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Constructing orthogonal orbitals from atomic orbitals

  1. Feb 26, 2009 #1
    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?
     
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  3. Feb 27, 2009 #2

    alxm

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    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.
     
  4. Feb 27, 2009 #3
    Here 'atomic orbitals' are understood exactly the same as those used for constructing molecular orbitals. For its meanings, you are refered to textbooks on quantum chemistry.
     
  5. Feb 27, 2009 #4

    alxm

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    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.
     
  6. Feb 27, 2009 #5
    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 spacial 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.
     
  7. Feb 27, 2009 #6

    alxm

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    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.
     
  8. Feb 28, 2009 #7
    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 spacial profile as the non-orthonormal one? I can only give an approximate argument in favor of this.
     
  9. Feb 28, 2009 #8

    alxm

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

    I don't know what you're asking. I'll leave it to someone else to attempt an answer.
     
  10. Mar 3, 2009 #9
    The orthogonal set is constructed from the non-orthogonal set of atomic orbitals. Pls read the example I gave.

    Thanks
     
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