Gamma-ray emission from microquasars: a numerical model for LSI+61º303

We explore the possible association between the microquasar LSI +61°303 and the EGRET source 2CG 135+01/3EG J0241+6103 by studying, with a detailed numerical model, whether this system can produce the emission and the variability detected by EGRET (>100 MeV) through inverse Compton (IC) scattering. Our numerical approach considers a population of relativistic electrons entrained in a cylindrical inhomogeneous jet, interacting with both the radiation and the magnetic fields, taking into account the Thomson and Klein-Nishina regimes of interaction. Our results reproduce the observed spectral characteristics and variability at γ-rays, thus strengthening the identification of LSI +61°303 as a high-energy γ-ray source.

Three-dimensional hydrodynamic numerical modelling of fold nappe formation, basement-cover deformation and slab detachment with applications to the helvetic nappe system (W Switzerland)

Many three-dimensional (3-D) structures in rock, which formed during the deformation of the Earth's crust and lithosphere, are controlled by a difference in mechanical strength between rock units and are often the result of a geometrical instability. Such structures are, for example, folds, pinch-and-swell structures (due to necking) or cuspate-lobate structures (mullions). These struc-tures occur from the centimeter to the kilometer scale and the related deformation processes con-trol the formation of, for example, fold-and-thrust belts and extensional sedimentary basins or the deformation of the basement-cover interface. The 2-D deformation processes causing these structures are relatively well studied, however, several processes during large-strain 3-D defor-mation are still incompletely understood. One of these 3-D processes is the lateral propagation of these structures, such as fold and cusp propagation in a direction orthogonal to the shortening direction or neck propagation in direction orthogonal to the extension direction. Especially, we are interested in fold nappes which are recumbent folds with amplitudes usually exceeding 10 km and they have been presumably formed by ductile shearing. They often exhibit a constant sense of shearing and a non-linear increase of shear strain towards their overturned limb. The fold axes of the Morcles fold nappe in western Switzerland plunges to the ENE whereas the fold axes in the more eastern Doldenhorn nappe plunges to the WSW. These opposite plunge direc-tions characterize the Rawil depression (Wildstrubel depression). The Morcles nappe is mainly the result of layer parallel contraction and shearing. During the compression the massive lime-stones were more competent than the surrounding marls and shales, which led to the buckling characteristics of the Morcles nappe, especially in the north-dipping normal limb. The Dolden-horn nappe exhibits only a minor overturned fold limb. There are still no 3-D numerical studies which investigate the fundamental dynamics of the formation of the large-scale 3-D structure including the Morcles and Doldenhorn nappes and the related Rawil depression. We study the 3-D evolution of geometrical instabilities and fold nappe formation with numerical simulations based on the finite element method (FEM). Simulating geometrical instabilities caused by sharp variations of mechanical strength between rock units requires a numerical algorithm that can accurately resolve material interfaces for large differences in material properties (e.g. between limestone and shale) and for large deformations. Therefore, our FE algorithm combines a nu-merical contour-line technique and a deformable Lagrangian mesh with re-meshing. With this combined method it is possible to accurately follow the initial material contours with the FE mesh and to accurately resolve the geometrical instabilities. The algorithm can simulate 3-D de-formation for a visco-elastic rheology. The viscous rheology is described by a power-law flow law. The code is used to study the 3-D fold nappe formation, the lateral propagation of folding and also the lateral propagation of cusps due to initial half graben geometry. Thereby, the small initial geometrical perturbations for folding and necking are exactly followed by the FE mesh, whereas