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Centrifugal distortion constants for diatomic molecules via the Prüfer phase function
JOURNAL OF MOLECULAR SPECTROSCOPY 124, 18% 189( 1987)
Centrifugal Distortion Constants for Diatomic Molecules via the Priifer Phase Function PETRI PAJUNEN Department of Chembry,
University of Oulu, Linnanmaa, 90570 Oulu, Finland
The perturbation theory approach to calculating centrifugal distortion constants is reformulated in terms of the Priifer phase function. The formulation does not involve wavefunctions and provides an efficient and stable method for all levels ofdiatomic molecule internuclear potentials. Numerical tests are performed and the method is compared with other computation techniques. 0 1987 Academic Press. Inc. 1. INTRODUCTION
The traditional energy level expression for a vibrating rotating diatomic molecule is
(1) E”J=lI$:‘+BJ(J+
l)-DJZ(J+
l)2+HJ3(J+
l)3+LJ4(J+
1)4+ - * -,
(1)
where E’,o’is the rotationless vibrational energy in vibrational state n, B, is the rotational constant, and D,, H,, and L, are centrifugal distortion constants. The centrifugal distortion constants are not independent parameters but are correlated (2) and the usual spectroscopic routine, to include enough centrifugal distortion constants for Eq. (1) to reproduce the data, often leads to difficulties in particular for highly excited vibrational levels for which the distortion constants have the largest magnitude. It is well known that the RKR equations (3-5) determine the potential energy curve from the Ei”’ and B, constants. Therefore, when fitting to spectroscopic data, the centrifugal distortion constants should be constrained to be consistent with E$“‘s and BGs. In practice, this may be done by determining preliminary vibrational and rotational constants, computing the approximate RKR curve, calculating the centrifugal distortion constants from this potential energy curve, and then holding these constants fixed in a fit to the data to yield improved Ei”‘s and Bv’s. From the improved constants, a new potential may be determined and the procedure iterated to convergence. Various methods have been devised to numerically calculate centrifugal distortion constants from the potential energy curve (6-14). The best method so far is a reformulation of the usual Rayleigh-Schriidinger perturbation approach solving the perturbation equation directly as a second-order inhomogeneous differential equation (1.3) thus avoiding all summations over some inevitably incomplete basis set. For each vibrational level, only the zeroth-order wavefunction and a few first perturbation wavefunctions are required, all of which are computed numerically. The method is stable even for vibrational levels near dissociation. In the present article, another variation of the above perturbation approach is presented. However, no wavefunctions are involved but the method is based on the Priifer 185
0022-2852187 $3.00 Copyright 0
1987 by Academic Press, Inc.
All rights of ~production in any fan” reSewed.
PETRI
186
PAJUNEN
phase function. In view of the known oscillation properties of the wavefunction the use of a phase function is expected to be computationally more efficient for very high quantum numbers. Furthermore, the present approach includes the phase integral approximation as a computational tool when appropriate, thus making the procedure computationally efficient for all vibrational levels. II. DERIVATION
OF THE
METHOD
From Eq. (l), the rotational constant B, and the centrifugal distortion constants &, H,, L,, - - - may be defined as derivatives of the energy with respect to J(J + 1) at constant V, all evaluated at J = 0 (15) B,=
aE [a.qJ+1)1
(2)
(4)
In the perturbation approach, the rotational and centrifugal distortion constants are associated with the first- and higher-order perturbation energies due to the centrifugal perturbation Hamiltonian. A general technique for calculating the perturbation energies by using the Prtlfer phase function approach is described elsewhere (16); in the present article a direct calculation of the energy derivatives, Eqs. (2)-(5) is presented. In the Priifer phase function approach (Z7-21), for each quantum number V, the corresponding eigenenergy E,(J(J + 1)) of the centrifugally perturbed potential l))= V(R)+&J(J+ CL is determined from the quantization condition
1)
v(R,J(J+
s R
Q(E,J(J+
l), v) =
