Abstract
Arguments are advanced to support the view that at present it is not possible to derive molecular structure from the full quantum mechanical Coulomb Hamiltonian associated with a given molecular formula that is customarily regarded as representing the molecule in terms of its constituent electrons and nuclei. However molecular structure may be identified provided that some additional chemically motivated assumptions that lead to the clamped nuclei Hamiltonian are added to the quantum mechanical account.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The work was completed in 1944 and was actually received by the journal in October 1948.
- 2.
An elementary example is afforded by the momentum operator \(\hat{p} = -i\hslash d/dq\), which is Hermitian on an appropriately defined class of \({L}^{2}\) functions \(\phi (q)\); for these functions it is self-adjoint on \(-\infty \leq q \leq +\infty \) but this property is lost if either of the \(\infty \) limits is replaced by any finite value a – see, for example, Thirring (1981).
- 3.
Some specifics of the implementation of permutational and rotational symmetry in quantum mechanics are discussed in section “The Symmetries of the Clamped Nuclei Electronic Hamiltonian.”
- 4.
A similar requirement must be placed on the denominator in Eq. 12 of Kutzelnigg (2007) for the equation to provide a secure definition.
- 5.
This means that the permutation and its inverse are always in the same class. A group with this property is said to be an ambivalent group.
- 6.
By “constant” here is meant simply that the elements of the matrix are not themselves dependent on the variables.
- 7.
It is sometimes convenient to think of the nuclear positions as defining a particular embedding for the basis vectors or coordinate frame.
- 8.
Notice that in this approximation the mass of the nucleus is of no consequence, only the charge matters.
- 9.
The operator in this form is clearly only possible for finite groups but similar operators are constructible for most infinite groups of interest.
- 10.
The Coulomb Hamiltonian for the electrons and nuclei specified by the molecular formula.
References
Born, M., & Huang, K. (1955). Dynamical theory of crystal lattices. Oxford: Oxford University Press.
Born, M., & Oppenheimer, J. R. (1927). Zur Quantentheorie der molekeln. Annalen der Physik, 84, 457.
Boys, S. F. (1950). Electronic wave functions. I. A general method of calculation for the stationary states of any molecular system. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 200, 542.
Cassam-Chenai, P. (2006). On non-adiabatic potential energy surfaces. Chemical Physics Letters, 420, 354.
Collins, M. A., & Parsons, D. F. (1993). Implications of rotation-inversion-permutation invariance for analytic molecular potential energy surfaces. The Journal of Chemical Physics, 99, 6756.
Combes, J. M., & Seiler, R. (1980). Spectral properties of atomic and molecular systems. In R. G. Woolley (Ed.), Quantum dynamics of molecules. NATO ASI B57 (p. 435). New York: Plenum.
Czub, J., & Wolniewicz, L. (1978). On the non-adiabatic potentials in diatomic molecules. Molecular Physics, 36, 1301.
Deshpande, V., & Mahanty, J. (1969). Born–Oppenheimer treatment of the hydrogen atom. American Journal of Physics, 37, 823.
Fock, V. (1930). Näherungsmethode zur Lösung des quantenmechanischen Mehrkörperproblems. Zeitschrift für Physik, 61, 126.
Frolov, A. M. (1999). Bound-state calculations of Coulomb three-body systems. Physical Review A, 59, 4270.
Hagedorn, G., & Joye, A. (2007). Mathematical analysis of Born–Oppenheimer approximations. In F. Gesztesy, P. Deift, C. Galvez, P. Perry, & W. Schlag. (Eds.), Spectral theory and mathematical physics: A festschrift in honor of Barry Simon’s 60th birthday (p. 203). London: Oxford University Press.
Handy, N. C., & Lee, A. M. (1996). The adiabatic approximation. Chemical Physics Letters, 252, 425.
Hartree, D. R. (1927). The wave mechanics of an atom with a non-Coulomb central field. Part I. Theory and methods. Mathematical Proceedings of the Cambridge Philosophical Society, 24, 89.
Hartree, D. R., & Hartree, W. (1936). Self-consistent field, with exchange, for beryllium. II. The (2s)(2p) \(^{3}\)P and \(^{1}\)P excited states. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 154, 588.
