G matrix elements

In Equation 5.25 the ratio of G matrix elements has been obtained using a diatomic approximation (Gi/G/) = [(1/12) + (l/2)]/[(l/12) + (1/1)]. Although in the gas phase the frequency of each isotopomer can be measured to high precision, say 0.05 cm-1 or better, such precision is impossible in the liquid because of inherent broadening caused by intermolecular forces. Except in special cases band centers cannot be located to better than 0.5 cm-1, that limit is imposed by the nature of the liquid state. There is an identical uncertainty for each isotopomer, so spectroscopic precision is about... 

The coefficients of the two velocity terms corresponding to these coordinates can be derived fairly easily from the rules for constructing the G matrix elements (Chapter 3) but the details will not be given here. The kinetic energy is... 

GF method calculations are simplified by the systematic behavior of the G matrix elements (Decius 1948). MUBFF calculations, however, are somewhat complicated by the force constants representing interactions between non-bonded atoms— these can be tedious to express in terms of internal coordinates. Computer programs have been written to partially automate calculations, thereby reducing the necessary effort and minimizing opportunities for errors (e.g., Schachtschneider 1964 Gale and Rohl 2003). 

The derivatives of these equations require one to obtain the first and second derivatives of the G matrix elements. This, in turn, requires to obtain the first, second and third derivatives of the d, R, a and P (equations 5) internal coordinates with respect to Y, and the first, second and third derivatives of the Cartesian coordinates with respect to the internal coordinates. 

For a given unitary irreducible representation r there will be nj matrix elements corresponding to each R. If the operations are R Rv Rtf. .. Rr, then the g matrix elements of a chosen i and j value,... 

If the atomic masses in the G matrix elements are expressed in atomic mass units rather than in grams and the frequencies are expressed in reciprocal centimeters (cm"1), 10.5-2 may be written... 

Figure AVI.l Diagrams showing how various common g-matrix elements are defined.

For the A vibrations of H20, the G-matrix elements are calculated by using the following parameters ... 

TABLE 8.1 G-matrix elements, their derivatives and potential energy coupling terms for the... 

From these frequencies and with the help of the corresponding G-Matrix elements (Wilson et al., 1955), the symmetry-adapted force constants (F) can be calculated directly. In the vibrations discussed here, F is a linear combination of stretching and interaction force constants / and f,r . 

The matrix elements (3,41) may be constructed if all three integrals are available at the same time. To achieve this, it is most profitable to compute them just in the order (ATlA ) (Xmatrix elements P y "Xg are then stored on a tape or disk together with the pair indices (/cv), (Xe"), They contribute to two G-matrix elements as follows ... 

The vibration-rotation interaction is the effect arising from coupling terms between angular and vibrational momenta as well as from the dependence of the rotational G-matrix elements (the /u-tensor) on the internal coordinates. The importance of this effect may to some extent be reduced provided an appropriate axis convention is used. The axis convention is the set of rules defining the orientation of the molecular axes, eg, g = x,y, z, relative to an arbitrary configuration as given by the position vectors, Ra, a. = 1, 2,... N. These rules can be expressed in three relations between the rag components, similar to the center of mass conditions(2.4). We shall refer to these relations as the axial constraints . Usually Eckart-condi-tions39 are imposed, but other possibilities may be considered. 

Nearly all of the error in the numerical int rals comes flx)m the interstitial region, hence the larger mesh density used for that integratioiL The numerical integration meshes are tested in the code segment IGTOFF by comparison of G matrix elements that are calculated analytically and numerically. If the calculated values differ too greatly, the various meshes can be enriched. 

The G (mass and geometry dependent) and F (mass and geometry independent) matrices are discussed by Wilson et al, (1955). For two bond stretches coupled by a shared atom, the relevant F and G matrix elements are defined by... 

If the G-matrix elements obtained are written for each combination of internal coordinates, there results... 

If such calculations are made for several types of molecules, it is immediately seen that the G-matrix elements themselves have many regularities. Decius [79] developed general formulas for writing G-matrix elements. Some of them are as follows ... 

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