The normal modes are basically ways the complex can move. The mathematical formalism of normal mode analysis makes deciphering the actual "normal modes" into physically meaningful observables somewhat difficult to picture. Think about a hydrogen molecule. The only way it can move (disregarding translation and rotation) is the two atoms either moving apart or closer. That is the one normal mode. Something excites that mode if it affects that motion. A pair of molecular pliers (solidifying hydrogen gas for example could have the same effect) might excite that mode since pliers can make things move closer together or farther apart. A molecular wind (e.g. diffusion) would not excite that mode since it only affects the movement of the molecule as a whole. Also instead of looking at the system from an outside point of view (placing both atoms at a certain x,y,z coordinate value) it helps to look at it from an internal point of view (one atom is x amount of distance from the other). Both coordinate representations suffice, but the internal coordinates, although somewhat more alien, make the normal mode mathematical equations simpler. To characterize the normal modes of more complex systems like proteins, obviously, one will analyze more normal modes than the hydrogen molecule. There are many more ways a protein can move than the one (closer or farther apart) way of the H2 molecule. If one explains all of these ways in terms of cartesian coordinates, it could be seen that different modes contain some of the same coordinates - i.e. they are not described in terms of an independent basis. In other words, if one coordinate is changed (atoms move closer or farther apart) then another coordinate has to also change (those two atoms cannot move farther apart without these three atoms changing their angle). It is like having to say how each of the three cartesian coordinates change when the H2 molecule vibrates rather than just saying how the one internal coordinate (distance between atoms) is changed. So just like that, normal coordinates make it easier to describe normal modes. If one changes coordinate representations from cartesian to normal coordinates (using the familiar and not so difficult mathematical methods to do so - any basic linear algebra book explains it) then it is seen that the coordinate fluctuations in the normal modes are independent of the fluctuations in the other modes. As seen with the H2 molecule, the cartesian coordinates are not independent when describing vibration. No matter which coordinate basis (cartesian or normal) is used, the modes are independent of each other - it is the actual coordinates used to describe the modes that may be dependant on each other - "if this distance changes then that angle must change". However, in normal modes, any single mode is seen to be a fluctuation of only a single normal coordinate. When one normal coordinate is changed, then other normal coordinates are not required to change. Then one can think of a normal mode of a system as described by a single coordinate rather than many coordinates at once. Interestingly, when one normal coordinate is changed, several x,y,z coordinates are changed. So the difficulty with understanding the independence of the normal coordinates seems to lie in the long familiarity of describing positions in terms of x,y,z, coordinates. However, when normal coordinates are used, each mode is described by the fluctuation of one coordinate about its mean. Thus, the frequency, is the frequency of this fluctuation and the eigenvalue of the mode then becomes is like the energy required to excite it. Lower frequency modes correspond to larger fluctuations. Hope this clears up some stuff. Personally, understanding the relationship between different coordinate bases helps illuminate what the normal modes actually are. It is not too difficult, then, to see how they relate to entropy and such.
One last thing. Normal mode analysis is just a different way of solving Newtons equations. The use of normal coordinates equates the entire vibration with a sum of individual vibrations along separate normal coordinates. If cartesian coordinates are used, then dividing up the vibration into the separate parts makes little sense because the vibrations then cannot be described as a sum of individual vibrations along separate coordinates - they are not separate - one mode will use the same coordinates as another mode - the two modes have to be considered together to make sense. So that's what normal mode analysis does for us - it allows us to break the total vibration into parts each of which is described a fluctuation of a single coordinate with no other mode vibrating along that same coordinate - all modes are orthogonal.
Last edited by jtibbitt; 05/30/04 01:58 AM.