Foundation of Multi-Configurational Quantum Chemistry
Giovanni Li Manni
Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany
Search for more papers by this authorKai Guther
Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany
Search for more papers by this authorDongxia Ma
Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany
Search for more papers by this authorWerner Dobrautz
Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany
Search for more papers by this authorGiovanni Li Manni
Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany
Search for more papers by this authorKai Guther
Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany
Search for more papers by this authorDongxia Ma
Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany
Search for more papers by this authorWerner Dobrautz
Max Planck Institute for Solid State Research, Heisenbergstrasse 1, 70569 Stuttgart, Germany
Search for more papers by this authorLeticia González
Institute of Theoretical Chemistry, Faculty of Chemistry, University of Vienna, Austria
Search for more papers by this authorRoland Lindh
Department of Chemistry – BMC, Uppsala University, Sweden
Search for more papers by this authorSummary
After introducing the fundamental goals—solving the Schrödinger equation—and the associated problems of quantum chemistry, we describe the basics of multiconfigurational approaches to solve the latter. As an exact—or full configuration interaction (FCI)—solution, even in a finite basis set, comes with an exponential scaling cost, the importance of an efficient representation in either a Slater determinant or configuration state function basis is discussed. With the help of such an efficient representation it is possible to apply iterative techniques, like the Davidson method, to obtain the exact solution of the most important low-lying eigenstates of the Hamiltonian, describing a quantum chemical system. As the exponential scaling still restricts these direct approaches to rather modest system sizes, we discuss in depth the multi-configurational extension of the self-consistent field method (MCSCF), which captures the static correlation of a problem and serves as a starting point for many more elaborate techniques. In addition, we present the complete active space approach—and the generalized and restricted extensions thereof—, which allows an intuitive construction of the chemically important reference space and enables a much more compact description of the important degrees of freedom of a problem at hand. We explain the state-specific and state-averaged approaches to obtain excited states within the MCSCF method and conclude this chapter by presenting stochastic Monte-Carlo approaches to solve the FCI problem for unprecedented active space sizes.
Bibliography
- Davidson, E.R. (1975). The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices. J. Comput. Phys. 17 (1): 87–94.
- Sleijpen, G.L.G. and Van der Vorst, H.A. (1996). A Jacobi-Davidson iteration method for linear eigenvalue problems. SIAM J. Matrix Anal. Appl. 17 (2): 401–425.
- Siegbahn, P.E.M. (1984). A new direct CI method for large CI expansions in a small orbital space. Chem. Phys. Lett. 109 : 417.
- Knowles, P.J. and Handy, N.C. (1984). A new determinant-based full configuration interaction method. Chem. Phys. Lett. 111 : 315.
- Handy, N.C. (1980). Multi-root configuration-interaction calculations. Chem. Phys. Lett. 74 (2): 280–283.
- Roos, B.O. (1972). New method for large-scale CI calculations. Chem. Phys. Lett. 15 (2): 153.
- Olsen, J., Roos, B.O., Jørgensen, P., and Jensen, H.J.Aa. (1988). Determinant based configuration interaction algorithms for complete and restricted configuration interaction spaces. J. Chem. Phys. 89 : 2185–2192.
-
Duch, W. (1986). GRMS or Graphical Representation of Model Spaces. Berlin: Springer.
10.1007/978-3-642-93347-9 Google Scholar
-
Shavitt, I. (1978). Matrix element evaluation in the unitary group approach to the electron correlation problem.
Int. J. Quantum Chem.
14
(S12): 5–32.
10.1002/qua.560140803 Google Scholar
- Shavitt, I. (1993). The history and evolution of Gaussian basis sets. Isr. J. Chem. 33 : 357.
- Moshinsky, M. (1968). Group Theory and the Many-body Problem. N.Y.: Gordon and Breach.
- Paldus, J. (1974). Group theoretical approach to the configuration interaction and perturbation theory calculations for atomic and molecular systems. J. Chem. Phys. 61 (12): 5321–5330.
-
Shavitt, I. (1977). Graph theoretical concepts for the unitary group approach to the many-electron correlation problem.
Int. J. Quantum Chem.
12
(S11): 131–148.
10.1002/qua.560120819 Google Scholar
- Cetlin, M.L. and Gel'fand, I.M. (1950). Finite-dimensional representations of the group of unimodular matrices. Doklady Akad. Nauk SSSR (N.S.) 71 : 825–828.
