Ever cut a piece of fishing line? Then you’ve broken chemical bonds using a blade. Heck, you’ve probably broken chemical bonds with your bare hands, if you’ve ever torn a piece of plastic. Most plastic objects are monomolecular.
I’m sorry to say that you are mostly wrong. Even a monomolecular fiber will disrupt the electrochemical bonds that bind “solid” objects together. Now, it may be with some substances you will get sufficient interface contact that they will re-weld, especially in near vacuum conditions, and of course with a substance like water ice you would fully expect the surfaces in contact to fuse back together due to heat transfer from adjacent ice causing any disruption to refreeze (just as two ice cubes put together will often freeze to one another). But if you cut a substance held together with long chain molecules or with an amormphous solid structure, like wood or glass, you would likely sufficiently disrupt those bonds enough that they would be effectively severed. Ditto for, say, someone’s arm; although a very clean cut might not initially leave much room for blood to escape, even the slightest shear load will cause the arm to separate.
Molecules in a solid substance are effectively as close together as they can be (for a given temperature) and you cannot just slide a mass of other molecules in between them readily; hence, why diffusion in solids is a slow, statistical process.
Stranger
So the take-away is the the Vorpal Blade doesn’t always go snicker-snack?
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A molecularly sharp blade would cut as efficiently as it’s possible to do so, but you still need an input of energy sufficient to break the bonds holding together what you’re cutting. You aren’t going to effortlessly make solid objects fall apart.
Stiffness is probably quite a useful property in something indestructably sharp - or at the very least, more skill would be required to ensure that an indestructable thin wire only cuts the things you want it to.
I could certainly be wrong, so I will switch to question mode.
Sure - they are as close as they can be under given conditions. But this is not ultradense exotic matter. Normal matter no matter how dense is still largely empty space. The structural strength of a material derives from bonding strength between adjacent atoms/molecules. Greater the bond strength, greater the energy required to disrupt it.
Imagine a perfect lattice arrangement of iron atoms. Adjacent atoms are spaced apart at regular, well-known distances. Now imagine your monomolecular blade (assuming the blade is made of atoms comparably small to iron atoms) swinging through the empty space between iron atoms in the lattice. Would it disrupt bonds? I cannot see how. The iron atoms are still right there and the lattice structure is still intact.
Conceivably the blade adge will hit some atoms, and disrupt the lattice there. But it will not hit every atom it needs to to disrupt the lattice and sever the object completely.
Equivalently, would ‘shooting’ a single atom with sufficient energy through an iron lattice disrupt that lattice even if it did not physically hit any atom on the lattice? That I guess is the key to understand this.
Normal matter does not contain any empty space at all. Most of the volume of an object contains very little mass, but what does that matter? It’s chock full of electromagnetic fields, and those are the basis for essentially all of the interactions we’re familiar with, anyway. This is the same reason why, when you slap your hand against your desk, it doesn’t go through.
As Chronos notes, substances are not full of “empty space”; they are filled with electromagnetic fields which are as dense as they can be without additional outside forces to compress them. Your blade is also filled with electromagnetic fields. When it passes through the substance to be cut, the fields of the blade will disrupt the fields of the substance, and unless they are capable of self repair (as with water ice) the substance will exhibit damage and severance. “…‘shooting’ a single atom with sufficient energy through an iron lattice” will disrupt the lattice, and in fact, this is exactly how ionizing radiation does damage. Even low energy ionizing radiation over a long duration can do significant damage, hence why polymers become embrittled when left out in sunlight for an extended duration.
Ah, one of my favorite threads: [THREAD=299054]“Why can’t my hand go through my desk?”[/THREAD]
Stranger
“Matter is mostly empty space” refers to the fact that most of an atom’s mass is concentrated in it’s nucleus. As far as the effective covalent/ionic radii of atoms go, they’re usually packed nearly solid.
ETA: what’s been said upthread.
Regarding monomolecular filament as a weapon, I’d like to know more about the field that keeps the wire rigid. If a filament one atom thick passes through a living body, I gather it would cause at least some destruction, even if it wasn’t instantly disruptive, but how about the force field? Wouldn’t that contribute to tissue trauma and the actual cleaving effect? Maybe a monofilament garrotte would be more indicative of the actual effect of one-atom-thick thread passing through living tissue.
If you had an ultra-thin, ultra-strong flexible filament, but all other constraints were as they are in the real world, then you could make a “blade” out of such a filament mounted in a frame similar to what one uses for a hacksaw or scroll saw. That’d allow you a few inches of penetration (or more, depending on the exact shape of the frame) before you hit the thick part, which should be plenty to mess up most targets.
Diamond CAN be sharpened. But to answer the OP, a sword wielded by a pair of human hands can’t really generate enough force to go through tank armor. Assuming your sword steel is harder and tougher than the tank, you still have several forces acting against you: the blade’s friction, the armor’s compression strength, the shortness of the stroke, and of course the tank’s defensive weapons. 
<makes note to cover my tanks with monofilament wires>