Possible deviations from the O(4) limit of the vibron model in diatomic molecules

Physics Letters A 316 (2003) 84–90 www.elsevier.com/locate/pla

Possible deviations from the O(4) limit of the vibron model in diatomic molecules Feng Pan a,b,∗ , Xin Zhang a , J.P. Draayer b a Department of Physics, Liaoning Normal University, Dalian 116029, PR China b Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803-4001, USA

Received 6 June 2003; accepted 15 July 2003 Communicated by B. Fricke

Abstract A U (3) ↔ O(4) transitional description of diatomic molecules in the U (4) vibron model is considered. The analysis includes the U (3) and O(4) limits of the theory. Applications to vibrational-like band-heads of several diatomic molecules indicate that there are significant deviations from the O(4) limit of the theory.  2003 Elsevier B.V. All rights reserved. PACS: 05.30.J; 33.20; 03.65.F Keywords: U (4) vibron model; U (3) ↔ O(4) transitional theory; O(4) limit; Vibrational spectra of diatomic molecules

The U (4) vibron model can be used to characterize relative motion of a dipole-deformed shape in threedimensional space and as a consequence provides a simple yet elegant framework for a description of rotational and vibrational spectra of diatomic molecules [1–4]. Dynamical symmetries play an important role in this approach, with its O(4) and U (3) limits corresponding to rigid and nonrigid structures (shapes), respectively. The shape (phase) transition of the vibron model refers to the process in which a system undergoes a change between the O(4) and U (3) limits of the theory. The O(4) limit has been used to describe diatomic molecules since it is equivalent to the result obtained from a quantum mechanical description with * Corresponding author.

E-mail address: [email protected] (F. Pan). 0375-9601/$ – see front matter  2003 Elsevier B.V. All rights reserved. doi:10.1016/S0375-9601(03)01144-7

a traditional Morse potential [2]. Transitional theory offers a more general parameterization, one in which the O(4) limit is a special case. Deviations from the O(4) limit vibron model can be use to study vibrationrotation motion of diatomic molecules. In the U (4) vibron model, elementary excitations are dipole p-bosons with spin and parity l P = 1− and scalar s-bosons with l P = 0+ . If the total number of bosons and angular momentum are assumed to be conserved quantities, the leading dynamical symmetry group is U (4). Two possible dynamical symmetry limits, U (3) and O(4), can be realized when the Hamiltonian of a system is exactly diagonal in basis of one of the following algebraic chains, U (4) ⊃ O(4) ⊃ O(3) ⊃ O(2) (I), U (4) ⊃ U (3) ⊃ O(3) ⊃ O(2)

(II).

(1)

F. Pan et al. / Physics Letters A 316 (2003) 84–90

85

A general Hamiltonian of the U (4) vibron model with only one- and two-body interactions can be expressed in terms of linear and quadratic invariants (Casimir operators) of all the subalgebras contained in (1), which can be written equivalently as [5]

As is shown in [5], in the transitional situation the Hamiltonian (2) can be diagonalized under the wavefunction
(ζ ) 
(ζ ) 
(ζ )  |k, ζ, J M = N S + x1 S + x2 · · · S + xk |lw ,

(2) Hˆ = gS0+ S0− + αS10 + ηCˆ O(3) ,

(6) where N is the normalization constant, ζ = 0, 1, . . . , k is an additional quantum number needed to distinguish different states with the same number of boson pairs k, angular momentum J and its third component M,

(2)

where c
† 2 1 † † + p ·p , s 2 2 c 2 1 − ˜ S0 = s + p˜ · p, 2 2

S0+ =

S10 =

    c2 † 1 3 1 † s s+ + p · p˜ + , 2 2 2 2

(3a)

(3b)

in which s + (s) and pµ+ (pµ ) are boson creation (2) (annihilation) operators, Cˆ O(3) is the Casimir operator of O(3), g, α, η, and c are real parameters. It can be shown that (2) is equivalent to a Hamiltonian in the O(4) limit when c = 1. The corresponding energy eigenvalues under the basis of chain (I) can be written as      N 3 N −g + E(v, J ) = α 1 + 2 2 4     1 1 2 + g(N + 2) v + −g v+ 2 2 + ηJ (J + 1), (4) where v = 0, 1, . . . , N/2 or (N − 1)/2 for N = even or odd, is the O(4) quantum number [1,2], J is the quantum number of the angular momentum. When c = 0, (2) is equivalent to the U (3) limit of the theory with energy eigenvalues given by 1 3 1 E(np , J ) = α + (α − g)np + gnp (np + 3) 4  2 4  1 + η − g J (J + 1) 4

(5)

in the basis of chain (II), where np is the U (3) quantum number. As c varies continuously within the closed interval [0, 1], the system undergoes a shape (phase) transition from U (3) when c = 0 to O(4) for c = 1, i.e., from a nonrigid shape to a rigid one. The transitional theory provides for all possible situations between and including the U (3) vibrational and O(4) rotational limits.