- Cetlin, M.L. and Gel'fand, I.M. (1950). Finite-dimensional representations of the group of orthogonal matrices. Doklady Akad. Nauk SSSR (N.S.) 71 : 1017–1020.Amer. Math. Soc. Transl. 64: 116 (1967).
- Gel'fand, I.M. (1950). The center of an infinitesimal group ring. Mat. Sb. (N.S.) 26 (68): 103–112.
- Louck, J.D. (1970). Recent progress toward a theory of tensor operators in the unitary groups. Am. J. Phys. 38 : 3–42.
-
Shavitt, I. (1981). The graphical unitary group approach and its application to direct configuration interaction calculations. In: The Unitary Group for the Evaluation of Electronic Energy Matrix Elements (ed. J. Hinze), 51–99. Berlin, Heidelberg: Springer Berlin Heidelberg.
10.1007/978-3-642-93163-5_2 Google Scholar
- Paldus, J. and Boyle, M.J. (1980). Unitary group approach to the many-electron correlation problem via graphical methods of spin algebras. Phys. Scr. 21 : 295.
- Drake, G.W.F. and Schlesinger, M. (1977). Vector-coupling approach to orbital and spin-dependent tableau matrix elements in the theory of complex spectra. Phys. Rev. A 15 : 1990–1999.
- Payne, P.W. (1982). Matrix element factorization in the unitary group approach for configuration interaction calculations. Int. J. Quantum Chem. 22 (6): 1085–1152.
- Jacobi, C.G.J. (1846). Über ein leichtes Verfahren, die in der Theorie der Säcularstörungen vorkommenden Gleichungen numerisch aufzulösen. Crelle's J. 30 : 51–94.
-
Mises, R.V. and Pollaczek-Geiringer, H. (1929). Praktische Verfahren der Gleichungsauflösung.
ZAMM - J. Appl. Math. Mech. / Zeitschrift für Angewandte Mathematik und Mechanik
9
(2): 152–164.
10.1002/zamm.19290090206 Google Scholar
- Lanczos, C. (1950). An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J. Nat. Bur. Stand. 45 : 225–282.
- Krylov, A.N. (1931). On the numerical solution of the equation by which in technical questions frequencies of small oscillations of material systems are determined. Izvestija AN SSSR (News of Academy of Sciences of the USSR), Otdel. mat. i estest. nauk 7 (4): 491–539.
-
Helgaker, T., Jørgensen, P., and Olsen, J. (2000). Molecular Electronic Structure Theory. Chichester, England: Wiley.
10.1002/9781119019572 Google Scholar
- Davidon, W.C. (1991). Variable metric method for minimization. SIAM J. Optim. 1 (1): 1–17.
- Roos, B.O., Taylor, P.R., and Siegbahn, P.E.M. (1980). A complete active space SCF method (CASSCF) using a density matrix formulated super-CI approach. Chem. Phys. 48 : 157–173.
-
Roos, B.O. (1980). The complete active space SCF method in a Fock-matrix-based super-CI formulation.
Int. J. Quantum Chem.
18
(S14): 175–189.
10.1002/qua.560180822 Google Scholar
- Siegbahn, P.E.M., Heiberg, A., Roos, B.O., and Levy, B. (1980). A comparison of the super-CI and the Newton-Raphson scheme in the complete active space SCF method. Phys. Scr. 21 : 323–327.
- Siegbahn, P.E.M., Almlöf, J., Heiberg, A., and Roos, B.O. (1981). The complete active space SCF (CASSCF) method in a Newton-Raphson formulation with application to the HNO molecule. J. Chem. Phys. 74 : 2384–2396.
- Ruedenberg, K., Schmidt, M.W., Gilbert, M.M., and Elbert, S.T. (1982). Are atoms intrinsic to molecular electronic wave functions? I. The FORS model. Chem. Phys. 71 : 41–49.
- Shepard, R. (1987). The multiconfiguration self-consistent field method. Adv. Chem. Phys. 69 : 63.
- Olsen, J., Yeager, D.L., and Jørgensen, P. (1983). Optimization and characterization of a multiconfigurational self-consistent field (MCSCF) state. Adv. Chem. Phys. 54 : 1.
- Aquilante, F., Autschbach, J., Carlson, R.K. et al. (2016). Molcas 8: new capabilities for multiconfigurational quantum chemical calculations across the periodic table. J. Comput. Chem. 37 (5): 506–541.
