Fock-space Coupled-cluster Theory Research Articles

Theoretical molecular spectroscopy of actinide compounds: the ThO molecule

The tiny-core generalised (Gatchina) relativistic pseudopotential (GRPP) model provides an accurate approximation for many-electron Hamiltonians of molecules containing heavy atoms, ensuring a proper description of the effects of non-Coulombian electron-electron interactions, electronic self-energy and vacuum polarisation. Combining this model with electron correlation treatment in the frames of the intermediate Hamiltonian Fock space coupled cluster theory employing incomplete main model spaces, one obtains a reliable and economical tool for excited state modelling. The performance of this method is assessed in applications to ab initio modelling of excited electronic states of the thorium monoxide molecule with term energies below 20,000 cm. Radiative lifetimes of excited states are estimated using truncated expansions of effective and metric operators in powers of cluster amplitudes.

Intermediate Hamiltonian Fock-space coupled-cluster theory for excitation energies, double ionization potentials, and double electron attachments with spin-orbit coupling.

The intermediate Hamiltonian Fock-space coupled-cluster methods at the singles and doubles level (IHFSCCSD) for excitation energies in the (1p, 1h) sector, double ionization potentials in the (0p, 2h) sector, and double electron attachments in the (2p, 0h) sector of the Fock space are implemented based on the CCSD method with spin-orbit coupling (SOC) included in the post-Hartree-Fock treatment using a closed-shell reference in this work. The active space is chosen to contain those orbitals that have the largest contribution to principal ionized or electron-attached states obtained from the equation-of-motion coupled-cluster calculations. Both time-reversal symmetry and spatial symmetry are exploited in the implementation. Our results show that the accuracy of IHFSCCSD results is closely related to the active space, and the sufficiency of the active space can be assessed from the percentage of transitions within the active space. In addition, unreasonable results may be encountered when the ionized or electron-attached states with a somewhat larger contribution from double excitations are included to determine the active space and cluster operators in the (0p, 1h) or (1p, 0h) sector of the Fock space. A larger active space may be required to describe SO splitting reliably than that in the scalar-relativistic calculations in some cases. The IHFSCCSD method with SOC developed in this work can provide reliable results for heavy-element systems when a sufficient active space built upon the principal ionization potential/electron affinity states is adopted.

Effect of the neutron quadrupole distribution in the TaO+ cation

We estimate the effect of the tensor parity nonconserving (PNC) interaction in the 181TaO+ molecular cation. It can be used to probe the unknown quadrupole distribution of the neutrons inside the Ta nucleus. To this end, we evaluate the constant which characterizes this interaction using the relativistic Fock space coupled cluster theory for electronic structure modelling. The 3Δ1 state of the TaO+ cation which can be used to measure the PNC effect is found to be the ground one in agreement with the previous theoretical study.

Partial fourth order schemes of triples in Fock-space coupled-cluster theory: Ionization potentials of ozone

In this paper, three lowest vertical ionization potentials of ozone molecule are presented using four different approximate triples-corrected methods, in addition to the full singles and doubles level, within the Fock-space coupled-cluster theory. One of them is third-order correction and the other three are fourth-order triples approximations, two of which are partial fourth-order corrections. Ozone represents a major challenge due to the multi-reference nature. The value of the partial fourth-order approximations has been highlighted. In particular, we identify a reliable and computationally cheap partial fourth-order scheme.

Fock-Space Coupled Cluster Theory: Systematic Study of Partial Fourth Order Triples Schemes for Ionization Potential and Comparison with Bondonic Formalism.

In this paper, we have made a systematic study of partial fourth order perturbative schemes due to triples to compute the ionization potential within Fock-space multi-reference coupled-cluster theory. In particular, we have obtained computationally less expensive correlation schemes due to fourth order triples. Prototype examples have been considered to explore the efficacy of the approximate methods mentioned, while the bondonic formalism supporting the bonding phenomenology is also respectively for the first time here advanced.

