Journal of Mathematical Chemistry

, Volume 44, Issue 4, pp 967–980 | Cite as

On the computation of excited states with MCSCF methods

  • Mathieu Lewin
Original Paper


We discuss the theoretical and practical problems arising when trying to compute excited states of nonrelativistic electrons in a molecular system, by multiconfiguration (MCSCF) methods. These nonlinear models approximate the linear Schrödinger theory and are a generalization of the well-known Hartree–Fock approach. Due to the MCSCF nonlinearity, a theoretical definition of what should be a MCSCF excited state is not clear at all, contrarily to the ground state case. We compare various definitions used in Quantum Chemistry. We in particular stress that some defects may lead to important computational problems, already observed in Quantum Chemistry (root flipping). We then present a definition of MCSCF excited states based on a solid mathematical ground and compare it with the most used methods. This new definition leads to a completely new algorithm for computing the first excited state, which was proposed and tested in a collaboration with Cancès and Galicher. Numerical results are provided for the simple case of two-electron systems, as an illustration of the possible issues which can arise as consequences of the nonlinearity of the MCSCF method.


Excited State Mountain Pass Hamiltonian Matrix Slater Determinant Symmetry Subspace 
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  1. 1.
    E. Cancès, M. Defranceschi, W. Kutzelnigg, C. Le Bris, Y. Maday, Computational quantum chemistry: a primer, in Handbook of Numerical Analysis, vol. X. (Elsevier, Amsterdam, 2003), pp. 3–270Google Scholar
  2. 2.
    Cancès É., Galicher H., Lewin M. (2006) Computing electronic structures: a new multiconfiguration approach for excited states. J. Comput. Phys. 212: 73–98CrossRefGoogle Scholar
  3. 3.
    Eade R., Robb M. (1981) Direct minimization in MCSCF theory. The quasi-Newton method, Chem. Phys. Lett. 83(2): 362–368Google Scholar
  4. 4.
    Friesecke G. (2003) The multiconfiguration equations for atoms and molecules: charge quantization and existence of solutions, Arch. Rat. Mech. Anal. 169: 35–71CrossRefGoogle Scholar
  5. 5.
    Golab J., Yeager D., Jørgensen P. (1985) Multiple stationary point representation in MCSCF calculations, Chem. Phys. 93: 83–100Google Scholar
  6. 6.
    Jørgensen P., Olsen J., Yeager D. (1981) Generalizations of Newton–Raphson and multiplicity independent Newton–Raphson approaches in multiconfigurational Hartree–Fock theory, J. Chem. Phys. 75(12): 5802–5815Google Scholar
  7. 7.
    Jørgensen P., Swanstrøm P., Yeager D. (1983) Guaranteed convergence in ground state multiconfigurational self-consistent field calculations. J. Chem. Phys. 78(1): 347–356CrossRefGoogle Scholar
  8. 8.
    Le Bris C. (1994) A general approach for multiconfiguration methods in quantum molecular chemistry. Ann. Inst. H. Poincaré Anal. Non linéaire 11(6): 441–484Google Scholar
  9. 9.
    Lewin M. (2004) Solutions of the multiconfiguration equations in quantum chemistry. Arch. Ration. Mech. Anal. 171(1): 83–114CrossRefGoogle Scholar
  10. 10.
    McCourt M., McIver J. (1987) On the SCF calculation of excited states: singlet states in the two-electron problem. J. Comput. Chem. 8(4): 454–458CrossRefGoogle Scholar
  11. 11.
    Olsen J., Jørgensen P., Yeager D. (1982) Multiconfigurational Hartree–Fock studies of avoided curve crossing using the Newton–Raphson technique, J. Chem. Phys. 76(1): 527–542Google Scholar
  12. 12.
    Roos B.O. (1987) The complete active space self-consistent field method and its applications in electronic structure calculation. Ab initio methods in quantum chemistry—II, Adv. Chem. Phys. 69: 399–446Google Scholar
  13. 13.
    Shepard R. (1987) The multiconfiguration self-consistent field method. Ab initio methods in quantum chemistry – II, Adv. Chem. Phys. 69: 63–200Google Scholar
  14. 14.
    Werner H.-J. (1987) Matrix-formulated direct multiconfiguration self-consistent field and multiconfiguration reference Configuration-Interaction methods. Ab initio methods in quantum chemistry—II, Adv. Chem. Phys. 69: 1–62Google Scholar
  15. 15.
    Werner H.-J., Knowles P. (1985) A second order multiconfiguration SCF procedure with optimum convergence. J. Chem. Phys. 82(11): 5053–5063CrossRefGoogle Scholar
  16. 16.
    Werner H.-J., Meyer W. (1981) A quadratically convergent MCSCF method for the simultaneous optimization of several states. J. Chem. Phys. 74(10): 5794–5801CrossRefGoogle Scholar
  17. 17.
    Yeager D., Lynch D., Nichols J., Jørgensen P., Olsen J. (1982) Newton–Raphson approaches and generalizations in multiconfigurational self-consistent field calculations. J. Phys. Chem. 86: 2140–2153CrossRefGoogle Scholar
  18. 18.
    Zhislin G.M. (1960) A study of the spectrum of the Schrödinger operator for a system of several particles, (Russian) Trudy Moskov. Mat. Obšč. 9: 81–120Google Scholar
  19. 19.
    G.M. Zhislin, A.G. Sigalov, The spectrum of the energy operator for atoms with fixed nuclei on subspaces corresponding to irreducible representations of the group of permutations, (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 29, 835–860 (1965). Transl. A.M.S. Ser. 2 91, 263–296 (English translation)Google Scholar
  20. 20.
    DALTON, a molecular electronic structure program. See

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© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Département de MathématiquesCNRS, Université de Cergy-PontoiseCergy-Pontoise CedexFrance

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