Perturbation potentials

Assuming the perturbing potential is pairwise additive, an argument virtnally identical to the calcnlation of = shows that... 

Consider first the polarization of the system in an electric field F along the z axis. The perturbing potential is... 

If we limit ourselves to spherically symmetric systems, the perturbing potential for the interaction of atoms A and B is... 

The eonclusion to be drawn from equation (6) is that the perturbation energy is equal to the value of the perturbing potential at the equilibrium separation plus terms which are proportional to the even derivatives of V(r) at the equilibrium separation, and also proportional to increasing powers of the mean square of the total deviation from this separation. It is via this mean square that the isotopic mass will affect the perturbation energy. 

The isotopic difference obviously depends exclusively on the even derivatives of the perturbing potential, which implies that a linear potential, corresponding to a constant force, gives rise to no isotope effect. This may be easily understood in the following way if two functions of a variable x, one parabolic (fi(x)) and the other linear (fz x)), i.e.,... 

To understand the fundamental differences between potentiometric and amperometric eleclroanalytical measurements, namely potentiometric measurements are those of the potential made at zero current (i.e. at equilibrium), while amperometric measurements are of the current in response to imposing a perturbing potential (dynamic, i.e. a non-equilibrium measurement). 

An alternative approach, which we will introduce here, is to apply a small perturbing potential across a cell or sample. An additional advantage of this approach over potentiometric analyses is that current is generated, since the potential is different from the equilibrium value - there is an overpotential q. An additional advantage over amperometric analyses is that the potential is only perturbing , and so any concentration changes within the cell or sample are minimized. 

Distortions need to have their symmetry contained in the direct product — ai + e + ti + t2 and we will consider only such displacements that leave the Cl-0 bonds invariant. This leaves only e-type and t2-type to be of concern. An e-deformation shortens or lengthens opposite edges of the tetrahedron and does little to split the ti degeneracy. The 2 modes will be classified as el-orbitals and the perturbing potential will have the form V — + rjV + t V y with the... 

The charge-density susceptibility is a linear response function it is nonlocal because a perturbing potential applied at any point r affects the charge density throughoutthe molecule. Quantum mechanically,x(r, r co) is specified by (2)... 

The predominant term in the perturbing potential V is of the form er, equal to the electric dipole moment operator. This is the origin of the selection rule that if ( 0, er i) = 0, the perturbed secular equation will not mix the states xpo and t/ i) so that the transition tpo ip i will not occur. 

Different pieces in Eq. (12) can be chosen to be the above perturbation potential, resulting in different LR functions, which are closely related to the... 

Pollutant Potential perturbation Potential atmospheric interactions... 

The second order perturbation theory term with two one-loop self-energy operators does not generate any logarithm squared contribution for the state with nonzero angular momentum since the respective nonrelativistic wave function vanishes at the origin. Only the two-loop vertex in Fig. 3.24 produces a logarithm squared term in this case. The respective perturbation potential determined by the second term in the low-momentum expansion of the two-loop Dirac form factor [111] has the form... 

The logarithmic contribution is induced only by the Uehling potential in Fig. 3.10, and may easily be calculated exactly in the same way as the logarithmic contribution induced by the radiative photon in (3.97). The only difference is that now the role of the perturbation potential is played by the kernel which corresponds to the polarization contribution to the Lamb shift of order a(Za) m... 

The respective perturbation potential is given by the form factor slope insertion in the external Coulomb potential (see (2.3))... 

The respective correction to the energy levels is given by the expectation value of this perturbation potential... 

The two-loop electron polarization contribution to the Lamb shift may be calculated exactly like the one-loop contribution, the only difference is that one has to use as a perturbation potential the two-loop correction to the Coulomb... 

The characteristic integration momenta in the matrix element of this perturbation potential between the Coulomb-Schrodinger wave functions are of the atomic scale mZa, and are small in comparison with the muon mass m. Hence, in the leading approximation the muon polarization may be approximated by the first term in its low-frequency expansion... 

Similarly to (7.1) it is easy to write a coordinate space representation for the perturbation potential corresponding to the diagram in Fig. 7.10... 

---