Abstract

A method is presented, based on recognition of the Cauchy dispersion equation as a series of Stieltjes, for the bounded extrapolation of long-wavelength refractivity data into the far ultraviolet. The extrapolation is performed with the [<i>n</i>, <i>n</i>-1] and [<i>n</i>,<i>n</i>] Padé approximants, which improve the convergence of the Cauchy equation and provide upper and lower bounds to its exact sum. Illustrative applications are given for atomic hydrogen, the inert gases, and molecular hydrogen, oxygen, and nitrogen. Comparison with the exact dispersion curve for atomic hydrogen and with available theoretical and experimental ultraviolet-dispersion data for the inert and diatomic gases indicates that the Padé approximants converge rapidly to accurate refractivity values. In addition, recognition of the Cauchy equation as a series of Stieltjes provides nontrivial constraints on the Cauchy coefficients; these afford a test of their accuracy and allow estimates of higher-order coefficients from measured refractivity data.

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  1. (He and Ne), C. Cuthbertson and M. Cuthbertson, Proc. Roy. Soc. (London) A135, 40 (1932); (Ar), E. R. Peck and J. Fisher, J. Opt. Soc. Am. 54, 1362 (1964), T. Larsen, Z. Physik 88, 389 (1934); (Kr and Xe), J. Kock, K. Fysiogr. Sallsk. Lund. 19, 173 (1949), W. Kronjager, Z. Physik 98, 17 (1935); (H2), J. Kock, Arkiv. Math. Astron. Fysik 8, 20 (1912), M. Kirn, Ann. Physik 64, 566 (1921); (N2), E. R. Peck and B. N. Khanna, J. Opt. Soc. Am. 56, 1059 (1966); (O2), E. Stoll, Ann. Physik 69, 81 (1922). For a review of earlier and additional work, see S. A. Korff and G. Breit, Rev. Mod. Phys. 4, 471 (1932).
  2. See, however, (Ar, Kr, Xe, H2, N2), D. W. O. Heddle, R. E. Jennings, and A. S. L. Parsons, J. Opt. Soc. Am. 53, 840 (1963), P. Gill and D. W. O. Heddle, 53, 847 (1963); (N2), P. G. Wilkinson, 50, 1002 (1960); (Ne, Ar), F. Marmo, Bull. Am. Phys. Soc. 9, 626 (1964); 10, 691 (1965).
  3. J. A. R. Samson, Adv. Atom. Mol. Phys. 2, 178 (1966); J. A. R. Samson and R. B. Cairns, J. Geophys. Res. 69, 4583 (1964); B. K. Ching, G. R. Cook, and R. A. Beeker, J. Quant. Spectry. Radiative. Transfer 7, 323 (1967); G. R. Cook and P. H. Metzger, J. Chem. Phys. 41, 321 (1964); R. E. Huffman, Y. Tanaka, and J. C. Larrabee, J. Chem. Phys. 39, 910 (1963).
  4. R. Migneron and J. S. Levinger, Phys. Rev. 139, A646 (1965); A. E. Kingston, J. Opt. Soc. Am. 54, 1145 (1964); A. Dalgarno, T. Degges, and D. A. Williams, Proc. Phys. Soc. (London) 92, 291 (1967); G. Liggett and J. S. Levinger, J. Opt. Soc. Am. 58, 109 (1968).
  5. G. A. Baker, Jr., in Advances in Theoretical Physics, K. A. Brueckner, Ed. (Academic Press Inc., New York, 1965), Vol. I, p. 1.
  6. H. S. Wall, Analytic Theory of Continued Fractions (D. Van Nostrand Co., Inc., Princeton, N. J., 1948), Chs. 17 and 20.
  7. M. Karplus and H. J. Kolker, J. Chem. Phys. 39, 1493 (1963).
  8. K. T. Chung, Phys. Rev. 166, 1 (1968); Y. M. Chan and A. Dalgarno, Proc. Phys. Soc. (London) 86, 777 (1965). The latter calculation is modified by a small empirical adjustment that slightly increases the resulting values; without this adjustment, the Chan and Dalgarno calculations would be slightly below the results of Chung.
  9. M. Karplus, J. Chem. Phys. 41, 880 (1964); G. A. Victor, J. C. Browne, and A. Dalgarno, Proc. Phys. Soc. (London) 92, 42 (1967).
  10. See, for example, J. C. Slater, Quantum Theory of Atomic Structure (McGraw-Hill Book Co., New York, 1960), Vol. I, p. 154.
  11. For explicit proof of this relation, see Ref. 5, pp. 15–16; we have dropped the absolute value sign given in that reference, because all of the derivatives are positive, by our definition of the series [Eqs. (3) and (4)] which differs by a minus sign from that of Baker; i.e., we do not explicitly introduce the variable z = -ω2.
  12. For detailed and explicit discussion of the Padé approximants to a series of Stieltjes for Rez ≥ 0, as well as of methods by which an arbitrary bounded function may be transformed to a series of Stieltjes, see Ref. 5, pp. 8–23.
  13. A. K. Common, J. Math. Phys. 9, 32 (1968).
  14. See also C. Schwartz, J. Comp. Phys. 1, 21 (1966).
  15. A. Dalgarno and A. E. Kingston, Proc. Rov. Soc. (London) A259, 424 (1960); the Kr and Xe Cauchy coefficients apparently do not satisfy Eq. (28) of the text.
  16. The necessary ω1 values for the atomic species are taken from, C. E. Moore, Atomic Energy Levels, Natl. Bur. Std. (U. S.) Circ. 467 (U. S. Gov't Printing Office, Waslington, D. C., 1949, 1952, 1958), Vols. I–III, while those for the molecular species are taken from G. Herzberg, Spectra of Diatomic Molecules (D. Van Nostrand Co., New York, 1950).
  17. J. A. Barker and P. J. Leonard, Phys. Letters 13, 127 (1964).
  18. W. Kolos and L. Wolniewicz, J. Chem. Phys. 46, 1426 (1967).
  19. The use of such constraining conditions to aid in evaluating the expansion coefficient αk. is similar to the recent work of F. Weinhold, J. Phys. A (Proc. Phys. Soc.) 1, 305 (1968). Since the αk coefficients are equal to the S(-2k - 2) sums, his proof that the gramian determinant formed from the sum rules is positive is closely related to Eq. (5) of the present paper.
  20. D. W. Marquardt, SIAM J. 11, 431 (1963).
  21. T. L. Saaty and J. Bram, Nonlinear Mathematics (McGraw-Hill Book Co., New York, 1964), p. 70.
  22. J. B. Rosen, SIAM J. 8, 181 (1960).

