Connection of the vibron model with the modified Pöschl–Teller potential in configuration space

Chemical Physics 283 (2002) 401–417 www.elsevier.com/locate/chemphys

Connection of the vibron model with the modified €schl–Teller potential in configuration space Po R. Lemus *, R. Bernal Instituto de Ciencias Nucleares, UNAM, A.P. 70-543, Circuito Exterior, C.U., 04510 Mexico, D.F., Mexico Received 18 February 2002

Abstract The modified P€ oschl–Teller (MPT) potential is analyzed by a realization of its dynamical group in configuration space. The expansions of the coordinate and momentum in terms of the creation and annihilation operators are obtained from this realization. A system of two MPT oscillators interacting via a potential coupling bilinear in the coordinate x is studied. This analysis allows to establish an exact quantum-mechanical connection between the suð2Þ vibron model and the traditional description of molecular vibrations. It is shown that the suð2Þ vibron model corresponds to taking the dominant Dv ¼ 1 interaction in an approximate fashion. The energy spectrum for the system of coupled oscillators is compared with both the suð2Þ vibron model and the more general approach of considering higher-order terms of the expansions in the algebraic space. In addition, a two oscillator system with an interacting potential in terms of the variables ui ¼ tanhðaxi Þ is analyzed. Comparison of the spectrum generated by this system with previous results in the space ðp; xÞ suggests that u is the natural variable to describe the P€ oschl–Teller systems. Ó 2002 Elsevier Science B.V. All rights reserved.

1. Introduction A considerable number of applications regarding the vibron model have been published since its appearance in the early eighties [1]. The vibron model, which was originated from the IBM in nuclear physics [2], is characterized by its algebraic structure and does not involve an explicit correspondence with the traditional rovibrational description in configuration space. The vibron model was first proposed to describe the rovibrational degrees of freedom of diatomic molecules [3], but

*

Corresponding author. Fax: +52-5-616-2233. E-mail address: [email protected] (R. Lemus).

later on was generalized to describe linear polyatomic molecules [4]. Due to its complexity, however, for arbitrary polyatomic molecules a simplified version was required. The one-dimensional version of the model, known as the su(2) vibron model, was proposed to describe the vibrational excitations of polyatomic molecules in a local mode scheme [5,6]. In the framework of this approach the stretching degrees of freedom are represented by Morse oscillators, while the bending modes may be associated either to a Morse or P€ oschl–Teller potentials, as long as isomerization and tunneling effects are not present. A relation between the SU(2) group and the Morse and P€ oschl–Teller systems can be directly established by means of certain coordinate trans-

0301-0104/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 ( 0 2 ) 0 0 6 3 0 - 4

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formations applied to the radial equation of a twodimensional harmonic oscillator [7]. This approach, however, only provides the action of the ladder operators on the corresponding functions, but no direct connection with the configuration space is given. The geometric connection can be established in the classical limit either by means of coherent states or intensive boson operators [1,5]. In the framework of this approach any algebraic Hamiltonian can be transformed into a classical one in terms of a set of coordinates, to which a transformation must be applied in order to identify the physical coordinates. Although the classical limit provides a systematic procedure to identify each term of the Hamiltonian, it is important to establish the quantum mechanical correspondence in order to obtain in univocal form the connection with configuration space. Concerning the Morse potential, this correspondence has been recently obtained by establishing the relation between the matrix elements of two interacting Morse oscillators and the corresponding su(2) matrix elements [8]. The su(2) vibron model was shown to correspond to an approximation where only the dominant Dv ¼ 1 interaction between Morse oscillators is taken into account. This analysis gave rise to a procedure to extend the algebraic model in order to systematically approach the Morse oscillators results [9]. This procedure is based on the expansion of the coordinate x and momentum p^ in terms of the su(2) generators; an expansion that was obtained in comparison with the matrix elements in both spaces. An alternative approach consists in establishing the creation and annihilation operators in terms of the physical space by means of the factorization method, which allows to study the relation between the ladder operators and the underlying dynamical group [10,11]. Formally the inversion of these operators leads to the expansion of x and p^, but in practice it is more convenient to follow the approach in [9]. For the stretching degrees of freedom, however, the appropriate variable to describe large oscillations is the Morse variable y ¼ 1  ebx , whose expansion in terms of creation (annihilation) operators has proved to lead to a much faster convergence [12]. In this work we shall present a detailed analysis of the MPT potential, in similar fashion to our

previous study of the Morse potential [8,9,12]. The Morse and MPT potentials are isospectral and present the same dynamical group SUð2Þ for the bound states, which is the reason why in the vibron model both have been contemplated. We obtain the expansions of x and p^ in terms of the creation and annihilation operators of the MPT. This task is achieved by inverting the expressions of the ladder operators, which have been previously obtained from the factorization method [13]. Based on these expansions we establish the approximation involved in the vibron model as well as the procedure to successively improve the description in the su(2) space. The description of the MPT oscillators in terms of the variable u ¼ tanhðaxÞ is also investigated. We show that the expansion of the interacting MPT potential in terms of u provides a better description, similar to the one played by the Morse variable y for the interacting Morse oscillator system. The present study allows to deal with both stretching and bending interactions simultaneously in the algebraic space of the dynamical group, in such a way that the correspondence with the traditional description is maintained. This paper is organized as follows. The wave functions for the MPT potential expressed by the Gegenbauer polynomials, as well as the suð2Þ creation and annihilation operators are presented in Section 2. In Section 3 the expansions of the coordinate x^ and momentum p^ from the su(2) generators are obtained. In Section 4 the vibron model is summarized, emphasizing the case of two interacting oscillator system. Section 5 is devoted to establishing the connection of the vibron model with the traditional approach in configuration space through the analysis of a system of two coupled MPT oscillators. In Section 6 the same analysis in terms of the variable u is also investigated. Finally, in Section 7 we present our summary and conclusions.

2. The modified P€oschl–Teller wave functions We start by presenting the eigenfunctions for the MPT problem. The MPT potential can be written as [14,15]

R. Lemus, R. Bernal / Chemical Physics 283 (2002) 401–417

V ðxÞ ¼ 

D ; cosh2 ðaxÞ

ð1Þ

where D is the depth of the well and a is related to the range of the potential, while x gives the relative distance from the equilibrium position. The Schr€ odinger equation associated to this potential is given by ! d2 Wqn ðxÞ 2l D þ 2 Eþ ð2Þ Wqn ðxÞ ¼ 0; dx2 h  cosh2 ðaxÞ where l is the reduced mass of the molecule and q is related with the depth of the potential as will be shown below. We now introduce the following definitions in accordance with [16]: rffiffiffiffiffiffiffiffiffiffiffiffi 2lE 2lD ¼ ; qðq þ 1Þ ¼ 2 2 ; 2 2 h h a a ð3Þ 1 q ¼ ð1 þ 2sÞ 2 and rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2lD þ s¼ ; 4 a2  h2

m ¼ 2s ¼ 2q þ 1

ð4Þ

with the constraint   q ¼ n, where n ¼ 0; 1; 2; . . . ; nmax . The quantum number m is introduced because it will be helpful for the identification of the ladder operators with the su(2) algebra. In terms of the variable u ¼ tanhðaxÞ, the solutions of equation (2) are given by =2

Wqn ðuÞ ¼ Nnq ð1  u2 Þ Cnqþð1=2Þn ðuÞ with the normalization constant sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi an!ðq  n  12Þ!ð2q  2nÞ! Nnq ¼ : p1=2 ðq  n  1Þ!ð2q  nÞ!

