Perturbation calculations, high-order

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]. 

To sum up, we have developed a general non-perturbative method that allows one to calculate the rate of relaxation processes in conditions when perturbation theory is not applicable. Theories describing non-radiative electronic transitions and multiphonon relaxation of a local mode, caused by a high-order anharmonic interaction have been developed on the basis of this method. In the weak coupling limit the obtained results agree with the predictions of the standard perturbation theory. 

Up to moderately high energy ( 179%) of the activation barrier for reactant product in the Are isomerization reaction, the fates of most trajectories can be predicted more accurately by Eq. (11) as the order of perturbation calculation increases, except just in the vicinity of the (approximate) stable invariant manifolds (e.g., see Eig. 5), and that the transmission coefficient K observed in the configurational space can also be reproduced by the dynamical propensity rule without any elaborate trajectory calculation (see Eig. 6). Our findings indicate that almost all observed deviations from unity of the conventional transmission coefficient k may be due to the choice of the reaction coordinate whenever the k arises from the recrossings, and most transitions in chemical... 

We have calculated the second- and fourth-order dipole susceptibilities of an excited helium atom. Numerical results have been obtained for the ls2p Pq-and ls2p f2-states of helium. For the accurate calculations of these quantities we have used the model potential method. The interaction of the helium atoms with the external electric held F is treated as a perturbation to the second- and to the fourth orders. The simple analytical expressions have been derived which can be used to estimate of the second- and higher-order matrix elements. The present set of numerical data, which is based on the Green function method, has smaller estimated uncertainties in ones than previous works. This method is developed to high-order of the perturbation theory and it is shown specihcally that the continuum contribution is surprisingly large and corresponds about 23% for the scalar part of polarizability. 

This statement refers to general-purpose implementations. For benchmark studies, codes developed for FCI calculations have been used to calculate terms in the perturbation expansion to very high orders [122-124],... 

The example of neon, where relativistic contributions account for as much as a0.5% of 711, shows that relativistic effects can turn out to be larger for high-order NLO properties and need to be included if aiming at high accuracy. Some efforts to implement linear and nonlinear response functions for two- and four-component methods and to account for relativity in response calculations using relativistic direct perturbation theory or the Douglas-Kroll-Hess Hamiltonian have started recently [131, 205, 206]. But presently, only few numerical investigations are available and it is unclear when it will become important to include relativistic effects for the frequency dispersion. 

Other interesting related phenomena, worthy of a quantum chemist s attention, are the study of the hyperpolarizabilities of molecules trapped in a cage, e.g. a zeolite [98] or on a surface [99-102], and the whole relevance of these properties to surface property calculations, where, at the local level, there are enormous electric fields. There is also the area of very high-order harmonic generation, such as can occur in very intense laser beams and where conventional perturbation theory breaks down [103]. Finally, there is magnetic non-linear optics and here the surface, computationally, has, so to speak, only been scratched. 

Since the symmetry-adapted perturbation theory provides the basis for the understanding of weak intermolecular interactions, it is useful to discuss the convergence properties of the sapt expansion. High-order calculations performed for model one-electron (Hj) (30), two-electron (H2) (14, 15), and four-electron (He and He-Hz) (31) systems show that the sapt series converges rapidly. In fact, already the second-order calculation reproduces the exact variational interaction energies with errors smaller than 4%. Several recent applications strongly indicate that this optimistic result holds for larger systems as well. 

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