- Email: [email protected]
Vibrational-rotational energy levels of diatomic molecules from the hill determinant method
Journal of Molecular Structure (Theochern), 166 (1988) 257-260 Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
257
VIBRATIONAL-ROTATIONAL ENERGYLEVELSOF DIATOMIC MOLECULESFROMTHE HILL DETERMINANTMETHOD D.A. ESTRIN, F.M. FERNANDEZand E.A. CASTRO Instituto de Investigaciones Fisicoquimicas Te6ricas y Aplicadas, DivisiSn Qu~mica Te6rica, Sucursal 4, Casilla de Correo 16, (1900) La Plata, Argentina
SUMMARY The Hill determinant method is applied to the perturbed-Morse-oscillator model. Accurate vibrational-rotational energies are obtained for the ground state of CO.
The Hill-determinant method proves to be suitable for computing highly accurate eigenvalues (refs. i-3). The procedure, f i r s t l y designed to treat onedimensional models with parity-invariant potentials (refs. 1,2) and central field problems (ref. 3), has recently been modified and applied to more general one-dimensional systems (ref. 4) and the vibrational-rotational spectra of diatomic molecules (ref. 5). The HDMconsists of writing the wavefunction ~(x) approximately as
~(x) = e- f ( x )
M ~ Cjx j+m j:O
,
(i)
where f(x) is properly chosen and the expansion coefficients Cj obey a recurrence relation obtained from the time-independent SchrBdinger equation (refs. I-5). The eigenvalues are then found to be the roots of either CM:O (refs. I-3) or CM_I=CM=O (refs. 4,5) for large enough M values. As shown by Huffaker (ref. 6) the potential energy function for a diatomic molecule can be accurately approximated by an expansion of the form N
V(r) : Vo(r) + D ~ j=4
bj(l-y) j
,
(2)
where y=exp{-a(r-ro)}, Vo(r)=D(y2-2y) is the Morse potential (ref. 7), and a, r 0 and D are its parameters. From now on the energy and length are given in units of M2/~r~ and r O, respectively. The eigenvalues of the radial SchrBdinger equation d2R/dr2 + 2{E - U(r)}R = O , 0166-1280/88/$03.50
© 1988El~vierSciencePubHshersB.V.
(3)
258
where U(r) is the effective potential V(r) + J(J+1)/2r 2 and J=0,1.... is the angular momentum quantum number, can be calculated through the HDM by expanding U(r) in power series of x=(r-ro)/r 0 (ref. 5). However, this procedure is not suitable when N in Eq. (2) is large enough. In this paper we show a modified HDM that applies to the perturbed-Morseoscillator (PMO) model (ref. 6) as given in Eq. (2). To this end we note that the eigenfunctions of the Morse oscillator can be written RtOJ(z)=e_Z/2zD/2F(z),,, L where z=2dy, d=(2D)l/2/a, b=2(-2E)1/2/a and F(z) is a polynomial function of z (ref. 7). Since the centrifugal term can be expanded in Taylor series around y=l as shown in the Appendix, we can write
(4)
U(r) = Vo(r) + D ~ uj(1-y) j j=l and R(z) = e-Z/2zb/2 jZO= Cj(1-z) j
(5)
The coefficients Cj are found to obey Cj+2 = I(j+1)(2j-2d+b+l)Cj+ I - {j(j-l)+j(b+l-4d)+d(2d-b-l)}Cj
J -d{2(j-1)-d+b+l}Cj_l+d 2 i ! 0 Cj_iu i
]
/{(j+l)(j+2)}
,
(6)
where j=0,1 .... and Cj=O i f j
(7)
The roots of this last equation are supposed to converge towards the actual eigenvalues as M tends to infinity (refs. 4,5,8). The low-lying rotational-vibrational energies for the ground electronic state of the CO molecule have been obtained using the parameters given in (ref. 7). Present results are compared in Table I with previous calculations (refs. 5,9) and experimental data (ref. 10). The HDMeigenvalues are supposed to be accurate up to the last figure and their slight disagreement with the experimental data is due to the inaccuracy of the PMO curve (ref. 7). As expected the HDM converges more quickly for the y-power series than for the expansion in powers of x
259
TABLE I V i b r a t i o n a l - r o t a t i o n a l eigenvalues (cm- I ) f o r the ground e l e c t r o n i c state of the CO molecule calcuated with the PMO model ( r e f .
