Gibbs ensemble Monte Carlo calculations are performed using a validated embedded-atom potential to obtain the vapor-liquid coexistence curve for elemental aluminum in good agreement with available experimental data up to the boiling point. These calculations are then extended to make a reliable prediction of the critical temperature, pressure, and density of Al, which have previously been known only with very large uncertainties. This demonstrates the ability of modern simulations to predict fundamental physical properties that are extremely difficult to measure directly.

We present a new hybrid meta exchange-correlation functional, called M05-2X, for thermochemistry, thermochemical kinetics, and noncovalent interactions. We also provide a full discussion of the new M05 functional, previously presented in a short communication. The M05 functional was parametrized including both metals and nonmetals, whereas M05-2X is a high-nonlocality functional with double the amount of nonlocal exchange (2X) that is parametrized only for nonmetals. In particular, M05 was parametrized against 35 data values, and M05-2X is parametrized against 34 data values. Both functionals, along with 28 other functionals, have been comparatively assessed against 234 data values:  the MGAE109/3 main-group atomization energy database, the IP13/3 ionization potential database, the EA13/3 electron affinity database, the HTBH38/4 database of barrier height for hydrogen-transfer reactions, five noncovalent databases, two databases involving metal-metal and metal-ligand bond energies, a dipole moment database, a database of four alkyl bond dissociation energies of alkanes and ethers, and three total energies of one-electron systems. We also tested the new functionals and 12 others for eight hydrogen-bonding and stacking interaction energies in nucleobase pairs, and we tested M05 and M05-2X and 19 other functionals for the geometry, dipole moment, and binding energy of HCN-BF3, which has recently been shown to be a very difficult case for density functional theory. We tested eight functionals for four more alkyl bond dissociation energies, and we tested 12 functionals for several additional bond energies with varying amounts of multireference character. On the basis of all the results for 256 data values in 18 databases in the present study, we recommend M05-2X, M05, PW6B95, PWB6K, and MPWB1K for general-purpose applications in thermochemistry, kinetics, and noncovalent interactions involving nonmetals and we recommend M05 for studies involving both metallic and nonmetallic elements. The M05 functional, essentially uniquely among the functionals with broad applicability to chemistry, also performs well not only for main-group thermochemistry and radical reaction barrier heights but also for transition-metal-transition-metal interactions. The M05-2X functional has the best performance for thermochemical kinetics, noncovalent interactions (especially weak interaction, hydrogen bonding, π···π stacking, and interactions energies of nucleobases), and alkyl bond dissociation energies and the best composite results for energetics, excluding metals.

Potential energy functions (PEFs) parametrized to bulk data are shown to perform poorly for small aluminum nanoparticles and clusters. In contrast, PEFs parametrized to a limited set of cluster and bulk data, but no nanoparticle data, perform well for nanoparticles. This validates a practical scheme for developing PEFs for nanoscale systems. Building on these findings, we optimized five PEFs by minimizing the error in the fit over a broad data set. Two of these PEFs have errors of less than or equal to 0.08 eV/atom for each of three categories of system sizes, i.e., for small clusters, for nanoparticles, and for bulk potential energies.

A new, parametrized many-body tight-binding model is proposed for calculating the potential energy surface for aluminum nanoparticles. The parameters have been fitted to reproduce the energies for a variety of aluminum clusters (Al{sub 2}, Al{sub 3}, Al{sub 4}, Al{sub 7}, Al{sub 13}) calculated recently by the PBE0/MG3 method as well as the experimental face-centered-cubic cohesive energy, lattice constant, and a small set of Al cluster ionization potentials. Several types of parametrization are presented and compared. The mean unsigned error per atom for the best model is less than 0.03 eV.