v=o, 1,2, * * *
#(R, E,.J(J+ 1))dR = 0,
L
(7)
for the Prtifer phase function B(R, E,J(J+ 1)) defined by
‘WV
(8)
by integrating the differential equation
W(R,E,J(J+ l))= -sin%(R, E,.J(J+ 1)) E-V(R)-+++
1) cos28(R,E,J(f+ 1)) (9)
CENTRIFUGAL DISTORTION CONSTANTS
187
between the two boundary points denoted by L and R with appropriate boundary conditions at those points obtained by substituting Eq. (8) into the Schriidinger equation. It is clear that the quantization condition and the Priifer phase function are natural functions of E and J. Therefore, the rotational constant B, which is defined as the energy derivative of Eq. (2) and is obtained from the total differential
by holding u constant
(11) may be computed by integrating the differential equations {‘(R, E) =
aO’(R,E, J(J+ 1)) aJ(J+ 1)
=&os”B(R,E)+$
50 E-V(R,J(J+l))-g
flR,E)2sine(R,E)cosB(R,E)
(12)
and x’(R, E) =
af?‘(R,E, J(J+ 1)) aE +$
E-
= - $cos:B(R, :=0 V(R,J(J+ 1))-g
E) x(R, E)2 sin d(R, E)cos B(R, E)
(13)
between the limits L and R with appropriate boundary conditions. A straightforward repetition of the above procedure yields expressions for the centrifugal distortion constants
etc. in terms of the various derivatives of the Pri.ifer phase function. Application of the higher-order phase integral approximations as a computational device was described in Ref. (21) for determination of the energy eigenvalues. In that procedure, each integration of the differential equations, Eqs. (8), (1 l), or (12) or similar equations for higher derivatives, is replaced by evaluation of a set of phase integrals (22, 2.3) which is computationally much less expensive for many cases (14). III. TEST CALCULATIONS
To illustrate the stability and efficiency of the present method, test calculations of rotational and centrifugal distortion constants were carried out for a Lennard-Jones potential used by Hutson and other workers previously (12-24). The potential is characterized by the dimensionless well capacity parameter
188
PETRI PAJUNEN TABLE I Rotational and Centrifugal Distortion Constants of the Model Lennard-Jones
v
0
4
8
12
16
20
104xBv/~
0.98488097
0.85719599
0.71546709
0.55711253
0.37960014
0.18132533
22
0.07438966
23
0.01778898
10gxD,/c
-1014 XHJE
Potential
-1019xLJE
0.29342704
0.3500817
0.7685739
0.2934270
0.3500817
0.7685734
0.29342704
0.350081
0.7687
0.2934271
0.3500820
0.7685757
0.46182058
0.9374192
3.439432
0.4618206
0.9374192
3.439463
0.46182061
0.93741
3.441
0.4618206
0.9374200
3.439470
0.75878140
2.631149
16.11840
0.7587814
2.631149
16.11894
0.75878144
2.63115
16.118
0.7587815
2.631153
16.11900
1.3321678
8.535332
97.11170
1.332168
8.535332
97.11507
1.3321679
8.53538
97.06
1.332168
8.535360
97.11617
2.6393822
38.73948
1063.238
2.639382
38.73948
1063.248
2.6393824
38.740
1063.
2.639385
38.74008
1063.308
7.2535704
482.2355
7.25357
402.224
66930.
7.25441
483.9
68000.
7.253722
482.3865
67010.66
20.23573
110.0471
6688.777
1189434.
66932.89
8569023.
26901610000.
Nofe. For each level, the first entry is the result obtained by the present method. For centrifugal distortion constants, the second entry is the value by Hutson from Ref. (13), the third value is by Kirschner as quoted in Ref. (IZ), and the fourth is the third-order phase integral value by Tromp (14).
2q~R:/h~=10000,
(15)
where t is the well depth and R, the equilibrium distance. The potential supports 24 bound vibrational levels. The calculations by Hutson (13) using 10 000 mesh points
CENTRIFUGAL
DISTORTION
CONSTANTS
189
range from u = 0 to o = 20. For higher levels than that, the outer turning point is at such a large distance that, while in principle stable, the Cooley algorithm for the vibrational wavefunctions becomes very laborious. The present results shown in Table I compared with those of Hutson (13), with those of Kirschner using an earlier energy-derivative fitting method (IO) as quoted in Barwell (12), and with those of Tromp ( 14) using third-order phase integral methods are extended for all levels of the potential. The agreement with the best previous results is essentially exact for the levels considered in those calculations and all the figures of the present results are believed to be significant, confirming the stability of the present method for all vibrational levels of realistic diatomic molecule internuclear potentials. RECEIVED:
May 28, 1986 REFERENCES
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.
