The dipole moment functions

The first set of data was obtained on a two-dimensional grid (the 2D grid in Ref. [4]) consisting of 420 geometrip of or D h symmetry, with bond distances R h between 0.60 and 1.65 A, and angles o hnh between 70 and 120°. The data points were more dense in the vicinity of the equilibrium geometry and the saddle point. The second set of data was determined on a six-dimensional 

The ab initio calculations give the components of the molecular dipole moment in a right-handed Cartesian axis system x y z with origin in the nitrogen nucleus. The Hj nucleus lies on the axis with a positive value of the z coordinate, and the plane defined hy the nitrogen nucleus and the protons Hj and H2 is the y z plane. Thus, in the x y z axis system, N has the coordinates (0,0,0), Hj has the coordinates (0,0,z)) with Zj 0, H2 has the coordinates (0,y2.Z2) H3 has the coordinates (x, y, Z2)- In general, all the coordinates y 2, Z2. 3, y, and Z3 are different from zero. 

The ab initio calculations produce values of fiy, i.e., the components of the electronically averaged dipole moment along the x y z axes defined above. In order to calculate molecular line strengths, however, we must determine, as functions of the vibrational coordinates, the dipole moment components along the molecule-fixed axes xyz (see equation (23)) defined by Eckart and Sayvetz conditions [1]. 

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]. 

The molecular bond (MB) representation We write I in the form given in Refs. [30,31]  

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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. 

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... 

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. 

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. 

We approximate the dipole moment function by a series expansion in the internal OH-stretching and SOH-bending displacement coordinates about the calculated equilibrium geometry. The dipole moment function is therefore written... 

The systematics found in the alkali hydride potentials suggest that it might be possible to model a simple ionic potential to reflect the behavior of the whole series of alkali hydrides. Here we construct a "practical" diabatic curve which reflects the physical properties, e.g. the dipole moment function.. We expect our diabatic curve to follow the ionic part of the adiabatic potential and to begin to deviate in the avoided crossing region. 

The parameters A and p are determined by a nonlinear least square fit of equation (1) to the RKR inner turning points of each of the alkali hydrides. This fitting procedure is justified by the fact that the magnitude of the dipole moment functions of LiH (25, 26) and NaH (33) at these internuclear distances are very close to those of opposite point charges a distance R apart. 

With the approximation of being electrically harmonic (eh), the transition moment reduces to a matrix element of the vibrational coordinate r times the slope of the dipole moment function at equilibrium ... 

The dipole polarizability can be used in place of the dipole moment function, and this will lead to Raman intensities. Likewise, one can compute electrical quadrupole and higher multipole transition moments if these are of interest. 

As a result of solving the Schrodinger equation for the model Hamiltonian (9.3) one gets anharmonic frequencies and relative IR intensities (if the dipole moment function is available) in the particular region of the IR spectrum [23-25]. It should be noted, however, that in many theoretical studies only the energy spectrum of the model 2D Hamiltonian was considered [26, 27]. 

As a result, a large number of a relatively intensive nonfim-damental transitions may appear in the IR spectra. In particular, the broad IR band associated with the asymmetric stretch of the O H O fragment is due to combinations with the O- O stretch. The change in the dipole moment function plays no role in the INS spectra and this is why the Vjs(OHO) bands in INS spectra are usually narrow and contain no combinations. 

The dipole moment functions for PF and PCI, calculated at the CASSCF and MRCI levels, are shown in Fig. 8. There is a significant difference between the CASSCF and MRCI functions at large R but little difference around equilibrium. This is illustrated in Figs. 9 and 10, which show the functions around the experimental equilibrium bond length for a variety of methods. These curves are fit to a fourth-... 

In the harmonic approximation, overtones and combination bands have zero intensity. To derive expressions for the overtone, it is necessary to consider both higher derivatives of the dipole moment function and an-harmonic terms in the potential. Detailed equations have been given by Overend. " It is also possible to find expressions for the intensities of particular lines within a band for example, Harman and Wallis consider the differences in the rotational lines in the separate branches of a rotation-... 

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