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Let us give a nice illustration (see [9] and references therein for more details) of the use of the previous formulas in the domain of molecular dynamics Let us consider a diatomic molecule (eg, O2 , NO, N2 ) hit by an inert atom (eg, He) The Hamiltonian of the system reads as H = H target + H projectile + H int , where H int (R, q) = V (R, q) V ( , q) , R being the relative position of the projectile with respect to the center of mass of the molecule, whereas q = q(t) is the position operator corresponding to the displacement of the intermolecular relative position with respect to the equilibrium Within the usual framework of the semiclassical approach, namely, the Born Oppenheimer approximation, the trajectory of the projectile R = R(t) is considered as classical, and we can deal with the series expansion of H int in terms of the small displacement q: 1 H int = h 0 (t) + h 1 (t)q + h 2 (t)q 2 + 2 By putting q = m (a+a ) 2 and neglecting terms beyond the quadratic ones, we get the following expression for the Hamiltonian: 1 H = M R2 + 2 + 2m + m B(t) a a + 1 2



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B(t)(a 2 + a )

(R25)

50,000 GDP per Capita at PPP*

(1030)

From this (approximate) Hamiltonian it becomes possible to determine, for instance, the probability of the target of changing its vibrational ground state to an excited state |n after the collision The calculation goes through the S-matrix formalism: | out = | (t = ) = S | in = S | (t = ) Thus the S matrix is the limit (in a certain functional sense) S = lim U (t, t 0 )

Internal Controls and Procedures Independent Auditing The Role of the User of Financial Statements Conclusion

(1031)

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of the evolution operator associated with the Hamiltonian through the equation i U (t, t 0 ) = H(t)U (t, t 0 ) , t U (t 0 , t 0 ) = I d (1032)

In dealing with this equation, one should consider the constraint resulting from the conservation of the classical energy:

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Defective Construction Work building has to be emptied while repairs are carried out, then this may result in loss of use, which may also be the subject of an award of compensation. Sometimes building contracts make provision for the payment of compensation in speci c circumstances (for example, the JCT conditions provide for the payment of direct loss and/or expense where the purchaser is responsible for a material effect on regular progress), but unless the contract contains clear words to the contrary the right to claim damages will not be excluded by the inclusion of such a provision (see Gilbert-Ash). At one time the approach to the assessment of damage to property was based on the diminution in value, but this approach has generally though not exclusively given way to assessment based on the cost of reinstatement. Where the cost of reinstatement is not the appropriate measure of damages, diminution in value may instead be the basis of assessment in appropriate cases. Nevertheless, there may be circumstances when both reinstatement and diminution are recoverable.

1 2M

(1033)

.

Fig 104 Transition probabilities for excitations of H2 by He scattering (a) Transition probabilities P 0 n f (n f = 0 to 6) as a function of the total energy E tot of the He + H2 system All curves are drawn smoothly through the data points of Gilmore and Yuan [149, 150] and denote results obtained by Gazdy and Micha [151, 152]

85000.00 120000.00 95000.00 75000.00 34000.00

(b) Transition probabilities P n i n f as a function of n f for E = 12 and 16 The three plots on the left are for E = 12 and those on the right are for E = 16 [149, 150] (reprinted with permission from [Zang, W-M, Feng, D-H, and Gilmore, R, Rev Mod Phys, 26, 867, 1990], American Institute of Physics)

The function R(t) is determined classically by taking into account a loss of kinetic energy suitably smoothed during the collision Once R(t) is known, one proceeds to the numerical integration of the 4 ~ 4-matrix version of (1032): i M[U (t, t 0 )] = M[H(t)] M[U (t, t 0 )] , t M[U (t 0 , t 0 )] = I 4 (1034)

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