A study of the ionisation and excitation energies of core electrons using a unitary group adapted state universal approach

Abstract

The unitary group adapted state universal multi-reference coupled cluster (UGA-SUMRCC) theory, recently developed by us using a normal-ordered multi-exponential wave operator Ansatz with spin-free excitations in the cluster operators, has the twin advantages of generating a spin-adapted coupled-cluster (CC) function and having a terminating expression of the so-called ‘direct term’ at the quartic power of cluster amplitudes. Not having any valence spectators, it also has the potentiality to describe orbital- and correlation-relaxation effectively. We illustrate this aspect by applying our formalism to study ionised/excited state energies involving core electrons. The high degree of orbital relaxation attendant on removal of a core electron and the consequent correlation relaxation of the ionised state are demonstrated to be captured very effectively with the Hartree–Fock orbitals for the neutral ground state.Three different formalisms have been used: (1) the ionisation potential/excitation energies (IP/EE) are computed as difference between state energies of the concerned states; (2) a quasi-Fock extension of the UGA-SUMRCC theory, which computes IP/EE directly and (3) use of a special model space to describe both the ground and the excited states such that EE is obtained directly. The model spaces for EE are incomplete and we have indicated the necessary modifications needed to have extensive energies. The results obtained by all the three approaches are very similar but they usually outperform the current CC-based allied theories with roughly the same computational effort.

Keywords:

• spin-free MRCC
• normal ordering
• core ionisation/excitation
• unitary group
• incomplete model space

Acknowledgements

DM acknowledges the Humboldt Award, the J. C. Bose Fellowship, the Indo-EU grant and the Indo-French project for research support. SS thanks the SPM fellowship of the CSIR (India) and AS thanks the DST (India) and CEFIPRA for generous financial support. It gives us great pleasure to dedicate this paper to Werner Kutzelnigg on the occasion of his 80th birthday. DM wishes Werner Kutzelnigg many more years of creative pursuit and excellent health, and cherishes many happy moments in the long and intense collaborations with him and his warm friendship.

Notes

Note: i, j,…, etc. denote inactive hole orbitals.

a, b,…, etc. denote inactive particle orbitals.

I, A denote active hole and active particle orbital respectively.

The amplitude, ‘t’, and the unitary generators, ‘E’, together constitute the cluster operators, ‘T’. Where a common amplitude is associated with a combination of two or more operators differing in their active orbital indices, the active orbital indices of the amplitude have been suppressed and replaced with a symbol, •, , etc. For excitations of the same rank (in terms of changes in occupancy of inactive orbitals), the different classes of excitation amplitudes (differing in changes of occupancy of active orbitals) are denoted by ¹T, ²T, etc.

Note: Geometry: R (O–H) = 0.9772 Å Θ (H–O–H) = 104.52°.

Energies are in eV.

Note: Geometries for G.S.:

R (C–H) = 1.087Å, Θ (H–C–H) = 104.3°; R (H–F) = 0.917Å; R (N–H) = 1.014Å, Θ (H–N–H) = 107.2°.

Geometries for ionised state:

R (C–H) = 1.039Å, Θ (H–C–H) = 104.3°; R (H–F) = 0.995Å; R (N–H) = 0.981Å, Θ (H–N–H) = 113.6°.

Energies are in eV.

Note: Geometry: R (C-O) = 1.1283 Å.

Energies are in eV.

Note: Geometry: R (O–H) = 0.9772 Å Θ H-O-H = 104.52°.

Energies are in eV.

Note: Geometry: R (N–N) = 2.068 au.

Energies are in eV.

xxx: Did not converge.

Note: Geometry of G.S.: R (C–H) = 1.087 Å.

Geometry of Excited State.: R (C–H) = 1.032 Å.

Energies are in eV.