Full CI Wave Functions Research Articles

The efficient conversion between linear wave function expansions and nonlinear graphically contracted function expansions

Several algorithms are examined for the transformation of a graphically contracted function (GCF) to a configuration state function (CSF) vector, denoted , and for the reverse transformation from CSF vector to GCF, denoted . The initial implementation of these algorithms is applied to a sequence of full-CI wave functions of similar molecules. The most efficient transformation is based on a depth-first search (DFS) and scales as where is an average facet count for the GCF wave function, is the dimension of the vector x, and is the dimension of the GCF basis. Two different transformations are proposed, one based on a pairwise recursive merge algorithm and the other based on a least-squares fitting algorithm. The recursive merge is a reliable algorithm, but our initial implementation does not display the expected scaling behaviour with respect to . The iterative least-squares fitting algorithm is based on an efficient DFS procedure to compute the error ε and its gradient g. However, the resulting iterative least-squares optimisation is slowly convergent, requiring excessive iterations to converge to a target error tolerance. For most of the calculations in this case study, the recursive merge procedure is found to be more efficient than the least-squares optimisation procedure.

Dynamical correlation within the Interacting Quantum Atoms method through coupled cluster theory

The Interacting Quantum Atoms (IQA) approach is an energy partition of an electronic system which has provided new understandings and insightful information about diverse chemical systems and processes from Hartree–Fock, CASSCF as well as truncated and full CI wavefunctions. Nonetheless, the unavailability of an implementation of IQA relying on size extensive correlated methods such as coupled cluster or Møller–Plesset perturbation theory, has prevented the application of this procedure to the study of molecular systems in which these models of electronic structure are necessary to obtain accurate results. This paper uses first and second-order density matrices consistent with a closed shell coupled cluster energy to carry out the IQA splitting. The presented developments are tested and discussed using H2, N2, H2O and CO as representative examples. It is our hope that the implementations discussed in this paper would give new insights in the study of chemical bonding, intermolecular interactions and other important phenomena in chemistry for which the consideration of electron correlation is critical.

Lossless compression of wave function information using matrix factorization: A “gzip” for quantum chemistry

We propose the use of the singular value decomposition to decrease the storage required for wave function information. The specific case considered is determinantal full configuration interaction, but the same technique is readily applicable to truncated configuration interaction and coupled-cluster calculations of various types; as we discuss this is a reformulation of approximate methods that have been in use for some time, but our approach eliminates those approximations. Numerical examples support the contention that considerable compression of the wave function is possible without significant loss of accuracy: as expected a considerable amount of the information contained in the full CI wave function is redundant.

On the formulation of a density matrix functional for Van der Waals interaction of like- and opposite-spin electrons in the helium dimer

Whereas a density functional that incorporates dispersion interaction has remained elusive to date, we demonstrate that in principle the dispersion energy can be obtained from a density matrix functional. In density matrix functional theory one tries to find suitable approximations to the two-particle reduced density matrix (2RDM) in terms of natural orbitals (NOs) and natural orbital occupation numbers (ONs). The total energy is then given as a function(al) of the NOs and ONs, i.e., as an implicit functional of the 1RDM. The left-right correlation in a (dissociating) bond, as well as various types of dynamical correlation, can be described accurately with a NO functional employing only J and K integrals (JK-only functional). We give a detailed analysis of the full CI wavefunction of the He(2) dimer, from which the dispersion part of the two-particle density matrix is obtained. It emerges that the entirely different physics embodied in the dispersion interaction leads to an essentially different type of exchange-correlation orbital functional for the dispersion energy (non-JK). The distinct NO functionals for the different types of correlation imply that they can be used in conjunction without problems of double counting. Requirements on the (primitive) basis set for Van der Waals bonding appear to be more modest than for other types of correlation.

