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A simple algebraic approach to a nonlinear quantum oscillator
Physics Letters A 308 (2003) 319–322 www.elsevier.com/locate/pla
A simple algebraic approach to a nonlinear quantum oscillator Adelio R. Matamala ∗ , Carlos R. Maldonado Facultad de Ciencias Químicas, Grupo QTC, Universidad de Concepción, Casilla 160-C, Concepción, Chile Received 10 December 2002; received in revised form 27 December 2002; accepted 6 January 2003 Communicated by P.R. Holland
Abstract Using the well-known ladder operators formalism, a simple algebraic approach to a one-dimensional quartic nonlinear oscillator is presented. Making a linear canonical transformation and deleting one term, the Hamiltonian can be reduced to a form permitting to obtain explicit analytic formula for energy eigenvalues with a good accuracy even for large nonlinearity coefficients. 2003 Elsevier Science B.V. All rights reserved. PACS: 03.65.F Keywords: Nonlinear oscillator; Ladder operators; Algebraic method
1. Introduction Nonlinear oscillator [1] is a central model in molecular physics. Certainly, anharmonicities must be included in all realistic description of vibrational dynamics of molecular systems. The most simple model to describe anharmonic spectra consists of a onedimensional harmonic oscillator perturbed by a term containing a quartic power in the coordinate. This nonlinear quantum oscillator with a quartic perturbation has been studied extensively in the past [2–12]. Recently a very interesting work about the calculation of energy eigenvalues for the quantum nonlinear oscillator with a polynomial potential has been reported [13]. * Corresponding author.
E-mail address: [email protected] (A.R. Matamala).
In general, the evaluation of accurate energy levels of a quantum mechanical quartic nonlinear oscillator has become the testing ground for new methods in quantum mechanics [14]. In this way, algebraic approaches have showed that one can obtain a great deal of information with relatively small computational effort [15,16]. In the present Letter, a simple approximate diagonalization of a one-dimensional nonlinear oscillator is presented. Using the well-known ladder operators formalism, a unitary transformation is defined in order to map the original Hamiltonian into effective one. Deleting one term, the Hamiltonian of the quartic oscillator can be reduced to a form permitting to obtain energy eigenvalues with impressive accuracy even for large nonlinearity coefficients. This approach was previously used in the algebraic study of the asymmetric rigid rotor [17].
0375-9601/03/$ – see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0375-9601(03)00075-6
320
A.R. Matamala, C.R. Maldonado / Physics Letters A 308 (2003) 319–322
2. Hamiltonian
3. Diagonalization
Let us consider the Hamiltonian of a one-dimensional harmonic oscillator perturbed by a term containing a quartic power in the coordinate:
If the last term in Hamiltonian (5), β(aˆ †4 + aˆ 4 ), is neglected for the time being, the nonzero matrix in the elements of the transformed Hamiltonian H basis of the number operator, {|n }, are given by
= 1 pˆ 2 + xˆ 2 + λxˆ 4 , H 2
(1)
where the mass and frequency of oscillation has been settled to unit and the canonical quantization [x, ˆ p] ˆ = i Iˆ is assumed with h¯ = 1. Introducing the well-known ladder operators aˆ and aˆ † through the following realization 1 aˆ = √ (xˆ + i p) ˆ 2
1 and aˆ † = √ (xˆ − i p), ˆ 2
(2)
the Hamiltonian (1) reads 1 † H = aˆ aˆ + 2 λ + 3Iˆ + 6 aˆ †2 + 2aˆ † aˆ + aˆ 2 4 + aˆ †4 + 4aˆ †3aˆ + 6aˆ †2aˆ 2 + 4aˆ † aˆ 3 + aˆ 4 . (3) Now, the action of the following linear canonical transformation θ †2 2 T = exp − aˆ − aˆ , (4) 2 where θ is a real parameter, upon the Hamiltonian (3) permits to obtain
†
H = T HT = α 2nˆ + Iˆ + 6β nˆ 2 + γ aˆ †2 + aˆ 2 + 4β aˆ †2nˆ + nˆ aˆ 2 + β aˆ †4 + aˆ 4 ,
β = λe /4, γ = e2θ − e−2θ + 6λe4θ /4. 4θ
= α(2n + 1) + 6βn2 , n = 0, 1, 2, . . ., (9) † m + 1|T H T |m − 1
= m(m + 1) γ + 4(m − 1)β , m = 1, 2, 3, . . . , T † |m + 1 m − 1|TH
= m(m + 1) γ + 4(m − 1)β ,
(10) m = 1, 2, 3, . . . . (11)
Therefore, the condition γ + 4βn = 0,
n = 0, 1, 2, . . .
