Stark matrix element

The high value of the electron density at the nucleus leads to the enhancement of the electron EDM in heavy atoms. The other possible source of the enhancement is the presence of small energy denominators in the sum over states in the first term of Eq.(29). In particular, this takes place when (Eo — En) is of the order of the molecular rotational constant. (It is imperative that a nonperturbative treatment be invoked when the Stark matrix element e z(v /0 z v / ) is comparable to the energy denominator (Eq En) [33].) Neglecting the second term of the right-hand side of Eq.(29), which does not contain this enhancement factor [8, 27], we get... 

When the A-doublet splitting and dipole moment are both known, as they are in the BC1 A1n state, the Stark-mixing experiment [excite R( J — 1), detect Stark-induced Q( J) fluorescence] provides a non-invasive method for measuring spatially and temporally varying electric fields. Since at low-J A J = 0, e <-> / Stark matrix elements are invariably large and A-doublet splittings are invariably small, low-J levels provide sensitivity to small electric fields ( 50 V/cm) [Moore, et al., 1984, Mandich, et al., 1985]. 

Moreover, Figure 22b indicates that the Stark matrix element H2sip has no significant effect on the formation of the molecular orbitals. It should be noted that this matrix element gains importance in LK systems. Nevertheless, it is difficult to test its values sensitively by the MO fitting procedure. 

Thus the IR active modes will be determined by the matrix elements of the polarlsablllty matrix and not by a combination of the surface selection rule and the normal IR selection rules l.e. all of the Raman active modes could become accessible. This effect has been formalized and Its significance assessed In a discussion (12) which compares Its magnitude for a number of different molecules. In the case of acrylonitrile adsorption discussed In the previous section, the Intensity of the C=N stretch band appears to vary with the square of the electric field strength as expected for the Stark effect mechanism. 

On the basis of these formulae one can convert measurements of area, which equals the integral in the latter formula, under spectral lines into values of coefficients in a selected radial function for electric dipolar moment for a polar diatomic molecular species. Just such an exercise resulted in the formula for that radial function [129] of HCl in formula 82, combining in this case other data for expectation values (0,7 p(v) 0,7) from measurements of the Stark effect as mentioned above. For applications involving these vibration-rotational matrix elements in emission spectra, the Einstein coefficients for spontaneous emission conform to this relation. 

In the following, we pay special attention to the connections among the spherical, Stark and Zeeman basis. Since in momentum space the orbitals are simply related to hyperspherical harmonics, these connections are given by orthogonal matrix elements similar (when not identical) to the elements of angular momentum algebra. 

Using the zero field n(m states we calculate the matrix elements (/i m Ez n m ) of the Stark perturbation to the zero field Hamiltonian. Writing the matrix element in spherical coordinates and choosing the z axis as the axis of quantization,... 

Since m is a good quantum number, each set of m states is independent of the others. We consider first the circular m = n - 1 states. They have no first order Stark shift since there are no other states of the same m and n. There are, however, two m=n - 2 states, which exhibit small linear Stark shifts. The Stark shifts are not large since the radial matrix element is small,1... 

The minus sign is for radial wavefunctions R e r) —I-r( as r — 0. On the other hand, the m = 0 states also include the low states for which the radial matrix element is large, and the extreme m = 0 states have Stark shifts of approximately 3n2E/2. 

By taking into account the matrix elements off diagonal in n, the second and higher order contributions to the Stark effect can be calculated. If the calculation is carried through second order, the energies are given by1... 

Fabre et a/.28 used a projection operator technique to describe the Stark shifts at fields below where low states of large quantum defects join the manifold. A less formal explanation is as follows. If, for example, the s and p states are excluded, as in Fig. 6.13 below 800 V/cm, effectively only the nearly degenerate (22 states are coupled by the electric field. The only differences among the m = 0,1, and 2 manifolds occur in the angular parts of the matrix element, i.e.1... 

Since Vd(r) is only nonzero near r = 0 the matrix element of Eq. (6.51) reflects the amplitude of the wavefunction of the continuum wave at r 0. Specifically, the squared matrix element is proportional to C, the density of states defined earlier and plotted in Fig. 6.18. From the plots of Fig. 6.18 it is apparent that the ionization rate into a continuum substantially above threshold is energy independent. However, as shown in Fig. 6.18, there is often a peak in the density of continuum states just at the threshold for ionization, substantially increasing the ionization rate for a degenerate blue state of larger This phenomenon has been observed experimentally by Littman et al.32 who observed a local increase in the ionization rate of the Na (12,6,3,2) Stark state where it crosses the 14,0,11,2 state, at a field of 15.6 kV/cm, as shown by Fig. 6.19. In this field the energy of the... 

