Exact Configuration Interaction Research Articles

Evaluation of the excitation spectra with diffusion Monte Carlo on an auxiliary bosonic ground state.

We aim to improve upon the variational Monte Carlo (VMC) approach for excitations replacing the Jastrow factor by an auxiliary bosonic (AB) ground state and multiplying it by a fermionic component factor. The instantaneous change in imaginary time of an arbitrary excitation in the original interacting fermionic system is obtained by measuring observables via the ground-state distribution of walkers of an AB system that is subject to an auxiliary effective potential. The effective potential is used to (i) drive the AB system's ground-state configuration space toward the configuration space of the excitations of the original fermionic system and (ii) subtract from a diffusion Monte Carlo (DMC) calculation contributions that can be included in conventional approximations, such as mean-field and configuration interaction (CI) methods. In this novel approach, the AB ground state is treated statistically in DMC, whereas the fermionic component of the original system is expanded in a basis. The excitation energies of the fermionic eigenstates are obtained by sampling a fermion-boson coupling term on the AB ground state. We show that this approach can take advantage of and correct for approximate eigenstates obtained via mean-field calculations or truncated interactions. We demonstrate that the AB ground-state factor incorporates the correlations missed by standard Jastrow factors, further reducing basis truncation errors. Relevant parts of the theory have been tested in soluble model systems and exhibit excellent agreement with exact analytical data and CI and VMC approaches. In particular, for limited basis set expansions and sufficient statistics, AB approaches outperform CI and VMC in terms of basis size for the same systems. The implementation of this method in current codes, despite being demanding, will be facilitated by reusing procedures already developed for calculating ground-state properties with DMC and excitations with VMC.

Importance of the completeness of the configuration interaction and close coupling expansions in R-matrix calculations for highly charged ions: electron-impact excitation of Fe20+

We have carried out two intermediate coupling frame transformation (ICFT) R-matrix calculations for the electron-impact excitation of -like , both of which use the same expansions for their configuration interaction (CI) and close-coupling (CC) representations. The first expansion arises from the configurations , , with n = 3, 4, for , which give rise to 564 CI/CC levels. The second adds configurations , for , which give rise to 590 CI/CC levels in total. Comparison of oscillator strengths and effective collision strengths from these two calculations demonstrates the lack of convergence in data for n = 4 from the smaller one. Comparison of results for the 564 CI/CC level calculation with an earlier ICFT R-matrix calculation which used the exact same CI expansion but truncated the CC expansion to only 200 levels demonstrates the lack of convergence of the earlier data, particularly for n = 3 levels. Also, we find that the results of our 590 CC R-matrix calculation are significantly and systematically larger than those of an earlier comparable DW-plus-resonances calculation. Thus, it is important still to take note of the (lack of) convergence in both atomic structural and collisional data, even in such a highly charged ion as Fe20+, and to treat resonances non-perturbatively. This is of particular importance for Fe ions given their importance in the spectroscopic diagnostic modelling of astrophysical plasmas.

Single-electron shell occupation and effectivegfactor in few-electron nanowire quantum dots

Nanowire double quantum dots occupied by an even number of electrons are investigated in the context of energy level structure revealed by electric dipole spin resonance measurements. We use numerically exact configuration interaction approach up to 6 electrons for systems tuned to Pauli spin blockade regime. We point out the differences between the spectra of systems with two and a greater number of electrons. For two-electrons the unequal length of the dots results in a different effective $g$-factors in the dots as observed by the recent experiments. For an increased number of electrons the $g$-factor difference between the dots appears already for symmetric systems and it is greatly amplified when the dots are of unequal length. We find that in the lowest energy states all except of two electrons form a frozen core of singlet shells that can be separated away for description of the properties of the system.

Quantum breathing mode of interacting particles in a one-dimensional harmonic trap

Extending our previous work, we explore the breathing mode---the [uniform] radial expansion and contraction of a spatially confined system. We study the breathing mode across the transition from the ideal quantum to the classical regime and confirm that it is not independent of the pair interaction strength (coupling parameter). We present the results of time-dependent Hartree-Fock simulations for 2 to 20 fermions with Coulomb interaction and show how the quantum breathing mode depends on the particle number. We validate the accuracy of our results, comparing them to exact Configuration Interaction results for up to 8 particles.

A comparison of Hartree–Fock and exact diagonalization solutions for a model two-electron system

We compare the restricted (RHF) and unrestricted (UHF) Hartree-Fock solutions with the exact configuration interaction (CI) solutions for the model problem of two interacting electrons confined within an infinite spherical potential of radius R. An investigation of the ground state energy, electron density, and a population analysis of the wave functions reveals that the UHF solution compares favorably with the CI one in both the small and large R limits. The Coulson-Fischer transition (the point at which the RHF and UHF solutions are no longer degenerate) occurs at an r(s) identical with R/2(1/3) approximately 6 a.u., close to the crossover r(s) where the exact CI density acquires a minimum at the sphere center; this behavior captures the onset of Wigner molecule formation. In this regime, however, the HF densities and energies are not quantitatively accurate when compared to the CI ones. This failure is further quantified by computing Ec/E to determine the ability of each method at capturing the exact correlation energy; we conclude that the UHF solutions can be reasonably accurate at both high and low densities, but not at intermediate r(s).

