Configuration Interaction Wavefunctions Research Articles

Fine-structure energy levels, oscillator strengths and lifetimes in Co XV

Excitation energies from ground state for 97 fine-structure levels as well as oscillator strengths and radiative decay rates for all electric-dipole-allowed and intercombination transitions among the fine-structure levels of the terms belonging to the (1s 22s 22p 6)3s 23p, 3s 3 p 2, 3s 23d, 3p 3, 3s3p3d, 3p 23d, 3s3d 2, 3s 24s, 3s 24p, 3s 24d, 3s 24f, and 3s3p4s configurations of Co XV are calculated, using extensive configuration-interaction (CI) wave functions, obtained with the CIV3 computer code of Hibbert. The important relativistic effects in intermediate coupling are included through the Breit-Pauli approximation via spin-orbit, spin-other-orbit, spin-spin, Darwin and mass correction terms. Small adjustments to the diagonal elements of the Hamiltonian matrices have been made. Our calculated excitation energies, including their ordering, are in excellent agreement with the experimental results and the experimentally compiled energy values of the National Institute for Standards and Technology (NIST) wherever available. The mixing among several fine-structure levels is found to be very strong, with most of the strongly mixed levels belonging to the (1s 22s 22p 6)3p 23d and 3s3d 2 configurations. The strong mixing among several fine-structure levels makes it very difficult to identify them uniquely. Perhaps, that may be the reason for the lack of both experimental and theoretical results for these levels. We believe that our extensive calculated values can guide experimentalists in identifying the fine-structure levels in their future work. From our radiative decay rates we have also calculated radiative lifetimes of some fine-structure levels. In this calculation we also predict new data for several fine-structure levels where no other theoretical and/or experimental results are available.

Transitions from the 3p6 4s state of Ti IV

This paper presents ab initio calculations for the electric dipole transition energies and A-values between the 3p64s 2Se (J = 1/2) state of Ti IV and the bound and resonance states 3p6np, 3p53d2 and 3p53d4s (J = 1/2 and 3/2). The calculations were carried out with configuration interaction wavefunctions and relativistic effects were included using the Breit–Pauli approximation. Generally for 2Se to 2P° transitions, the present calculated and the measured transition energies of Ryabtsev et al (2005 Opt. Spectrosc. 98 519–27) agree to 0.002 au. There is a difference of up to 40% between the present theoretical A-values for 2S to 2P° transitions and those of Ryabtsev et al (2005 Opt. Spectrosc. 98 519–27). The paper also compares the present theoretical transition energies from the 2So state to the 3p53d2 and 3p53d4s 2S°, 2P°, 4S°, 4P°, 4D° and 4F° states with the semiempirical calculations of Ryabtsev et al (2005 Opt. Spectrosc. 98 519–27).

Relativistic multi-configuration calculations of Kα and Kβ X-ray transitions for highly ionized Mo ions

A relativistic multi-configuration Dirac–Fock technique has been used for computing the transition wavelengths, transition probabilities, absorption oscillator strengths, and line strengths for the Kα and Kβ line transitions of He-like to Ne-like molybdenum ions. The contributions from the Breit interaction, quantum electrodynamic corrections, and nuclear mass corrections to the initial and final levels have been taken into account. Transitions from the ground state to the n = 2 and 3 states of He-like and Li-like molybdenum have been calculated using two sets of configuration-interaction wavefunctions. One set of wavefunctions was generated using the fully relativistic GRASPVU code and the other was obtained using GRASP 2, the calculated transition wavelength, transition probabilities, and absorption strengths obtained by these two independent methods are in very good agreement and there is good agreement between these results and recent theoretical and experimental results. These data provide reference values for the level lifetimes, charge state distributions, and average charge of molybdenum plasmas.

Metastable levels in Sc II: lifetime measurements and calculations

The lifetime of the metastable level 3d2 1G4 in a singly charged scandium has been experimentally investigated at the ion storage ring CRYRING. A laser probing technique has been used for the studies. We also report calculated lifetimes and transition rates for 10 metastable levels in Sc II, including the 3d2 1G4 state, by calculations using configuration interaction wave functions. The lifetime of the 1G4 level is experimentally determined to be τexp=2.64±0.18 s and is in good agreement with the theoretical calculation τth=2.84 s.