(12)
yields the θn values that allow an approximate diago. From Eqs. (7) and (8), nalization of Hamiltonian H and using the new variable Z = exp(−2θn ), condition (12) reads Z 3 − Z − (6 + 4n)λ = 0.
(13)
Eq. (13) can be solved in explicit analytical form using Cardano’s formula, obtaining 1 θn (λ) = − ln cn (λ) , 2 where
(14)
cn (λ) = cn+ (λ) + cn− (λ),
(15)
and
(5)
in which the number operator nˆ = aˆ † aˆ has been introduced and the α, β and γ parameters are given by α = e2θ + e−2θ + 3λe4θ /4,
T † |n n|TH
(6) (7) (8)
cn± (λ) = (2n + 3)1/3
× λ1/3 1 ± 1 −
1 27(2n + 3)2 λ2
1/3 . (16)
And, from Eqs. (6), (7) and (9), the approximate energy eigenvalues are given by 1 1 En (λ) = cn (λ)3 + cn (λ) + 3λ (2n + 1) 2 cn (λ) 4 3 + λn2 , n = 0, 1, 2, . . . . (17) 2
A.R. Matamala, C.R. Maldonado / Physics Letters A 308 (2003) 319–322
The excellent concordance between exact energy levels and those calculated by Eq. (17) for several values of λ parameter is showed in Table 1. In order to check
321
the validity of deleting the last term in Eq. (5), the actual value of β for the chosen values of n and λ has been included in Table 1.
Table 1 Energy for a quartic nonlinear oscillator obtained from Eq. (17). Exact energy values have been taken from Ref. [4] (exact)
%εr
0.559146 0.803771 1.50497 3.13138 6.69422
0.2076 1.0860 1.7462 1.9499 1.9977
n
λ
β
En (λ)
En
0
0.1 1.0 10 100 1000
0.016764 0.062500 0.156250 0.348160 0.755607
0.560307 0.812500 1.53125 3.19244 6.82795
1
0.1 1.0 10 100 1000
0.014246 0.046895 0.112531 0.248342 0.537836
1.77339 2.75994 5.38213 11.3249 24.2722
1.76950 2.73789 5.32161 11.1873 23.9722
0.2198 0.8054 0.8054 1.2300 1.2514
2
0.1 1.0 10 100 1000
0.012550 0.038498 0.090472 0.198706 0.429889
3.13934 5.17732 10.3371 21.8818 46.9615
3.13862 5.17929 10.3471 21.9069 47.0173
0.0229 0.0380 0.0966 0.1146 0.1187
3
0.1 1.0 10 100 1000
0.011309 0.033110 0.076805 0.168191 0.363639
4.62712 7.92882 16.0513 34.0936 73.2250
4.62888 7.94240 16.0901 34.1825 73.4191
0.0380 0.1710 0.2411 0.2601 0.2645
4
0.1 1.0 10 100 1000
0.010350 0.029302 0.067362 0.147214 0.318143
6.21765 10.9465 22.3619 47.6009 102.284
6.22030 10.9636 22.1088 47.7072 102.516
0.0426 0.1560 1.1448 0.2228 0.2253
5
0.1 1.0 10 100 1000
0.009580 0.026440 0.060378 0.131753 0.284639
7.89791 14.1890 29.1721 62.1915 133.682
7.89977 14.2031 29.2115 62.2812 133.877
0.0235 0.0993 0.1349 0.1440 0.1456
6
0.1 1.0 10 100 1000
0.008946 0.024195 0.054965 0.119803 0.258757
9.65835 17.6277 36.4165 77.7230 167.106
9.65784 17.6340 36.4369 77.7708 167.212
0.0053 0.0357 0.0560 0.0615 0.0634
7
0.1 1.0 10 100 1000
0.008411 0.022377 0.050625 0.110240 0.238056
11.4916 21.2420 44.0478 94.0922 202.353
11.4873 21.2364 44.0401 94.0780 202.311
0.0374 0.0264 0.0175 0.0151 0.0208
322
A.R. Matamala, C.R. Maldonado / Physics Letters A 308 (2003) 319–322
Table 2 Values of εn calculated by Eq. (19) and those reported in Ref. [4] n
εn (Eq. (19))
εn (Ref. [4])
%εr
0 1 2 3 4 5 6 7 8 9 10
0.681420222 2.423739027 4.691169765 7.316236392 10.22107529 13.35971785 16.70137472 20.22375309 23.90983826 27.74614126 31.72165752
0.66798621 2.39364402 4.69679539 7.33573001 10.2443085 13.3793366 16.7118896 20.2208495 23.8899937 27.7063935 31.6594566
2.0111 1.2573 0.1198 0.2657 0.2268 0.1466 0.0629 0.0146 0.0831 0.1435 0.1965
Now, working Eq. (17) in the large λ regime, we have En (λ) ∼ εn λ1/3 ,
λ 1,
This work was supported by Fundación ANDES (CHILE), project C-13760. Authors thank valuable comments of the anonymous referee.
