What is the role of the spin variable in quantum chemistry?

[Amok]

In various quantum chemistry books and course booklets I came across spin wave functions (usually referred to as alpha and beta for spin and up and down, respectively) that depend on a so called spin-variable. They are usually used to construct slater determinants. An example of this is Modern Quantum Chemistry (a great book). My problem is that sometimes, when matrix elements are computed, integrations over the spin-variable are made (e.g http://en.wikipedia.org/wiki/Franck–Condon_principle). My teacher told me this makes no sense since spin is supposed to be a discrete variable, so it should be a sum. What is this spin variable? Does it have anything to do with the actual 'spin' observable? I think it is a bit of a construct for when you actually have to write down matrix elements as integrals and not as scalar products using braket notation. Can anyone clarify this for me?

 

[Physics Monkey]

I suspect this is simply slightly confusing notational choice. It is not uncommon to use the integral sign abstractly to represent a large variety of sums and integrals. Unless one is integrating over spin coherent states (an overcomplete basis), the sum over spin degrees of freedom is always discrete.

 

[cgk]

The spin variable is one of the four non-relativistic coordinates of an electron, the other three being (for example) spatial positions:
$$\vec x = (\vec r, s)$$
where s is either alpha or beta (or up or down) -- the spin variable. The electronic wave function of N electrons is a function of N such x variables and antisymmetric with respect to the exchange of them (of full x'es, not only r's or s's individually).

"Spin integration" is used as a synonym for "spin summation", and means summing over the two possible states of each spin variable. This is just a form for avoiding cumbersome notation by artificially splitting up the x = (r,s) integrations into r-integrations and s-summations. No advanced magic going on here.

 

[Amok]

The spin variable is one of the four non-relativistic coordinates of an electron, the other three being (for example) spatial positions:
$$\vec x = (\vec r, s)$$
where s is either alpha or beta (or up or down) -- the spin variable. The electronic wave function of N electrons is a function of N such x variables and antisymmetric with respect to the exchange of them (of full x'es, not only r's or s's individually).

"Spin integration" is used as a synonym for "spin summation", and means summing over the two possible states of each spin variable. This is just a form for avoiding cumbersome notation by artificially splitting up the x = (r,s) integrations into r-integrations and s-summations. No advanced magic going on here.

I get the notation when you integrate over x = (r,s), but in the book I mentioned the actually integrate over the spin-variable (over s).