- Li Manni, G., Smart, S.D., and Alavi, A. (2016). Combining the complete active space self-consistent field method and the full configuration interaction quantum Monte-Carlo within a super-CI framework, with application to challenging metal-porphyrins. J. Chem. Theory Comput. 12 (3): 1245–1258.
- Zgid, D. and Nooijen, M. (2008). The density matrix renormalization group self-consistent field method: orbital optimization with the density matrix renormalization group method in the active space. J. Chem. Phys. 128 : 144116.
- Ghosh, D., Hachmann, J., Yanai, T., and Chan, G.K.-L. (2008). Orbital optimization in the density matrix renormalization group, with applications to polyenes and β-carotene. J. Chem. Phys. 128 : 144117.
- Fosso-Tande, J., Nguyen, T.-S., Gidofalvi, G., and DePrince, A.E. III (2016). Large-scale variational two-electron reduced-density-matrix-driven complete active space self-consistent field methods. J. Chem. Theory Comput. 12 : 2260–2271.
- Li Manni, G., Kats, D., Tew, D.P., and Alavi, A. (2019). Role of valence and semi-core electron correlation on spin gaps in Fe(II)-porphyrins. J. Chem. Theory Comput.
- Malmqvist, P.-Å., Rendell, A., and Roos, B.O. (1990). The restricted active space self-consistent-field method, implemented with a split graph unitary-group approach. J. Phys. Chem. 94 (14): 5477–5482.
- Ma, D., Li Manni, G., and Gagliardi, L. (2011). The generalized active space concept in multiconfigurational self-consistent field methods. J. Chem. Phys. 135 : 044128.
-
Levy, B. and Berthier, G. (1968). Generalized Brillouin theorem for multiconfigurational SCF theories.
Int. J. Quantum Chem.
2
(2): 307–319.
10.1002/qua.560020210 Google Scholar
-
Levy, B. and Berthier, G. (1968). Generalized Brillouin theorem for multiconfigurational SCF theories.
Int. J. Quantum Chem.
3
(2): 247–247.
10.1002/qua.560030213 Google Scholar
- Banerjee, A. and Grein, F. (1977). Multiconfiguration wave functions for excited states. Selection of optimal configurations: the b 1Σ+ and d 1Σ+ states of NH. J. Chem. Phys. 66 : 1054–1062.
- Ruedenberg, K., Cheung, L.M., and Elbert, S.T. (1979). MCSCF optimization through combined use of natural orbitals and the Brillouin-Levy-Berthier theorem. Int. J. Quantum Chem. 16 : 1069–1101.
- Werner, H.-J. and Meyer, W. (1980). A quadratically convergent multiconfiguration-self-consistent field method with simultaneous optimization of orbitals and CI coefficients. J. Chem. Phys. 73 : 2342.
-
Shepard, R. and Simons, J. (1980). Multiconfigurational wave function optimization using the unitary group method.
Int. J. Quantum Chem.
18
: 211.
10.1002/qua.560180825 Google Scholar
- Dalgaard, E. and Jørgensen, P. (1978). Optimization of orbitals for multiconfigurational reference states. J. Chem. Phys. 69 : 3833.
- Yeager, D.L. and Jørgensen, P. (1979). Convergency studies of second and approximate second order multiconfigurational Hartree-Fock procedures. J. Chem. Phys. 71 : 755.
- Dalgaard, E. (1979). A quadratically convergent reference state optimization procedure. Chem. Phys. Lett. 65 : 559.
- Lengsfield, B.H. III (1980). General second order MCSCF theory: a density matrix directed algorithm. J. Chem. Phys. 73 : 382.
- Werner, H.-J. and Meyer, W. (1981). A quadratically convergent MCSCF method for the simultaneous optimization of several states. J. Chem. Phys. 74 : 5794.
- Olsen, J. and Jørgensen, P. (1982). Update methods in multiconfigurational self-consistent field calculations. J. Chem. Phys. 77 : 6109.
- Olsen, J., Jørgensen, P., and Yeager, D.L. (1982). Multiconfigurational Hartree-Fock studies of avoided curve crossing using the Newton-Raphson technique. J. Chem. Phys. 76 : 527.
- Werner, H.-J. and Knowles, P.J. (1985). A second order multiconfiguration SCF procedure with optimum convergence. J. Chem. Phys. 82 : 5053.