ð5Þ

ð6Þ

The eigenvalues can be determined by the condition q   ¼ n and expressed as En ¼ 

~ x h 2 ðq  nÞ ; m

ð7Þ

a2 h m: 2l

note that for integer q the state associated to null energy is not normalizable. In this case the last bound state corresponds to q  n ¼ 1, which implies that nmax ¼ q  1 ¼ ðm  3Þ=2. With the factorization method it is possible to establish the creation and annihilation operators for the functions (5) [13]. For the annihilation operators we find rffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi d u þ1 2 ^  b¼ 1u þ ; ð9Þ 2 du 1  u m where we remark that 2 ¼ m  2n  1 ¼ 2q  2n. The action of the operator (9) on the wave functions is given by b^Wqn ðuÞ ¼ k Wqn1 ðuÞ;

ð10Þ

where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ¼ nð1  n=mÞ:

ð11Þ

As we can see, this operator annihilates the ground state Wq0 ðuÞ, as expected for a lowering operator. For the creation operator we have rffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi d u 1 y 2 þ  b^ ¼ 1  u  ð12Þ 2 du 1  u m with the following effect on the wave functions b^y Wqn ðuÞ ¼ kþ Wqnþ1 ðuÞ;

ð13Þ

where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð14Þ kþ ¼ ðn þ 1Þð1  ðn þ 1Þ=mÞ: y Since b^ is a raising operator, it is expected that it annihilates the last bound state. Indeed, for this state  ¼ 1 and the square root in (12) causes the operator to vanish. The operators fby ; bg, together with the number operators n^, satisfy the commutation relations 2^ nþ1 ; ½b^; b^y ¼ 1  m

½^ n; b^y ¼ b^y ;

½^ n; b^ ¼ b^y ; ð15Þ

where ~¼ x

403

ð8Þ

The number of bound states is determined by the dissociation limit  ¼ q  n ¼ 0. We should

which can be identifed with the usual su(2) commutation relations by p introducing ffiffiffi pthe ffiffiffi set of transformations fby ¼ J^ = m; b ¼ J^þ = m; n^ ¼ ðm  1Þ=2  J^0 g, where Jl satisfy the usual ‘‘angular momentum’’ commutation relations [17]. In

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terms of these operators, the MPT Hamiltonian acquires the simple form

~ y hx  b b þ bby ; ð16Þ H^ ¼ 2

for large D. In this limit the operators b^y and b^ are reduced to the harmonic operators a^y and a^ satisfying the bosonic commutation relations

whose eigenvalues, when compared with the MPT energies

as expected.

EM ðvÞ ¼ xe ðv þ 1=2Þ  xe xe ðv þ 1=2Þ

2

ð17Þ

lead to the identification ~ x h : ð18Þ m In the space generated by these operators, the wave functions acquire a polynomial form ~; xe ¼  hx

xe x e ¼

n

Wqn ðuÞ ¼ Nmn ðb^y Þ Wq0 ðuÞ;

ð19Þ

where the normalization constant is obtained through the commutation relations (15), and turns out to be sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mn ðm  n  1Þ! m Nn ¼ ; ð20Þ n!ðm  1Þ! q=2

and Wq0 ðuÞ ¼ N0q ð1  u2 Þ is the ground state. The suð2Þ label j for the irreducible representations is related with m in the following way: j¼

m1 : 2

ð21Þ

The harmonic limit is recovered by taking a ! 0 and D ! 1, but keeping the product k ¼ 2a2 D finite, so that sffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffi lx h d  y ¼ a^y ; lim b^ ¼ ð22aÞ x m!1 2 h 2lx dx sffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffi lx h d  ¼ a^ lim b^ ¼ xþ m!1 2 h 2lx dx

ð22bÞ

with sffiffiffi k x¼ ; l

ð23Þ

where we have made use of the approximation rffiffiffiffiffiffiffiffiffi 2lD m=2 ¼ s ’ h2 a2 

½^ a; a^y ¼ 1;

½^ a; a^ ¼ ½^ ay ; a^y ¼ 0;

ð24Þ

3. Expansion of x and p in terms of the su(2) generators In this section we derive the expansions of the coordinate x and momentum p^ in terms of the operators (9) and (12). This task can in principle be achieved by two methods, either by a comparison of the matrix elements, or by means of the inversion of the ladder operators. The first alternative was followed for the Morse system [9]. In this case, however, the matrix elements for the coordinate and momentum are not known in analytic simple form and, consequently, we have chosen the second approach. Extracting the coordinate x and the derivative ðd=dxÞ from (9) and (12), we obtain 1 sinhðaxÞ ¼ pffiffiffi ðb^y fn þ b^gn Þ  y^; 2 m

ð25Þ

pffiffiffi d a m ¼ ðb^qn  b^y hn Þ; dx 2 coshðaxÞ

ð26Þ

where for convenience we have defined the operator y^, as well as the following diagonal operators: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 m2 ; gn ¼ ; ð27aÞ fn ¼ ð  1Þ ð þ 1Þ rffiffiffiffiffiffiffiffiffiffiffi  hn ¼ ; 1

rffiffiffiffiffiffiffiffiffiffiffi  : qn ¼ þ1

ð27bÞ

We recall that the n-dependence of these variables is given through 2 ¼ m  2n  1. In the harmonic limit the set of functions (27) acquire the following values: lim fn ¼ lim gn ¼ 2;

m!1

m!1

lim hn ¼ lim qn ¼ 1

m!1

m!1

ð28Þ

and consequently lim y^ ¼ 0:

m!1

ð29Þ

R. Lemus, R. Bernal / Chemical Physics 283 (2002) 401–417

We shall first proceed to obtain the expansion of x. As suggested by the result (29) we expand the function arcsinhð^ y Þ about zero arcsinhð^ y Þ ¼ y^ 

y5 y^3 3^ þ þ : 6 40

ð30Þ

This expression, together with (25), leads to a 1=m expansion for x: sffiffiffiffiffiffiffiffiffi 1 h  1 1 ^y 3 x¼ ðb^y fn þ b^gn Þ  ðb fn þ b^gn Þ ~ 2 2lx 24 m 3 1 ^y ^gn Þ5    ; þ ð b f þ b ð31Þ n 640 m2 where the following relation sffiffiffiffiffiffiffiffiffi 1 1 h  pffiffiffi ¼ ~ 2a m 2 2lx

