Exploring biorthonormal transformations of pair-correlation functions in atomic structure variational calculations

Abstract

Multiconfiguration expansions frequently target valence correlation and correlation between valence electrons and the outermost core electrons. Correlation within the core is often neglected. A large orbital basis is needed to saturate both the valence and core-valence correlation effects. This in turn leads to huge numbers of configuration state functions (CSFs), many of which are unimportant. To avoid the problems inherent to the use of a single common orthonormal orbital basis for all correlation effects in the multiconfiguration Hartree-Fock (MCHF) method, we propose to optimize independent MCHF pair-correlation functions (PCFs), bringing their own orthonormal one-electron basis. Each PCF is generated by allowing single-and double-excitations from a multireference (MR) function. This computational scheme has the advantage of using targeted and optimally localized orbital sets for each PCF. These pair-correlation functions are coupled together and with each component of the MR space through a low dimension generalized eigenvalue problem. Nonorthogonal orbital sets being involved, the interaction and overlap matrices are built using biorthonormal transformation of the coupled basis sets followed by a counter-transformation of the PCF expansions. Applied to the ground state of beryllium, the new method gives total energies that are lower than the ones from traditional complete active space (CAS)-MCHF calculations using large orbital active sets. It is fair to say that we now have the possibility to account for, in a balanced way, correlation deep down in the atomic core in variational calculations.

Hyperfine structures and Landé gJ-factors for n=2 states in beryllium-, boron-, carbon-, and nitrogen-like ions from

“…The CSFs that describe polarization effects are however energetically unimportant, and a large energy optimized orbital basis is needed to converge the contributions [32]. Although not used in the present work, non-orthogonal orbitals, specifically targeted for describing polarization effects, drastically improve convergence [33]. In addition to being affected by polarization effects, hyperfine structures are know to be sensitive also to higher ordercorrelation effects.…”

Accurate transition probabilities from large-scale multiconfiguration calculations - A tribute to Charlotte Froese Fischer

“…Due to the rapid increase of the number of CSFs in the correlation function this scheme soon gets impracticable. Verdebout et al [43] proposed to split the originally large problem into several smaller problems ("Divide and Conquer"). This amounts to splitting the correlation function Λ into several smaller functions…”

“…To illustrate the effectiveness of the method we look at the ground state of Be [43]. Using a multireference {1s 2 2s 2 , 1s 2 2p 2 , 1s 2 3s 2 , 1s 2 3d 2 } and three separately optimized PCFs Λ v , Λ cv and Λ cc describing, respectively, valence, core-valence, and core-core correlation, we obtain rapid energy convergence and lower total energies than for a very large ordinary MCHF calculation (see table 3).…”

Doublet-quartet energy separation in boron: A partitioned-correlation-function-interaction method

“…Relativistic energies relative to the ground state have been reported by Safronova et al [10] P 5/2 excitation energy, both in the MCHF + BP method tailored to the neutral atom, and results from a partitioned-correlation-function-interaction (PCFI) method that allows nonorthogonal orbital bases for the inclusion of different correlation effects [12,13]. Relativistic and finite mass corrections are included.…”