Baker, Jr., G. A.

G. A. Baker, Jr., in Advances in Theoretical Physics, K. A. Brueckner, Ed. (Academic Press Inc., New York, 1965), Vol. I, p. 1.

Barker, J. A.

J. A. Barker and P. J. Leonard, Phys. Letters 13, 127 (1964).

Bram, J.

T. L. Saaty and J. Bram, Nonlinear Mathematics (McGraw-Hill Book Co., New York, 1964), p. 70.

Chung, K. T.

K. T. Chung, Phys. Rev. 166, 1 (1968); Y. M. Chan and A. Dalgarno, Proc. Phys. Soc. (London) 86, 777 (1965). The latter calculation is modified by a small empirical adjustment that slightly increases the resulting values; without this adjustment, the Chan and Dalgarno calculations would be slightly below the results of Chung.

Common, A. K.

A. K. Common, J. Math. Phys. 9, 32 (1968).

Cuthbertson, C.

(He and Ne), C. Cuthbertson and M. Cuthbertson, Proc. Roy. Soc. (London) A135, 40 (1932); (Ar), E. R. Peck and J. Fisher, J. Opt. Soc. Am. 54, 1362 (1964), T. Larsen, Z. Physik 88, 389 (1934); (Kr and Xe), J. Kock, K. Fysiogr. Sallsk. Lund. 19, 173 (1949), W. Kronjager, Z. Physik 98, 17 (1935); (H2), J. Kock, Arkiv. Math. Astron. Fysik 8, 20 (1912), M. Kirn, Ann. Physik 64, 566 (1921); (N2), E. R. Peck and B. N. Khanna, J. Opt. Soc. Am. 56, 1059 (1966); (O2), E. Stoll, Ann. Physik 69, 81 (1922). For a review of earlier and additional work, see S. A. Korff and G. Breit, Rev. Mod. Phys. 4, 471 (1932).

