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Fast, accurate RKR computations
JOURNAL
OF
MOLECUL.ZR
SPECTROSCOPY
Fast,
Accurate
44, 194-196
RKR
(1972)
Computations’
In the RKR method (1) of determining potential curves for diatomic molecules, the turning points of the rotationless curve are related to experimental data through integrals of form
ft'=
Cl
C” [G(v) -
G(d)]-"* dv’,
J”min
g* = l/c,
(1)
*” B(v’)[G(v) I Urni”
- G(v’)]-“2 dv’,
where cl is a constant which depends on the molecular reduced mass, and G(V) and B(v) are the conventional expressions for the vibrational energy and the rotational constant. The problem of accurately evaluating these integrals has cut a wide swath in the literature, as a number of authors have uaed a variety of methods to treat the singularity which occurs at the upper limit of integration (I-12). Several of these authors have alluded to the “impossibility” of employing a numerical quadrature over the entire range of integration. This impossibility notwithstanding, the quadrature scheme described here allows one to compute the required integrals with six-$gure accuracy by evaluating the integrand at only four poinls. The quadrature method of Gauss is ideally suited for estimating the above integrals, because this method does not require one to assess the integrand at the limits of integration (19). However, the singularity is still a source of possible difficulty which can cause the numerical integration to converge very slowly, unless this singularity is effectively removed by a properly chosen weight function. The general quadrature formula is
’ w(x)F,(x)dz M c HiF,(Zi). s-1 n
(2)
The first step in Eq. (2) is simply the transformation to standard form, involving the substitution v = (b + a)/2 + x(b - a)/2. In the second definition, w(x) is an arbitrary weight function, which can be chosen to remove any untoward behavior in F(z). The integral is then approximated by a sum over n points. In the method of Gauss the abscissae (5;) and weights (Hi) are chosen so as to give the highest order of accuracy for a polynomial form of FW(x), consistent with the selected weight function. The final approximation in (2) becomes exact when F,(x) is a polynomial of order 2n - 1 or lower. A commonly used form of Eq. (2) is the case W(X) = 1. Here the abscissae are roots of the Legendre polynomials, and the weights are related to these roots in a straightforward way (1s). Zare used the Gauss-Legendre quadrature in his original program (14); but since this quadrature ignores the singularity at v, Zare’s routine converges slowly. A moment’s reflection on the nature of the integrands in Eqs. (1) suggests that one utilize a weight func1 This work was supported by the New Zealand Universities Research Committee, the U. S. Air Force Office of Scientific Research (Grant AF-AFOSR-71-2134, University of Canterbury), and the Office of Naval Research (Grant N66014-67-A-0285-6091). 194 Copyright
0
1972 by Academic
All rights of reproduction
Press, Inc.
in any form reserved.
195
NOTES tion of form
(1 -
CC)-~~~,
s’
j(v')dv'
s
= cz l (1 -1
Vmill
x)-
F,(x) dx.
(3)
In fact, for a purely harmonic potential, F,(x) = 1, and the quadrature is exact. For the real potential curves descriptive of molecular states, F,(z) is expected to remain a slowly varying, well-behaved function, so that the sum in Eq. (2) should still represent the integral to a high degree of accuracy. The forms w(x) = 1, (1 - x)-i/z are special cases of the more general Mehler’s quadrature, w(x)
= (1 -
x)a(l
+ x)@,
ff, B > -1,
(4)
for which the (zi] and (HiJ in (2) are extracted from the Jacobi polynomials (19). Specifically, for fl = 0, (Y = -M, one obtains for n = 4 the following abscissae and weights: 21 = -0.844
313 216 98,
HI = 0.286 317 537 84,
~2 = -0.269
354 952 47,
HP = 0.628 988 549 88,
x3 =
0.447 631 372 26,
Hs = 0.887 296 386 41,
x4 =
0.932 703 463 86,
H4 = 1.025 824 650 61
(5)
