Rotational correction term matrix

Eventually, qiplying Eq. (3.11) separately for each synunetiy vibrational coordinate we obtain the rotational correction term matrix Rg (in units of D/A or D/rad)... 

With the addition of the Ppp term the atomic polar tensor matrix becomes mass independent since it refers to a space-fixed coordinate system. Thus, the transformation of measured infrared intensities of different isotopic species of a molecule widi identical symmetry will result in proximately the same Px mahix, within the experimental uncertainties. This is an important feature of the atomic polar tensor representation of infrared band intensities. The troublesome problem of rotational correction terms is treated in a straightforward and general way. The treatment, however, introduces some difficulties in the physical interpretation of the elements of atomic polar tensors. These will be discussed later in conjunction with some examples of calculations. 

Experimental intensity data, reference Cartesian system, internal coordinates, force fields and L matrices for methyl chloride are the same as given in section m.C. Bond displacement coordinates are defined in Fig. 4.6. Rotational corrections to the dipole moment derivatives with respect to symmetry coordinates are evaluated using the heavy isotope method [34]. The rotational correction terms are given in Table 4.8. To illustrate the calculations in more detail the entire V matrix of methyl chloride is presented in Table 4.9. To remove die rotational terms fi om the sets of linear equations for symmetiy... 

Cartesian reference systems, geometric parameters and symmetiy coordinates for H2O and NH3 are given in Chapter 3. Dipole moment derivatives with respect to symmetiy coordinates for H2O, evaluated in analyzing experimental absolute infrared intensities, are also presented there. dp/dSj dipole moment derivatives for ammonia used in the present calculations were taken from Ref. [147] and are presented in Table 4.13. The signs of these quantities have been fixed with the aid of ab initio MO calculations [147]. Elements of the respective matrices for both molecules were evaluated by employing the heavy isotope mediod [34], weighting the respective heavy atoms by a factor of 1000. The rotational correction terms for ammonia are tabulated in Table 3.3. The Rs matrix for H2O has the following form (in D A l or D rad )... 

The q = 1 components of the first term on the right-hand side of this equation represents a correction to the so-called rotational distortion term, whose matrix elements were given in equations (8.363) and (8.364). In the light of this term we should replace —2B by (—2B + y) in the off-diagonal elements (8.364), add nothing to the energy of the 2n3/2 component, and subtract y from the energy of the 2ni/2 component. 

Obsoved intensities, com-dinate definitions and force fields used in evaluating die elements of P, are as alrea given in section III.C. In constructing the linear equations (4.126) corrected for rotational contribution Pg matrix is used. The heavy isotope created to eliminate rotational terms in the right-hand side of Eq. (4.126) refers to a molecule with the mass of the oxygen multiplied by 1000. The same isotope is, of course, used to evaluate the rotational correction to the pj derivative. 

S is the matrix of rotational correction expressed in terms of symmetry coordinates [Eqs. (3.5) and (3.11)]. The elements of Px(v) are determined by purely vibrational distortions. From the rotation-free atomic polar tensor an invariant with respect to Cartesian axes reorientation can be deduced from the trace of the product Px( )(v). P x( >(v)... 

It can be shown that det C = 1 (Problem 2.13.1). The norm of C, or trace of C, or sum of its diagonal terms, is 2 + 2y. Since det C = 1, we can consider the Lorentz transformation matrix X like the four-dimensional analog of the Eulerian rotation in 3-space. We now seek quantities that are "covariant with the Lorentz transformation"—that is, are "relativistically correct". We next define in this new four-space a few essential quantities ... 

H2. Indeed in fitting the term values to energy level expressions, it is necessary to take into account this correlation matrix otherwise, incorrect values for the standard deviations are obtained and the rotational constants may be in error by the order of one (correct) standard deviation. 

The next major advance in understanding came with a paper by Secrest [24], another student of Hirschfelder. This work follows even more closely the approach used by Curtiss, in that it essentially involves the approximate diagonalization of the rotational part of the Hamiltonian using the transformation that diagonalizes the potential matrix. However, unlike Curtiss, there was no attempt to write the exact solution in terms of the approximate ones, so there was no systematic way to compute corrections to the sudden approximations. However, Secrest s paper was much more accessible to the general audience and thus increased interest and activity in the methods. 

For applications at intermediate energies it seems more appropriate to use Lorentz transformations to obtain the relative momenta in the NN COM frame in terms of the momenta (k, p k p ), rather than rely on the nonrelativistic results in eq. (3.13). The effective NN COM energy, e, in eq. (3.14) also should be computed relativistically. Wigner rotations should also be included. However, at present it is not clear how to do such a calculation, since the correct Lorentz invariant form of the fully off-shell t-matrix is unspecified. 

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