the initial large perturbation describing a half graben is defined by a contour line inter-secting the finite elements. Further, the 3-D algorithm is applied to 3-D viscous nacking during slab detachment. The results from various simulations are compared with 2-D resulats and a 1-D analytical solution. -- On retrouve beaucoup de structures en 3 dimensions (3-D) dans les roches qui ont pour origines une déformation de la lithosphère terrestre. Ces structures sont par exemple des plis, des boudins (pinch-and-swell) ou des mullions (cuspate-lobate) et sont présentés de l'échelle centimétrique à kilométrique. Mécaniquement, ces structures peuvent être expliquées par une différence de résistance entre les différentes unités de roches et sont généralement le fruit d'une instabilité géométrique. Ces différences mécaniques entre les unités contrôlent non seulement les types de structures rencontrées, mais également le type de déformation (thick skin, thin skin) et le style tectonique (bassin d'avant pays, chaîne d'avant pays). Les processus de la déformation en deux dimensions (2-D) formant ces structures sont relativement bien compris. Cependant, lorsque l'on ajoute la troisiéme dimension, plusieurs processus ne sont pas complètement compris lors de la déformation à large échelle. L'un de ces processus est la propagation latérale des structures, par exemple la propagation de plis ou de mullions dans la direction perpendiculaire à l'axe de com-pression, ou la propagation des zones d'amincissement des boudins perpendiculairement à la direction d'extension. Nous sommes particulièrement intéressés les nappes de plis qui sont des nappes de charriage en forme de plis couché d'une amplitude plurikilométrique et étant formées par cisaillement ductile. La plupart du temps, elles exposent un sens de cisaillement constant et une augmentation non linéaire de la déformation vers la base du flanc inverse. Un exemple connu de nappes de plis est le domaine Helvétique dans les Alpes de l'ouest. Une de ces nap-pes est la Nappe de Morcles dont l'axe de pli plonge E-NE tandis que de l'autre côté de la dépression du Rawil (ou dépression du Wildstrubel), la nappe du Doldenhorn (équivalent de la nappe de Morcles) possède un axe de pli plongeant O-SO. La forme particulière de ces nappes est due à l'alternance de couches calcaires mécaniquement résistantes et de couches mécanique-ment faibles constituées de schistes et de marnes. Ces différences mécaniques dans les couches permettent d'expliquer les plissements internes à la nappe, particulièrement dans le flanc inver-se de la nappe de Morcles. Il faut également noter que le développement du flanc inverse des nappes n'est pas le même des deux côtés de la dépression de Rawil. Ainsi la nappe de Morcles possède un important flanc inverse alors que la nappe du Doldenhorn en est presque dépour-vue. A l'heure actuelle, aucune étude numérique en 3-D n'a été menée afin de comprendre la dynamique fondamentale de la formation des nappes de Morcles et du Doldenhorn ainsi que la formation de la dépression de Rawil. Ce travail propose la première analyse de l'évolution 3-D des instabilités géométriques et de la formation des nappes de plis en utilisant des simulations numériques. Notre modèle est basé sur la méthode des éléments finis (FEM) qui permet de ré-soudre avec précision les interfaces entre deux matériaux ayant des propriétés mécaniques très différentes (par exemple entre les couches calcaires et les couches marneuses). De plus nous utilisons un maillage lagrangien déformable avec une fonction de re-meshing (production d'un nouveau maillage). Grâce à cette méthode combinée il nous est possible de suivre avec précisi-on les interfaces matérielles et de résoudre avec précision les instabilités géométriques lors de la déformation de matériaux visco-élastiques décrit par une rhéologie non linéaire (n>1). Nous uti-lisons cet algorithme afin de comprendre la formation des nappes de plis, la propagation latérale du plissement ainsi que la propagation latérale des structures de type mullions causé par une va-riation latérale de la géométrie (p.ex graben). De plus l'algorithme est utilisé pour comprendre la dynamique 3-D de l'amincissement visqueux et de la rupture de la plaque descendante en zone de subduction. Les