The electric field as a perturbation

1 THE ELECTRIC FIELD AS A PERTURBATION The inhomogeneous field at a slightly shifted point 

Assume the electric field vector (ro) is measured at a point indicated by the vector i-Q. What will we measure at a point shifted by a small vector r = (x, y, z) with respect to tq The components of the electric field intensity represent smooth functions in space and this is why we may compute the electric field from the Taylor expansion (for each of the components x, y, z separately, all the derivatives are computed at point ro)  

These two electric field intensities (at points tq and tq + r) have been calculated in order to consider the energy gain associated with the shift r of the electric point charge Q. Similar to the ID case just considered, we have the energy gain A = -Q r. There is only one problem which of the two electric field intensities is 

The components of such moments in general are not independent. The three components of the dipole moment are indeed independent, but among the quadru-pole components we have the obvious relations 0 / = from their definition, which reduces the number of independent components from 9 to 6. This however is not all. From the Maxwell equations (see Appendix G, p. 962), we obtain the Laplace equation, AF = 0 (A means the Laplacian), valid for points without electric charges. Since S = -W and therefore -V = AF we obtain 

We have therefore only five independent moments that are quadratic in coordinates. For the same reasons we have only seven (among 27) independent moments with the third power of coordinates. Indeed, ten original components 0,q qi q with (q, q, q ) = xxx, yxx, yyx, yyy, zxx, zxy, zzx, zyy, zzy, zzz correspond to all permutationally non-equivalent moments. We have, however, three relations these components have to satisfy. They correspond to the three equations, each obtained from the differentiation of eq. (12.5) over x,y, z, respectively. This results in only seven independent components q,q, q - 

---

The Electric Field as a Perturbation The Homogeneous Electric Field... 

Stark effect The splitting of lines in the spectra of atoms due to the presence of a strong electric field. It is named after the German physicist Johannes Stark (1874-1957), who discovered it in 1913. Like the normal Zeeman effect, the Stark effect can be understood in terms of the classical electron theory of Lorentz. The Stark effect for hydrogen atoms was also described by the Bohr theory of the atom. In terms of quantum mechanics, the Stark effect is described by regarding the electric field as a perturbation on the quantum states and energy levels of an atom in the absence of an electric field. This application of perturbation theory was its first use in quantum mechanics. 

It is worthwhile recalling at this stage the basic assumptions made so far. First of all, the perturbation must be sufficiently weak in order to allow first-order treatment. Secondly, the pulse must be so long that the molecule can recognize the electric field as a periodic perturbation. On the other hand the transition probability must remain small compared to unity because otherwise the first-order approximation breaks down. A more detailed and illuminating discussion of the applicability of (2.22) is given by Cohen-Tannoudji, Diu, and Laloe (1977 ch.XIII). 

Buckingham and Pople refer to the effect of the electric field as a paramagnetic term, and it has the dependence of the second term in equation (5), Although equation (5) has the virtue of attempting to describe the true electronic environment of the proton, it has the disadvantages of intractability. The electric field perturbation model is mathematically simple but an extreme approximation. Since these two treatments lead to the same functional dependence on p, perhaps the electric field model provides a useful approximation to the more complete description of equation (5), Whether this proves to be true or whether the characteristic arbitrariness of the electrostatic model will deprive the model of more than qualitative predictive value is not yet clear. In any event, the two treatments do concur in shifting attention from the p" term to the p term with its opposite sign. 

A third approach, the one most important for the current discussions, is to treat the electric-field as a source of perturbation of the total molecular energy using real and virtual excited state transitions. This approach uses electronic wave functions either for all of the electrons of the molecule (ab initio calculations) or for only the valence electrons (so-called semiempirical theories). Semiempirical Hamiltonians may ignore electron interactions completely (Huckel theory). They may assume one jt-orbital per carbon and assume no overlap between adjacent electron orbitals (Pariser-Parr-Pople or PPP). Or, they may include both the a and... 

Since the electric field is a polar vector, it acts to break the inversion synnnetry and gives rise to dipole-allowed sources of nonlinear polarization in the bulk of a centrosymmetric medium. Assuming that tire DC field, is sufficiently weak to be treated in a leading-order perturbation expansion, the response may be written as... 

Since we normally deal with electrostatic field perturbations that are small, we can profitably expand the electric moments as a Taylor series in the external field ... 

If the perturbation is a homogeneous electric field F, the perturbation operator P i (eq. (10.17)) is the position vector r and P2 is zero. As.suming that the basis functions are independent of the electric field (as is normally the case), the first-order HF property, the dipole moment, from the derivative formula (10.21) is given as (since an HF wave function obeys the Hellmann-Feynman theorem)... 

In addition to these external electric or magnetic field as a perturbation parameter, solvents can be another option. Solvents having different dielectric constants would mimic different field strengths. In the recent past, several solvent models have been used to understand the reactivity of chemical species [55,56]. The well-acclaimed review article on solvent effects can be exploited in this regard [57]. Different solvent models such as conductor-like screening model (COSMO), polarizable continuum model (PCM), effective fragment potential (EFP) model with mostly water as a solvent have been used in the above studies. 

Nuclear size corrections of order (Za) may be obtained in a quite straightforward way in the framework of the quantum mechanical third order perturbation theory. In this approach one considers the difference between the electric field generated by the nonlocal charge density described by the nuclear form factor and the field of the pointlike charge as a perturbation operator [16, 17]. 

Considering the crystal electric field as a first-order perturbation, the mixing of states with higher energy and opposite parity, nla"[S"l"]J"M") may be represented by iff") with A) and Z ) defined as... 

Noise spectrography is an efficient technique allowing observation of electrical phenomena in systems which are liable to be perturbed by the application of an external electric field. As a consequence, noise measurements are convenient for the study of electrochemical or bioelec-trochemical systems such as electrolytes, biopolyelectrolytes, electrolyte-membrane interfaces, etc. 

A well-known nonlinear process taking place in the liquid state of anisotropic molecules is the optical-field induced birefringence (optical Kerr effect ). This nonlinearity results from the reorientation of the molecules in the electric field of a light beam. In the isotropic phase the optical field perturbs the orientational distribution of the molecules. In the perturbed state more molecules are aligned parallel to the electric field than perpendicularly to it and as a consequence the medium becomes birefringent. On the other hand in liquid crystals the orientational distribution of the molecules is inherently anisotropic. The optical field, just as a d.c. electric or magnetic field, induces a collective rotation of the molecules. This process can be described as a reorientation of the director. 

---