Here the integral, if interpreted as a functional on compactly supported functions on positive reals, makes sense. $$\int _$, and $\chi (x)= \mid x \mid ^s$then the above "integral" is the Gamma function. Suppose $\phi$ is a homomorphism of the additive group $K$ into $S^1$ and suppose $\chi $ is a homomorphism of the multiplicative group $K^*$ into $S^1$. The case originally considered by Carl Friedrich Gauss was the quadratic Gauss sum, for R the field of residues modulo a prime number p, and the Legendre symbol.In this case Gauss proved that G() p 1 2 or ip 1 2 for p congruent to 1 or 3 modulo 4 respectively (the quadratic Gauss sum can also be evaluated by Fourier analysis as well as by contour integration). The contribution of the group to a molecular orbital was calculated using Mulliken population analysis.
The po-tential energy curves between the dihedral angles and their corre-sponding energies are given in Fig. Besides of these calculations, the group contributions of the molecular orbitals and to prepare TDOS (or DOS), PDOS and OPDOS spectra GaussSum 2.2 38 was used. In other words, according to scan results, there are two possible conformers (S1 and S2).
Suppose $K$ is a locally compact second countable field, and $K^*$ the multiplicitative group of nonzero elements of $K$. analysis of the molecule showed that there are two local minima near 0° (or 360°) and 180° for E-6-ClN molecule.