Cuthbertson, M.

(He and Ne), C. Cuthbertson and M. Cuthbertson, Proc. Roy. Soc. (London) A135, 40 (1932); (Ar), E. R. Peck and J. Fisher, J. Opt. Soc. Am. 54, 1362 (1964), T. Larsen, Z. Physik 88, 389 (1934); (Kr and Xe), J. Kock, K. Fysiogr. Sallsk. Lund. 19, 173 (1949), W. Kronjager, Z. Physik 98, 17 (1935); (H2), J. Kock, Arkiv. Math. Astron. Fysik 8, 20 (1912), M. Kirn, Ann. Physik 64, 566 (1921); (N2), E. R. Peck and B. N. Khanna, J. Opt. Soc. Am. 56, 1059 (1966); (O2), E. Stoll, Ann. Physik 69, 81 (1922). For a review of earlier and additional work, see S. A. Korff and G. Breit, Rev. Mod. Phys. 4, 471 (1932).

Dalgarno, A.

A. Dalgarno and A. E. Kingston, Proc. Rov. Soc. (London) A259, 424 (1960); the Kr and Xe Cauchy coefficients apparently do not satisfy Eq. (28) of the text.

Heddle, D. W. O.

See, however, (Ar, Kr, Xe, H2, N2), D. W. O. Heddle, R. E. Jennings, and A. S. L. Parsons, J. Opt. Soc. Am. 53, 840 (1963), P. Gill and D. W. O. Heddle, 53, 847 (1963); (N2), P. G. Wilkinson, 50, 1002 (1960); (Ne, Ar), F. Marmo, Bull. Am. Phys. Soc. 9, 626 (1964); 10, 691 (1965).

Jennings, R. E.

See, however, (Ar, Kr, Xe, H2, N2), D. W. O. Heddle, R. E. Jennings, and A. S. L. Parsons, J. Opt. Soc. Am. 53, 840 (1963), P. Gill and D. W. O. Heddle, 53, 847 (1963); (N2), P. G. Wilkinson, 50, 1002 (1960); (Ne, Ar), F. Marmo, Bull. Am. Phys. Soc. 9, 626 (1964); 10, 691 (1965).

Karplus, M.

M. Karplus, J. Chem. Phys. 41, 880 (1964); G. A. Victor, J. C. Browne, and A. Dalgarno, Proc. Phys. Soc. (London) 92, 42 (1967).

M. Karplus and H. J. Kolker, J. Chem. Phys. 39, 1493 (1963).

Kingston, A. E.

A. Dalgarno and A. E. Kingston, Proc. Rov. Soc. (London) A259, 424 (1960); the Kr and Xe Cauchy coefficients apparently do not satisfy Eq. (28) of the text.

Kolker, H. J.

M. Karplus and H. J. Kolker, J. Chem. Phys. 39, 1493 (1963).

Kolos, W.

W. Kolos and L. Wolniewicz, J. Chem. Phys. 46, 1426 (1967).

Leonard, P. J.

J. A. Barker and P. J. Leonard, Phys. Letters 13, 127 (1964).

Levinger, J. S.

R. Migneron and J. S. Levinger, Phys. Rev. 139, A646 (1965); A. E. Kingston, J. Opt. Soc. Am. 54, 1145 (1964); A. Dalgarno, T. Degges, and D. A. Williams, Proc. Phys. Soc. (London) 92, 291 (1967); G. Liggett and J. S. Levinger, J. Opt. Soc. Am. 58, 109 (1968).

Marquardt, D. W.

D. W. Marquardt, SIAM J. 11, 431 (1963).

Migneron, R.

