Perturbative calculation of

Dalgarno A and Lewis J T 1956 The representation of long-range forces by series expansions. II. The complete perturbation calculation of long-range forces Proc. Phys. Soc. A 69 59... 

Wesolowski T A and Warshel A 1994 Ab initio free energy perturbation calculations of solvation free energy using the frozen density functional approach J. Phys. Chem. 98 5183... 

FIGURE 3.2. A free-energy perturbation calculation of the free energy associated with an adiabatic charging of an Na+ ion in water (Ref. 10). The parameter A transforms the solute from it uncharged (A = 0) to its charged form (A = 1). 

According to Bartell (1961a), the relative motion of the interacting non-bonded atoms is described by means of a harmonic oscillator when the two atoms are bonded to the same atom, and by means of two superimposed harmonic oscillators when the atoms are linked to each other via more than one intervening atom. It is the second case which is of interest in connection with the biphenyl inversion transition state. The non-bonded interaction will of course introduce anharmonicity, but since a first-order perturbation calculation of the energy only implies an... 

Wesolowski, T. A. and A. Warshel. 1994. Ab initio Free Energy Perturbation Calculations of Solvation Free Energy Using Frozen Density Functional Approach. J. Phys. Chem. 98, 5183. 

Florian, J. Goodman, M.F. Warshel, A., Free energy perturbation calculations of DNA destabilization by base substitutions the effect of neutral guanine-thymine, adenine-cytosine and adenine-difluorotoluene mismatches, J. Phys. Chem. B 2000,104, 10092-10099. 

For vanadocene second-order perturbation calculations of the spin-orbit contribution to the zero field splitting, DLS, showed the II ( 2) level to lie below A ( - ), but the estimated value, 1.4 to 1.5 cm-1, constitutes only about half of the observed result. How-... 

Finally, in many of the perturbation calculations of the effect of substituents and other structural changes, an important tacit assumption is made and it is far from obvious that it is always fulfilled. As already discussed, the physical argument on which the calculation is based is that the value of the initial slope, or the height of a small barrier along the way, determine the rate at which the photochemical reaction occurs. However, the experimental value with which comparison is made usually is not the reaction rate but the quantum yield, which of course also depends on rates of other competing processes and these may be affected by substitution as well. For instance, the rate at which fluorescence occurs is related to the absorption intensity of the first transition, the rate of intersystem crossing may be affected by introduction of heavy atoms... 

S. Yoneda and H. Umeyama, Free energy perturbation calculations of multiple mutation... 

Freed et al. [42,43], among others [44,45] have performed RG perturbation calculations of conformational properties of star chains. The results are mainly valid for low functionality stars. A general conclusion of these calculations is that the EV dependence of the mean size can be expressed as the contribution of two terms. One of them contains much of the chain length dependence but does not depend on the polymer architecture. The other term changes with different architectures but varies weakly with EV. Kosmas et al. [5] have also performed similar perturbation calculations for combs with branching points of different functionalities (that they denoted as brushes). Ohno and Binder [46] also employed RG calculations to evaluate the form of the bead density and center-to-end distance distribution of stars in the bulk and adsorbed in a surface. These calculations are consistent with their scaling theory [27]. 

Daggett, V. D., Kollman, P. A. and Kuntz, I. D. (1989) Free energy perturbation calculations of charge interactions with the helix dipole, Chemica Scripta. 29A, 205-215. 

For a perturbative calculation of the spectrum of (10.4.1) we assume that electron 1 is in the single-particle SSE state m) and electron 2 is in a highly excited state, much more weakly bound than electron 1. In this situation electron 2 spends most of its time in regions with X2 xi. Therefore, we assume that the eigenfunctions of (10.4.1) are approximately of the form... 

The 5s manifold shows great complexity. For the lowest state S23.4(5s) = 0.37. This value is considerably lower than many structure calculations predict, but the perturbation calculation of Kheifets and Amusia (1992) obtains 0.384. The orbital energy ess (11.18) is 27.6+0.3eV, which is to be compared to the Dirac—Fock value 27.49 eV. The Hartree—Fock value is 25.70 eV. The criterion for the strength of the perturbation, given by the ratio of the standard deviation to the mean of the 5s manifold is 0.18. The ratios S29.i(5s) S23.4(5s) and S23.4(5s) Z/S/(5s) are compared at different momenta in fig. 11.10. The condition for the validity of the weak-coupling binary-encounter approximation is completely satisfied within experimental error. 

Although an analytical expression of k has not been obtained yet, a systematic method for its calculation has been given [32, 41]. The k is proportional to in its perturbational calculation of the lowest-order in k X), but it becomes proportional to with 0 < a < 1, when the perturbational calculation diverges [42]. To be more exact, k( has a dependence as... 

C. Amovilli and R. McWeeny, Perturbation calculation of molecular interaction energies an example, HF...HF, Cbem. Phys. Lett., 128 (1986) 11-17. 

S. Diner, J.P. Malrieu. and P. Claverie, Localized bond orbitals and the correlation problem. 1. The perturbation calculation of the ground state energy. Theor. Chim. Acta, 163 (1987) 269. 

S. Hirata, H. Torii M. Tasumi (1995). J. Chem. Phys., 103, 8963-8979. Vibrational analyses of tra 5-polyacetylene based on ab initio second-order Moller-Plesset perturbation calculations of tranj-oligoenes. 

This expression is much more convenient for practical evaluations since only one state has to be evolved in time, namely the state Xn,+ referring to the final electron state under consideration. By these means the equation above may describe pair production with electron capture. It also yields ionization and electron-transfer probabilities of a hydrogen-like target. These probabilities are obtained by projecting Xn.+ onto the target and projectile states. Non-perturbative calculations of such processes require a fully numerical time-evolution of the Dirac equation. 

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