Homogeneous 22-shifts

The classification of homogeneous scalar weighted shifts is known. Recently, Kor\'{a}nyi obtained a large class of inequivalent irreducible homogeneous bi-lateral $2$-by-$2$ block shifts. In this paper, we construct two distinct classes of examples not in the list of Kor\'{a}nyi. It is then shown that these new examples of irreducible homogeneous bi-lateral $2$-by-$2$ block shifts, together with the ones found earlier by Kor\'{a}nyi, account for every unitarily inequivalent irreducible homogeneous bi-lateral $2$-by-$2$ block shift.Comment: 26 page

Collective excitations in liquid DMSO : FIR spectrum, Low frequency vibrational density of states and ultrafast dipolar solvation dynamics

Valuable dynamical and structural information about neat liquid DMSO at ambient conditions can be obtained through study of low frequency vibrations in the far infrared (FIR), that is, terahertz regime. For DMSO, collective excitations as well as single molecule stretches and bends have been measured by different kinds of experiments such as OHD-RIKES and terahertz spectroscopy. In the present work we investigate the intermolecular vibrational spectrum of DMSO through three different computational techniques namely (i) the far-infra red spectrum obtained through Fourier transform of total dipole moment auto time correlation function, (ii) from Fourier transform of the translational and angular velocity time autocorrelation functions and a (iii) quenched normal mode analysis of the parent liquid at 300K. The three spectrum, although exhibit differences among each other, reveal similar features which are in good, semi-quantitative, agreement with experimental results. Study of participation ratio of the density of states obtained from normal mode analysis shows that the broad spectrum around 100 cm-1 involves collective oscillations of 300-400 molecules. Dipolar solvation dynamics exhibit ultrafast energy relaxation (dipolar solvation dynamics) with initial time correlation function around 140 fs which can be attributed to the coupling to the collective excitations. We compare properties of DMSO with those of water vis-a-vis the existence of the low frequency collective modes. Lastly, we find that the collective excitation spectrum exhibits strong temperature dependence.Comment: 24 pages,8 figure