Superposition state dipole moment

The generalization to the control of the dynamics of a molecule with n electronic states is straightforward. For the purpose of deducing the control conditions we will examine the extreme case in which every possible pair of these electronic states is connected via the radiation field and a nonzero transition dipole moment. If the molecule is coupled to a radiation field that is a superposition of individual fields, each of which is resonant with a dipole allowed transition between two surfaces, the density operator of the system can be represented in the form... 

Equation (2.35)]. Equation (15.6b) is formally equivalent to (2.66) with the exception that in the present case the outgoing channel also includes, in addition to the vibrational state, the particular electronic state. It is important to realize that because of the nonadiabatic coupling both excited electronic states and both electronic product channels are populated, even if one transition dipole moment is exactly zero for all nuclear geometries. Furthermore, the superposition of two complex-valued amplitudes in the case that both transition moments are non-zero can lead to interesting interference patterns. 

Here 7 is the average radiative line half-width at half maximum (HWHM) of bound levels (which has been introduced phenomenologically), a>m = (Em — En)/ti, and < = (EM h Em). In obtaining this result, we have assumed that (a) the cw fields are turned on at t — —00, at which time the system is in its initial superposition state [Eq. (6.14)], (b) the medium has no permanent dipole moment, and (c) (El tl ZT2) = 0 due to the fact that IE)) and E2) are assumed to have the same parity. 

From the experimental values of dipole moments it is possible, in a number of cases, to make a semi-quantitative evaluation of the weights of the various valence bond structures contributing to a bond (see Chapter 18). These calculations must be regarded as only approximate since the bond is described in terms of the Heider-London theory with the superposition of ionic states. The results cannot, therefore, be more precise than is permitted by the Heitler-London approximation. Nevertheless, the calculations are of significance since they permit an assessment to be made of the more important structures contributing to the bond and thus assist in predicting and explaining the reactivity of bonds. 

Dipole moment offree radicals— It has already been pointed out in Chapter 5 that in the stabilization of free radicals containing oxygen and nitrogen atoms, the superposition of ionic states plays a significant part. This view is supported by the values of the dipole moments of a,a -diphenyl j3-picrylhydrazine, /, /i, — 4-92 D... 

In bonds between different atoms the dipole moment may have intermediate values owing to the superposition of covalent and ionic states. In such cases the bond is described by the function... 

The effect of quantum interference on spontaneous emission in atomic and molecular systems is the generation of superposition states that can be manipulated, to reduce the interaction with the environment, by adjusting the polarizations of the transition dipole moments, or the amplitudes and phases of the external driving fields. With a suitable choice of parameters, the superposition states can decay with controlled and significantly reduced rates. This modification can lead to subnatural linewidths in the fluorescence and absorption spectra [5,10]. Furthermore, as will be shown in this review, the superposition states can even be decoupled from the environment and the population can be trapped in these states without decaying to the lower levels. These states, known as dark or trapped states, were predicted in many configurations of multilevel systems [11], as well as in multiatom systems [12],... 

The CPT effect and its dependence on quantum interference can be easily explained by examining the population dynamics in terms of the superposition states. v) and a). Assume that a three-level A-type atom is composed of a single upper state 3) and two ground states 1) and 2). The upper state is connected to the lower states by transition dipole moments p31 and p32. After introducing superposition operators 5+ = (S ) = 3)(.v and 5+ = (Sa) = 3)(a, where. v) and a) are the superposition states of the same form as Eqs. (107) and (108), the Hamiltonian (65) can be written as... 

As we have shown in Sec. III. A, the second-order correlation function of the fluorescence field depends on correlation functions of the atomic dipole moments (S+(f)S+(f + x)Sy(t)Sj (t)), which correspond to different processes including photon emissions from a superposition of the excited levels. Therefore, we write the correlation functions G (R, t) and G (R, t R, t + x) in terms of the symmetric and antisymmetric superposition states as... 

A mixing of atomic or molecular states can be implemented by applying external fields. To illustrate this method, we consider a V-type atom with the upper states connected to the ground state by perpendicular dipole moments (p12 p32). When the two upper states are coupled by a resonant microwave field, the states become a linear superposition of the bare states... 

Calculated DFT properties listed in Table 1 were obtained from the fit of the ground-state potential energy curves to 12 points calculated around the energy minimum [32]. Dissociation energy has been corrected for basis set superposition error by a standard counterpoise technique. The local approximation to the exchange and correlation gives the best fit to bond distances, theoretical values differ by no more than 0.03 A (4%) from the experimental ones (see Table 1). Vibrational frequencies are also predicted to lie within 1 % off the experiment. One should remember, however, that other advanced quantum chemical methods give equally satisfactory results for these, basicaly one-electron quantities and that inclusion of nonlocal effects does not improve the DFT predictions. The dipole moment, fi, is much more sensitive... 

It is therefore possible to construct a superposition of states of the dipole, such that the effective dipole moment of the molecule bobs up and down in time. The amount that the dipole bobs, relative to the constant component, can be controlled by the relative population in the two states. Moreover, the degree of polarization of the molecule plays a significant role. For a fully polarized molecule, when 81/2 = 0, only the dc portion of the dipole persists, although even it can vanish if there is equal population in the two states, dipole up and dipole down. ... 

As before, the molecule can also be in a superposition state. No matter how complicated this superposition is, the expectation value of the dipole must instantaneously point in some direction, since the only available angular dependence resides in the C q functions, which yield only dipoles. In other words, no superposition can generate the field pattern of a quadrupole moment, for example. 

To see that the dipole moment of the superposition state oscillates with time, consider the same states at time t = (1 /2)h/ Eb — Ea). For the particle in a box, the energy of the second eigenstate is four times that of the first, Eb = AEa (Eq. 2.24), so ll2)hl Eb — Ea) = /6)h/Ea- The individual wavefunctions at this time have both real and imaginary parts. For the lower-energy state, we find by using the relationship exp(—/0) = cos (0) — i sin (0) that... 

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