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Geometries, Binding Energies, Ionization Potentials, and Electron Affinities of Metal Clusters: Mg , n= 1-7 n0,+-1
Kaining Duanmu, Orlando Roberto-Neto, Francisco Bolivar Correto Machado, Jared A. Hansen, Jun Shen, Piotr Piecuch, and Donald G. Truhlar J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b03080 • Publication Date (Web): 27 May 2016 Downloaded from http://pubs.acs.org on May 31, 2016
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Revised for J. Phys. Chem. C, 5/27/2016
Geometries, Binding Energies, Ionization Potentials, and Electron Affinities of Metal Clusters: Mgn0,±1 , n = 1−7 Kaining Duanmu,1 Orlando Roberto-Neto,*,2 Francisco B. C. Machado,3 Jared A. Hansen,4 Jun Shen,4 Piotr Piecuch,*,4,5 and Donald G. Truhlar*,1 1
Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, 55455-043, Minnesota, USA 2 Divisão de Aerotermodinâmica e Hipersônica, Instituto de Estudos Avançados, São José dos Campos, 12228-001, São Paulo, Brazil 3 Departamento de Química – Instituto Tecnológico da Aeronáutica, DCTA - São José dos Campos, 12228-900, São Paulo, Brazil 4 Department of Chemistry, Michigan State University, East Lansing, Michigan 48824, USA 5 Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA _________________________________ ABSTRACT. Equilibrium geometries, binding energies, adiabatic ionization potentials, and adiabatic electron affinities for neutral and singly charged magnesium clusters, , n = 1–7, have been computed by using 39 exchange-correlation (XC) Mg 0,±1 n functionals in Kohn-Sham density functional theory and several coupled-cluster methods with single, double, and triple excitations, including CCSD(T) for all species, CCSD(2)T and CR-CC(2,3) for species with n = 1–3, and CCSDt, CC{t;3}, and CCSDT for species with n = 1 and 2. We have used augmented polarized-valence and polarizedcore-valence correlation-consistent basis sets. We have found that the geometry and binding energy of the weakly bound Mg2 dimer requires a robust treatment of connected triple excitations, represented in this work by the CR-CC(2,3), CC{t;3}, and full CCSDT methods, which are more accurate than the popular quasi-perturbative CCSD(T) approximation, but CCSD(T) is sufficiently accurate to be applied to other Mg clusters. We have also demonstrated that for all Mg clusters examined in this study, hybrid XC functionals generally have higher accuracy than local ones, with PW6B95, SOGGA11-X, M11, and PWB6K being the most accurate, both for the geometries and for the binding energies, ionization potentials, and electron-detachment energies. Keywords: anions, binding energies, cations, coupled-cluster theory, electron affinities, ionization potentials, Kohn-Sham density functional theory, magnesium clusters, metal clusters, molecular structures
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1. Introduction Understanding the dependence of structural, electronic, energetic, optical, and magnetic properties of atomic and molecular clusters on the system size is one of the most important aspects of cluster science.1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 This, in particular, applies to clusters formed by divalent metals, which have been of fundamental interest for quite some time.16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32 In this category, clusters of magnesium atoms, which have an s2 valence electron configuration, are especially interesting, since the smallest Mg cluster, namely Mg2, is a system characterized by the very weak binding of the van der Waals type governed by a delicate balance of dispersion and exchange repulsion forces,33,34 which cannot be captured by the HartreeFock or other low-order theory treatments, whereas larger Mgn species are covalently bound and characterized by an increase in the degree of s-p hybridization with the number of atoms in the cluster,29 while having significant contributions from pairwise non-additive many-body interactions.34,35,36,37,38,39,40,41 Furthermore, one observes an unusually large and not always monotonic variation of bonding properties when going from the smaller magnesium clusters toward the larger ones and the bulk limit, and this non-monotonic trend constitutes a challenge for electronic structure methods. Another challenge is the rapidly varying cohesive energy per atom in the smaller Mgn species; it is about 4 and 10 times larger in the Mg3 and Mg4 clusters than in Mg2.34 Another aspect that makes the magnesium and other Group II elements challenging for quantum chemistry is that their ground states display significant multi-configurational character due to the quasi-degeneracy of the valence s and p subshells, which is further enhanced by cluster formation, as in the classic case of the Be atom, and the dimer 42 and trimer 43 , 44 , 45 , 46 , 47 species of Be, which require a high-level treatment of electron correlation effects. Similarly strong multi-configurational character of the ground-state wave function was also found in our earlier studies of the neutral and charged Al2-9 clusters.48 Several experimental and theoretical studies of magnesium clusters of varying size have already been reported. Experimental structures and electronic properties of smallto-large magnesium clusters have been obtained using gas-phase spectroscopy,33,49 Raman matrix spectroscopy,50 photoelectron and mass spectroscopy techniques,51,52,53 electron ionization in supersonic expansion chambers, 54 and high-resolution transmission electron microscopy.55 These experiments have probed a number of neutral clusters containing up to over 100 atoms and anions of up to 35 atoms. However, the only accurate, isomer-specific information provided by the experiments is the bond length and binding energy of Mg2 and the photoelectron spectra of small magnesium anions; the existing spectra of larger clusters cannot be attributed to individual isomers. Theoretical investigations of magnesium clusters reported to date involve numerous Kohn-Sham density functional theory (DFT) 56 calculations exploiting various exchange-correlation (XC) functionals16,18,21,50,57,58,59,60,61,62,63,64,65,66,67,68 and several ab initio wave function computations using Hartree–Fock theory,40 Møller-Plesset (MP) perturbation theory, 69 including the second-order MP237,38,39 and fourth-order MP4 treatments,29,39,41 complete-active-space self-consistent-field (CASSCF)70 approach,35,71
