+ –
M
b a
+ t2
M - molecule with dipole moment
surface
+ t1
x
I+
1. Molecule, M, weakly phys. absorbed on surface. Not translating or rotating. (Example, CO on Cu surface.) 2. Dipole moment points out of wall. Interaction with wall very weak; can be ignored. 3. When not interacting with ion – vibrations harmonic. 4. M has – side to right.
opposite
+ t2
+ t1 x
I+
Qualitatively Correct Model
+ M
M +
–
b a
+ t2
surface
Ion causes cubic perturbation of molecule Correct symmetry, odd.
– end of M always closer to I+ than positive end of M.
Bond stretch energy lowered – closer to I+. + further from I+.
+ M
M +
–
b a
surface
Bond contracted Energy raised.
0 n t
These terms equal. Unperturbed problem. They cancel. Canceling gives
0 0 i c c H n n n n n n
Have
0 0 i c c H n n n n n n
m n
Approximations Usually start in a particular state 0 Dealing with weak perturbation. System is not greatly changed by perturbation. Assume:
* c c 1
Time Dependent Perturbation Theory Time dependent Schrö dinger Equation
H i
t
H H H
0
Take
time independent
time dependent will treat as time dependent perturbation H0 time independent - solution are
H 0 i
0
0 t
iEn t / 0 0 q , t q e n n 0 n q
complete set of time independent orthonormal eigenfunctions of H0.
time dependent phase factors
e iEn t /
To solve
H i
t
Expand
q, t cn t 0 n H H
Substitute expansion
0
used derivative product rule
0 0 i c 0 i c c H c H n n n n n n n n n n n
At any time, t, I+ to M distance = a.
a b 2 (Vt )2
b = distance of closest approach (called impact parameter). V = Ion velocity.
Ion flies by molecule Coulomb interaction perturbs vibrational states of M. Model for Interaction
+ M
M +
–
b a
+ t2
surface
+ t1 x
I+
Positively charged ion, I+, flies by M. I+ starts infinitely far away at t Passes by M at t = 0. I+ infinitely far away at t
With these assumptions:
cm

i 0 (q, t ) 0 ( q , t ) H m
No longer coupled equations
Grazing Collision of an Ion and a Dipolar Molecule - Vibrational Excitation
Time independent.
0 never changes significantly.
The probability of being in the initial state
The probability of being in any other state never gets much bigger than zero.
Left multiply by
0 m
0 0 i c c H m n m n n
Zeroth order eigenkets are orthonormal.
Therefore,
cm

i 0 cn 0 m H n n
Exact to this point. Set of coupled differential equations. In Time Dependent Two State Problem (Chapter 8) two coupled equations 0 H 0