I have tested this quadrature on several molecular states of 1~ , LiD and CO, and found its performance extraordinary in every case. Its reliability was verified by computing the curve for the X(iP) state of CO, for which a very high precision RKR curve has been published (12). The results of the 4-point summation fell between the values calculated by two methods (labeled “I” and “II”) for all levels v = O-28 given in Table I of Ref. (la). The A (IX+) states of the alkali hydrides are somewhat anomalous in that their potential curves are “superharmonic” (negative w,z,) over a portion of the energy range. Again, using the spectroscopic parameters of Crawford and Jorgensen (16) for LID, I found the 4point result to agree within 1 ppm with the value obtained by subdividing the integration interval 32 times. (When the interval was subdivided, each segment was evaluated over 4 points, using the Gauss-Legendre quadrature for the first m - 1 segments and the constants in Eq. (5) for the last segment.) The LiD computation was taken too = 25, although the experimental parameters are valid to only v - 16. As a particularly severe challenge to the method, I computed the B[O+u(~II)] curve of 1%to v = 75, at which point the energy is within 0.3% of the dissociation limit. The vibrational constants of this state (16) are good to only v N 70, but they predict a reasonable AG curve up to v - 75. At these high levels the method began to falter: It was necessary to subdivide the interval into 8 segments to achieve the desired ppm convergence (again by comparison with the 16- and 32-segment values). However, the 4-point, quadrature did give ppm precision for v 5 60. It seems likely that extending the (1 - x)- 1’2 weighted quadrature to 12 or 16 points would yield better results than subdividing the interval as was done here, but his modification has not yet been tried. This integration scheme enables one to generate entire RKR potential curves in about 1 set of computational time on a fast computer. For accurate assessment of eigenvalues, such as the work of Ref. (la), it may prove advantageous to produce the RKR curve at many more points, say intervals of size Av = 0.1 or 0.01, and thereby remove some of the burden on the interpolation routines used in solving the Schrodinger equation and obtaining the wavefunctions and eigenvalues. Also, with the accuracy now achievable with the latter methods, it should be feasible to derive higher-order rotation-vibration distortion parameters in a purely computational manner, by starting from the rotationless RKR curve,
NOTES
1%
which depends only on G(v) and H(c), and calculstiug eigeuvalues and appropriate espectation values for the effective poteutial ?‘(r, J) = T’(r, J = 0) + J(J + 1) jr1.2 REFERENCES 1. R. RYDRERG,2. Phys. 73, 376 (1931); 80, 514 (1933); 0. KLEIN, Z. Phys. 76, 22G (1932); A. L. G. REES, Proc. Phys. Sac. 69,998 (1947). 2. J. T. VANDERSLICE, E. A. MASON,W. G. M.~Is~H,ANDE. R. LIPPINCOTT, J. illal. Spectrosc. 3, 17 (1959). 8. R. N. ZARE,J. Chem. Phys. 40,1934 (1964). 4. W. G. RICHARDSANDR. F. BARROW,Proc. Phys. Sac. 83,1045 (1964). 6. S. WEISSMAN, J. T. VANDERSLICE, ANDR. J. BATTINO,J. Chem. Phys. 39,2226 (1965). 6. R. J. SPINDLER, J. f&ant. Spectrosc. Radiat. Transfer 6, 165 (1965). 7. F. L. ZELEZNIK,J. Chem. Phys. 42,2836 (1965). 8. F. R. GILMORE,J. Quant. Spectrosc. Radiat. Transfer 6,369 (1965). 9. N. I. ZHIRNOVANDA. S. VASILEVSKII,Opt. Spectrosc. (Engl. Trans.) 26, 13 (1968). 10. W. R. JARMAIN,J. f&ant. Spectrosc. Radiat. Transfer 11,421 (1971). 11. J. A, COXON,J. Quant. Spectrosc. Radiat. Transfer 11,443 (1971). Id. A. W. MANTZ,J. K. G. WATSON,K. N~R.%H~RI Rao, D. L. ALBRITTON,A. L. SCHMELTEKOPF,.&ND R. N. ZARE,J. Mol. Spectrosc. 39,180 (1971). 19. Z. KOPAL, “Numerical Analysis,” Chapt . VII, Chapman and Hall, London, 1961. 14. R. N. ZARE,Lawrence Radiation Laboratory Report UCRL-16925 (1963). 16. F. H. CRAWFORD ANDT. JORGENSEN, Phys. Rev. 47,358 (1935). 16. J. TELLINGHUISMN, Resolution of the visible-infrared absorption spectrum of Is into three contributing transitions, J. Chem. Phys. Vol. 68, (1973), in press. JOEL TELLINGHUISEN Laboratory of Molecular Structure and Spectra, Department Received:
of Physics, University April 83, 1972
of Chicago,
Chicago,
Illinois
60637
2 Since I first submitted this paper, I have learned of similar work by A. S. Dickinson, also submitted to this journal. Dickinson has mentioned the advantages of the GaussMehIer quadrature with appropriate weight functions but has not tested the (1 - %)-1’2weighted quadrature discussed here.
