Approaching the Complete Basis Set Limit for Spin-State Energetics of Mononuclear First-Row Transition Metal Complexes

Abstract

Convergence to the complete basis set (CBS) limit is analyzed for the problem of spin-state energetics in mononuclear first-row transition metal (TM) complexes by taking under scrutiny a benchmark set of 18 energy differences between spin states for 13 chemically diverse TM complexes. The performance of conventional CCSD(T) and explicitly correlated CCSD(T)-F12a/b calculations in approaching the CCSD(T)/CBS limits is systematically studied. An economic computational protocol is developed based on the CCSD-F12a approximation and (here proposed) modified scaling of the perturbative triples term (T#). This computational protocol recovers the relative spin-state energetics of the benchmark set in excellent agreement with the reference CCSD(T)/CBS limits (mean absolute deviation of 0.4, mean signed deviation of 0.2, and maximum deviation of 0.8 kcal/mol) and enables performing canonical CCSD(T) calculations for mononuclear TM complexes sized up to ca. 50 atoms, which is illustrated by application to heme-related metalloporphyrins. Furthermore, a good transferability of the basis set incompleteness error (BSIE) is demonstrated for spin-state energetics computed using CCSD(T) and other wave function methods (MP2, CASPT2, CASPT2/CC, NEVPT2, and MRCI + Q), which justifies efficient focal-point approximations and simplifies the construction of multimethod benchmark studies.

Figure 1

Structures of 13 small TM complexes making up the benchmark set (L^– = N₂C₃H₅^–, Cp^– = C₅H₅^–). Superscripts give multiplicities of the investigated spin states.

Figure 2

Box-plot of basis set incompleteness errors (BSIEs) of the FC-CCSD and FC-(T) terms in conventional and explicitly correlated calculations relative to the CBS limits from Table 3. Each box represents 50% of the population (with the median marked in the middle), and the whiskers extend from the minimum to the maximum. Annotated values are MADs (mean absolute deviations). See Supporting Information for numeric data of individual complexes (Tables S7 and S8) and the timings of representative calculations (Figure S1).

Figure 3

Box-plot of basis set incompleteness errors (BSIEs) of the Δ3s3p term in conventional and explicitly correlated CCSD(T) calculations relative to the CBS limits from Table 3. The graphical convention is identical to that used in Figure 2 (but mind the different scale); the annotated values are MADs. See Supporting Information for numerical data of individual complexes (Table S9) and the timings of representative calculations (Figure S1).

Figure 4

Box-plot of basis set incompleteness errors (BSIEs) in FC-CCSD-F12a calculations with several basis sets of triple-ζ (or lower) quality relative to the CBS limits of Table 3. Individual data are represented with points. Annotated values are MADs. See Supporting Information for numeric data (Table S10) and the timings of representative calculations (Figure S2).

Figure 5

Graphical representation of the composite basis set cT(D) for the example of one of the small models studied here (Fe^IIL₂, left) and Fe^II porphyrin complex (FeP, right). The colored areas represent the assignment of atomic bases: triple-ζ for Fe and coordinated N atoms (orange) and double-ζ for the other atoms (green). Note that the number of atoms in the triple-ζ region does not increase with expansion of the ligand.

Figure 6

Basis set incompleteness error (BSIE) of the FC-(Tα) term of eq 10 for the spin-state energetics computed at the CCSD(T)-F12a/cT(D) level. The mean absolute deviation (MAD) and mean signed deviation (MSD) from the reference values are plotted as functions of α, and statistical box-plots are shown for the three representative values: α = 0 corresponding to the unscaled (T) correction, α = 1 corresponding to the Marchetti–Werner (T*) correction, and corresponding to the (T#) correction proposed in this work. For numerical data of individual complexes, see Table S13, Supporting Information.

Figure 7

Basis set incompleteness errors (BSIEs) for the spin-state energetics calculated at the CCSD(T#)-F12a/cT(D) level with respect to the reference CBS limits of Table 3: (a) for the FC-CCSD, FC-(T#), and Δ3s3p terms separately and (b) for the final CCSD(T) energy estimates, i.e., sums of the above three terms. Values annotated to the right are MADs.

Figure 8

Relation between the computation time and basis set incompleteness error (BSIE) resulting from calculations of the quintet–singlet energy difference for at the CCSD(T) level: (a) the CCSD(T#)-F12a/cT(D) protocol of eq 11 (b) the CCSD(T)-F12b/a[T:Q] + Δ3s3p-F12b/acT protocol of eq 9 and (c) the reference CBS extrapolation ac[Q:5]/Q, and some other conventional CCSD(T) calculations and extrapolations (see Section 2.4 for the notation of basis sets and extrapolations). The reported times are obtained by summing the wall clock times needed to compute the energies of the two spin states. All times were measured using Molpro 2019.2 running 12 MPI processes, each allocating up to 24 GB of RAM, on an Intel Xeon 6146 system equipped with triple RAID-0 of Intel P4600 solid-state drives.

Figure 9

Statistical distribution of (a) differential energies between various methods and CCSD(T) in the CBS limit and (b) the corresponding differential BSIEs obtained using the cT(D) basis set for the studied benchmark set of spin-state energetics. Mind different energy scales for (a,b). For numerical data of (b), see Table S18, Supporting Information.

Figure 10

Basis set incompleteness errors (BSIEs, kcal/mol) for the CCSD(T) spin-state energetics of the benchmark set calculated using computational protocols P₁–P₆ (see text for discussion). Annotated values are MADs.

Figure 11

CCSD(T) doublet–sextet splitting for MnCp₂ with different choices of reference orbitals (HF, M06, TPSSh, and BP86), including scalar-relativistic effects: (a) DLPNO–CCSD(T₁) results from ref (64), (b) CCSD(T) results extrapolated using modified def2 basis sets analogously as in ref (64), and (c) best estimates of CCSD(T)/CBS limits (see text).

Figure 12

Structures of (a) Fe^IIP(NH₃) and (b) .