9832 | Phys. Chem. Chem. Phys., 2021, 23, 9832–9842 This journal is © the Owner Societies 2021 Cite this: Phys. Chem. Chem. Phys., 2021, 23, 9832 Quantifying electron-correlation effects in small coinage-metal clusters via ab initio calculations V. G. de Pina, a B. G. A. Brito, b G.-Q. Hai c and L. Ca ˆ ndido * a We investigate many-electron correlation effects in neutral and charged coinage-metal clusters Cu n , Ag n , and Au n (n = 1–4) via ab initio calculations using fixed-node diffusion Monte Carlo (FN-DMC) simulations, density functional theory (DFT), and the Hartree–Fock (HF) method. From very accurate FN-DMC total energies of the clusters and the HF results in the infinity large complete-basis-set limit, we obtain correlation energies in these strongly correlated many-electron clusters involving d orbitals. The obtained bond lengths of the clusters, atomic binding and dissociation energies, ionization potentials, and electron affinities are in satisfactory agreement with the available experiments. In the analysis, the electron correlation effects on these observable physical quantities are quantified by relative correlation contributions determined by the difference between the calculated FN-DMC and HF results. We show that the correlation contribution is not only significant for the quantities related to electronic structures of the coinage-metal clusters, such as electron affinity, but it is also essential for the stability of the atomic structures of these clusters. For example, the electron correlation contribution is responsible for more than 90% of the atomic binding energies of the small neutral copper clusters. We also demonstrate the orbital-occupation dependence of the correlation energy and electron pairing of the valence electrons in these coinage-metal clusters from the electron correlation-energy gain and spin-multiplicity change in the electron addition processes, which are reflected in their ionization potentials and electron affinities. 1 Introduction The study of small metallic clusters has significantly pro- gressed over the last few decades. 1,2 The transition metal clusters of high-Z atoms such as those formed by noble metals Cu, Ag, and Au, known as coinage-metal clusters are of great interest since they have essential characteristics suitable for industrial applications, 3 healthcare, 4 and electronic devices. 5–13 Besides the technological interest, these materials attract a great deal of attention in physics because of the particular electronic structures of the coinage-metal atoms. The Cu, Ag, and Au atoms have completely filled 3d, 4d, and 5d orbitals, respectively, with 10 electrons, and the valence shell contains a single s electron. The valence s electrons in the coinage-metal clusters are close to the d electrons in real space and energy, and their interactions and correlations are strong and subtle. Such facts have significant implications for their structural, electronic, optical, and chemical properties. 14–16 In general, electron correlations in atoms, molecules, and atomic clusters contribute less than 1% in their total ground- state energies, except in some very light ones such as the He atom and the H 2 molecule. The relative contributions of the electron-correlation energy in the ground states of atoms decrease with the increasing atomic number. For example, in 3d transition-metal atoms, the electron-correlation energy is only about 0.1% of the total ground-state energy. Despite being a tiny part of the total energies of atoms and molecules, the electron-correlation energy and consequent correlation effects are always important and sometimes decisive in their physical and chemical properties. In the cases of the coinage-metal atoms Cu, Ag, and Au, the energy differences between the nd and (n + 1)s orbitals (n = 3, 4, and 5) are small and the electrons in these orbitals are strongly correlated. Therefore, careful treatment of electron correlation in the d and s valence orbitals is required in any theoretical investigation of these atoms and related molecules and clusters. Although the properties of coinage-metal clusters such as atomic binding energy, ioniza- tion potential, electron affinity, and electronic state energy levels have been studied both experimentally and theoretically, most of the theoretical calculations are at the level of quantum chemistry methods suffering from the electronic complexity of these clusters. 17–25 Quantities such as ionization potential (IP) and electron affinity (EA) are determined by the differences a Instituto de Fı ´sica, Universidade Federal de Goia ´s, 74.001-970, Goiaˆnia, GO, Brazil. E-mail: ladir@ufg.br b Instituto de Cie ˆncias Exatas, Naturais e Educaça ˜o, UFTM, 38.025-180, Uberaba, MG, Brazil c Instituto de Fı ´sica de Sa ˜o Carlos, Universidade de Sa ˜o Paulo, 13560-970, Sa ˜o Carlos, SP, Brazil Received 15th December 2020, Accepted 19th March 2021 DOI: 10.1039/d0cp06499h rsc.li/pccp PCCP PAPER