Detailed theoretical and experimental description of normal Auger decay in O2

The normal Auger electron spectrum of the O2 molecule is assigned in detail on the basis of ab initio valence configuration interaction (CI) wavefunctions. Potential energy curves of the ground state, the core-ionized states and the doubly charged final states are calculated and Auger decay rates are obtained with the one-centre approximation. Using the lifetime vibrational interference method, band shapes are obtained for all contributions to the Auger spectrum. The calculated Auger electron spectrum allows us to identify all features observed experimentally. Significant differences to previous assignments are reported. A quantitative simulation of the spectrum is given on the basis of a curve-fitting procedure, in which the energetic positions and intensities of the theoretical bands were optimized. Besides providing a basis for a refined analysis of the spectrum, the fit allows us to assess the accuracy of the calculation. As expected for this level of theory, the absolute accuracy of the valence CI energies is found to be about 0.3 eV. The inherent error of the one-centre transition rates is less than 5% of the most intense transition in the spectrum. The frequently questioned one-centre Auger transition rates are shown to be rather appropriate if applied with reasonable wavefunctions and if the vibrational band structure of the molecular spectrum is properly taken into account.

Ab Initio Explanation of Tunneling Line Shapes for the Kondo Impurity State

We report an ab initio study of the Kondo states formed from a Co adatom on Cu(111) and Cu(100). The model consists of a CoCun cluster ( n = 5-19) embedded in (111) and (100) Cu slabs. An embedding potential derived from density functional theory treats the interaction between the periodic crystal surroundings and the CoCun cluster, while strong electron correlations within CoCun are explicitly accounted for via configuration interaction (CI) methods. Analysis of the embedded CI wave function provides insight into the nature of the Kondo state, specifically into the influence of the crystal host on the Co d-electronic structure. We predict that different d-orbitals are preferentially singly occupied in Co on Cu(111) versus Cu(100) as a result of the different crystalline environment. We propose that these variations in the local d-electronic structure on Co, not accounted for in previous theories, are responsible for the drastically different Kondo resonance line shapes observed in scanning tunneling microscopy experiments on these two surfaces.

Pfaffian pairing and backflow wavefunctions for electronic structure quantum Monte Carlo methods

We investigate pfaffian trial wave functions with singlet and triplet pair orbitals by quantum Monte Carlo methods. We present mathematical identities and the key algebraic properties necessary for efficient evaluation of pfaffians. Following upon our previous study \cite{pfaffianprl}, we explore the possibilities of expanding the wave function in linear combinations of pfaffians. We observe that molecular systems require much larger expansions than atomic systems and linear combinations of a few pfaffians lead to rather small gains in correlation energy. We also test the wave function based on fully-antisymmetrized product of independent pair orbitals. Despite its seemingly large variational potential, we do not observe additional gains in correlation energy. We find that pfaffians lead to substantial improvements in fermion nodes when compared to Hartree-Fock wave functions and exhibit the minimal number of two nodal domains in agreement with recent results on fermion nodes topology. We analyze the nodal structure differences of Hartree-Fock, pfaffian and essentially exact large-scale configuration interaction wave functions. Finally, we combine the recently proposed form of backflow correlations \cite{drummond_bf,rios_bf} with both determinantal and pfaffian based wave functions.

Ligand field effects on the multiplet structure of the U4f XPS of UO2

Ab initio, fully relativistic four component theory was used to determine atomic and interatomic many-body effects for the U4f X-ray photoelectron spectrum of an embedded UO8-12 cluster representing UO2. Many-body effects were included through the use of configuration interaction wavefunctions that allow the mixing of XPS allowed and XPS forbidden configurations. Charge transfer configurations were not included. This work extends our earlier studies on simulations of the 4f XPS for the free U4+ cation. While the main XPS features are similar in both cases, ligand field effects changed the multiplet structure in important ways that better simulated experimental data for UO2. Neither initial nor final state covalency significantly reduced the 4f–5f exchange integrals, and the differences between the atom and cluster model were due to ligand field splitting of the 5f band and increased distributions of intensity from XPS allowed to XPS forbidden peaks. The prominent 7eV satellites associated with UO2 were absent in the simulations, which provided further evidence that these satellites are due to charge transfer and not other interatomic effects.