R. Migneron and J. S. Levinger, Phys. Rev. 139, A646 (1965); A. E. Kingston, J. Opt. Soc. Am. 54, 1145 (1964); A. Dalgarno, T. Degges, and D. A. Williams, Proc. Phys. Soc. (London) 92, 291 (1967); G. Liggett and J. S. Levinger, J. Opt. Soc. Am. 58, 109 (1968).

Moore, C. E.

The necessary ω1 values for the atomic species are taken from, C. E. Moore, Atomic Energy Levels, Natl. Bur. Std. (U. S.) Circ. 467 (U. S. Gov't Printing Office, Waslington, D. C., 1949, 1952, 1958), Vols. I–III, while those for the molecular species are taken from G. Herzberg, Spectra of Diatomic Molecules (D. Van Nostrand Co., New York, 1950).

Parsons, A. S. L.

See, however, (Ar, Kr, Xe, H2, N2), D. W. O. Heddle, R. E. Jennings, and A. S. L. Parsons, J. Opt. Soc. Am. 53, 840 (1963), P. Gill and D. W. O. Heddle, 53, 847 (1963); (N2), P. G. Wilkinson, 50, 1002 (1960); (Ne, Ar), F. Marmo, Bull. Am. Phys. Soc. 9, 626 (1964); 10, 691 (1965).

Rosen, J. B.

J. B. Rosen, SIAM J. 8, 181 (1960).

Saaty, T. L.

T. L. Saaty and J. Bram, Nonlinear Mathematics (McGraw-Hill Book Co., New York, 1964), p. 70.

Samson, J. A. R.

J. A. R. Samson, Adv. Atom. Mol. Phys. 2, 178 (1966); J. A. R. Samson and R. B. Cairns, J. Geophys. Res. 69, 4583 (1964); B. K. Ching, G. R. Cook, and R. A. Beeker, J. Quant. Spectry. Radiative. Transfer 7, 323 (1967); G. R. Cook and P. H. Metzger, J. Chem. Phys. 41, 321 (1964); R. E. Huffman, Y. Tanaka, and J. C. Larrabee, J. Chem. Phys. 39, 910 (1963).

Schwartz, C.

See also C. Schwartz, J. Comp. Phys. 1, 21 (1966).

Slater, J. C.

See, for example, J. C. Slater, Quantum Theory of Atomic Structure (McGraw-Hill Book Co., New York, 1960), Vol. I, p. 154.

Wall, H. S.

H. S. Wall, Analytic Theory of Continued Fractions (D. Van Nostrand Co., Inc., Princeton, N. J., 1948), Chs. 17 and 20.

Wolniewicz, L.

W. Kolos and L. Wolniewicz, J. Chem. Phys. 46, 1426 (1967).

Other

(He and Ne), C. Cuthbertson and M. Cuthbertson, Proc. Roy. Soc. (London) A135, 40 (1932); (Ar), E. R. Peck and J. Fisher, J. Opt. Soc. Am. 54, 1362 (1964), T. Larsen, Z. Physik 88, 389 (1934); (Kr and Xe), J. Kock, K. Fysiogr. Sallsk. Lund. 19, 173 (1949), W. Kronjager, Z. Physik 98, 17 (1935); (H2), J. Kock, Arkiv. Math. Astron. Fysik 8, 20 (1912), M. Kirn, Ann. Physik 64, 566 (1921); (N2), E. R. Peck and B. N. Khanna, J. Opt. Soc. Am. 56, 1059 (1966); (O2), E. Stoll, Ann. Physik 69, 81 (1922). For a review of earlier and additional work, see S. A. Korff and G. Breit, Rev. Mod. Phys. 4, 471 (1932).

See, however, (Ar, Kr, Xe, H2, N2), D. W. O. Heddle, R. E. Jennings, and A. S. L. Parsons, J. Opt. Soc. Am. 53, 840 (1963), P. Gill and D. W. O. Heddle, 53, 847 (1963); (N2), P. G. Wilkinson, 50, 1002 (1960); (Ne, Ar), F. Marmo, Bull. Am. Phys. Soc. 9, 626 (1964); 10, 691 (1965).