Diagonal and off-diagonal hyperfine structure matrix elements in KCs within the relativistic Fock space coupled cluster theory

The four-component relativistic Fock space coupled cluster method is used to describe the magnetic hyperfine interaction in low-lying electronic states of the KCs molecule. Both diagonal and off-diagonal matrix elements as functions of the internuclear separation R are calculated within the finite-field scheme. The resulting matrix elements exhibit very weak dependence on R for the separations exceeding 8 Å, whereas in the vicinity of the ground-state equilibrium the deviation of molecular HFS matrix elements from the atomic values reaches 15%. The dependence of the computed HFS couplings on the level of core correlation treatment is discussed.

Electronic Transition Dipole Moments in Relativistic Coupled-Cluster Theory: the Finite-Field Method

The determination of the matrix elements of single-electron operators for the transitions between electronic states (primarily, the dipole moments of electron transitions) is one of the critical problems in simulating the atomic and molecular spectra. In this study, the efficiency of a simple version of the finite-field method for calculating these values within the Fock-space coupled-cluster theory has been investigated.

Analysis of the fine structure ofSn11+–Sn14+ions by optical spectroscopy in an electron-beam ion trap

We experimentally re-evaluate the fine structure of Sn$^{11+...14+}$ ions. These ions are essential in bright extreme-ultraviolet (EUV) plasma-light sources for next-generation nanolithography, but their complex electronic structure is an open challenge for both theory and experiment. We combine optical spectroscopy of magnetic dipole $M1$ transitions, in a wavelength range covering 260\,nm to 780\,nm, with charge-state selective ionization in an electron beam ion trap. Our measurements confirm the predictive power of \emph{ab initio} calculations based on Fock space coupled cluster theory. We validate our line identification using semi-empirical Cowan calculations with adjustable wavefunction parameters. Available Ritz combinations further strengthen our analysis. Comparison with previous work suggests that line identifications in the EUV need to be revisited.

Perturbative order analysis of the similarity transformed Hamiltonian in Fock-space coupled cluster theory: difference energy and electric response properties

A perturbative analysis of the ground-state similarity transformed Hamiltonian and its effect on the various Fock-space coupled cluster (FSCC) sectors is presented through calculation of ionisation potential, electron affinity, excitation energies and response properties. Various truncation schemes of the effective Hamiltonian are presented with explicit form of the defining equations. Based on such a truncation, the approximate methods are labelled as FSCC(n), where n represents the correlation energy of the ionised, electron attached or excited states corrected at least up to nth order within coupled cluster singles and doubles scheme (CCSD). A lower scaling CC2 type of approach (abbreviated as FS-CC2) is compared against the group of FSCC(n) methods for energies. Electric response properties have been compared and contrasted for the two lower scaling methods: FSCC(2) and FS-CC2. The various truncated methods are tested for a number of small molecules. The results obtained from a range of truncated methods are compared against full FSCCSD calculations.

Electronic structure theory of the superheavy elements

High-accuracy calculations of atomic properties of the superheavy elements (SHE) up to element 122 are reviewed. The properties discussed include ionization potentials, electron affinities and excitation energies, which are associated with the spectroscopic and chemical behavior of these elements, and are therefore of considerable interest. Accurate predictions of these quantities require high-order inclusion of relativity and electron correlation, as well as large, converged basis sets. The Dirac–Coulomb–Breit Hamiltonian, which includes all terms up to second order in the fine-structure constant α, serves as the framework for the treatment; higher-order Lamb shift terms are considered in some selected cases. Electron correlation is treated by either the multiconfiguration self-consistent-field approach or by Fock-space coupled cluster theory. The latter is enhanced by the intermediate Hamiltonian scheme, allowing the use of larger model (P) spaces. The quality of the calculations is assessed by applying the same methods to lighter homologs of the SHEs and comparing with available experimental information. Very good agreement is obtained, within a few hundredths of an eV, and similar accuracy is expected for the SHEs. Many of the properties predicted for the SHEs differ significantly from what may be expected by straightforward extrapolation of lighter homologs, demonstrating that the structure and chemistry of SHEs are strongly affected by relativity. The major scientific challenge of the calculations is to find the electronic structure and basic atomic properties of the SHE and assign its proper place in the periodic table. Significant recent developments include joint experimental–computational studies of the excitation spectrum of Fm and the ionization energy of Lr, with excellent agreement of experiment and theory, auguring well for the future of research in the field.

