Configuration Interaction Wavefunctions Research Articles

All single excitations in multiconfigurational wave functions

An alternative to the Hartree-Fock wave function is proposed in the form of a Slater determinant which interacts as strongly as possible with all its singly excited configurations. It can be found through a multiconfigurational procedure where the configuration-interaction wave function is set up to include explicitly all singly excited configurations to a principal determinant.

Natural orbitals in momentum space and correlated radial momentum distributions: I. The 1S ground state of li+

Natural orbitals (NO) in momentum space, as well as correlated and independent particle model (IPM) radial momentum distributions, are presented for the 1S ground state of the Li+ ion. These are obtained by Fourier transformation of the position space one-particle (reduced) density matrix. The NO expansion of an accurate (97.6% of the correlation energy) configuration interaction (CI) wave function is used for determining the correlated distribution, while the IPM radial momentum distribution is obtained from the “best” analytical Restricted Hartree–Fock (RHF) orbitals of Roothaan, Sachs and Weiss.

Results of an analytic calculation for the coherent atomic scattering factor for the ground state of the helium isoelectronic sequence

The analytical method developed by the authors for the determination of the expectation value of single-particle operators W = Σi W(ri) correct to second order is employed to obtain an analytic expression for the coherent atomic scattering factor F(k) for the ground state of the helium isoelectronic sequence valid for all values of momentum transfer. The trial wave function ϕot employed is an energy minimized Hartree product of hydrogenic states. For helium the results for the form factor are within 1.2% of the highly accurate values calculated using a 120 parameter configuration interaction wave function and have an accuracy equivalent to that of an analytical Hartree-Fock treatment. This error is further reduced as the atomic number is increased. The calculations are extended to the infinite momentum transfer range and the accuracy of the results are studied in this limit by employing the Kato electron-nucleus cusp condition for the exact ground state wave function of two-electron atomic systems. We observe that in the infinite momentum transfer limit our results for helium are in error by 0.57% and that this error is further diminished for each successive element of the isoelectronic sequence. In addition, we note that the cusp condition is exactly satisfied and is independent of the variational parameter employed in the trial wave function.

Concepts useful in the graphical representation of the coulomb hole: Application to the description of the ground state of the four-electron boron ion b+

The representable reference density matrices introduced in an earlier article are applied to the description of the Coulomb hole in a configuration interaction wave function for the ground state of the four-electron boron ion B+. It is found that a satisfactory picture of the Coulomb hole emerges only when the symmetry of the correlated wave function is recognized in the construction of the reference full density matrix. It is conjectured, but not yet fully established, that particle number symmetry is the most important symmetry to be recognized.

Direct calculation of density matrices: Natural orbitals and occupation numbers of model conjugated molecules

The occupation numbers and natural orbitals of several model π-electron molecules are presented and analyzed, which are directly calculated by employing a theory developed earlier by one of the authors. Computational details are discussed briefly, and the errors which arise in our calculations are estimated through a comparison with the density matrix of a full configuration interaction wave function for the pyridine molecule.

The calculation of the electric dipole moment of CO

The electric dipole moment of CO determined by the molecular Hartree–Fock function of Huo is calculated to be +0.15 D (C+O−), while a configuration interaction wave function built from this molecular Hartree–Fock function plus 138 doubly substituted and 62 singly substituted configurations gives an electric dipole moment of –0.17 D (C−O+). This value is in good agreement with the experimentally observed value of –0.12 D (C−O+) and confirms the sign of the dipole moment which was deduced from a microwave experiment by Rosenblum, Nethercot and Townes. In perturbation language we have calculated the second-order contribution to ∞z∞ (the first-order contribution is zero). The contribution coming from doubly substituted configurations is small; the contribution of the singly substituted configurations is relatively more important. The relatively large error given by the Hartree–Fock function is due to the near cancellation of electronic and nuclear contributions to the electric dipole moment of CO.

Sum-over-states calculations of the polarizabilities of the ground and first excited states of the lithium atom

Approximate moments S(k) of the oscillator strength distribution are calculated by a finite sum over states. Small configuration interaction (CI) wave functions arc constructed in such a manner that the length and velocity formulas for S(−2), the static polarizability, give essentially identical results. The oscillator strengths are used to obtain the frequency dependent polarizability. Both the static and dynamic results for the lithium ground state compare favorably with values obtained with extensive CI wave functions. The method is also applied to the first excited (2P) state of lithium.