7) and the HDM. v and J are the
v i b r a t i o n a l and r o t a t i o n a l quantum numbers, respectively, a: present, b: r e f . 5, c: v a r i a t i o n a l ( r e f . 9), d: h y p e r v i r i a l perturbative ( r e f . 9), e: numerical i n tegration ( r e f . 9), f : experimental ( r e f .
I0).
v=O J=O
J--2
J=10
a
1 081.777 8
1 093.312 6
1 293.180 04
b
1 081.777 8
1 093.312 6
1 293.180 04
c
1 081.777 7
1 093.312 5
1 293.179 9
d
1 081.777 7
1 093.326 3
1 293.571 0
e
1 081.777
1 093.321 5
1 293.188 8
f
1 081. 777 8 v=2 J=O
J=2
J=lO
a
3 225.050 17
3 236.480 04
3 434.526 81
b
3 225.050 24
3 236.480 09
3 434.526 81
f
3 225.050 2
3 236.480 09 v=4
J=O
J=2
J=lO
a
9 496.238 97
9 507.353 64
9 699.939 03
b
9 496.238 92
9 507.353 68
9 699.938 92
c
9 496.236 4
9 507.351 1
9 699.937 0
d
0 496.242 16
9 507.465 1
9 702.069 5
e
9 496.227
9 507.429 7
9 700.014 8
( r e f . 5). In f a c t , i t is found that the former requires about h a l f the number of terms to y i e l d the same accuracy. I t is also concluded that the Rayleigh-Ritz v a r i a t i o n a l method is more accurate than the other algorithms used by Requena et al ( r e f . 9). REFERENCES I
S.N. Biswas, Ko Datta, R.P. Saxena, P.K. Srivastava and V.S. Varma, Phys.Rev. D 4, 3617 (1971); J.Math.Phys. 14, 1190 (1973). 2 K. Banerjee, Lett.Math. Phys. 1, 3--23 (1976); Proc.R.Soc. London A 364, 265 (1978); K. Banerjee and S.P. Bhatnagar, Phys.Rev. D 18, 4767 (1978). 3 J. K i l l i n g b e c k , J.Phys. A 18, LI025 (1985); i b i d 18, 245 (1985); i b i d 19, 2903 (1986).
260
4 D.A. Estr~n, F.M. FernAndez and E.A. Castro (submitted for publication). 5 D.A. Estr~n, F.M. FernAndez and E.A. Castro (submitted for publication). 6 J.N. Huffaker, J.Mol.Spectrosc. 71, 160 (1978); J.Chem.Phys., 64, 8 (1976). 7 P. Morse, Phys.Rev. 34, 57 (1929). 8 F.M. FernAndez, J.F. Ogilvie and R.H. Tipping, J.Chem.Phys. 85, 5850 (1986). 9 A. Requena, J. ZuBiga and R. Pe~a, An. QuCmica, Ser. A 80, 169 (1984). 10 A.W. Mantz, J.K.G. Watson, K.N. Rao, D. Albritton, A.L. Schmeltekopf and R.N. Zare, J.Mol.Spectrosc. 39, 180 (1971).
Appendix The expansion of F(r)=(ro/r) 2 in powers of q=l-y, namely F(r)=Fo+Flq+F2q2+.... is required in order to apply the HDMto the PMOmodel. The analytical procedure proposed by Huffaker (ref. 6) becomes rather tedious as j in Fj increases. Instead, a most convenient recurrence relation for the coefficients F. is d e v ~ 1 2 j +1 _j + ed in what follows. Since @F/Bq = -2(ar)-l(1-q)-lF and ar=aro+q+~q +... ~q . . . , then
Fn+ 1 = _
[
n 1
a i
j=O
j+l Fj+I - 2 ~ Fn-j ~-~
The starting point is FO=I.
j~O
]
/{aro(n+l )}
. . .n=O,l ...
