A correlation-energy formula due to Colle and Salvetti [Theor. Chim. Acta 3'7, 329 (1975)], in which the correlation energy density is expressed in terms of the electron density and a Laplacian of the second-order Hartree-Fock density matrix, is restated as a formula involving the density and local kinetic-energy density. On insertion of gradient expansions for the 1ocal kinetic-energy density, density-functional formulas for the correlation energy and correlation potential are then obtained. Through numerical calculations on a number of atoms, positive ions, and molecules, of both openand closed-shell type, it is demonstrated that these formulas, like the original Colle-Salvetti formulas, give correlation energies within a few percent.

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The first experimental determination of the chemical shielding tensor of a singlet carbene center, 1,3,4,5tetramethylimidazol-2-ylidene (1), is reported. Detailed ab initio theoretical calculations of the chemical shielding tensors are reported for local density functional theory (LDFT) and the Hartree-Fock levels (HF) in the LORG and IGLO frameworks. The chemical shielding anisotropy is quite large, which is characteristic of the lowest-energy-singlet carbene centers. The anisotropy at the carbene center in 1 is ~240 ppm. This is one of the largest anisotropies observed for carbon in a strictly organic framework. The calculations indicate that the orientation of the chemical shielding tensor of the carbene is such that the most shielded component is perpendicular to the molecular plane, with the intermediate component oriented approximately along the direction of the lone pair, and the least shielded component perpendicular to the other two. This orientation, as well as the relative size of the anisotropy, agrees with calculations on the singlet carbenes 1, imidazol-2-ylidene (2), and :CF2. The size of the anisotropy is predicted to be enhanced for the parent carbene :CH2-

Different sizes of water clusters from a dimer to twenty water molecules are studied using density functional theory. The binding energies of water clusters are calculated, and a relationship in terms of a simple function has been found between binding energy and the size of the water clusters. The interpolation of this correlation function reproduces the binding energies for the other water clusters to an accuracy within 1 kcal/mol. The extrapolation of the function gives the binding energy, −11.38 kcal/mol, which agrees very well with the experimental binding energy of ice, −11.35 kcal/mol. We also find small water clusters composed of mainly planar four membered rings to be more stable, implying the existence of magic numbers for water clusters with sizes of 4, 8, and 12.

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