Constructing orthogonal orbitals from atomic orbitals

[hiyok]

Imagine there is a molecule which consists of several atoms, and for each atom there is an effective orbital, phi_i, which are not orthogonal. Now we want to construct from them a set of orthogonal orbitals, psi_i. Of course there are many ways to do this. Let W be the matrix that realizes our requirement, i.e., \sum_jW_{ij}phi_j=psi_i. The question is, can one assert that, it is always possible to get a W whose off-diagonal elements are unanimously much smaller that its diagonal ones?

 

[alxm]

Atomic orbitals? 'Orbitals' normally refers to the solutions to the _electronic_ Schrödinger equation. And they are taken to be orthonormal, which is valid insofar as the Born-Oppenheimer approximation holds.

 

[hiyok]

Here 'atomic orbitals' are understood exactly the same as those used for constructing molecular orbitals. For its meanings, you are referred to textbooks on quantum chemistry.

 

[alxm]

Here 'atomic orbitals' are understood exactly the same as those used for constructing molecular orbitals. For its meanings, you are referred to textbooks on quantum chemistry.

If you're talking about 'atomic orbitals that construct molecular orbitals' then you're talking about MO theory.
That's not something you'd find in a modern textbook on quantum chemistry, that's something you'd find in an introductory chemistry textbook. Because, it's not a quantum description, in terms of being a solution to the wave equation. It's a model to rationalize chemical bonding in terms of single-electron (hydrogenlike), non-interacting orbitals. The interaction (forming 'molecular orbitals') is regarded as a linear sum of 'atomic' ones.

MO theory is a model of the quantum-mechanical description. The actual quantum mechanical orbitals do not identify any particular electron as "belonging" to any particular atom, and can't be easily separated into them.

 

[hiyok]

Let me be clearer. Consider two protons (and one electron), located at R1 and R2 respectively. At low energy, only 1s orbitals, \psi(r-R1) and \psi(r-R2), which are centered about R1 and R2 respectively, need be considered to span an effective Hilbert space {psi(r-R1), psi(r-R2)}. It is obvious that these two orbitals are linearly independent, but they are not orthogonal. So, we can construct from them two orthogonal orbitals which have similar spatial profiles as the original s orbitals. The question is, can we extend this statement to a crystal, that is, can we find a set of orthogonal orbitals that resemble the original atomic orbitals? One way is to use Wanier orbitals, which, although localized on a unit cell, may not necessarily be localized on a single site, if a unit cell constitutes of several atoms.

 

[alxm]

But that's what is done in quantum chemistry. You form a basis in orthonormal one-electron orbitals, which describe the total wavefunction as a Slater determinant.

The description of those orbitals is a question of basis sets. If you use single-electron (hydrogenlike) orbitals as your basis set, that's a Slater-type basis. Mostly, gaussian basis sets are used. The individual basis functions are not necessarily orthonormal in themselves.

 

[hiyok]

I did not say that one cannot use non-orthonormal set of basis. I just ask， is it possible to always find an othonormal set that has similar spatial profile as the non-orthonormal one? I can only give an approximate argument in favor of this.

 

[alxm]

I just ask， is it possible to always find an othonormal set that has similar spatial profile as the non-orthonormal one? I can only give an approximate argument in favor of this.

An orthonormal set of what which has the same spatial profile as what?

I don't know what you're asking. I'll leave it to someone else to attempt an answer.

 

[hiyok]

The orthogonal set is constructed from the non-orthogonal set of atomic orbitals. Pls read the example I gave.

Thanks