Electric dipole moment operator

The electric dipole moment operator jx has components along the cartesian axes ... 

A Ionic strength Electric dipole moment operator Superscript designating radioactive... 

The predominant term in the perturbing potential V is of the form er, equal to the electric dipole moment operator. This is the origin of the selection rule that if ( 0, er i) = 0, the perturbed secular equation will not mix the states xpo and t/ i) so that the transition tpo ip i will not occur. 

The integrations over the electronic coordinates contained in < >, as well as the integrations over vibrational degrees of freedom yield "expectation values" of the electric dipole moment operator because the electronic and vibrational components of i and [Pg.287]

The electric dipole moment operator is independent of spin and, in the absence of spin-orbit coupling, the spin part of the magnetic moment operator will have no influence on MCD. We shall therefore neglect the spin part of Eq. (8) until we come to spin-orbit-induced MCD in Section II.A.4. 

The term maia a(1) is the first-order correction to the integral of the electric dipole moment operator in the a direction over orbitals a and i. The perturbed integral will depend on the change of the orbitals in the presence of a magnetic field or spin-orbit coupling. 

From Eq. (72) we see that the contribution to the MCD intensity from the perturbation to the transition density can be identified with the MCD due to the mixing of the excited state J with other excited states. The remainder of the MCD intensity from terms and spin-orbit-induced C terms is due to the perturbation of the integrals over the electric dipole moment operator (Eq. 52). The perturbed integrals thus include the contribution to the MCD from the mixing of excited states with the ground state. The perturbed integrals are written in terms of unperturbed orbitals (Eqs. 53 and 54) rather than unperturbed states or transition densities as this form is much easier to compute. With some further effort the contribution to the MCD from the perturbed integrals can also be analyzed in terms of transitions. 

Total electric dipole moment operator corresponding to left (-) and right (+) circularly polarized light. 

The operator d is the electric-dipole-moment operator. The sum is over electrons and nuclei ... 

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 ... 

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 probability of a transition being induced by interaction with electromagnetic radiation is proportional to the square of the modulus of a matrix element of the form where the state function that describes the initial state transforms as F, that describing the final state transforms as Tk, and the operator (which depends on the type of transition being considered) transforms as F. The strongest transitions are the El transitions, which occur when Q is the electric dipole moment operator, — er. These transitions are therefore often called electric dipole transitions. The components of the electric dipole operator transform like x, y, and z. Next in importance are the Ml transitions, for which Q is the magnetic dipole operator, which transforms like Rx, Ry, Rz. The weakest transitions are the E2 transitions, which occur when Q is the electric quadrupole operator which, transforms like binary products of x, v, and z. 

Only one of these (Elu) contains a representation to which the electric dipole moment operator belongs. Therefore only one of the three possible transitions is symmetry allowed, and for this one the radiation must be polarized in the (x, v) plane (see Table 5.2). 

We now collect together terms in Po, Pr and pt. We shall need the electric dipole moment operator which we define by... 

The difference between the definitions of the shift operators J and the spherical tensor components T, (./) should be noted because it often causes confusion. Because J is a vector and because all vector operators transform in the same way under rotations, that is, according to equation (5.104) with k = 1, it follows that any cartesian vector V has spherical tensor components defined in the same way (see table 5.2). There is a one-to-one correspondence between the cartesian vector and the first-rank spherical tensor. Common examples of such quantities in molecular quantum mechanics are the position vector r and the electric dipole moment operator pe. 

Here i//0 is the ground vibrational wave function and ij/ is the wavefunction corresponding to the first excited vibrational state of the th normal mode /< is the electric dipole moment operator Qj is the normal coordinate for the /th vibrational mode the subscript 0 at derivative indicates that the term is evaluated at the equilibrium geometry. The related rotational strength or VCD intensity is determined by the dot product between the electric dipole and magnetic dipole transition moment vectors, as given in (2) ... 

Since the electric dipole moment operator is a vector operator, the electric dipole transition moment will also be a vector quantity. The probability of an electric dipole transition is given by the square of the scalar product between the transition moment vector in the molecule and the electric field vector of the light, and is therefore proportional to the squared cosine of the angle between these two vectors. Thus, an orientational dependence results for the absorption and emission of linearly polarized light. The orientation of the transition moment with respect to the molecular system of axes is... 

---