OF
MOLECUL.ZR
SPECTROSCOPY
Fast,
Accurate
44, 194-196
RKR
(1972)
Computations’
In the RKR method (1) of determining potential curves for diatomic molecules, the turning points of the rotationless curve are related to experimental data through integrals of form
ft'=
Cl
C” [G(v) -
G(d)]-"* dv’,
J”min
g* = l/c,
(1)
*” B(v’)[G(v) I Urni”
- G(v’)]-“2 dv’,
where cl is a constant which depends on the molecular reduced mass, and G(V) and B(v) are the conventional expressions for the vibrational energy and the rotational constant. The problem of accurately evaluating these integrals has cut a wide swath in the literature, as a number of authors have uaed a variety of methods to treat the singularity which occurs at the upper limit of integration (I-12). Several of these authors have alluded to the “impossibility” of employing a numerical quadrature over the entire range of integration. This impossibility notwithstanding, the quadrature scheme described here allows one to compute the required integrals with six-$gure accuracy by evaluating the integrand at only four poinls. The quadrature method of Gauss is ideally suited for estimating the above integrals, because this method does not require one to assess the integrand at the limits of integration (19). However, the singularity is still a source of possible difficulty which can cause the numerical integration to converge very slowly, unless this singularity is effectively removed by a properly chosen weight function. The general quadrature formula is
’ w(x)F,(x)dz M c HiF,(Zi). s-1 n
(2)
The first step in Eq. (2) is simply the transformation to standard form, involving the substitution v = (b + a)/2 + x(b - a)/2. In the second definition, w(x) is an arbitrary weight function, which can be chosen to remove any untoward behavior in F(z). The integral is then approximated by a sum over n points. In the method of Gauss the abscissae (5;) and weights (Hi) are chosen so as to give the highest order of accuracy for a polynomial form of FW(x), consistent with the selected weight function. The final approximation in (2) becomes exact when F,(x) is a polynomial of order 2n - 1 or lower. A commonly used form of Eq. (2) is the case W(X) = 1. Here the abscissae are roots of the Legendre polynomials, and the weights are related to these roots in a straightforward way (1s). Zare used the Gauss-Legendre quadrature in his original program (14); but since this quadrature ignores the singularity at v, Zare’s routine converges slowly. A moment’s reflection on the nature of the integrands in Eqs. (1) suggests that one utilize a weight func1 This work was supported by the New Zealand Universities Research Committee, the U. S. Air Force Office of Scientific Research (Grant AF-AFOSR-71-2134, University of Canterbury), and the Office of Naval Research (Grant N66014-67-A-0285-6091). 194 Copyright
0
1972 by Academic
All rights of reproduction
Press, Inc.
in any form reserved.
195
NOTES tion of form
(1 -
CC)-~~~,
s’
j(v')dv'
s
= cz l (1 -1
Vmill
x)-
F,(x) dx.
(3)
In fact, for a purely harmonic potential, F,(x) = 1, and the quadrature is exact. For the real potential curves descriptive of molecular states, F,(z) is expected to remain a slowly varying, well-behaved function, so that the sum in Eq. (2) should still represent the integral to a high degree of accuracy. The forms w(x) = 1, (1 - x)-i/z are special cases of the more general Mehler’s quadrature, w(x)
= (1 -
x)a(l
+ x)@,
ff, B > -1,
(4)
for which the (zi] and (HiJ in (2) are extracted from the Jacobi polynomials (19). Specifically, for fl = 0, (Y = -M, one obtains for n = 4 the following abscissae and weights: 21 = -0.844
313 216 98,
HI = 0.286 317 537 84,
~2 = -0.269
354 952 47,
HP = 0.628 988 549 88,
x3 =
0.447 631 372 26,
Hs = 0.887 296 386 41,
x4 =
0.932 703 463 86,
H4 = 1.025 824 650 61
(5)