References
14n2 + 22n + 9 . 22/34(2n + 3)2/3
(19)
This analytic result for εn is in good agreement with those calculated numerically in Ref. [4]. Although Eq. (19) falls to describe εn in the limit n → ∞, Table 2 shows that the present approach is a good approximation even for large nonlinearity coefficient λ. On the other hand, if 27(2n + 3)2 λ2 < 1, from the resolution of cubic equation (13) the coefficients cn (λ) can be written in the following form √ 1 2 cn (λ) = √ cos cos−1 27 λ(2n + 3) . (20) 3 3 Taking the limit λ → 0 in Eq. (20) we obtain cn (λ) = 1, and replacing this value in Eq. (17) the well-known energy levels of a harmonic oscillator are obtained: 1 lim En (λ) = n + , λ→0 2
Acknowledgements
(18)
where εn =
diagonalize the Hamiltonian of a one-dimensional quartic nonlinear oscillator. The present approach has two important advantages: (a) analytic formula for energy levels, and (b) a wide range of validity of the model. The method is simple and straightforward for obtaining energy levels with a very good accuracy even for large nonlinear parameter. Evaluation of matrix elements and transition probabilities are in progress.
n = 0, 1, 2, . . . .
(21)
4. Conclusions Using the ladder operators of harmonic oscillator a unitary transformation was defined in order to
[1] In the present Letter we adopt the term nonlinear oscillator instead of the traditional term anharmonic oscillator to designate an oscillator having a nonparabolic potential energy curve. A complete discussion about it is given by N.H. Fletcher, Am. J. Phys. 70 (2002) 1205. [2] C.M. Bender, T.T. Wu, Phys. Rev. 184 (1969) 1231. [3] K. Bhaumik, B. Dutta-Roy, J. Math. Phys. 16 (1975) 1131. [4] F.T. Hioe, D. MacMillen, E.W. Montroll, Phys. Rep. 43 (1978) 305. [5] C.S. Hsue, J.L. Chern, Phys. Rev. D 29 (1984) 643. [6] S.C. Chahjlany, D.A. Letov, V.N. Malney, J. Phys. A: Math. Gen. 24 (1991) 2731. [7] B. Bacus, Y. Meurice, A. Soemadi, J. Phys. A: Math. Gen. 28 (1995) L381. [8] S.M. McRae, E.R. Vrscay, J. Math. Phys. 38 (1997) 2899. [9] S. Mandal, J. Phys. A: Math. Gen. 31 (1998) L501. [10] I.A. Ivanov, J. Phys. A: Math. Gen. 31 (1998) 6995. [11] A. Pathak, S. Mandal, Phys. Lett. A 286 (2001) 261. [12] S. Mandal, Phys. Lett. A 299 (2002) 531. [13] M. Jafarpour, D. Afshar, J. Phys. A: Math. Gen. 35 (2002) 87. [14] F.M. Fernández, Introduction to Perturbation Theory, CRC Press, Boca Raton, 2000. [15] J. Récamier, R. Jáuregui, Int. J. Quantum Chem. Symp. 26 (1992) 153. [16] F.M. Fernández, E.A. Castro, Algebraic Methods in Quantum Chemistry and Physics, CRC Press, Boca Raton, 1996. [17] A.R. Matamala, J. Planelles, Int. J. Quantum Chem. 77 (2000) 704.