ð32Þ

was taken into account. The powers involved in (31) can be expanded and rearranged by commuting the diagonal operators. Considering up to the second term in (31) we obtain sffiffiffiffiffiffiffiffiffi 1 h ^y  1 1 ^y ^y ^y x¼ ðb b b F1;n þ b^y b^y b^F2;n b fn  ~ 2 2lx 24 m þ b^y b^b^y F3;n þ b^b^y b^y F4;n Þ þ    þ hc; ð33Þ where hc refers to the hermitian conjugate contribution and F1;n ¼ fnþ2 fnþ1 fn ;

ð34aÞ

F2;n ¼ fn fn1 gn ;

ð34bÞ

F3;n ¼ fn fn gnþ1 ;

ð34cÞ

F4;n ¼ fn fnþ1 gnþ2 :

ð34dÞ

The commutation relations (15) allow to further simplify the expression (33) to sffiffiffiffiffiffiffiffiffi 1 h ^y 1 1 ^y ^y ^y b Q0;n  ðb b b F1;n x¼ ~ 2 2lx 24 m y ^y ^ ^ þ b b bQ1;n Þ þ    þ hc; ð35Þ

405

where

  1 2n þ 1 1 F3;n Q0;n ¼ fn  24m m   2n þ 2 þ2 1 F4;n ; m Q1;n ¼ F2;n þ F3;n þ F4;n :

ð36aÞ ð36bÞ

The expansion (35) provides successive approximations for x in the su(2) space. The simplest approximation consists in taking the first term but considering the harmonic limit (28) for the diagonal contributions sffiffiffiffiffiffiffiffiffi h ^y ^ ðb þ bÞ: x ð37Þ ~ 2lx This approximation will be referred to as the vibron approximation, a name that will be justified in Section 5. To improve the expansion we can consider the full linear term sffiffiffiffiffiffiffiffiffi 1 h ^y ðb fn þ b^gn Þ x ð38Þ ~ 2 2lx or the extended approximation given by the expansion (35) up to cubic terms. Although the power expansion in 1=m suggests a fast convergence, the effect of the diagonal contributions could be significant. In order to clarify this point we shall analyze an oscillator with the following set of parameters ~ ¼ 1000 cm1 ; hx pffiffiffiffiffiffiffiffiffiffiffi 2  : l ¼ 1 amu; a ¼ 2:578 A m ¼ 23;

ð39Þ

The maximum number of quanta for this oscillator is nmax ¼ ðm  3Þ=2 ¼ 10. The small parameter m ¼ 23 was chosen in order to emphasize the anharmonic effects. In Table 1 we present a comparison of the exact matrix elements hWqn jxjWqnþ1 i with those obtained considering the different approximations. We have named the approximations (37), (38) and (35) as vibron, linear and extended approximations, respectively. We note from the table that, surprisingly, the vibron approximation provides the best overal description for the whole range of states. This behavior is illustrated in Fig. 1. Even though both linear and extended

406

R. Lemus, R. Bernal / Chemical Physics 283 (2002) 401–417

Table 1 Comparison of the exact matrix elements hWqn jxjWqnþ1 i with the vibron, linear and extended approximations n

Exact

Vibron

Linear

Extended

0 1 2 3 4 5 6 7 8 9

0.13582 0.20173 0.26066 0.31931 0.38132 0.45006 0.52975 0.62681 0.75226 0.92128

0.12701 0.17549 0.29075 0.236068 0.25689 0.27348 0.28657 0.29663 0.30396 0.30874

0.13926 0.21273 0.28427 0.36277 0.45585 0.57420 0.73692 0.98475 1.42705 2.51064

0.13539 0.19943 0.25280 0.297277 0.32479 0.307611 0.153526 )0.516997 )4.28114 )6.48583 Fig. 2. Diagonal operators fn ; gn ; sn and rn as a function of the quantum number n. Table 2 Comparison betwen the exact matrix elements hWqn jxjWqnþ3 i and the results provided by the extended approximation (35)

Fig. 1. Deviations Dx  jhWqn jxjWqnþ1 iexact  hWqn jxjWqnþ1 iapprox j as a function of the quantum number n for the vibron (j), linear (d) and extended (N) approximations.

approximations improve the description for low number of quanta, a considerable deviation appears for the high energy region. The slow convergence is explained by the behavior of the functions fn and gn , which is displayed in Fig. 2. An important deviation from the harmonic limit value of 2 is manifest even for small quantum numbers. The large values of these functions for high number of quanta enhance the differences derived from the various contributions to the matrix elements in the expansion (31). A similar situation occurs for the matrix elements hWqn jxjWqnþ3 i, where important deviations ocurr for the highest number of quanta, as seen in Table 2. We next proceed to obtain the expansion for the momentum operator. From the identity cosh2 ðaxÞ  sinh2 ðaxÞ ¼ 1

ð40Þ

n

Exact

Extended

0 1 2 3 4 5 6 7

)0.002815 )0.006306 )0.0112607 )0.018153 )0.027658 )0.04070 )0.05825 )0.0783198

)0.0036186 )0.0094263 )0.020199 )0.040800 )0.08287 )0.17904 )0.44496 )1.51595

we obtain 1 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi coshðaxÞ 1 þ y^2

ð41Þ

whose expansion about y^ ¼ 0 is given by sechðaxÞ ¼ 1 

y^2 3 4 þ y^ þ    : 2 8

ð42Þ

The substitution of this expression into (26) leads to the expansion for the momentum " # i pffiffiffiffiffiffiffiffiffiffiffi y^2 3 4 ~ 1  þ y^ þ    ðb^y hn  b^qn Þ: 2hlx p^ ¼ 2 2 8 ð43Þ Because of the form of the operator y^, this expression is given in terms of powers of the parameter 1=m, similarly to the case of x. Hence we could attempt to carry out successive approximations. In this case, however, this approach cannot be applied in a straightforward way since the operator p^ is not

R. Lemus, R. Bernal / Chemical Physics 283 (2002) 401–417

Hermitian when the series is cut. If we still decide to continue in this way it is necessary to consider the operator p^ ¼ ð^ py þ p^Þ=2 before taking the approximation. Proceeding in this manner we obtain  i pffiffiffiffiffiffiffiffiffiffiffi ^y 1 ~ b sn þ ðb^y b^y b^y G1;n þ b^y b^y b^G2;n 2 hlx p^ ¼ 2 8m y ^^y y ^y ^ ^ ^ þ b bb G3;n þ bb b G4;n Þ þ    þ hc ð44Þ with the following definition for sn sn ¼ ðhn þ qnþ1 Þ=2;

ð45Þ

while for the G’s G1;n ¼ ðfnþ2 fnþ1 hn þ qnþ3 gnþ2 gnþ1 Þ=2;

ð46aÞ

G2;n ¼ ðfn fn1 qn þ qnþ1 gn fn1 Þ=2;

ð46bÞ

G3;n ¼ ðfn gnþ1 hn þ qnþ1 fn gnþ1 Þ=2;

ð46cÞ

G4;n ¼ ðgnþ2 fnþ1 hn  hnþ1 gnþ2 gnþ1 Þ=2:

ð46dÞ

Again the expansion (44) can be simplified to  i pffiffiffiffiffiffiffiffiffiffiffi ^y 1 1 ^y ^y ^y ~ b W0;n þ 2 hlx ðb b b G1;n p^ ¼ 2 8m þ b^y b^y b^W1;n Þ þ    þ hc; ð47Þ where W0;n

  1 2n þ 1 1 ¼ sn þ G3;n 8m m   2n þ 2 þ2 1 G4;n ; m

W1;n ¼ G2;n þ G3;n þ G4;n :

ð48aÞ ð48bÞ

Note that because of the symmetry of the MPT potential the expansions of x and p^ involve only odd powers of the operators b^y ðb^Þ, since x and p^ are odd functions under inversion. It is worth noting that in the harmonic limit the expressions (35) and (47) reduces to sffiffiffiffiffiffiffiffiffi 1 h  ð^ ay þ a^Þ; lim x ¼ ð49aÞ m!1 2 2lx lim p^ ¼

m!1

i pffiffiffiffiffiffiffiffiffiffiffi y 2 hlxð^ a  a^Þ 2

ð49bÞ

as expected, with the harmonic frequency (23).

