Closed-shell molecular orbitals calculations

Most ab initio quantum chemical molecular orbital calculations involve, in some form, the solution of the Hartree-Fock equations. Following Roothaan (13,14) these equations are usually given in a matrix form that for a closed shell molecule takes the deceivingly simple form ... 

Crystal-field spectroscopy of dn ions is being extended nowadays to the near-infrared region with interesting results. Vibronic structure is nowadays used to obtain information on the deformation of the excited state. Also in case of the closed-shell d° complexes the excited state appears to be strongly distorted. Among the latter class especially the linear species show efficient luminescence. Molecular-orbital calculations are in progress and will probably yield interesting results. 

This rule is derived from the Hiickel MO (molecular orbital) calculation on planar monocyclic conjugated hydrocarbons (CH)m where m is an integer equal to or greater than 3, according to which (An + 2) n-electrons are contained in a closed-shell system. Examples of systems that obey the Hiickel rule include ... 

H. Halim, N. Heinrich, W. Koch, J. Schmidt, and G. Frenking, /. Comput. Chem., 7, 93 (1986). MlNDO/3 and MNDO Calculations of Closed- and Open-Shell Cations Containing C, H,N, andO. D. B. Boyd, D. W. Smith,/. J. P. Stewart, and E. Wimmer,/. Comput. Chem., 9,387 (1988). Numerical Sensitivity of Trajectories Across Conformational Energy Hypersurfaces from Geometry Optimized Molecular Orbital Calculations AMI, MNDO, and MINDO/3. 

A number of approximate molecular orbital calculations on ferrocene were reviewed by Cotton and Wilkinson in 1959 and the result of an ab-initio calculation was published in 1972 The latter work also contains references to approximate molecular orbital calculations published between 1959 and 1972. It seems that the particular stability of ferrocene is due to its closed-shell structure all occupied molecular orbitals are bonding between the metal and the ligands, the cyclopentadienyl rings function as five-electron ligands, and the iron atom attains an inert gas configuration of 18 electrons in the valence shell. 

Moreover, the Hartree-Fock Eqs. (50) implemented in Eqs. (51) and (52) are known as Roothaan equations [109] and constitute the basis for closed-shell (or restricted Hartree-Fock, RHF) molecular orbitals calculations. Their extension to spin effects provides the equations for the open shell (or unrestricted Hartree-Fock, UHF), which are also known as the Pople-Nesbet Unrestricted equations [118]. 

Here we give the molecule specification in Cartesian coordinates. The route section specifies a single point energy calculation at the Hartree-Fock level, using the 6-31G(d) basis set. We ve specified a restricted Hartree-Fock calculation (via the R prepended to the HF procedure keyword) because this is a closed shell system. We ve also requested that information about the molecular orbitals be included in the output with Pop=Reg. 

The simplest antisymmetric function that is a combination of molecular orbitals is a determinant. Before forming it, however, we need to account for a factor we ve neglected so far electron spin. Electrons can have spin up i+Vi) or down (-V2). Equation 20 assumes that each molecular orbital holds only one electron. However, most calculations are closed shell calculations, using doubly occupied orbitals, holding two electrons of opposite spin. For the moment, we will limit our discussion to this case. 

Electron propagator theory generates a one-electron picture of electronic structure that includes electron correlation. One-electron energies may be obtained reliably for closed-shell molecules with the P3 method and more complex correlation effects can be treated with renormalized reference states and orbitals. To each electron binding energy, there corresponds a Dyson orbital that is a correlated generalization of a canonical molecular orbital. Electron propagator theory enables interpretation of precise ab initio calculations in terms of one-electron concepts. 

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