Open-shell molecular orbitals calculations

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. 

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]. 

The fact is that the molecular orbitals describing the resulting cation may well be quite different from those of the parent molecule. We speak of electron relaxation, and so we need to examine the problem of calculating accurate HF wavefunctions for open-shell systems. 

The calculation of the indices requires the overlap matrix S of atomic orbitals and the first-order density (or population) matrix P (in open-shell systems in addition the spin density matrix Ps). The summations refer to all atomic orbitals /jl centered on atom A, etc. These matrices are all computed during the Hartree-Fock iteration that determines the molecular orbitals. As a result, the three indices can be obtained... 

All electron calculations were carried out with the DFT program suite Turbomole (152,153). The clusters were treated as open-shell systems in the unrestricted Kohn-Sham framework. For the calculations we used the Becke-Perdew exchange-correlation functional dubbed BP86 (154,155) and the hybrid B3LYP functional (156,157). For BP86 we invoked the resolution-of-the-iden-tity (RI) approximation as implemented in Turbomole. For all atoms included in our models we employed Ahlrichs valence triple-C TZVP basis set with polarization functions on all atoms (158). If not noted otherwise, initial guess orbitals were obtained by extended Hiickel theory. Local spin analyses were performed with our local Turbomole version, where either Lowdin (131) or Mulliken (132) pseudo-projection operators were employed. Broken-symmetry determinants were obtained with our restrained optimization tool (136). Pictures of molecular structures were created with Pymol (159). 

Ab initio calculations were carried out for all the low-lying non-Rydberg states of the systems N2, 02, NO, Of, and NO+. In N2, for example, there are 102 molecular states that result from nitrogen atoms in the lowest 4S, 2D, and 2P states. These states were all uniformly described using VCI wave functions constructed as described in Section II. Minimum basis, double-f basis and double-f-plus-polarization basis sets were employed for these studies. For the minimum basis-set calculations, which were always carried out first, the VCI wave functions represent full Cl projections with the constraint that the K shells were kept frozen for all states. However, no constraint on the 2og and 2ou orbitals was made since a Cl among these orbitals is necessary to ensure proper description of the hole states in these molecules, such as C3n of N2. The calculations all have the property of asymptotically connecting with the correct atomic states. This computational method has previously been applied, with reliable results, to both closed- and open-shell systems.6 9 11... 

Finally we describe several methods that combine molecule-dependent empirical parameters with a moderate level ab initio molecular orbital method. The BAC-MP4 method of Melius and coworkers115-118 combines a computationally inexpensive molecular orbital method with a bond additivity correction. This procedure uses a set of accurate experimental data to obtain a correction for bonds of different types that is then used to adjust calculated thermochemical data such as enthalpies of formation. Quite accurate results can be obtained if suitable reference molecules are available and if the errors in the calculation are systematic. The computational methodology is based on an MP4/6-31G(d,p)//HF/6-/31G(d) calculation. A pairwise additive empirical bond correction is derived for different bonds from fitting to experimental enthalpies of formation or in some cases to high quality ab initio computations. In addition, for open-shell molecules an additional correction is needed to compensate for spin contamination of the wavefunction from higher spin states in the unrestricted Hartree-Fock (UHF) method. 

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