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Natural orbital symmetries

  1. Feb 23, 2014 #1
    Dear experts,

    Could anyone please confirm or disprove my guess that each Lowdin natural orbital
    of an isolated many-electron atom, obtained from its exact non-relativistic wavefunction,
    can be expanded into the series over spherical harmomics, where the spherical
    harmonics with one and only one value of orbital number [itex]l[/itex] are present, i.e.
    \psi_i(r) = R_i(r) \cdot \sum_{m} { C_i^{l,m} \cdot Y_{l,m}(\theta,\phi)}
    (where [itex]R(r)[/itex] is a radial part and [itex] Y_{l,m}(\theta,\phi) [/itex] are spherical harmonics)

    Any reference on the subject is highly appreciated.

    Thank you in advance!

    P.S. Lowdin natural orbitals [P.-O. Löwdin, Phys. Rev. 1955, 97, 1474–1489] are defined as follows. Suppose [itex]\Psi(x_1,...,x_N)[/itex] is
    an (exact) wavefunction (obtained by the full configuration interaction method, for example). A spinless reduced density matrix is then defined as
    \gamma(r,r') = \sum_{\sigma_1 ,...,\sigma_N} \int {\Psi^{\sigma_1 ,...,\sigma_N *}(r,r_2,...,r_N) \Psi^{\sigma_1 ,...,\sigma_N}(r',r_2,...,r_N) \cdot dr_2...dr_N}
    (where [itex]\sigma_i[/itex] are spin variables, [itex]r_i[/itex] are coordinates, and [itex]x = (r,\sigma)[/itex])
    Then Lowdin natural orbitals comprise such a set of mutually orthogonal, unity-normalized
    functions [itex]\psi(x)[/itex] that brings the spinless reduced density matrix to the 'diagonal'
    form, i.e.
    \gamma \left( {x,x'} \right) = \sum\limits_i^{} {n_i \psi _i ^* (x)\psi _i (x')}
    where [itex]n_i[/itex] are expansion coefficients (also known as the 'occupation numbers').
    (it can be easily shown that such a set of functions can always be built).
    Last edited: Feb 23, 2014
  2. jcsd
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