Calculation of transition matrix elements by nonsingular orbital transformations

A general strategy is described for the evaluation of transition matrix elements between pairs of full class CI wave functions built up from mutually nonorthogonal molecular orbitals. A new method is proposed for the counter-transformation of the linear expansion coefficients of a full CI wave function under a nonsingular transformation of the molecular-orbital basis. The method, which consists in a straightforward application of the Cauchy–Binet formula to the definition of a Slater determinant, is shown to be simple and suitable for efficient implementation on current high-performance computers. The new method appears mainly beneficial to the calculation of miscellaneous transition matrix elements among individually optimized CASSCF states and to the re-evaluation of the CASCI expansion coefficients in Slater-determinant bases formed from arbitrarily rotated (e.g., localized or, conversely, delocalized) active molecular orbitals. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

Molecular one‐electron properties using the multireference Hartree–Fock CI method

AbstractThe recently introduced multireference Hartree–Fock configuration interaction (MRHFCI) method has been applied to the calculation of the dipole moment of the LiH, BH, FH, CO, and H2O molecules. The results obtained indicate that MRHFCI wave functions, much more compact but of the same quality of the orthogonal full CI ones, can provide better dipole moments than the corresponding full CI wave functions. The value of the dipole moments obtained with the MRHFCI wave functions is quite insensitive to the choice of the HF references but the same is not true for the electronic energy. Therefore, further studies are necessary to develop a criterion for selecting a set of HF references which could provide the best values of energy and dipole moment. Also, it would be important to verify if other one‐electron properties can be computed with the same degree of accuracy obtained for the dipole moments. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008

Anatomy of Bond Formation. Domain-Averaged Fermi Holes as a Tool for the Study of the Nature of the Chemical Bonding in Li2, Li4, and F2

Domain-averaged Fermi hole (DAFH) analysis represents a relatively new strategy for extracting useful new insights into electronic structure and bonding from correlated wave functions. We analyze a full-valence CASSCF description of the Li4 rhombus, in order to discern the role played by the domains of the non-nuclear attractors in the sharing of the valence electrons. Similarly we examine the electron reorganization that accompanies the bond dissociation process in the Li2 molecule, which also features such a non-nuclear attractor for a significant range of nuclear separations. Full-CI wave functions for H2, for a wide range of bond lengths, are used to determine how robust are the DAFH descriptions from full-valence CASSCF wave functions to the incorporation of dynamical electron correlation. Comparisons are made, for H2 and Li4, with a much cheaper strategy in which restricted Kohn-Sham orbitals from B3LYP calculations are inserted into a simplified DAFH expression which applies at the restricted Hartree-Fock level. We also investigate the breaking of the relatively weak F-F bond in F2, in order to determine the extent to which the DAFH analysis of such a system differs from that of a more conventional homopolar bond, such as the one in H2.

Theoretical molecular Auger spectra with electron population analysis

An approximation method is proposed to simulate the molecular Auger spectra. Auger transition rates are estimated with atomic populations of valence orbitals on an excited atom by Mulliken and Löwdin population analyses. Normal Auger energies and relative rates for H 2O and NH 3 molecules are evaluated using a full CI wave functions among the valence orbitals constructed from SCF orbitals for the neutral ground state and for the core-hole excited state. Theoretical spectra show fairly good correspondence with the experimental spectra. It is demonstrated that the present approach is applicable for qualitative assignment and analysis of Auger spectra for larger molecules.

Simultaneous optimization of Gaussian type function exponents for electron and positron with full-CI wavefunction – application to ground and excited states of positronic compounds with multi-component molecular orbital approach

In order to obtain the best wavefunction of positronic compound with molecular orbital (MO) treatment, the full-configuration interaction (full-CI) fully variational MO (FVMO) method is proposed for multi-component systems, in which all the variational parameters in electronic and positronic wavefunctions are optimized under the full-CI scheme. We have applied the full-CI multi-component FVMO method to the ground and positronic-excited states of [H −;e +] system. Our treatment gives good improvement in the basis functions for positronic compounds owing to the extension of flexibility in the variational space, though the convergence of electron–positron correlation term is slower than that of conventional electron correlation.

