Index k saddles and dividing surfaces in phase space, with applications to isomerization dynamics

In this paper we continue our studies of the phase space geometry and dynamics associated with index k saddles (k > 1) of the potential energy surface. Using normal form theory, we give an explicit formula for a "dividing surface" in phase space, i.e. a co-dimension one surface (within the energy shell) through which all trajectories that "cross" the region of the index k saddle must pass. With a generic non-resonance assumption, the normal form provides k (approximate) integrals that describe the saddle dynamics in a neighborhood of the index k saddle. These integrals provide a symbolic description of all trajectories that pass through a neighborhood of the saddle. We give a parametrization of the dividing surface which is used as the basis for a numerical method to sample the dividing surface. Our techniques are applied to isomerization dynamics on a potential energy surface having 4 minima; two symmetry related pairs of minima are connected by low energy index one saddles, with the pairs themselves connected via higher energy index one saddles and an index two saddle at the origin. We compute and sample the dividing surface and show that our approach enables us to distinguish between concerted crossing ("hilltop crossing") isomerizing trajectories and those trajectories that are not concerted crossing (potentially sequentially isomerizing trajectories). We then consider the effect of additional "bath modes" on the dynamics, which is a four degree-of-freedom system. For this system we show that the normal form and dividing surface can be realized and sampled and that, using the approximate integrals of motion and our symbolic description of trajectories, we are able to choose initial conditions corresponding to concerted crossing isomerizing trajectories and (potentially) sequentially isomerizing trajectories.Comment: 49 pages, 12 figure

Intramolecular vibrational energy redistribution as state space diffusion: Classical-quantum correspondence

We study the intramolecular vibrational energy redistribution (IVR) dynamics of an effective spectroscopic Hamiltonian describing the four coupled high frequency modes of CDBrClF. The IVR dynamics ensuing from nearly isoenergetic zeroth-order states, an edge (overtone) and an interior (combination) state, is studied from a state space diffusion perspective. A wavelet based time-frequency analysis reveals an inhomogeneous phase space due to the trapping of classical trajectories. Consequently the interior state has a smaller effective IVR dimension as compared to the edge state.Comment: 5 pages, 3 figure

Genealogical and family history of the state of New Hampshire : a record of the achievements of her people in the making of a commonwealth and the founding of a nation. Vol. I

Paged continuously.; Spine title: Genealogy New Hampshire.; Includes index

On Mountin Sculpture in the Sierra Nevada, and the Method of Glacial Erosion.