- Knowles, P.J. and Werner, H.-J. (1985). An efficient second-order MCSCF method for long configuration expansions. Chem. Phys. Lett. 115 : 259.
- Werner, H.-J. (1987). Matrix-formulated direct multiconfiguration self-consistent field and multiconfiguration reference configuration-interaction methods. Adv. Chem. Phys. 69 : 1.
- Gyorffy, W., Shiozaki, T., Knizia, G., and Werner, H.-J. (2013). Analytical energy gradients for second-order multireference perturbation theory using density fitting. J. Chem. Phys. 138 (10): 104104.
- Aquilante, F., Malmqvist, P.-Å., Pedersen, T.B. et al. (2008). Cholesky decomposition-based multiconfiguration second-order perturbation theory (CD-CASPT2): application to the spin-state energetics of Co-III(diiminato)(NPh). J. Chem. Theory Comput. 4 : 694.
- Aquilante, F., Pedersen, T.B., and Lindh, R. (2007). Low-cost evaluation of the exchange Fock matrix from Cholesky and density fitting representations of the electron repulsion integrals. J. Chem. Phys. 126 : 194106.
- Ma, Y., Knecht, S., Keller, S., and Reiher, M. (2017). Second-order self-consistent-field density-matrix renormalization group. J. Chem. Theory Comput. 13 : 2533–2549.
- Menezes, F., Kats, D., and Werner, H.-J. (2016). Local complete active space second-order perturbation theory using pair natural orbitals (PNO-CASPT2). J. Chem. Phys. 145 : 124115.
- Péter, G., Szalay, T.M., Gidofalvi, G. et al. (2012). Multiconfiguration self-consistent field and multireference configuration interaction methods and applications. Chem. Rev. 112 (1): 108–181.
- Banerjee, A. and Grein, F. (1976). Convergence behavior of some multiconfiguration methods. Int. J. Quantum Chem. 10 : 123–134.
- Hinze, J. (1973). MCSCF. I. The multi-configuration self-consistent-field method. J. Chem. Phys. 59 (12): 6424–6432.
- Hoffmann, M.R., Fox, D.J., Gaw, J.F. et al. (1984). Analytic energy second derivatives for general MCSCF wave functions. J. Chem. Phys. 80 : 2660.
- Vogiatzis, K.D., Ma, D., Olsen, J. et al. (2017). Pushing configuration-interaction to the limit: towards massively parallel MCSCF calculations. J. Chem. Phys. 147 (18): 184111.
- Li Manni, G., Ma, D., Aquilante, F. et al. (2013). Split-GAS method for strong correlation and the challenging case of Cr2 . J. Chem. Theory Comput. 9 : 3375–3384.
- Vogiatzis, K.D., Li Manni, G., Stoneburner, S.J. et al. (2015). Systematic expansion of active spaces beyond the CASSCF limit: a GASSCF/Split-GAS benchmark study. J. Chem. Theory Comput. 11 : 3010–3021.
- Booth, G.H., Thom, A.J.W., and Alavi, A. (2009). Fermion Monte Carlo without fixed nodes: A game of life, death and annihilation in Slater determinant space. J. Chem. Phys. 131 : 054106.
- Cleland, D., Booth, G.H., and Alavi, A. (2010). Survival of the fittest: accelerating convergence in full configuration-interaction quantum Monte Carlo. J. Chem. Phys. 132 : 041103.
- Cleland, D., Booth, G.H., and Alavi, A. (2011). A study of electron affinites using the initiator approach to full configuration interaction quantum Monte Carlo. J. Chem. Phys. 134 : 024112.
- Overy, C., Booth, G.H., Blunt, N.S. et al. (2014). Unbiased reduced density matrices and electronic properties from full configuration interaction quantum Monte Carlo. J. Chem. Phys. 141 : 244117.
- Thomas, R.E., Sun, Q., Alavi, A., and Booth, G.H. (2015). Stochastic multiconfigurational self-consistent field theory. J. Chem. Theory Comput. 11 (11): 5316–5325.
- Li Manni, G. and Alavi, A. (2018). Understanding the mechanism stabilizing intermediate spin states in Fe(II)-porphyrin. J. Phys. Chem. A 122 : 4935–4947.
- White, S.R. (1992). Density matrix formulation for quantum renormalization groups. Phys. Rev. Lett. 69 : 2863–2866.
