Is the goal here to generate a set of random numbers which are apparently normally distributed, but with a sum of 1? Or is the goal to generate a set of numbers that have the SHAPE of a normal distribution PDF, yet have a sum of 1?
You seem to be asking for the former, yet your own accepted answer shows you generated a set of points off a Gaussian PDF, then shifted it so the sum was 1. As you should see, the result of what you did in your accepted answer is not a random set at all. A random set of numbers would not be perfectly symmetrical around the center element of the vector.
So IF you really want pseudo-random numbers with the property that the sum is 1, it is easy. Even trivially so. Use randn to generate a random vector.
We should see the sum was not 1. We now just make it so.
Are they now pseudo-normally distributed? Is the sum as desired?
So the sum is now 1, and we have a vector which is indeed pseudo-randomly normally distributed. For a larger sample (so the histogram will look smooth), we can see it does indeed have the necessary properties.
So while it is not clear what you are asking to do, but the title of your question was:
Gaussian distribution with sum 1
Your other comments also suggest that you may be confused as to what you are asking.