Multiconfigurational Self-consistent Field Procedure Research Articles

Electron correlation within the relativistic no-pair approximation.

This paper addresses the definition of correlation energy within 4-component relativistic atomic and molecular calculations. In the nonrelativistic domain the correlation energy is defined as the difference between the exact eigenvalue of the electronic Hamiltonian and the Hartree-Fock energy. In practice, what is reported is the basis set correlation energy, where the "exact" value is provided by a full Configuration Interaction (CI) calculation with some specified one-particle basis. The extension of this definition to the relativistic domain is not straightforward since the corresponding electronic Hamiltonian, the Dirac-Coulomb Hamiltonian, has no bound solutions. Present-day relativistic calculations are carried out within the no-pair approximation, where the Dirac-Coulomb Hamiltonian is embedded by projectors eliminating the troublesome negative-energy solutions. Hartree-Fock calculations are carried out with the implicit use of such projectors and only positive-energy orbitals are retained at the correlated level, meaning that the Hartree-Fock projectors are frozen at the correlated level. We argue that the projection operators should be optimized also at the correlated level and that this is possible by full Multiconfigurational Self-Consistent Field (MCSCF) calculations, that is, MCSCF calculations using a no-pair full CI expansion, but including orbital relaxation from the negative-energy orbitals. We show by variational perturbation theory that the MCSCF correlation energy is a pure MP2-like correlation expression, whereas the corresponding CI correlation energy contains an additional relaxation term. We explore numerically our theoretical analysis by carrying out variational and perturbative calculations on the two-electron rare gas atoms with specially tailored basis sets. In particular, we show that the correlation energy obtained by the suggested MCSCF procedure is smaller than the no-pair full CI correlation energy, in accordance with the underlying minmax principle and our theoretical analysis. We also show that the relativistic correlation energy, obtained from no-pair full MCSCF calculations, scales at worst as X(-2) with respect to the cardinal number X of our correlation-consistent basis sets optimized for the two-electron atoms. This is better than the X(-1) scaling suggested by previous studies, but worse than the X(-3) scaling observed in the nonrelativistic domain. The well-known 1/Z- expansion in nonrelativistic atomic theory follows from coordinate scaling. We point out that coordinate scaling for consistency should be accompanied by velocity scaling. In the nonrelativistic domain this comes about automatically, whereas in the relativistic domain an explicit scaling of the speed of light is required. This in turn explains why the relativistic correlation energy to the lowest order is not independent of nuclear charge, in contrast to nonrelativistic theory.

A MCSCF method for ground and excited states based on full optimizations of successive Jacobi rotations.

A new multiconfigurational self-consistent field (MCSCF) method based on successive optimizations of Jacobi rotation angles is presented. For given one- and two-particle density matrices and an initial set of corresponding integrals, a technique is developed for the determination of a Jacobi angle for the mixing of two orbitals, such that the exact energy, written as a function of the angle, is fully minimized. Determination of the energy-minimizing orbitals for given density matrices is accomplished by successive optimization and updating of Jacobi angles and integrals. The total MCSCF energy is minimized by alternating between CI and orbital optimization steps. Efficiency is realized by optimizing CI and orbital vectors quasi-simultaneously by not fully optimizing each in each improvement step. On the basis of the Jacobi-rotation based approach, a novel MCSCF procedure is formulated for excited states, which avoids certain shortcomings of traditional excited-state MCSCF methods. Applications to specific systems show the practicability of the developed methods.

Multiconfiguration self-consistent field procedure employing linear combination of atomic-electron distributions

We present a multiconfiguration self-consistent field (MCSCF) procedure employing recently developed approximations for electron repulsion integrals. Molecular charge distributions in the two-electron interaction part are expanded into atomic-electron distributions without linear dependencies, and the four-center quantities of electron repulsion integrals are reduced to two- and three-center quantities. The method is plugged into the approximate second-order MCSCF procedure and applied to calculations of the HNO molecule. This procedure enables us to reduce the CPU time of the integral transformation step which is usually the most time consuming. It is also shown that the present approximation is very accurate not only for the ground state but also for the low-lying excited states, even after a substantial reduction of the number of required integrals.

Analytical energy gradients for multiconfiguration self-consistent field wave functions with frozen core orbitals

A method to calculate analytical energy gradients for multiconfiguration self-consistent field (MCSCF) wave functions with frozen core orbitals is presented. Since the core orbitals, which are taken from a closed shell SCF calculation, are not variationally optimized in the MCSCF procedure, it is necessary to determine their derivatives by solving a set of coupled perturbed Hartree–Fock (CPHF) equations. The technique is similar to the calculation of energy gradients for CI wave functions, but is complicated by the fact that the SCF and MCSCF orbitals are different. This makes it necessary to perform a transformation between the two orbital basis sets at an intermediate stage. The CPHF equations are solved by an iterative method, in which optionally part of the Hessian matrix can be constructed and inverted explicitly. Some applications of the method are presented. For the molecule P2S, optimized geometries for two isomers and a saddle point are compared for MCSCF wave functions with frozen and fully optimized core orbitals. It is demonstrated that in both cases virtually identical results are obtained and that the frozen-core approximation leads to significant savings in computer time. Some preliminary results are also reported for tetrasilabicyclo[1.1.0]butane, Si4H6.