- Schollwöck, U. (2005). The density-matrix renormalization group. Rev. Mod. Phys. 77 : 259–315.
- Chan, G.K.-L. and Sharma, S. (2011). The density matrix renormalization group in quantum chemistry. Annu. Rev. Phys. Chem. 62 (1): 465–481.
- Marti, K.H. and Reiher, M. (2010). The density matrix renormalization group algorithm in quantum chemistry. Z. Phys. Chem. 224 (3): 583–599.
- Olivares-Amaya, R., Hu, W., Nakatani, N. et al. (2015). The ab-initio density matrix renormalization group in practice. J. Chem. Phys. 142 (3): 034102.
- Zgid, D. and Nooijen, M. (2008). Obtaining the two-body density matrix in the density matrix renormalization group method. J. Chem. Phys. 128 : 144115.
- Malmqvist, P.-Å., Roos, B.O., and Schimmelpfennig, B. (2002). The restricted active space (RAS) state interaction approach with spin–orbit coupling. Chem. Phys. Lett. 357 : 230–240.
- Malmqvist, P.-Å., Pierloot, K., Shahi, A.R.M. et al. (2008). The restricted active space (RAS) followed by second-order perturbation theory method: theory and application to the study of CuO2 and Cu2O2 systems. J. Chem. Phys. 128 : 204109.
- Ma, D., Li Manni, G., Olsen, J., and Gagliardi, L. (2016). Second-order perturbation theory for generalized active space self-consistent-field wave functions. J. Chem. Theory Comput. 12 (7): 3208–3213.
- Yaffe, L.G. and Goddard, W.A. (1976). Orbital optimization in electronic wave functions; equations for quadratic and cubic convergence of general multiconfiguration wave function. Phys. Rev. A 13 : 1682.
- Roos, B.O., Walch, S.P., Bauschlicher, C.W., and Nelin, C.J. (1983). Theoretical evidence for multiple 3d bonding in the V2Cr2 molecules. Chem. Phys. Lett. 103 : 175.
- Nakano, H. and Hirao, K. (2000). A quasi-complete active space self-consistent field method. Chem. Phys. Lett. 317 : 90.
- Panin, A.I. and Sizova, O.V. (1996). Direct CI method in restricted configuration spaces. J. Comput. Chem. 17 (2): 178–184.
- Panin, A.I. and Simon, K.V. (1996). Configuration interaction spaces with arbitrary restrictions on orbital occupancies. Int. J. Quantum Chem. 59 : 471.
- Ivanic, J. (2003). Direct configuration interaction and multiconfigurational self-consistent-field method for multiple active spaces with variable occupations. I. Method. J. Chem. Phys. 119 : 9364.
- Fleig, T., Olsen, J., and Marian, C.M. (2001). The generalized active space concept for the relativistic treatment of electron correlation. I. Kramers-restricted two-component configuration interaction. J. Chem. Phys. 114 : 4775.
- Liu, Y., Fang, L., Shen, X. et al. (2000). Absorption, resonance Raman and Raman excitation spectra of lanthanum dimers in argon matrices. Chem. Phys. 262 : 25.
- Golub, G.H., Luk, F.T., and Overton, M.L. (1981). A block Lanczos method for computing the singular values and corresponding singular vectors of a matrix. ACM Trans. Math. Softw. 7 (2): 149–169.
- Austin, B.M., Zubarev, D.Y., and Lester, W.A. (2012). Quantum Monte Carlo and related approaches. Chem. Rev. 112 (1): 263–288.PMID: 22196085.
- Petruzielo, F.R., Holmes, A.A., Changlani, H.J. et al. (2012). Semistochastic projector Monte Carlo method. Phys. Rev. Lett. 109 : 230201.
- Blunt, N.S., Smart, S.D., Kersten, J.A.-F. et al. (2015). Semi-stochastic full configuration interaction quantum Monte Carlo: developments and application. J. Chem. Phys. 142 : 184107.
- Blunt, N.S., Smart, S.D., Booth, G.H., and Alavi, A. (2015). An excited-state approach within full configuration interaction quantum Monte Carlo. J. Chem. Phys. 143 (13).
- Blunt, N.S., Booth, G.H., and Alavi, A. (2017). Density matrices in full configuration interaction quantum Monte Carlo: excited states, transition dipole moments, and parallel distribution. J. Chem. Phys. 146 (24): 244105.