Finite perturbation calculation

J(B, C) coupling constants for a number of organoboranes [e.g. l-MeBsHg, B(CH=CH2) 3, BH3CN ] are in good agreement with values calculated from an INDO-SCF finite perturbation calculation. This only took into account the Fermi contact contribution to /(B, C). ... 

These compounds have been the subject of several theoretical [7,11,13,20)] and experimental[21] studies. Ward and Elliott [20] measured the dynamic y hyperpolarizability of butadiene and hexatriene in the vapour phase by means of the dc-SHG technique. Waite and Papadopoulos[7,ll] computed static y values, using a Mac Weeny type Coupled Hartree-Fock Perturbation Theory (CHFPT) in the CNDO approximation, and an extended basis set. Kurtz [15] evaluated by means of a finite perturbation technique at the MNDO level [17] and using the AMI [22] and PM3[23] parametrizations, the mean y values of a series of polyenes containing from 2 to 11 unit cells. At the ab initio level, Hurst et al. [13] and Chopra et al. [20] studied basis sets effects on and y. It appeared that diffuse orbitals must be included in the basis set in order to describe correctly the external part of the molecules which is the most sensitive to the electrical perturbation and to ensure the obtention of accurate values of the calculated properties. 

Wood, R. H. Muhlbauer, W. C. F. Thompson, P. T., Systematic errors in free energy perturbation calculations due to a finite sample of configuration space. Sample-size hysteresis, J. Phys. Chem. 1991, 95, 6670-6675... 

From our experimental results and different models used in theoretical calculations using either CND0/2 (23-25, 37>38) and PCIL0 methods (26,27), or the electric field effect by IND0 finite perturbation theory (28), the following models can be supposed ... 

The main lines of the Prigogine theory14-16-17 are presented in this section. A perturbation calculation is employed to study the IV-body problem. We are interested in the asymptotic solution of the Liouville equation in the limit of a large system. The resolvent method is used (the resolvent is the Laplace transform of the evolution operator of the N particles). We recall the equation of evolution for the distribution function of the velocities. It contains, first, a part which describes the destruction of the initial correlations this process is achieved after a finite time if the correlations have a finite range. The other part is a collision term which expresses the variation of the distribution function at time t in terms of the value of this function at time t, where t > t t—Tc. This expresses the fact that the system has a memory because of the finite duration of the collisions which renders the equations non-instantaneous. 

The experimental 3J(HH) spin-spin coupling constants for Ha, Hp, and Hp for the -isomer 22 were quite satisfactorily reproduced (A = 0.1 - 1 Hz) by calculations, using a finite perturbation method (FPT level (26), Perdew/IGLO-III at a MP2/6-31G(d)) geometry for the model structure (E)-1 -cyclopropyl-2-(trimethylsilyl)ethyl cation. The calculations confirm the /rexperimentally observed carbocation 22 (20, 27, 28). The (E)- 1-cyclopropy 1-2-... 

To confirm equation (4), we used the FPT (Finite Perturbation Theory) INDO (Intermediate Neglect of Differential Overlap) method (39) to calculate the Jqjj for various values of torsion angles. A comparison of the experimental and calculated values is plotted in Figure 5. 

Figure 2.8 is the energy schematic of the combined system. As the tip and the sample approach each other with a finite bias V, the potential 1/ in the barrier region becomes different from the potentials of the free tip and the free sample. To make perturbation calculations, we draw a separation surface between the tip and the sample, then define a pair of subsystems with potential surfaces Us and Un respectively. As we show later on, the exact position of the separation surface is not important. As shown in Fig. 2.8, we define the potentials of the individual systems to satisfy two conditions. First, the sum of the two potentials of the individual systems equals the potential of the combined system, that is. 

In most DFT coupling calculations the FC contribution is calculated using the finite perturbation theory, FPT, using either a single36 or a... 

The possibility of a finite value for aM is an intriguing idea worth studying. Indeed, it was discussed very early by Gell-Mann and Low in their classic and seminal paper QED at small distances [67], in which they showed that it is something to be seriously considered. However, they could not decide from their analysis whether enj is finite or infinite. The standard QED statement that it is infinite was established later on the basis of perturbative calculations. Nevertheless, and contrary to an extended belief, the alternative presented by Gell-Mann and Low has not been really settled. It is still open, in spite of the many attempts to clarify this question. 

Most numerical methods for calculating molecular hyperpolarizability use sum over states expressions in either a time-dependent (explicitly including field dependent dispersion terms) or time-independent perturbation theory framework [13,14]. Sum over states methods require an ability to determine the excited states of the system reliably. This can become computationally demanding, especially for high order hyperpolarizabilities [15]. An alternative strategy adds a finite electric field term to the hamiltonian and computes the hyperpolarizability from the derivatives of the field dependent molecular dipole moment. Finite-field calculations use the ground state wave function only and include the influence of the field in a self-consistent manner [16]. 

A DFT-based third order perturbation theory approach includes the FC term by FPT. Based on the perturbed nonrelativistic Kohn-Sham orbitals spin polarized by the FC operator, a sum over states treatment (SOS-DFPT) calculates the spin orbit corrections (35-37). This approach, in contrast to that of Nakatsuji et al., includes both electron correlation and local origins in the calculations of spin orbit effects on chemical shifts. In contrast to these approaches that employed the finite perturbation method the SO corrections to NMR properties can be calculated analytically from... 

The experimentally measured coupling constant J(HaHy)exp. = 5.5 Hz is reproduced by calculations for the endo-si y substituted model structure 44 using a finite perturbation level (FPT) 26) (Perdew/IGLO-III) approach J(HaHy)caic. = 5.9 Hz, whereas only 1.2 Hz is calculated for the HaHy-coupling constant in the exo-si y substituted model structure 45. Finally the bonding orbital of the bridging C-C-bond between Cq and Cy in these type of bicyclobutonium ions can be visualized by calculations of the natural bond orbitals (NBO s) (Figure 13). 

It is ensured that the NHIMs, if they exist, survive under arbitrary perturbation to maintain the property that the stretching and contraction rates under the linearized dynamics transverse to dominate those tangent to In practice, we could compute the only approximately with a finite-order perturbative calculation. Therefore, the robustness of the NHIM against perturbation (referred as to structurally stable [21,53]) is expected to provide us with one of the most appropriate descriptions of a phase-space bottleneck of reactions, if such an approximation of the Ji due to a finite order of the perturbative calculation can be regarded as a perturbation. One of the questions arising is, How can the NHIMs composed of a reacting system in solutions survive under the influence of solvent molecules (This is closely relevant to the subject of how the system and bath should be identified in many-body systems.)... 

Calculations in terms of the self-consistent finite perturbation theory (SCPT) and analysis of contributions of localized molecular orbitals in terms of the polarization propagator theory (CLOPPA), conducted by Krivdin and Kuznetsova, indicate additivity of coupling constants in saturated, sterically strained heterocycles. Their... 

Beer and Grinter used finite perturbation theory to calculate J(Si-H), J(Si-H), (161) and J(Si-C) (143) as well as the analogous phosphorus couplings. The best results are obtained for J(Si-Q (Table XXI) for which the correlation coefficient of the best fit of calculated to observed values is 0-985. The various calculated contributions to J(Si-C) are in Table XXI. Clearly, the orbital and spin-dipolar terms, which are small and also of opposite sign, have little effect on the calculated value of J. It is concluded that the Fermi contact term is probably sufficient for the calculation of J(Si-Q and that inclusion of silicon d orbitals is not required. 

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