407

As for the position matrix elements, several approximations can be proposed for the momentum. In the vibron approximation we have i pffiffiffiffiffiffiffiffiffiffiffi ^y ^ ~ ðb  bÞ; 2hlx ð50Þ p 2 while for the full linear term i pffiffiffiffiffiffiffiffiffiffiffi ^y ~ ðb sn  b^rn Þ; 2hlx p ð51Þ 2 where rn ¼ ðqn þ hn1 Þ=2:

ð52Þ

Finally, the third possibility corresponds to the extended approximation given by the expansion (47) up to cubic terms. In Table 3 we present a comparison of the exact values for the matrix elements     d i q hWn j^ pjWqnþ1 i ¼ Wqn  Wqnþ1 h dx with the results provided by the different approximations. Again, the vibron approximation provides a fairly good description for the whole range of quantum numbers. Similar results are obtained in the linear approximation. In the extended approximation the matrix elements become closer to the exact values for small quanta, while for high number of quanta the deviation is considerable. This fact can be appreciated graphically in Fig. 3. In Table 4 we include the comparison of the matrix elements ði=hÞhWqn j^ pjWqnþ3 i provided by the extended approximation. As expected, important differences are found for a large number of quanta. In Fig. 1 the functions hn and qn are also displayed. Table 3 Comparison of the exact matrix elements ði=hÞhWqn j^ pjWqnþ1 i with the vibron, linear and extended approximations n

Exact

Vibron

Linear

Extended

1 2 3 4 5 6 7 8 9 10

3.6767 4.94067 5.71199 6.17392 6.38991 6.38149 6.14566 5.65519 4.84834 3.5626

3.7655 5.20279 6.21852 6.99871 7.6161 8.10796 8.49612 8.79432 9.01149 9.15341

3.76978 5.21001 6.22931 7.01432 7.63873 8.14167 8.54905 8.88545 9.19732 9.70866

3.8744 5.5356 6.9189 8.2807 9.8348 11.9217 15.3171 22.4358 45.9866 80.09657

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R. Lemus, R. Bernal / Chemical Physics 283 (2002) 401–417

ters when we deal with a system of interacting MPT oscillators. Before analyzing this system we shall present briefly the basic ideas behind the suð2Þ vibron model. This will allow us to discuss its connection with the configuration space in a selfcontained presentation.

4. The su(2) vibron model

Fig. 3. Deviations Dp  ði=hÞjhWqn jpjWqnþ1 iexact hWqn jpjWqnþ1 i approx j as a function of the quantum number n for the vibron (j), linear (d) and extended (N) approximations.

Table 4 pjWqnþ3 i Comparison betwen the exact matrix elements ði=hÞhWqn j^ and the results provided by the extended approximation (35) n

Exact

Extended

3 4 5 6 7 8 9 10

)0.20687 )0.41459 )0.653179 )0.91257 )1.1765 )1.4167 )1.5769 )1.5143

)0.2658 )0.61967 )1.17166 )2.0510 )3.525 )6.2311 )12.044 )29.3108

We note that these functions do not differ appreciable from the harmonic value, the slow convergence of the expansion (47) is due to the behavior of the functions fn and gn . From this analysis, we conclude that due to the poor quality of the approximation for high number of quanta the expansions (35) and (47) are not suited to describe the matrix elements in the whole range of quantum numbers. We expect that this behavior will be reflected in the energy calculation, as will be shown in Section 5. A surprising result is that the simplest approximation gives rise in average to the best agreement with the exact results for the matrix elements. In general the convergence of x and p^ will depend not only on the numbers of terms included in 1=m expansion, but also on the coupling parame-

The SU(2) approach to the description of the molecular vibrational degrees of freedom is based on the isomorphism between the (one-dimensional) P€ oschl–Teller or Morse eigenfunctions and states associated to the SUð2Þ  SOð2Þ chain. This correspondence can be established through the SU(2) generators and a realization from a two-dimensional harmonic oscillator [7]. Because of its connection with the Morse potential this model was originally proposed to describe the stretching degrees of freedom, but later on was extended to include the bending modes, whose associated potential was identified with the P€ oschl–Teller potential [6]. In the latter identification, however, one should be careful since the MPT potential is symmetric and includes a dissociation limit. Because of these properties only the symmetric local modes may be represented, with the proviso that the depth of the potential should be high enough to exceed the dissociation limit of the stretching modes. It should also be clear that this potential can be used to approximate anharmonic effects outside the range of the isomerization region. For asymmetric bending potentials it may still be possible to utilize the vibron model by associating either a Morse potential or a double Morse potential, the latter in case tunneling effects are present [18]. In this section we briefly describe the general procedure which was originally proposed to apply the vibron model to a molecular system. In particular we concentrate on the MPT potential. A detailed description of this model can be found in [1,19]. The generators of the U(2) group can be realized in the form G ¼ fN^ ; J^þ ; J^ ; J^0 g, which satisfy the commutation relations [17] ½J^0 ; J^ ¼ J^ ;

½J^þ ; J^ ¼ 2J^0 ;

½N^ ; J^l ¼ 0; ð53Þ

R. Lemus, R. Bernal / Chemical Physics 283 (2002) 401–417

where l ¼ 0; . Here Jl satisfy the usual ‘‘angular momentum’’ commutation relations of SU(2) and N^ is the number operator related to J^2 through 1 ð54Þ J^2 ¼ N^ ðN^ þ 2Þ; 4 which implies that j ¼ N =2. A realization of the Morse Hamiltonian in terms of the su(2) algebra is given by ^ ¼ hx0 ðJ^2  J^2 Þ: H 0 N The Hamiltonian (55) SUð2Þ  SOð2Þ basis SUð2Þ # jj

 ;

ð55Þ is

diagonal

SOð2Þ # mi;

in

the

ð56Þ

where j and m are the quantum numbers that characterize the eigenvalues of the operators J^2 and J^0 , respectively. In order to see the association of the eigenvalues of (55) with the ones of the P€ oschl–Teller oscillator we introduce the quantum number [1,7] v ¼ j  m ¼ 0; 1; 2; . . . ; ½N =2 ;