(ζ )  S + xi =

c (ζ ) 2(1 − c2 xi )


† 2 s +

1 (ζ )

2(1 − xi )

p† · p† ,

(7) |lw is the boson pair vacuum state with total boson number N = J + νs (νs = 0 or 1) and total p-boson number np = J satisfying s 2 |lw = 0,

p˜ · p|lw ˜ = 0.

(8)

Energy eigenvalues E (k,ζ ) of the Hamiltonian (2) can be expressed as E (k,ζ ) = h(k,ζ ) + αΛ01 + ηJ (J + 1),

(9)

where h(k,ζ ) =

k  α (ζ )

i=1

and Λ01 =

,

(10)

xi

     3 1 2 1 c νs + + J+ . 2 2 2

(11)

The quantum number k is related to the total boson number N by N = 2k + νs + J . The ζ th set of (ζ ) variables xi is determined by
 
gc2 νs + 12 g J + 32 α = + (ζ ) (ζ ) (ζ ) xi 1 − c 2 xi 1 − xi −

k 

2g

(ζ ) j =i xi

− xj

(ζ )

for i = 1, 2, . . . , k.

(12)

In what follows, we apply the U (3) ↔ O(4) transitional theory of the U (4) vibron model to a description of the spectra of diatomic molecules, with the O(4) limit with phase parameter c = 1 considered to be a special case. For simplicity, only vibrational band-head configurations are considered since higher

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F. Pan et al. / Physics Letters A 316 (2003) 84–90

order terms including vibration–rotation couplings are not included in the interaction. This simplification provides for a minimal model with up to two-body interactions to test possible deviations from the O(4) limit of the theory in the description of vibrational spectra of diatomic molecules. All such vibrational levels below dissociation are considered. It is well known that the O(4) limit of the theory is an reasonable approximation since it is equivalent to quantum mechanical description with the conventional Morse potential. However, as for the Morse form, the O(4) limit is highly perscriptive and hence can only be used to describe low-lying levels to a certain level of accuracy. In the O(4) limit the Hamiltonian contains only one free parameter, namely g. It can be shown that (4) can be rewritten as a two-term Dunham expansion,     1 1 2 + Y20 v + E(v, J = 0) = const + Y10 v + 2 2 (13) with a corresponding relation N =−

Y10 − 2. Y20

(14)

The maximum number of bound vibrational states in the theory is restricted by the value of N where N = 2vm

or 2vm + 1

(15)

for N even or odd, respectively. Hence, using (14), we can estimate the N value of the model from experimental values of Y10 = ωe and Y20 = −ωe xe , where ωe and ωe xe are the standard symbols used in spectroscopy. The experimental vibrational levels are then fit by adjusting the parameter g in the O(4) limit of the theory. Consider, for example, the electronic ground state X1 Σ + of HCl. The experimental vibration constants are ωe = 2990.9463 cm−1 , ωe xe = 52.8186 cm−1 , from which N = ωe /ωe xe − 2 = 54.6268. We take the nearest integer 55 as the total vibron number of the system. The vibrational band will stop at vm = 27 based on this assumption. Then, the whole vibrational spectrum is fit by using (4) with g = 52.4728 cm−1 . The results show that the fit is reasonable only for lowlying levels. There are significant errors for higherlying excited levels. The deviation is also shown clearly in dissociation energy calculated from the

O(4) limit, which gives E(vm = 27) = 41086.1 cm−1 , while the experimental value is D0 = 35759.1 cm−1 . Alternatively, as is well known, the theory can be extended by including all possible higher order terms according to the Dunham expansion with energy eigenvalue    l 1 k E(v, J ) = (16) J (J + 1) , Ykl v + 2 kl

which can be obtained by adding a series of products of powers of Casimir operators to the O(4) limit of vibron model Hamiltonian,   (2) k  (2) l ykl CO(4) CO(3) , H= (17) kl

which yields the same form as that given by Dunham expansion shown in (16). In addition, the eigenvalues of Hamiltonian (17) is equivalent to those obtained from the Schrödinger equation with a potential in the form of an expansion in powers of a Morse potential function [6], 