I have tested this quadrature on several molecular states of 1~ , LiD and CO, and found its performance extraordinary in every case. Its reliability was verified by computing the curve for the X(iP) state of CO, for which a very high precision RKR curve has been published (12). The results of the 4-point summation fell between the values calculated by two methods (labeled “I” and “II”) for all levels v = O-28 given in Table I of Ref. (la). The A (IX+) states of the alkali hydrides are somewhat anomalous in that their potential curves are “superharmonic” (negative w,z,) over a portion of the energy range. Again, using the spectroscopic parameters of Crawford and Jorgensen (16) for LID, I found the 4point result to agree within 1 ppm with the value obtained by subdividing the integration interval 32 times. (When the interval was subdivided, each segment was evaluated over 4 points, using the Gauss-Legendre quadrature for the first m - 1 segments and the constants in Eq. (5) for the last segment.) The LiD computation was taken too = 25, although the experimental parameters are valid to only v - 16. As a particularly severe challenge to the method, I computed the B[O+u(~II)] curve of 1%to v = 75, at which point the energy is within 0.3% of the dissociation limit. The vibrational constants of this state (16) are good to only v N 70, but they predict a reasonable AG curve up to v - 75. At these high levels the method began to falter: It was necessary to subdivide the interval into 8 segments to achieve the desired ppm convergence (again by comparison with the 16- and 32-segment values). However, the 4-point, quadrature did give ppm precision for v 5 60. It seems likely that extending the (1 - x)- 1’2 weighted quadrature to 12 or 16 points would yield better results than subdividing the interval as was done here, but his modification has not yet been tried. This integration scheme enables one to generate entire RKR potential curves in about 1 set of computational time on a fast computer. For accurate assessment of eigenvalues, such as the work of Ref. (la), it may prove advantageous to produce the RKR curve at many more points, say intervals of size Av = 0.1 or 0.01, and thereby remove some of the burden on the interpolation routines used in solving the Schrodinger equation and obtaining the wavefunctions and eigenvalues. Also, with the accuracy now achievable with the latter methods, it should be feasible to derive higher-order rotation-vibration distortion parameters in a purely computational manner, by starting from the rotationless RKR curve,
NOTES
1%
which depends only on G(v) and H(c), and calculstiug eigeuvalues and appropriate espectation values for the effective poteutial ?‘(r, J) = T’(r, J = 0) + J(J + 1) jr1.2 REFERENCES 1. R. RYDRERG,2. Phys. 73, 376 (1931); 80, 514 (1933); 0. KLEIN, Z. Phys. 76, 22G (1932); A. L. G. REES, Proc. Phys. Sac. 69,998 (1947). 2. J. T. VANDERSLICE, E. A. MASON,W. G. M.~Is~H,ANDE. R. LIPPINCOTT, J. illal. Spectrosc. 3, 17 (1959). 8. R. N. ZARE,J. Chem. Phys. 40,1934 (1964). 4. W. G. RICHARDSANDR. F. BARROW,Proc. Phys. Sac. 83,1045 (1964). 6. S. WEISSMAN, J. T. VANDERSLICE, ANDR. J. BATTINO,J. Chem. Phys. 39,2226 (1965). 6. R. J. SPINDLER, J. f&ant. Spectrosc. Radiat. Transfer 6, 165 (1965). 7. F. L. ZELEZNIK,J. Chem. Phys. 42,2836 (1965). 8. F. R. GILMORE,J. Quant. Spectrosc. Radiat. Transfer 6,369 (1965). 9. N. I. ZHIRNOVANDA. S. VASILEVSKII,Opt. Spectrosc. (Engl. Trans.) 26, 13 (1968). 10. W. R. JARMAIN,J. f&ant. Spectrosc. Radiat. Transfer 11,421 (1971). 11. J. A, COXON,J. Quant. Spectrosc. Radiat. Transfer 11,443 (1971). Id. A. W. MANTZ,J. K. G. WATSON,K. N~R.%H~RI Rao, D. L. ALBRITTON,A. L. SCHMELTEKOPF,.&ND R. N. ZARE,J. Mol. Spectrosc. 39,180 (1971). 19. Z. KOPAL, “Numerical Analysis,” Chapt . VII, Chapman and Hall, London, 1961. 14. R. N. ZARE,Lawrence Radiation Laboratory Report UCRL-16925 (1963). 16. F. H. CRAWFORD ANDT. JORGENSEN, Phys. Rev. 47,358 (1935). 16. J. TELLINGHUISMN, Resolution of the visible-infrared absorption spectrum of Is into three contributing transitions, J. Chem. Phys. Vol. 68, (1973), in press. JOEL TELLINGHUISEN Laboratory of Molecular Structure and Spectra, Department Received:
of Physics, University April 83, 1972
of Chicago,
Chicago,
Illinois
60637
2 Since I first submitted this paper, I have learned of similar work by A. S. Dickinson, also submitted to this journal. Dickinson has mentioned the advantages of the GaussMehIer quadrature with appropriate weight functions but has not tested the (1 - %)-1’2weighted quadrature discussed here.