Heitler, W., & London, F. (1927). Wechselwirkung neutraler Atome und homöopolare Bindung nach der Quantenmechanik. Zeitschrift für Physik, 44, 455.
Herrin, J., & Howland, J. S. (1997). The Born–Oppenheimer approximation: Straight-up and with a twist. Reviews in Mathematical Physics, 9, 467.
Hinze, J., Alijah, A., & Wolniewicz, L. (1998). Understanding the adiabatic approximation; The accurate data of H\(_{2}\) transferred to H\(_{3}^{+}\). Polish Journal of Chemistry, 72, 1293.
Hohenberg, P., & Kohn, W. (1964). Inhomogeneous electron gas. Physical Review, 136, 864.
Hunter, G. (1975). Conditional probability amplitudes in wave mechanics. International Journal of Quantum Chemistry, 9, 237.
Hunter, G. (1981). Nodeless wave functions and spiky potentials. International Journal of Quantum Chemistry, 19, 755.
Kato, T. (1951). On the existence of solutions of the helium wave equation. Transactions of the American Mathematical Society, 70, 212.
Klein, M., Martinez, A., Seiler, R., & Wang, X. P. (1992). On the Born-Oppenheimer expansion for polyatomic molecules. Communications in Mathematical Physics, 143, 607.
Kołos, W., & Wolniewicz, L. (1963). Nonadiabatic theory for diatomic molecules and its application to the hydrogen molecule. Reviews of Modern Physics, 35, 473.
Kutzelnigg, W. (2007). Which masses are vibrating or rotating in a molecule? Molecular Physics, 105, 2627.
Longuet-Higgins, H. C. (1963). The symmetry groups of non-rigid molecules. Molecular Physics, 6, 445.
McWeeny, R. (1950). Gaussian approximations to wave functions. Nature, 166, 21.
Mohallem, J. R., & Tostes, J. G. (2002). The adiabatic approximation to exotic leptonic molecules: Further analysis and a nonlinear equation for conditional amplitudes. Journal of Molecular Structure: Theochem, 580, 27.
Mulliken, R. S. (1931). Bonding power of electrons and theory of valence. Chemical Reviews, 9, 347.
Nakai, H., Hoshino, M., Miyamato, K., & Hyodo, S. (2005). Elimination of translational and rotational motions in nuclear orbital plus molecular orbital theory. The Journal of Chemical Physics, 122, 164101.
Pauling, L. (1939). The nature of the chemical bond. Ithaca: Cornell University Press.
Roothaan, C. C. J. (1951). New developments in molecular orbital theory. Reviews of Modern Physics, 23, 69.
Slater, J. C. (1930). Note on Hartree’s method. Physical Review, 35, 210.
Sutcliffe, B. T. (2000). The decoupling of electronic and nuclear motions in the isolated molecule Schrödinger Hamiltonian. Advances in Chemical Physics, 114, 97.
Sutcliffe, B. T. (2005). Comment on “Elimination of translational and rotational motions in nuclear orbital plus molecular orbital theory”. The Journal of Chemical Physics, 123, 237101.
Sutcliffe, B. T. (2007). The separation of electronic and nuclear motion in the diatomic molecule. Theoretical Chemistry Accounts, 118, 563.
Thirring, W. (1981). Quantum mechanics of atoms and molecules. A course in mathematical physics (Vol. 3, E. M. Harrell, Trans.). Berlin: Springer.
Wilson, E. B. (1979). On the definition of molecular structure in quantum mechanics. International Journal of Quantum Chemistry, 13, 5.
Woolley, R. G., & Sutcliffe, B. T. (1977). Molecular structure and the Born–Oppenheimer approximation. Chemical Physics Letters, 45, 393.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media B.V.
About this entry
Cite this entry
Sutcliffe, B., Woolley, R.G. (2012). The Position of the Clamped Nuclei Electronic Hamiltonian in Quantum Mechanics. In: Leszczynski, J. (eds) Handbook of Computational Chemistry. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0711-5_2
Download citation
DOI: https://doi.org/10.1007/978-94-007-0711-5_2
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-0710-8
Online ISBN: 978-94-007-0711-5
eBook Packages: Chemistry and Materials ScienceReference Module Physical and Materials ScienceReference Module Chemistry, Materials and Physics