résultats obtenus sont comparés à des modèles 2-D et à la solution analytique 1-D. -- Viele drei dimensionale (3-D) Strukturen, die in Gesteinen vorkommen und durch die Verfor-mung der Erdkruste und Litosphäre entstanden sind werden von den unterschiedlichen mechani-schen Eigenschaften der Gesteinseinheiten kontrolliert und sind häufig das Resulat von geome-trischen Istabilitäten. Zu diesen strukturen zählen zum Beispiel Falten, Pich-and-swell Struktu-ren oder sogenannte Cusbate-Lobate Strukturen (auch Mullions). Diese Strukturen kommen in verschiedenen Grössenordungen vor und können Masse von einigen Zentimeter bis zu einigen Kilometer aufweisen. Die mit der Entstehung dieser Strukturen verbundenen Prozesse kontrol-lieren die Entstehung von Gerbirgen und Sediment-Becken sowie die Verformung des Kontaktes zwischen Grundgebirge und Stedimenten. Die zwei dimensionalen (2-D) Verformungs-Prozesse die zu den genannten Strukturen führen sind bereits sehr gut untersucht. Einige Prozesse wäh-rend starker 3-D Verformung sind hingegen noch unvollständig verstanden. Einer dieser 3-D Prozesse ist die seitliche Fortpflanzung der beschriebenen Strukturen, so wie die seitliche Fort-pflanzung von Falten und Cusbate-Lobate Strukturen senkrecht zur Verkürzungsrichtung und die seitliche Fortpflanzung von Pinch-and-Swell Strukturen othogonal zur Streckungsrichtung. Insbesondere interessieren wir uns für Faltendecken, liegende Falten mit Amplituden von mehr als 10 km. Faltendecken entstehen vermutlich durch duktile Verscherung. Sie zeigen oft einen konstanten Scherungssinn und eine nicht-lineare zunahme der Scherverformung am überkipp-ten Schenkel. Die Faltenachsen der Morcles Decke in der Westschweiz fallen Richtung ONO während die Faltenachsen der östicher gelegenen Doldenhorn Decke gegen WSW einfallen. Diese entgegengesetzten Einfallrichtungen charakterisieren die Rawil Depression (Wildstrubel Depression). Die Morcles Decke ist überwiegend das Resultat von Verkürzung und Scherung parallel zu den Sedimentlagen. Während der Verkürzung verhielt sich der massive Kalkstein kompetenter als der Umliegende Mergel und Schiefer, was zur Verfaltetung Morcles Decke führ-te, vorallem in gegen Norden eifallenden überkippten Schenkel. Die Doldenhorn Decke weist dagegen einen viel kleineren überkippten Schenkel und eine stärkere Lokalisierung der Verfor-mung auf. Bis heute gibt es keine 3-D numerischen Studien, die die fundamentale Dynamik der Entstehung von grossen stark verformten 3-D Strukturen wie den Morcles und Doldenhorn Decken sowie der damit verbudenen Rawil Depression untersuchen. Wir betrachten die 3-D Ent-wicklung von geometrischen Instabilitäten sowie die Entstehung fon Faltendecken mit Hilfe von numerischen Simulationen basiert auf der Finite Elemente Methode (FEM). Die Simulation von geometrischen Instabilitäten, die aufgrund von Änderungen der Materialeigenschaften zwischen verschiedenen Gesteinseinheiten entstehen, erfortert einen numerischen Algorithmus, der in der Lage ist die Materialgrenzen mit starkem Kontrast der Materialeigenschaften (zum Beispiel zwi-schen Kalksteineinheiten und Mergel) für starke Verfomung genau aufzulösen. Um dem gerecht zu werden kombiniert unser FE Algorithmus eine numerische Contour-Linien-Technik und ein deformierbares Lagranges Netz mit Re-meshing. Mit dieser kombinierten Methode ist es mög-lich den anfänglichen Materialgrenzen mit dem FE Netz genau zu folgen und die geometrischen Instabilitäten genügend aufzulösen. Der Algorithmus ist in der Lage visko-elastische 3-D Ver-formung zu rechnen, wobei die viskose Rheologie mit Hilfe eines power-law Fliessgesetzes beschrieben wird. Mit dem numerischen Algorithmus untersuchen wir die Entstehung von 3-D Faltendecken, die seitliche Fortpflanzung der Faltung sowie der Cusbate-Lobate Strukturen die sich durch die Verkürzung eines mit Sediment gefüllten Halbgraben bilden. Dabei werden die anfänglichen geometrischen Instabilitäten der Faltung exakt mit dem FE Netz aufgelöst wäh-rend die Materialgranzen des Halbgrabens die Finiten Elemente durchschneidet. Desweiteren wird der 3-D Algorithmus auf die Einschnürung während der 3-D viskosen Plattenablösung und Subduktion angewandt. Die 3-D Resultate werden mit 2-D Ergebnissen und einer 1-D analyti-schen Lösung verglichen.

