Stark tensor

In molecular quantum mechanics, we often find ourselves manipulating expressions so that one of a pair of interacting operators is expressed in laboratory-fixed coordinates while the other is expressed in molecule-fixed. A typical example is the Stark effect, where the molecular electric dipole moment is naturally described in the molecular framework, but the direction of an applied electric field is specified in space-fixed coordinates. We have seen already that if the molecule-fixed axes are obtained by rotation of the space-fixed axes through the Euler angles (, 6, /) = >, the spherical tensor operator in the laboratory-fixed system Tkp(A) can be expressed in terms of the molecule-fixed components by the standard transformation... 

It is well-known that the electron repulsion perturbation gives rise to LS terms or multiplets (also known as Russell-Saunders terms) which in turn are split into LSJ spin-orbital levels by spin-orbit interaction. These spin-orbital levels are further split into what are known as Stark levels by the crystalline field. The energies of the terms, the spin-orbital levels and the crystalline field levels can be calculated by one of two methods, (1) the Slater determinantal method [310-313], (2) the Racah tensor operator method [314-316]. 

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. 

Rydberg states of Ba and Sr in an external magnetic field have been considered by Halley, Delande, and Taylor (35), by means of the R-matrix complex rotation method. Seipp and Taylor (36) used the same method for the Stark and Stark-Zeeman problem of Rydberg states of Na. Themelis and Nicolaides (96) investigated the ls 2s 2p 3s 5, 3p 4s 5, and 3d bound states of Na. They used the CESE method to compute tunneling rates and scalar and tensor polarizabilities and hyperpolarizabilities. Medikeri, Nair, and Mishra (145,146) considered shape resonances in Be", Mg" and Ca" in two-particle-one-hole-Tamm-Dancoff approximation. Photodetachment rate for Cl" described by one-electron model was computed by Yao and Chu (73)... 

FIGURE 15.4 Hyperfine levels. Left field-free levels F = 0,1. Centre electric-field-induced shift of levels and tensor Stark splitting of the triplet F = 1. Right Zeeman splitting of the doubletF =, Mp = 1. 

Oq and 02 being the scalar and tensor polarizability constants, respectively, can be determined experimentally and calculated theoretically. For J = 0 or 1/2, for which the formula breaks down, there is only a scalar effect. The Stark effect can be seen as an admixture of other states into the state under study. Perturbing states are those for which there are allowed electric dipole transitions (Sect.4.2) to the state under study. Energetically close-lying states have the greatest influence. A theoretical calculation of the constants Oq and 02 involves an evaluation of the matrix elements of the electric dipole operator (Chap.4). Investigations of the Stark effect are therefore, from a theoretical point of view, closely related to studies of transition probabilities and lifetimes of excited states. (Sects. 4.1 and 9.4.5). In Fig. 2.13 an example of the Stark effect is given different aspects of this phenomenon have been treated in [2.31]. 

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... 

An example of the Stark effect is given in Fig.9.39. Note, that while only the tensor Stark interaction constant 03 (Sect.2.5.2) can be determined in an LC experiment (Sect.7.1.5 Fig.9.18), the scalar interaction constant ag can be obtained in this type of experiment as well as 03. In the same way, isotope shifts can be measured by direct optical high-resolution methods while resonance methods and quantum-beat spectroscopy can only be used for measurements of splittings within the same atom. 

The notion [136], and accumulating experimental evidence [139], that the the BPni-isotropic coexistence line might terminate in a critical point has stimulated recent theoretical work by Lubensky and Stark [148], Letting Q = Qij denote the customary order parameter tensor, they assume a new, pseudoscalar order parameter formed from the chiral term in the free energy... 

Because the vibrations that underlie IR absorption spectra must affect the electric dipole of a molecule, we would expect the frequencies of these modes to be sensitive to local electric fields, and this is indeed the case. Shifts in vibration frequencies caused by external electric fields can be measured in essentially the same manner as electronic Stark shifts, by recording oscillations of the IR transmission in the presence of oscillating fields. The Stark tuning rate is defined as S = dv/dE , where v is the wavenumber of the mode and is the projection of the field (E) on the normal coordinate [88, 89]. To a first approximation, is given by —u Aft + E Aa)/hc, where ti is a unit vector parallel to the normal coordinate, Afi is the difference between the molecule s dipole moments in the excited and groxmd states, and Aa is the difference between the polarizability tensors in the two states (Sect. 4.13, Box 4.15 and Box 12.1). However, anharmonicity and geometrical distortions caused by the field also can contribute to vibrational Stark effects [90, 91]. 

The Stark effect of a rotational level is determined by the permanent electric dipole moment ju, the electric polarizability tensor a of the molecule and the applied electric field E. The Hamiltonian of these interactions can be given as ... 

In addition to the Stark effect produced by the permanent dipole moment, small electric dipole moments are induced by the electric field, and this effect is characterized by the electric-polarizability tensor a. This is important only with very large electric fields and will not be discussed further. 

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