A rigid rod with mass $m$ is initially held at an angle with respect to a frictionless plane by a string. Then the string is cut. What happens to the rod? My intuition suggests that the rod should slide to the left but I cannot figure out how this is supposed to happen when the only forced involved are vertical (reaction force and weight). Moreover, while initially the reaction force should balance the weight, the reaction force causes the rod to rotate about its centre of mass. This would cause the end of the rod in contact with the surface to leave it. But then the reaction force disappears and the weight of the rod places it back on the surface again. I am not sure how to deal with this "jerky motion" either.
I'm guessing from your description that the left end of the rod is the end on contact with the ground, and the right end is the in the air at the point that the string is cut.
I think that what will happen is this: When the string is cut, gravity will cause the rod to fall, such that its centre of gravity falls vertically. As the rod falls, it will rotate to become more horizontal (it must be horizontal when it lands on the ground). That will of course cause the left end of the rod, which is already on the ground, to move to the left (your intuition is correct in that regard), but the right end will move to the right, ensuring no horizontal movement of the centre of gravity.
What are the forces that cause this movement of the ends? They will be internal forces within the rod (caused because the external forces are acting to compress the rod, and the rod reacts to that). Those internal forces will act along the rod and so have a horizontal component. If you sum/integrate these horizontal components along the whole rod, they'll cancel out and so add up to zero, but at any particular point along the rod, the horizontal force acting on that bit of rod will be nonzero.
The left end of the rod will not leave the ground while the rod is falling: The rod is simultaneously falling and rotating, and the vertical component of the two motions will exactly cancel at the left end, so that the left end will only move horizontally along the ground. (The entire rod may of course bounce off the ground immediately after falling, as a result of the collision with the ground, but I don't think that's what you were asking about).
The only force applied is vertical. So centre of mass's horizontal position can't change. Say left edge doesn't slide to left(in this case center of mass will shift to right). It says it all.
If (the surface on which the rod is falling/or the rod) is elastic, only then there will be a jerky motion and a rebound and hence possible displacement of horizontal position of centre of mass.