In this regime, where the levels are discrete, it is possible to calculate the intensities of the transitions by matrix diagonalization, just as the energies are calculated. It is simply a matter of computing the eigenvectors of the Hamiltonian as well as its eigenvalues. For example, to calculate the intensities in the spectra shown in Fig. 8.12 we calculate the rcp amplitude in each of the Stark states and multiply it by the matrix element connecting the 3s state to the n p state,... 

The central problem is to calculate the field required to drive the n — n + 1 transition via an electric dipole transition. In the presence of an electric field, static or microwave, the natural states to use are the parabolic Stark states. While there is no selection rule as strict as the M = 1 selection rule for angular momentum eigenstates, it is in general true that each n Stark state has strong dipole matrix elements to only the one or two n + 1 Stark states which have approximately the same first order Stark shifts. Red states are coupled to red states, and blue to blue. Explicit expressions for these matrix elements between parabolic states have been worked out,25 and, as pointed out by Bardsley et al.26, the largest matrix elements are those between the extreme red or blue Stark states. These matrix elements are given by (n z n + 1) = n2/3.26... 

Since the extreme n and n + 1 Stark states have the largest coupling matrix elements, it seems reasonable to assume that the n— n + 1 transitions occur through these states and calculate the field required to drive the transition between this pair of levels. While this is an approximation, it is useful, just as considering the extreme Stark states gives a reasonable description of the ionization of Na. 

To observe a 7s — 9 transition requires that there be a 9p admixture in the 9 state. For odd this admixture is provided by the diamagnetic interaction alone, which couples states of and 2, as described in Chapter 9. For even states the diamagnetic coupling spreads the 9p state to all the odd 9( states and the motional Stark effect mixes states of even and odd (. Due to the random velocities of the He atoms, the motional Stark effect and the Doppler effect also broaden the transitions. Together these two effects produce asymmetric lines for the transitions to the odd 9t states, and double peaked lines for the transitions to even 9( states. The difference between the lineshapes of transitions to the even and odd 9i states comes from the fact that the motional Stark shift enters the transitions to the odd 9( states once, in the frequency shift. However, it enters the transitions to the even 9( states twice, once in the frequency shift and once in the transition matrix element. Although peculiar, the line shapes of the observed transitions can be analyzed well enough to determine the energies of the 9( states of >2 quite accurately.25... 

Because of the quantum origin of the Stark effect (see Section 5.2), the evolution of the density matrix elements /mm must be considered for the ground state molecules. In the field region we must choose the... 

To simplify the analysis, it is reasonable to divide the field strength range into "normal" and "strong". Normal should be taken to mean those fields for which the condition x = B/irhS oc 1 is satisfied, i.e. the Stark shift of the 2s and 2pj/2 levels, caused by the fields, proves to be of the same order as the Lamb shift (here is the matrix element of the 2sj/2 - 2p1/2 transition, E is the field strength and S is the Lamb shift). 

The energies of the levels in an electric field can be calculated by numerical diagonalisation of the above matrix for different values of the electric field and the J, M quantum numbers. However, perturbation theory has also often been used and we may readily derive an expression for the second-order Stark energy using the above matrix elements. The result is as follows ... 

We now apply these general results to the specific problem of HF. Apart from the Stark effect, we shall otherwise ignore the very small matrix elements which are off-diagonal in J, and evaluate the terms for J = 1, Mj = 0, 1, /H = h = 1 /2. As Weiss pointed out, the total magnetic component Mz = Mj + MF + Mh is a good quantum number and may be used to set up a decoupled representation in which states of different Mz value are diagonalised separately. For J = 1 there are twelve primitive basis states, which we write below in the form MF, Mh, Mj). 

This is, in essence, the result needed to construct figure 8.47. There is more to be done, however, because it is necessary to use (8.430) to derive matrix elements for the parity-conserved functions, and then to take note of the rotational distortion which mixes the fine-structure states. This mixing can be represented by an effective 12 value, which is designated 12 eff in table 8.28, where the results of the Stark experiments are listed. 

In this appendix we collect together tables of matrix elements of r, r2, r3, z, z2, rz, and r(r2 — z2) in the scaled hydrogenic basis and perturbation corrections to the ground-state energy for the LoSurdo-Stark, Zeeman, screened Coulomb, and charmonium Hamiltonians and the one-electron diatomic ion. 

Matrix Elements of the LoSurdo Stark Perturbation rz in the Scaled Hydrogenic Basis"... 

For high quantum numbers n, the matrix element (7.2.11) is very close to the matrix element (7.2.10). This fact is consistent with the elongated nature of the extremal Stark states (see Fig. 7.4), which renders them quasi-one-dimensional as discussed above. This correspondence holds for the off-diagonal matrix elements as well. Therefore, and as fax as the dipole matrix elements are concerned, the SSE states are good approximations to the extremal Staxk states. Since both the SSE states and the... 

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