Fix nuclear geometry

Both current methods share the limitation of being conducted for fixed nuclear geometry—typically the ground-state equilibrium geometry. However,... 

This interpretation of fi shows that the polarization changes in the electron distribution (responses to the external potential displacements) can be determined from the external softness properties calculated for the fixed nuclear geometry (external potential). This very property is used in determining the mapping relations between the modes of the electron populations and the nuclear positions (see Sect. 2.3). 

Single point calculation with fixed nuclear geometry Rhs and different LS electron distributions with an integer SD occupations, resulting the energies EHS,LS,min and i Hs,Ls,Ts. Energies for the different distributions should be equal... 

In the second part of this section (3.1.2.), with one exception, I will limit my survey to other calculations which have used ab initio techniques to determine frequency-dependent hyperpolarizabilities. It is unfortunate that, again with one exception, none of these calculations takes account of nuclear vibrations, not even to the extent of zero-point vibrational averaging (i.e. a fixed nuclear geometry is assumed). Any close agreement with experiment, which doesn t happen often, must therefore be considered coincidental. To redress (somewhat) the balance of this section, I will also report on an excellent paper dealing with a set of non-ab-initio calculations. 

This expression looks just like the continuous-wave differential cross-section for a fixed nuclear geometry in a single electronic state (1), weighted by the probability distribution of nuclear geometries in the populated electronic states, summed up over the duration of the incoming X-ray pulse and, finally, weighted by the... 

It seems that intruders can be avoided almost entirely for states computed at fixed nuclear geometries by a judicious choice of the incomplete model space [27,28]. The situation is far less satisfactory, however, for generating potential surfaces. Here different intruder states plague the computations at different regions of the potential surface, and there is neither a unique nor a natural choice of an incomplete model space which avoids intruders at all geometries. It thus becomes necessary to switch over to different model spaces for describing the different regions of the surface [29]. 

To make our analysis of G more clear we report here a compact formulation of the energy of a cluster at fixed nuclear geometry... 

Fig. 8.14 Snapshots of the time evolution of an electronic wavepacket prepared close to Coln-1 for fixed nuclear geometry. The underlying electronic wavepacket is created as the normaUzed superposition of the CASSCF-wavefunctions of ground and first excited state. The depicted motion of the electron density takes place in 1.6 ps...

The Born-Oppenheimer approximation is probably the approximation that has been most successfully applied in Theoretical Chemistry. In general terms, we could say that it allows the decomposition of the Schrddinger equation in two parts one part that describes the electronic wavefunction for a fixed nuclear geometry, and the other part that describes the nuclear wavefunction, where the energy from the electronic wavefunction plays the role of a potential energy. This is, however, a very generalized description of this approximation, so let us go into a more detailed description of it. 

However, one should keep in mind that this relation only holds for a given fixed nuclear geometry, while the measured rotational g tensor is for a particular vibrational state. Direct application of it will therefore neglect the possibly large contributions from vibrational motion of the nuclei (see e.g. Lutnaes et al. (2009)), as discussed in Chapter 8. For an accurate determination of the magnetizability it is therefore necessary to correct explicitly for the vibrational corrections included in the measured rotational g tensor. 

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