Inner-shell excited quartet states in Li-like O5+ and Ne7+ ions

The Rayleigh-Ritz variational method and a configuration-interaction wave function are used to obtain the energies, fine structures and hyperfine structures of high-lying core excited quartet states \(1s2lnl^\prime\,\!^{4}{\rm P}^{0}(n)~(n=1\!-\!5)\) and \(1s2pnp\,\!^{4}{\rm P}(m)~(m=1\!-\!5)\) in O5+ and Ne7+ ions, including the mass polarization and relativistic corrections. Restricted variational method is carried out to extrapolate a better energy. The wavelengths, oscillator strengths, and lifetimes are also explored. The results compared with other theoretical and experimental data in the literature.

Theoretical study of electric moments, polarizabilities, and fine and hyperfine coupling constants of the B3Πg, C3Πu, A′5Σg+, and C″5Πu states of N2This article is part of a Special Issue on Spectroscopy at the University of New Brunswick in honour of Colan Linton and Ron Lees.This paper is dedicated to Professors Ronald M. Lees and Colan Linton from the Department of

Several properties are calculated for the first time for the B3Πg, C3Πu, A′5Σg+, and C″5Πu states of N2, i.e., quadrupole (Θ) and hexadecapole (Φ) moments, static dipole polarizability (α), and hyperfine structure (hfs) data such as the electric field gradient (qii) and magnetic Frosch–Foley parameters (b, c, d). These quantities are evaluated with the density functional theory (DFT) method B3LYP/aug-cc-pVQZ. The variation with geometry is also considered for all parameters above. The B(σgπg) and C(σuπg) triplet Π states have similar values of the anisotropy P|2|(2) ∝ (Pxx − Pyy), with P = Θii, qii, or Tii (dipolar hfs tensor). The first-order spin-orbit coupling constant AΛ and the second-order electron-spin g-shift Δg⊥ are obtained with a 6-311+G(2d) basis and multireference configuration interaction (MRDCI) wave functions. Values of AΛ calculated for W3Δu and (B, C, C′, C″) Π states reproduce well the experimental data (C and C′ are two different minima of 13Πu). The variation of AΛ with R(N-N) is regular for all states studied, except for AΠ(13Πu), which changes abruptly between the C and C′ minima (from 38 to 2 cm−1). The Δg⊥ values — evaluated via sum over states expansions — are calculated for A′5Σg+ as well as A3Σu+ and B′3Σu−. The spin-rotation constant γΣ is derived via Curl’s equation γΣ = (−2B)Δg⊥. We find γΣ(A′) to be negative and to decrease steadily with vibrational excitation, in line with experimental observations. Both Δg⊥ and γΣ of A′5Σg+ are determined by the coupling of this weakly bound state with the strongly repulsive 15Πg state.

Electron collisions with theNO2radical using theR-matrix method

The $R$-matrix method is used to calculate the elastic integral, differential, momentum transfer cross sections, and the excitation cross sections from the ground state $X\text{ }{^{2}A}_{1}$ to the five low-lying electronically excited states $A\text{ }{^{2}B}_{1}$, $B\text{ }{^{2}B}_{2}$, $C\text{ }{^{2}A}_{2}$, ${^{4}A}_{2}$, and ${^{4}B}_{2}$ of the ${\text{NO}}_{2}$ radical. Twenty-one target states are included in the close-coupling expansion of the scattering system, where each target state is represented by configuration-interaction wave function. The calculations give two shape resonances of ${^{3}B}_{1}$ and ${^{1}B}_{1}$ symmetries located at 1.18 and 2.33 eV, respectively, with a common configuration $6{a}_{1}\text{ }2{b}_{1}$. We also detect eight core-excited shape resonances and one Feshbach resonance. Partial elastic, total, and electronic excitation cross sections are presented and compared with the limited previous work. The dissociative nature of the ${^{3}B}_{1}$ and ${^{1}B}_{1}$ shape resonances is explored by performing scattering calculations in which one N-O bond is stretched while the other bond and the bonding angle are frozen. These resonances support dissociative attachment yielding NO and ${\text{O}}^{\ensuremath{-}}$ in their respective ground states.

Electric quadrupole and magnetic dipole transitions among 3d6 levels of Fe III

Radiative rates for electric quadrapole (E2) and magnetic dipole (M1) transitions among the lowest 32 levels belonging to the 3d6 configuration of doubly charged iron (Fe III) are calculated using our recently generated (Deb and Hibbert 2007 J. Phys. B: At. Mol. Opt. Phys. 40 F251) J-dependent configuration interaction (CI) wavefunctions. These results are then compared with other available theoretical calculations. It is found that there are large disagreements among various calculations for many of the transitions, with the present A-values for E2 and M1 transitions often significantly higher than the corresponding values in other calculations.