Description and implementation of cut calculation-method in a sawmill

Työn tavoitteena oli kuvata ja ottaa käyttöön sahauseräkohtaisen kannattavuuden laskentamenetelmä sahalle, sekä tehdä laskentamalli menetelmän tueksi. Sahauksen peruskäsitteiden jälkeen työssä on esitelty sahan tuotantoprosessi. Tuotantoprosessi on kuvattu kirjallisuuden ja asiantuntijoiden haastattelujen perusteella. Seuraavaksi kartoitettiin hyötyjä ja vaikutuksia, mitä laskentamenetelmältä odotetaan.. Kustannuslaskennan teoriaa selvitettiin kirjallisuuslähteitä käyttäen silmälläpitäen juuri tätä kehitettävää laskentamenetelmää. Lisäksi esiteltiin Uimaharjun sahalla käytettävät ja laskentaan liittyvät laskenta- ja tietojärjestelmät.Nykyisin sahalla ei ole minkäänlaista menetelmää sahauseräkohtaisen tuloksen laskemiseksi. Pienillä muutoksilla sahan tietojärjestelmään ja prosessikoneisiin voidaan sahauserä kuljettaa prosessin läpi niin, että jokaisessa prosessin vaiheessa sille saadaan kohdistettua tuotantotietoa. Eri vaiheista saatua tietoa käyttämällä saadaan tarkasti määritettyä tuotteet, joita sahauserä tuotti ja paljonko tuotantoresursseja tuottamiseen kului. Laskentamalliin syötetään tuotantotietoja ja kustannustietoa ja saadaan vastaukseksi sahauserän taloudellinen tulos.Toimenpide ehdotuksena esitetään lisätutkimusta tuotantotietojen automaattisesta keräämisestä manuaalisen työn ja virheiden poistamiseksi. Suhteellisen pienillä panoksilla voidaan jokaiselle sahauserälle kerätä tuotantotiedot täysin automaattisesti. Lisäksi kehittämäni laskentamallin tilalle tulisi hankkia sovellus, joka käyttäisi paremmin hyväksi nykyisiä tietojärjestelmiä ja poistaisi manuaalisen työvaiheen laskennassa.

Radiaalikompressorin spiraalivirtauksen numeerinen mallinnus

Diplomityössä mallinnetaan numeerisesti radiaalikompressorin spiraalin virtaus. Spiraalin tarkoituksena radiaalikompressorissa on kerätä tasaisesti virtaus diffuusorin kehältä. Spiraaliin on viime aikoina kiinnitetty enemmän huomiota, koska on havaittu, että kompressorin hyötysuhdetta voidaan parantaa spiraalia optimoimalla. Spiraalin toimintaa tarkastellaan kolmella eri massavirralla. Työn alussa käsitellään spiraalin toimintaperiaatteita. Numeerisena ratkaisijana käytetään Teknillisessä korkeakoulussa kehitettyä FIN-FLO -koodia. FINFLO -laskentaohjelmassa ratkaistaan Navier-Stokes yhtälöt kolmeulotteiselle laskenta-alueelle. Diskretointi perustuu kontrollitilavuus menetelmään. Työssä käsitellään laskentakoodin toimintaperiaatteitta. Turbulenssia mallinnetaan algebrallisella Baldwin-Lomaxin ja kahden yhtälön Chienin k-e turbulenssimalleilla. Laskentatuloksia verrataan Lappeenrannan teknillisessä korkeakoulussa tehtyihin mittauksiin kyseessä olevasta kompressorin spiraalista. Myös eri turbulenssimalleilla ja hilatasoilla saatuja tuloksia verrataan keskenään. Laskentatuloksien jälkikäsittelyä varten ohjelmoitiin neljä eri tietokoneohjelmaa. Laskennalla pyritään saamaan lisäselvyyttä virtauksen käyttäytymiseen spiraalissa ja erityisesti ns. kielen alueella. Myös kahden eri turbulenssimallin toimivuutta kompressorin numeerisessa mallinnuksessa tutkitaan.

Mesoscale numerical analysis of the historical November 1982 heavy precipitation event over Andorra (Eastern Pyrenees)

From 6 to 8 November 1982 one of the most catastrophic flash-flood events was recorded in the Eastern Pyrenees affecting Andorra and also France and Spain with rainfall accumulations exceeding 400 mm in 24 h, 44 fatalities and widespread damage. This paper aims to exhaustively document this heavy precipitation event and examines mesoscale simulations performed by the French Meso-NH non-hydrostatic atmospheric model. Large-scale simulations show the slow-evolving synoptic environment favourable for the development of a deep Atlantic cyclone which induced a strong southerly flow over the Eastern Pyrenees. From the evolution of the synoptic pattern four distinct phases have been identified during the event. The mesoscale analysis presents the second and the third phase as the most intense in terms of rainfall accumulations and highlights the interaction of the moist and conditionally unstable flows with the mountains. The presence of a SW low level jet (30 m s-1) around 1500 m also had a crucial role on focusing the precipitation over the exposed south slopes of the Eastern Pyrenees. Backward trajectories based on Eulerian on-line passive tracers indicate that the orographic uplift was the main forcing mechanism which triggered and maintained the precipitating systems more than 30 h over the Pyrenees. The moisture of the feeding flow mainly came from the Atlantic Ocean (7-9 g kg-1) and the role of the Mediterranean as a local moisture source was very limited (2-3 g kg-1) due to the high initial water vapour content of the parcels and the rapid passage over the basin along the Spanish Mediterranean coast (less than 12 h).

