Chapter 8. Application 2 â Modeling Molecular Dissociation with ...
Chapter 8. Application 2 â Modeling Molecular Dissociation with ...
Chapter 8. Application 2 â Modeling Molecular Dissociation with ...
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49and X is the bond displacement coordinate,The term€€δ t − nTX = r − r e . (<strong>8.</strong>4)δ( t − nT) in Equation (<strong>8.</strong>2) is known as a delta function and has the following behavior:( ) =⎧ ∞, if t = nT⎫⎨⎬⎩ 0, otherwise ⎭ . (<strong>8.</strong>5)This function represents the laser pulse. It is assumed that the duration of the laser pulse is veryshort compared to the time T between laser pulses, so that each pulse appears as a big (infinite)spike. The system€evolves in the potential V ( X) between laser pulses. The laser pulse effectivelygives a big kick to the bond and can lead to dissociation.Discretizing Newton’s equations of motion for this system leads to a nonlinear map of theform€P n+1 = P n − T dV ( X)dXX n+1 = X n + T (<strong>8.</strong>6)µ P n+1 .In the case of the He-I 2van der Waals complex, the potential is assumed to be a Morse oscillator,€V ( X) = D( 1 − e −aX) 2 . (<strong>8.</strong>7)The map may be simplified somewhat by redefining the variables,€p = aT P and q = aX . (<strong>8.</strong>8)µThis leads to a nonlinear van der Waals map given by€p n+1 = p n + d e −2q n − e −q n( )q n+1 = q n + p n+1 .(<strong>8.</strong>9)In Equation (<strong>8.</strong>9), the parameter d is defined as€d = 2a2 T 2 D. (<strong>8.</strong>10)µ€
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