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One‐ and Two‐Center Expansions of the Breit‐Pauli Hamiltonian
dc.contributor.authorFontana, Peter R.en_US
dc.contributor.authorMeath, William J.en_US
dc.date.accessioned2010-05-06T22:37:54Z
dc.date.available2010-05-06T22:37:54Z
dc.date.issued1968-09en_US
dc.identifier.citationFontana, Peter R.; Meath, William J. (1968). "One‐ and Two‐Center Expansions of the Breit‐Pauli Hamiltonian." Journal of Mathematical Physics 9(9): 1357-1364. <http://hdl.handle.net/2027.42/70736>en_US
dc.identifier.urihttps://hdl.handle.net/2027.42/70736
dc.description.abstractThe orbit‐orbit, spin‐spin, and spin‐orbit Hamiltonians of the Breit‐Pauli approximation are expressed in terms of irreducible tensors. One‐ and two‐center expansions are given in a form in which the coordinate variables of the interacting particles are separated. In the one‐center expansions of the orbit‐orbit and spin‐orbit Hamiltonians the use of the gradient formula reduces some of the infinite sums to finite ones. Two‐center expansions are discussed in detail for the case of nonoverlapping charge distributions. The angular parts of the matrix elements of these Hamiltonians are evaluated for product wavefunctions.en_US
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dc.publisherThe American Institute of Physicsen_US
dc.rights© The American Institute of Physicsen_US
dc.titleOne‐ and Two‐Center Expansions of the Breit‐Pauli Hamiltonianen_US
dc.typeArticleen_US
dc.subject.hlbsecondlevelPhysicsen_US
dc.subject.hlbtoplevelScienceen_US
dc.description.peerreviewedPeer Revieweden_US
dc.contributor.affiliationumPhysics Department, University of Michigan, Ann Arbor, Michiganen_US
dc.contributor.affiliationotherTheoretical Chemistry Institute, University of Wisconsin, Madison, Wisconsinen_US
dc.contributor.affiliationotherTheoretical Chemistry Institute, University of Wisconsin, Madison, Wisconsinen_US
dc.description.bitstreamurlhttp://deepblue.lib.umich.edu/bitstream/2027.42/70736/2/JMAPAQ-9-9-1357-1.pdf
dc.identifier.doi10.1063/1.1664722en_US
dc.identifier.sourceJournal of Mathematical Physicsen_US
dc.identifier.citedreferenceE. A. Power and S. Zienau, J. Franklin Inst. 263, 331 (1957).en_US
dc.identifier.citedreferenceW. J. Meath and J. O. Hirschfelder, J. Chem. Phys. 44, 3197, 3210 (1966).en_US
dc.identifier.citedreferenceH. B. L. Casimir and D. Polder, Phys. Rev. 73, 360 (1948); M. R. Aub, E. A. Power, and S. Zienau, Phil. Mag. 2, 571 (1957); E. A. Power and S. Zienau, Nuovo Cimento 6, 7 (1957); I. E. Dzialoshinskii, J. Exptl. Theoret. Phys. 3, 977 (1957); C. Mavroyannis and M. J. Stephen, Mol. Phys. 5, 629 (1962).en_US
dc.identifier.citedreferenceM. E. Rose, Elementary Theory of Angular Momentum (John Wiley & Sons, Inc., New York, 1957).en_US
dc.identifier.citedreferenceA. R. Edmonds, Angular Momentum in Quantum Mechanics (John Wiley & Sons, Inc., New York, 1957).en_US
dc.identifier.citedreferenceM. Blume and R. E. Watson, Proc. Roy. Soc. (London) A270, 127 (1962).en_US
dc.identifier.citedreferenceR. M. Pitzer, C. W. Kern, and W. N. Lipscomb, J. Chem. Phys. 37, 267 (1962); M. Geller and R. W. Griffith, J. Chem. Phys. 40, 2309 (1964); D. M. Schrader, J. Chem. Phys. 41, 3266 (1964).en_US
