Coupling operator approximation

By introducing the simplest semi-classical approximation to the propagators, in which the nuclear motion kinetic energy is assumed to commute with the anion and neutral potential energy functions and with the non BO coupling operators, one obtains... 

Indeed it is easy to see that, in general, the symmetry of the model will not be recovered by the variational solution since, if any one of the R departs from the symmetry of H, then the coupling operators Vrs will destroy the symmetry of the other departures from symmetry will quickly propogate throughout the model solution of the form (9) will have rather complicated behaviour in the variational process, for example each single-configuration approximation should show characteristic saddle point behaviour when variations 5 R are admitted. The minimum in the variational expression when the S R are constrained to have the correct symmetry should also be a local maximum with respect to symmetry-breaking variations S R. 

Most common among the approximate spin-orbit Hamiltonians are those derived from relativistic effective core potentials (RECPs).35-38 Spin-orbit coupling operators for pseudo-potentials were developed in the 1970s.39 40 In the meantime, different schools have devised different procedures for tailoring such operators. All these procedures to parameterize the spin-orbit interaction for pseudo-potentials have one thing in common The predominant action of the spin-orbit operator has to be transferred from... 

Second, it is a good approximation for an ion like HeAr+ to assume that the spin-orbit coupling operator is the same as that for the free Ar+ ion, L S, where f is the atomic spin orbit coupling constant. If the basis functions are confined to those arising from the 2P3/2 and 2Pi/2 states, the spin-orbit operator is also diagonal in a case (e) basis... 

The method works as follows. The mass velocity, Darwin and spin-orbit coupling operators are applied as a perturbation on the non-relativistic molecular wave-functions. The redistribution of charge is then used to compute revised Coulomb and exchange potentials. The corrections to the non-relativistic potentials are then included as part of the relativistic perturbation. This correction is split into a core correction, and a valence electron correction. The former is taken from atomic calculations, and a frozen core approximation is applied, while the latter is determined self-consistently. In this way the valence electrons are subject to the direct influence of the relativistic Hamiltonian and the indirect effects arising from the potential correction terms, which of course mainly arise from the core contraction. 

The electron-phonon coupling constant as a function of the doping level in silicon is presented in Fig. 2. The coupling is approximately directly proportional to the electron carrier concentration in the heavily doped silicon, but the electrical resistance of the silicon only slightly depends on the carrier concentration i n this range. This can be used for optimization of thermal characteristics of different microdevices operating at low tenqieratures. 

Here, Ho is the molecular Hamiltonian in the BO approximation, and V is the nonadiabatic coupling operator. U(t) = —/x e(r)cos(a)/t) is the pulse excitation operator. Here, e t) is the amplitude of the laser pulse with photon polarization vector e, and coi is laser central frequency. In Eq. 6.18, i] denotes the parameter depending on photon polarization direction of the linearly polarized laser pulse = 1 for the polarization vector e+, while r] = -l for e. ... 

In the centrifugal sudden approximation (CSA), which is also known as the coupled states approximation, the operator j j is replaced by a value such as [j +j - 2 J j ] such that there is no coupling between different" 2 states 07 4,25]. The S matrix elements are then of the form... 

Totally symmetric modes are not subject to symmetry restrictions. Their potentials may contain odd and even terms in Q so that the harmonic-oscillator approximation imposes unwarranted symmetry restrictions. Similarly, the corresponding vibronic coupling operator may contain both odd and even terms so that the distinction between pseudo-Jahn-Teller and pseudo-Renner-Teller coupling disappears. Since the potential energy minimum of a totally symmetric mode is different in different electronic states, the pseudo-Jahn Teller/Renner-Teller limit is quite different from the limiting cases discussed in Section I V,B,C. Finally, the transition moments... 

To express the vibronic coupling strength quantitatively, we require an analytical expression for the coupling operator. It is standard practice to assume this operator to be a linear function of the vibrational coordinate. In other words, the analytical form of this operator is approximated by the first nonvanishing term in a Taylor series expansion. It may appear that this approximation is consistent with the harmonic oscillator approximation, which also amounts to a Taylor series expansion truncated after the first nonvanishing Q-dependent term. However, this analogy does not hold, as follows directly from the form of the (two-state) vibronic Hamiltonian matrix, namely,... 

From Eq. (4.171) we understand that Eq. (6.29) already features a separation into radial r and angular, cp) variables. All angular variables are contained in the operator product a 1), which is essentially the spin-orbit coupling operator known from the Pauli approximation. The remaining operators can all be expressed by the radial variable r alone. 

We have already noted in chapter 13 that the Pauli approximation produces a spin-orbit coupling operator that maybe employed in essentially one-component, i.e., nonrelativistic or scalar-relativistic, methods via perturbation theory. Of course, this is an approximation compared with fully fledged four-component methods, but it can be a very efficient one that requires less computational effort without significant loss of accuracy. 

F. Neese. Efficient and Accurate Approximations to the Molecular Spin-Orbit Coupling Operator and their use in Molecular g-Tensor Calculations. /. Chem. Phys., 122 (2005) 034107. 

Here a) represents a wave packet belonging to channel a and Vf measures the coupling between the resonance i) and the decay channel a. Note that the states a) span a finite m-dimensional subspace of the collision space. Expression (20) is approximate since the wave packet a) should depend on the index i. It means that we are considering only m states a) instead of nx m doorway states [19]. However, this limitation is not severe since it is always possible to extend the number of quasi-bound states in the model space. Using (20) the coupling operator (19) can be transformed into... 

Neglect of these particular operators removes the coupling in basis functions with different Q. quantum numbers and drastically reduces the size of die basis set[7,25]. This is the centrifugal sudden approximation (CSA) and is often known as the coupled-states approximation, although that is a term we prefer not to use as it is easily confused with "close-coupling". The basis set expansion for a CSA calculation of vibrational-rotational excitation cross sections would thus be... 

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