Relativistic unitary coupled cluster theory and applications

Abstract

We present the first formulation and application of relativistic unitary coupled cluster theory to atomic properties. The remarkable features of this theory are highlighted, and it is used to calculate the lifetimes of 5 2 D 3/2 and 6 2 P 3/2 states of Ba + and P b + respectively. The results clearly suggest that it is very well suited for accurate ab initio calculations of properties of heavy atomic systems.PACS numbers: 31.15. Ar, 31.15.Dv, 31.25.Jf, 32.10.Fn There have been a number of attempts to modify coupled-cluster (CC) theory [1], despite its spectacular success in elucidating the properties of a wide range of many-body systems [1,2,3,4,5]. One interesting case in point is unitary coupled-cluster (UCC) theory which was first proposed by Kutzelnigg [6]. In this theory, the effective Hamiltonian is Hermitian by construction and the energy which is the expectation value of this operator in the reference state is unlike in CC theory, an upper bound to the ground state energy [5]. Another attractive feature of this theory which we shall discuss later is that at a given level of approximation it incorporates certain higher order excitations that are not present in CC theory. Furthermore, it is well suited for the calculation of properties where core relaxation effects are important [7,8]. In spite of the aforementioned advantages, there have been relatively few studies based on this method [9,10]. This work is the first relativistic formulation of unitary coupled cluster (UCC) theory and also the first application of this theory to atomic properties. In this letter, we first present the formal aspects of relativistic UCC theory and then apply it to calculate the lifetimes of the 5 2 D 3/2 and 6 2 P 3/2 states of Ba + and P b + respectively; which depend strongly on both relativistic and correlation effects. The comparison of the results of these calculations with accurate experimental data would constitute an important test of this theory.The exact wave function for a closed shell state in CC theory is obtained by the action of the operator exp(T ) on the reference state |Φ . However, in UCC theory [10], it is written aswhere σ = T − T † ; and T and T † are the excitation and deexcitation operators respectively. σ is clearly antiHermitian, since σ † = −σ. Using this unitary ansatz for the correlated wave function, the relativistic UCC equation in the Dirac-Coulomb approximation can be written aswhere H is the Dirac-Coulomb HamiltonianUsing the normal ordered Hamiltonian, Eq.(2) can be rewritten aswhere the normal ordered Hamiltonian is defined as H N = H − Φ| H |Φ and ∆E = E − Φ| H |Φ . . The choice of the operator σ makes the effective Hamiltonian H N = exp(−σ)H N exp(σ) Hermitian.The effective Hamiltonian is expressed by the Hausdorff expansion in CC theory as

Ba+quadrupole polarizabilities: Theory versus experiment

“…These matrix elements were also calculated by us earlier using the same CCSD(T) method but with different basis functions [16,29], and all the results are in good agreement. There are also other calculations available for these matrix elements using different variants of the RCC methods and basis functions [30][31][32][33]; all the calculated results seem to be in reasonable agreement with each other. Moreover, the accuracies of these matrix elements can be verified by using them to estimate the lifetimes of the 5D states and comparing them with the measurements as stated in the Introduction.…”

“…The main reason for the difference between the SDpT method and our results is that the calculated values of the E2 matrix elements of the 6S 1/2 → 5D 3/2 and 6S 1/2 → 5D 5/2 transitions for the two cases are different. It is necessary to emphasize that the precision of the theoretical determination of the lifetimes of the 5D states depends critically on the accuracies of these matrix elements [29,30,[32][33][34]. A precise measurement of the quadrupole polarizability of the ground state in Ba + could test the accuracies of these E2 matrix elements that are calculated by different many-body methods.…”

Relativistic coupled-cluster study of RaF as a candidate for the parity- and time-reversal-violating interaction

“…All these methods are known in the literature as the variational coupledcluster (VCC) method [25]. These VCC methods are well established in the non-relativistic framework but are not that much popular in the relativistic domain, a few are documented in the literature like relativistic UCC by Sur et al [26,27], applicable only for the purpose of atomic calculations. Recently, Sasmal et al implemented ECC in the four-component relativistic domain to calculate the magnetic hyperfine structure (HFS) constants of both atoms and molecules in their open-shell ground state configuration [28].…”

Static electric dipole moment of the francium atom induced by axionlike particle exchange

“…where Γ is Euler Γ-function, and 1 F 1 is Kummer confluent hypergeometric function, and N is positive integer. Study of the atomic electronic structure was performed within the coupled cluster approach [81][82][83][84][85]. It is based on the exponential ansatz of the wavefunction:…”