Potential energy polarizability

Here, auir) is the mean polarizability of a pair of species separated by distance r, while pn is their electric dipole moment and U the potential energy, p o is the permanent electric dipole moment for the two species at infinity. 

Polarizabilities of atoms, 75, 76, 125 Polarization of bonds, 207 Potential energy surfaces, see Potential surfaces... 

These results have been recently confirmed by more elaborate ab initio methods including electron correlation and polarizability functions. Structure [7] (Scheme 6c) is found to be more stable than [9] by only 1.5 kcal mol-1. Structure [8] does not exist as a minimum on the potential energy surface and spontaneously collapses to [7]. 

The QM/MM methodology [1-7] has seen increasing application [8-16] and has been recently reviewed [17-19], The classical solvent molecules may also be assigned classical polarizability tensors, although this enhancement appears to have been used to date only for simulations in which the solute is also represented classically [20-30], The treatment of the electronic problem, whether quantal, classical, or hybrid, eventually leads to a potential energy surface governing the nuclear coordinates. 

The (nonlocal) polarizabilities are important DFT reactivity descriptors. But, how are polarizabilities related to chemistry As stated above, an essential ingredient of the free energy surface is the potential energy surface and, in particular, its gradients. In a classical description of the nuclei, they determine the many possible atomic trajectories. Thanks to Feynman, one knows a very elegant and exact formulation of the force between the atoms namely [22,23]... 

Spectroscopic applications usually require us to go beyond single-point electronic energy calculations or structure optimizations. Scans of the potential energy hypersurface or at least Taylor expansions around stationary points are needed to extract nuclear dynamics information. If spectral intensity information is required, dipole moment or polarizability hypersurfaces [202] have to be developed as well. If multiple relevant minima exist on the potential energy hyper surface, efficient methods to explore them are needed [203, 204],... 

Although the potential energy functions can be made to reproduce thermodynamic solvation data quite well, they are not without problems. In some cases, the structure of the ion solvation shell, and in particular the coordination number, deviates from experimental data. The marked sensitivity of calculated thermodynamic data for ion pairs on the potential parameters is also a problem. Attempts to alleviate these problems by introducing polarizable ion-water potentials (which take into account the induced dipole on the water caused by the ion strong electric field) have been made, and this is still an active area of research. 

Molecular dynamic studies used in the interpretation of experiments, such as collision processes, require reliable potential energy surfaces (PES) of polyatomic molecules. Ab initio calculations are often not able to provide such PES, at least not for the whole range of nuclear configurations. On the other hand, these surfaces can be constructed to sufficiently good accuracy with semi-empirical models built from carefully chosen diatomic quantities. The electric dipole polarizability tensor is one of the crucial parameters for the construction of such potential energy curves (PEC) or surfaces [23-25]. The dependence of static dipole properties on the internuclear distance in diatomic molecules can be predicted from semi-empirical models [25,26]. However, the results of ab initio calculations for selected values of the internuclear distance are still needed in order to test and justify the reliability of the models. Actually, this work was initiated by F. Pirani, who pointed out the need for ab initio curves of the static dipole polarizability of diatomic molecules for a wide range of internuclear distances. 

Calculated from the CAS ° ° polarizability radial functions and the oooocAS potential energy curve. 

In Table 1 the predicted dipole and quadmpole polarizability tensor components ay and C,y for the vibrational states with quantum number v are given. They were calculated for all vibrational states supported by the potential energy function as expectation values of the polarizability radial functions a(R) and C(R) over the vibrational wavefunction (equation (14)). The latter were obtained from... 

Table 2. LiH dipole and quadrupole polarizability (in atomic units) for the vibrational ground state u = 0 calculated with different response theory methods. P(Pe) is the value at the minimum of the potential energy curve, Pq o is the value in the vibrational ground state and ZPVC = Pq o P(Pe) is the corresponding zero-point-vibrational correction...

Fig. 5. HF dependence of the dipole and quadrupole polarizability tensor (in atomic units) on the vibrational quantum number v. Calculated from the ioo > 9552 polarizability radial function and the potential energy curve.

Calculated from the polarizability radial functions and the potential energy... 

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