12.2 Characteristics of the Implementation

The pnoccsd program works¹ with localized molecular occupied orbitals (LMOs). Their local nature is exploited to truncate for each LMO pair the number of interacting virtual orbitals and thereby the number of required two-electron integrals. PNO-MP2 belongs to the family of local correlation methods which have in common the key idea that the correlation energy can be written as a sum of pair contributions \[\begin{align} E^{\text{MP2}}_{corr} = \sum_{i\le j} e^{ij} = \frac{1}{4} \sum_{iajb} t_{ab}^{ij} \langle ij||ab\rangle \end{align}\](12.1) and that with localized molecular orbitals the pair contributions \(e^{ij}\) decrease fast with the distance \(|r_{ij}|\) between the LMOs \(i\) and \(j\) (approximately with \(1/|r_{ij}|^6\)) so that for large molecules the number of pairs that need to be included to compute the correlation energies within a given accuracy increases only linearly with the number of atoms. Furthermore, only such virtual orbitals contribute significantly to the correlation energy \(e^{ij}\) of a pair \(ij\) which are spatially close to \(i\) and \(j\).

The OSV-PNO hybrid approach implemented in the pnoccsd program starts with the determination of truncated subsets of projected atomic orbitals (PAOs) and orbital-specific virtual (OSV) orbitals for each LMO \(i\) as the eigenvectors of diagonal density contributions \(D^{ii}\) computed from (approximate) MP2 doubles amplitudes: \[\begin{align} D^{ii}_{ab} = 2 \sum_c t^{ii}_{ac} t^{ii}_{cb} = \sum_{\tilde{c}} d^{ii}_{a\tilde{c}} n^{ii}_{\tilde{c}} d^{ii}_{b\tilde{c}} \end{align}\](12.2) with occupation numbers \(n^{ii}_{\tilde{c}}\) not smaller than a predefined threshold tolosv. The OSVs are first used to calculate for each pair \(ij\) estimates for the pair correlation energy contribution \(e^{ij}\) and to set up a list of pairs \(ij\) for which significant contributions to the total correlation energy are expected. Only for this list of pairs the OSVs of the two LMOs are merged together to an orthonormal set of pre-PNOs \(\tilde{c}_{ij}\) which are used to compute the contribution of the pair \(ij\) to the density matrix via approximated MP2 amplitudes and as eigenvectors of these densities the coefficients of the pair natural orbitals: \[\begin{align} D^{ij}_{\tilde{a}_{ij}\tilde{b}_{ij}} = 2 \sum_{\tilde{c}_{ij}} t^{ij}_{\tilde{a}_{ij}\tilde{c}_{ij}} t^{ij}_{\tilde{c}_{ij}\tilde{b}_{ij}} = \sum_{\bar{c}} d^{ij}_{a\bar{c}} n^{ij}_{\bar{c}} d^{ij}_{b\bar{c}} ~. \end{align}\](12.3) Similar as for the OSVs also the PNOs with occupation numbers \(n^{ij}_{\bar{c}}\) below the PNO trunction threshold \(tolpno\) are discarded.

For OSV-PNO-MP2 the MP2 equations are then solved with the amplitudes and two-electron integals projected onto the so determined set of pair natural orbitals. For the explicitly-correlated variant OSV-PNO-MP2-F12 similar sets of orbital and pair specific auxiliary orbitals (OSX) are determined for the occupied and the complementary virtual orbital spaces which appear in the calculation of the three-electron integrals for the additional contributions from the geminals.

The thresholds for discarding PNOs and OSVs and the subset of auxiliary orbitals for the local RI approximation for computing two-electron integrals play an important role for the accuracy and the computational costs of calculations with the pnoccsd program. Per default all thresholds are coupled to the PNO truncation threshold tolpno such that the errors due to the OSV, local RI and pair truncation approximations and due to the screening of (AO) integrals and contributions to the residual are about an order of magnitude smaller than the PNO truncation error. The default value for tolpno is \(10^{-7}\) which gives PNO truncation errors in the correlation energy \(< 1\%\). Test calculations showed that the relative errors in the correlation energy decrease typically linear with \(\sqrt{\texttt{tolpno}}\).

All approximations in the CCSD energy are equivalent for UHF and RHF references such that a UHF-based calculation on a closed shell species gives the same energy as the RHF-based calculation to within numerical precision. In V7.4.2, the approximations involved in the (T) energy lead to small differences between UHF and RHF based calculations of a closed shell system.

12.2.1 Strong pair approximation

Beyond MP2 the PNO implementation makes per default use of a strong pair approximation:

• The 3.rd order CCD ladder terms and corresponding exchange terms: \[\begin{align} \sum_{kl} t^{kl}_{ab} (ki\tilde{|}lj) + \sum_{cd} t^{ij}_{cd} (ac\tilde{|}cd) - \sum_{ck} \Big( (ki\tilde{|}bc) t^{kj}_{ac} + (kj\tilde{|}ac)t^{ki}_{bc} \Big) \end{align}\](12.4) decay together as \(R^{-6}\) with the distance between the centers of the LMOs \(i\) and \(j\) and for the first term also with the distance between the centers of the LMOs \(k\) and \(l\). These terms are only evaluated if \(ij\) is a strong pair. For the last two terms in addition \(ki\) and \(kj\) and for the first term \(kl\), \(ki\), and \(lj\) are required to be strong pairs.

• The Coulomb integrals \((ki\tilde{|}ca)\) in the ring terms are neglected if \(ki\) is a distant or neglected pair.

Per default the threshold for \(T_{\text{strg}} = \sqrt{10 \times T_{\text{PNO}}}\) and \(T_{\text{weak}} = T_{\text{pair}}^{2/3} \times T_{\text{strg}}^{1/3}\). Above \((pq\tilde{|}rs)\) denote \(t_1\)-dressed two-electron integrals.