dc.identifier.citedreferenceThe starting point for this Hamiltonian is the Breit‐Hamiltonian: G. Breit, Phys. Rev. 34, 553 (1929); 36, 383 (1930); 39, 616 (1932).en_US
dc.identifier.citedreferenceH. A. Bethe and E. E. Salpeter, Quantum Mechanics of One‐ and Two‐Electron Atoms (Academic Press Inc., New York, 1957), p. 170.en_US
dc.identifier.citedreferenceThis Hamiltonian has recently been derived using quantum electrodynamics by T. Itoh, Rev. Mod. Phys. 37, 159 (1965).en_US
dc.identifier.citedreferenceJ. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, The Molecular Theory of Gases and Liquids (John Wiley & Sons, Inc., New York, 1954), p. 1044.en_US
dc.identifier.citedreferenceE. P. Wigner, Z. Phys. 43, 624 (1927); C. Eckart, Rev. Mod. Phys. 2, 305 (1930).en_US
dc.identifier.citedreferenceThe phase convention we use for the Ylm(0,φ)Ylm(0,φ) is the same as that used, for example, in E. U. Condon and G. H. Shortley, Theory of Atomic Spectra (Cambridge University Press, London, 1935), and in Refs. 4 and 5.en_US
dc.identifier.citedreferenceSee, for example, Ref. 4, p. 61.en_US
dc.identifier.citedreferenceClosed form expressions for these coefficients are available (see Refs. 4 and 5), and they are tabulated in Ref. 13. The 3‐j symbols, which are closely related to the Clebsch‐Gordan coefficients, have been tabulated in detail by M. Rotenberg, R. Bivins, N. Metropolis, and J. K. Wooten, Jr., The 3‐j and 6‐j Symbols (The Technology Press, Cambridge, Mass., 1959).en_US
dc.identifier.citedreferenceP. R. Fontana, Phys. Rev. 125, 220 (1962).en_US
dc.identifier.citedreferenceM. E. Rose, J. Math. & Phys. 37, 215 (1958).en_US
dc.identifier.citedreferenceR. A. Sack, J. Math. Phys. 5, 245 (1964); 5, 252 (1964).en_US
dc.identifier.citedreferenceP. R. Fontana, J. Math. Phys. 2, 825 (1961); Y. N. Chiu, J. Math. Phys. 5, 283 (1964).en_US
dc.identifier.citedreferenceThe transformed Hamiltonian has the following form HLL=−12∑k>j1rjk3[2r2k(pj⋅pk)−(rj×pk)(pj×pk)−(rjk×pk)⋅Ij−(rkj×pj)⋅Ik]. The terms of the form (rjk×pk)⋅Ik.(rjk×pk)⋅Ik. represent the coupling of the angular momentum of electron k relative to electron j with the angular momentum of electron j.en_US
dc.identifier.citedreferenceM. E. Rose, Multipole Fields (John Wiley & Sons, Inc., New York, 1955), p. 28.en_US
dc.identifier.citedreferenceR. J. Beuhler and J. O. Hirschfelder, Phys. Rev. 83, 628 (1951); 85, 149 (1952).en_US
dc.identifier.citedreferenceP. R. Fontana, Phys. Rev. 123, 1865 (1961).en_US
dc.identifier.citedreferenceR. A. Sack, J. Math. Phys. 5, 260 (1964).en_US
dc.identifier.citedreferenceThe result for n  =  1n=1 agrees with the previous work of R. C. Carlson and L. S. Rushbrooke, Proc. Cambridge Phil. Soc. 46, 626 (1950) and Refs. 17 and 22.en_US
dc.identifier.citedreferenceSee appendices 1.A–1.C of W. J. Meath, The University of Wisconsin Theoretical Chemistry Institute Technical Report WIS‐TCI‐75, April, 1965. For explicit expressions through (1/R3)(1∕R3) see W. J. Meath and J. O. Hirschfelder, J. Chem. Phys. 44, 3197 (1966).en_US
dc.identifier.citedreferenceSee, for example, Ref. 4, p. 62.en_US
dc.identifier.citedreferenceSee, for example, Ref. 4, p. 89.en_US
dc.owningcollnamePhysics, Department of


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