ð57Þ

which corresponds to the number of quanta in the oscillator. In terms of the quantum numbers v and N, the eigenstates (56) can be expressed as !v rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðN  vÞ! J^ pffiffiffiffi j½N ; 0i: ð58Þ j½N ; v P N !v! N The eigenvalues of the Hamiltonian (55) in the basis (58) are simply obtained from (57) x0  h hx0 ðN þ 1Þ ðv þ 1=2Þ þ N 4N hx0 2  ðv þ 1=2Þ ; N

J^þ pffiffiffiffi j½N ; vi ¼ N

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi 1v j½N ; v  1i; v 1þ N

J^ pffiffiffiffi j½N ; vi ¼ N

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi  v j½N ; v þ 1i: ðv þ 1Þ 1  N

The harmonic limit is obtained by taking the limit N ! 1: J^þ lim pffiffiffiffi ¼ a; N

ð62aÞ

J^ lim pffiffiffiffi ¼ ay : N !1 N

ð62bÞ

N !1

In the vibron model the number of bosons N arises as a natural parameter connected with the depth of the potential. On the other hand, the variable m introduced in Section 2 is a measure of the same property, and consequently the limits (22) and (62) are equivalent. Since the parameters N and m are related through the representation label j: N ¼m1

xe x e ¼

x 0 h : N

ð63Þ

the normalization of the ladder operators in (61) could have been carried out with m instead of N. The choice of N in (61), however, is not consistent with Eqs. (12) and (14). It is thus convenient to introduce the definitions J^ J^þ ðm  1Þ ^  J0 by ¼ pffiffiffi ; b ¼ pffiffiffi ; v^ ¼ ð64Þ 2 m m in terms of m. The action on the SOð2Þ basis now takes the form pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b^y j½N ; vi ¼ ðv þ 1Þð1  ðv þ 1Þ=mÞj½N ; v þ 1i; ð65aÞ

ð59Þ

which when compared with the MPT eigenvalues (17) leads to the identifications ðN þ 1Þ ; N

ð61aÞ

ð61bÞ

EðvÞ ¼ 

xe ¼  hx ¼  hx 0

409

ð60Þ

We nowpturn ffiffiffiffi to the action of the rescaled operators J^ = N on the states (58):

b^j½N ; vi ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vð1  v=mÞj½N ; v  1i;

ð65bÞ

which is consistent with (11) and (14). With the new normalization the commutation relations acquire the same form as (15) and the Hamiltonian takes the form (16) with the same identification (18). In the same way the eigenstates (58) are v

j½N ; vi ¼ Nmn ðb^y Þ j½N ; 0i with the same normalization constant (20).

ð66Þ

410

R. Lemus, R. Bernal / Chemical Physics 283 (2002) 401–417

In the framework of the vibron model we interpret (62) in the opposite sense, i.e. as a way to construct the anharmonic representation of harmonic operators [19]. Any function of a; ay can be mapped into the same function of J^þ ; J^ through the correspondence J^þ a ! b ¼ pffiffiffi ; m

J^ ay ! by ¼ pffiffiffi : m

ð67Þ

This interpretation suggests a procedure to ‘‘anharmonize’’ the vibrational description of molecular system. We may start from a well-established Hamiltonian expanded in terms of bosonic local operators ayi ðai Þ and then carry out the substitution (67). The new Hamiltonian will contain the physical interactions which are known to be relevant, but it must now be diagonalized in a space associated to the local basis (66). The description is then given in terms of the operators byi ðbi Þ, which take into account the anharmonicity of the local oscillators. Let us study two interacting oscillators in the framework of the suð2Þ vibron model. Considering the simplest interaction conserving the total number of quanta, we have the Hamiltonian 2 h X  ~ ðb^y1 b^2 þ b^1 b^y2 Þ; ~ Hsuð2Þ ¼ x ðb^y b^i þ b^i b^yi Þ þ khx 2 i¼1 i

ð68Þ where k is the interaction parameter. The first term corresponds to the independent oscillators which we identify as MPT potentials with eigenvalues (17). The interaction term suð2Þ

~ ðb^y1 b^2 þ b^1 b^y2 Þ hx Hint: ¼ k

ð69Þ

gives rise to the matrix elements suð2Þ

hN ; v1 þ 1; N ; v2  1jHint: jNv1 ; Nv2 i sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðv1 þ 1Þ h v2 i ~ v2 ðv1 þ 1Þ ; 1 1 ¼ k hx m m ð70Þ where vi stands for the number of quanta of the ith oscillator. In the suð2Þ vibron model the local functions are denoted by jN ; vi i. We note that corrections of order 1=m appear. When m ! 1 the matrix elements (70) reduce to

suð2Þ lim hv1 þ 1; v2  1jH^int: jv1 ; v2 i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ khx v2 ðv1 þ 1Þ

N !1

ð71Þ

as expected, since in this limit all the harmonic results are recovered. In order to interpret the correction appearing in (70), we shall analyze the case of two interacting MPT oscillators from the point of view of the dynamical group identified in Section 4.

5. Connection of the vibron model to configuration space In Section 2 we showed that the su(2) algebra can be identified as the dynamical group for the MPT potential; the wave functions can be expressed in terms of the repeated action of the creation operator b^y on the ground state. On the other hand, in the framework of the vibron model the connection of the MPT potential with the SU(2) group was established through a twodimensional harmonic oscillator. The approach followed in the vibron model consists in expanding the Hamiltonian in terms of su(2) generators associated to local oscillators (internal coordinates). It is not obvious, however, what is the relation of these generators to the creation and annihilation operators presented in Section 2. The aim of this section is to establish this connection through the analysis of two interacting MPT potentials. A simple Hamiltonian for two identical, interacting MPT oscillators in configuration space is given by ! 2 X 1 2 1 H¼ p^  D 2l i cosh2 ðaxi Þ i¼1 ! p^1 p^2 ~ 2 x1 x2 : þk ð72Þ þ lx l The first two terms correspond to two MPT independent oscillators, while the third is a particular interaction chosen to correspond in the limit N ! 1 to the interaction h:o: H^int: ¼ hkxð^ ay1 a^2 þ a^1 a^y2 Þ

ð73Þ

R. Lemus, R. Bernal / Chemical Physics 283 (2002) 401–417

for two harmonic oscillators. In configuration space Eq. (73) takes the equivalent form ! p^1 p^2 h:o: 2 Hint: ¼ k ð74Þ þ lx x1 x2 : l Note that the matrix elements of the interaction (73) in a harmonic basis coincide with the matrix elements in (71). We now turn our attention to the problem of computing the non-diagonal contributions to the interaction term ! p^1 p^2 MPT 2 ~ x1 x2 : Hint: ¼ k ð75Þ þ lx l To this end we introduce the bosonic operators sffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffi ~ l x 1 p^ ; ð76aÞ xi  i c^yi ¼ ~l i 2 h 2hx

which can be rearranged as c^y ’ b^y zn þ b^f0n ;

ð79aÞ

c^ ’ b^y z0n þ b^fn ;

ð79bÞ

where we have introduced the functions     1 fn 1 fn zn ¼ þ sn ; z0n ¼  sn ; 2 2 2 2 fn ¼

