Curvilinear co-ordinates

It is nice to have a distinctive notation for the curvilinear co-ordinates, which emphasizes their difference from and yet their one-to-one correlation with the Rt co-ordinates. Most authors reporting anharmonic calculations do not in fact make any distinction they denote the curvilinear co-ordinates by the same symbols customarily used to denote the corresponding rectilinear coordinates in harmonic calculations. For many purposes this is satisfactory, particularly since the harmonic force constants are not altered by the change from rectilinear to curvilinear co-ordinates. However, in a general discussion it is important to distinguish the two sets, and so for the remainder of this section we shall follow Hoy et al.12 and write the curvilinear co-ordinates with the symbol Hi. 

The discussion so far may be summarized as follows. There are two reasons for using curvilinear co-ordinates to represent the anharmonic force field of a polyatomic molecule, despite their apparent complexity. The first is that it is only in this way that we obtain cubic and quartic force constants which are independent of isotopic substitution. The second is that in terms of curvilinear bond-stretching and angle-bending co-ordinates we obtain the simplest expression for the force field, in the sense that cubic and quartic interaction terms are minimized. The first reason is compulsive the second reason is not compulsive, but it does make the curvilinear co-ordinates very desirable. 

The remainder of this section is devoted to formulating the algebra of the transformation from the curvilinear co-ordinates fh to the normal coordinates Qr, and to making the corresponding transformation in the representation of the potential energy. 

The quadratic force constants in the normal co-ordinates are, of course, diagonal, i.e. H = eor<5J<5 .) The expansion (49) is then obtained by substituting (42) and (44) into (48). In fact it is possible to write closed formulae for the force constants in terms of the force constants /, as is done in equations (11) of ref. 12. It is an important property of this transformation that a purely quadratic force field in the curvilinear co-ordinates 8 gives rise to quadratic,... 

Symmetry, and the Number of Independent Force Constants.—As in harmonic calculations, the rather general discussion of the preceding section can be simplified in particular cases by making use of symmetry, as discussed by Hoy et a/.12 Thus we may choose the curvilinear co-ordinates Jfin linear combinations that span the irreducible representations of the point group we denote such symmetrized curvilinear co-ordinates by the symbol S, and we define them by means of a U matrix exactly analogous to that used for rectilinear coordinates ... 

Finally, we should note that in cases where a redundant set of curvilinear co-ordinates Hi are defined, the transformation to curvilinear symmetry coordinates becomes more complicated. This difficulty is discussed briefly in ref. 12, but it will not be developed here. 

Kuchitsu and co-workers5 7 were the first to introduce what is perhaps the simplest and most generally useful model, in which they assume all anharmonic force constants in curvilinear co-ordinates to be zero with the exception of cubic and quartic bond-stretching constants. These may be estimated from the corresponding diatomics, or from a Morse function, or they may be adjusted to give the best fit to selected spectroscopic constants to which they make a major contribution. This is often called the valence-force model. It is clear from the results on general anharmonic force fields quoted above that this model is close to the truth, and in fact summarizes 80 % of all that we have learnt so far about anharmonic force fields. 

Braaten, M.E. and Shyy, W. (1987), Study of pressure correction methods with multigrid for viscous flow calculations in non-orthogonal curvilinear co-ordinates, Numerical Heat Transfer, 11,417 42. 

True curvilinear internal co-ordinates L tensor. R-L Q Normal co-ordinates... 

Table 6 Anharmonic force fields in curvilinear internal co-ordinates for COa and for CS2 ...

Table 10 Force constants60 in curvilinear internal co-ordinates and H2Se ° for HaO, H2S,...

Table 13 Anharmonic force fields of some bent triatomic molecules in curvilinear internal co-ordinates, up to cubic terms only ...

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