J. A. R. Samson, Adv. Atom. Mol. Phys. 2, 178 (1966); J. A. R. Samson and R. B. Cairns, J. Geophys. Res. 69, 4583 (1964); B. K. Ching, G. R. Cook, and R. A. Beeker, J. Quant. Spectry. Radiative. Transfer 7, 323 (1967); G. R. Cook and P. H. Metzger, J. Chem. Phys. 41, 321 (1964); R. E. Huffman, Y. Tanaka, and J. C. Larrabee, J. Chem. Phys. 39, 910 (1963).

R. Migneron and J. S. Levinger, Phys. Rev. 139, A646 (1965); A. E. Kingston, J. Opt. Soc. Am. 54, 1145 (1964); A. Dalgarno, T. Degges, and D. A. Williams, Proc. Phys. Soc. (London) 92, 291 (1967); G. Liggett and J. S. Levinger, J. Opt. Soc. Am. 58, 109 (1968).

G. A. Baker, Jr., in Advances in Theoretical Physics, K. A. Brueckner, Ed. (Academic Press Inc., New York, 1965), Vol. I, p. 1.

H. S. Wall, Analytic Theory of Continued Fractions (D. Van Nostrand Co., Inc., Princeton, N. J., 1948), Chs. 17 and 20.

M. Karplus and H. J. Kolker, J. Chem. Phys. 39, 1493 (1963).

K. T. Chung, Phys. Rev. 166, 1 (1968); Y. M. Chan and A. Dalgarno, Proc. Phys. Soc. (London) 86, 777 (1965). The latter calculation is modified by a small empirical adjustment that slightly increases the resulting values; without this adjustment, the Chan and Dalgarno calculations would be slightly below the results of Chung.

M. Karplus, J. Chem. Phys. 41, 880 (1964); G. A. Victor, J. C. Browne, and A. Dalgarno, Proc. Phys. Soc. (London) 92, 42 (1967).

See, for example, J. C. Slater, Quantum Theory of Atomic Structure (McGraw-Hill Book Co., New York, 1960), Vol. I, p. 154.

For explicit proof of this relation, see Ref. 5, pp. 15–16; we have dropped the absolute value sign given in that reference, because all of the derivatives are positive, by our definition of the series [Eqs. (3) and (4)] which differs by a minus sign from that of Baker; i.e., we do not explicitly introduce the variable z = -ω2.

For detailed and explicit discussion of the Padé approximants to a series of Stieltjes for Rez ≥ 0, as well as of methods by which an arbitrary bounded function may be transformed to a series of Stieltjes, see Ref. 5, pp. 8–23.

A. K. Common, J. Math. Phys. 9, 32 (1968).

See also C. Schwartz, J. Comp. Phys. 1, 21 (1966).

A. Dalgarno and A. E. Kingston, Proc. Rov. Soc. (London) A259, 424 (1960); the Kr and Xe Cauchy coefficients apparently do not satisfy Eq. (28) of the text.

The necessary ω1 values for the atomic species are taken from, C. E. Moore, Atomic Energy Levels, Natl. Bur. Std. (U. S.) Circ. 467 (U. S. Gov't Printing Office, Waslington, D. C., 1949, 1952, 1958), Vols. I–III, while those for the molecular species are taken from G. Herzberg, Spectra of Diatomic Molecules (D. Van Nostrand Co., New York, 1950).

J. A. Barker and P. J. Leonard, Phys. Letters 13, 127 (1964).

W. Kolos and L. Wolniewicz, J. Chem. Phys. 46, 1426 (1967).

The use of such constraining conditions to aid in evaluating the expansion coefficient αk. is similar to the recent work of F. Weinhold, J. Phys. A (Proc. Phys. Soc.) 1, 305 (1968). Since the αk coefficients are equal to the S(-2k - 2) sums, his proof that the gramian determinant formed from the sum rules is positive is closely related to Eq. (5) of the present paper.

D. W. Marquardt, SIAM J. 11, 431 (1963).

T. L. Saaty and J. Bram, Nonlinear Mathematics (McGraw-Hill Book Co., New York, 1964), p. 70.

J. B. Rosen, SIAM J. 8, 181 (1960).

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