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Javier Junquera

Introducci ón a la asignatura de Computación Avanzada Grado en Física. Javier Junquera. Datos identificativos de la asignatura. Bibliography:. M. P. Allen and D. J. Tildesley Computer Simulation of Liquids Oxford Science Publications ISBN 0 19 855645 4. How to reach me.

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Javier Junquera

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  1. Introducción a la asignatura de ComputaciónAvanzada Grado en Física Javier Junquera

  2. Datos identificativos de la asignatura

  3. Bibliography: M. P. Allen and D. J. Tildesley ComputerSimulation of Liquids Oxford SciencePublications ISBN 0 19 855645 4

  4. How to reach me Javier Junquera Ciencias de la Tierra y Física de la Materia Condensada Facultad de Ciencias, Despacho 3-12 E-mail: javier.junquera@unican.es URL: http://personales.unican.es/junqueraj In the web page, you can find: - The program of the course - Slides of the different lecture - The code implementing the simulation of a liquid interacting via a Lennard-Jones potential Office hours: - At the end of each lecture - At any moment, under request by e-mail

  5. Physical problem to be solved during this course Given a set of classicalparticles (atomsormolecules) whosemicroscopicstatemay be specified in terms of: - positions - momenta Note thattheclassicaldescription has to be adequate. Ifnotwe can notspecify at thesame time thecoordinates and momenta of a givenmolecule and whoseHamiltonianmay be written as the sum of kinetic and potentialenergyfunctions of the set of coordinates and momenta of eachmolecule Solvenumerically in thecomputertheequations of motionwhichgovernsthe time evolution of thesystem and allitsmechanicalproperties

  6. Note about the generalized coordinates May be simplythe set of cartesiancoordinates of eachatomornucleus in thesystem Sometimesitis more usefultotreatthemolecule as a rigidbody. In this case, willconsist of: - theCartesiancoordinates of the center of mass of eachmolecule - togetherwith a set of variables thatspecifythe molecular orientation In any case, stands fortheappropriate set of conjugatemomenta

  7. Kinetic and potential energy functions Usuallythekineticenergytakestheform molecular mass runsoverthedifferent components of themomentum of themolecule Thepotentialenergycontainstheinterestinginformationregarding intermolecular interactions

  8. Potential energy function of an atomic system Consider a systemcontainingatoms. Thepotentialenergymay be dividedintotermsdependingonthecoordinates of individual, pairs, triplets, etc. Expectedto be small Onebodypotential Particleinteractions Onebodypotential Representstheeffect of anexternalfield (including, forexample, thecontainedwalls) Pairpotential Dependsonlyonthemagnitude of thepairseparation Threeparticlepotential Significant at liquiddensities. Rarelyincluded in computersimulations (very time consumingon a computer) Thenotationindicates a summationoverall distinctpairswithoutcontaininganypairtwice. Thesamecaremust be takenfortriplets, etc.

  9. The effective pair potential Thepotentialenergymay be dividedintotermsdependingonthecoordinates of individual, pairs, triplets, etc. Thepairwiseapproximationgives a remarkablygooddescriptionbecausetheaveragethreebodyeffects can be partiallyincludedbydefiningan “effective” pairpotential Thepairpotentialsappearing in computersimulations are generallyto be regarded as effectivepairpotentials of thiskind, representingmany-bodyeffects A consequence of thisapproximationisthattheeffectivepairpotentialneededto reproduce experimental data mayturnouttodependonthedensity, temperature, etc. whilethe true two-bodypotentialdoesnot.