k  V (r) = (18) Vk 1 − exp −β(r − re ) . k

It should be clear that the Dunham coefficients obtained from (17) are not independent of one another due to the fact that the total number of bosons is a conserved quantity. This feature can be regarded as an advantage of the algebraic model since the number of parameters in the model are reduced. In this Letter, however, only the minimal model is tested against the O(4) limit of the theory, with all possible two-body interactions taken into account. When all the parameters in the model are nonzero, the system deviates from its O(4) dynamical symmetry limit and the energy eigenvalue formula is highly nonlinear, going beyond the quartic form given in (4). Though it may be difficult to find the exact form of the corresponding effective potential in such cases, an improved fit to experiment suggests that the whatever that potential may be it is closer a realistic one than the idealized Morse form. Since all vibrational levels up to the dissociation energy are to be considered, the N value needs to be determined accordingly. From the definition of the dissociation energy, D0 = E(vm ) − E(v = 0),

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F. Pan et al. / Physics Letters A 316 (2003) 84–90

one can evaluate the maximum vibrational band frequency vm from the corresponding experimental value. Furthermore, a Dunham expansion formula with 4–6 parameters is accurate enough to describe vibrational levels including high-lying confiburations. The N value determined with this scheme seems suitable for the whole vibrational spectrum. For the diatomic molecule HCl with 4 Dunham coefficients, Y10 = 2990.9463 cm−1 , Y20 = −52.8186 cm−1 , Y30 = 0.22437 cm−1 , Y40 = −0.01218 cm−1 , and dissociation energy D0 = 35759.1 cm−1 , one determines that vm = 18 and N = 37. Sixteen diatomic molecules, each with sufficient experimental data to describe the whole vibrational spectrum, were chosen. All experimental vibrational levels, including some not reported below the dissociation energy, were obtained by using the Dunham expansion formula with a set of vibrational spectroscopy constants fixed by experiment. This set is denoted as E exp . Only electronic ground states of these molecules are taken into account. The experimental data are listed in Table 1. A least-square fitting routine was used with the quality of the fits measured by the rootmean-square deviation

N 1 
cal exp 2 Ei − Ei σ= N

1/2 ,

(20)

i

which was calculated for each molecule, where N is the total number of vibrational levels in the electronic ground state, namely, N = vm + 1. The transitional theory has three parameters, g, α, and c, while the O(4) limit has only one, g. In order to compare theories with different number of parameters, the quality of the fits may also be measured by the quantity S=

N 1  cal exp  E i − E i , N −n

(21)

i

where n is the number of parameters in the theory. Hence, n = 1 for the O(4) limit, and n = 3 for the transitional theory. The parameters used in the fitting are given in Table 2, where gI represents the parameter of the O(4) limit, and {gII , α, c} are parameters of the transitional theory. The calculated results and errors in the fits are shown in Table 3.

87

From Table 3, it can be seen that the root-meansquare deviations of the transitional theory are less than those of the O(4) limit for all cases considered. In considering the number of parameters by using the quantity S, the results of the transitional theory are still better than those of the O(4) limit, except for CsH. For CsH, deviations of the fits are σI = 331.565 cm−1 , σII = 321.75 cm−1 , SI = 310.242 cm−1 , and SII = 325.395 cm−1 , respectively. The results show that the fit of the transitional theory for CsH is better than that of the O(4) limit if a least-square deviation is adopted as the criterion of the fit, while the O(4) limit is better than the transitional theory when the absolute average deviation is adopted as the criterion. In this sense CsH can be regarded as a typical molecule in the O(4) limit. In order to measure deviations from the O(4) limit of the theory, we introduce a quantity called rigidity creditability, which is defined as

R1 =

σII , σI

R2 =

SII . SI

(22)

If Ri (i = 1, 2) is greater than or equal to 1, the corresponding molecule is best described by the O(4) limit of the theory, otherwise a transitional theory with c = 1 is best. Using these measures, one can effectively analyze deviations from the O(4) limit in the model. In addition, it is important to note that the phase parameter c plays a very significant role in the transitional theory. In most cases, the O(4) limit is an approximate treatment, in which the phase parameters of all diatomic molecules are fixed to be 1. However, the results show that the phase parameter may deviate a little from the O(4) point for most diatomic molecules. Thus, we come to a conclusion that most diatomic molecules are quasi-rigid. In order to illustrate the deviations from the O(4) limit with appropriate phase parameter c, rigidity creditabilities R1 and R2 for these typical molecules plotted against the phase parameter c are shown in Figs. 1 and 2. From these figures, one can see that most rigidity creditabilities of these molecules are in the range of 0.6–0.8. The values for CsH and KH are relatively large, and that for HI is the smallest. In short, the results of the fit show that the transitional theory is obviously better than those of the O(4) limit.

