Graduate Computational Physics: Trivial & Non-Trivial Solutions, LCAO

[sams]

Dear Everyone,

1. Could anyone please explain what is meant by trivial and non-trivial solutions?

2. LCAO:
Ψ = ∑c_rΦ_r
Ψ: Molecular orbitals
c: coefficients
Φ: Atomic orbitals
When we talk about basis sets, do we mean here the coefficients or the atomic functions?

Thanks a lot...

 

[blue_leaf77]

1. Could anyone please explain what is meant by trivial and non-trivial solutions?

http://www.mathwords.com/t/trivial.htm
As far as my experience is concerned, trivial solution is the zero function or vector.

When we talk about basis sets, do we mean here the coefficients or the atomic functions?

That depends on which you are talking about, molecule or atom. If the former, then the molecular orbitals are the basis if the latter it's the atomic orbital.

 

[ExplosivePete]

When we solve for solutions to differential equations, in physics we are often only interested in nontrivial solutions. I'm not certain of this, but it is my intuition that the space containing the solutions contain some form of a zero element. Physically, we are uninterested in these trivial solutions. So we disregard them.