A Lagrange multiplier approach for excited state properties through intermediate Hamiltonian formulation of Fock space multireference coupled-cluster theory

In this paper, we present a formulation based on Lagrange multiplier approach for efficient evaluation of excited state energy derivatives in Fock space coupled cluster theory within the intermediate Hamiltonian framework. The formulation is applied to derive the explicit generic expressions up to second order energy derivatives for [1, 1] sector of Fock space with singles and doubles approximation. Its advantage, efficiency, and interconnection in comparison to the Lagrange multiplier approach in traditional formulation of Fock space, which is built on the concept of Bloch equation based effective Hamiltonian, has been discussed. Computational strategy for their implementation has also been discussed in some detail.

Application of relativistic Fock-space coupled-cluster theory to study Li and Li-like ions in plasma

The ionization potentials, transition energies, and oscillator strengths of Li and Li-like C${}^{3+}$ and Al${}^{10+}$ are computed at different plasma environments with the Fock-space multireference coupled-cluster theory to examine the parametric dependence of these properties on plasma density and/or temperature. The results presented here show that the ionization and transition energies as well as the oscillator strengths are very sensitive to the plasma environment. It further shows that the spectral lines corresponding to $\ensuremath{\Delta}n=0$ transitions for Li-like C${}^{3+}$ and Al${}^{10+}$ are blueshifted, whereas the lines associated to $\ensuremath{\Delta}n\ensuremath{\ne}0$ are redshifted ($n$ is the principal quantum number).

Constrained variational approach for energy derivatives in Intermediate Hamiltonian Fock-space coupled-cluster theory

Fock-space multi-reference coupled cluster theory is an efficient method for electronic structure of nearly degenerate cases. However, it suffers from the intruder state problem for large model space. The intermediate Hamiltonian formulation eliminates intruder state problem and helps in the convergence of equations. In this paper we have implemented intermediate Hamiltonian approach in Fock-space coupled cluster method for the response properties. We test our method for the dipole moments of doublet radicals. We report dipole moments of CN, SF and NS radicals.

Kαtransition probabilities for fluorinelike ions from neon to gold:Ab initiorelativistic coupled-cluster calculations

We have applied the relativistic Fock-space coupled-cluster theory to calculate $K\ensuremath{\alpha}$ transition properties of F-like ions for a number of elements throughout the periodic table. A state-specific form of the coupled-cluster method for electron detachment has been used to calculate the $K\ensuremath{\alpha}$ energies and line strengths. The computed energies are used to determine the transition probabilities for more than 30 elements to study the $Z$ dependencies for $Z=10--79$. This application of a variant of coupled-cluster theories to study the $K\ensuremath{\alpha}$ transitions should be the most accurate determination of these transition probabilities to date. The most important finding in this paper is the crossover of $K{\ensuremath{\alpha}}_{1}$ and $K{\ensuremath{\alpha}}_{2}$ at high $Z$. In general, for lighter ions, $K{\ensuremath{\alpha}}_{2}$ is higher in magnitude than $K{\ensuremath{\alpha}}_{1}$, but at $Z=41--42$ they are switched. In addition, the results demonstrate the variation with $Z$ of the $K{\ensuremath{\alpha}}_{2}$ $A$ values along the Periodic Table. The astrophysical significance in the analysis of $K\ensuremath{\alpha}$ lines, important in x-ray astronomy, is pointed out.