The calculation of electron localization and delocalization indices at the Hartree-Fock, density functional and post-Hartree-Fock levels of theory

Localization, λ(A), and delocalization indices, δ(A,B), as defined in the atoms in molecules theory, are a convenient tool for the analysis of molecular electronic structure from an electron-pair perspective. These indices can be calculated at any level of theory, provided that first- and second-order electron densities are available. In particular, calculations at the Hartree–Fock (HF) and configuration interaction (CI) levels have been previously reported for many molecules. However, λ(A) and δ(A,B) cannot be calculated exactly in the framework of Kohn–Sham (KS) density functional theory (DFT), where the electron-pair density is not defined. As a practical workaround, one can derive a HF-like electron-pair density from the KS orbitals and calculate approximate localization and delocalization indices at the DFT level. Recently, several calculations using this approach have been reported. Here we present HF, CI and approximate DFT calculations of λ(A) and δ(A,B) values for a number of molecules. Furthermore, we also perform approximate CI calculations using the HF formalism to obtain the electron-pair density. In general, the approximate DFT and CI results are closer to the HF results than to the CI ones. Indeed, the approximate calculations take into account Coulomb electron correlation effects on the first-order electron density but not on the electron-pair density. In summary, approximate DFT and CI localization and delocalization indices are easy to calculate and can be useful in the analysis of molecular electronic structure; however, one should take into account that this approximation increases systematically the delocalization between covalently bonded atoms, with respect to the exact CI results.

Magnetic properties of quantum dots and rings

Exact many-body methods as well as current-spin-density functional theory are used to study the magnetism and electron localization in two-dimensional quantum dots and quasi-one-dimensional quantum rings. Predictions of broken-symmetry solutions within the density functional model are confirmed by exact configuration interaction (CI) calculations: In a quantum ring the electrons localize to form an antiferromagnetic chain which can be described with a simple model Hamiltonian. In a quantum dot the magnetic field localizes the electrons as predicted with the density functional approach.

The separation of the dynamical and non-dynamicalelectron correlation effects

Electron pair densities and their moments associated with a set ofmulticonfigurational wavefunctions that systematically includenondynamic electron correlation have been constructed for theberyllium isoelectronic sequence. The obtained two-electronquantities are compared to the Hartree-Fock and nearly exactconfiguration interaction results. It is shown that theelectron-electron counterbalance density identifies thenon-dynamical (long-range) electron correlation. Hence, the wavefunction that maximizes the electron-electron counterbalancedensity contains all the non-dynamical electron correlation.This statement constitutes the proposed separation criterion.Explicitly calculated dynamical and non-dynamical Coulomb holesfor the isoelectronic series of beryllium illustrate thefeasibility of this approach.

On the accuracy of the potential harmonic functions

AbstractThe matrix elements of the correlation function between symmetric potential harmonics am simplified into the analytical summations of the grand angular momenta by dy using the recurrence and coupling relations of the potential harmonics. 'The correlation‐function potential‐harmonic and generalized‐Laguerre‐function method (CFPHGLF), recently developed by us, is applied to the S states of the helium‐like systems for Z = 2 to 9. The results exhibit good convergence with the bases in tern of both the angular and radial directions. The final eigen‐energies agree excellently with the best s‐limits of the variational configuration interaction (CI) method for the involved low‐lying S states. The accuracy of the potential harmonic (PH) expansion scheme is discussed relative to the exact Hylleraas CI results (HCI), and Hartree‐Fock results. Moreover, suggestion is given for the future improvement of the PH scheme.

Reduced multireference CCSD method: An effective approach to quasidegenerate states

Standard multireference (MR) coupled cluster (CC) approaches are based on the effective Hamiltonian formalism and generalized Bloch equation. Their implementation, relying on the valence universal or state universal cluster Ansatz, is very demanding and their practical exploitation is often plagued with intruder state and multiple solution problems. These problems are avoided in the so-called state selective or state specific (SS) MR approaches that concentrate on one state at a time. To preserve as much as possible the flexibility and generality offered by the general MR CC approaches, yet obtaining a reliable and manageable algorithm, we propose a novel SS strategy providing a size-extensive CC formalism, while exploiting the MR model space and the corresponding excited state manifold. This strategy involves three steps: (i) The construction of a variational configuration interaction (CI) wave function within the singly (S) and doubly (D) excited state manifold, (ii) the cluster analysis of this CI wave function providing the information about the higher than pair cluster amplitudes, and (iii) the exploitation of these amplitudes in the so-called externally corrected CCSD procedure. This approach is referred to as the reduced MR (RMR) SS CCSD method and is implemented at the ab initio level and applied to several model systems for which the exact full CI results are available. These include two four electron H4 systems (usually referred to as the H4 and S4 models), an eight electron H8 model and the singlet-triplet separation problem in CH2. It is shown that the RMR CCSD approach produces highly accurate results, is free from intruder state problems, is very general and effective and applicable to both closed and open shell systems.

