Dipole moment functions

Dubai H-R, Ha T-K, Lewerenz M and Quack M 1989 Vibrational spectrum, dipole moment function, and potential energy surface of the CH chromophore In CHXg molecules J. Chem. Phys. 91 6698-713... 

Hollenstein H, Marquardt R, Quack M and Suhm M A 1994 Dipole moment function and equilibrium structure of methane In an analytical, anharmonic nine-dimenslonal potential surface related to experimental rotational constants and transition moments by quantum Monte Carlo calculations J. Chem. Phys. 101 3588-602... 

In the present work, we must carry out transformations of the dipole moment functions analogous to those descrihed for triatomic molecules in Refs. [18,19]. Our approach to this problem is completely different from that made in Refs. [18,19]. We do not transform analytical expressions for the body-fixed dipole moment components (/Zy, fiy, fi ). Instead we obtain, at each calculated ab initio point, discrete values of the dipole moment components fi, fiy, fif) in the xyz axis system, and we fit parameterized, analytical functions of our chosen vibrational coordinates (see below) through these values. This approach has the disadvantage that we must carry out a separate fitting for each isotopomer of a molecule Different isotopomers with the same geometrical structure have different xyz axis systems (because the Eckart and Sayvetz conditions depend on the nuclear masses) and therefore different dipole moment components (/Z, fiy, fij. We resort to the approach of transforming the dipole moment at each ab initio point because the direct transformation of analytical expressions for the body-fixed dipole moment components (/Zy, fiyi, fi i) is not practicable for a four-atomic molecule. The fact that the four-atomic molecule has six vibrational coordinates causes a huge increase in the complexity of the transformations relative to that encountered for the triatomic molecules (with three vibrational coordinates) treated in Refs. [18,19]. 

Equations (4.30) and (4.31) have been developed and dehned within a time-dependent framework. These equations are identical to Eqs. (35) and (32), respectively, of Ref. 80. They differ only in that a different, more appropriate, normalization has been used here for the continuum wavefunction and that the transition dipole moment function has not been expanded in terms of a spherical harmonic basis of angular functions. All the analysis given in Ref. 80 continues to be valid. In particular, the details of the angular distributions of the various differential cross sections and the relationships between the various possible integral and differential cross sections have been described in that paper. 

Figure 10.5. The dipole moment function from the MCVB calculation of BeH. The vertical dotted line marks the calculated equihbrium intemuclear distance.

Figure 12.1 shows the dipole moment functions in terms of intemuclear distance of CO, BF, and BeNe, calculated with our conventional 6-3IG basis arrangement. 

Figure 12.1. The dipole moment functions for CO, BF, BeNe calculated at a number of distances with the conventional 6-3IG basis arrangement.

We also calculated the dipole moment functions for CO, BF, and BeNe with an ST03G basis, and it can be seen in Fig. 12.2 that there are real difficulties with the minimal basis. We have argued that the numerical value and sign of the electric... 

An additional point that should be considered is that in the harmonic oscillator approximation, the selection mle for transitions between vibrational states is Ay = 1, where v is the vibrational quantum number and Ay > 1, that is, overtone transitions, which involve a larger vibrational quantum number change, are forbidden in this approximation. However, in real molecules, this rule is slightly relaxed due to the effect of anharmonicity of the oscillator wavefunction (mechanical anharmonicity) and/or the nonlinearity of the dipole moment function (electrical anharmonicity) [55], affording excitation of vibrational states with Ay > 1. However, the absorption cross sections for overtone transitions are considerably smaller than for Ay = 1 transitions and sharply decrease with increasing change in Av. 

Since both ipel and d depend on the nuclear configuration, the dipole moment d is a function of the nuclear configuration, and can be called the dipole-moment function. Ordinarily, one determines d averaged over the molecular zero-point vibrations. 

The ground-state potential curve and dipole-moment function for NO have been recently calculated within a Cl framework.68 Configuration-interaction studies of excited stales have been carried out by Thulslrup.69 An earlier Cl study of the X2ll and A2S,+ states has been reported by Green.70... 

Henry s group is also involved in theoretical studies to determine sources of local mode overtone intensity. These investigators have developed a very successful approach that uses their harmonically coupled anharmonic oscillator local mode model to obtain the vibrational wavefunctions, and ab initio calculations to obtain the dipole moment functions. The researchers have applied these calculations to relatively large molecules with different types of X-H oscillator. Recently they have compared intensities from their simple model to intensities from sophisticated variational calculations for the small molecules H20 and H2CO. For example, for H2CO they generated a dipole moment function in terms of all six vibrational degrees of freedom.244 This comparison has allowed them to determine the quality of basis set needed to calculate dipole moment... 

The transition dipole moment function of the parent molecule, pi. 

The transition dipole moment functions are — like the potentials — functions of Q. Their magnitudes determine the overall strength of the electronic transition ki —> kf. If the symmetry of the electronic wavefunc-tions demands likfki to be exactly zero, the transition is called electric-dipole forbidden. The calculation of transition dipole functions belongs, like the calculation of the potential energy surfaces, to the field of quantum chemistry. However, in most cases the fikfkt are unknown, especially their coordinate dependence, which almost always forces us to replace them by arbitrary constants. 

The intermolecular term has the same general form as the absorption cross section in the case of direct photodissociation, namely the overlap of a set of continuum wavefunctions with outgoing free waves in channel j, a bound-state wavefunction, and a coupling term. For absorption cross sections, the coupling between the two electronic states is given by the transition dipole moment function fi (R,r, 7) whereas in the present case the coupling between the different vibrational states n and n is provided by V (R, 7) = dVi(R, r, 7)/dr evaluated at the equilibrium separation r = re. 

If only the first derivatives in tie dipole-moment function and the second derivatives (k=j) in the potential function are retained, the strict harmonic-oscillator-linear-dipole-moment approximation, the selection rules are strict ... 

Avk=(vk+jic)= Avj=(vj+jj)=0, j k. A photon-induced transition via an external field occurs only for a given fundamental frequency, for w = between adjacent vibrational states, Vk and (vk l). Non-vanishing terms, to higher order in Eq. (28), mechanical anharmonicity, and non-linear terms in the dipole-moment function, Eq. (29), lead to breakdown in these strict selection rules, and frequencies other than the set, a n l>u2> u... . . to, become observable in the dipole-allowed spectrum. 

The difference in electric dipole moment (0.007 72 D) between the proton and deuteron species is discussed by Muenter and Klemperer [87] and attributed to the difference in zero-point amplitude averaged over the same dipole moment function. If the difference is purely vibrational in origin, the dipole moment of the vibrationless molecule is calculated to be 1.7965 D, which compares with a theoretical value of 1.942 D obtained from Hartree Fock calculations by Huo [94], The nuclear spin-rotation constants of non-rigid diatomic molecules have been discussed theoretically by Hindermann and Cornwell [95]. 

Clearly, if the matrix elements in this equation, and the vibrational wave functions, are known, the dipole moment function M(R) can be obtained. The square of the matrix... 

Kaiser [90] showed how the sign ambiguity above can be resolved, and the absolute signs of the M matrix elements determined the interested reader is referred to his paper. By combining the M values with the theoretical vibrational wave functions, he was able to use equation (8.339) to derive values of the dipole moment function at a range of values of R R,.. He then fitted the dipole moment function to a sixth-order polynomial in R-Re, from which he was able to calculate the distance derivatives given in table 8.21. 

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