 1  gn þ rn ; 2 2

ð76bÞ

We have used a different notation from the usual fay ; ag in order to emphasize that the matrix elements will be computed in the MPT basis. In terms of these operators the interaction (75) acquires the simple form MPT ~ kð^ Hint: ¼ hx cy1 c^2 þ c^1 c^y2 Þ:

ð77Þ

In order to establish the relation between this expression and the interaction associated to the su(2) vibron model given by (69), we proceed to express (77) in terms of the creation and annihilation operators given by (12) and (9), respectively. This task will be achieved by means of the expansions (35) and (47). The substitution of (35) and (47) into (76) leads to an expansion of cy ðcÞ in terms of powers of 1=m. As a first approximation, however, we shall consider only the linear contributions in by ðbÞ. If this is the case the operators (76) take the approximate form 1 1 c^y ’ ðb^y fn þ b^gn Þ þ ðb^y sn  b^rn Þ; 4 2 1 1 c^ ’ ðb^y fn þ b^gn Þ  ðb^y sn  b^rn Þ; 4 2

f0n ¼

 1  gn  rn : 2 2

ð80aÞ

ð80bÞ

From (28), the harmonic limit of these functions is obtained in a straightforward way lim zn ¼ lim fn ¼ 1;

m!1

m!1

lim z0 m!1 n

¼ lim f0n ¼ 0 m!1

ð81Þ

and consequently lim c^y ¼ b^y ¼ a^y ;

m!1

sffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffi ~ lx 1 p^ : xi þ i c^i ¼ ~l i 2 h 2hx

411

lim c^ ¼ b^ ¼ a^

m!1

ð82Þ

as expected. An intermediate situation is present when we consider the low lying region of the spectrum. In this case we have lim c^y ¼ b^y ;

nnmax

lim c^ ¼ b^

nnmax

ð83Þ

but keeping the MPT basis. Let us now return to the two-oscillator system. Considering the approximation (79), we obtain ð^ cy1 c^2 þ c^1 c^y2 Þ ’ b^y1 b^2 A^ðn1 ; n2 Þ þ b^1 b^y2 B^ðn1 ; n2 Þ ^ ðn1 ; n2 Þ; þ b^y b^y C^ðn1 ; n2 Þ þ b^1 b^2 D 1 2

ð84Þ where A^ðn1 ; n2 Þ ¼ zn1 fn2 þ z0n1 f0n2 ;

ð85aÞ

B^ðn1 ; n2 Þ ¼ f0n1 z0n2 þ zn2 fn1 ;

ð85bÞ

C^ðn1 ; n2 Þ ¼ zn1 z0n2 þ z0n1 zn2 ;

ð85cÞ

^ ðn1 ; n2 Þ ¼ f0 fn þ fn f0 : D n1 2 1 n2

ð85dÞ

ð78aÞ

From the discussion for one oscillator, a reasonable approximation for small n1 and n2 corresponds to (81) and consequently

ð78bÞ

A^ðn1 ; n2 Þ ¼ B^ðn1 ; n2 Þ ’ 1; ^ ðn1 ; n2 Þ ’ 0; C^ðn1 ; n2 Þ ¼ D

ð86Þ

412

R. Lemus, R. Bernal / Chemical Physics 283 (2002) 401–417

which implies MPT ~ kð^ ¼ hx cy1 c^2 þ c^1 c^y2 Þ Hint:

~ kðb^y1 b^2 þ b^1 b^y2 Þ: ’ hx

ð87Þ

This situation correspond to the vibron approximation discussed in Section 3, as it can be checked by direct substitution of (37) and (50) into (76). We thus conclude that the su(2) vibron model very nearly corresponds to taking the dominant Dn ¼ 1 interaction between coupled MPT oscillators, as long as we exclude the neighborhood of the dissociation energy. Based on the previous analysis worked out for the matrix elements of x and p^, we expect to improve the description of the interaction (77) only for the low lying region of the spectrum when considering the linear and the extended approximations discussed in Section 3. In order to evaluate the expansions we shall proceed to compare the exact diagonalization of the Hamiltonian (72) with the different results when x and p^ are substituted into (75) for the vibron, linear and extended approximations. To this end we shall chose again the set of parameters (39) for each oscillator with an interaction parameter k ¼ 0:01. This value was chosen in order to keep the polyad number P ¼ n1 þ n 2 ¼ v 1 þ v 2

ð88Þ

as an approximately well defined quantity for the whole spectrum. The parameter k provides a measure of the interaction between the local oscillators. For small k there is a slight mixture of the basis functions jm; n1 ; n2 i  Wqn1 ðu1 ÞWqn2 ðu2 Þ

i correspond to the energies from the where Eexp exact diagonalization of the MPT Hamiltonian i (72), Eth refers to the energies obtained from the approximation and nP stands for the number of states in the polyad P. In Fig. 4 we present the average polyad deviation obtained when the different approximations – vibron, linear and extended – are considered. We note the moderate deviation found over the whole range of polyads in the vibron approximation. We only include up to polyad 14 in order to display clearly the differences among the various approximations. The results displayed in Fig. 4 are in accordance with the corresponding results for the matrix elements for x and p^. For the linear case a decrease in the deviation is not detected, but for the extended approximation a small improvement appears for small number of quanta, while a significant increase in the deviation is obtained starting from polyad 8. The expansions (35) and (47) are not suitable to describe the whole range of quantum numbers. Since in general this is the situation we are interested in, it is worth exploring an approach which improves upon the vibron model. Suppose we were able to compute all the terms in (35) and (47) which contribute to the matrix elements hWqn jxj Wqnþ1 i and hWqn j^ pjWqnþ1 i. In this case we can write the linear approximation as sffiffiffiffiffiffiffiffiffi 1 h ^y ðb Fn þ b^Gn Þ; x ð91aÞ ~ 2 2lx

ð89Þ

and the quantum number P is nearly preserved. As k increases the eigenfunctions cease being dominated by components with a well defined polyad number. Since for k ¼ 0:01 it is possible to assign a (dominant) polyad number to each state, we can calculate the energy deviation per polyad rp with the following definition: ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P i i 2 i ðEexp  Eth Þ rp ¼ ; ð90Þ nP

Fig. 4. Energy deviation per polyad rp for the vibron (j), linear (d) and extended (N) approximations in the fx; pg space.

