Dipole moment integral/operator

The derivation of the second line of equation (A.81) follows the same reasoning as was used to obtain the one-electron part of the electronic energy [equation (A.21)], since both fi and h are sums of single-particle operators. The dipole moment integrals over basis functions in the last line of equation (A.81) are easily evaluated. Within the HF approximation, dipole moments may be calculated to about 10% accuracy provided a large basis set is used. 

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... 

Of more general interest are the selection rules for S, L, and /, as these quantum numbers describe an atomic state with greater accuracy. To the approximation that we neglect spin in the Hamiltonian operator, the spin wave functions are independent of the coordinate wave functions. The dipole moment integral will vanish because of the orthogonality of the spin functions unless the spin quantum numbers match in the initial and final states. To this approximation, we thus have the selection rule AS == 0 that is, only transitions between terms of the same multiplicity are allowed. The selection rules for L and J cannot be derived so simply the results are ... 

The electronic contribution to the dipole moment is thus determined from the density matrix and a series of one-electron integrals J dr< (-r)0. The dipole moment operator, r, h.)-components in the x, y and z directions, and so these one-electron integrals are divided into their appropriate components for example, the x component of the electronic contribution to the dipole moment would be determined using ... 

The integrations over the eleetronie eoordinates eontained in I)f p , as well as the integrations over vibrational degrees of freedom yield "expeetation values" of the eleetrie dipole moment operator beeause the eleetronie and vibrational eomponents of i and f are identieal ... 

This integral of the electronic dipole moment operator is a function of a nuclear coordinate Q. The integral may be expanded in a Taylor series with respect to Q (equation 4) and... 

For identical volumes of integration, s (H) can be written as the sum of a position-independent and a position-dependent term. When y is the dipole-moment operator, this is accomplished as follows. 

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. 

Evaluate required integrals over the electric and magnetic dipole moment and spin-orbit coupling operators as well as GIAO contributions, if needed. 

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. 

Note that H here is the complete Hamiltonian, that is, it presumably includes new terms dependent on the nature of X. It is occasionally the case that the integral on the r.h.s. of Eq. (9.33) can be readily evaluated. Indeed, it is choice of X = E that leads to the definition of the dipole moment operator presented in Eq. (9.1). 

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.)... 

Transition Moment.—A wave-mechanical quantity whiph is proportional to the square root of the intensity of a transition, and is given by the integral f wave functions of the initial and final states. The dipole moment vector, M is given by M = Ser where r is the radius vector from the center of gravity of the positive charge to the electron. M is also known as the dipole moment operator. 

Here the summation of charges times position vectors is replaced by the integral over the total wavefunction T (the square of the wavefunction is a measure of charge) of the dipole moment operator (the summation over all electrons of the product of an electronic charge and the position vectors of the electrons). To perform an ab initio calculation of the dipole moment of a molecule we want an expression for the moment in terms of the basis functions r/j, their coefficients c, and the geometry (for a molecule of specified charge and multiplicity these are the only variables in an ab initio calculation). The Hartree-Fock total wavefunction T is composed of those component orbitals i// which are occupied, assembled into a Slater determinant (Section 5.2.3.1), and the i// s are composed of basis functions and their coefficients (Sections 5.3). Equation (5.206), with the inclusion of the contribution of the nuclei to the dipole moment, leads to the dipole moment in Debyes as (ref. [lg], p. 41)... 

For the respective quantum mechanical description of a molecule in a stationary state, a few additional aspects need to be addressed. First, the system state is characterized by a wavefunction VP, and system properties, such as the total energy or dipole moment, are calculated through integration of VP with the relevant operator in a distinct way. Note that an operator is simply an instruction to do some mathematical operation such as multiplication or differentiation, and generally (but not always) the order in which such calculations are performed affects the final result. Second, the wavefunctions V obey the Schrodinger equation ... 

Mote also that the dipole-moment operator, being a vector, must invert its sign fttinder inversion J. Hence, with respect to I, the dipole moment is always antisymmetric. Titus, for the integrals in Eq. (8.9) to be nonzero also requires that is3) and ij ) be of opposite symmetry with respect to inversion. Given the extant conditions Sp the behavior of E3) and E ) with respect to the reflection ah, the symmetry j fe(j[uirements with respect to X are most easily accommodated through the rotational components of the li3) and l ) states. 

After multiplication with the dipole operators and integration over space, the perturbations of one of the components (u) of the electric and magnetic dipole moment by the v-component of the perturbing field read... 

Note also that the dipole moment operator, being a vector, must reverse its sign under inversion /. Hence, with respect to /, the dipole moment is always antisymmetric. Thus for the integrals in Eq. (19) to be nonzero also requires that... 

It can be calculated from an integral taken over the product of the wavefunctions of the initial (m) and final (n) states of a spectral transition and the appropriate dipole moment operator (D) of the electromagnetic radiation. 

---