(A.I)
257
VIBRATIONAL-ROTATIONAL ENERGYLEVELSOF DIATOMIC MOLECULESFROMTHE HILL DETERMINANTMETHOD D.A. ESTRIN, F.M. FERNANDEZand E.A. CASTRO Instituto de Investigaciones Fisicoquimicas Te6ricas y Aplicadas, DivisiSn Qu~mica Te6rica, Sucursal 4, Casilla de Correo 16, (1900) La Plata, Argentina
SUMMARY The Hill determinant method is applied to the perturbed-Morse-oscillator model. Accurate vibrational-rotational energies are obtained for the ground state of CO.
The Hill-determinant method proves to be suitable for computing highly accurate eigenvalues (refs. i-3). The procedure, f i r s t l y designed to treat onedimensional models with parity-invariant potentials (refs. 1,2) and central field problems (ref. 3), has recently been modified and applied to more general one-dimensional systems (ref. 4) and the vibrational-rotational spectra of diatomic molecules (ref. 5). The HDMconsists of writing the wavefunction ~(x) approximately as
~(x) = e- f ( x )
M ~ Cjx j+m j:O
,
(i)
where f(x) is properly chosen and the expansion coefficients Cj obey a recurrence relation obtained from the time-independent SchrBdinger equation (refs. I-5). The eigenvalues are then found to be the roots of either CM:O (refs. I-3) or CM_I=CM=O (refs. 4,5) for large enough M values. As shown by Huffaker (ref. 6) the potential energy function for a diatomic molecule can be accurately approximated by an expansion of the form N
V(r) : Vo(r) + D ~ j=4
bj(l-y) j
,
(2)
where y=exp{-a(r-ro)}, Vo(r)=D(y2-2y) is the Morse potential (ref. 7), and a, r 0 and D are its parameters. From now on the energy and length are given in units of M2/~r~ and r O, respectively. The eigenvalues of the radial SchrBdinger equation d2R/dr2 + 2{E - U(r)}R = O , 0166-1280/88/$03.50
© 1988El~vierSciencePubHshersB.V.
(3)
258
where U(r) is the effective potential V(r) + J(J+1)/2r 2 and J=0,1.... is the angular momentum quantum number, can be calculated through the HDM by expanding U(r) in power series of x=(r-ro)/r 0 (ref. 5). However, this procedure is not suitable when N in Eq. (2) is large enough. In this paper we show a modified HDM that applies to the perturbed-Morseoscillator (PMO) model (ref. 6) as given in Eq. (2). To this end we note that the eigenfunctions of the Morse oscillator can be written RtOJ(z)=e_Z/2zD/2F(z),,, L where z=2dy, d=(2D)l/2/a, b=2(-2E)1/2/a and F(z) is a polynomial function of z (ref. 7). Since the centrifugal term can be expanded in Taylor series around y=l as shown in the Appendix, we can write
(4)
U(r) = Vo(r) + D ~ uj(1-y) j j=l and R(z) = e-Z/2zb/2 jZO= Cj(1-z) j
(5)
The coefficients Cj are found to obey Cj+2 = I(j+1)(2j-2d+b+l)Cj+ I - {j(j-l)+j(b+l-4d)+d(2d-b-l)}Cj
J -d{2(j-1)-d+b+l}Cj_l+d 2 i ! 0 Cj_iu i
]
/{(j+l)(j+2)}
,
(6)
where j=0,1 .... and Cj=O i f j
(7)
The roots of this last equation are supposed to converge towards the actual eigenvalues as M tends to infinity (refs. 4,5,8). The low-lying rotational-vibrational energies for the ground electronic state of the CO molecule have been obtained using the parameters given in (ref. 7). Present results are compared in Table I with previous calculations (refs. 5,9) and experimental data (ref. 10). The HDMeigenvalues are supposed to be accurate up to the last figure and their slight disagreement with the experimental data is due to the inaccuracy of the PMO curve (ref. 7). As expected the HDM converges more quickly for the y-power series than for the expansion in powers of x
259
TABLE I V i b r a t i o n a l - r o t a t i o n a l eigenvalues (cm- I ) f o r the ground e l e c t r o n i c state of the CO molecule calcuated with the PMO model ( r e f .
