Computing electronic structures: a new multiconfiguration approach for excited states. (English) Zbl 1078.81078
Summary: We present a new method for the computation of electronic excited states of molecular systems. This method is based upon a recent theoretical definition of multiconfiguration excited states due to one of us [see M. Lewin, Solutions of the multiconfiguration equations in quantum chemistry, Arch. Ration. Mech. Anal. 171, 83–114 (2004; Zbl 1063.81102)]. Our algorithm, dedicated to the computation of the first excited state, always converges to a stationary state of the multiconfiguration model, which can be interpreted as an approximate excited state of the molecule.
The definition of this approximate excited state is variational. An interesting feature is that it satisfies a non-linear Hylleraas-Undheim-MacDonald type principle: the energy of the approximate excited state is an upper bound to the true excited state energy of the \(N\)-body Hamiltonian. To compute the first excited state, one has to deform paths on a manifold, like this is usually done in the search for transition states between reactants and products on potential energy surfaces. We propose here a general method for the deformation of paths which could also be useful in other settings.
We also compare our method to other approaches used in Quantum Chemistry and give some explanation of the unsatisfactory behaviour which are sometimes observed when using the latter.
Numerical results for the special case of two-electron systems are provided: we compute the first singlet excited state potential energy surface of the \(H_{2}\) molecule.
The definition of this approximate excited state is variational. An interesting feature is that it satisfies a non-linear Hylleraas-Undheim-MacDonald type principle: the energy of the approximate excited state is an upper bound to the true excited state energy of the \(N\)-body Hamiltonian. To compute the first excited state, one has to deform paths on a manifold, like this is usually done in the search for transition states between reactants and products on potential energy surfaces. We propose here a general method for the deformation of paths which could also be useful in other settings.
We also compare our method to other approaches used in Quantum Chemistry and give some explanation of the unsatisfactory behaviour which are sometimes observed when using the latter.
Numerical results for the special case of two-electron systems are provided: we compute the first singlet excited state potential energy surface of the \(H_{2}\) molecule.
MSC:
| 81V55 | Molecular physics |
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 81V70 | Many-body theory; quantum Hall effect |
| 81Q15 | Perturbation theories for operators and differential equations in quantum theory |
| 81-08 | Computational methods for problems pertaining to quantum theory |
Keywords:
multiconfiguration method; excited state; time-independent Schrödinger equation; quantum chemistry; mountain pass method; minimax principle; Hartree-Fock theory; configuration-interaction methodCitations:
Zbl 1063.81102References:
| [1] | Head-Gordon, M., Quantum chemistry and molecular processes, J. Phys. Chem., 100, 13213-13225 (1996) |
| [2] | Löwdin, P., Quantum theory of many-particle systems. III. Extension of the Hartree-Fock scheme to include degenerate systems and correlation effects, Phys. Rev., 97, 6, 1509-1520 (1955) · Zbl 0065.44907 |
| [3] | Löwdin, P., Correlation problem in many-electron quantum mechanics. I. Review of different approaches and discussion of some current ideas, Adv. Chem. Phys., 2, 207-322 (1959) |
| [4] | Shepard, R., The multiconfiguration self-consistent field method. Ab initio methods in quantum chemistry - II, Adv. Chem. Phys., 69, 63-200 (1987) |
| [5] | Lewin, M., Solutions of the multiconfiguration equations in quantum chemistry, Arch. Rat. Mech. Anal., 171, 1, 83-114 (2004) · Zbl 1063.81102 |
