Perturbation theory third-order

The G3SX method based on the third-order perturbation theory, G3SX(MP3), is especially noteworthy in that it has a mean absolute deviation of 1.04 kcal/mol for the 376 energies in the G3/99 test set and 0.90 kcal/mol for the 222 enthalpies of formation. In this respect, it is as accurate as G3 theory and much less expensive. All of the G3SX methods have the advantage of being suitable for studies of potential energy surfaces. 

Nuclear size corrections of order (Za) may be obtained in a quite straightforward way in the framework of the quantum mechanical third order perturbation theory. In this approach one considers the difference between the electric field generated by the nonlocal charge density described by the nuclear form factor and the field of the pointlike charge as a perturbation operator [16, 17]. 

A tedious third order perturbation theory calculation [16, 17] produces some additional state-independent terms with the net result being a few percent different from the naive result above. The additional state-independent contribution beyond the naive result above has the form [20]... 

Using a syllogistic approach analogous to the earlier construction of the Pshg perturbation summation (Equation 4), we can "derive" the general third order perturbation theory expression(l),... 

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... 

In a similar way we can show that the probability amplitude for three-photoq absorption, obtained from the third-order perturbation theory term, is given as > ... 

An extensive account of the theory of the Spin Hamiltonian parameters for low spin Co(II) complexes has been given hy McGarvex (50). This approach is based on third order perturbation theory, involving all excited states giving a spin-orbit contribution to the I az + b(x - y ), Aj), or the yz, A2) ground state respectively. The formulae obtained for the g and A tensors in case of non axial symmetry depend upon a large number (12) of adjustable parameters, and a judicious choice of assumptions is required to make use of these equations. A simpler approach is therefore indicated. 

Curtiss, L.A., Redfern, P.C., Raghavachari, K., and Pople, A.J., Gaussian-3 theory a variation based on third order perturbation theory and an assessment for the contribution of core-related correlation, Chem. Phys. Lett., 313, 600-607 (1999). 

The nonlinear susceptibility is evaluated using third-order perturbation theory, and resonant enhancement is readily demonstrated to occur. Four-wave mixing is a useful experimental technique to extend the energy range available to tunable dye lasers [468]. It is also of interest that processes involving excitation by three photons allow transitions between even and odd parity states to be excited, as do single-photon transitions. 

The terms on the first line of Eq. (81) describe single and double excitations of the closed core, while those on the second line describe single and double excitations of the atom where the valence orbital is also excited. Substituting Eq. (81) into the Schrodinger equation one obtains a set of coupled equations for the expansion coefficients that can be found in Ref. [44]. The first and second iterations of the equations for the expansion coefficients leads to results that are identical to first- and second-order perturbation theory. In third-order perturbation theory, terms associated with triple excitations contribute to the energy. These terms have no counterpart in the iterative solution to the equations under consideration. 

Using the third order perturbation theory developed by McGarvey and Hitchman ... 

SimFonia Spectra of radicals in fluid solution are calculated using up to third order perturbation theory for isotropic hyperfine couplings. Parameters for the simulation are given in an easy-to-use graphical interface. Instrumental effects such as modulation amplitude and receiver time constant can be included to account for line shape distortions or line broadening. Spectra from SimFonia can be transferred to WinEPR software for editing and output. The software is a commercial product but one version is available free of charge at http //www.bruker-biospin.de/EiPR/software/emx.html. 

The dispersion interaction in the third-order perturbation theory contributes to the three-body non-additivity and is called the Axilrod-Teller energy. The term represents a correlation effect. Note that the effect is negative for three bodies in a linear configuration. 

The detailed calculation, using third-order perturbation theory [1257,1263], demonstrates that the power absorbed in the third field zone leads to the signal... 

More subtle are the spin-orbit-induced heavy-atom chemical shifts at the atom nearest to the heavy one, or at more remote nuclei. If one uses the same Hamiltonians as Ramsey, one must go to third-order perturbation theory, with one Zeeman, one hyperfine, and one spin-orbit matrix element [22]. For a recent discussion on the nature of this shift, see Ref. [23]. It was also noted that an analogous effect, a heavy-atom shift on the heavy atom can occur, for instance on the Pb(II) nucleus in PbR compounds. The early semiempirical calculations suggested that the Zeeman-SO-Fermi contact cross term, zero in Ramsey s theory, could then become the dominant contribution to the Pb chemical shift [24]. 

As will be outlined in more detail below, the much higher complexity of the d3mamics at conical intersections calls for a new strategy for the calculation of absorption and emission signals, which differs from the established formalism of nonlinear optics, based on higher-order (t3 ically third-order) perturbation theory in the laser-matter interaction. Independently of... 

The trouble is that, without saying so, we have assumed that only molecule A feels the effects of other molecules. But we know better. Each molecule polarizes its neighbor, if only a little bit, and that affects how the neighbor molecule will then interact with the next one. The Axilrod-Teller correction estimates the effect of this indirect interaction by applying third-order perturbation theory to the dispersion energy of three spherical molecules. In Section 10.1 we see how second-order perturbation theory predicts the attraction between two polarizable particles. It takes third-order terms to treat the effect of a third particle, and the derivation is lengthy, so let s just jump to the result this time ... 

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