Dipole moment integral

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

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... 

In the program it is only necessary to modify the matrix elements of the one-electron part of tire Hamiltonian h j by adding the dipole moment integrals ... 

The selection rule for l, discussed in Section 40d, allows only transitions with Al = +1 for the hydrogen atom. The lines of the Lyman series, with lower state that with n = 1 and l — 0, are in consequence due to transitions from upper states with 1 = 1. The radial electric dipole moment integral... 

The finite field procedure is the most often used procedure because of two main advantages (1) it is very easy to implement, and (2) it can be applied to a wide range of quantum mechanical methods. To calculate the energy in the presence of a uniform electric field of strength F, an F-r term needs to be added to the one-electron Hamiltonian. This interaction term can be constructed from just the dipole moment integrals over the basis set. Any ab initio or semiem-pirical method can then be used to solve the problem, with or without electron correlation. It is the ability to obtain properties from highly correlated methods that makes finite field calculations usually the most accurate available. 

Combination and difference bands Besides overtones, anharmonicity also leads to the appearance of combination bands and difference bands in the IR spectrum of a polyatomic molecule. In the harmonic case, only one vibration may be excited at a time (the transition dipole moment integral vanishes when the excited state is given by a product of more than one Hermite polynomial corresponding to different excited vibrations). This restriction is relaxed in the anharmonie case and one photon can simultaneously excite two different fundamentals. A weak band appears at a frequency approximately equal to the sum of the fundamentals involved. (Only approximately because the final state is a new one resulting from the anharmonie perturbation to the potential energy mixing the two excited state vibrational wave functions.)... 

Using eqn [61] for a fundamental vibration Qp it can be seen that the band intensity is proportional to the following transition dipole moment integral ... 

This is just the formula (29) for the dipole derivative expressed in the MO basis. In this context the h" are the dipole moment integrals and the h" are the derivatives of the dipole integrals, t has been assumed here and at most points in this chapter that the basis set used does not consist of functions with an explicit dependence upon the external electric field—if this is not the case then the full formula for the second derivative of the energy (48) must be used for the dipole derivative as well.) Taking the perturbations in the other order,... 

However, it is not necessary to solve these equations, as use may be made of an exchange theorem, which can be expressed most simply using the technique of Handy and Schaefer. The coefficients U would multiply elements of the dipole moment integrals

, but these integrals are the right-hand side of a set of simultaneous equations which have already been solved, i.e. 

The IR selection rule depends on the fact that the transition dipole moment integral must be nonzero in order to observe the transition as an IR absorption band. This means that the integrand for Md must have Aj symmetry. 

In the main text we introduced the selection rules for IR spectroscopy via the transition dipole moment integral. This appendix gives a little more detail on the origin of the selection rules, with explicit formulae for the vibrational wavefunctions. This also allows a more complete explanation of the observation that absorption due to transitions involving neighbouring levels (e.g. n = 0 to n = 1) are more easily observed than overtones which involve transitions to higher levels in the ladder of vibrational states. 

Using Eq. 8.35, the transition dipole moment integral is given by the relation... 

If a diatomic molecule is represented as a rigid rotor, the transition dipole moment integral for a rotational transition is... 

When transitions are observed between vibrational energy levels, infrared radiation is emitted or absorbed. The vibrational selection rules are derived in the Bom-Oppenheimer approximation by evaluating the transition dipole moment integral... 

In order for a given normal mode of a polyatomic molecule to give rise to a vibrational band (be infrared active ), the transition dipole moment integral for the two vibrational wave functions of the normal modes must be nonzero. This integral can be studied by group theory. However, it is often possible by inspection of the normal modes to identify those that modulate the dipole moment of the molecule. 

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 ... 

Of course, the total electronic dipole moment integral (6.16) must include contributions from core (CR) and lone pair (LP) as well as bond (BD) NBOs of the Lewis structure. The near-spherical core orbitals normally make insignificant contributions to the dipole integral, but the contributions of valence lone pairs usually cannot be ignored at any reasonable level of approximation. Thus, the superficial freshman-level sum of bond dipoles picture (even more superficially, with bond dipoles envisioned in terms of isolated point charges at each atomic nucleus) cannot give a realistic description of the molecular dipole moment of most chemical species. 

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