| [6] | Friesecke, G., The multiconfiguration equations for atoms and molecules: charge quantization and existence of solutions, Arch. Rat. Mech. Anal., 169, 35-71 (2003) · Zbl 1035.81069 |
| [7] | Atkins, P.; Friedman, R., Molecular Quantum Mechanics (1997), Oxford University Press: Oxford University Press Oxford |
| [8] | Cancès, E.; Defranceschi, M.; Kutzelnigg, W.; Le Bris, C.; Maday, Y., Computational quantum chemistry: a primer, Handbook of Numerical Analysis, vol. X (2003), Elsevier: Elsevier Amsterdam, pp. 3-270 · Zbl 1070.81534 |
| [9] | Born, M.; Oppenheimer, R., Quantum theory of molecules, Ann. Phys., 84, 457-484 (1927) · JFM 53.0845.04 |
| [10] | Zhislin, G. M., Discussion of the spectrum of Schrodinger operators for systems of many particles, Trudy Moskovskogo matematiceskogo obscestva, 9, 81-120 (1960), (in Russian) |
| [11] | Löwdin, P., Quantum theory of many-particle systems. II. Study of the ordinary Hartree-Fock approximation, Phys. Rev., 97, 6, 1490-1508 (1955) · Zbl 0065.44907 |
| [12] | Kohn, W.; Sham, L., Self-consistent equations including exchange and correlation effects, Phys. Rev., 140, A1133-A1138 (1965) |
| [13] | Dreizler, R.; Gross, E., Density Functional Theory (1990), Springer-Verlag: Springer-Verlag Berlin · Zbl 0723.70002 |
| [14] | Löwdin, P., Quantum theory of many-particle systems. I. Physical interpretations by mean of density matrices, natural spin-orbitals, and convergence problems in the method of Configurational Interaction, Phys. Rev., 97, 6, 1474-1489 (1955) · Zbl 0065.44907 |
| [15] | Lieb, E.; Simon, B., The Hartree-Fock theory for Coulomb systems, Commun. Math. Phys., 53, 185-194 (1977) |
| [16] | Lions, P.-L., Solutions of Hartree-Fock equations for Coulomb systems, Commun. Math. Phys., 109, 33-87 (1987) · Zbl 0618.35111 |
| [17] | Le Bris, C., A general approach for multiconfiguration methods in quantum molecular chemistry, Ann. Inst. H. Poincaré Anal. Non linéaire, 11, 6, 441-484 (1994) · Zbl 0837.35005 |
| [18] | Werner, H.-J., Matrix-formulated direct multiconfiguration self-consistent field and multiconfiguration reference Configuration-Interaction methods. Ab initio methods in quantum chemistry - II, Adv. Chem. Phys., 69, 1-62 (1987) |
| [19] | Werner, H.-J.; Meyer, W., A quadratically convergent multiconfiguration-self-consistent field method with simultaneous optimization of orbitals and CI coefficients, J. Chem. Phys., 73, 5, 2342-2356 (1980) |
| [20] | Werner, H.-J.; Knowles, P., A second order multiconfiguration SCF procedure with optimum convergence, J. Chem. Phys., 82, 11, 5053-5063 (1985) |
| [21] | Eade, R.; Robb, M., Direct minimization in MCSCF theory. The quasi-Newton method, Chem. Phys. Lett., 83, 2, 362-368 (1981) |
| [22] | Frisch, M.; Ragazos, I.; Robb, M.; Schlegel, H., An evaluation of three direct MCSCF procedures, Chem. Phys. Lett., 189, 6, 524-528 (1992) |
| [23] | Jørgensen, P.; Olsen, J.; Yeager, D., Generalizations of Newton-Raphson and multiplicity independent Newton-Raphson approaches in multiconfigurational Hartree-Fock theory, J. Chem. Phys., 75, 12, 5802-5815 (1981) |
| [24] | Yeager, D.; Lynch, D.; Nichols, J.; Jørgensen, P.; Olsen, J., Newton-Raphson approaches and generalizations in multiconfigurational self-consistent field calculations, J. Phys. Chem., 86, 2140-2153 (1982) |
| [25] | Jørgensen, P.; Swanstrøm, P.; Yeager, D., Guaranteed convergence in ground state multiconfigurational self-consistent field calculations, J. Chem. Phys., 78, 1, 347-356 (1983) |
| [26] | Jensen, H., Electron correlation in molecules using direct second order MCSCF, Relativistic and Electron Correlation Effects in Molecules and Solids (1994), Plenum Press: Plenum Press New York, pp. 179-206 |
| [27] | Roos, B. O., 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, 399-446 (1987) |