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F. Pan et al. / Physics Letters A 316 (2003) 84–90

Table 1 The experimental data of vibrational spectroscopy constants (Dunham coefficients Yi0 ) and dissociation energies (in cm−1 ) of electronic ground states of sixteen different diatomic molecules Moleculesa Y10 Y20 Y30 Y40 Y50 Y60 D0

N2 –X1 Σg+ 2358.57 −14.324 −2.26 × 10−3 −2.4 × 10−4

78714.2

Moleculesa Y10 Y20 Y30 Y40 Y50 Y60 D0

HF–X1 Σ + 4138.32 −89.88 0.9 −0.011 −6.7 × 10−4 47336.3

Moleculesb Y10 Y20 Y30 Y40 Y50 Y60 D0

15318

Y10 Y20 Y30 Y40 Y50 Y60 Y70 Y80 Y90 D0

CO–X1 Σ +

559.7507 −2.694271 −3.32527 × 10−3 −2.27337 × 10−4 3.92041 × 10−6 −6.02984 × 10−8 19997.3

6 LiH–X1 Σ +

1419.68479 −23.15436 −0.061875 0.0562386 −8.1416 × 10−3 6.5495 × 10−4 −3.1102 × 10−5 7.8811 × 10−7 −8.6223 × 10−9 19582.6

NO–X2 /r

2169.81358 −13.28831 0.010511 5.74 × 10−5 9.83 × 10−7 −3.166 × 10−8 89462.3

HCl–X1 Σ +

1904.405 −14.1870 0.024 −9.3 × 10−4

52399.8

HBr–X1 Σ +

2990.95 −52.8186 0.22437 −0.01218

HI–X1 Σ +

2648.975 −45.2175 −2.9 × 10−3

35759.1

NaH–X1 Σ + 1171.75909 −19.52352 0.12131 −0.59 × 10−3 −2.235 × 10−4

Moleculesc

Cl2 –X1 Σg+

2309.01 −39.6435 −0.02 0.01621

30310.1

KHd –X1 Σ + 986.65055 −15.844615 0.38533062 −0.09217627 0.018413172 −2.385268 × 10−3 14282.6 6 LiD–X1 Σ +

1074.40587 −13.67905 0.170767 −0.0292128 4.4598 × 10−3 −3.8432 × 10−4 1.8147 × 10−5 −4.4433 × 10−7 4.3772 × 10−9 19759.2

24632.8

RbH–X1 Σ +

CsH–X1 Σ +

937.1046 −14.2777 0.09658 −8.62 × 10−4

891.465 −12.943 0.1053 −3.19 × 10−3 1.81 × 10−5 −6.21 × 10−6 14348.2

14115 7 LiH–X1 Σ +

7 LiD–X1 Σ +

1405.07781 −22.68035 −0.059985 0.0539595 −7.7313 × 10−3 6.1554 × 10−4 −2.8930 × 10−5 7.2552 × 10−7 −7.8559 × 10−9 19589.8

1055.00696 −13.18954 0.161683 −0.0271594 4.0715 × 10−3 −3.4452 × 10−4 1.5974 × 10−5 −3.8406 × 10−7 3.7152 × 10−9 19768.8

a b c d

Data taken from Ref. [7]. Data taken from Ref. [8]. Data taken from Ref. [9]. Additional coefficients of KH are 103 Y70 = 0.20131181, 104 Y80 = −0.11081649, 106 Y90 = 0.38314085, 108 Y10,0 = −0.75651294, 1010 Y11,0 = 0.64835269.

To demonstrate the systematic fitting quality, the average total deviations for these diatomic molecules σ¯ =

Nt 1  σi , Nt i=1

Nt 1  S¯ = Si Nt i=1

(23)

are used, where Nt is the total number of molecules in the fit. The results are σ¯ I = 1170.62 cm−1 ,

σ¯ II = 788.585 cm−1 ,

S¯I = 1051.22 cm−1 ,

S¯II = 785.818 cm−1 .

