Computational Physics

9.4 Orthonormalisation; conjugate gradient and RM-DIIS techniques 283 Now we may proceed in two ways: (i) Calculate as a first g ...

284 Quantum molecular dynamics does not preserve orthonormality if we are not at the energy minimum; therefore, in the simulatio ...

9.4 Orthonormalisation; conjugate gradient and RM-DIIS techniques 285 descent direction: ζk=− 2 [ Hψk− ∑ l 〈ψk|H|ψl〉ψk ] . (9.75 ...

286 Quantum molecular dynamics system and that the negative charge cloud would therefore oscillate more and more violently aroun ...

9.4 Orthonormalisation; conjugate gradient and RM-DIIS techniques 287 feature ofEq. (9.81)for us is that the correction factor c ...

288 Quantum molecular dynamics The resulting states|ψnJ〉are not yet orthogonal; an orthogonalisation of the states must be perfo ...

9.5 Implementation of the Car–Parrinello technique for pseudopotential DFT 289 nuclear positions remain fixed, the method scales ...

290 Quantum molecular dynamics parts of the pseudopotential, and the electrostatic energy. The expressions are: ∇RnElocal=− ∑ K ...

Exercises 291 and the normalisation constraint 〈ψ|ψ〉= ∑ rs CrCs〈χr|χs〉= ∑ rs CrCsSrs are given by μC ̈r= 2 ∑ s (HrsCs− SrsCs). W ...

292 Quantum molecular dynamics must be minimised subject to the normalisation constraint ∑ rs CrCsSrs=1. The Lagrangian function ...

References 293 Apply the conjugate gradients method to this problem and compare the results with the matrix diagonalisation meth ...

294 Quantum molecular dynamics [22] T. A. Arias, J. D. Joannopoulos, and M. C. Payne, ‘Ab initio molecular-dynamics techniques e ...

10 The Monte Carlo method 10.1 Introduction In Chapter 8 we saw how a classical many-particle system can be simulated by the MD ...

296 The Monte Carlo method Direct Monte Carlo is a powerful method which can be applied to a wide variety of problems inside and ...

10.2 Monte Carlo integration 297 exact integral. We calculate the variance in the result: σ^2 = 〈( b−a N ∑N i= 1 fi ) 2 〉 − (〈 b ...

298 The Monte Carlo method Several methods have been devised to reduce the error of the Monte Carlo integ- rationmethod;foradisc ...

10.3 Importance sampling through Markov chains 299 does not affect the sum. In fact, it is possible to generate artificial numbe ...

300 The Monte Carlo method from uncorrelated chains – it is defined in terms of the transition probabilityT(X→ X′)for having the ...

10.3 Importance sampling through Markov chains 301 Let us introduce the functionρ(X,t)which gives us the probability of occurren ...

302 The Monte Carlo method In the first stage, given a stateX, we propose a new stateX′with a probability given byωXX′. In the s ...