Transition moment integral

A related measure of the intensity often used for electronic spectroscopy is the oscillator strengdi,/ This is a dimensionless ratio of the transition intensity to tliat expected for an electron bound by Hooke s law forces so as to be an isotropic hanuonic oscillator. It can be related either to the experimental integrated intensity or to the theoretical transition moment integral ... 

Equation (Bl.1,1) for the transition moment integral is rather simply interpreted in the case of an atom. The wavefiinctions are simply fiinctions of the electron positions relative to the nucleus, and the integration is over the electronic coordinates. The situation for molecules is more complicated and deserves discussion in some detail. 

Here each < ) (0 is a vibrational wavefiinction, a fiinction of the nuclear coordinates Q, in first approximation usually a product of hamionic oscillator wavefimctions for the various nomial coordinates. Each j (x,Q) is the electronic wavefimctioii describing how the electrons are distributed in the molecule. However, it has the nuclear coordinates within it as parameters because the electrons are always distributed around the nuclei and follow those nuclei whatever their position during a vibration. The integration of equation (Bl.1.1) can be carried out in two steps—first an integration over the electronic coordinates v, and then integration over the nuclear coordinates 0. We define an electronic transition moment integral which is a fimctioii of nuclear position ... 

This last transition moment integral, if plugged into equation (B 1.1.2). will give the integrated intensity of a vibronic band, i.e. of a transition starting from vibrational state a of electronic state 1 and ending on vibrational level b of electronic state u. 

Equation (B1.1.10) and equation (B1.1.11) are the critical ones for comparing observed intensities of electronic transitions with theoretical calculations using the electronic wavefiinctions. The transition moment integral... 

Einstein derived the relationship between spontaneous emission rate and the absorption intensity or stimulated emission rate in 1917 using a thennodynamic argument [13]. Both absorption intensity and emission rate depend on the transition moment integral of equation (B 1.1.1). so that gives us a way to relate them. The symbol A is often used for the rate constant for emission it is sometimes called the Einstein A coefficient. For emission in the gas phase from a state to a lower state j we can write... 

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. 

The applications to selection rules work as follows. Intensities depend on the values of the transition moment integral of equation (Bl.l.lT... 

The intensity of absorption or emission associated with a vibrational transition is proportional to the square of the transition moment integral (Appendix),... 

In order for a transition to occur, the transition moment integral must be an even function. This integral is the product of the wave function corresponding to the ground state Tff, an operator R, and the wave function of the excited state Ye. Since R is antisymmetric, the product of the wave... 

The transition moment integrals (Ip) can be expressed in terms of the ethylene transition moment M as... 

Similar expressions are obtained for the component of the transition moment integral in y and z directions. Therefore, for an isotropic radiation of frequency vm the transition probability is given as... 

The intensity of absorption for an electronic transition is the probability of absorption between two given energy states. It can be theoretically computed by using the expression forthe transition moment integral. The transition moment integral MU1 between the ground and the first excited state is expressed as... 

The strength or intensity of absorption is related to the dipole strength of transition D or square of the transition moment integral M m , and is pressed in terms of oscillator strength / or integrated molar extinction jfe Jv. A transition with /= 1, is known as totally allowed transition. But the transitions between all the electronic, vibrational or rotational states are not equally permitted. Some are forbidden which can become allowed under certain conditions and then appear as weak absorption bands. The rules which govern such transitions are known as selection rules. For atomic energy levels, these selection rules have been empirically obtained from a comparison between the number of lines theoretically... 

The same conclusions can be drawn from the quantum mechanical condition that the square of the transition moment integral Mum must be nonzero for a transition to be induced by electromagnetic radiation. That is... 

Figure 3.9 Schematic representation of transition moment integral in. pyridine and formaldehyde. (Ref. 3.4).

Figure 3.18 Schematic representation of transition moment integral for monophotonic and biphotonic transitions in naphthalene. (A) Transition forbidden by one photon process (B) Allowed by two photon process.

The transition moment is. measured by the transition moment integral which for a transition between the energy states m- n is expressed as... 

A) commutes with 7 (A) and with Tx(X) (Problem 8.10), so that the transition-moment integral vanishes unless the states k and / have the same eigenvalue "(A) [Eq. (1.51)]. Thus the coupling constants JAA, which occur in "(A), do not appear in the expression for the transition frequencies. The same argument shows the transition frequencies to be independent of JX X/ Q.E.D. 

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.)... 

Certain transition moment integrals for electronic transitions in homopoiar diatomic molecules can be obtained very simply from overlap integrals,12 lor example ... 

JAig - Eu). These are transitions from molecular orbitals essentially localized on the ligands to molecular orbitals essentially localized on the metal atom. An examination of the transition moment integrals for these two transitions reveals that, for any reasonable molecular orbitals, the Aig - Eu transition will be more intense than Ai - A2u. 

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