A simple algebraic approach to a nonlinear quantum oscillator Adelio R. Matamala ∗ , Carlos R. Maldonado Facultad de Ciencias Químicas, Grupo QTC, Universidad de Concepción, Casilla 160-C, Concepción, Chile Received 10 December 2002; received in revised form 27 December 2002; accepted 6 January 2003 Communicated by P.R. Holland
Abstract Using the well-known ladder operators formalism, a simple algebraic approach to a one-dimensional quartic nonlinear oscillator is presented. Making a linear canonical transformation and deleting one term, the Hamiltonian can be reduced to a form permitting to obtain explicit analytic formula for energy eigenvalues with a good accuracy even for large nonlinearity coefficients. 2003 Elsevier Science B.V. All rights reserved. PACS: 03.65.F Keywords: Nonlinear oscillator; Ladder operators; Algebraic method
1. Introduction Nonlinear oscillator [1] is a central model in molecular physics. Certainly, anharmonicities must be included in all realistic description of vibrational dynamics of molecular systems. The most simple model to describe anharmonic spectra consists of a onedimensional harmonic oscillator perturbed by a term containing a quartic power in the coordinate. This nonlinear quantum oscillator with a quartic perturbation has been studied extensively in the past [2–12]. Recently a very interesting work about the calculation of energy eigenvalues for the quantum nonlinear oscillator with a polynomial potential has been reported [13]. * Corresponding author.
E-mail address: [email protected] (A.R. Matamala).
In general, the evaluation of accurate energy levels of a quantum mechanical quartic nonlinear oscillator has become the testing ground for new methods in quantum mechanics [14]. In this way, algebraic approaches have showed that one can obtain a great deal of information with relatively small computational effort [15,16]. In the present Letter, a simple approximate diagonalization of a one-dimensional nonlinear oscillator is presented. Using the well-known ladder operators formalism, a unitary transformation is defined in order to map the original Hamiltonian into effective one. Deleting one term, the Hamiltonian of the quartic oscillator can be reduced to a form permitting to obtain energy eigenvalues with impressive accuracy even for large nonlinearity coefficients. This approach was previously used in the algebraic study of the asymmetric rigid rotor [17].
0375-9601/03/$ – see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0375-9601(03)00075-6
320
A.R. Matamala, C.R. Maldonado / Physics Letters A 308 (2003) 319–322
2. Hamiltonian
3. Diagonalization
Let us consider the Hamiltonian of a one-dimensional harmonic oscillator perturbed by a term containing a quartic power in the coordinate:
If the last term in Hamiltonian (5), β(aˆ †4 + aˆ 4 ), is neglected for the time being, the nonzero matrix in the elements of the transformed Hamiltonian H basis of the number operator, {|n }, are given by
= 1 pˆ 2 + xˆ 2 + λxˆ 4 , H 2
(1)
where the mass and frequency of oscillation has been settled to unit and the canonical quantization [x, ˆ p] ˆ = i Iˆ is assumed with h¯ = 1. Introducing the well-known ladder operators aˆ and aˆ † through the following realization 1 aˆ = √ (xˆ + i p) ˆ 2
1 and aˆ † = √ (xˆ − i p), ˆ 2
(2)
the Hamiltonian (1) reads 1 † H = aˆ aˆ + 2 λ + 3Iˆ + 6 aˆ †2 + 2aˆ † aˆ + aˆ 2 4 + aˆ †4 + 4aˆ †3aˆ + 6aˆ †2aˆ 2 + 4aˆ † aˆ 3 + aˆ 4 . (3) Now, the action of the following linear canonical transformation θ †2 2 T = exp − aˆ − aˆ , (4) 2 where θ is a real parameter, upon the Hamiltonian (3) permits to obtain
†
H = T HT = α 2nˆ + Iˆ + 6β nˆ 2 + γ aˆ †2 + aˆ 2 + 4β aˆ †2nˆ + nˆ aˆ 2 + β aˆ †4 + aˆ 4 ,
β = λe /4, γ = e2θ − e−2θ + 6λe4θ /4. 4θ
= α(2n + 1) + 6βn2 , n = 0, 1, 2, . . ., (9) † m + 1|T H T |m − 1
= m(m + 1) γ + 4(m − 1)β , m = 1, 2, 3, . . . , T † |m + 1 m − 1|TH
= m(m + 1) γ + 4(m − 1)β ,
(10) m = 1, 2, 3, . . . . (11)
Therefore, the condition γ + 4βn = 0,
n = 0, 1, 2, . . .
