Computational Physics

9 Quantum molecular dynamics 9.1 Introduction In the previous chapter we considered systems of interacting particles. They were ...

264 Quantum molecular dynamics Energy (a.u.) 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 X –1.13 –1.125 –1.12 –1.115 –1.11 –1.105 –1.1 –1. ...

9.1 Introduction 265 becomesνvibr =13.64× 1013 Hz, to be compared with the experimental value νvibr=12.48× 1013 Hz[1].^1 The har ...

266 Quantum molecular dynamics The (classical) force on nucleusnis given as the negative gradient∇nof the energy with respect to ...

9.2 The molecular dynamics method 267 method by recalling the energy functionals of the Hartree–Fock and the density functional ...

268 Quantum molecular dynamics In density functional theory the energy can be written as a function of the ground state density, ...

9.2 The molecular dynamics method 269 so that the energy can be written in terms of theCrkandS: Etot=Etot({Crk},S). (9.19) As th ...

270 Quantum molecular dynamics have the opportunity to adapt themselves to the changing nuclear configuration at any time), ther ...

9.2 The molecular dynamics method 271 of the Fock matrix (seeSection 4.5.2and above): forψ ̈k=0, Eq. (9.21) reduces to an eigenv ...

272 Quantum molecular dynamics molecule 9.3 An example: quantum molecular dynamics for the hydrogen In this subsection we work o ...

9.3 An example: quantum molecular dynamics for the hydrogen molecule 273 5 10 15 20 25 30 35 40 45 Energy –2.1 –2.05 –2 –1.95 –1 ...

274 Quantum molecular dynamics ofX=1 the energy tends to−2.078 547 6 a.u., the same value as was found in Problem 4.9. 9.3.2 The ...

9.3 An example: quantum molecular dynamics for the hydrogen molecule 275 Taking the derivative with respect toXwe find d dX 〈1s, ...

276 Quantum molecular dynamics with t= (α+β)(γ+δ) α+β+γ+δ (PQ)^2 , (9.44) withRPas given above and RQ= γRC+δRD γ+δ , (9.45) and ...

9.3 An example: quantum molecular dynamics for the hydrogen molecule 277 1.35 1.36 1.37 1.38 1.39 1.4 1.41 1.42 0 50 100 150 200 ...

278 Quantum molecular dynamics 1.35 1.36 1.37 1.38 1.39 1.4 1.41 1.42 1.43 0 50 100 150 200 250 300 X Time Figure 9.4. The chang ...

9.4 Orthonormalisation; conjugate gradient and RM-DIIS techniques 279 these values depends on the particular integration algorit ...

280 Quantum molecular dynamics the smooth evolution of the orbitals! However, if the rotation is always close to the unit transf ...

9.4 Orthonormalisation; conjugate gradient and RM-DIIS techniques 281 The equation of motion then reads μψ ̈k= ∑ l (^) klψl (9.5 ...

282 Quantum molecular dynamics which is repeated until the orbitals do not change any more. By inspection of this algorithm it i ...