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## Homework Statement

Solve for the wavefuntions using the Secular Determinant, of cyclopropene anion/cation/whocares.

## Homework Equations

Secular determinant:

| x 1 1|

| 1 x 1|

|1 1 x |

## The Attempt at a Solution

I'm literally understanding every aspect of this concept, but what I cannot understand is the math behind the solutions. It looks very un-math like, more of an intuitive answer but I want to know how to solve the math out, because quite frankly its making me angry that something so simple is stumping me.

Anyway: Solving the matrix yields x = -2,1,1

Psi 1:

-2c1 + c2 + c3 = 0

c1 + -2c2 + c3 = 0

c1 + c2 + -2c3 = 0

c1^2 + c2^2 + c3^2 = 1

The solution for the secular polynomial of x=-2 is simple: you can solve it out for c1=c2=c3 and the only solution that satisfies that is c = (√3 /3)

My problem now lies with solving for the psi 2 and psi 3 solutions. I know the answers are

psi2: c1 = 0 c2 = +√2 / 2 and c3 = - √2 / 2

psi3: c1 = 2/√6 c2 = c3 = -1/√6

I have no idea how the math from this system can solve for those variables. I'm completely missing something here.

c1 + c2 + c3 = 0

c1 + c2 + c3 = 0

c1 + c2 + c3 = 0

c1^2 + c2^2 + c3^2 = 1

Thank you.