Time dependent quantum theory of reactive molecular collisions
This lecture describes, in a pedagogic manner, the basic theory used in time dependent quantum reactive scattering theory. The calculations are based on propagating a prepared wavepacket forward in time using the time dependent Schrödinger equation. The a
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Summary. This lecture describes, in a pedagogic manner, the basic theory used in time dependent quantum reactive scattering theory. The calculations are based on propagating a prepared wavepacket forward in time using the time dependent Schrodinger equation. The advantages of the theory lie in the fact that a single time dependent propagation of the wavepacket provides the cross sections over a wide range of energies.
1. Wavepacket propagation: Solving the time-dependent Schrodinger equation The time-dependent Schrodinger equation (TDSE) is
a
A
(1.1 )
in at p(x, t) = HlJf(x, t). If the hamiltonian equation becomes
lJf(x, t
+ T)
II is not dependent on the time t, the solution of the above
= exp
-iih) (-n-
p(x, t).
(1.2)
The equation enables us to relate the wavefunction at time t + T to that at time t. It is only valid if the hamiltonian is not an explicit function of time. The most straightforward way to see what is implied by this equation is to expand the exponential on the right hand side of eq. (1.2). exp
-iIIT) ( -n- lJf(x, t)
('~ T) _;, (~T )'
{ 1_ i
+ 3!
HT HT A)3 1 (A)4} (n + 4f n ... lJf(x, t). (1.3)
This expansion shows clearly what is needed to solve the TDSE. We must be able to operate with the hamiltonian operator on the wavefunction to generate a new function. W. Jakubetz (ed.), Methods in Reaction Dynamics © Springer-Verlag Berlin Heidelberg 2001
2
Gabriel G. Balint-Kurti
4>(x, t)
=
iItJt(x, t).
(1.4)
If we are able to perform this basic operation then we can solve the TDSE.
2. Expansion of the propagator Kosloff [1] showed how we can propagate the wavefunction forward in time in a more efficient way than in eq. (1.3) above. Instead of expanding the exponential operator in a Taylor series he proposed that it be expanded in a "Chebyshev" series. This series has the form exp
-iiIT) ( -Ii-
exp ( x
-i
(¥- +Ii Vmin ) T)
t,(2 - 6no )Jn (i12~T) Pn(-iiI).
(2.1)
The Pn are Chebyshev polynomials of complex argument. They obey the recursion formula (2.2)
Hnorm is a "normalised" hamiltonian. It is normalised in such a way as to limit its spectrum to lie between -1 and +1. The spectrum of the hamiltonian is the range of possible eigenvalues it can have. This normalisation is performed by finding the range of the hamiltonian operator, (2.3)
and
Hnorm =
c.E
H - I (-2A
A
A
c.E
-2-
+ Vmin)
(2.4)
.
The first few Chebyshev polynomials are
POe -ix) = 1;
PI (-ix) = -2ix;
P2 ( -ix) = -4x 2
+ 1;
(2.5)
The I n (in eq. 2.1) are Bessel functions. These playa very important role in the convergence of the expansion. For n values greater than the argument, LlT/21i, these Bessel functions decrease exponentially in value. We can therefore predict that the number of terms needed in the expansion is approximately
N ~ LlEr. (2.6) 21i This is in fact an important conclusion. The number of terms required to expand the time evolution operator is proportional to the range of the hamiltonian operator. Or equivalently, this is the number of operations of
Time dependent quantum theory
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