Hessian Matrix Method Research Articles

Theoretical Study on the NO2 + NO2− Electron Transfer Reaction

AbstractThe NO2 + NO2− electron transfer reaction was studied with DFT‐B3LYP method at 6‐311 + G* basis set level for the eight selected structures: four species favor the structure of “head to head”. The geometry of transition state was obtained by the linear coordinate method. Three parameters, non‐adiabatic activation energy (Ead), coupling matrix element (Hif) and reorganization energy (λ) for electron transfer reaction can be calculated. According to the reorganization energy of the ET reaction, the values obtained from George‐Griffith‐Marcus (GGM) method (the contribution only from diagonal elements of force constant matrix) are larger than those obtained from Hessian matrix method (including the contribution from both diagonal and off‐diagonal elements), which suggests that the coupling interactions between different vibrational modes are important to the inner‐sphere reorganization energy for the ET reactions in gaseous phase. The value of rate constant was obtained by using above three activation parameters.

Generalized approach for accelerated maximum likelihood based image restoration applied to three-dimensional fluorescence microscopy.

For deconvolution applications in three-dimensional microscopy we derived and implemented a generic, accelerated maximum likelihood image restoration algorithm. A conjugate gradient iteration scheme was used considering either Gaussian or Poisson noise models. Poisson models are better suited to low intensity fluorescent image data; typically, they show smaller restoration errors and smoother results. For the regularization, we modified the standard Tikhonov method. However, the generic design of the algorithm allows for more regularization approaches. The Hessian matrix of the restoration functional was used to determine the step size. We compared restoration error and convergence behaviour between the classical line-search and the Hessian matrix method. Under typical working conditions, the restoration error did not increase over that of the line-search and the speed of convergence did not significantly decrease allowing for a twofold increase in processing speed. To determine the regularization parameter, we modified the generalized cross-validation method. Tests that were done on both simulated and experimental fluorescence wide-field data show reliable results.

Study on the structure and property for the NO2+NO2− electron transfer system

The structures and the properties of the NO2+NO2− electron transfer system are studied with ab initio calculations at the B3LYP/6-311+G∗ basis set level for the three selected structures: three species favor the structure of ‘head to head’. The nitrogen dioxide molecule separation distances computed using the DFT/B3LYP method were found to agree with the second order of Møller–Plesset perturbation theory lever (MP2) results. The 351.1nm (3.532eV) photoelectron spectrum of the nitrite anion (NO2−) is obtained. The electron affinity of NO2 is found to be 2.273±0.005eV, which leads to the heat of formation ΔfH00(NO2−)=−183.4±0.9kJ/mol. It shows there are relatively strong interactions between them. Additionally, the geometry of transition state is also obtained by the linear coordinate method. From the analysis of the charge on the transition state and the isolated state, the reaction kinetics mechanism was derived. The activation energy and the coupling matrix element of the rate constant of the ET reaction are also calculated. According to the reorganization energy of the ET reaction, the values obtained from the George–Griffith–Marcus (GGM) method (the contribution only from diagonal elements of force constant matrix) are larger than those obtained from the Hessian matrix method (including the contribution from both diagonal and off-diagonal elements), which suggests that the coupling interactions between different vibrational modes are important to the inner-sphere reorganization energy for the ET reactions in gaseous phase.

Studies on density functional theory for the electron-transfer reaction mechanism between M-C6H6 and M+-C6H6 complexes in the gas phase

Density functional theory (DFT) is used to theoretically investigate the electron-transfer (ET) reactions between M (Li, Na, Mg)–C6H6 and M+–C6H6 complexes in the gas phase. The geometry optimization of the metal–benzene complexes and the encounter state in the process of ET reaction was performed at the 6-31G basis set level. The metal atoms (or metal ions)–benzene molecule separation distances computed using DFT method were found to agree with second-order Moller–Plesset (MP2) results. The precursor complex has C6 symmetry, the distances between acceptor and donor is about 3.0–3.6 A, which yields a bonding energy of approximately 0.9–1.5 eV. It shows there are relatively strong interactions between them. Additionally, the geometry of transition state is also obtained by the linear coordinate method. From the analysis of the charge on the transition state and the isolated state, the reaction mechanism was derived. Also the activation energy and the coupling matrix element of the rate constant of the ET reaction are calculated. According to the reorganization energy of the ET reaction, the values obtained from George–Griffith–Marcus (GGM) method (the contribution only from diagonal elements of force constant matrix) are larger than those obtained from Hessian matrix method (including the contribution from both diagonal and off-diagonal elements), which suggests that the coupling interactions between different vibrational modes are important to the inner-sphere reorganization energy for the ET reactions in gaseous phase. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 78: 186–194, 2000

Theoretical study of inner-sphere reorganization energy of the electron transfer reactions between M–C6H6 and M+–C6H6 complexes in the gas phase: an ab initio computation

An ab inito computation of reorganization energy for the electron transfer (ET) reactions between metal–benzene and metal ion–benzene complexes is presented. The geometry optimization of the metal–benzene complexes was performed. The metal atoms (or metal ions)– benzene molecule separation distances computed using an ab initio method were found to agree with earlier reported results. Values of reorganization energies using George-Griffith Marcus (GGM) method (the contribution from only diagonal elements of force constant matrix) and Hessian matrix method (including the contribution from both diagonal and off-diagonal elements) were computed. Results of reorganization energy show that the GGM method gives much lower values compared to those obtained using the Hessian method, suggesting that the coupling interactions between different vibrational modes are important to the inner-sphere reorganization energy for the ET reactions in gaseous phase.