Abstract
The ansatz Ψ= (1+1/2r12)Φ+χ with Φ the bare nuclear (or screened nuclear) wave function and χ expanded in products of one-electron functions is explored for second-order perturbation theory and for variational calculations of the ground state of Helium-like ions.
The energy increments E (2)l corresponding to the partial wave expansion of χ go asymptotically as l−8, while conventional partial wave increments go as l−4. χ is coupled to Φ by a “residual” interaction U12 that has no singularity for r12=0. With the present ansatz it is sufficient to include l-values up to 5 in order to get the second-order energy accurate to one microhartree. For the same accuracy l≤4 is sufficient in a “CI with correlated reference function” while in conventional CI one must go to l∼50. The surprisingly faster convergence of the variational approach as compared to second-order perturbation theory is explained. The slow convergence of the traditional partial wave expansion is entirely due to the attempt to represent the quantity 1=〈Φ¦r12r12 −1¦Φ〉 by its partial wave expansion. The best reference function Φ shows very little shielding and resembles closely the eigenstate of the bare nuclear Hamiltonian. The generalization to arbitrary systems is discussed and it is pointed out that the calculation of “difficult” integrals can be avoided without a significant loss in accuracy.
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References and footnotes
For a review of the CI method see Shavitt, I. in Modern Theoretical Chemistry, III, Methods of Electronic Structure Theory (H. F. Schaefer III, Ed.), New York; Plenum, 1977. Many landmark papers on the CI method are reviewed in H. F. Schaefer III, Quantum Chemistry, Clarendon, Oxford 1984
It is not sufficient that the basis is complete with respect to an ordinary Hilbert space norm, but with respect to some Soboliev norm, the so-called convergence in energy space, as discussed by Klahn, B., Bingel, W. A.: Theoret. Chim. Acta (Berl.) 44, 9, 27 (1977)
Kutzelnigg, W.: Theoret. Chim. Acta 1, 327 343 (1963); Ahlrichs, R., Kutzelnigg, W.: J. Chem. Phys. 48, 1819 (1968)
Edmiston, C., Krauss, M.: J. Chem. Phys. 45, 1833 (1966)
For reviews see Meyer, W.: in Modern Theoretical Chemistry III, Methods of Electronic Structure Theory (H. F. Schaefer, III Ed., New York: Plenum 1977 and Kutzelnigg, W. ibidem
Bender, C. F., Davidson, E. R.: J. Chem. Phys. 70, 2675 (1966)
Carrol, D. P., Silverstone, H. J., Metzger, R. M.: J. Chem. Phys. 71, 4142 (1979); see also Bunge, C. F.: Theoret. Chim. Acta (Berl.) 16, 126 (1970); and Weiss, A.: Phys. Rev. 122, 1826 (1961)
Schwanz, C.: Phys. Rev. 126, 1015 (1962); Meth. Comp. Phys. 2, 241 (1963)
Lakin, W.: J. Chem. Phys. 43, 2954 (1965)
Kato, T.: Commun. Pure Appl. Math. 10, 151 (1957); see also Pack, R. T., Byers-Brown, W.: J. Chem. Phys. 45, 556 (1966)
Hylleraas, E. A.: Z. Phys. 54, 347 (1929), Z. Phys. 65, 209 (1930)
Kinoshita, T.: Phys. Rev. 105, 1490 (1957)
Pekeris, C. L.: Phys. Rev. 112, 1649 (1958); 115, 1216 (1959); 126, 1470 (1962); Frankowski, K., Pekeris, C. L.: Phys. Rev. 146, 46 (1966)
Larson, S.: Phys. Rev. 169, 49 (1968); A6, 1786 (1972); see also Perkins, J. F.: Phys. Rev. A5, 514 (1972) A8, 700 (1973)
Sims, J. S., Hagstrom, S.: Phys. Rev. A4, 908 (1971), 11, 418 (1975); Int. J. Quantum Chem. 9, 149 (1975); Sims, J. S., Hagstrom, S., Rumble, J. R.: Int. J. Quantum Chem. 10, 853 (1976)
Clary, D. C., Handy, N. C.: Chem. Phys. Letters 51, 483 (1977); Clary, D. C.: Mol. Phys. 34, 793 (1977)
Muszyńska, J., Papierowska, D., Woźnicki, W.: Chem. Phys. Letters 76, 136 (1980); Preiskorn, A., Woźnicki, W.: Chem. Phys. Lett. 86, 369 (1982); Mol. Phys. 52, 1291 (1984)
Longstaff, J. V., Singer, K.: Theoret. Chim. Acta (Berl.) 2, 265 (1964); Lester, W. A., Krauss, M.: J. Chem. Phys. 41, 1407 (1964); King, H. F.: J. Chem. Phys. 46, 705 (1967); Pan, K. C., King, H. F.: J. Chem. Phys. 53, 4397 (1970); 56, 4667 (1972); Handy, N. C.: Mol. Phys. 26, 169 (1973); Salmon, L., Poshusta, R. D.: J. Chem. Phys. 59, 3497 (1973); Adamowitz, L., Sadlej, A.: J. Chem. Phys. 67, 4298 (1977); 69, 3992 (1978)
Jeziorski, B., Szalewicz, K.: Phys. Rev. A19, 2360 (1979); Szalewicz, K., Jeziorski, B., Monkorst, H. J., Zabolitzky, J. G.: J. Chem. Phys. 78, 1420 (1983); 79, 5543 (1983); 81, 368 (1984); Chem. Phys. Letters 91, 169 (1982)
Green, L. C., Mulder, M. M., Milner, P. C.: Phys. Rev. 91, 35 (1953)
Schmidt, H. M., v. Hirschhausen, H.: Phys. Rev. A28, 3179 (1983); see also Byron, F. W., Joachain, C. J.: Phys. Rev. 157, 1 (1967)
Chandrasekhar, S.: Astrophys. J. 100, 176 (1944)
Grein, F., Tseng, T. J.: Chem. Phys. Lett. 7, 506 (1970); see also Hultgren, G. O., Kern, C. W.: Chem. Phys. Lett. 10, 233 (1971); and Löwdin, P. O., Redei, L.: Phys. Rev. 114, 752 (1959)
Boys, S. F.: Proc. Roy. Soc. A309, 195 (1968); Boys, S. F., Handy, N. C.: Proc. Roy. Soc. A310, 43, 63, A311, 309 (1969)
Scherr, C. W., Knight, R. E.: Revs. Mod. Phys. 35, 436 (1963)
Midtal, J.: Phys. Rev. 138, 1010 (1965)
A recent discussion on various definitions of the partial wave increments can be found in: Jankowski, K., Zaharewitz, D. W., Silverstone, H. J.: J. Chem. Phys. 82, 1969 (1985)
Kutzelnigg, W.: Isr. J. Chem. 19, 193 (1980); Schindler, M., Kutzelnigg, W.: J. Chem. Phys. 76, 1919 (1982)
Wallmeier, H., Kutzelnigg, W.: Chem. Phys. Letters 78, 341 (1981); Wallmeier, H.: Phys. Rev. A29, 2993 (1984)
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Kutzelnigg, W. r 12-Dependent terms in the wave function as closed sums of partial wave amplitudes for large l . Theoret. Chim. Acta 68, 445–469 (1985). https://doi.org/10.1007/BF00527669
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DOI: https://doi.org/10.1007/BF00527669