Geology and Natural History. 615 eated layer was then placed at such an angle that the re- Sriion of the sound-wave was sent through a second tin the same dimensions as the above), and its action rendered Syits causing a second sensitive flame, placed at the end STtube, to become violently affected. This action continued \u27\u27ill\u27.the heated layer intervened; but upon its withdrawal, \ mentioned sensitive flame, receiving the whole of the direct became again violently agitated, and at the same moment ond sensitive flame, ceasing to be affected returned to its anquility.—Nature, ix, 334. e. c. p. IP 41. L IL Geology and Natubal History. Mountain Sculpture in the Sierra Nevada, and the *d of glacial erosion; by E S. Carr.—After speaking of visional planes in the granites of the Sierra Nevada, Prof. remarks as follows.—The greatest check to the free play and rolling power of these divisional planes is the occurrence, in ignse numbers and size, of domes, cones, and round wave-ridges, ijgether with an innumerable brood of modified forms and com- K binations. The curved cleavage which measures and determines Kpae rounded forms may be designated the dome cleavage, inas- Oiuch as the dome is apparently the most perfect typical form of , s tie group. KjDomes of close-grained siliceous granite are admirably calculated .withstand the action of atmospheric and mechanical forces. IpTo other rock-form can compare with it in strength; no other sTqffgred so unflinching a resistance to the tremendous pressure of l;tbe glaciers. A dam of noble domes extends across the head of jpToseniite Valley, from Mount Starr-King to North Dome, which | was effectually broken through by the combined force of the EHoffman and Tenaya glaciers; but the great South Lyell glacier, jwhich entered the valley betwoen Starr King and Half Dome, was ;; unable to force the mighty barrier, and the approach of the long summer which terminated the glacial epoch found it still mazing 2 and swedging compliantly among the strong unflinching bosses, . just as the winds are compelled to do at the present time. | \u27, The Starr-King group of domes is perhaps the most interesting L.of the Merced basin. The beautiful conoid, Starr King, the loftiest I and most perfect of the group, was one of the first to emerge from I the Glacial sea. * * * |i There appear to be no positive limits to the extent of dome I structure in the granites of the Sierra, when considered in all its numerous modifications. Rudimentary domes exist everywhere, - waiting their development, to as great a depth as observation can , reach. The western flank was formerly covered with slates, which have evidently been carried off by glacial denudation from the middle and upper regions; small patches existing on the summits and spurs of the Hoffman and Merced Mountains are all that are now left. When a depth of two or three thousand feet below the Am. Jour. Soi.—Third Seribs. Vol. VII, No. 41.—Mat, 1874. 26 516 i Scientific Intelligence. bottom of the slates is reached, the dome structure prev *1 to the exclusion of others. As we proceed southward mo** ward along the chain from the region adjacent to YosemtaV Mt,K dome forms gradually become less perfect. * * Va*y,* Glacial erosion.—No matter how, abundant the glaci l r a vertical precipice can not be produced unless its cleavaqe A tical, nor a dome without dome structure in the rock acted ** Therefore, when we say that the glacial ice-sheet and kvm glaciers molded the mountains, we must remember that th \u27 molding power upon hard granite possessing a strong physical structure is comparatively slight. In such hard, strongly built granite regions, glaciers do not so much mold and shape as dim. inter forms already conceived and ripe. The harder the rock and the better its specialized cleavage planes are developed\u27 tlie \u27 greater will be tne degree of controlling power possessed \u27by it over its own forms, as compared with that of the disinterring glacier; and the softer the rock and more generally developed iti cleavage planes, the less able will it be to resist ice action and maintain its own forms. In general, tha grain of a rock determines its surf ace forms ; yet it would matter but little what the grain might be—straight, curved, or knotty—if the excavating and sculpturing tool were sharp, because in that case it would cut without reference to the grain. Every carpenter knows that only a dull tool will follow the grain of wood. Such a tool is the glacier, gliding with tremendous pressure past splitting precipices and smooth-swelling domes, flexible as the wind, yet hard-tempered as steel. Mighty as its effects appear to us, it has only developed the predestined forms of mountain beauty which were ready and waiting to receive the baptism of light.— Overland Monthly,\u27 May, 1874. 2. Note on the recent Volcanic Action in Hawaii; by T. Coan, from a letter to J. D. Dana, dated Hilo, Hawaii, Jan. 6th, 1874.— You are aware that the great summit crater of Mauna Loa, Mokuaweoweo, has, for a number of years, shown but few and feeble symptoms of activity, until the past year. For a few days in August, 1872, there was a brilliant light in the crater; and again on the 6th and 7th of Jan., 1873, there were vivid demonstrations, which roused the attention of many witnesses. But it was not until the 20th of April, 1873, that a continuous exhibition of mountain pyrotechnics commenced. From that day to the present, now almost nine months, the action within the great cauldron has not remitted. Most of the time the boiling has been vehement, and the scene was never more brilliant than a_ few nights ago. Sustained jets of molten-rock were constantly rising 50 to 200 feet within the mural caldron, and the surgings, puffings and roarings have been heard low down the sides of the mountain, and, as some testify, as far as Reed\u27s Ranch, probably fifteen miles. But the great marvel of this eruption is its duration. We have seen nothing like it before in this crater. The eruption of 1855-6https://scholarlycommons.pacific.edu/jmb/1003/thumbnail.jp

History of the town of Rindge, New Hampshire, from the date of the Rowley Canada or Massachusetts charter, to the present time, 1736-1874, with a genealogical register of the Rindge families.

Genealogical register : p. [419]-775