Induced moment correlation

London-van der Waals forces generally are multipole (dipole-dipole or dipole-induced dipole) interactions produced by a correlation between fluctuating induced multipole (principal dipole) moments in two nearly uncharged polar molecules. Even though the time-averaged, induced multipole in each molecule is zero, the correlation between the two induced moments does not average to zero. As a result an attractive interaction between the two is produced at very small molecular distances. 

Fig. 3.51. Logarithmic plot of the normalized induced dipole moment correlation function, C(t), for hydrogen-argon mixtures at 165 K. Measurements at 90 amagat ( ) 450 amagat ( ) and 650 amagat (o). The broken lines at small times represents the portion of C(t) affected by the smoothing of the wings of the spectral profiles. Reproduced with permission by the National Research Council of Canada from [109].

Hyperpolarizabilities can be calculated in a number of different ways. The quantum chemical calculations may be based on a perturbation approach that directly evaluates sum-over-states (SOS) expressions such as Eq. (14), or on differentiation of the energy or induced moments for which (electric field) perturbed wavefunctions and/or electron densities are explicitly calculated. These techniques may be implemented at different levels of approximation ranging from semi-empirical to density functional methods that account for electron correlation through approximations to the exact exchange-correlation functionals to high-level ab initio calculations which systematically include electron correlation effects. 

From Eqs. (3.30) and (3.37) it is seen that at low field strengths both the contributions of the induced moment Af( ) and of the permanent dipole moment are linear in E. This correlates well with the continuum expression for M as discussed in the context of Eq. (3.2). 

There is an implicit sum in Eq. (34), as usual, over the repeated suffixes a, a b, b t, t u and u. A typical term in this sum can be represented by Fig. 1 e, and can be interpreted in terms of correlated fluctuations. A fluctuation in moment t at site a produces a change in the field component u at site b in the other molecule, and because molecule B is polarizable this induces a change in moment u at site b. Site b may in fact be the same as b in which case we have a local polarization, or it may be a different one, in which case the polarization is non-local. The induced moment u produces a field component t at site a in the first molecule, which interacts with the original fluctuation via the polarizability. Again a may be the same as a or different, corresponding to a local or non-local polarization of molecule A. 

The average cloud is spherically synnnetric with respect to the nucleus, but at any instant of time there may be a polarization of charge givmg rise to an instantaneous dipole moment. This instantaneous dipole induces a corresponding instantaneous dipole in the other atom and there is an interaction between the instantaneous dipoles. The dipole of either atom averages to zero over time, but the interaction energy does not because the instantaneous and induced dipoles are correlated and... 

For forbidden transitions in atoms and molecules this phenomenon may be experimentally observed in spectra induced by collisions. As is known, the selection rules on some transitions may be cancelled during collision. The perturbers are able to induce a dipole moment of transition having the opposite direction in successive collisions due to intercollisional correlation. Owing to this, the induced spectra do involve the gap (Fig. 1.7), the width of the latter being proportional to the gas density [46, 47], Theorists consider intercollisional correlation to be responsible for the above phenomenon [48, 49, 50]. 

London-van der Waals forces, which are multipole interactions produced by correlation between fluctuating induced multipole moments in two nearly uncharged polar molecules. These forces also include dispersion forces that arise from the correlation between the movement of electrons in one molecule and those of neighboring molecules. The van der Waals dispersion interaction between two molecules is generally very weak, but when many groups of atoms in a polymeric structure act simultaneously, the van der Waals components are additive. 

One of the more profound manifestations of quantum mechanics is that this curve does not accurately describe reality. Instead, because the motions of electrons are correlated (more properly, the electronic wave functions are correlated), the two atoms simultaneously develop electrical moments that are oriented so as to be mutually attractive. The force associated with tills interaction is referred to variously as dispersion , the London force, or the attractive van der Waals force. In the absence of a permanent charge, the strongest such interaction is a dipole-dipole interaction, usually referred to as an induced dipole-induced dipole interaction, since the moments in question are not permanent. Such an interaction has an inverse sixtli power dependence on the distance between the two atoms. Thus, the potential energy becomes increasingly negative as the two noble gas atoms approach one another from infinity. 

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