  10. Example of ideal effective pair potentials Hard-spherepotential Square-wellpotential Discovery of non-trivial phasetransitions, notevidentjustlookingtheequations Softspherepotential (ν=1) Softspherepotential (ν=12) No attractivepart No attractivepart

  11. It is useful to divide realistic potentials in separate attractive and repulsive components Attractiveinteraction Van der Waals-London orfluctuatingdipoleinteraction Classicalargument C. Kittel Introductionto Solid StatePhysics (3rd Edition) John Wiley and sons Electric fieldproducedbydipole 1 on position 2 Instantaneousdipoleinducedbythisfieldon 2 Potentialenergy of thedipolemoment Alwaysattractive J. D. Jackson ClassicalElectrodynamics (Chapter 4) John Wiley and Sons Istheunitvector directedfrom 1 to 2

  12. It is useful to divide realistic potentials in separate attractive and repulsive components Attractiveinteraction Van der Waals-London orfluctuatingdipoleinteraction Quantumargument Hamiltonianfor a system of twointeractingoscillators Wheretheperturbativetermisthedipole-dipoleinteraction Fromfirst-orderperturbationtheory, we can compute thechange in energy C. Kittel Introductionto Solid StatePhysics (3rd Edition) John Wiley and sons

  13. It is useful to divide realistic potentials in separate attractive and repulsive components Repulsiveinteraction As thetwoatoms are broughttogether, theirchargedistributiongraduallyoverlaps, changingtheenergy of thesystem. Theoverlapenergyisrepulsiveduetothe Pauli exclusionprinciple: No twoelectrons can havealltheir quantum numbersequal Whenthecharge of thetwoatomsoverlapthereis a tendeyncforelectronsfromatom B tooccupy in partstates of atom A alreadyoccupiedbyelectrons of atom A and viceversa. Electrondistribution of atomswithclosedshells can overlaponlyifaccompaniedby a partialpromotion of electronstohigherunoccpiedlevels Electronoverlapincreasesthe total energy of thesystem and gives a repulsivecontributiontotheinteraction

  14. The repulsive interaction is exponential Born-Mayer potential

  15. It is useful to divide realistic potentials in separate attractive and repulsive components Buckingham potential F. Jensen IntroductiontoComputationalChemistry John Wiley and Sons Becausetheexponentialterm converges to a constant as , whiletheterm diverges, the Buckingham potential“turnsover” as becomessmall. Thismay be problematicwhendealingwith a structurewithvery short interatomicdistances

  16. Comparison of effective two body potentials Buckingham potential Lennard-Jones Morse F. Jensen IntroductiontoComputationalChemistry John Wiley and Sons

  17. The Lennard-Jones potential Thewelldepthisoftenquoted as in units of temperature, whereistheBoltzmann’sconstant Forinstance, tosimulateliquidArgon, reasonablevalues are: Wemustemphasizethatthese are notthevalueswhichwouldapplytoanisolatedpair of argonatoms

  18. The Lennard-Jones potential Thewelldepthisoftenquoted as in units of temperature, whereistheBoltzmann’sconstant Suitableenergy and lengthparametersforinteractionsbetweenpairs of identicalatoms in differentmolecules WARNING: Theparameters are notdesignedto be transferable: the C atomparameters in CS2 are quite differentfromthevaluesappropriateto a C in graphite

  19. Is realistic the Lennard-Jones potential? Dashed line: 12-6 effectiveLennard-Jones potentialforliquid Ar Solid line: Bobetic-Barker-Maitland-Smith pairpotentialforliquid Ar (derivedafterconsidering a largequantity of experimental data) Lennard-Jones Attractivetail at largeseparations, duetocorrelationbetweenelectroncloudssurroundingtheatoms Steeplyrisingrepulsivewall at short distances, dueto non-bondedoverlapbetweentheelectronclouds Optimal

  20. Separation of the Lennard-Jones potential into attractive and repulsive components Attractivetail at largeseparations, duetocorrelationbetweenelectroncloudssurroundingtheatoms Steeplyrisingrepulsivewall at short distances, dueto non-bondedoverlapbetweentheelectronclouds

  21. Separation of the Lennard-Jones potential into attractive and repulsive components: energy scales repulsive attractive

  22. Beyond the two body potential: the Axilrod-Teller potential Axilrod-Tellerpotential: Threebodypotentialthatresultsfrom a third-orderperturbationcorrectiontotheattractive Van der Waals-London dipersioninteractions

  23. For ions or charged particles, the long range Coulomb interaction has to be added Where are thechargesof ions and , and isthepermittivity of free space

  24. Calculating the potential

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