G. HERZBERG,“Spectra of Diatomic Molecules,” Van Nostrand, Princeton, NJ, 1950. J. L. DUNHAM,P&. Rev.41,721-731 (1932). R. RYDBERG, 2. Phys. 73,376-385 (1931); 80,514-524 (1932). 0. KLEIN, Z. Phyx 76,226-235 (1932). A. L. G. REES, Phys. Sot. 59,998-1008 (1947). C. L. BECKEL,J. Chem. Phys. 50,2372-2375 (1969). J. TELLINGHUISEN, Chem. Phys. Left. l&544-548 (1973). J. WEI AND J. TELLINGHUISEN, J. Mol. Spectrosc. 50, 3 17-332 (1974). D. L. ALBRITTON,W. J. HARROP, A. L. SCHMELTENKOPF, AND R. N. ZARE, J. Mol. Spectrosc. 46,2536 (1973). S. M. KIRSCHNERAND J. K. WATSON, J.Mol. Specwosc. 47,234-242 (1973). J. TELLINGHUISEN AND D. L. ALBRITTON,J. Mol. Spectrosc. 57, 160-163 (1975). M. G. BARWELL,M.Sc. thesis, University of Waterloo, 1976. J. M. HUTSON,J. Phys. B. 14, 851-857 (1981). J. W. TROMP, M.Sc. thesis, University of Waterloo, 1982. R. J. LE ROY, in ‘Semiclassical Methods in Molecular Scattering and Spectroscopy” (M. S. Child, Ed.) pp. 109- 126, Reidel, Dordrecht, 1980. P. PAJUNEN,J. Chem. Phys., submitted for publication. H. PROFER,Math. Ann. 95,499-5 18 (1926). P. B. BAILEY,SIAM J. Appl. Math. 14,242-249 (1966). B. A. HARGRAVE,J. Compul. Phys. 20,381-396 (1976). P. B. BAILEY, M. K. GORDON, AND L. F. SHAMPINE,ACM Trans. Math. So/ware 4, 193-208 (1978). P. PAJUNENAND J. TIENARI,J. Cornput. Phys. 65, 159-169 (1986). J. LUPPIAND P. PAJUNEN,J. Chem. Phys. 77, 1505-1511 (1982). J. LUPPIAND P. PAJUNEN,J. Chem. Phys. 81, 1836-I 840 (1984).
Centrifugal Distortion Constants for Diatomic Molecules via the Priifer Phase Function PETRI PAJUNEN Department of Chembry,
University of Oulu, Linnanmaa, 90570 Oulu, Finland
The perturbation theory approach to calculating centrifugal distortion constants is reformulated in terms of the Priifer phase function. The formulation does not involve wavefunctions and provides an efficient and stable method for all levels ofdiatomic molecule internuclear potentials. Numerical tests are performed and the method is compared with other computation techniques. 0 1987 Academic Press. Inc. 1. INTRODUCTION
The traditional energy level expression for a vibrating rotating diatomic molecule is
(1) E”J=lI$:‘+BJ(J+
l)-DJZ(J+
l)2+HJ3(J+
l)3+LJ4(J+
1)4+ - * -,
(1)
where E’,o’is the rotationless vibrational energy in vibrational state n, B, is the rotational constant, and D,, H,, and L, are centrifugal distortion constants. The centrifugal distortion constants are not independent parameters but are correlated (2) and the usual spectroscopic routine, to include enough centrifugal distortion constants for Eq. (1) to reproduce the data, often leads to difficulties in particular for highly excited vibrational levels for which the distortion constants have the largest magnitude. It is well known that the RKR equations (3-5) determine the potential energy curve from the Ei”’ and B, constants. Therefore, when fitting to spectroscopic data, the centrifugal distortion constants should be constrained to be consistent with E$“‘s and BGs. In practice, this may be done by determining preliminary vibrational and rotational constants, computing the approximate RKR curve, calculating the centrifugal distortion constants from this potential energy curve, and then holding these constants fixed in a fit to the data to yield improved Ei”‘s and Bv’s. From the improved constants, a new potential may be determined and the procedure iterated to convergence. Various methods have been devised to numerically calculate centrifugal distortion constants from the potential energy curve (6-14). The best method so far is a reformulation of the usual Rayleigh-Schriidinger perturbation approach solving the perturbation equation directly as a second-order inhomogeneous differential equation (1.3) thus avoiding all summations over some inevitably incomplete basis set. For each vibrational level, only the zeroth-order wavefunction and a few first perturbation wavefunctions are required, all of which are computed numerically. The method is stable even for vibrational levels near dissociation. In the present article, another variation of the above perturbation approach is presented. However, no wavefunctions are involved but the method is based on the Priifer 185
0022-2852187 $3.00 Copyright 0
1987 by Academic Press, Inc.
