A quadratically convergent multiconfiguration–self‐consistent field method with simultaneous optimization of orbitals and CI coefficients
01 Sep 1980-Journal of Chemical Physics (American Institute of PhysicsAIP)-Vol. 73, Iss: 5, pp 2342-2356
TL;DR: In this article, a quadratically convergent MC-SCF procedure is described, which is based on the direct minimization of the energy, and the convergence radius is much improved by taking into account in the energy expansion those parts of third and higher order terms which account exactly for the orthonormality constraints imposed on the orbitals.
Abstract: A quadratically convergent MC–SCF procedure is described which is based on the direct minimization of the energy. In comparison to the Newton–Raphson technique, which has previously been applied by several authors for orbital optimization, the convergence radius is much improved by taking into account in the energy expansion those parts of third and higher order terms which account exactly for the orthonormality constraints imposed on the orbitals. The nonlinear equations which define the improved orbitals are solved iteratively by a simple adaption of the Gauss–Seidel method. The coefficients of the configuration expansion can be optimized simultaneously with the orbitals, a necessary requirement for over‐all quadratic convergence. The removal of redundant variables as well as useful approximations for the optimization of core orbitals are discussed. The convergence of the method is demonstrated to be much superior to classical Fock operator techniques and MC–SCF methods which are based on the generalized Brillouin theorem. The formalism is carried down to matrix operations and shows a simple structure.
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TL;DR: In this article, an MCSCF procedure is described which is based on the direct minimization of an approximate energy expression which is periodic and correct to second order in the changes in the orthonormal orbitals.
Abstract: An MCSCF procedure is described which is based on the direct minimization of an approximate energy expression which is periodic and correct to second order in the changes in the orthonormal orbitals Within this approximation, the CI coefficients are fully optimized, thereby accounting for the coupling between orbital rotations and CI coefficients to higher order than in previous treatments Additional transformations among the internal orbitals and their associated one‐ and two‐electron integrals are performed which amounts to treating the rotations among internal orbitals to higher than second order These extra steps are cheap compared to the four index transformation performed in each iteration, but lead to a remarkable enhancement of convergence and overall efficiency In all calculations attempted to date, convergence has been achieved in at most three iterations The energy has been observed to converge better than quadratically from the first iteration even when the initial Hessian matrix has many negative eigenvalues
2,739 citations
TL;DR: In this paper, a second-order optimisation procedure for general complete active space (CAS) wavefunctions is described. But this method is restricted to very long complete active spaces.
2,365 citations
1,209 citations
TL;DR: The multiconfiguration self-consistent field method offers the most general approach to the computation of chemical reactions and multiple electronic states and localized orbitals are convenient both for selection of the appropriate active space and for understanding the computed results.
Abstract: The multiconfiguration self-consistent field (MCSCF) method offers the most general approach to the computation of chemical reactions and multiple electronic states. This review discusses the design of MCSCF wavefunctions for treating these problems and the interpretation of the resulting orbitals and configurations. In particular, localized orbitals are convenient both for selection of the appropriate active space and for understanding the computed results. The computational procedures for optimizing these wavefunctions and the techniques for recovery of dynamical correlation energy are reviewed.
612 citations
TL;DR: A multidimensional approach, based on the measurement and accurate theoretical description of both even and odd harmonic orders, enabled us to reconstruct both quantum amplitudes and phases of the electronic states with a resolution of ~100 attoseconds.
Abstract: The ultrafast motion of electrons and holes after light-matter interaction is fundamental to a broad range of chemical and biophysical processes. We advanced high-harmonic spectroscopy to resolve spatially and temporally the migration of an electron hole immediately after ionization of iodoacetylene while simultaneously demonstrating extensive control over the process. A multidimensional approach, based on the measurement and accurate theoretical description of both even and odd harmonic orders, enabled us to reconstruct both quantum amplitudes and phases of the electronic states with a resolution of ~100 attoseconds. We separately reconstructed quasi-field-free and laser-controlled charge migration as a function of the spatial orientation of the molecule and determined the shape of the hole created by ionization. Our technique opens the prospect of laser control over electronic primary processes.
448 citations
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30 Nov 1961
TL;DR: In this article, the authors propose Matrix Methods for Parabolic Partial Differential Equations (PPDE) and estimate of Acceleration Parameters, and derive the solution of Elliptic Difference Equations.
Abstract: Matrix Properties and Concepts.- Nonnegative Matrices.- Basic Iterative Methods and Comparison Theorems.- Successive Overrelaxation Iterative Methods.- Semi-Iterative Methods.- Derivation and Solution of Elliptic Difference Equations.- Alternating-Direction Implicit Iterative Methods.- Matrix Methods for Parabolic Partial Differential Equations.- Estimation of Acceleration Parameters.
5,317 citations
1,673 citations
TL;DR: In this paper, the general multi-configuration self-consistent field (MC•SCF) method is presented with no restrictions on the types of configurations participating in the expansion of the total wave function.
Abstract: The general multi‐configuration self‐consistent‐field (MC‐SCF) method is presented with no restrictions on the types of configurations participating in the expansion of the total wavefunction. The general coupled Fock‐like equations for the ``best'' orbitals to be used in such a multi‐configuration wavefunction are derived. Formally these coupled nonlinear equations are decoupled with the use of projection operators and transformed into a pseudoeigenvalue problem. Several general methods, based on orbital transformations and on the use of the generalized Brillouin theorem, are presented for solving the coupled nonlinear Fock like equations for the determination of the MC‐SCF orbitals. The formalism presented is applicable not only to the ground state of a given system, but also to any excited state, yielding an upper bound to the true energy of the desired state.
295 citations
TL;DR: The matrix elements of the total Hamiltonian between a multiconfigurational SCF wave function and some well-defined linear combinations of excited Slater determinants are equal to zero as discussed by the authors.
Abstract: The matrix elements of the total Hamiltonian between a multiconfigurational SCF wave function and some well-defined linear combinations of excited Slater determinants are equal to zero. By means of this generalized Brillouin theorem it is possible to estimate the improvements to be expected from a subsequent configuration-interaction treatment. The expression of the effective potential for the orbitals can be also derived in the frame of a given multiconfigurational theory. As an example, the case of the CMC-SCF method recently suggested [9] is examined.
Les elements de matrice de 1'Hamiltonien construits sur une fonction d'onde SCF multiconfigurationnelle et certaines combinaisions lineaires bien definies de determinants de Slater excites sont nuls. A I'aide de ce theoreme de Brillouin generalise, on peut prevoir 1'importance relative d'une interaction de configuration ulterieure et determiner 1'expression due potential effectif donnant les orbitals. A titre d'exemple, on examine le cas de la methode CMC-SCF proposee recemment [9].
Die matrixelemente des Hamiltonoperators zwischen einer “Multikonfiguration-SCF-Funktion” und bestimmten Linearkombinationen von angeregten Slaterdeterminanten verschwinden. Mit Hilfe dieses verallgemeinerten Brillouin-schen Theorems ist es moglich die durch Konfigurationwechselwirkung zu erwartende Verbesserung abzuschatzen und den Ausdruck des effektiven Potentials fur die Einelektronenfunktionen abzuleiten. Das wird am Beispiel der Kurzlich vorgeschlagenen “CMC-SCF” Methode [9] demonstriert.
272 citations