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the coupled-cluster (CC) method72,73 using a popular quasiperturbative CCSD(T)74,75 approximation,29,34,36,38,40,71 and the multireference configuration interaction (MRCI) method.35 A van der Waals potential for Mg2, in which damped dispersion contributions based on C6, C8, and C10 coefficients were added to a repulsive potential based on Hartree–Fock calculations, has been reported as well.76 A few of the past computational studies have attempted to examine structural and energetic properties of the neutral and charged magnesium clusters as functions of the number of Mg atoms in the cluster. For example, Ref. 40 reported calculations of cohesive energies of Mgn clusters with n = 2−22 using a combination of Hartree–Fock theory and the method of increments based on CCSD(T) computations for small clusters, and Refs. 63 and 64 reported DFT studies of structures and cohesive energies of neutral and anionic Mgn clusters with n = 2−22. Reference 68 reported analogous DFT calculations of neutral and cationic Mgn clusters with n up to 30. However, none of the previously published studies have considered high-level (CCSD(T) or better) wave function calculations for magnesium clusters with more than four atoms, and none of the previous DFT studies of magnesium clusters have attempted to systematically calibrate the DFT results against high-level CC calculations. In this article, we apply several CC methods with single, double, and triple excitations and larger correlation-consistent basis sets to compute accurate structures of neutral and singly charged Mg2-4 clusters and to determine energetic properties, including cohesive energies, adiabatic ionization potentials, and adiabatic electron affinities of neutral and singly charged clusters Mg 0,±1 , n = 1−7. The size range of the n magnesium clusters encompasses the transition from the van der Waals, dispersion-type bonding in Mg2 to covalent bonding in Mg4 and larger clusters. It also encompasses the transition from 2D to 3D structures, which occurs around Mg4. The results of our highlevel CC calculations and the available experimental data provide us with the necessary reference information to assess the performance of 39 XC functionals, so that we can make useful recommendations about which functionals are best suited for calculating the structural and energetic properties of magnesium clusters. 2. Computational Methods Most of the wave function computations that serve in this article as reference data for benchmarking DFT are based on the widely used CCSD(T) method, in which a quasiperturbative noniterative correction due to connected triply excited clusters that represents an improvement over the earlier CCSD+T(CCSD) ≡ CCSD[T] approach77 is added to the energy obtained in CCSD78,79 calculations. This method – sometimes called the gold standard of quantum chemistry – offers reasonably high accuracy for the majority of closed-shell and high-spin, open-shell polyatomic systems near their equilibrium geometries. However, we have found that in the case of Mg2, the bond length calculated by CCSD(T) and, to a lesser degree, the CCSD(T) binding energy are not in good agreement with the available experimental values. To understand whether these discrepancies are due to the noniterative quasiperturbative treatment of the connected triple excitations by the CCSD(T) approach and to examine if the inadequacy
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of CCSD(T) in the case of the magnesium dimer extends to other neutral and charged Mgn species, we have performed a number of additional CC calculations with singly, doubly, and triply excited clusters, ranging from the iterative treatments of singles, doubles, and triples offered by the full CCSDT80,81 and active-space CCSDt82 methods, to the robust non-iterative corrections due to all or some triples to the CCSD and CCSDt energies defining the CR-CC(2,3) 83 , 84 , 85 , 86 and CC(t;3) 87 , 88 , 89 approaches. The full CCSDT and active-space CCSDt calculations, and the CC(t;3) computations have been limited to the magnesium dimer. They have been used in this work to judge the performance of the more practical CR-CC(2,3) method, which can be applied to larger molecular clusters, and to provide insights into problems encountered in the CCSD(T) calculations for Mg2. The most accurate CR-CC(2,3) calculations, represented by variant D of CR-CC(2,3), which we elaborate on some more below and which are of the nearly full CCSDT or CC(t;3) quality in the all-electron calculations for Mg2, have been performed for the magnesium dimer and trimer, and their ions, allowing us to calibrate the CCSD(T) results, which we have subsequently obtained for all Mgn systems with n = 1-7 and their ions. The active-space CCSDt approach is based on selecting the dominant triply excited amplitudes within the full CCSDT scheme using active orbitals, following arguments originating from state-specific multi-reference CC considerations employing a singlereference formalism. 90 , 91 This enables one to replace the prohibitively expensive computational steps of full CCSDT that scale as no3nu5, where no and nu are, respectively, the numbers of occupied and unoccupied orbitals used in the post-SCF calculations, by more manageable NoNuno2nu4 steps, where No (< no) and Nu (
