On Rüdenberg's integral approximations and their unrestricted and combined use in molecular orbital theories of Hartree-Fock type
Rüdenberg's well‐known letter of 1951 contains two implications which still have not been employed so far in numerical computations: (1) Whenever all types of attraction and repulsion integrals are subject to the Rüdenberg approximations in its simplest form partially known already from Mullike...
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| Published in | International journal of quantum chemistry Vol. 76; no. 2; pp. 148 - 160 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
New York
John Wiley & Sons, Inc
15.01.2000
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0020-7608 1097-461X |
| DOI | 10.1002/(SICI)1097-461X(2000)76:2<148::AID-QUA5>3.0.CO;2-R |
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| Abstract | Rüdenberg's well‐known letter of 1951 contains two implications which still have not been employed so far in numerical computations: (1) Whenever all types of attraction and repulsion integrals are subject to the Rüdenberg approximations in its simplest form partially known already from Mulliken, the attractive, the Coulomb, as well as the exchange part of the restricted Hartree–Fock–Roothaan equation can be led back to the calculation of corresponding diagonal elements, only. Using Rüdenberg's more elaborate concepts, which are invariant with respect to the rotation of local coordinate axes, the complete Fock‐matrix representation can be constructed equivalently from only its own diagonal blocks, each belonging to one atom. Similar formulas are valid for the unrestricted Hartree–Fock theory of Pople and Nesbet. (2) If, however, one prefers to calculate all types of one‐ and two‐center integrals exactly as suggested in Rüdenberg's headline, the original simplicity of both representations is lost. Instead, one is led to more complicated expressions, which arise from the fact that Rüdenberg's integral formulas, when applied to certain kinds of three‐center repulsion integrals, imply considerable oversimplifications. In spite of this critical result, Rüdenberg's ideas offer an extension together with an interpretation of the semiempirical Wolfsberg and Helmholz recipe (better known from Hoffmann's “extended Hückel” theory), on the one hand, and of the “neglect of differential overlap” schemes ZDO and NDDO, on the other, from a common point of view. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 76: 148–160, 2000 |
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| AbstractList | Rüdenberg's well‐known letter of 1951 contains two implications which still have not been employed so far in numerical computations: (1) Whenever all types of attraction and repulsion integrals are subject to the Rüdenberg approximations in its simplest form partially known already from Mulliken, the attractive, the Coulomb, as well as the exchange part of the restricted Hartree–Fock–Roothaan equation can be led back to the calculation of corresponding diagonal elements, only. Using Rüdenberg's more elaborate concepts, which are invariant with respect to the rotation of local coordinate axes, the complete Fock‐matrix representation can be constructed equivalently from only its own diagonal blocks, each belonging to one atom. Similar formulas are valid for the unrestricted Hartree–Fock theory of Pople and Nesbet. (2) If, however, one prefers to calculate all types of one‐ and two‐center integrals exactly as suggested in Rüdenberg's headline, the original simplicity of both representations is lost. Instead, one is led to more complicated expressions, which arise from the fact that Rüdenberg's integral formulas, when applied to certain kinds of three‐center repulsion integrals, imply considerable oversimplifications. In spite of this critical result, Rüdenberg's ideas offer an extension together with an interpretation of the semiempirical Wolfsberg and Helmholz recipe (better known from Hoffmann's “extended Hückel” theory), on the one hand, and of the “neglect of differential overlap” schemes ZDO and NDDO, on the other, from a common point of view. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 76: 148–160, 2000 |
| Author | Koch, Wolfhard |
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| Notes | istex:47E353841AD4E9F4BB85CCAA37FE7E028F10DFE2 ArticleID:QUA5 ark:/67375/WNG-69RWQJ9J-L Dedicated to Professor Klaus Rüdenberg. |
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| References | Rüdenberg, K. J Chem Phys 1951, 19, 1433. (a) Billingsley, F. P., II; Bloor, J. E. Chem Phys Lett 1969, 4, 48. Cook, D. B. Structures and Approximations for Electrons in Molecules; Ellis Horwood, Chichester, 1978. (e) Newton, M. D.; Lathan, W. A.; Hehre, W. J.; Pople, J. A. J Chem Phys 1969, 51, 3927. (c) Newton, M. D.; Ostlund, N. S.; Pople, J. A. J Chem Phys 1968, 49, 5192. Nicholson, B. J. Adv Chem Phys 1970, 18, 249. Pople, J. A.; Beveridge, D. L. Approximate Molecular Orbital Theory; McGraw-Hill: New York, 1970, and references therein. (b) Harris, F. W.; Trautwein, A.; Delhalle, J. Chem Phys Lett 1980, 72, 315. Wolfsberg, M.; Helmholz, L. J Chem Phys 1952, 20, 837. Ruttink, P. J. A. Theor Chim Acta (Berl) 1966, 6, 83. (b) Billingsley, F. P., II; Bloor, J. E. J Chem Phys 1971, 55, 5178. (d) Newton, M. D. J Chem Phys 1969, 51, 3917. Mulliken, R. S. J Chim Phys 1949, 46, 500, 521. Parr, R. G. J Chem Phys 1952, 20, 1499. Zerner, M. C. In Reviews in Computational Chemistry; Lipkowitz, K. B.; Boyd, D. B., Eds.; VCH: New York, 1991; Vol. 2, p 313, and references therein. Pople, J. A.; Santry, D. P.; Segal, G. A. J Chem Phys 1965, 43, S 129. Roothaan, C. C. J. Rev Mod Phys 1951, 23, 69. Bouferguene, A.; Fares, M.; Hoggan, P. E. Int J Quantum Chem 1996, 57, 801. Blyholder, G.; Coulson, C. A. Theor Chim Acta (Berl) 1968, 10, 316. (f) Harris, F. E.; Rein, R. Theor Chim Acta (Berl) 1966, 6, 73. Szabo, A.; Ostlund, N. S. Modern Quantum Chemistry; Macmillan: New York, 1982. Hoffmann, R. J Chem Phys 1963, 39, 1368. Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes; Cambridge University Press: Cambridge, 1986. Ayed, O.; Bernard, E.; Silvi, B. J Mol Struct (Theochem) 1986, 135, 159. (g) Brown, R. D.; Burden, F. R.; Williams, G. R. Theor Chim Acta (Berl) 1970, 18, 98. Pople, J. A.; Nesbet, R. K. J Chem Phys 1954, 22, 571. (a) Newton, M. D.; Boer, F. P.; Lipscomb, W. N. J Am Chem Soc 1966, 88, 2353, 2361, 2367. Koch, W. Int J Quantum Chem 1999, 00, 000. 1952; 20 1949; 46 1965; 43 1986; 135 1954; 22 1951; 19 1966; 6 1969 1971 1968 1969 1969 1966 1970; 4 55 49 51 51 6 18 1966 1980; 88 72 1970 1991; 2 1951; 23 1986 1999; 00 1982 1996; 57 1970; 18 1968; 10 1963; 39 1978 |
| References_xml | – reference: Roothaan, C. C. J. Rev Mod Phys 1951, 23, 69. – reference: (a) Newton, M. D.; Boer, F. P.; Lipscomb, W. N. J Am Chem Soc 1966, 88, 2353, 2361, 2367. – reference: Mulliken, R. S. J Chim Phys 1949, 46, 500, 521. – reference: (a) Billingsley, F. P., II; Bloor, J. E. Chem Phys Lett 1969, 4, 48. – reference: Ruttink, P. J. A. Theor Chim Acta (Berl) 1966, 6, 83. – reference: Pople, J. A.; Nesbet, R. K. J Chem Phys 1954, 22, 571. – reference: Zerner, M. C. In Reviews in Computational Chemistry; Lipkowitz, K. B.; Boyd, D. B., Eds.; VCH: New York, 1991; Vol. 2, p 313, and references therein. – reference: Parr, R. G. J