Electric moment integrals

The important result of the analysis above is that the calculation of spectroscopic line intensities and the determination of selection rules is reduced to an assessment of the electric moment integrals given in (6.300). We now consider the main examples in more detail. 

After expanding the required binomial expansions, the electric moment integral can be written as ... 

Thus, the CETO electric moment integrals are simple combinations of overlap integrals. A similar behavior as tl one encountered in the GTO framework [68b] or in the STO case [18]. 

As a result of the foregoing considerations, the wave-mechanical calculation of the intensities of spectral lines and the determination of selection rules are reduced to the consideration of the electric-moment integrals defined in Equation 40-11. We shall discuss the results for special problems in the following sections. 

Since interchange of the nuclei does not affect the electric moment integral for molecules with like nuclei, and since x is unaffected by this... 

Transition intensities are detennined by the wavefiinctions of the initial and final states as described in the last sections. In many systems there are some pairs of states for which tire transition moment integral vanishes while for other pairs it does not vanish. The temi selection rule refers to a simnnary of the conditions for non-vanishing transition moment integrals—hence observable transitions—or vanishing integrals so no observable transitions. We discuss some of these rules briefly in this section. Again, we concentrate on electric dipole transitions. 

One-electron electric dipole moment integral over orbitals p and q. 

For an electric-dipole transition between two molecular states, the transition-moment integral is... 

Recall that homonuclear diatomic molecules have no vibration-rotation or pure-rotation spectra due to the vanishing of the permanent electric dipole moment. For electronic transitions, the transition-moment integral (7.4) does not involve the dipole moment d hence electric-dipole electronic transitions are allowed for homonuclear diatomic molecules, subject to the above selection rules, of course. [The electric dipole moment d is given by (1.289), and should be distinguished from the electric dipole-moment operator d, which is given by (1.286).] Analysis of the vibrational and rotational structure of an electronic transition in a homonuclear diatomic molecule allows the determination of the vibrational and rotational constants of the electronic states involved, which is information that cannot be provided by IR or microwave spectroscopy. (Raman spectroscopy can also furnish information on the constants of the ground electronic state of a homonuclear diatomic molecule.)... 

The first term in the square bracket in this equation is the electric monopole moment, which is equal to the nuclear charge, Ze. The second term in the square bracket is the electric dipole moment while the third term in the square bracket is the electric quadmpole moment. For a quantum mechanical system in a well-defined quantum state, the charge density p is an even function, and because the dipole moment involves the product of an even and an odd function, the corresponding integral is identically zero. Therefore, there should be no electric dipole moment or any other odd electric moment for nuclei. For spherical nuclei, the charge density p does not depend on 0, and thus the quadmpole moment Q is given by... 

Mm I2 is sometimes written as R m 2, where R is known as the Imatrix element of the electric dipole moment and has the same meaning as Vie transition moment integral. There are a few other related quantities which are used forexpressing the strength of an electronic transition. f)ipole strength of transition... 

This links the transition matrix element to the transition moment integrals (b ri a) (first moments of the electron distribution) along the direction of electric field of the emitted or absorbed photon ... 

In order to compute these kind of integrals, let us allow any electric moment operator, belonging to the c-th order operator set, be referred to the center C and be written as ... 

Fluorescence is defined simply as the electric dipole tranation from an excited electronic state to a lower state, usually the ground state, of the same multiplicity. Mathematically, the probability of an electric-dipole induced electronic transition between specific vibronic levels is proportional to R f where Rjf, the transition moment integral between initial state i and final state f is given by Eq. (1), where represents the electronic wavefunction, the vibrational wavefunctions, M is the electronic dipole moment operator, and where the Born-Oppenheimer principle of parability of electronic and vibrational wavefunctions has been invoked. The first integral involves only the electronic wavefunctions of the stem, and the second term, when squared, is the familiar Franck-Condon factor. 

The multipole expansion has already been used in certain quantum chemical calculations [59-65]. As localized orbitals are concentrated in certain spatial region, they can also be represented by their multipole moments. In the following we investigate whether the Coulomb integrals in terms of localized orbitals can be substituted by the multipole expansion of electric moments. 

We decompose the charge distribution of the whole electron system into sum of contributions from localized orbitals. If the localized orbitals do not overlap and their electric moments can be considered as transferable, than it is expected that the Coulomb integrals can be approached by the sum of interaction energies... 

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