Hydrogen atom polarizability

Table 7-1 lists some comparisons between experimental mean molecular polarizabilities and those estimated by Eq. (6). In this scheme, the estimation of mean molecular polarizability for acetic add needs five values, values for sp -C, for sp -C, for sp -O, for sp -O, and for a hydrogen atom. 

In this approximation the mean polarizability a is given in atomic units and h is the number of hydrogen atoms. They founda mean absolute error of 1.2% and... 

This result indicates that in strictly theoretical calculations, the f functions may almost as well be omitted unless they can be optimized for the London energy itself. For the purpose of semi-empirical calculations, however, the /A functions from the polarizability must be retained for the substitution in the London energy. The error for hydrogen atoms is only about 4 per cent, however, and there does not appear to be any reason that it would increase greatly in more complex systems. 

The Slater-Kirkwood equation (Eq. 39) was selected with N = 4 for carbon and N = 1 for hydrogen. The success of the equivalent calculation for the intermolecular interaction of CH4 molecules was mentioned in the previous section. Atoms, rather than bonds, were chosen as the basis for the calculation because the location of the atom centers is unambiguous and the approximation of isotropic polarizability is better for an atom than for a bond. Possible deviations from isotropic polarizability are discussed in Section V. Ketelaar19 gives for the atomic polarizabilities of hydrogen and carbon a = 0.42 and 0.93x 10-24 cm3, respectively. The resulting equation for the London energy is... 

In order to demonstrate the efficiency of the g f) function in the calculation of the polarizability. Rerat et al. (13) have carried out the calculation of the polarizability for the ground state of the hydrogen atom. This computation has been made with aff N)) and without ai, N)) the dipolar factor, versus the number of the spectral l n) states involved in the calculation. The convergence of such series aif N) and ai (N) leads to discrete values of 4.4018 and 3.6632 (i.e. the result of Tarmer and Thakkar) corresponding respectively to 97.8% and 81.4% of the exact value. This result illustrates the fact that a large part of the continuum contribution is simulated through the use of the dipolar factor. Moreover the convergence of the series aif N) is faster as we can see on table 1. 

An important addition to the model was the inclusion of virtual particles representative of lone pairs on hydrogen bond acceptors [60], Their inclusion was motivated by the inability of the atom-based electrostatic model to treat interactions with water as a function of orientation. By distributing the atomic charges on to lone pairs it was possible to reproduce QM interaction energies as a function of orientation. The addition of lone pairs may be considered analogous to the use of atomic dipoles on such atoms. In the model, the polarizability is still maintained on the parent atom. In addition, anisotropic atomic polarizability, as described in Eq. (9-28), is included on hydrogen bond acceptors [65], Its inclusion allows for reproduction of QM polarization response as a function of orientation around S, O and N atoms and it facilitates reproduction of QM interaction energies with ions as a function of orientation. 

In this respect, the solvatochromic approach developed by Kamlet, Taft and coworkers38 which defines four parameters n. a, ji and <5 (with the addition of others when the need arose), to evaluate the different solvent effects, was highly successful in describing the solvent effects on the rates of reactions, as well as in NMR chemical shifts, IR, UV and fluorescence spectra, sol vent-water partition coefficients etc.38. In addition to the polarity/polarizability of the solvent, measured by the solvatochromic parameter ir, the aptitude to donate a hydrogen atom to form a hydrogen bond, measured by a, or its tendency to provide a pair of electrons to such a bond, /, and the cavity effect (or Hildebrand solubility parameter), S, are integrated in a multi-parametric equation to rationalize the solvent effects. 

As shown in Fig. 7.1, at large distances, the system can be considered as a neutral hydrogen atom plus an isolated proton. The field of the proton polarizes the hydrogen atom to induce a dipole. The interaction between the proton and the induced dipole generates a van der Waals force. The van der Waals force can be treated as a classical phenomenon by introducing a phenomenological polarizability a ... 

The polarizability of the hydrogen atom can be calculated accurately using perturbation theory (Pauling and Wilson, 1935). The result is... 

The result in (6.28) is obtained neglecting the contributions connected with the polarizability of the constituent nucleons in the deuterium atom, and the polarizability contribution of the proton in the hydrogen atom. Meanwhile, as may be seen from (6.21), proton polarizability contributions are comparable to the accuracy of the polarizability contribution in (6.28), and cannot be ignored. The deuteron is a weakly bound system and it is natural to assume that the deuteron polarizability is a sum of the polarizability due to... 

The dimensions of the polarizability a are those of volume. The polarizability of a metallic Bphere is equal to the volume of the sphere, and we may anticipate that the polarizabilities of atoms and ions will be roughly equal to the atomic or molecular volumes. The polarizability of the normal hydrogen atom is found by an accurate quantum-mechanical calculation to be 4.5 ao that is, very nearly the volume of a sphere with radius equal to the Bohr-orbit radius a0 (4.19 a ). 

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