All rights of ~production in any fan” reSewed.
PETRI
186
PAJUNEN
phase function. In view of the known oscillation properties of the wavefunction the use of a phase function is expected to be computationally more efficient for very high quantum numbers. Furthermore, the present approach includes the phase integral approximation as a computational tool when appropriate, thus making the procedure computationally efficient for all vibrational levels. II. DERIVATION
OF THE
METHOD
From Eq. (l), the rotational constant B, and the centrifugal distortion constants &, H,, L,, - - - may be defined as derivatives of the energy with respect to J(J + 1) at constant V, all evaluated at J = 0 (15) B,=
aE [a.qJ+1)1
(2)
(4)
In the perturbation approach, the rotational and centrifugal distortion constants are associated with the first- and higher-order perturbation energies due to the centrifugal perturbation Hamiltonian. A general technique for calculating the perturbation energies by using the Prtlfer phase function approach is described elsewhere (16); in the present article a direct calculation of the energy derivatives, Eqs. (2)-(5) is presented. In the Priifer phase function approach (Z7-21), for each quantum number V, the corresponding eigenenergy E,(J(J + 1)) of the centrifugally perturbed potential l))= V(R)+&J(J+ CL is determined from the quantization condition
1)
v(R,J(J+
s R
Q(E,J(J+
l), v) =
v=o, 1,2, * * *
#(R, E,.J(J+ 1))dR = 0,
L
(7)
for the Prtifer phase function B(R, E,J(J+ 1)) defined by
‘WV
(8)
by integrating the differential equation
W(R,E,J(J+ l))= -sin%(R, E,.J(J+ 1)) E-V(R)-+++
1) cos28(R,E,J(f+ 1)) (9)
CENTRIFUGAL DISTORTION CONSTANTS
187
between the two boundary points denoted by L and R with appropriate boundary conditions at those points obtained by substituting Eq. (8) into the Schriidinger equation. It is clear that the quantization condition and the Priifer phase function are natural functions of E and J. Therefore, the rotational constant B, which is defined as the energy derivative of Eq. (2) and is obtained from the total differential
by holding u constant
(11) may be computed by integrating the differential equations {‘(R, E) =
aO’(R,E, J(J+ 1)) aJ(J+ 1)
=&os”B(R,E)+$
50 E-V(R,J(J+l))-g
flR,E)2sine(R,E)cosB(R,E)
(12)
and x’(R, E) =
af?‘(R,E, J(J+ 1)) aE +$
E-
= - $cos:B(R, :=0 V(R,J(J+ 1))-g
E) x(R, E)2 sin d(R, E)cos B(R, E)
(13)
between the limits L and R with appropriate boundary conditions. A straightforward repetition of the above procedure yields expressions for the centrifugal distortion constants
etc. in terms of the various derivatives of the Pri.ifer phase function. Application of the higher-order phase integral approximations as a computational device was described in Ref. (21) for determination of the energy eigenvalues. In that procedure, each integration of the differential equations, Eqs. (8), (1 l), or (12) or similar equations for higher derivatives, is replaced by evaluation of a set of phase integrals (22, 2.3) which is computationally much less expensive for many cases (14). III. TEST CALCULATIONS
To illustrate the stability and efficiency of the present method, test calculations of rotational and centrifugal distortion constants were carried out for a Lennard-Jones potential used by Hutson and other workers previously (12-24). The potential is characterized by the dimensionless well capacity parameter
188
PETRI PAJUNEN TABLE I Rotational and Centrifugal Distortion Constants of the Model Lennard-Jones
v
0
4
8
12
16
20
104xBv/~
0.98488097
0.85719599
0.71546709
0.55711253
0.37960014
0.18132533
22
0.07438966
23
0.01778898
10gxD,/c
-1014 XHJE
Potential
-1019xLJE
0.29342704
0.3500817
0.7685739
0.2934270
0.3500817
0.7685734
0.29342704
0.350081
0.7687
0.2934271
0.3500820
0.7685757
0.46182058
0.9374192
3.439432
0.4618206
0.9374192
3.439463
0.46182061
0.93741
3.441
0.4618206
0.9374200
3.439470
0.75878140
2.631149
16.11840
0.7587814
2.631149
16.11894
0.75878144
2.63115
16.118
0.7587815
2.631153
16.11900
1.3321678
8.535332
97.11170
1.332168
8.535332
97.11507
1.3321679
8.53538
97.06
1.332168
8.535360
97.11617
2.6393822
38.73948
1063.238
2.639382
38.73948
1063.248
2.6393824
38.740
1063.