Ab initiocalculations of forbidden transition amplitudes and lifetimes of the low-lying states inV4+

We report electric quadrupole $(E2)$ and magnetic dipole $(M1)$ transition amplitudes of the first few low-lying states of quadruply ionized vanadium $({\mathrm{V}}^{4+})$, which are important in various experimental applications and astrophysics. To our knowledge, most of these presented results are determined for the first time in the literature. A relativistic multireference Fock-space coupled-cluster theory with single $(S)$, double $(D)$, and partial triple $(T)$ excitations is employed to compute the forbidden transition probabilities and lifetimes of the low-lying states in ${\mathrm{V}}^{4+}$. Estimations of different correlation effects arising through the above formalism have been highlighted by investigating core and valence electron excitations. A long lifetime is found for the first excited $3d\phantom{\rule{0.2em}{0ex}}^{2}D_{5∕2}$ state, which suggests that ${\mathrm{V}}^{4+}$ may be one of the useful candidates for many important studies.

Relativistic multireference Fock-space coupled-cluster calculation of the forbidden6s2S01→6s5dD13magnetic-dipole transition in ytterbium

We report the forbidden 6s{sup 2} {sup 1}S{sub 0}{yields}6s5d {sup 3}D{sub 1} magnetic-dipole transition amplitude computed using multireference Fock-space coupled-cluster theory. Our computed transition matrix element (1.34x10{sup -4}){mu}{sub B} is in excellent agreement with the experimental value (1.33x10{sup -4}){mu}{sub B}. This value in combination with other known quantities will be helpful in determining the parity-nonconserving amplitude for the 6s{sup 2} {sup 1}S{sub 0}{yields}6s5d {sup 3}D{sub 1} transition in atomic Yb. To our knowledge, this calculation is the most accurate to date, and can be very important in the search for physics beyond the standard model. We further report the 6s6p {sup 3}P{sub 0}{yields}6s6p {sup 1}P{sub 1} and 6s5d {sup 3}D{sub 1}{yields}6s6p {sup 3}P{sub 0} transition matrix elements, which are also in good agreement with the earlier theoretical estimates.

Brueckner based generalized coupled cluster theory: Implicit inclusion of higher excitation effects

A generalization of the single reference Coupled Cluster parameterization for the ground state wave function is proposed that includes substitution operators that annihilate the reference determinant, but which act nontrivially on the correlated part of the wave function. It is shown that an inclusion of such two-body operators can mimic the effect of conventional connected triple and higher excitation operators. Results obtained with Brueckner based Generalized Coupled Cluster Doubles theory (BGCCD-version x) are found to be comparable in accuracy to CCSD(T) and CCSDT for a number of difficult test cases. In the current version of the BGCCD approach we obtain correlated ionization potentials and electron affinities as a by-product of a ground state calculation. This multistate nature of the BGCCD-X approach can give rise to problems with intruder states similar as in Fock Space Coupled Cluster theory.

Many-body similarity transformations generated by normal ordered exponential excitation operators

Normal ordered exponential operators have been used extensively in open-shell formulations of coupled cluster theory. The inverse of such an operator is known to exist, but a closed form explicit expression for the inverse is not available. We will address the evaluation of many-body similarity transformations generated by normal ordered exponential transformation operators without explicit use of the inverse. The similarity transform can be obtained as the solution of a linear system of equations that can be solved trivially using backward substitution. In addition a closed form diagrammatic expression for the similarity transformed operator is presented. Using the many-body similarity transformation strategy a simple and more general formulation of Fock space coupled cluster theory is presented which is akin in spirit to the formulation by Stolarczyk and Monkhorst [Phys. Rev. A 32, 725, 743 (1985); 37, 1908, 1926 (1988)], but which on the other hand is completely equivalent to the conventional wave operator formulation of Fock space coupled cluster theory (under suitable conditions). Other possible applications of the many-body similarity transformation will be briefly discussed.

Size-extensive effective hamiltonian formalisms using quasi-Hilbert and quasi-Fock space strategies with incomplete model spaces

An open-shell coupled cluster (CC) theory is developed using an incomplete model space (IMS) that works entirely within one particular n-valence Hilbert space sector. It is proved that the theory is size-extensive since both H eff and the energies obtained therefrom are connected. The method may be viewed as the Fock space CC theory for an IMS projected onto a given Hilbert space, and is denoted here as a “quasi-Hilbert” space theory. A “quasi-Fock“ formalism is also outlined where additional information on the lower valence model spaces are deliberately retained.