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Water molecule39Lagrange multipliersJ.P. Ryckaert, G. Ciccotti and H.J.C. Berendsen, 1977Let us just consider fixed bond lengths for adiatomic molecule (e.g. water).Atom i (i = 1,2,3) follows the equation:mx f gi i i iforces from other moleculesforces toimposeconstraints40
Lagrange multipliersConstraints that bonds have length d 12 and d 23 :rd2 212 12 12rd2 223 23 2300with rij r , r x xij ij i j 1 1 gk 12 xk 12 23xk 232 2 where λ 12 and λ 23are the yet undetermined Lagrange multipliers.g r , g r r , g r1 12 12 2 23 23 12 12 3 23 2341Lagrange multipliersExecute theVerlet algorithmin two steps:insert g i ! 2 fixi( tt) 2 xi( t) xi( tt) tmi ! 2 gixi( tt) xi( tt) tm x ( tt) x ( tt) t r ( t)! 2 121 1 12m1 x ( tt) x ( tt) t r ( t) t r ( t)! 2 23 2 122 2 23 12m2 m2 x ( tt) x ( tt) t r ( t)! 2 233 3 23m3i42
Page 1 and 2: Guidance on Impact AssessmentsDocum
Page 3 and 4: Potentialshard core interaction 0.3
Page 5 and 6: PotentialsLenard Jones potentialε
Page 7 and 8: Contact timeone dimensionenergy:122
Page 9 and 10: Verlet method• One needs to store
Page 11 and 12: Total energy21Small perturbationdiv
Page 13 and 14: TricksMost time consuming loop is c
Page 15 and 16: Verlet tablesThe Verlet table LIST
Page 17 and 18: Linked cell methodexamplecell 1 cel
Page 19: MoleculesMolecules are composed of
Page 23 and 24: Lagrange multipliersFirst solve ter
Page 25 and 26: Contact Dynamics (CD)P. Lötstedt (
Page 27 and 28: Example: two particles (1D)Matrix H
Page 29 and 30: Example: two particles (1D)No overl
Page 31 and 32: Contact DynamicsCoulomb-Graph:Frict
Page 33 and 34: Contact Dynamics ext1F 1 1 1
Page 35 and 36: Contact DynamicsPerform direct inve
Page 37 and 38: Non-spherical objectsUp to now: ato
Page 39 and 40: Two dimensionsTime evolution of the
Page 41 and 42: 3d rigid body rotationangular momen
Page 43 and 44: Euler anglesRotations in three dime
Page 45 and 46: AOne can show:Quaternions 2 2
Page 47 and 48: Long-range potentialsLong-range are
Page 49 and 50: Ewald summationThe Ewald sum is onl
Page 51 and 52: Particle - Mesh algorithmJ.W. Eastw
Page 53 and 54: Particle - Mesh algorithmPM algorit
Page 55 and 56: Multipole expansion„Fast Multipol
Page 57 and 58: Canonical EnsembleTill now we consi
Page 59 and 60: Velocity rescalingThe main problem
Page 61 and 62: Constraint methodThis method is sim
Page 63 and 64: Nosé - Hoover thermostatThis new d
Page 65 and 66: Nosé - Hoover thermostatCoupling t
Page 67 and 68: Nosé - Hoover thermostatnormalener
Page 69 and 70: Constant pressureKeep pressure pfix
Page 71 and 72:
givingequationsof motion:NPT ensemb
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