R. Lemus, R. Bernal / Chemical Physics 283 (2002) 401–417

i pffiffiffiffiffiffiffiffiffiffiffi ^y ~ ðb Sn  b^Rn Þ 2 hlx 2 with the diagonal contributions given by rffiffiffiffiffiffiffiffiffi ~ hWqnþ1 jxjWqn i 2lx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi; Fn ¼ 2 h  ðn þ 1Þð1  ðn þ 1Þ=mÞ p

ð91bÞ

ð92aÞ

rffiffiffiffiffiffiffiffiffi ~ hWqn1 jxjWqn i 2lx pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; Gn ¼ 2 h  nð1  n=mÞ

ð92bÞ

hWqnþ1 j^ pjWqn i 2i ffi; Sn ¼  pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ ðn þ 1Þð1  ðn þ 1Þ=mÞ 2lx

ð92cÞ

2i hWqn1 j^ pjWqn i ffi: Rn ¼  pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ nð1  n=mÞ 2lx

ð92dÞ

The functions involved in (91) contains the information of all the contributions of type ^ jWq i, O ^ ¼ x; p^, appearing in the expansions hWqn1 jO n (35) and (47) and consequently a considerable change is expected. In Fig. 5 we present the average polyad deviation for the vibron model as well as for the approximations (91). From this plot is evident that one must take into account all the ^ jWq i. contributions to the matrix element hWqn1 jO n We can further improve the description by taking the full second-order matrix elements ^ jWq i in the expansions (35) and (47). In hWqn3 jO n Fig. 5 we also show the resulting behavior for this approximation. As can be seen, a significant decrease in the deviation is found for the whole energy range.

413

From this analysis we conclude that taking into account the main contribution to the MPT matrix ^ jWq i leads to significant imelements hWqn1 jO n provement in the description of the interacting oscillators. In this work we have evaluated these contributions directly from the exact matrix elements, but work is in progress to derive the matrix elements from (35) and (47). This would have the advantage of providing analytical expressions for the matrix elements. 6. Interacting MPT oscillators in the {u,p} space In the previous sections the MPT oscillators were analyzed in terms of the fx; pg space. Although fairly good results are obtained for the vibron model, and significant improvement is obtained with (91), it is important to investigate the possibility of getting a better description in a different space. This argument is based on the fact that in the analysis of Morse oscillators the natural variable y ¼ 1  ebx turns out to be the best choice to expand the potential. This variable remains finite for large x, a fact that considerable improves the convergence. Since this fact suggests that the same situation may be present in the case of the MPT oscillators, we now search for the appropriate variable to expand this potential. The Morse variable y satisfies two properties. First, when the two oscillator potential is expanded about the equilibrium position, the potentials for the two independent oscillators are reproduced. Secondly, the variable y does not diverge for large x. Consider the properties of the variable u ¼ tanhðaxÞ in the case of the MPT potential. Concerning the behavior of u as a function of x we have lim tanhðaxÞ ¼ 1;

x!1

ð93Þ

which ensure the first condition. On the other hand, the expansion of the potential for the two oscillator system up to second-order is  2  2  2  X oV oV 2 u þ u1 u2 ; ð94Þ V ¼ i 2 oui e ou1 u2 e i¼1 Fig. 5. Energy deviation per polyad rp for the vibron approximation (j), the approximation (91) (d) and when both matrix ^ jWq i, O ^ ¼ x; p^; b ¼ 1; 3 are considered (N). elements hWqn jO nþb

which in terms of the structure constants frr and frr0 takes the form

414

V ¼

R. Lemus, R. Bernal / Chemical Physics 283 (2002) 401–417 2 X

frr a2 u2i þ frr0 a2 u1 u2 :

ð95Þ

i¼1

Since u2 is related with the MPT potential in the form 1 u2 ¼ 1  cosh2 ðaxÞ

ð96Þ

the first terms of (95) give rise to the potential of two independent MPT oscillators. Since both conditions are satisfied, we propose the following interaction Hamiltonian in terms of the variables ui : 2   X HiMPT þ k grr0 p^1 p^2 þ frr0 a2 u1 u2 : H¼ ð97Þ i¼1

As we intend to compare the spectrum generated by this Hamiltonian with the previous analysis, we shall be interested in recovering the interaction (73) in the harmonic limit. This is achieved by choosing grr0 ¼ 1=l;

~2: frr0 ¼ lx

ð98Þ

We now proceed to compare the exact diagonalization of (97) with the different approximation derived from the expansion of u in the su(2) space. From (41) we obtain in a straightforward way the expansion 1 3 u ¼ tanhðaxÞ ¼ y^  y^3 þ y^5 þ    : 2 8

ð99Þ

We can thus obtain the expansions associated to the different approximations. For the vibron approximation, we obtain 1 u ¼ pffiffiffi ðb^y þ b^Þ; m

  1 2n þ 1 1 F3;n Q00;n ¼ fn  8m m   2n þ 2 þ2 1 F4;n : m

ð103Þ

Since this expansion differs from (35) only in the term a and the fraction 1=8m, we do not expect to obtain a better convergence than for x. Indeed, in Table 5 we compare the exact matrix elements hWqnþ1 jujWqn i with the three approximations considered. We appreciate that even for low energy no improvement is found. This fact is illustrated in Fig. 6. As in the previous case the vibron approximation provides the best overall description for the whole range of quantum numbers. This result may suggest that u is not the natural variable. We must remark, however, that Fig. 6 shows the quality of the various approximations in the expansion (102), but does not give us a criterion to discard u as the natural variable. We still have to compare the predictions for the energy spectrum taking into account all the contributions of the matrix elements. Let us first consider the vibron approximation. Substitution of (50) and (99) into (96) leads to H¼

2 X

~ ðb^y1 b^2 þ b^1 b^y2 Þ: HiMPT þ khx

ð104Þ

i¼1

This Hamiltonian coincides with the su(2) Hamiltonian (68). In this approximation the spectrum generated is the same as the one obtained in the fx; pg space. However, the exact spectrum is

ð100Þ Table 5 Comparison of the exact matrix elements hWqn j tanhðaxÞjWqnþ1 i with the vibron, linear and extended approximations

while for the linear case 1 u ¼ pffiffiffi ðb^y fn þ b^gn Þ: 2 m

where

ð101Þ

n

Exact

Vibron

Linear

Extended

The extended approximation will be given by taking up to cubic terms in (99) sffiffiffiffiffiffiffiffiffi a h ^y 0  1 1 ^y ^y ^y b Q0;n  ðb b b F1;n þ b^y b^y b^Q1;n Þ u¼ ~ 2 2lx 8m þ    þ hc; ð102Þ

0 1 2 3 4 5 6 7 8 9

0.2081 0.29408 0.35949 0.41389 0.46076 0.50163 0.53700 0.56640 0.58714 0.58497

0.2039 0.281771 0.33678 0.379035 0.41247 0.43910 0.460131 0.47628 0.48804 0.49572

0.22360 0.34156 0.45643 0.58248 0.73192 0.92195 1.18322 1.58114 2.29189 4.03113

0.2049 0.2775 0.304834 0.2669 0.10064 )0.362196 )1.62692 )5.6525 )25.20420 )39.30350

R. Lemus, R. Bernal / Chemical Physics 283 (2002) 401–417

a u 2

415

sffiffiffiffiffiffiffiffiffi h ^y 0 ^ 0 ðb Fn þ bGn Þ ~ 2lx

ð105Þ

with the diagonal contributions given by

jhWqn j tanhðaxÞjWqnþ1 iexact

hWqn j

Fig. 6. Deviations Du   tanhðaxÞjWqnþ1 iapprox j as a function of the quantum number n for the vibron (j), linear (d) and extended (N) approximations.

