# Natural orbital symmetries

1. Feb 23, 2014

### timntimn

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 $l$ are present, i.e.
$\psi_i(r) = R_i(r) \cdot \sum_{m} { C_i^{l,m} \cdot Y_{l,m}(\theta,\phi)}$
(where $R(r)$ is a radial part and $Y_{l,m}(\theta,\phi)$ are spherical harmonics)

Any reference on the subject is highly appreciated.

P.S. Lowdin natural orbitals [P.-O. Löwdin, Phys. Rev. 1955, 97, 1474–1489] are defined as follows. Suppose $\Psi(x_1,...,x_N)$ 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 $\sigma_i$ are spin variables, $r_i$ are coordinates, and $x = (r,\sigma)$)
Then Lowdin natural orbitals comprise such a set of mutually orthogonal, unity-normalized
functions $\psi(x)$ 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 $n_i$ 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