Structure of the exact wave function

We investigate the structure of the exact wave function as a solution of the Schrödinger equation, aiming the singles and doubles description of the exact wave function. The basis is that the Hamiltonian involves only one and two body operators. We first present two theorems that indicate a possibility of the singles and doubles description of the exact wave function. We then examine the exponential ansatz, as this theorem implies it to be a possible structure of the exact wave function. Variational CCS (singles) wave function is shown to be certainly exact for one particle Hamiltonian. Thouless transformation plays an important role in the formulation. The conventional CCSD (singles and doubles) function is restrictive, even if it is solved variationally. A wider coupled cluster function with general singles and doubles substitution operators (CCGSD) is also not exact for the existence of noncommuting operators. We then analyze some formal properties of the full CI wave function, and finally, we propose an ansatz of the exact wave function and describe the method of solution. It involves successive solutions of the secular equations of the order of singles and doubles. It is variational and we can calculate both ground and excited states.

Hubbard and Heisenberg Models for Four-Site Four-Electron Systems. Group-Theoretical Interrelationships and Applications to Multinuclear Transition-Metal Clusters

Abstract Hubbard models for four-site four-electron {4,4} systems are analytically solved to elucidate group-theoretical interrelationships between effective model Hamiltonians for radical clusters, metal clusters, Cu4O4, and so on. The group operations used for this purpose are permutation group (SN), spatial symmetry (PN), spin rotation (S), and time-reversal (T). The magnetic (color) group (T × S) is utilized for determination of the magnetic symmetry of spin structures obtained on the basis of the classical Heisenberg model; namely a vector representation of radical spins. The magnetic double group theory (T × S × PN) is used for characterization of general Hartree–Fock solutions (GHF) involving axial, helical, and torsional spin density waves for {4,4} systems. The spin-optimized SCF solutions are constructed from the spin projection of the GHF solutions (T × S × PN) by the use of the permutation symmetry (SN). The SN group is also used for the quantum Heisenberg model for {4,4} systems. In order to show the interrelationships between these model Hamiltonians, the electronic states of {4,4} systems with the D4h, Td, and D3h symmetries are constructed by the use of the magnetically ordered general spin orbitals which have been determined by the GHF calculations. The spin-optimized SCF solutions for {4,4} systems examined here are equivalent to the full CI wavefunctions satisfying both spatial and spin symmetries. The relative contributions of the spin polarization (SP) and doubly excited configurations in the GHF, projected GHF and SO-SCF wavefunctions are clarified to elucidate possible mechanisms of spin alignments and antiferromagnetic spin correlations. Implications of these computational results are discussed in relation to quantum and classical representations of spin alignments in molecular magnetic materials such as iron–sulfur clusters. Molecular magnets having helical and torsional spin structures are designed and discussed in relation to the most general spin alignment in the species.

A procedure to obtain an accurate approximation to a full CI wavefunction

Based on Jacobi elementary rotations, a simple, elegant procedure to obtain approximate CI wavefunctions is discussed. Essentially, a sequence of (2 x 2) matrices is builtup, and the eigenvector attached to the lowest eigenvalue is used to construct a stepwise set of coefficients, which become a very good approximation to theexact Cl result. Full CI calculations could easily be reached in this way. An example formed by some atoms of the He isoelectronic sequence is provided in order to test the flexibility and accuracy of the procedure. A Fortran 90 code is available.

Electronic Spin Alignment in Free Radical Reactions

Abstract Classical spin alignment in radical reactions is studied by the GSF-HF wave function in H3 model system. The GSF-HF wave function introduces a simple classical picture of electronic spins directly and leads to the classical Heisenberg model. The instability of various spin alignments is studied on the GSF-HF electronic energy surface. In compensation for the classical spin freedom, the GSF-HF method does not satisfy quantum mechanical spin-symmetry. The GSF-HF wave function is analyzed with a projection operator constructed by a full-CI wave function.

Spin-projected Extended Hartree-Fock using a Valence Bond approach

We describe spin-projected Extended Hartree-Fock calculations, performed with a Valence Bond Self-Consistent Field program. Potential energy curves are given for BH, BeH, and N2. For BH the EHF function ranks well with the corresponding Spin-coupled and full CI wave functions. For BeH, the EHF function introduces spin contamination in the separated Be atom due to the rigidity of the wave function. This results in an inferior potential energy curve compared to Spin-coupled and full CI. The triple bond breaking in N2 is again nicely described by EHF. The Extended Hartree-Fock method as suggested by Lowdin can be a feasible tool in describing bond breaking.

