Frozen orbital analysis

Rules for Excited States of Degenerate Systems Interpretation by Frozen Orbital Analysis... 

Frozen orbital analysis Excited state Degenerate system Ab initio... 

The frozen-orbital approximation is explained in the standard texts of quanmm chemistry, such as A. Szabo, N. S. Ostlund, Modem Quantum Cchemistry Introduction to Advanced Electronic Structure Theory (McGraw-Hill, New York, 1989) the concept of the frozen-orbital analysis was first proposed in the following papen H. Nakai, H. Morita, H. Nakatsuji, J. Phys. Chem. 100, 15753 (1996)... 

In chemical shift calculations for acylium ions, it was not necessary to model the ionic lattice to obtain accurate values. These ions have tetravalent carbons with no formally empty orbitals, as verified by natural bond orbital calculations (89). Shift calculations for simple carbenium ions with formally empty orbitals may require treatment of the medium. We prepared the isopropyl cation by the adsorption of 2-bromopropane-2-13C onto frozen SbF5 at 223 K and obtained a 13C CP/MAS spectrum at 83 K (53). Analysis of the spinning sidebands yielded experimental values of = 497 ppm, 822 = 385 ppm, and (%3 = 77 ppm. The isotropic 13C shift, 320 ppm, is within 1 ppm of the value in magic acid solution (17). Other NMR evidence includes dipolar dephasing experiments and observation at higher temperature of a scalar doublet ( c-h = 165 Hz) for the cation center. 

We focus on the nonlinear dynamics for the collinear configuration which we treat as part of the full multidimensional system. This is to be contrasted with two-degree-of-freedom models where the molecule is assumed to be frozen in some angular configuration, such that the bending degree of freedom is excluded from the dynamics. In our analysis, bending is taken into account in terms of linearized dynamics, which allows us to extend the results for the collinear situation to the full three-dimensional system. The restriction we must be aware of is that the three-dimensional system may have periodic orbits that are not of collinear type. 

In Figure 11 we demonstrate the application of the contour CSM analysis on the lone-pair orbital of a distorted water molecule (perhaps a frozen moment of a vibration, or a water molecule in a matrix of amorphous ice, or a water molecule trapped in a micropore). The ratio of length of the two O-H bonds is 0.9 (instead... 

All calculations in Ref. [22] were performed utilizing the Gaussian-98 code [30]. The potential energy scan was performed by means of the Mqller-Plesset perturbation theory up to the fourth order (MP4) in the frozen core approximation. The electronic density distribution was studied within the population analysis scheme based on the natural bond orbitals [31,32], A population analysis was performed for the SCF density and MP4(SDQ) generalized density determined applying the Z-vector concept [33]. 

The prediction of molecular Rydberg spectra, as well as the analysis of the available laboratory data, has constituted a challenge on the theory. A number of ab initio calculations have been carried out on transition probabilities of Rydberg molecules, such as the frequently quoted Hartree-Fock (HF) study of H3 by King and Morokuma (15), the self-consistent-field frozen-core calculation with floating-spherical-Slater-type orbitals (FSSO) on the second-row Rydberg... 

For hydrogenic atoms, where the scattering wavefunctions are known (these are hypergeometric functions), this singularity can be identified and handled analytically [115,116]. On the other hand, the SSEA uses numerical scattering orbitals as a basis, computed within a term-dependent, frozen-core HF scheme. This is done so as to be able to handle arbitrary many-electron structures for small as well as for large values of energy. Therefore, a differ-enf fype of analysis was needed and was implemented in Ref. [54] (See also Ref. [105]). If was shown fhaf the Cy(E,E ) can be written as... 

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