In the simple problem you state I think it would be the SQRT(Sa^2+Sb^2).  

The distance between them is d, which is the difference between the positions xa and xb.  d=xa-xb, where the uncertainty in xa is sa and the uncertainty in xb is sb.  Applying uncertainty propagation to that gives sd=SQRT(sa^2*(partial d/partial xa)^2+sb^2*(partial d/partial xb)^2) where partial / partial x_i is the partial derivative.  partial d/partial xa is the derivative of d with respect to xa which is just 1, and partial d/partial xb is the derivative of d with respect to xb which is -1.  Put it all together and you get SQRT(sa^2+sb^2), if sa=sb then sd = SQRT(2)*sa.

Thanks Brian for your response. Your proposal gives me sd=0.19. I know (let's say just that, just to be brief) that the result must be about 0.43 so I don't think it is ok.

I also have a theory, let me tell you:

sa includes ~68% of the surface of the gaussian distribution. 2*sa includes ~95% of it. 3*sa includes 99% and so on. So, to calculate how far an atom can vibrate from the mean position (A), I can use, for instance:

A=SQRT(2*sa)

sa is 0.135 so A turns out to be ~0.5 Angs. Note that I've used 2*sa arbitrarily

Since the two atoms may vibrate in opposite sense, the maximum oscillation of the distance is 1.0 Angs. In other words, the distance oscillates from 4.6 (3.6+1) to 2.6 (3.6-1). I should have said that the mean value of d is 3.6. 

From what I know, this is quite accurate, I've run a molecular dynamics simulation and found that the distance oscillates between these two values. So, I know the mean value (3.6) and I know the maximum (4.6) and the minimum value (2.6) of that the distribution should describe. Can I extract the sigma of the distance from this info?

If I apply the previous equation:

A = SQRT(2*sd) ; 1.0 = SQRT(2*sd) , then sd=0.5

It's not far from the good result, which is 0.43, but if one compares two gaussian distribution with s=0.5 and s=0.43, you'll notice that it is an important difference.

As you have seen, i have obtained the value of A assuming 2sa and, when I've used the same equation to obtain sd, I have used again 2sd inside the square root. I'm afraid that this is an arbitrary assumption, because I can also use 3sd or 4sd in both cases without any problem. I know 'A', but I don't know whether it corresponds to 3sd or 4sd or 2sd. Any idea?

Thanks!

You seem too focused on the value that you want to obtain instead of the value that you have. If the numbers for sa (sb == sa?) and sd are correct, then they simply do not match. Are you sure that both of them are rms deviations, and not FWHM (for example)?

Hi Ulf,

I want to obtain sd from logical thinking. As I said before, I already have it based on the results of a simulation, but I'd like to get it without doing this simulation because it takes 1 month in a supercomputer and I intuitively think that sd can be extracted somehow from sa and sb.

sa and sb are 0.135 and are experimental values. In fact, those are called thermal ellipsoids in a crystal structure (U) but are equivalent to the standard deviation. From the simulation I get sa=sb=0.128.

In any case, the simulation is fine for multiple reasons, I just need a model to explain the results so, logically, I'm focused on explaining the results. 

Hi Brian, 

Now I think you're right, and that the disagreement comes from the fact that I'm considering it is a 1D problem. In fact I have Sx,Sy and Sz for each atom, what would be then Sd?

Thanks

Hi Sergi,

The distance between two atoms in free space is just d=SQRT((x2-x1)^2+(y2-y1)^2+(z2-z1)^2).  Apply the uncertainty propagation formula (assuming no covariance between quantities) gives sd=(1/d)*SQRT((sx1^2+sx2^2)(x2-x1)^2+(sy1^2+sy2^2)(y2-y1)^2+(sz1^2+sz2^2)(z2-z1)^2).  

I hope that works out for you.