| [28] | Golab, J.; Yeager, D.; Jørgensen, P., Multiple stationary point representation in MCSCF calculations, Chem. Phys., 93, 83-100 (1985) |
| [29] | Hylleraas, E.; Undheim, B., Numerische Berechnung der \(2 s\)-Terme von Orthound Par-Helium, Z. Phys., 65, 759-772 (1930) · JFM 56.1311.06 |
| [30] | MacDonald, J., Successive approximations by the Rayleigh-Ritz variation method, Phys. Rev., 43, 830-833 (1933) · Zbl 0007.11803 |
| [31] | Olsen, J.; Jørgensen, P.; Yeager, D., Multiconfigurational Hartree-Fock studies of avoided curve crossing using the Newton-Raphson technique, J. Chem. Phys., 76, 1, 527-542 (1982) |
| [32] | Cheung, L.; Elbert, S.; Ruedenberg, K., MCSCF optimization through combined use of natural orbital and the Brillouin-Levy-Berthier theorem, Int. J. Quantum Chem., 16, 1069-1101 (1979) |
| [33] | Lewis, A.; Overton, M., Eigenvalue optimization, Acta Numer., 5, 149-190 (1996) · Zbl 0870.65047 |
| [34] | Werner, H.-J.; Meyer, W., A quadratically convergent MCSCF method for the simultaneous optimization of several states, J. Chem. Phys., 74, 10, 5794-5801 (1981) |
| [35] | Reed, M.; Simon, B., Methods of modern mathematical physics, Analysis of Operators, vol. IV (1978), Academic Press: Academic Press New York · Zbl 0401.47001 |
| [36] | Docken, K.; Hinze, J., LiH potential curves and wavefunctions, J. Chem. Phys., 57, 11, 4928-4936 (1972) |
| [37] | McCourt, M.; McIver, J., On the SCF calculation of excited states: singlet states in the two-electron problem, J. Comput. Chem., 8, 4, 454-458 (1987) |
| [38] | DALTON, a molecular electronic structure program, See http://www.kjemi.uio.no/software/dalton/dalton.html; DALTON, a molecular electronic structure program, See http://www.kjemi.uio.no/software/dalton/dalton.html |
| [39] | Jensen, H.; Jørgensen, P., A direct approach to second-order MCSCF calculations using a norm extended optimization scheme, J. Chem. Phys., 80, 3, 1204-1214 (1984) |
| [40] | Schlegel, H., Optimization of equilibrium geometries and transition structures, Adv. Chem. Phys., 67, 249-286 (1987) |
| [41] | Quapp, W.; Heidrich, D., Analysis of the concept of minimum energy path on the potential energy surface of chemically reaction systems, Theoret. Chim. Acta, 66, 245-260 (1984) |
| [42] | Henkelman, G.; Jóhannesson, G.; Jónsson, H., (Schwartz, S. D., Methods for Finding Saddle Points and Minimum Energy Paths (2000), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht) |
| [43] | Henkelman, G.; Jónsson, H., A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives, J. Chem. Phys., 111, 15, 7010-7022 (1999) |
| [44] | Henkelman, G.; Jónsson, H.; Uberuaga, B., A climbing image nudged elastic band method for finding saddle points and minimum energy paths, J. Chem. Phys., 113, 22, 9901-9904 (2000) |
| [45] | Henkelman, G.; Jónsson, H., Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points, J. Chem. Phys., 113, 22, 9978-9985 (2000) |
| [46] | Chiu, S.-L.; McDouall, J.; Hiller, I., Prediction of whole reaction paths for large molecular systems, J. Chem. Soc., Faraday Trans., 90, 12, 1575-1579 (1994) |
| [47] | Elber, R.; Karplus, M., A method for determining reaction paths in large molecules: application to myoglobin, Chem. Phys. Lett., 139, 5, 375-380 (1987) |
| [48] | Jasien, P.; Shepard, R., A general polyatomic potential energy surface fitting method, Int. J. Quantum Chem., 22, 183-198 (1988) |
| [49] | Quapp, W.; Hirsch, H.; Imig, O.; Heidrich, D., Searching for saddle points of potential energy surfaces by following a reduced gradient, J. Comput. Chem., 19, 9, 1087-1100 (1998) |
| [50] | Czerminski, R.; Elber, R., Self-avoiding walk between two fixed points as a tool to calculate reaction paths in large molecular systems, Int. J. Quantum Chem., 24, 167-185 (1990) |
| [51] | E, W.; Ren, W.; Vanden-Eijnden, E., String method for the study of rare events, Phys. Rev. B, 66, 052301 (2002) |
