How Are Coefficients of Secular Equations in LCAO Evaluated?

[sams]

Dear Everyone,

I would like to ask a question regarding the evaluation of the coefficients in Linear Combination of Atomic Orbitals (LCAO). In Molecular Quantum Mechanics book authored by Peter Atkins and Ronald Friedman (Fourth Edition ISBN 0199274983), we are trying to calculate the coefficients of the LCAO of H₂⁺ (First atom: H_A and Second atom: H_B) molecule-ion.

After solving the 2x2 secular determinant of this system,
|α-E β-ES|
|β-ES α-E|
the energies are obtained as:
E₊ = (α+β)/(1+S)
E_- = (α-β)/(1-S)
Note: α and β are the expectation values of the Hamiltonian.

Solving the secular equations:
∑c_r{H_rs - ES_rs} = 0
We will obtain the real coefficients of the normalized wavefunctions:
c_A = c_B, c_a = 1/{2(1+S)}^½ for E₊ = (α+β)/(1+S)
and
c_A = -c_B, c_a = 1/{2(1-S)}^½ for E_- = (α-β)/(1-S)

1. I have proved that c_A = c_B and c_A = -c_B by matrix multiplication of the secular equations and inserting E₊ and E_- in the equations. But may anyone please advise me how can we obtain c_a = 1/{2(1+S)}^½ and c_a = 1/{2(1-S)}^½ ?

2. Another question regarding the overlap integral S in the above secular determinant. Solving this integral leads to the following relation: S = <A|B> = {1 + (ZR/a_o) + 1/3(ZR/a_o)²}e^-ZR/a_o
How can we calculate Z? Is it the Z_eff for a compound or mixture found in the following link https://en.wikipedia.org/wiki/Effective_atomic_number ?

Thanks a lot for the help and support...

 

[DrClaude]

But may anyone please advise me how can we obtain c_a = 1/{2(1+S)}^½ and c_a = 1/{2(1-S)}^½ ?

Write down the condition for normalization.

2. Another question regarding the overlap integral S in the above secular determinant. Solving this integral leads to the following relation: S = <A|B> = {1 + (ZR/a_o) + 1/3(ZR/a_o)²}e^-ZR/a_o
How can we calculate Z? Is it the Z_eff for a compound or mixture found in the following link https://en.wikipedia.org/wiki/Effective_atomic_number ?

Z is the actual atomic number, so Z=1 if you are considering H$_2^+$.

 

[sams]

Write down the condition for normalization.

1. ∫Ψ*Ψ=1
∫(c_AΦ_A + c_BΦ_B)(c_AΦ_A + c_BΦ_B) = 1

Let: ∫Φ_AΦ_B = ∫Φ_BΦ_A = S
If: c_A = c_B then: c_A = 1/{2(1+S)}½

Z is the actual atomic number, so Z=1 if you are considering H$_2^+$.

2. But in the case of heteronuclear diatomic molecule where we have different atomic numbers, shouldn't we calculate the Z_eff of the compound as a whole as given in the link https://en.wikipedia.org/wiki/Effective_atomic_number in second section "For a compound or mixture"?

Thank DrClaude for your advice.

 

[DrClaude]

2. But in the case of heteronuclear diatomic molecule where we have different atomic numbers, shouldn't we calculate the Z_eff of the compound as a whole as given in the link https://en.wikipedia.org/wiki/Effective_atomic_number in second section "For a compound or mixture"?

The secular equation you are using is valid for single-electron diatomic molecules. The equation for S given above is for a homonuclear diatomic molecule of hydrogenic atoms, such that wave functions $\Phi_A$ and $\Phi_B$ are identical except for their origin. For a heteronuclear molecule, you will have to recalculate that overlap integral.

An effective atomic number is only necessary if one of the atoms is only approximately a single-electron atom, such as an alkali atom.

What system do you want to apply this to, and what for?

 

[sams]

There is no specific system I want to apply this. I just wanted to ask about the case of heteronuclear compounds. The idea is much clearer now. Thanks again DrClaude...