different. In Fig. 7 we display the average polyad deviation from the exact diagonalization for the vibron approximation. We should note the decrease in the deviation, which is reflected in the fact that rp  12 cm1 for polyad 18 , while in Fig. 4 the same deviation is reached for polyad 11. In the same figure the linear as well as the extended approximation are included. In both cases the deviation increase is considerably grater than the previous results for the space fx; pg. This behavior is ascribable to the factor ð1=8mÞ in (102), since in (35) the equivalent factor is ð1=24mÞ. In order to obtain definite conclusions we must consider the results when the complete matrix el^ jWq i, O ^ ¼ u; p^, are taken into acements hWqn1 jO n count. It is only in this context that we can evaluate the advantage of the space fu; pg. In this case we can write the linear approximation for u as

pffiffiffi hWqnþ1 jujWqn i ffi; Fn0 ¼ 2 m pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn þ 1Þð1  ðn þ 1Þ=mÞ

ð106aÞ

pffiffiffi hWqn1 jujWqn i ffi: G0n ¼ 2 m pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nð1  n=mÞ

ð106bÞ

Although closed analytic results without involving sums are not known for the matrix elements appearing in (106), it is possible to obtain expressions in term of double sums in a straightforward way [20]. Substitution of the Gegenbauer polynomials Cnqþð1=2Þn ðuÞ ¼

½n=2 X 1 ð1Þm Cðq þ 12  nÞ m¼0



Cð12 þ n  mÞ n2m ð2uÞ m!ðn  2mÞ!

ð107Þ

into the matrix elements of u and p^ allows to obtain 1 q Nnq Cðq  n þ 1=2Þ22n1 hWqn1 jujWqn i ¼ Nn1 a ½n=2 ½ðn1Þ=2 X X pþs  ð1Þ 22p2s p¼0

s¼0

Cðq  p þ 1=2ÞCðq  s þ 1=2Þ  p!s!ðn  2pÞ!ðn  2s  1Þ! Cðn  p  s þ 3=2ÞCðq  n þ 1=2Þ  ; Cðq  p  s þ 2Þ ð108Þ while for the momentum operator i q hW jp^jWqn i h n1 q Nnq 22n1 ¼ Nn1

½n=2 ½ðn1Þ=2 X X p¼0

ð1Þ

pþs 2p2s

2

s¼0

Cðq  p þ 1=2ÞCðq  s þ 1=2Þ p!s!ðn  2pÞ!ðn  2s  1Þ!  Cðn  p  s  1=2ÞCðq  n þ 3=2Þ  ðn  2pÞ Cðq  p  s þ 1Þ  Cðn  p  s þ 1=2ÞCðq  n  1=2Þ  ðq  pÞ : Cðq  p  s þ 1Þ



Fig. 7. Energy deviation per polyad rp for the vibron (j), linear (d) and extended (N) approximations in the fu; pg space.

ð109Þ

416

R. Lemus, R. Bernal / Chemical Physics 283 (2002) 401–417

Fig. 8. Energy deviation per polyad rp for the vibron approximation (j) as well as the approximations when only the matrix ^ jWq i, O ^ ¼ u; p^; b ¼ 1; 3 are considered. The elements hWqn jO nþb symbol (d) is used when only the elements with b ¼ 1 are taken into account, while (N) refers to the case when both b ¼ 1; 3 are considered.

These expressions have the clear disadvantage of involving a double sum and do not reflect the 1=m ^ jWq i, dependence of the matrix elements hWqnb jO n b ¼ 1; 3. The derivation of simpler results for these matrix elements is in progress. In Fig. 8 we compare the deviation of the vibron model with the results obtained for the linear approximation given by (91b) and (105). We also show the deviation ^ jWq i are included. If we when the elements hWqn3 jO n compare Figs. 5 and 8, which display the corresponding deviations in the fx; p^g and fu; p^g spaces, respectively, we can appreciate a remarkable reduction in the deviation in Fig. 8. While in the fu; p^g space the energy deviation for the vibron approximation in polyad 18 is around 12 cm1 , the same deviation is reached for polyad 10 in the fx; p^g space. A similar reduction is obtained for the linear approximations. This results allows to conclude that u as the natural variable to deal with MPT oscillators. In particular, when the MPT oscillators are approximated by the vibron model, the spectroscopic constants should be identified in the framework of the potential expansion of type (94).

7. Conclusions In this paper we have analyzed the MPT potential in the context of the su(2) dynamical

group. Starting with a realization of the dynamical group in terms of the configuration space variables we have established the expansions of the coordinate and momentum in terms of creation and annihilation operators of MPT wave functions. Three approximations are considered: vibron, linear and extended approximations. The exact matrix elements of x and p^ are compared with those provided by the different approximations. We have also analyzed the theoretical relationship between the su(2) vibron model and the coupled MPT system. It has been shown that the standard su(2) approach to molecular vibrational excitations (vibron model) is equivalent to considering MPT oscillators via the dominant Dn ¼ 1 selection rule. This approximation arises naturally from an expansion of the MPT operators c^y ð^ cÞ in terms of the su(2) generators b^y ðb^Þ, which in turn allows to establish an extended su(2) model when the next terms in the expansion are taken into account. Concerning the expansions (35) and (47), the vibron approximation turns out to provide the best approximation for the whole range of states. In contrast, both linear and extended approximations deviate considerably as the polyad number increases. The reason for this behavior is the slow convergence of the series. In order to improve the description it is necessary to take into account all contributions to the matrix ele^ jWq i in the different powers of the ments hWqn1 jO n expansions. Although we have not obtained simple closed analytic expression for such contributions, we have estimated their effect by studying the consequences of successively con^ jWq i, sidering the exact matrix elements hWqnb jO n b ¼ 1; 3. The results show a remarkable improvement of the description compared with the vibron approximation, as for the case of the Morse potential [12]. We have also analyzed the expansion of the interaction MPT potential in terms of the variable u. The three cases, vibron, linear and extended approximations were considered. The convergence of the expansion of u in terms of the creation and annihilation operators turned out to be worse than the expansion of x. This result, however, was a consequence of the particular

R. Lemus, R. Bernal / Chemical Physics 283 (2002) 401–417

expansion and should not be used to obtain definite conclusions concerning the suitability of the fu; p^g space to describe interacting MPT oscillators. A comparison of the average deviation of energies provided by the vibron approximation in this space with the results in terms of fx; pg, did lead to the conclusion that u is the natural variable for the MPT potential. This conclusion was further supported by the analysis based on the successive inclusion of the exact matrix ele^ jWq i, b ¼ 1; 3. These results are ments hWqnb jO n similar to the case of the Morse potential where y turns out to be the appropriate variable to expand the potential. This work together with the previous analysis of the Morse potential allows to carry out a complete description (including bending and stretching degrees of freedom) of molecular vibrational excitations in the framework of the su(2) dynamical group, while keeping direct connection with the molecular structure constants. We expect to apply these results to calculate the potential constants from the spectroscopic parameters, which in turn will lead to study isotopic effects within the Born–Oppenheimer approximation.

Acknowledgements We are indebted to A.Frank for invaluable discussions and suggestions. This work is supported in part by CONACyT, Mexico, under project 32397-E.

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