Stark Polarizability

To translate between the actual measurement of PNC, which involves the ratio of PNC to the induced transition between 6s and 7s in the presence of an electric field, we used f3 = 27.00(20)aQ, determined entirely theoretically. In the second reference of [5], a slightly different but significantly more accurate value of 0 was quoted, 

Since the publication of this result, however, a measurement at Notre Dame has indicated that there may be a problem with this number [56]. The ideal resolution of this issue would be a recalculation of 0 along with that of PNC with more powerful theoretical methods that reach the tenth of percent level without the need for semiempirical corrections. 

In the theoretical prediction for Qw, we have used radiative corrections calculated by Marciano and Rosner [37]. These radiative corrections are one-loop, which normally would be expected to dominate. However, recently Bednkikov et. al. [59] considered vacuum polarization corrections to the photon line in the ladder diagram in which there is one photon and one Z exchange. Note that when there is no vacuum polarization this correction requires care, since part of the photon is associated with binding however, with vacuum polarization it is simply one of a large set of two-loop corrections. A surprisingly large value of 0.4 percent was found. 

Another part of the radiative correction, in which a radiative photon corrects the coupling of the Z with an electron, is simply 

However, it is known from studies of radiative corrections to the hyperfine splitting of hydrogen that similar graphs, which in lowest order contribute 

---

We will conclude this section by illustrating how the tensor Stark polarizability constant can be measured using level-crossing spectroscopy. As illustrated in Fig.9,18, for the case of the potassium 5d 03/2 state the unknown Stark effect is measured in terms of the well-known Zeeman effect... 

The perturbation theory is the convenient starting point for the determination of the polarizability from the Schrodinger equation, restricted to its electronic part and the electric dipole interaction regime. The Stark Hamiltonian —p. describes the dipolar interaction between the electric field and the molecule represented by its... 

The variational theorem which has been initially proved in 1907 (24), before the birthday of the Quantum Mechanics, has given rise to a method widely employed in Qnantnm calculations. The finite-field method, developed by Cohen andRoothan (25), is coimected to this method. The Stark Hamiltonian —fi.S explicitly appears in the Fock monoelectronic operator. The polarizability is derived from the second derivative of the energy with respect to the electric field. The finite-field method has been developed at the SCF and Cl levels but the difficulty of such a method is the well known loss in the numerical precision in the limit of small or strong fields. The latter case poses several interconnected problems in the calculation of polarizability at a given order, n ... 

Classical anharmonic spring models with or without damping [9], and the corresponding quantum oscillator models seem well removed from the molecular problems of interest here. The quantum systems are frequently described in terms of coulombic or muffin tin potentials that are intrinsically anharmonic. We will demonstrate their correspondence after first discussing the quantum approach to the nonlinear polarizability problem. Since we are calculating the polarization of electrons in molecules in the presence of an external electric field, we will determine the polarized molecular wave functions expanded in the basis set of unperturbed molecular orbitals and, from them, the nonlinear polarizability. At the heart of this strategy is the assumption that perturbation theory is appropriate for treating these small effects (see below). This is appropriate if the polarized states differ in minor ways from the unpolarized states. The electric dipole operator defines the interaction between the electric field and the molecule. Because the polarization operator (eq lc) is proportional to the dipole operator, there is a direct link between perturbation theory corrections (stark effects) and electronic polarizability [6,11,12]. 

Polarization, polarizability, perturbation theory, and stark effects... 

The 2S state has a large electron polarizability due to the proximity of the 2P state. Motion through the magnetic field produces an electric field E , = (v/c)B. This field can lead to the quenching of the 2S state, and a Stark shift The quenching rate is rs s V 2/T2p, where B2= <2S ezE 2P> 2, and is the radiative decay rate of the 2P state. At a temperature of lmK, r,= 10-3s-1. The Stark shift is smaller by a factor of approximately ten, and is also negligible. 

In many practical cases, the factors f i are very close to unity and can be omitted. The parameters So, and mo,- are then equal to their gas-phase values oto, and mo,. Equation [100] then gives the polarizability change in terms of spectroscopic moments and gas-phase solute dipoles. Experimental measurement and theoretical calculation of Aao = aoi - ocoi is still challenging. Perhaps the most accurate way to measure Akq presently available is that by Stark spectroscopy,which also gives Awq. Equation [100] can therefore be used as an independent source of Aao, provided all other parameters are available, or as a consistency test for the band shape analysis. 

Middendorf, T. R., Mazzola, L. T., Lao, K., Steffen, M. A., and Boxer, S. G., 1993, Stark effect (electroabsorption) spectroscopy of photosynthetic reaction centers at 1.5 K Evidence that the special pair has a large excited-state polarizability. Biochim. Biophys. Acta, 1143 223fi234. 

It must be stressed that the polarizability gradient da/dQk also appears in the equation for Raman intensities [175], as indicated also by Lambert [176]. Thus, in view of Eq. (25), we can extend the consequences of the static electric field to vibrations which are forbidden by the surface selection rule the high electric field in the double layer can induce a dipole moment component in the direction of the field on permanent dipoles which are parallel to the surface. Thus the effect of orientation due to the electric field is just a manifestation of the Stark effect. 

Infrared investigations of the Stark effect can afford data on the molecular properties of adsorbates (polarizability, dipole moment gradients, and polarizability gradients). However such studies are at their very beginning and require to know the actual value of the electric field at the interface. Data on the potentials of zero charge for solid electrode materials would be welcome. 

Excited state properties of molecules are often important parameters in different models of interacting systems and chemical reactions. For example, excited state polarizabilities are key quantities in the description of electrochromic and solva-tochromic shifts [99-103]. In gas phase there has been a series of experiments were excited state polarizabilities have been determined from Laser Stark spectroscopy by Hese and coworkers [104-106]. However, in the experiments most often not all the tensor components can be determined uniquely without extra information from either theory or other experiments. 

The polarizability of some neutral atoms (H, Li, K, Cs) has been recently determined by a method similar to the Stern-Gerlach experiment ( 7, p. 166), namely, by measuring the deflexion of a beam of atoms in an inhomogeneous electric field (Stark, 1936). The results do not agree very well with theoretical computations from atomic models. 

The value of the polarizability a of an atom or molecule can be calculated by evaluating the second-order Stark effect energy — %aF2 by the methods of perturbation theory or by other approximate methods. A discussion of the hydrogen atom has been given in Sections 27a and 27e (and Problem 26-1). The helium atom has been treated by various investigators by the variation method, and an extensive approximate treatment of many-electron atoms and ions based on the use of screening constants (Sec. 33a) has also been given.3 We shall discuss the variational treatments of the helium atom in detail. 

---