Transition dipole moment functions, electronic

The transition dipole moment functions are — like the potentials — functions of Q. Their magnitudes determine the overall strength of the electronic transition ki —> kf. If the symmetry of the electronic wavefunc-tions demands likfki to be exactly zero, the transition is called electric-dipole forbidden. The calculation of transition dipole functions belongs, like the calculation of the potential energy surfaces, to the field of quantum chemistry. However, in most cases the fikfkt are unknown, especially their coordinate dependence, which almost always forces us to replace them by arbitrary constants. 

The intermolecular term has the same general form as the absorption cross section in the case of direct photodissociation, namely the overlap of a set of continuum wavefunctions with outgoing free waves in channel j, a bound-state wavefunction, and a coupling term. For absorption cross sections, the coupling between the two electronic states is given by the transition dipole moment function fi (R,r, 7) whereas in the present case the coupling between the different vibrational states n and n is provided by V (R, 7) = dVi(R, r, 7)/dr evaluated at the equilibrium separation r = re. 

Photodissociation involves electronic transitions initiated by the absorption of light. The key molecular property that mediates the interaction with light is the transition dipole moment. The electronic transition dipole moment between the jth and kth electronic state is defined by the integral over the electronic degrees of freedom for the operator where the dipole moment is sandwiched between the two electronic wave functions given by... 

In the following paragraphs we give selected examples of the use of our wavefunctions and potential curves to predict or confirm various spectroscopic features of the alkalis. We know of plans to observe Li2 spectra in at least two laboratories (23, 24) so some predictions regarding the spectra appear to be in order. Julienne (25) has used our wavefunctions for LI2 to calculate the electronic transition dipole moment function corres-... 

Potential energy curves and transition dipole moment functions for the NH molecule have been computed by Goldfield and Kirby (1987). The photodissociation cross sections into the excited and H states give rise to an unshielded rate of about 5 x 10 s (Kirby and Goldfield 1988), which is comparable to that of OH, and about a factor of two smaller than that of CH (van Dishoeck 19876). The photodissociation rate of NH is based on cross sections calculated by van Dishoeck (1986). Although many of the exdted electronic potentials are repulsive, most of them have vertical excitation energies larger than 13.6 eV, so that the destruction by interstellar radiation is not very rapid. 

Another related issue is the computation of the intensities of the peaks in the spectrum. Peak intensities depend on the probability that a particular wavelength photon will be absorbed or Raman-scattered. These probabilities can be computed from the wave function by computing the transition dipole moments. This gives relative peak intensities since the calculation does not include the density of the substance. Some types of transitions turn out to have a zero probability due to the molecules symmetry or the spin of the electrons. This is where spectroscopic selection rules come from. Ah initio methods are the preferred way of computing intensities. Although intensities can be computed using semiempirical methods, they tend to give rather poor accuracy results for many chemical systems. 

The measurements are predicted computationally with orbital-based techniques that can compute transition dipole moments (and thus intensities) for transitions between electronic states. VCD is particularly difficult to predict due to the fact that the Born-Oppenheimer approximation is not valid for this property. Thus, there is a choice between using the wave functions computed with the Born-Oppenheimer approximation giving limited accuracy, or very computationally intensive exact computations. Further technical difficulties are encountered due to the gauge dependence of many techniques (dependence on the coordinate system origin). 

State averaging gives a wave function that describes the first few electronic states equally well. This is done by computing several states at once with the same orbitals. It also keeps the wave functions strictly orthogonal. This is necessary to accurately compute the transition dipole moments. 

The results presented here follow the analysis presented in Ref. 84. It is possible to define the functions (r, R, 9, t = 0) only when we have specified whether the electronic transition involved is perpendicular or parallel (i.e., whether the transition dipole moment is perpendicular to or lies in the molecular plane). 

In addition, group theory can be used to assess when transition dipole moments must be zero. The product of the irreducible representations of the two wave functions and the dipole moment operator within the molecular point group symmetry must contain the totally symmetric representation for the matrix element to be non-zero (note that, if the molecule does not contain an inversion center, the operator r does not belong to any single irrep, except for the trivial case of Ci symmetry see Appendix B for more details). A consequence of this consideration is that, for instance, electronic transitions between states of the same symmetry are forbidden in molecules possessing inversion centers. 

At this point, it is of interest to discuss the relationship between MO theory and the intensity of electronic transitions. The oscillator strength of an electronic absorption band is proportional to the square of the transition dipole moment integral, ( /gM I/e) where /G and /E are the ground- and excited-state wave functions, and r is the dipole moment operator. In a one-electron approximation, (v(/G r v(/E) 2= K Mrlvl/fe) 2> where v /H and /fe are the two MOs involved in the one-electron promotion v /H > v / ,. Metal-ligand covalency results in MO wave... 

Assume that we have computed CASSCF wave function for two different electronic states. Now we want to compute transition properties, for example, the transition dipole moment. How can we do that. The two states will in general be described by two non-orthonormal sets of MOs, so the normal Slater rules cannot be applied. Let us start by considering the case where two electronic states fi and v are described by the same set of MOs. The transition matrix element for a one-electron operator A is then given by the simple expression ... 

The TDSE requires an initial condition, i.e. one must specify the wave packet at f = 0. Based on the assumption that the nuclei do not move during an electronic transition but only the electrons, the photo-excitation can be described as a Franck-Condon transition where the wave packet is excited vertically to the excited electronic state. Here fixj is the electronic transition dipole moment of the transition between the ground state X and the th electronically excited state, whereas Xx is a wave function of the electronic groimd state, typically the lowest vibrational state. Assuming a vertical electronic transition the initial wave packet on the excited state PES is given by... 

The electronic-transition dipole moment for the G E transition is defined by Mge = ( g A/ ge1 e> where the are the state wave functions and A/ ge is the dilference in dipole moment of the ground and excited states [22]. The intensity of the transition is proportional to Mge - The broad absorption bands usually observed in transition metal systems are composed of progressions in the vibrational modes that correlate with the differences in nuclear coordinates between the vibrationally equilibrated ground and excited state. Since the energy difference between the donor and acceptor is generally solvent-dependent, the distribution of solvent environments that is characteristic of solutions may also contribute to the bandwidth (see further discussion of this point in the sections below). If the validity of the Born Oppenheimer approximation is assumed, the intensity of each of these vibronic components is given by Eq. 11,... 

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