Fine structure effective collision strengths for the electron impact excitation of

Electron impact excitation collision strengths are required for the analysis and interpretation of stellar observations. To date no evaluations exist for the ion. An 18-state R-matrix calculation has been performed. The target states are represented by configuration interaction wavefunctions and consist of the 18 lowest LS states, having configurations 3s23p3, 3s3p4 and 3s23p23d. These target states give rise to 39 fine structure levels and 741 possible transitions. The fine structure collision strengths were obtained by transforming to a jj-coupling scheme using the JAJOM program of Saraph and have been determined at a sufficiently fine energy mesh to delineate properly the resonance structure. The effective collision strengths are calculated by averaging the electron collision strengths over a Maxwellian distribution of velocities. The nonzero effective collision strengths for transitions between the fine structure levels are given for electron temperatures (Te) in the range log10 Te(K)=4.5–6.5. Data for transitions among the five fine structure levels arising from the 3s23p3 ground state configurations are considered in some detail in this paper. These calculations are the first evaluations of collision strengths and effective collision strengths for this ion.

Configuration interaction calculations of annihilation rates for positronic complexes of alkali hydrides

Ab initio multireference single- and double-excitation configuration interaction (MRD-CI) wave functions have been employed to compute the annihilation rates (AR) of positronic molecular complexes of four alkali hydrides. The first step in these calculations is the evaluation of integrals of the two-particle Dirac delta function δ +− over pairs of electronic and positronic basis functions. MRD-CI wave functions calculated with the same basis are then employed to obtain expectation values of the δ +− operator ( Z eff), which in turn are proportional to the corresponding annihilation rates (AR) of the associated many-particle states. The importance of removing near-linear dependencies in the basis sets employed is stressed as well as the advisability of placing diffuse (small-exponent) functions in the basis only at the most electronegative center of the molecule. A tendency to underestimate the Z eff values is noted because of the impracticality of including sufficiently high-l basis functions in the basis for general molecular systems. However, comparison with the relatively accurate values for the four-electron e +LiH complex obtained by Quantum Monte-Carlo (QMC) and other methods indicates that the fractional error is nearly constant over a large range of internuclear distance, consistent with the expectation that missing correlation effects in the MRD-CI treatment are predominantly atomic in nature. A scaling procedure based on the asymptotic δ +− value, which is the same for all four alkali hydrides, is then shown to produce good agreement with the QMC AR data for e +LiH. The same procedure has been applied to the δ +− values for the positronic complexes of the heavier alkali hydrides for which no other theoretical results are available. Trends in the variation of the AR results with bond distance are discussed.

Hyperfine induced ${\sf 1 s}{\sf 2 s}\ ^{\sf 1}$ S $_{\sf 0} \rightarrow {\sf 1 s}^{\sf 2}\ ^{\sf 1}$ S0 M1 transition of He-like ions

Hyperfine induced \(1s2s ^1\)S\(_0 \rightarrow 1s^2 ^1\)S0 M1 transition probabilities of He-like ions have been calculated from relativistic configuration interaction wavefunctions including the frequency independent Breit interaction and QED effects. Present results for 151Eu and 155Gd are in good agreement with previous calculations [L.N. Labzowsky et al., Phys. Rev. A 63, 054105 (2001)]. Electronic data are given in terms of a general scaling law in Z that, given isotopic nuclear spin and magnetic moment, allows hyperfine induced decay rates to be estimated for any isotope. The results should be helpful for future experimental investigations on QED and parity non-conservation effects.

Oscillator strengths for allowed transitions in neutral oxygen

The B-spline box-based R-matrix method in the Breit–Pauli formulation has been used to calculate oscillator strengths for allowed transitions among the n=2–4 levels and from the n=2 levels to higher excited levels up to the n=11 in neutral oxygen. The close-coupling configuration–interaction wavefunctions are generated to accurately represent the inner-core and core–valence correlation effects. The term dependence of wavefunctions has been accounted for by non-orthogonal sets of one-electron radial functions. The relativistic corrections are included through the one-body mass correction, Darwin and spin–orbit operators in the Breit–Pauli Hamiltonian. The accuracy of our oscillator strengths is evaluated by comparing present results with other available reliable calculations and experiments for the low-lying transitions. A very good agreement with available other theoretical and experimental results is generally noted. There is also a good agreement between the length and velocity values of oscillator strengths.

Breit-PauliR-matrix calculation of fine-structure effective collision strengths for the electron impact excitation of Mg V

Context. Electron impact excitation collision strengths are required for analysing and interpreting stellar observations. Aims. This calculation aims to provide fine-structure effective collision strengths for the Mg ix ion using a method that includes contributions from resonances. Methods. A 26-state Breit-Pauli R-matrix calculation has been performed. The target states are represented by configuration interaction wavefunctions and consist of the 26 lowest LS states, having configurations 2s 2 , 2s2p, 2p 2 , 2s3s, 2s3p, 2s3d, 2p3s, 2p3p, and 2p3d. These target states give rise to 46 fine structure levels and 1035 possible transitions. The effective collision strengths are calculated by averaging the electron collision strengths over a Maxwellian distribution of electron velocities. Results. The non-zero effective collision strengths for transitions between the fine structure levels are given for electron temperatures (Te) in the range log10 Te(K) = 3.0–7.0. Values for selected transitions are given in this paper and links provided to the entire data set.