Electronic states of BP, BP +, BP −, B 2P 2, [formula omitted] and [formula omitted

Using augmented sextuple zeta basis sets and internally contracted multireference configuration interaction (MRCI) wavefunctions, potential energy, electric dipole and transition moments have been computed for the X 3Π, a 1Σ +, b 1Π and A 3Σ − states of BP, X 2Σ + and A 2Π states of BP − and X 4Σ − and A 4Π states of BP +. From these data spectroscopic constants, radiative transition probabilities and photoelectron spectra of BP − and BP have been evaluated. The non-vanishing spin–orbit coupling elements between the four low lying triplet and singlet states of the neutral BP have also been calculated from MRCI wavefunctions. The treatment of the corresponding perturbations in the manifold of dense rovibrational states in the three lowest states would require a precise knowledge of the electronic excitation energies. Our best singlet–triplet separations (X–a) are calculated to be 2412 cm −1 (MRCI) and 2482 cm −1 (restricted coupled cluster with perturbative triples (RCCSD(T))) with an estimated error bound of about ±200 cm −1. All three states have long radiative lifetimes with cascading among the rovibrational levels of different states. The ionization energy IE e of BP is calculated to be 9.22 eV (MRCI) and 9.48 eV (RCCSD(T)), the electron affinity EA e 2.51 eV (MRCI) and 2.74 eV (RCCSD(T)). The photoelectron spectra of BP and BP − have been obtained from the Franck–Condon factors of the MRCI potentials. For the UV spectroscopy the dipole allowed radiative transition probabilities are given for A 3Σ − ↔ X 3Π, b 1Π ↔ a 1Σ + of BP, A 2Π ↔ X 2Σ + of BP − and A 4Π ↔ X 4Σ − of BP +. The ionization energy IE e of B 2P 2 of 8.71 eV and the electron affinity EA e of 2.34 eV have been calculated by the RCCSD(T)/aVQZ approach. Also the harmonic vibrational wavenumbers for the electronic ground states of the ions B 2 P 2 + and B 2 P 2 - are given.

Simultaneous ionization and excitation of helium by electron impact

We present numerical results for He (1s2) (e,2e) He+ reaction process for transitions to the n=1, 2 and 3 states of He+ for noncoplanar symmetric geometry at incident energies of 1000 and 1600eV. The calculations are performed using the plane wave impulse approximation (PWIA) and the 3C method (also called the Brauner, Briggs and Klar (BBK) model) that includes post collision interaction and multiple scattering effects. In both the methods we have used the highly correlated configuration interaction wave function for the ground state of helium. A comparison of the present theoretical cross sections with the recent measured data of Ren et al. [X.G. Ren, C.G. Ning, J.K. Deng, G.L. Su, S.F. Zhang, Y.R. Huang, G.Q. Li, Phys. Rev. A 72 (2005) 042718] shows reasonably good agreement.

Quantum Entanglement and Electron Correlation in Molecular Systems

AbstractWe study the relation between quantum entanglement and electron correlation in quantum chemistry calculations. We prove that the Hartree–Fock (HF) wave function does not violate Bell's inequality, and thus is not entangled, whereas the configuration interaction (CI) wave function is entangled since it violates Bell's inequality. Entanglement is related to electron correlation and might be used as an alternative measure of the electron correlation in quantum chemistry calculations. As an example we show the calculations of entanglement for the H2 molecule and how it correlates with the traditional electron correlation, which is the difference between the exact and the HF energies.

Matrix-covariant representation of high-order configuration interaction and coupled cluster theories

We present the closed form of the reduced density matrices (RDMs) of arbitrary order for configuration interaction (CI) wave functions at any excitation level, up to the full CI. A special operator technique due to Bogoliubov is applied and extended. It focuses on constructions of matrix-covariant expressions independent of the basis set used. The corresponding variational CI equations are given in an explicit form containing the matrices related to conventional excitation operators. A subsequent transformation of the latter to an irreducible form makes it possible to generate the matrix-covariant representation for coupled cluster (CC) models. Here this transformation is performed for a simplified high-order CC scheme somewhat reminiscent of the quadratic CI model. A generalized spin-flip approximation closely related to high-order CI and CC models is presented, stressing on a possible inclusion of nondynamical and dynamical correlation effects for multiple bond breaking. A derivation of the full CI and simple CC models for systems involving effective three-electron interactions is also given, thereby demonstrating the capability of the proposed method to deal with complicated many-body problem. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008