7) and the HDM. v and J are the
v i b r a t i o n a l and r o t a t i o n a l quantum numbers, respectively, a: present, b: r e f . 5, c: v a r i a t i o n a l ( r e f . 9), d: h y p e r v i r i a l perturbative ( r e f . 9), e: numerical i n tegration ( r e f . 9), f : experimental ( r e f .
I0).
v=O J=O
J--2
J=10
a
1 081.777 8
1 093.312 6
1 293.180 04
b
1 081.777 8
1 093.312 6
1 293.180 04
c
1 081.777 7
1 093.312 5
1 293.179 9
d
1 081.777 7
1 093.326 3
1 293.571 0
e
1 081.777
1 093.321 5
1 293.188 8
f
1 081. 777 8 v=2 J=O
J=2
J=lO
a
3 225.050 17
3 236.480 04
3 434.526 81
b
3 225.050 24
3 236.480 09
3 434.526 81
f
3 225.050 2
3 236.480 09 v=4
J=O
J=2
J=lO
a
9 496.238 97
9 507.353 64
9 699.939 03
b
9 496.238 92
9 507.353 68
9 699.938 92
c
9 496.236 4
9 507.351 1
9 699.937 0
d
0 496.242 16
9 507.465 1
9 702.069 5
e
9 496.227
9 507.429 7
9 700.014 8
( r e f . 5). In f a c t , i t is found that the former requires about h a l f the number of terms to y i e l d the same accuracy. I t is also concluded that the Rayleigh-Ritz v a r i a t i o n a l method is more accurate than the other algorithms used by Requena et al ( r e f . 9). REFERENCES I
S.N. Biswas, Ko Datta, R.P. Saxena, P.K. Srivastava and V.S. Varma, Phys.Rev. D 4, 3617 (1971); J.Math.Phys. 14, 1190 (1973). 2 K. Banerjee, Lett.Math. Phys. 1, 3--23 (1976); Proc.R.Soc. London A 364, 265 (1978); K. Banerjee and S.P. Bhatnagar, Phys.Rev. D 18, 4767 (1978). 3 J. K i l l i n g b e c k , J.Phys. A 18, LI025 (1985); i b i d 18, 245 (1985); i b i d 19, 2903 (1986).
260
4 D.A. Estr~n, F.M. FernAndez and E.A. Castro (submitted for publication). 5 D.A. Estr~n, F.M. FernAndez and E.A. Castro (submitted for publication). 6 J.N. Huffaker, J.Mol.Spectrosc. 71, 160 (1978); J.Chem.Phys., 64, 8 (1976). 7 P. Morse, Phys.Rev. 34, 57 (1929). 8 F.M. FernAndez, J.F. Ogilvie and R.H. Tipping, J.Chem.Phys. 85, 5850 (1986). 9 A. Requena, J. ZuBiga and R. Pe~a, An. QuCmica, Ser. A 80, 169 (1984). 10 A.W. Mantz, J.K.G. Watson, K.N. Rao, D. Albritton, A.L. Schmeltekopf and R.N. Zare, J.Mol.Spectrosc. 39, 180 (1971).
Appendix The expansion of F(r)=(ro/r) 2 in powers of q=l-y, namely F(r)=Fo+Flq+F2q2+.... is required in order to apply the HDMto the PMOmodel. The analytical procedure proposed by Huffaker (ref. 6) becomes rather tedious as j in Fj increases. Instead, a most convenient recurrence relation for the coefficients F. is d e v ~ 1 2 j +1 _j + ed in what follows. Since @F/Bq = -2(ar)-l(1-q)-lF and ar=aro+q+~q +... ~q . . . , then
Fn+ 1 = _
[
n 1
a i
j=O
j+l Fj+I - 2 ~ Fn-j ~-~
The starting point is FO=I.
j~O
]
/{aro(n+l )}
. . .n=O,l ...
(A.I)