Effect of different subsets on convergence patterns of hyperspherical harmonic expansion for theS states of the helium atom

Effects of different subsets on convergence patterns of hyperspherical harmonic (HH) expansions for the low-lying 1S and 3S states of the helium atom have been investigated with the correlation-function-hyperspherical-harmonic-generalized-Laguerre-function (CFHHGLF) method by successively introducing HH subsets with the fixed three-dimensional angular momentums (l) into the atomic wave functions. The eigenenergies given by the HH subsets of l=0, 1, 2, and 3 are in good agreement with the best s-, sp-, spd-, and spdf limits of variational configuration interaction (CI) calculations, respectively. The final eigenenergies of the ground state as well as the examined low-lying excited 1S and 3S states are quite close to the exact Hylleraas CI (HCI) values at the sixth decimal place. Moreover, l=0 and l≠0 expansion results also tell us that it is not necessary to take into account too many HHs at the given l, especially for higher l, and that the more the absolute electron correlation energies the bigger l it takes to obtain precise eigenenergies. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 64: 661–668, 1997

Molecular quadrupole moment functions of HF and N2. I. Ab initio linear-response coupled-cluster results

The orthogonally spin-adapted linear-response coupled-cluster (LRCC) theory with singly and doubly excited clusters (CCSD) was employed to calculate quadrupole moment functions of the HF and N2 molecules in their ground electronic states. We also calculated several potential energy curves for both systems using various CC and non-CC methods, ranging from the limited and full configuration interaction (CI) and first-order CI (FOCI) to finite-order many-body perturbation theory. FOCI and related complete active space self-consistent field (CASSCF) methods were used in both energy and quadrupole moment calculations. Most of the calculations were performed using the medium-size basis set of TZ+2P quality devised by Sadlej [A. Sadlej, Coll. Czech. Chem. Commun. 53, 1995 (1988)] for high-level ab initio calculations of electrostatic molecular properties. In addition, a number of model CC calculations using small basis sets were performed, for which the exact full CI results, both for the energy and multipole moments, are available. It was demonstrated that the CCSD approach provides a realistic description of quadrupole moment functions, for all relevant geometries in the case of HF and for internuclear separations up to 1.5 times the equilibrium bond length for N2. The results of this study will be used for the analysis of the rovibrational dependence of quadrupole moments and for the calculation of quadrupole transition moments for both HF and N2.

Resolution of the sign problem in quantum Monte Carlo simulations of annulenes

The applicability of Green's function (GF) and Feynman path-integral quantum Monte Carlo (QMC) methods for the simulation of cyclic networks with (4n + 2) and 4n (n = 1, 2, 3, …) electrons is analysed. Both QMC techniques are employed in simulations on the basis of the simple Huckel Hamiltonian which is exclusively defined by nearest-neighbour hopping elements. In addition we have used the Pariser-Parr-Pople (PPP) Hamiltonian to perform GF QMC simulations. The electronic energies E derived by the QMC methods are compared either with Huckel molecular orbital (HMO) results or exact configuration interaction data where (π) electronic correlations are fully taken into account. A sign problem occurs in QMC simulations of 4n annulenes. This leads to an error in the total energy in the standard formulations of the employed QMC techniques, which is enhanced with decreasing ring size. A simple modification in the QMC formalisms is suggested to avoid the numerical uncertainties caused by the sign problem in 4n annu...

Accurate static and dynamic polarizabilities of Li-

Using multiconfiguration linear-response theory, accurate predictions are given for the static and dynamic dipole polarizabilities of ${\mathrm{Li}}^{\mathrm{\ensuremath{-}}}$. Benchmark full configuration-interaction (CI) results are also presented. Virtually exact full CI results are obtained for a very compact wave function keeping 1s doubly occupied and describing the correlated movement of the remaining two electrons with three orbitals: 2s, 3s, and 2p. d functions are found to be crucial for the calculation of the dipole polarizability and for obtaining gauge invariance. The static polarizability is predicted to be 798\ifmmode\pm\else\textpm\fi{}5 a.u.

Approximate Wave Functions for Atomic Be

A configuration interaction calculation, involving thirty-seven configurations and including the ${(1s)}^{2}{(2s)}^{2}$ Hartree-Fock function, has been done for the ground state of atomic Be. Approximately ninety percent of the correlation energy has been incorporated into the final total energy. The results indicate that the correlation energy is associated with two effects, namely that of the "correlation hole" as has been observed for He and that of "orbital degeneracy" (which does not appear in the two-electron He case). The former effect is best handled by the Hylleraas approach and the latter by the configuration interaction method, and the results suggest that an admixture of the two methods would lead to the most rapidly convergent description of the exact four-electron wave function. The errors introduced by handling "high-lying" configurations by second-order perturbation theory rather than by exact configuration interaction are also investigated.