Analytic derivative calculation of electronic g-tensors based on multireference configuration interaction wavefunctions

A computer program has been written to perform analytic derivative calculations of magnetic response properties based on uncontracted multireference configuration interaction (MRCI) wavefunctions. At this point, orbital relaxation is neglected in this treatment and the basis functions are assumed to be independent of the perturbation. It is therefore equivalent to an untruncated sum-over-states (SOS) approach to the g-tensor. Hence, the treatment is more rigorous than all previous MRCI g-tensor calculations that have used truncated SOS formulations. In the present calculations, MRCI calculations were performed on top of full valence complete-active space self-consistent field (CASSCF) reference wavefunctions without truncation of the reference or correlation spaces. Results are reported for two types of spin–orbit coupling operators, a simple effective nuclear charge approximation and the spin–orbit mean field (SOMF) approximation. The effects of orbital relaxation are discussed. The values obtained can be used in future calibration studies.

On the vibronic coupling approximation: A generally applicable approach for determining fully quadratic quasidiabatic coupled electronic state Hamiltonians

In this report we introduce an iterative procedure for constructing a quasidiabatic Hamiltonian representing N(state)-coupled electronic states in the vicinity of an arbitrary point in N(int)-dimensional nuclear coordinate space. The Hamiltonian, which is designed to compute vibronic spectra employing the multimode vibronic coupling approximation, includes all linear terms which are determined exactly using analytic gradient techniques. In addition, all [N(state)][N(int)] quadratic terms, where [n]=n(n+1)/2, are determined from energy gradient and derivative coupling information obtained from reliable multireference configuration interaction wave functions. The use of energy gradient and derivative coupling information enables the large number of second order parameters to be determined employing ab initio data computed at a limited number of points (N(int) being minimal) and assures a maximal degree of quasidiabaticity. Numerical examples are given in which quasidiabatic Hamiltonians centered around three points on the C(3)H(3)N(2) potential energy surface (the minimum energy point on the ground state surface and the minimum energy points on the two- and three-state seams of conical intersection) were computed and compared. A method to modify the conical intersection based Hamiltonians to better describe the region of the ground state minimum is introduced, yielding improved agreement with ab initio results, particularly in the case of the Hamiltonian defined at the two-state minimum energy crossing.

Fine-structure energy levels, oscillator strengths and lifetimes in Mn XIII

We have performed large scale CIV3 calculations of excitation energies from ground state for 98 fine-structure levels as well as of oscillator strengths and radiative decay rates for all electric-dipole-allowed and intercombination transitions among the fine-structure levels of the terms belonging to the (1s22s2 2p6)3s2 3p, 3s3p2, 3s23d, 3p3, 3s3p3d, 3p2 3d, 3s3d2, 3s2 4s, 3s2 4p, 3s24d, 3s2 4f and 3s3p4s configurations of Al-like manganese. These states are represented by very extensive configuration-interaction (CI) wavefunctions obtained with the CIV3 computer code of Hibbert. The important relativistic effects in intermediate coupling are incorporated by means of the Breit–Pauli Hamiltonian. In order to keep our calculated energy splittings as close as possible to the experimental values, we have made small adjustments to the diagonal elements of the Hamiltonian matrices. Our calculated excitation energies, including their ordering, are in excellent agreement with the available experimental results. From our radiative decay rates, we have also calculated radiative lifetimes of some fine-structure levels. Differences between our calculated lifetimes and those of Froese Fisher et al (2006 At. Data Nucl. Data Tables 92 607) for several fine-structure levels are discussed.

