User:Kinkreet/Protein Science/Sandbox

Electron transfer
Electron transfer for chemicals occur as the donor and acceptor collide; this rarely happens in biology because the donor and acceptor are often large proteins or complexes which are retarded by the medium they are in. If biological electron transfer relies on these random collisions, then reaction rates will be too slow to sustain life. Instead, redox centres are found confined to complexes, and these are transported to close proximity to aid transfer. But still, the electrons must travel through an insulating medium for a few Å.

Marcus theory is a theory aimed to describe the rates of electron transfer based on kinetics.

Nuclear Vibration modelled with Hooke's Law
In a biatomic molecule, the bond between the two atoms vibrate. The higher the energy of the bond, the higher its vibrational state, and the bigger the maximum displacement. Provided displacement from the typical separation is small, the bond can be modelled as a spring, where the energy of the bond relates to displacement according to Hooke's Law - the larger the displacement, the higher the restoring force.

Hooke's law states that F=-kx, where F is the restoring force exerted by the spring on that end (N or kg·m/s2); k is the spring constant (N/m or kg/s2); and x is the displacement from its equilibrium position (m). As work done equals force times distance, W=-kx2. So Hooke's law states that the energy of the bond is proportional to the magnitude of the square of the displacement. This produces a parabolic curve when plotted energy vs. displacement.

With increasing temperature and thermal fluctuation, heat energy may cause the bond to have bigger and more extreme displacements, allowing the nuclei involved to come extremely close and/or further away than normal. This rearrangement of the nuclei may provide the activation energy required for electron transfer.

In more complicated molecules with more than one bond, a single function, called the nuclear coordinate, is used to represent the sum of all the energies of all the bonds. This is a massive simplification.

Frank-Condon Principle
There are two types of motions involved when electrons transfer, the motion of the electron and the movement of the nuclei. The electron move because the nuclei and solvent molecules surrounding it have rearranged, with energy provided by thermal fluctuations. These rearrangements cause orbitals and wavefunctions to overlap, and so electrons can jump from donor to acceptor.

Because the electron virtually have no mass, the movement of the electron is almost an instantaneous process. The movement of the nuclei is much slower, this is due to its relatively enormous mass compared to the electron, and thus larger inertia. Therefore, during electron transfer, the nuclei are effectively stationary, and only moves after the electron have completed its movement. Thus we can assume that electron transfer occurs in a stationary nuclear framework. This is known as the Frank-Condon principle.

Conservation of energy
Therefore, during electron transfer, the displacement of the nuclei of the donor and acceptor must remain the same.

At the lowest vibrational state of a molecule at its ground state, the nuclei are at their equilibrium locations and experience no net forces from the electrons and other nuclei in the molecule.

When an electron transfer, the charge distribution/density is changed, and there is now a net force on the nuclei from the electrostatic repulsion/attraction of electrons and neighbourning nuclei in this new distribution. Thus, the nuclei will start to vibrate around their new equilibrium position.

Because the original 'stationary' equilibrium position of the nuclei in the initial electronic state is where the nucleus must move from, it becomes a turning point in the vibrational movements in its new vibrational higher electronic state, this transition is known as a vertical transition, and assumes the nuclei do not move.

In quantum mechanics, particles are not viewed as particles in the classical sense, but rather a probability distribution. So instead of saying the nuclei stay at the same location, we say that the nuclei retain their initial dynamic state. In quantum mechanics, the dynamic state is represented by the wave function, and so it can also be said that the nuclear wave function does not change during electronic transition.

Initially, at the ground state, the nucleus have a bell-shaped wave function centred around the equilibrium position. When it is excited to its new electronic state, it will enter into a vibrational state which most resembles its initial state. This vibrational state will have its wave function centred around the new equilibrium position, but one of its peaks will superimpose relatively well with the wave function of the peak of the ground state wave function, because this transition corresponds to the nuclear dynamic state which is least changed.

However, there are many wavefunctions which peaks superimpose well, and so many transitions to different vibrational states are observed, each with different intensities (higher intensities with higher resemblance).

The Frank-Condon factor is a measure of this resemblance. In word, it basically states "the intensity of a vibronic transition is proportional to the square of the overlap integral between the vibrational wavefunctions of the two states that are involved in the transition".

Due to thermal fluctuation, the solvent must rearrange. It is this rearrangement which causes electron transfer.