Convergence Problems and Energetic Regions in π Excited States of Certain Conjugated Molecules

1980 ◽  
Vol 35 (10) ◽  
pp. 1006-1012
Author(s):  
Myriam S. de Giambiagi ◽  
Mario Giambiagi ◽  
Henrique G. P. Lins de Barros
Keyword(s):  
Excited States ◽  
Variation Method ◽  
Bond Orders ◽  
Convergence Problem ◽  
Conjugated Molecules ◽  
Particular Solutions ◽  
Convergence Parameter ◽  
Convergence Problems

AbstractWhen calculating π bond orders of excited and superexcited states of conjugated molecules, difficulties arise in applying the variation method; besides, the convergence problems involved are well known. For pyridazine, chosen for discussion, 27 states are considered; the convergence problem is envisaged through two criteria in the choice of a parameter introduced in the compromise Hamiltonian. This convergence parameter is related to the variation method. There exist three particular solutions for bond orders, which divide the 27 states into energetical regions.

A Modified Reliability Index Approach for Reliability-Based Design Optimization

2009 ◽  
Author(s):  
Po Ting Lin ◽  
Hae Chang Gea ◽  
Yogesh Jaluria
Keyword(s):  
Reliability Index ◽  
Reliability Based Design Optimization ◽  
Convergence Problem ◽  
Analysis Problem ◽  
Numerical Examples ◽  
Reliability Based Design ◽  
Index Approach ◽  
Convergence Problems ◽  
Definition Of ◽  
Performance Constraint

RBDO problems have been intensively studied for many decades. Since Hasofer and Lind defined a measure of the second-moment reliability index, many RBDO methods utilizing the concept of reliability index have been introduced as the Reliability Index Approach (RIA). In the RIA, a reliability analysis problem is formulated to find the reliability index for each performance constraint and the solutions are used to evaluate the failure probability. However, the traditional RIA suffers from inefficiency and convergence problems. In this paper, we revisited the definition of the reliability index and revealed the convergence problem in the traditional RIA. Furthermore, a new definition of the reliability index is proposed to correct this problem and a modified Reliability Index Approach based on this definition is developed. Numerical examples using both the traditional RIA and the modified RIA are compared and discussed.

Some Features of Excited States Density Matrix Calculation and Their Pairing Relations in Conjugated Systems

1983 ◽  
Vol 38 (5) ◽  
pp. 595-600
Author(s):  
Myriam Segre de Giambiagi ◽  
Mario Giambiagi
Keyword(s):  
Density Matrix ◽  
Excited States ◽  
Simple Approach ◽  
Alternative Formulation ◽  
Density Matrices ◽  
Convergence Problem ◽  
Conjugated Systems ◽  
Self Consistent ◽  
Natural Separation ◽  
Matrix Calculation

Direct PPP-type calculations of self-consistent (SC) density matrices for excited states are described and the corresponding “thawn” molecular orbitals (MO) are discussed. Special atten­tion is addressed to particular solutions arising in conjugated systems of a certain symmetry, and to their chemical implications. The U(2) and U(3) algebras are applied, respectively, to the 4- electron and 6-electron cases; a natural separation of excited states in different cases follows. A simple approach to the convergence problem for excited states is given. The complementarity relations, an alternative formulation of the pairing theorem valid for heteromolecules and non-alternant systems, allow some fruitful experimental applications. Together with the extended pairing relations shown here, they may help to rationalize general trends.

SEMIEMPIRICAL SCF CALCULATIONS ON AZULENE AND ITS SINGLY CHARGED IONS

1965 ◽  
Vol 43 (11) ◽  
pp. 3026-3038 ◽  
Author(s):  
J. E. Bloor
Keyword(s):  
Dipole Moment ◽  
Excited States ◽  
Ground State ◽  
Basis Set ◽  
Bond Orders ◽  
Electron Charge Density ◽  
Charge Densities ◽  
Bond Lengths ◽  
Set Agreement ◽  
Singly Charged

SCF MOs for azulene have been obtained by the semiempirical Pariser, Parr, Pople procedure using the Nishimoto–Mataga method of calculating repulsion integrals and the assumption that nearest neighbor resonance integrals are independent of interatomic distance. Excited states calculated from these MOs by a CI calculation are in very good agreement with experiment. Ground state charge densities, bond orders, and the dipole moment are similar to other SCFMO calculations and reveal no disadvantage in adopting a constant resonance integral for all bonds. It is shown that estimates of the π-electron charge density by n.m.r. methods are not compatible with direct dipole moment measurements and it is suggested that the interpretation of the n.m.r. measurements suffers from inaccuracies in estimating ring currents. Doubt is also thrown on the use of simple relationships between calculated π-bond orders and bond lengths obtained by X-ray crystallographic measurements on the solid state, particularly since all the bond lengths in azulene are predicted to be longer than in benzene whereas experiment shows some to be shorter. Calculations on spin densities and charge densities of the singly charged azulene anion and cation have been performed by a restricted Hartree–Fock perturbation method in which the matrix elements for the interaction between singly excited states and the ground state are calculated using the closed shell SCFMOs of azulene as the basis set. Agreement with experiment for the anion is fairly good. For the cation our results are in severe disagreement with recent VB calculations, but there are no experimental results available to decide between the two methods.

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