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F. Pan et al. / Physics Letters A 316 (2003) 84–90

89

Table 2 Parameters used in the fits with the units of gI , gII and α are in cm−1 N gI gII α c

N gI gII α c

N2

Cl2

CO

NO

HF

HCl

HBr

HI

95 29.7652 29.3752 20200 0.986

109 6.1028 6.0418 10200 0.995

145 15.9206 16.0506 7100 0.987

81 27.8146 27.8946 7300 0.973

39 112.205 113.275 12500 0.980

37 90.8709 92.5809 6500 0.961

33 90.625 91.585 33000 0.986

29 96.0843 92.5843 32000 0.988

NaH

KH

RbH

CsH

6 LiH

6 LiD

7 LiH

7 LiD

41 32.1402 32.0292 7100 0.985

49 22.0261 22.0461 6900 0.993

45 24.0484 23.6884 7200 0.984

49 19.9897 20.0097 7800 0.996

47 32.6832 32.8332 6500 0.990

57 21.3903 21.3303 6500 0.985

47 32.5151 32.6351 6700 0.989

59 20.2052 20.1452 6600 0.986

Table 3 Deviations of the fits measured in cm−1 N2 σI σII SI SII

σI σII SI SII

Cl2

CO

NO

HF

HCl

HBr

HI

787.334 605.233 708.62 574.396

2014.55 1639.17 1772.66 1513.4

2338.18 1475.79 2072.38 1428.33

1375.86 996.265 1244.55 1033.14

1422.24 895.821 1287.78 946.42

1050.41 660.034 964.058 724.524

1431.0 641.184 1300.2 709.803

NaH

KH

RbH

CsH

6 LiH

6 LiD

7 LiH

7 LiD

592.636 400.742 539.072 417.216

410.826 381.076 384.991 382.62

685.428 367.157 610.681 375.717

331.565 321.75 310.242 325.395

528.988 483.787 493.672 490.87

773.3 528.183 695.252 525.316

558.245 493.225 518.335 497.981

752.947 526.35 677.342 519.115

3676.43 2201.59 3239.61 2108.84

Fig. 1. Rigidity creditabilities R1 and the corresponding phase parameters c for different diatomic molecules.

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F. Pan et al. / Physics Letters A 316 (2003) 84–90

Fig. 2. Rigidity creditabilities R2 and the corresponding phase parameters c of the different diatomic molecules.

The results again show that the transitional theory is systematically better than the O(4) limit for a description of vibrational spectra of diatomic molecules. In conclusion, we have presented an algebraic transitional description of vibration spectra of diatomic molecules within the framework of U (4) vibron model. In particular, we have concentrated on possible deviations from the O(4) limit of the theory. By design the discussion was limited to a minimal model that did not include high-order terms which correspond to more than two-body terms in the interaction. The analysis shows that an algebraic Hamiltonian with two-body interactions can provide with reasonable description of the whole vibration spectrum when the phase parameter c is not taken to be exactly 1. Further improvements by including highorder interactions are clearly possible and a suggested next step. A least-square fit was performed for both the transitional theory and O(4) limit of the theory. Accuracy and the creditability of the various approximation were discussed. The results show that there is obvious deviations from the O(4) limit for most diatomic molecules, and that the transitional theory with c = 1 is better for a description of vibrational spectra of diatomic molecules, though the O(4) limit of the theory can still be regarded as a very good approximation for low-lying spectra.

Acknowledgements This work was supported by the US National Science Foundation (9970769 and 0140300) and by the Natural Science Foundation of China (10175031).

References [1] F. Iachello, Chem. Phys. Lett. 78 (1981) 581. [2] F. Iachello, R.D. Levine, J. Chem. Phys. 77 (1982) 3046. [3] O.S. van Roosmalen, A.E.L. Dieperink, F. Iachello, Chem. Phys. Lett. 85 (1982) 32. [4] O.S. van Roosmalen, F. Iachello, R.D. Levine, A.E.L. Dieperink, J. Chem. Phys. 79 (1983) 2515. [5] F. Pan, X. Zhang, J.P. Draayer, J. Phys. A 35 (2002) 7173. [6] S. Oss, Adv. Chem. Phys. 93 (1996) 455. [7] K.P. Huber, G. Herzberg, Molecular Spectra and Molecular Structure (IV). Constants of Diatomic Molecules, Van Nostrand Reinhold, New York, 1979. [8] W.C. Stwalley, W.T. Zemke, S.C. Yang, J. Phys. Chem. Ref. Data 20 (1991) 153. [9] W.C. Stwalley, W.T. Zemke, J. Phys. Chem. Ref. Data 22 (1993) 89.