Atomic many-body effects in the 4f XPS of the U5+ and U4+ cations: part II: consequences of orbital relaxation

Ab initio, fully relativistic four component theory was used to determine atomic many-body effects for the 4f X-ray photoelectron spectra (XPS) of U5+ and U4+ cations. Many-body effects were included through the use of configuration interaction (CI) wavefunctions, WF‘s, that allow the mixing of XPS allowed and XPS forbidden configurations. This work extends our earlier study of the U 4f XPS in that the orbitals for the final, ionic states of the cations are allowed to relax in the presence of the 4f core-hole. In the earlier work, orbitals optimized for the initial state were frozen and also used for the final, ionic states. While the main XPS features are similar in both cases, using relaxed orbitals for the ionic states introduces changes in the multiplet splitting and in the 4f5/2 and 4f7/2 spin–orbit splitting. The extent of configuration mixing for the U5+ and U4+ final state WF’s is characterized by the magnitude of the intensity lost by the main peaks to satellites. Overall, the use of relaxed orbitals improves the agreement between the theoretical XPS for the U4+ cation and the experimental measurements for UO2.

Oscillator strengths of allowed and intercombination lines in Si II using non-orthogonal wavefunctions

The importance of valence–shell, core–valence and core–core correlation and interactions between the members of 3s2nd 2D Rydberg series and between the Rydberg series and 3s3p22D perturber state in singly ionized silicon has been examined using term-dependent non-orthogonal orbitals in the multiconfiguration Hartree–Fock approach. Large sets of spectroscopic and correlation non-orthogonal functions have been chosen to adequately describe the term dependence of wavefunctions, various correlation corrections and strong interactions in Rydberg series. The relativistic corrections are included through the one-body mass correction, Darwin and spin–orbit operators and two-body spin–other-orbit operator in the Breit–Pauli Hamiltonian. Extensive configuration-interaction wavefunctions have been used in the representation of Si II levels to calculate oscillator strengths and transition probabilities. The accuracy of present oscillator strengths is evaluated by the agreement between the length and velocity formulations combined with the agreement between the calculated and measured transition energies. The present results have been compared with previous calculations, experimental measurements and astronomical observations.

A study of some atomic properties for He-like selected ions

The atomic properties have been studied for He-like ions (He atom, Li+, Be2+ and B3+ions). These properties included, the atomic form factor f(S), electron density at the nucleus , nuclear magnetic shielding constant and diamagnetic susceptibility ,which are very important in the study of physical properties of the atoms and ions. For these purpose two types of the wave functions applied are used, the Hartree-Fock (HF) waves function (uncorrelated) and the Configuration interaction (CI) wave function (correlated). All the results and the behaviors obtained in this work have been discussed, interpreted and compared with those previously obtained.

Energy levels and radiative rates for transitions in Mg-like iron, cobalt and nickel

Energy levels and radiative rates for electric dipole (E1) transitions among the lowest 141 levels of the (1s 22s 22p 6) 3ℓ 2, 3ℓ3ℓ′, and 3ℓ4ℓ configurations of Fe XV, Co XVI, and Ni XVII are calculated through the CIV3 code using extensive configuration-interaction (CI) wavefunctions. The important relativistic effects are included through the Breit–Pauli approximation. In order to keep the calculated energy splittings close to the experimental values, we have made small adjustments to the diagonal elements of the Hamiltonian matrices. The energy levels, including their orderings, are in excellent agreement with the available experimental results for all three ions. However, experimental energies are only available for a few levels. Since mixing among some levels is found to be very strong, it becomes difficult to identify these uniquely. Additionally, some discrepancies with other theoretical work (particularly for Ni XVII) are very large. Therefore, in order to confirm the level ordering as well as to assess the accuracy of energy levels and radiative rates, we have performed two other independent calculations using the GRASP and FAC codes. These codes are fully relativistic, but the CI in the calculations is limited to the basic (minimum) configurations only. This enables us to assess the importance of including elaborate CI for moderately charged ions. Additionally, we report results for electric quadrupole (E2), magnetic dipole (M1), and magnetic quadrupole (M2) transitions, and list lifetimes for all levels. Comparisons are made with other available experimental and theoretical results, and the accuracy of the present results is assessed.

Calculation of transition moments between internally contracted MRCI wave functions with non-orthogonal orbitals†

We present an efficient algorithm for calculating transition properties between internally contracted multireference configuration interaction (MRCI) wave functions using different orbitals for both states. The method is applied to low lying electronic states of aluminium monoxide, and transition moment functions computed with the same or different orbital sets for the individual states are compared. The difference between transition moments obtained with state-averaged or state-specific orbitals is found to be rather small at the MRCI level, but substantial at the CASSCF level. †Dedicated to Professor Pavel Rosmus.