Stationary perturbation theory

After a short recapitulation of the basic concepts of stationary perturbation theory, this is applied to a many-electron Hamiltonian, with or without an external field, given in a Fock space formulation in terms of a finite basis, the exact eigenfunctions of which are the full-CI wave functions. The Lie algebra L c (n) of the variational group corresponding to this problem is presented. It has an important subalgebra L c (1) of one-particle transformations. Hartree-Fock and coupled Hartree-Fock (also uncoupled Hartree-Fock) as well as MC-SCF and coupled MC-SCF are outlined in this framework

Full-CI calculations of alternant cyclic polyenes (CH) N, N=2, 4, 6, ƒ 18, in the PPP approximation

Full-CI calculations are performed on the PPP model of alternant cyclic polyenes using a recently developed direct full-CI algorithm based on an expansion in Slater determinants. By exploiting the cyclic symmetry and a partial spin adaptation, we have been able to compute a full-CI wavefunction and energy up to N = 18 on the CRAY YMP 8/432 of the Italian University Computing Centre CINECA. The wavefunctions and energies for the series N = 4ν and N = 4ν + 2 are discussed and their behaviour as functions of N is analyzed.

The dissociation energy of He 2+

Ab initio potential energy curves for He 2 + are computed using full CI wavefunctions in conjunction with large Gaussian and Slater basis sets. The computed curves yield rotational-vibrational transitions with errors of less than 1 cm −1. They also yield quasibound levels with energies and lifetimes in reasonable agreement with translational spectroscopy measurements. The best D e is estimated to be 2.470 eV. Green's function Monte Carlo calculations yield a very similar value, 2.466 ± 0.005.

On the SCF calculation of excited states: Singlet states in the two‐electron problem

AbstractThe problem of determining SCF wave functions for excited electronic states is examined for singlet states of two‐electron systems using a Lowdin natural orbital transformation of the full CI wave function. This analysis facilitates the comparison of various SCF methods with one another. The distribution of the full CI states among the natural orbital MCSCF states is obtained for the S states of helium using a modest Gaussian basis set. For SCF methods that are not equivalent to the full CI wave functions, it is shown that the Hartree‐Fock plus all single excitation wave functions are equivalent to that of Hartree‐Fock plus one single excitation. It is further shown that these wave functions are equivalent to the perfect pair or TCSCF wave functions in which the CI expansion coefficients are restricted to have opposite signs. The case of the natural orbital MCSCF wave function for two orbitals is examined in greater detail. It is shown that the first excited state must always be found on the lower natural orbital MCSCF CI root, thus precluding the use of the Hylleras‐Undeim‐MacDonald (HUM) theorem in locating this state. It is finally demonstrated that the solution obtained by applying the HUM theorem (minimizing the upper MCSCF CI root with respect to orbital mixing parameters) is an artifact of the MCSCF method and does not correspond to any of the full CI states.

The effect of electronic correlation on molecular wavefunctions

The minimal basis set Hartree-Fock and full-CI wavefunctions of some molecules (N 2, CO, BH 2, CH 4) are analyzed in terms of their valence-bond components. The internal valence correlation has dramatic effects on this distribution. The coefficients of the ionic forms decrease very rapidly with ionicity; the largest coefficients concern neutral determinants with maximum spreading of the p electrons which are multiplied by important factors (3 to 10) by the CI. The neutral components with two electrons is the same p AO are increased to a lesser extent. Correlation decreases the charge fluctuation in a dramatic way. On the other hand the atomic spin fluctuation increases under correlation. When the main atomic configuration is not the ground state one (as in BH 2 or CH 4) the correlation does not increase the components of the atomic ground states, and has little effect on hybridization. These various trends reflect a compromise between the necessity of bond building between electrons of opposite spins and the tendency of the atoms to increase their components in their low energy configurations.

Static dipole polarizability of electronically excited molecules: H 2(B 1Σ u+)

The dipole polarizability of H 2(B 1Σ u +) is computed using extended Gaussian basis sets and Hartree-Fock, multiconfiguration Hartree-Fock, and configuration interaction wavefunctions. Electron correlation contributions are found to be significant (≈ 25%) with the largest contribution arising from angular correlation. With a full CI wavefunction, the components of the dipole polarizability were computed to be (in au): α ⊥ = 50 and α | = 257.