2.639385
38.74008
1063.308
7.2535704
482.2355
7.25357
402.224
66930.
7.25441
483.9
68000.
7.253722
482.3865
67010.66
20.23573
110.0471
6688.777
1189434.
66932.89
8569023.
26901610000.
Nofe. For each level, the first entry is the result obtained by the present method. For centrifugal distortion constants, the second entry is the value by Hutson from Ref. (13), the third value is by Kirschner as quoted in Ref. (IZ), and the fourth is the third-order phase integral value by Tromp (14).
2q~R:/h~=10000,
(15)
where t is the well depth and R, the equilibrium distance. The potential supports 24 bound vibrational levels. The calculations by Hutson (13) using 10 000 mesh points
CENTRIFUGAL
DISTORTION
CONSTANTS
189
range from u = 0 to o = 20. For higher levels than that, the outer turning point is at such a large distance that, while in principle stable, the Cooley algorithm for the vibrational wavefunctions becomes very laborious. The present results shown in Table I compared with those of Hutson (13), with those of Kirschner using an earlier energy-derivative fitting method (IO) as quoted in Barwell (12), and with those of Tromp ( 14) using third-order phase integral methods are extended for all levels of the potential. The agreement with the best previous results is essentially exact for the levels considered in those calculations and all the figures of the present results are believed to be significant, confirming the stability of the present method for all vibrational levels of realistic diatomic molecule internuclear potentials. RECEIVED:
May 28, 1986 REFERENCES
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.
G. HERZBERG,“Spectra of Diatomic Molecules,” Van Nostrand, Princeton, NJ, 1950. J. L. DUNHAM,P&. Rev.41,721-731 (1932). R. RYDBERG, 2. Phys. 73,376-385 (1931); 80,514-524 (1932). 0. KLEIN, Z. Phyx 76,226-235 (1932). A. L. G. REES, Phys. Sot. 59,998-1008 (1947). C. L. BECKEL,J. Chem. Phys. 50,2372-2375 (1969). J. TELLINGHUISEN, Chem. Phys. Left. l&544-548 (1973). J. WEI AND J. TELLINGHUISEN, J. Mol. Spectrosc. 50, 3 17-332 (1974). D. L. ALBRITTON,W. J. HARROP, A. L. SCHMELTENKOPF, AND R. N. ZARE, J. Mol. Spectrosc. 46,2536 (1973). S. M. KIRSCHNERAND J. K. WATSON, J.Mol. Specwosc. 47,234-242 (1973). J. TELLINGHUISEN AND D. L. ALBRITTON,J. Mol. Spectrosc. 57, 160-163 (1975). M. G. BARWELL,M.Sc. thesis, University of Waterloo, 1976. J. M. HUTSON,J. Phys. B. 14, 851-857 (1981). J. W. TROMP, M.Sc. thesis, University of Waterloo, 1982. R. J. LE ROY, in ‘Semiclassical Methods in Molecular Scattering and Spectroscopy” (M. S. Child, Ed.) pp. 109- 126, Reidel, Dordrecht, 1980. P. PAJUNEN,J. Chem. Phys., submitted for publication. H. PROFER,Math. Ann. 95,499-5 18 (1926). P. B. BAILEY,SIAM J. Appl. Math. 14,242-249 (1966). B. A. HARGRAVE,J. Compul. Phys. 20,381-396 (1976). P. B. BAILEY, M. K. GORDON, AND L. F. SHAMPINE,ACM Trans. Math. So/ware 4, 193-208 (1978). P. PAJUNENAND J. TIENARI,J. Cornput. Phys. 65, 159-169 (1986). J. LUPPIAND P. PAJUNEN,J. Chem. Phys. 77, 1505-1511 (1982). J. LUPPIAND P. PAJUNEN,J. Chem. Phys. 81, 1836-I 840 (1984).