High-Speed Solid-Rotor Induction Machine – Electromagnetic Calculation and Design

Within the latest decade high-speed motor technology has been increasingly commonly applied within the range of medium and large power. More particularly, applications like such involved with gas movement and compression seem to be the most important area in which high-speed machines are used. In manufacturing the induction motor rotor core of one single piece of steel it is possible to achieve an extremely rigid rotor construction for the high-speed motor. In a mechanical sense, the solid rotor may be the best possible rotor construction. Unfortunately, the electromagnetic properties of a solid rotor are poorer than the properties of the traditional laminated rotor of an induction motor. This thesis analyses methods for improving the electromagnetic properties of a solid-rotor induction machine. The slip of the solid rotor is reduced notably if the solid rotor is axially slitted. The slitting patterns of the solid rotor are examined. It is shown how the slitting parameters affect the produced torque. Methods for decreasing the harmonic eddy currents on the surface of the rotor are also examined. The motivation for this is to improve the efficiency of the motor to reach the efficiency standard of a laminated rotor induction motor. To carry out these research tasks the finite element analysis is used. An analytical calculation of solid rotors based on the multi-layer transfer-matrix method is developed especially for the calculation of axially slitted solid rotors equipped with wellconducting end rings. The calculation results are verified by using the finite element analysis and laboratory measurements. The prototype motors of 250 – 300 kW and 140 Hz were tested to verify the results. Utilization factor data are given for several other prototypes the largest of which delivers 1000 kW at 12000 min-1.

Calculation analysis of energy saving tools for fan and pump applications

This thesis analyses the calculation of FanSave and PumpSave energy saving tools calculation. With these programs energy consumption of variable speed drive control for fans and pumps can be compared to other control methods. With FanSave centrifugal and axial fans can be examined and PumpSave deals with centrifugal pumps. By means of these programs also suitable frequency converter can be chosen from the ABB collection. Programs need as initial values information about the appliances like amount of flow and efficiencies. Operation time is important factor when calculating the annual energy consumption and information about it are the length and profile. Basic theory related to fans and pumps is introduced without more precise instructions for dimensioning. FanSave and PumpSave contain various methods for flow control. These control methods are introduced in the thesis based on their operational principles and suitability. Also squirrel cage motor and frequency converter are introduced because of their close involvement to fans and pumps. Second part of the thesis contains comparison between results of FanSave’s and PumpSave’s calculation and performance curve based calculation. Also laboratory tests were made with centrifugal and axial fan and also with centrifugal pump. With the results from this thesis the calculation of these programs can be adjusted to be more accurate and also some new features can be added.

CNDOL: A fast and reliable method for the calculation of electronic properties of very large systems. Applications to retinal binding pocket in rhodopsin and gas phase porphine.

Very large molecular systems can be calculated with the so called CNDOL approximate Hamiltonians that have been developed by avoiding oversimplifications and only using a priori parameters and formulas from the simpler NDO methods. A new diagonal monoelectronic term named CNDOL/21 shows great consistency and easier SCF convergence when used together with an appropriate function for charge repulsion energies that is derived from traditional formulas. It is possible to obtain a priori molecular orbitals and electron excitation properties after the configuration interaction of single excited determinants with reliability, maintaining interpretative possibilities even being a simplified Hamiltonian. Tests with some unequivocal gas phase maxima of simple molecules (benzene, furfural, acetaldehyde, hexyl alcohol, methyl amine, 2,5 dimethyl 2,4 hexadiene, and ethyl sulfide) ratify the general quality of this approach in comparison with other methods. The calculation of large systems as porphine in gas phase and a model of the complete retinal binding pocket in rhodopsin with 622 basis functions on 280 atoms at the quantum mechanical level show reliability leading to a resulting first allowed transition in 483 nm, very similar to the known experimental value of 500 nm of "dark state." In this very important case, our model gives a central role in this excitation to a charge transfer from the neighboring Glu(-) counterion to the retinaldehyde polyene chain. Tests with gas phase maxima of some important molecules corroborate the reliability of CNDOL/2 Hamiltonians.