(12)
yields the θn values that allow an approximate diago. From Eqs. (7) and (8), nalization of Hamiltonian H and using the new variable Z = exp(−2θn ), condition (12) reads Z 3 − Z − (6 + 4n)λ = 0.
(13)
Eq. (13) can be solved in explicit analytical form using Cardano’s formula, obtaining 1 θn (λ) = − ln cn (λ) , 2 where
(14)
cn (λ) = cn+ (λ) + cn− (λ),
(15)
and
(5)
in which the number operator nˆ = aˆ † aˆ has been introduced and the α, β and γ parameters are given by α = e2θ + e−2θ + 3λe4θ /4,
T † |n n|TH
(6) (7) (8)
cn± (λ) = (2n + 3)1/3
× λ1/3 1 ± 1 −
1 27(2n + 3)2 λ2
1/3 . (16)
And, from Eqs. (6), (7) and (9), the approximate energy eigenvalues are given by 1 1 En (λ) = cn (λ)3 + cn (λ) + 3λ (2n + 1) 2 cn (λ) 4 3 + λn2 , n = 0, 1, 2, . . . . (17) 2
A.R. Matamala, C.R. Maldonado / Physics Letters A 308 (2003) 319–322
The excellent concordance between exact energy levels and those calculated by Eq. (17) for several values of λ parameter is showed in Table 1. In order to check
321
the validity of deleting the last term in Eq. (5), the actual value of β for the chosen values of n and λ has been included in Table 1.
Table 1 Energy for a quartic nonlinear oscillator obtained from Eq. (17). Exact energy values have been taken from Ref. [4] (exact)
%εr
0.559146 0.803771 1.50497 3.13138 6.69422
0.2076 1.0860 1.7462 1.9499 1.9977
n
λ
β
En (λ)
En
0
0.1 1.0 10 100 1000
0.016764 0.062500 0.156250 0.348160 0.755607
0.560307 0.812500 1.53125 3.19244 6.82795
1
0.1 1.0 10 100 1000
0.014246 0.046895 0.112531 0.248342 0.537836
1.77339 2.75994 5.38213 11.3249 24.2722
1.76950 2.73789 5.32161 11.1873 23.9722
0.2198 0.8054 0.8054 1.2300 1.2514
2
0.1 1.0 10 100 1000
0.012550 0.038498 0.090472 0.198706 0.429889
3.13934 5.17732 10.3371 21.8818 46.9615
3.13862 5.17929 10.3471 21.9069 47.0173
0.0229 0.0380 0.0966 0.1146 0.1187
3
0.1 1.0 10 100 1000
0.011309 0.033110 0.076805 0.168191 0.363639
4.62712 7.92882 16.0513 34.0936 73.2250
4.62888 7.94240 16.0901 34.1825 73.4191
0.0380 0.1710 0.2411 0.2601 0.2645
4
0.1 1.0 10 100 1000
0.010350 0.029302 0.067362 0.147214 0.318143
6.21765 10.9465 22.3619 47.6009 102.284
6.22030 10.9636 22.1088 47.7072 102.516
0.0426 0.1560 1.1448 0.2228 0.2253
5
0.1 1.0 10 100 1000
0.009580 0.026440 0.060378 0.131753 0.284639
7.89791 14.1890 29.1721 62.1915 133.682
7.89977 14.2031 29.2115 62.2812 133.877
0.0235 0.0993 0.1349 0.1440 0.1456
6
0.1 1.0 10 100 1000
0.008946 0.024195 0.054965 0.119803 0.258757
9.65835 17.6277 36.4165 77.7230 167.106
9.65784 17.6340 36.4369 77.7708 167.212
0.0053 0.0357 0.0560 0.0615 0.0634
7
0.1 1.0 10 100 1000
0.008411 0.022377 0.050625 0.110240 0.238056
11.4916 21.2420 44.0478 94.0922 202.353
11.4873 21.2364 44.0401 94.0780 202.311
0.0374 0.0264 0.0175 0.0151 0.0208
322
A.R. Matamala, C.R. Maldonado / Physics Letters A 308 (2003) 319–322
Table 2 Values of εn calculated by Eq. (19) and those reported in Ref. [4] n
εn (Eq. (19))
εn (Ref. [4])
%εr
0 1 2 3 4 5 6 7 8 9 10
0.681420222 2.423739027 4.691169765 7.316236392 10.22107529 13.35971785 16.70137472 20.22375309 23.90983826 27.74614126 31.72165752
0.66798621 2.39364402 4.69679539 7.33573001 10.2443085 13.3793366 16.7118896 20.2208495 23.8899937 27.7063935 31.6594566
2.0111 1.2573 0.1198 0.2657 0.2268 0.1466 0.0629 0.0146 0.0831 0.1435 0.1965
Now, working Eq. (17) in the large λ regime, we have En (λ) ∼ εn λ1/3 ,
λ 1,
This work was supported by Fundación ANDES (CHILE), project C-13760. Authors thank valuable comments of the anonymous referee.
