I summarized the math in the last post. Here is what you can do with it.
I2 can be described with Morse potentials. Here we are looking at the electronic ground state and the second excited state (because that is the bright state). Energy is plotted against I-I distance. I guess it is pretty much straight forward to solve the stationary Schrödinger equation of such a system to get the vibrational levels and wave functions. The lowest 20 ground state levels and 80 excited state levels are shown.
This is the textbook situation of a larger equilibrium bond distance and lower binding energy in the excited state. Vertical excitation from the ground state will put the wave packet on the left edge of the potential in a highly excited vibrational state. In the classical picture you can say the molecule will start vibrating because the two atoms are closer together than the new equilibrium distance.
Excitation from the v=0 vibronic ground state would require about 509 nm excitation energy and would lead to an almost dissociative state. In the simulation we started with v=1 where you have more density at a larger bond distance and a vertical excitation of 588 nm from the second maximum. v=1 is a realistic situation because this level is just 213/cm or 2.5 kJ/mol above the ground state, so we have 36% in this state (relative to 100% v=0). The situation works together with what we observe. 588 nm corresponds to yellow light. Yellow absorption gives a purple appearance.
In the simulation there was a 300 fs Gauss pulse (simulated with numerical Runge-Kutta integration) and then the wave packet evolved like I explained in the last post. With this ultrashort pulse, excitation of a coherent wave packet is possible. This wave packet is formed above the second maximum of the ground state function, then it oscillates back and forth. These oscillations occur with a period of 333 fs on average, this fits very well with the energy gap between the two levels where the wave packet should be mostly localized according to the excitation energy.
The classical explanation for the oscillations is that since the pulse is only about the length of a vibrational period or shorter, you can excite coherent motion.
You get the same with quantum mechanics but with a different explanation. First you look at the energy uncertainty of the pulse:
Another formula on Wikipedia looks almost the same, the energy of a harmonic oscillator:
In other words: The energy uncertainty of a pulse corresponds to the energy gap between two levels of a system with an oscillatory period equal to the pulse length. If the pulse is much longer than the oscillatory period, excitation will be sharp - one eigenstate will be excited and all subsequent motion is only a phase factor in the complex plane. If it is the same length or shorter there will be excitation into several levels and temporal evolution like I explained last time.
The software used is: Fortran scripts for the numerical integration, the Python Pylab package for creation of graphics, Video Mach for making a movie out of the pictures.
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