A multiconfiguration self-consistent-field (MCSCF) study of the bent and linear conformations of HCCN

An MCSCF procedure of the complete active space (CAS) genre has been used to predict the structures and energetics of the HCCN molecule. Full configuration interaction (CI) within the π-electron space (1πx, 1πy, 2πx, 2πy, 3πx, 3πy orbitals, six electrons) was used in these MCSCF studies. With double zeta plus polarization (DZP) and triple zeta plus double polarization (TZ2P) basis sets, the bent structure lies 6.2 and 5.7 kcal, respectively, below the linear conformation. Multireference CI wave functions reduce this energy difference significantly.

Self-consistent-field (SCF) and multiconfiguration (MC) Hartree-Fock (HF) methods in atomic calculations: Numerical integration approaches

The self-consistent field (SCF) and multiconfiguration (MC) Hartree-Fock (HF) methods are reviewed. The emphasis is on the specification, design and implementation of these methods as a part of an atomic structure software package, dealing with wavefunction determination. Numerical integration approaches will be used throughout. After outlining a class of MCHF problems, the derivation of the MCHF equations is reviewed and some theory developed on the variation of the energy functional with respect to a rotation of the orbital basis among orbitals of the same symmetry. Various procedures are proposed and evaluated for dealing with cases where the radial basis for a given total energy is not unique. A numerical MC SCF procedure is described using a top-down approach. The overall algorithm will be outlined first with numerical details provided later. Finally some test cases are proposed.

Generalization of analytic configuration interaction (CI) gradient techniques for potential energy hypersurfaces, including a solution to the coupled perturbed Hartree–Fock equations for multiconfiguration SCF molecular wave functions

Analytic configuration interaction gradient techniques for multiconfiguration self-consistent field (MCSCF) wave functions have been developed by solving the coupled perturbed equations for the generalized Hartree–Fock operator. The method has been implemented in conjunction with the loop-driven graphical unitary group approach for the determination of correlated wave functions. This theoretical approach is in principle applicable to any type of configuration interaction (CI) wave function constructed from orbitals determined from a rigorous SCF or MCSCF procedure. Initial applications have been to CI wave functions involving all single and double excitations relative to an open-shell singlet restricted Hartree–Fock reference function. The specific example chosen was the first excited singlet state (b 1B1) of CH2, for which the equilibrium geometry and harmonic vibrational frequencies have been predicted with a double zeta plus polarization basis set.

Multiconfiguration self-consistent-field study of the importance of triply and quadruply excited electronic configurations in the water molecule

The importance of triple excitations, configurations differing by three electrons from the Hartree–Fock reference configuration, is of considerable interest in electronic structure theory. A simple double zeta basis set O(9s 5p/4s 2p), H(4s/2s) has been used to address this problem for the water molecule. Configuration interaction (CI) and multiconfiguration self-consistent-field (MCSCF) wave functions including up to all single, double, triple, and quadruple (SDTQ) excitations (a total of 17 678 1A1 configurations) have been obtained for this purpose. An interesting result is that the MCSCF wave function including only single excitations yields 52.3% of the comparable correlation energy obtained with all single and double excitations. Using canonical SCF orbitals, triple excitations are found to contribute only 0.8% of the correlation energy. However, the MCSCF procedure increases this correlation energy fraction by more than a factor of 5. The MCSCF energies for the wave functions including SD and SDTQ excitations are less than 0.001 hartree lower than the comparable straight CI energies.

MCSCF wave functions for excited states of polar molecules: Application to BeO

A previously reported multiconfiguration self-consistent field (MCSCF) algorithm based on the generalized Brillouin’s theorem is extended in order to treat excited states of polar molecules. In particular, the algorithm takes into account the proper treatment of nonorthogonality in the space of single excitations and invokes, when necessary, a constrained optimization procedure to prevent variational collapse of exited states. In addition, a configuration selection scheme (suitable for use in conjunction with extended configuration interaction methods) is proposed for the MCSCF procedure. The algorithm is used to study the low-lying singlet states of BeO, a system which has not previously been studied using an MCSCF procedure. MCSCF wave functions are obtained for three 1Σ+ and two 1Π states. The 1Σ+ results are juxtaposed with comparable results for MgO in order to assess the generality of description presented here.