It can seen that to calculate the matrix elements it is enough to hae information about: ) the coefficients a i and b i (including in factors j ( m, ) m (see Eq. (4)); ) the matrix elements only on the wae functions of the ground ibrational state (see Eq. (7)); 3) the p order deriaties of function f( y ) in a point y 0 (see Eq. (8)), where p m. Matrix elements 0 ( y y0) 0 can be determined analytically for the special cases of potentials. ( p) The alues of the deriaties f ( y 0 ) occurring in Eq. (5) can be found semi-empirically or using the results of ab initio calculations. It is important to note that each differentiation of the function is associated with appearance of some scale factor that depends on molecule parameters. This leads to a certain hierarchy of the factors ( m, ) m. Consideration of this hierarchy allows limiting the number of terms in Eq. (7) by seeral terms. As an example of efficiency of obtained algebraic expressions consider onedimensional harmonic oscillator and obtain a formula for the ibrational dependences of matrix elements for the case. Using Eqs. (8) (0), (4) and the data gien in Table we obtain: m j j m ( )! m i i i j m m ( )... ( j m )! m! i j i j i 3 j i j j. (9) 3

14 It is not difficult to show that the number of terms defined by the j-sums in Eq. (9) equals the binomial coefficient C j. This corresponds to the number of all possible different paths m ector from point A towards point D on Fig. Thus, we obtain the required formula m ( j m) f ( r) m m!( m m )! m 0 f ( r) 0 j 0 j j m j!( j m )!( m j)!. We get another form of this expression expanding f() r in a Taylor series with regard to Eq. (6). Taing into account that the function 0 ( y) of the harmonic oscillator is een (see Table) and replacing the integration oer the interal 0 r on the integration on the interal r (since the difference between these integrals is negligible for the real potentials of molecules) the following formula is finally obtained j m m f ( r ) f ( r) m!( )! e m m m 0 0 j m j j m ( j m )!( m j)! j!!. (0) In this it is used the following relation for the harmonic oscillator ( )! ( ( 0 )) y y dy!, here () z is the Gamma function. It should be noted that Eq. (0) for the case =, r e =0 and m 0 was obtained earlier by using hyperirial theorem and a technique of differentiation with respect to the parameter [] and the binomial formula for operators with Cauchy's integral theorem and Baer-Campbell-Hausdorff theorem [3]. It should be noted that there is a misprint in formula (4.3) from []: in this formula the ( j n m r) f (0) must be replaced by the ( j n m r) f (0). This misprint has been found in [3]. A wide range of physical and chemical problems (molecular plasma physics, plasma chemistry, etc.) requires information about the total set of matrix elements for all possible alues of ibrational quantum numbers of the combining states. An important consequence of the expressions obtained in present paper is the possibility obtaining these date calculating 4

15 only ( max ) matrix elements (see Eq. (5)) 4 instead the computation ( max ) ( max ) independent alues of matrix elements taing place at numerically salation of the Schrödinger equation. Thus, using the algebraic expressions obtained in present paper can reduce significantly (approximately max 4 times) the number of necessary calculations of matrix elements compared to the number of calculations using numerical methods. Een for comparatiely light diatomic molecules (except the hydrogen molecule and hydrides) using the obtained expressions allows reduce the number of necessary calculations more than one order of magnitude. 5. Conclusion The diagram technique describing the actions of the annihilation operator at the consideration of ibrational dependences of matrix elements has been deeloped. The algebraic formulas for the matrix elements with arbitrary alues of the ibrational quantum numbers of the combining states for arbitrary sufficiently differentiable functions of the internuclear distance are obtained in terms of matrix elements on the wae functions of the ground ibrational state. These formulas for the case of anharmonic potential cures were preiously absent in the literature nown to us. It is important to emphasize that proposed method for determining of ibrational dependences of matrix elements is a direct, simple and intuitie unlie the techniques deeloped earlier and does not require using any special theories of quantum mechanics (hyperirial, Hellmann-Feynman and others) and hypergeometric functions. The only required information reduces to nowing the explicit form of the annihilation operator and the wae function of the ground ibrational state of the molecule. All necessary data for such algebraic calculations of the ibrational dependencies of matrix elements for the harmonic oscillator and the Morse potential are gien. The efficiency of the diagram technique method is illustrated by the example of matrix elements for the harmonic oscillator. Obtained formulas can be used to determine the absolute alues and the ibrational dependence of arious radiatie, electrical, magnetic and other characteristics of the ibrational states of diatomic molecules as well as at the study of arious spectroscopic characteristics of polyatomic molecules, for example, using the local mode model [3]. These formulas gie the opportunity for obtaining analytical expressions for the ibrational 4 This result has been pointed out by Prof. V. V. Smirno. 5

16 dependence of matrix elements on the functions that hae more complex form than those studied preiously, for example, the functions haing the form of Padé approximation. Functions of this type are widely used to describe the dipole moments of the hydrogen halides and other diatomic molecules (see [3] and references therein). It should be noted that obtained formulas can be used to analyze the ibrational dependences of arious characteristics not only for the ground electronic states but also the electronically excited states of molecules (including the branching ratios of spontaneous emission, radiatie lifetimes, -doubling, g-factors, etc.) for example with the help of the sum rule formulas [33]. Anowlegement The author is grateful to Prof. V. V. Smirno for his aluable comments and obserations (see footnote 4). References [] H. Ishiawa, J. Phys. A. 35, 4453 (00). [] R. J. Le Roy, LEVEL 8.0: A Computer Program for Soling the Radial Schrödinger Equation for Bound and Quasibound Leels, Uniersity of Waterloo Chemical Physics Research Report CP-663 (007) ( [3] A. Fran, P. an Isacer, Algebraic Methods in Molecular and Nuclear Structure Physics. (Wiley, New Yor, 994). [4] F. Iachello, R. D. Leine, Algebraic Theory of Molecules. (Uniersity Press, New Yor, Oxford, 995). [5] S.-H. Dong, Factorization Methods in Quantum Mechanics. (Springer, New Yor, 007). [6] L. Infeld, T. E. Hull, Re. Mod. Phys. 3, (95). [7] F. Iachello, S. Oss, J. Chem. Phys. 04, 6956 (996). [8] F. Cooper, A. Khare, U. Suhatme, Phys. Rep. 5, 67 (995). [9] S. A. Astasheich, B. P. Laro, A. V. Modin, and I. S. Umrihin, Rus. J. Phys. Chem. B.. 6 (008). [0] S. A. Astasheich, Opt. Spectrosc. 0, 75 (007). [] R. M. Wilcox, J. Chem. Phys. 45, 33 (966). [] J. Morales, J. Lopez-Bonilla, A. Palma, J. Math. Phys. 8, 03 (987). [3] J. Morales, A. Flores-Rieros, J. Math. Phys. 30, 393 (989). 6

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