| [52] | Choi, Y.; McKenna, P., A mountain pass method for the numerical solution of semilinear elliptic problems, Nonlinear Anal., 20, 4, 417-437 (1993) · Zbl 0779.35032 |
| [53] | Choi, Y.; McKenna, P.; Romano, M., A mountain pass method for the numerical solution of semilinear wave equations, Numer. Math., 64, 4, 487-509 (1993) · Zbl 0796.65109 |
| [54] | Liotard, D.; Penot, J., Study of Critical Phenomena (1981), Springer: Springer Berlin, p. 213 |
| [55] | Edelman, A.; Arias, T.; Smith, S., The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl., 20, 2, 303-353 (1998) · Zbl 0928.65050 |
| [56] | Quapp, W., Reaction pathways and projection operators: application to string methods, J. Comput. Chem., 25, 10, 1277-1285 (2004) |
| [57] | Stachó, L.; Bán, M., Theor. Chim. Acta, 83, 433-440 (1992) |
| [58] | Stachó, L.; Bán, M., Theor. Chim. Acta, 84, 535-543 (1993) |
| [59] | Stachó, L.; Bán, M., Procedure for determining dynamically defined reaction path, Comp. Chem., 17, 1, 21-25 (1993) |
| [60] | Bán, M.; Dömötör, G.; Stachó, L., Dynamically defined reaction path (DDRP) method, J. Mol. Struct.: THEOCHEM, 311, 29 (1994) |
| [61] | Landau, L.; Lifchitz, E., Quantum Mechanics (1977), Pergamon Press: Pergamon Press Oxford |
| [62] | Hehre, W.; Radom, L.; Schleyer, P.; Pople, J., Ab initio Molecular Orbital Theory (1986), Wiley: Wiley New York |
| [63] | Löwdin, P.-O.; Shull, H., Natural orbitals in the quantum theory of two-electron systems, Phys. Rev., 101, 6, 1730-1739 (1956) · Zbl 0070.23502 |
| [64] | Ando, T., Properties of fermions density matrices, Rev. Mod. Phys., 35, 3, 690-702 (1963) |
| [65] | Coleman, A., Structure of fermions density matrices, Rev. Mod. Phys., 35, 3, 668-689 (1963) |
| [66] | Coleman, A.; Yukalov, V., Reduced Density Matrices: Coulson’s Challenge (2000), Springer-Verlag: Springer-Verlag Berlin · Zbl 0998.81506 |
| [67] | Gomez, C.; Bunks, C.; Chancelier, J.; Delebecque, F.; Goursat, M.; Nikoukhah, R.; Steer, S., Engineering and Scientific Computing with Scilab (1999), Birkauser · Zbl 0949.68606 |
| [68] | Frisch, M.; Trucks, G.; Schlegel, H.; Scuseria, G.; Robb, M.; Cheeseman, J.; Zakrzewski, V.; Montgomery, J.; Stratmann, R.; Burant, J.; Dapprich, S.; Millam, J.; Daniels, A.; Kudin, K.; Strain, M.; Farkas, O.; Tomasi, J.; Barone, V.; Cossi, M.; Cammi, R.; Mennucci, B.; Pomelli, C.; Adamo, C.; Clifford, S.; Ochterski, J.; Petersson, G.; Ayala, P.; Cui, Q.; Morokuma, K.; Malick, D.; Rabuck, A.; Raghavachari, K.; Foresman, J.; Cioslowski, J.; Ortiz, J.; Stefanov, B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Gomperts, G.; Martin, R.; Fox, D.; Keith, T.; Al-Laham, M.; Peng, C.; Nanayakkara, A.; Gonzalez, C.; Challacombe, M.; Gill, P.; Johnson, B.; Chen, W.; Wong, M.; Andres, J.; Head-Gordon, M.; Replogle, E.; Pople, J., Gaussian 98 (Revision A.7) (1998), Gaussian Inc.: Gaussian Inc. Pittsburgh, PA |
| [69] | H.-J. Werner, P.J. Knowles, R. Lindh, M. Schütz, P. Celani, T. Korona, F.R. Manby, G. Rauhut, R.D. Amos, A. Bernhardsson, A. Berning, D.L. Cooper, M.J.O. Deegan, A.J. Dobbyn, F. Eckert, C. Hampel, G. Hetzer, A.W. Lloyd, S.J. McNicholas, W. Meyer, M.E. Mura, A. Nicklass, P. Palmieri, R. Pitzer, U. Schumann, H. Stoll, A.J. Stone, R. Tarroni, T. Thorsteinsson, Molpro, version 2002.6, A package of ab initio programs, 2003. Available from: <http://www.molpro.net>; H.-J. Werner, P.J. Knowles, R. Lindh, M. Schütz, P. Celani, T. Korona, F.R. Manby, G. Rauhut, R.D. Amos, A. Bernhardsson, A. Berning, D.L. Cooper, M.J.O. Deegan, A.J. Dobbyn, F. Eckert, C. Hampel, G. Hetzer, A.W. Lloyd, S.J. McNicholas, W. Meyer, M.E. Mura, A. Nicklass, P. Palmieri, R. Pitzer, U. Schumann, H. Stoll, A.J. Stone, R. Tarroni, T. Thorsteinsson, Molpro, version 2002.6, A package of ab initio programs, 2003. Available from: <http://www.molpro.net> |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