Qualidade de medições em química analítica. Estudo de caso: determinação de cádmio por espectrofotometria de absorção atômica com chama

Calculation of uncertainty of results represents the new paradigm in the area of the quality of measurements in laboratories. The guidance on the Expression of Uncertainty in Measurement of the ISO / International Organization for Standardization assumes that the analyst is being asked to give a parameter that characterizes the range of the values that could reasonably be associated with the result of the measurement. In practice, the uncertainty of the analytical result may arise from many possible sources: sampling, sample preparation, matrix effects, equipments, standards and reference materials, among others. This paper suggests a procedure for calculation of uncertainties components of an analytical result due to sample preparation (uncertainty of weights and volumetric equipment) and instrument analytical signal (calibration uncertainty). A numerical example is carefully explained based on measurements obtained for cadmium determination by flame atomic absorption spectrophotometry. Results obtained for components of total uncertainty showed that the main contribution to the analytical result was the calibration procedure.

Calculation of the variance in surveys of the economic climate

Public opinion surveys have become progressively incorporated into systems of official statistics. Surveys of the economic climate are usually qualitative because they collect opinions of businesspeople and/or experts about the long-term indicators described by a number of variables. In such cases the responses are expressed in ordinal numbers, that is, the respondents verbally report, for example, whether during a given trimester the sales or the new orders have increased, decreased or remained the same as in the previous trimester. These data allow to calculate the percent of respondents in the total population (results are extrapolated), who select every one of the three options. Data are often presented in the form of an index calculated as the difference between the percent of those who claim that a given variable has improved in value and of those who claim that it has deteriorated.

Eddy-current losses in rotor surface mounted permanent magnets

The aim of the work is to study the existing analytical calculation procedures found in literature to calculate the eddy-current losses in surface mounted permanent magnets within PMSM application. The most promising algorithms are implemented with MATLAB software under the dimensional data of LUT prototype machine. In addition finite elements analyze, utilized with help of Flux 2D software from Cedrat Ltd, is applied to calculate the eddy-current losses in permanent magnets. The results obtained from analytical methods are compared with numerical results.

Numerical Dominance of the Argentine Ant vs Native Ants and Consequences on Soil Resource Searching in Mediterranean Cork-Oak Forests (Hymenoptera: Formicidae)

This study is focused on the dominance exerted by the invasive Argentine ant over native ants in a coastal Mediterranean area. Theimpact of this invasive ant on native ant assemblages and its consequences on total ant biomass and on the intensity of habitat explorationwere evaluated. Foraging ants were observed and their trajectories recorded during 5-minute periods in two study zones, one invaded andthe other non-invaded. Ant species detected, ant worker abundance, ant biomass and the intensity of soil surface searching done by antswere compared between the two zones. The Argentine ant invasion provoked a drastic reduction of the ant species richness. Apparentlyonly one native ant species is able to coexist with the Argentine ant, the cryptic Plagiolepis pygmaea. Ant worker abundance was also modified after the invasion: the number of Argentine ant workers detected, which represented 92% of the invaded zone, was two times higher than the number of native ant workers detected in the non-invaded zone. The total ant biomass was inversely affected, becoming four times lower in the invaded zone highly dominated by Linepithema humile. The higher number of Argentine ant workers and their fast tempo of activity implied an alteration of the intensity of soil surface searching: scanning by the Argentine ants in the invaded zone was higher than that done by the native ants in the non-invaded zone, and the estimated time for a complete soil surface scan was 64 minutes in the invaded zone and 108 minutes in the non-invaded zone. Consequently, resources will be discovered faster by ants in the invaded zone than in the non-invaded zone. The increase of the mean temperature and the decrease of the relative humidity from May to August reduced the ant activity in the two study zones but this reduction was greater in the invaded zone

New approximation to the third-order density : application to the calculation of correlated multicenter indices

In this work we present the formulas for the calculation of exact three-center electron sharing indices (3c-ESI) and introduce two new approximate expressions for correlated wave functions. The 3c-ESI uses the third-order density, the diagonal of the third-order reduced density matrix, but the approximations suggested in this work only involve natural orbitals and occupancies. In addition, the first calculations of 3c-ESI using Valdemoro's, Nakatsuji's and Mazziotti's approximation for the third-order reduced density matrix are also presented for comparison. Our results on a test set of molecules, including 32 3c-ESI values, prove that the new approximation based on the cubic root of natural occupancies performs the best, yielding absolute errors below 0.07 and an average absolute error of 0.015. Furthemore, this approximation seems to be rather insensitive to the amount of electron correlation present in the system. This newly developed methodology provides a computational inexpensive method to calculate 3c-ESI from correlated wave functions and opens new avenues to approximate high-order reduced density matrices in other contexts, such as the contracted Schrödinger equation and the anti-Hermitian contracted Schrödinger equation