I don't know if there's something for screws in wood, as wood's a rather strange material.
For bolts into steel, however, there's the AISC Manual (no prices listed, but expect it to be in the $300+ range); there used to be a separate book on joints, so you could calculate the strength based on the size of the fastener, bolt pattern, etc. There was also a section for calculating the strength of different weld patterns.
Now, the rules for the optimal strengthy are going to be similar -- further apart will support a greater moment, so when the contact patch is a square, you want the first two to be in opposing corners. If using four, fill all of the corners. However, because it's wood, you have a chance of spliting, so you don't want to go too close to the edges. (if you have to; pre-drill).
The other thing to remember is that with screws and bolts, the fasteners shouldn't be taking the full load -- they're pulling the structural material together, so that the load's transfered as friction ... this means if you see a gap between the two pieces, you need more fasteners. If you have a really large contact patch, drop another screw in the middle.
The standard practice is a 30 degree scarf joint preferably joined on a stud. I like to miter my two end with the same setting so I know the angle is perfect(when possible) then take my pieces to their wall a little over sized and scribe them to the floor (making sure the top line of the trim is on plane) and mark out for your end cuts. Once fit, position your pieces and join them with glue and micro pins(skip this if you don't have a micro pinner, brads just make a mess). Let it tack up for a minute and the start securing it with finishing nails. Try to crisscross nails at the scarf without getting too close to the joint. Cheers!
Best Answer
There are a lot of ways you could go here.
Conceal The Screws
If you just make a simple butt joint like you have in your picture, you can do as @AlaskaMan says in the comments - countersink the screws and put a plug or putty over the screw head. That will hide the screw; the plug / putty will be visible but less noticeable than the screw.
Improved Screws
Pocket screws mentioned by @JimStewart in the comments are usually stronger than straight screws. The pilot holes are drilled at an angle and the screw is countersunk; you cut an oval shaped plug if you want to conceal the screws. You could insert the screws from the inside of the frame, depending on what you're making here, they may not be visible on that side, even without the plugs.
You could use corner brackets, which are very easy to use, and in some configurations pretty strong. If you can put them on one side of the assembly that's out of sight, they may not be visible.
Join without Screws
One sure way to make sure you don't see the fasteners is to not use fasteners. These methods are important enough that there's a name for it - the craft of joinery. You could write a book about it, in fact many books have been written about it - here's one:
Woodwork Joints: How They Are Set Out, How Made and Where Used. by William Fairham
There are far more joints in that book than I have ever seen in actual use. One of the simplest is a simple half lap corner joint. This one just about anyone could make with just a hand saw, and with good glue used correctly, it will be stronger than a screwed joint.
But the half lap as shown will be stronger on a joint where the pieces are flat, you get more glue surface that way. With the pieces standing, in the configuration in your photo, a slightly more difficult joint would be better, a bridle joint:
A mortise and tenon joint is similar, a little harder to make, but with less visible seam:
There are many other variations and possibilities - box joints, doweling, biscuits, etc. etc. but hopefully this gives you some ideas.