Wavefunction quality and error estimation of single- and multi-reference coupled-cluster and CI methods: The [formula omitted] model system

Different diagnostics for the approximation quality of wavefunction methods are discussed: the weighted T 1 and D 1 norm, the deviation from the full configuration interaction wavefunction, and the variance of the trial wavefunction. All diagnostics are evaluated for the H 4 model system. Various wavefunction based approaches are analyzed. The wavefunction variance performs as a very reliable quantity to estimate the accuracy of the wavefunction and the energy expectation value. Although significantly more expensive than the evaluation of a plain energy expectation value the calculation of the wavefunction variance seems to be a comprehensive substitute for a full configuration interaction calculation.

Electric dipole transitions among low lying levels of Fe IV

We report a large-scale configuration interaction (CI) calculation for the fine-structure levels of FeIV belonging to the 3d5 3d44s and 3d44p configurations. In order to generate well-represented CI wave functions, we use the Hartree-Fock functions for 1s,2s,2p,3s,3p and 3d orbitals of the 3d5–6 S level, augmented by seven more orbitals (4s,4p, 4d,4f, 5s, 5p and 5d), optimised on different energies: 4s and 4p are optimised on spectroscopic levels, while the remaining orbitals are optimised to account for the effect of different 3d4 cores and correlation effects. Relativistic effects are included through the Breit-Pauli operators. Excitation energies of all the fine-structure levels of the three lowest configurations and oscillator strengths for all E1 transitions among these levels are calculated in the present investigation. A number of these oscillator strengths, for transitions between some low-lying sextet levels, are presented and compared with other available calculations.

Determining quasidiabatic coupled electronic state Hamiltonians using derivative couplings: A normal equations based method

A self-consistent procedure for constructing a quasidiabatic Hamiltonian representing N(state) coupled electronic states in the vicinity of an arbitrary point in nuclear coordinate space is described. The matrix elements of the Hamiltonian are polynomials of arbitrary order. Employing a crude adiabatic basis, the coefficients of the linear terms are determined exactly using analytic gradient techniques. The remaining polynomial coefficients are determined from the normal form of a system of pseudolinear equations, which uses energy gradient and derivative coupling information obtained from reliable multireference configuration interaction wave functions. In a previous implementation energy gradient and derivative coupling information were employed to limit the number of nuclear configurations at which ab initio data were required to determine the unknown coefficients. Conversely, the key aspect of the current approach is the use of ab initio data over an extended range of nuclear configurations. The normal form of the system of pseudolinear equations is introduced here to obtain a least-squares fit to what would have been an (intractable) overcomplete set of data in the previous approach. This method provides a quasidiabatic representation that minimizes the residual derivative coupling in a least-squares sense, a means to extend the domain of accuracy of the diabatic Hamiltonian or refine its accuracy within a given domain, and a way to impose point group symmetry and hermiticity. These attributes are illustrated using the 1 (2)A(1) and 1 (2)E states of the 1-propynyl radical, CH(3)CC.

Excited electronic and ionized states of the nitramide molecule, H2NNO2, studied by the symmetry-adapted-cluster configuration interaction method

Symmetry-adapted-cluster configuration interaction (SAC-CI) wave functions were employed to compute 16 singlet and 13 triplet vertical transitions, and 14 ionized states including relative intensities of the nitramide molecule, H2NNO2. This molecule is the simplest neutral closed-shell molecule which has an N–NO2 bond and is a member of the nitramine family, R,R′N(NO2), an important class of energetic materials with practical applications. The present nitramide results showed strong similarities with the ones of the N, N-dimethylnitramine molecule, which has also an N–NO2 bond and was previously studied using the SAC-CI method. Experimental ultraviolet and photoelectron band spectra of the nitramide molecule could be successfully assigned. All the singlet transitions have valence character. The computed singlet and triplet transitions, excepting a singlet one, result from excitation originating in the four highest occupied molecular orbitals, which have close energies. Most of the singlet and triplet transitions involved mixing of singly excited configurations. The strongest computed transition, at 6.8 eV, is a mixture of two nπNO2 → π* configurations corresponding to excitations from the highest occupied molecular orbital (HOMO) to the first two virtual orbitals and has an optical oscillator strength value of 0.2665. The computed ionized states described the whole measured spectrum, have excellent agreement when compared with the measured ionization potentials and revealed an inversion of the ordering of the first states not expected according to Koopmanns’ theorem, thereby showing the limitations of the latter.