What Determines Bonding Energy in Atomic Systems?

[RedX]

I have some basic questions on bonds.

Take two identical particles in their ground state. When separated, the Hamiltonian can be written as the matrix:

1 0
0 1

where 1 is the energy.

When brought close together, there is the possibility of transitions through off-diagonal terms, and the matrix becomes:

1 .1
.1 1

Computer calculations give the eigenvalues of this matrix to be .7 and 1.3.

Therefore bonding occurs since the two particles will be in the .7 state instead of the 1 state, for a binding energy difference of 2*1-2*.7=.6.

So would it be correct to say that chemical bonding only happens because electrons have spin? For otherwise one electron would be in a state of .7, the other 1.3, and there would be no binding energy difference (adding off-diagonal terms doesn't change the trace, so the total energy would be the same if the off-diagonal terms were there or not).

Now if you increase the diagonal elements to say:

2 .1
.1 2

then the energy levels go up (as can be verified by computer calculations). But the original energy levels of:

2 0
0 2

where higher to begin with. In general, if you increase the diagonal elements to say:

10 .1
.1 10

does the difference in energy between this and this:

10 0
0 10

increase? That is, does the binding energy in general increase when your atoms are in high energy valence states when initially separated (in this particular case, ten times the energy of the ground state considered earlier)?

I also noticed that the more you increase the off-diagonal elements, the higher the binding energy you get, so that:

1 100
100 1

has a high binding energy. Also, it doesn't matter if 100 is -100: the sign doesn't seem to change the eigenvalues (as 100^2=(-100)^2). My question is what determines the size of the off-diagonal elements? The diagonal elements come from the Schrodinger equation for hydrogen: these are the energy shells. What is the origin of the off-diagonal elements? Is it electromagnetic?

Lastly, I noticed that if you increase the size of the matrix to 3x3:

1 .2 0
.2 1 .2
0 .2 1

the binding energy increases. Is this the reason that solids are formed? The bigger the solid, the less the binding energy? Is there a point of diminishing returns?

You can't form a solid with hydrogen, since each atom only has one electron so can only form a bond with 1 nearest neighbor. So what about helium? Would helium look like:

2 .0001 0
.0001 2 .0001
0 .0001 2

I assumed the off-diagonal elements to be small so that there is no significant bonding, since helium I recall from chemistry likes to be alone. Nevertheless, shouldn't helium form a solid, since even though the off-diagonal elements are small, they nevertheless are non-zero, so that it should be energetically favorable for helium to form a solid?

In fact, why can't you make helium a triangle with:2 .0001 .0001
.0001 2 .0001
.0001 .0001 2

Here are some computer calculations:

http://www.wolframalpha.com/input/?i=eigenvalues[{{1,.2,0},{.2,1,0},{0,0,1}}]

http://www.wolframalpha.com/input/?i=eigenvalues[{{1,.2,0},{.2,1,.2},{0,.2,1}}]

http://www.wolframalpha.com/input/?i=eigenvalues[{{1,.2,0,0},{.2,1,.2,0},{0,.2,1,.2},{0,0,.2,1}}]

 

[DrDu]

What you are doing goes under the name Hueckel theory or tight binding.
You are assuming that the atomic orbitals are orthogonal. How would your results change if you include the non-orthogonality of the orbitals?