References
14n2 + 22n + 9 . 22/34(2n + 3)2/3
(19)
This analytic result for εn is in good agreement with those calculated numerically in Ref. [4]. Although Eq. (19) falls to describe εn in the limit n → ∞, Table 2 shows that the present approach is a good approximation even for large nonlinearity coefficient λ. On the other hand, if 27(2n + 3)2 λ2 < 1, from the resolution of cubic equation (13) the coefficients cn (λ) can be written in the following form √ 1 2 cn (λ) = √ cos cos−1 27 λ(2n + 3) . (20) 3 3 Taking the limit λ → 0 in Eq. (20) we obtain cn (λ) = 1, and replacing this value in Eq. (17) the well-known energy levels of a harmonic oscillator are obtained: 1 lim En (λ) = n + , λ→0 2
Acknowledgements
(18)
where εn =
diagonalize the Hamiltonian of a one-dimensional quartic nonlinear oscillator. The present approach has two important advantages: (a) analytic formula for energy levels, and (b) a wide range of validity of the model. The method is simple and straightforward for obtaining energy levels with a very good accuracy even for large nonlinear parameter. Evaluation of matrix elements and transition probabilities are in progress.
n = 0, 1, 2, . . . .
(21)
4. Conclusions Using the ladder operators of harmonic oscillator a unitary transformation was defined in order to
[1] In the present Letter we adopt the term nonlinear oscillator instead of the traditional term anharmonic oscillator to designate an oscillator having a nonparabolic potential energy curve. A complete discussion about it is given by N.H. Fletcher, Am. J. Phys. 70 (2002) 1205. [2] C.M. Bender, T.T. Wu, Phys. Rev. 184 (1969) 1231. [3] K. Bhaumik, B. Dutta-Roy, J. Math. Phys. 16 (1975) 1131. [4] F.T. Hioe, D. MacMillen, E.W. Montroll, Phys. Rep. 43 (1978) 305. [5] C.S. Hsue, J.L. Chern, Phys. Rev. D 29 (1984) 643. [6] S.C. Chahjlany, D.A. Letov, V.N. Malney, J. Phys. A: Math. Gen. 24 (1991) 2731. [7] B. Bacus, Y. Meurice, A. Soemadi, J. Phys. A: Math. Gen. 28 (1995) L381. [8] S.M. McRae, E.R. Vrscay, J. Math. Phys. 38 (1997) 2899. [9] S. Mandal, J. Phys. A: Math. Gen. 31 (1998) L501. [10] I.A. Ivanov, J. Phys. A: Math. Gen. 31 (1998) 6995. [11] A. Pathak, S. Mandal, Phys. Lett. A 286 (2001) 261. [12] S. Mandal, Phys. Lett. A 299 (2002) 531. [13] M. Jafarpour, D. Afshar, J. Phys. A: Math. Gen. 35 (2002) 87. [14] F.M. Fernández, Introduction to Perturbation Theory, CRC Press, Boca Raton, 2000. [15] J. Récamier, R. Jáuregui, Int. J. Quantum Chem. Symp. 26 (1992) 153. [16] F.M. Fernández, E.A. Castro, Algebraic Methods in Quantum Chemistry and Physics, CRC Press, Boca Raton, 1996. [17] A.R. Matamala, J. Planelles, Int. J. Quantum Chem. 77 (2000) 704.