7.36. Excited States using PNO-based coupled cluster

Despite the successes of the DLPNO-CC approximation for ground states, the use of PNOs for excited states has been less fruitful. It is not straightforward to define a PNO-based scheme for excited states, which will maintain the balance between speed and accuracy, as observed for the ground state. As an intermediate solution, the basis for ground state DLPNO quantities is transformed back to the canonical basis and are used within the canonical EOM routine. This procedure is justified, as the main bottle neck of the EOM-CCSD or STEOM-CCSD methods comes from the ground state calculation. Approximating the ground state CCSD amplitudes with MP2 amplitudes is also possible, as done in the EOM-CCSD(2) approach. However, it is not reliable and can lead to large errors, when the reference HF wave function does not provide a reasonable zeroth order approximation to the ground state wave function. Note that the back-transformed PNO scheme (bt-PNO) described here is available for both open-shell (UHF (QROs) or ROHF reference) and closed-shell (RHF reference) systems.

7.36.1. General Description

The back transformation of the ground state DLPNO-CCSD amplitudes to the virtual space involves three steps. The \(T\) amplitudes in the PNO basis are first converted into the PAO basis, then subsequently to the atomic orbital (AO) basis, and finally to the canonical MO basis[234]. For example, in the closed-shell case, we have

\[d_{\tilde{\mu}\tilde{a}_{ij} }^{ij}T^{ij}_{\tilde{a}_{ij}\tilde{b}_{ij} }d_{\tilde{b}_{ij}\tilde{\mu} }^{ij} \Rightarrow L_{\mu\tilde{\mu} }^{ij}T^{ij}_{\tilde{\mu}\tilde{\nu} }L_{\tilde{\nu}\nu}^{ij} \Rightarrow C_{a\mu }^{ij}T^{ij}_{\mu\nu}C_{\nu b}^{ij} \Rightarrow T^{ij}_{ab},\]
\[d_{\tilde{\mu}\tilde{a}_{ii} }^{i}T^{i}_{\tilde{a}_{ii} } \Rightarrow L_{\mu\tilde{\mu} }^{i}T^{i}_{\tilde{\mu} } \Rightarrow C_{a\mu }^{i}T^{i}_{\mu} \Rightarrow T^{i}_{a},\]

The AO basis functions are denoted as \(\mu,\nu,\ldots\), while \(\tilde{\mu},\tilde{\nu},\ldots\) refers to PAOs. The missing pairs are treated using MP2 amplitudes. If all the thresholds are set to zero, the back-transformed amplitudes match exactly with the canonical RI-EOM-CCSD ones. On the other hand, when all the thresholds are made infinitely tight, one obtains the EOM-CCSD(2) results. This PNO-based excited state approach is available for all the flavors of EOM-CCSD and for STEOM-CCSD in both open- and closed-shell systems.

Below, we list all the parameters that influence the DLPNO-CCSD-based excited state calculations

%mdci
#bt-PNO-EOM and STEOM parameters - defaults displayed
DoEOMMP2 true        # MP2 correction for missing pairs
DoRECAN  true        # recanonicalization of the occupied
                     # orbitals before the excited state calculation
                     #(only relevant for the RHF implementation)
end

7.36.2. Reference State Energy

Here, it should be noted that the reference energy for PNO-based EOM-CCSD or STEOM-CCSD is slightly different from that printed for a converged ground state DLPNO-CCSD calculation, as it includes the perturbative correction for different truncated quantities.

----------------------
COUPLED CLUSTER ENERGY
----------------------

E(0)                                       ...   -113.913498239
E(CORR)(strong-pairs)                      ...     -0.401457078
E(CORR)(weak-pairs)                        ...     -0.000339627
E(CORR)(corrected)                         ...     -0.401796705
E(TOT)                                     ...   -114.315294944

In the bt-PNO-EOM-CCSD scheme, the CI-like excited state treatment of the reference state is defined by back-transformed DLPNO amplitudes (or MP2 amplitudes for the weak pairs). The energy corresponding to this set of amplitude is printed at the beginning of the EOM calculations.

Dressing integrals for EOM-CCSD            ... 
Reference state energy for EOM-DLPNO-CCSD     ...   -114.314954945

done (    0.4)

Therefore, to calculate the total energy of an excited (ionized or electron attached) state, one needs to add the excitation energy to the reference state energy in bt-PNO-EOM-CCSD.

7.36.3. Use of Local Orbitals

The use of local orbitals makes it difficult to follow a particular guess vector in the Davidson digonalization process in EOM-CC and STEOM-CC. Therefore, it is advisable to recanonicalize the occupied orbitals after the ground state DLPNO-CCSD calculation by setting DoRECAN to true (i.e. only relevant for the closed-shell RHF implementation). It should be noted that the recanonicalization does not change the EOM-CCSD energies. However, the STEOM-CC energies are not invariant to orbital rotations and differ slightly for local and canonical orbitals. In the open-shell bt-PNO implementation, we follow a different procedure in that all quantities are transformed to the delocalized basis before proceeding with the back transformation and the excited state calculation.

7.36.4. Some tips and tricks for bt-PNO calculations

  • The bt-PNO scheme with tightPNO settings gives results, which are within 0.01 eV of the canonical EOM-CCSD numbers, at a fraction of the computational cost[234]. So, use of bt-PNO scheme is always preferable over canonical calculations.

  • In the case of an RHF reference, one should set ‘DLPNOLINEAR true’ and ‘NEWDOMAINS true’ in the mdci block input to use the 2015 fully linear scaling implementation, which is more robust than the 2013 implementation used as default in bt-PNO scheme.

  • The transition moment in bt-PNO-EOM (RHF only) and bt-PNO-STEOM (RHF, UHF (QROs) or ROHF) is only available using the linear approximation.