Nineteen analytic potential energy functions (PEFs) for aluminum (three pairwise additive ones, six nonpairwise additive ones with three-body terms, and ten embedded atom-type PEFs) were obtained from the literature. The PEFs were tested and reparametrized using a diverse training set that includes 20 potential energy curves and a total of 224 geometries for five aluminum clusters Al N (N = 2, 3, 4, 7, and 13) computed using hybrid density functional theory, as well as the experimental face-centered cubic cohesive energy and lattice constant. The best PEFs from the literature have mean unsigned errors (MUEs) over the clusters in the data set of ∼0.12 eV/atom. The best reparametrized PEFs from the literature have MUEs of 0.06 eV/atom. The data set is also used to develop, parametrize, and systematically study the effectiveness of several functional forms designed specifically to model many-body effects in clusters, including bond angle, screening, and coordination number effects; a total of eighteen new PEFs are proposed and tested. The best potential overall has an MUE of 0.05 eV/atom, explicitly includes screening and coordination number effects, features linear scaling, and incorporates the accurate two-body and bulk limits.

In previous work we have shown that the PBE0 hybrid density functional method with the MG3 all-electron basis set is an accurate method for calculating the atomization energies of small aluminum clusters (Al2-Al7). However, the MG3 basis set is very expensive for molecules much larger than Al13; therefore, we have developed a new effective core potential (ECP) method for aluminum to reduce the cost of obtaining accurate results for nanoparticles. Our method involves a hybrid of the Stuttgart semiempirical effective core potential and the compact effective potential (CEP) potential, and it uses a newly optimized polarized valence triple-ζ basis set. The combination of the new ECP and the new polarized valence triple-ζ basis set for Al is called the Minnesota effective core (MEC) method for Al. The method was optimized with a training set of atomization energies and geometries for AlH, AlC, AlO, AlCCH, Al2H, Al2C, Al2O, and Al3 and atomization energies of three Al13 structures, and we tested it on six test sets composed of 20 atomization energies for systems as large as Al13. We also present an improved all-electron polarized triple split basis set for oxygen, called 6-311+G(d*,p). For the test sets, the mean unsigned error (MUE) of the new method with respect to PBE0/MG3 is 0.06 eV for atomization energies and 0.007 Å for bond lengths, which constitutes a very significant improvement over the quality of the results that can be obtained with effective core potentials and valence basis sets in the literature (of the eight methods in the literature, the best previous method had average errors of 0.63 eV and 0.036 Å). We have also tested the MEC method with a variety of hybrid density functional theory, hybrid meta density functional theory, and pure GGA and meta GGA functionals and found that the average MUE, relative to each functional with all-electron basis sets, is 0.04 eV for atomization energies and 0.009 Å for bond lengths for the new effective core method and 0.16-0.20 eV and 0.013-0.033 Å for effective core potential and valence basis sets in the literature.

We calculated the atomization energy of aluminum clusters (Al-2-Al-7) with several multilevel methods, including MCG3/3 and G3X, that have been previously shown to provide high accuracy for atomization energies. We used the results to test a number of hybrid density functional theory (HDFT) methods and found that the PBE0 method is in best agreement with the accurate methods. We then used the PBE0/MG3 method to develop a more extensive data set for the energies of small aluminum clusters (Al-2-Al-13), and this was used to test a number of semiempirical methods, in particular Austin model 1 (AM1), modified neglect of differential overlap (MNDO), modified symmetric-orthogonalized intermediate neglect of differential overlap (MSINDO) with and without d-functions, parametrized model 3 (PM3), and the tight-binding total energy (TBTE) method, for geometries, energies, and multiplicities of At clusters. The AMI model and MSINDO model are the most accurate of the semiempirical methods for energetics, and PM3 is the most accurate method for geometries.

We optimized the coefficients for 11 multi-coefficient correlation methods (MCCMs) against 54 atomization energies for molecules composed of C, H, and O atoms and containing 343 bonds. The methods included two scaling-all-correlation (SAC) methods, a multi-coefficients SAC (MCSAC) method, three Utah methods, two Colorado methods, MC-QCISD, MCG3, and G3S. The mean unsigned errors are (on average) 36% lower than for calculations with general parameters. The mean unsigned error per bond for the most highly recommended methods is only 0.1−0.2 kcal/mol.