Chem Phys 1952, 20, 1499. – reference: (d) Newton, M. D. J Chem Phys 1969, 51, 3917. – reference: Ayed, O.; Bernard, E.; Silvi, B. J Mol Struct (Theochem) 1986, 135, 159. – reference: Nicholson, B. J. Adv Chem Phys 1970, 18, 249. – reference: Rüdenberg, K. J Chem Phys 1951, 19, 1433. – reference: Pople, J. A.; Santry, D. P.; Segal, G. A. J Chem Phys 1965, 43, S 129. – reference: Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes; Cambridge University Press: Cambridge, 1986. – reference: Wolfsberg, M.; Helmholz, L. J Chem Phys 1952, 20, 837. – reference: Blyholder, G.; Coulson, C. A. Theor Chim Acta (Berl) 1968, 10, 316. – reference: (c) Newton, M. D.; Ostlund, N. S.; Pople, J. A. J Chem Phys 1968, 49, 5192. – reference: Cook, D. B. Structures and Approximations for Electrons in Molecules; Ellis Horwood, Chichester, 1978. – reference: (e) Newton, M. D.; Lathan, W. A.; Hehre, W. J.; Pople, J. A. J Chem Phys 1969, 51, 3927. – reference: (g) Brown, R. D.; Burden, F. R.; Williams, G. R. Theor Chim Acta (Berl) 1970, 18, 98. – reference: Hoffmann, R. J Chem Phys 1963, 39, 1368. – reference: (b) Billingsley, F. P., II; Bloor, J. E. J Chem Phys 1971, 55, 5178. – reference: Pople, J. A.; Beveridge, D. L. Approximate Molecular Orbital Theory; McGraw-Hill: New York, 1970, and references therein. – reference: (b) Harris, F. W.; Trautwein, A.; Delhalle, J. Chem Phys Lett 1980, 72, 315. – reference: Bouferguene, A.; Fares, M.; Hoggan, P. E. Int J Quantum Chem 1996, 57, 801. – reference: (f) Harris, F. E.; Rein, R. Theor Chim Acta (Berl) 1966, 6, 73. – reference: Szabo, A.; Ostlund, N. S. Modern Quantum Chemistry; Macmillan: New York, 1982. – reference: Koch, W. Int J Quantum Chem 1999, 00, 000. – volume: 10 start-page: 316 year: 1968 publication-title: Theor Chim Acta (Berl) – volume: 2 start-page: 313 year: 1970 1991 – year: 1986 – volume: 135 start-page: 159 year: 1986 publication-title: J Mol Struct (Theochem) – volume: 20 start-page: 837 year: 1952 publication-title: J Chem Phys – year: 1982 – volume: 88 72 start-page: 2353 315 year: 1966 1980 publication-title: J Am Chem Soc Chem Phys Lett – volume: 22 start-page: 571 year: 1954 publication-title: J Chem Phys – volume: 57 start-page: 801 year: 1996 publication-title: Int J Quantum Chem – volume: 4 55 49 51 51 6 18 start-page: 48 5178 5192 3917 3927 73 98 year: 1969 1971 1968 1969 1969 1966 1970 publication-title: Chem Phys Lett J Chem Phys J Chem Phys J Chem Phys J Chem Phys Theor Chim Acta (Berl) Theor Chim Acta (Berl) – volume: 43 start-page: S 129 year: 1965 publication-title: J Chem Phys – volume: 23 start-page: 69 year: 1951 publication-title: Rev Mod Phys – volume: 00 start-page: 000 year: 1999 publication-title: Int J Quantum Chem – volume: 18 start-page: 249 year: 1970 publication-title: Adv Chem Phys – volume: 19 start-page: 1433 year: 1951 publication-title: J Chem Phys – volume: 39 start-page: 1368 year: 1963 publication-title: J Chem Phys – year: 1978 – volume: 20 start-page: 1499 year: 1952 publication-title: J Chem Phys – volume: 6 start-page: 83 year: 1966 publication-title: Theor Chim Acta (Berl – volume: 46 start-page: 500 year: 1949 publication-title: J Chim Phys |
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| SubjectTerms | extended Hückel theory (EHT) integral approximations according to Mulliken and Rüdenberg neglect of diatomic differential overlap (NDDO) unrestricted (and restricted) Hartree-Fock molecular orbitals zero differential overlap (ZDO) |
| Title | On Rüdenberg's integral approximations and their unrestricted and combined use in molecular orbital theories of Hartree-Fock type |
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