E_{ia} = 〈Ψ_{ex}| a^{†}_{i}a_{a}| Ψ_{ex}〉 |
(14.1) |

can be brought to a diagonal form through a singular value decomposition (SVD) of the excitation amplitudes

[O^{†}EV]_{ij} = δ_{ij}_{i} |
(14.2) |

The columns of the matrices

From excitation amplitudes computed with the `ricc2` program NTOs and their weights
(the singular values) can be calculated with `ricctools`

. E.g. using the
right eigenvectors for the second singlett excited state in irrep 1:

`ricctools -ntos CCRE0-2--1---1`

`ntos_occ`

and `ntos_vir`

.
Note that the NTO analysis ignores for the correlated methods (CIS(D), ADC(2), CC2, CCSD, etc.)
the double excitation contributions and correlation contributions to the ground state.
This is no problem for single excitation dominated transition out of a
``good'' single reference ground state, in particular if only a qualitative picture is wanted,
but one has to be aware of these omission when using NTOs for states with large
double excitation contributions or when they are used for quantitative comparisons.