Dipole approximation matrix element

According to the results of the last section, if the integrals in (3.48) all vanish, then the probability for a transition between states m and n is zero. Actually, (3.47) is the result of several approximations, and even if the electric dipole-moment matrix elements vanish, there still might be some probability for the transition to occur. 

One such link between semiempirical theory and experiment that appeared about that time was the development of calculational methods for optical rotatory dispersion. Moffitt s theoretical work with Kronig—Kramers transforms coupled with Djerassi s experimental data on steroids gave rise to rules for the prediction of the sign of optical rotation. Computer calculations with semiempirical methods played a role. i Wavefunctions of at least an approximate sort were needed for the dipole and dipole velocity matrix elements of the theory. 

For vibrational transitions it is not appropriate to use the Born-Oppenheimer (BO) approximation because the ground and excited states in the context of vibrational transitions have the same electronic wavefunction and differ only in the nuclear wavefunctions, a consequence of which is that the electronic contribution to the magnetic dipole transition matrix element vanishes in the BO approximation. In order to include the important electronic contribution to magnetic dipole transition moments, one has to choose either to make further approximations to the magnetic dipole operator yielding effective non-vanishing magnetic transition moments, or to go beyond the BO approximation. Various approximate models and exact a priori methods have resulted in the last 25 years. 

The Einstein coefficient Ba in equation (98) can thus be related to the square of the transition dipole moment matrix element by equation (99), in the electric dipole approximation ... 

This matrix element represents the coupling of states K and S through terms in the Hamiltonian, which has previously been neglected in the Bom-Oppenheimer approximation. Since we have expanded the dipole moment matrix element, which is responsible for the intensities of optical transitions, we can see that the effect of this expansion is to allow states to borrow intensity from one another. A transition which may be forbidden in the zero-order theory may obtain intensity from a nearby state to which a transition is allowed. Note that in Eq. (46), the sum runs over all the excited states S, except the state K. The are the zero-order energies, and we may write the denominator as hiws - o>k) - Clearly, efficient borrowing... 

The faet that the El approximation to af i eontains matrix elements of the eleetrie dipole operator between the initial and final states makes it elear why this is ealled the eleetrie dipole eontribution to afwithin the El notation, the E stands for eleetrie moment and the 1 stands for the first sueh moment (i.e., the dipole moment). 

Without loss of generality y = y can be assumed. If the dipole moment can be assumed to be a linear function of coordinate within the spread of the frozen Gaussian wave packet, the matrix element (gy,q,p, Pjt(r) Y,q, p ) can be evaluated analytically. Since the integrand in Eq. (201) has distinct maxima usually, we can introduce the linearization approximation around these maxima. Namely, the Taylor expansion with respect to bqp = Qq — Qo and 8po = Po — Po is made, where qj, and pj, represent the maximum positions. The classical action >5qj, p , ( is expanded up to the second order, the final phase-space point (q, p,) to the first order, and the Herman-Kluk preexponential factor Cy pj to the zeroth order. This approximation is the same as the ceUularization procedure used in Ref. [18]. Under the above assumptions, various integrations in U/i(y, q, p ) can be carried out analytically and we have... 

The probability of finding the system in the state vRjn> has an oscillatory time dependence. For off-resonance conditions, the system presents a line width at half maximum equal to 4 ll/fi. This matrix element can be expanded in a multipolar expansion, the first term being the electric dipole approximation [45, 152, 154],... 

Judd (1962) and Ofelt (1962) demonstrated that, for electric dipole f f transitions and under certain approximations, the square of the matrix element in Equation (6.6) can be written as follows ... 

Grimmeiss et al. (1974) and Morgan (1975) j0"1 K2 No Dipole matrix element assumed constant or Matrix elements approximated, with emphasis on symmetry aspects Parabolic... 

Asymptotic formulae. For a discussion of induced dipoles in highly polarizable species, it is often sufficient to consider the so-called classical multipole induction approximation in its simplest form (i.e., neglecting field gradients and hyperpolarizabilities). In such a case, one needs to know only the vibrational matrix elements of the multipole moments,... 

To deduce whether a transition is allowed between two stationary states, we investigate the matrix element of the electric dipole-moment operator between those states (Section 3.2). We will use the Born-Oppenheimer approximation of writing the stationary-state molecular wave functions as products of electronic and nuclear wave functions ... 

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... 

Here 0) is the frequency of the radiation, i denotes the initial state of the molecule, and f labels the final state of the photofragments, dv = Pj dE, where is the density of final states. Usually, the wavelength of the radiation considerably exceeds the size of the molecule, and one can use the dipole approximation (see, e.g., ref. 17). Then Hf d. fi (d is the component of the dipole moment along the external electric field) and the problem reduces to the analysis of the dipole matrix element... 

This result is an exact expression for the transition matrix element. Physically we have a dipole interaction with the vector potential and a dipole interaction with the magnetic field modulated by a phase factor. The problem is that this integral is difficult to compute. An approximation can be invoked. The wavevector has a magnitude equal to 1 /X. The position r is set to the position of an atom and is on the order of the radius of that atom. Thus K r a/X. So if the wavelength of the radiation is much larger than the radius of the atom, which is the case with optical radiation, we may then invoke the approximation e k r 1 + ik r. This is commonly known as the Bom approximation. This first-order term under this approximation is also seen to vanish in the first two terms as it multiplies the term p e. A further simplification occurs, since the term a (k x e) has only diagonal entries, and our transition matrix evaluates these over orthogonal states. Hence, the last term vanishes. We are then left with the simplified variant of the transition matrix ... 

The statement that there is good agreement between theoretical and experimental data is conventionally the conclusion drawn from the results presented. However, a deeper insight can be obtained when the dipole matrix elements involved are determined experimentally, and compared with results from different theoretical approximations of the influences of electron-electron interactions. In order to make such a comparison, the experimental matrix elements will be derived first for the case of 2p photoionization in magnesium at 80 eV photon energy. From equs. (5.14), (5.15), and (5.16) one can calculate DS, Dd and A from the measured observables. The results are listed in Table 5.1, together with data from several theoretical calculations. 

The previous formulation for the photoionization process provides the starting point for theoretical calculations. For simplicity, and because the conditions are well fulfilled, in many applications the dipole approximation is often used. (For extensions and derivations, relevant in the present context of photoionization studies with synchrotron radiation, see [KJG95] and references therein.) This approximation is based on a special property of the matrix element ... 

Within the dipole approximation, one can have different forms for the dipole matrix element (see [BSa57]). The form presented so far is called the momentum form (or the velocity form) because the relevant operator contains the momentum p ... 

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