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Theoryof Molccular Dynamics Similations

http://www.ch.cmbnctorg/MD_tutoria!/pagcs/MD.Partl .html

r(t + dt) = r(/) + v(t)dt + -a(t)d^

Zł

r(t - dt) = r(t) - \{t)dt + ^a(t)dt²

Zł

Summing ihcsc two cquabons. one obUms

r(t + <5f) i 2r(t) - r(t - ót) + a(t)ót²

The Vcriet algorithm uses poskions and acceieralions at time t and the poskions from timc t-dt to calcu latc ncw poskions at tkne t+dt. The Verlct algorithm uses no cxphcit vclocilics. The advantages of the Vcrlcl algorillim arc, I) i u straightforward, and II) the storagc rcquircmcnts aic modest . The disadvantagc a that the algpnthm is of modemie preckion.

The Lcap-frog algorithm

r(t + ót) = r(t)+im + - Ót j Ót

+—ót j    - — 0/ j + a(t)ót

In thk algorithm, the vcłockics arc firat calcuhled at time t+l/2dt\ thcsc arc uscd to calcuhtc the poskions, r, at tkne t+dt. In thk way, the vebckics leap over the poskions, then the poskions leap over the velockies. The advantagc of thk algpnthm k that the veiocibes arc cxpBdtly calcuhled. howcvcr, the dkadvantagc k that they arc not caicuhted at the same time as the poskions. The yelpddcs at tkne t can be approxknatcd by the rchtnnship:

The Yclocity Verlet algorithm

Thk algorithm yiclds poskions, velockies and accelcrabons at time t. Thcrc k no compromke on preckion.

11    r(t i ót) i r(t) i v(t)ót + —a(t)ót*

k i ót) 1 gil I ~[rt(f) | a(t i <5f )]<fr 2

8*9

2009-03-05 08:57