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Diagnostics for double excitations:

As pointed out in ref. [12], the %T1 diagnostic (or %T2 = 100 - %T1 ) which is evaluated directly from the squared norm of the single and double excitation part of the eigenvectors %T1 = 100*T1/(T1 + T2) with Ti = $ \sum_{{\mu_i}}^{}$Eμi2 where the excitation amplitudes are for spin-free calculations in a correspoding spin-adapted basis (which is not necessarily normalized) has the disadvantage that the results depend on the parameterization of the (spin-adapted) excitation operators. This prevents in particular a simple comparison of the results for singlet and triplet excited states if the calculations are carried out in a spin-free basis. With the biorthogonal representation for singlet spin-coupled double excitations [112] results for %T1 also differ largely between the left and right eigenvectors and are not invariant with respect to unitary transformations of the occupied or the virtual orbitals.

The ricc2 module therefore uses since release 6.5 an alternative double excitation diagnostic, which is defined by %$ \cal {T}$1 = 100*$ \cal {T}$1/($ \cal {T}$1 + $ \cal {T}$2) with $ \cal {T}$1 = $ \sum_{{ai}}^{}$Eai2 and $ \cal {T}$2 = $ \sum_{{i>j}}^{}$$ \sum_{{a>b}}^{}$Eaibj2 with Eai and Eaibj in the spin-orbital basis. They are printed in the summaries for excitation energies under the headings %t1 and %t2. For spin-adapted excitation amplitudes $ \cal {T}$1 and $ \cal {T}$2 have to be computed from respective linear combinations for the amplitudes which reproduce the values in the spin-orbital basis. For ADC(2), which has a symmetric secular matrix with identical left and right normalized eigenvectors $ \cal {T}$1 and $ \cal {T}$2 are identical with the contributions from the singles and doubles parts for the eigenvectors to the trace of the occupied or virtual block of the (orbital unrelaxed) difference density between the ground and the excited state, i.e. the criterium proposed in ref. [12]. Compared to the suggestion from ref. [12] $ \cal {T}$1 and $ \cal {T}$2 have the additional advantage of that they are for all methods guaranteed to be postive and can be evaluated with the same insignificantly low costs as T1 and T2 . They are invariant with respect to unitary transformations of the occupied or the virtual orbitals and give by construction identical results in spin-orbital and spin-free calculations. For CC2 and CIS(D ) the diagnostics $ \cal {T}$1 and $ \cal {T}$2 agree for left and right eigenvectors usually with a few 0.01% , for CIS(D) and ADC(2) they are exactly identical. For singlet excitations in spin-free calculations, %$ \cal {T}$2 is typically by a factors of 1.5-2 larger than %T2 . The second-order methods CC2, ADC(2), CIS(D ) and CIS(D) can usually be trusted for %$ \cal {T}$2≤15% .

For compatibility, the program can be switched to use of the old %$ \cal {T}$1 and %$ \cal {T}$2 diagnostics (printed with the headers ||T1|| and ||T2||) by setting the flag oldnorm in the data group $excitations. Note that the choice of the norm effects the individual results left and right one- and two-photon transition moments, while transition strengths and all other observable properties independent of the individual normalization of the right and left eigenvectors.

The %$ \cal {T}$2 and %T2 diagnostics can not be monitored in the output of the (quasi-) linear solver. But it is possible to do in advance a CIS(D) calculation. The CIS(D) results for the %$ \cal {T}$2 and %T2 correlate usually well with the results for this diagnostic from the iterativ second-order models, as long as there is clear correspondence between the singles parts of the eigenvectors. Else the DIIS solver will print the doubles diagnostics in each iteration if the print level is set > 3 . States with large double excitation contributions converge notoriously slow (a consequence of the partitioned formulation used in the ricc2 program). However, the results obtained with second-order methods for doubly excited states will anyway be poor. It is strongly recommended to use in such situations a higher-level method.


next up previous contents index
Next: Visualization of excitations: Up: Calculation of Excitation Energies Previous: Trouble shooting:   Contents   Index
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