Multi-reference perturbation theory with Cholesky decomposition for the density matrix renormalization group

We present a second-order N-electron valence state perturbation theory (NEVPT2) based on a density matrix renormalization group (DMRG) reference wave function that exploits a Cholesky decomposition of the two-electron repulsion integrals (CD-DMRG-NEVPT2). With a parameter-free multireference perturbation theory approach at hand, the latter allows us to efficiently describe static and dynamic correlation in large molecular systems. We demonstrate the applicability of CD-DMRG-NEVPT2 for spin-state energetics of spin-crossover complexes involving calculations with more than 1000 atomic basis functions. We first assess in a study of a heme model the accuracy of the strongly- and partially-contracted variant of CD-DMRG-NEVPT2 before embarking on resolving a controversy about the spin ground state of a cobalt tropocoronand complex.


I. ACTIVE SPACE SELECTION
A. model 2 complex  (14,18)), including four π, π * pairs on the ligands (denoted π 1 to π 4 and π * 1 to π * 4 ), five metal 3d orbitals and five second 3d shells (denoted 3d ) to account for the double-shell effect. A hybrid procedure including manual orbital inspection and selection on top of our automated active space selection protocol 1 was applied for the choice of this active space: first, a DMRG[256](52,72)#CAS(6,10)-SCF calculation on the quintet state was performed (i. e. a DMRG-CI (52,72) calculation with m = 256 employing orbitals from a CAS(6,10)-SCF calculation following the nomenclature of Ref. 1. The CAS(6,10) comprised five metal 3d orbitals and five second 3d shells in the active space). Then, the automatic selection protocol was followed (with an orbital selection threshold of 45% of the largest single-orbital entropy s(1) value 2,3 ), however, out of the automatically selected orbitals, only the ligand antibonding π * and the corresponding bonding π pairs have been included, resulting in the π 1 to π 4 and π * 1 to π * 4 orbitals. The π, π * space has then been augmented manually by the metal 3d orbitals and second 3d shells to form the final active space. This step was necessary, as the majority of the d orbitals, which would be needed to ensure the universality of the active space across all spin states, was not selected by the automated selection scheme applied to only a single spin state. Whereas our automated selection was based on an initial calculation for the quintet state only, the union of automatically selected orbitals for all spin states (as proposed in Ref. 1) would in fact result in an active space similar to the one selected here semi-automatically.

B. Cobalt tropocoronand complex
The active space chosen in calculations of [Co(TC-3,3)(NO)] is shown in Fig. S2 FIG. S1. Active orbitals employed in NEVPT2 calculations on the model 2 complex.
π NO,xy and π * NO,xy ), and a σ orbital (denoted σ 3d x 2 −y 2 ) that forms a bonding-antibonding pair with the 3d x 2 −y 2 orbital, which is unoccupied in the S 0 state.
To obtain the final active space, orbitals selected by the automatic selection protocol (with the orbital selection threshold of 20% of the largest s(1) value) was augmented by two tropocoronand π orbitals (complementary to the automatically selected π * orbitals) and missing metal d orbitals and 3d double shells (except for the 3d x 2 −y 2 double shell, which showed a very low s(1) value).  Table I in the main paper.
In Table S1, the 5 B 2 DMRG-SCF calculation with m = 1024/256 (marked with † ) has obviously converged to a local minimum, which can be seen by comparison with energies obtained for smaller m values. Such local energy minima may be encountered in DMRG calculations. Although they can be easily identified in sequences of calculations with different m values (which is always recommended), it is instructive to see whether such a technical problem affects the results. A converged calculation to the global energy minimum yields a lower energy (−1798.8379677 a. u.) but with an almost unchanged wavefunction character. The updated energy affects the relative energies by less than 2 kcal/mol. The relative energy is then within the range of the relative energies calculated for the other m values (cf . Table S10) and we may therefore use the data of Table S1 nevertheless.    calculation converged in a local minimum: its total electronic energy (see Table S1) is by 0.001447 a. u. higher than the energy obtained from a m = 256 calculation, although one would expect an energy lowering due to the variational principle. Interestingly enough, the relative CD-DMRG-SC-NEVPT2 energies (see Table I in the paper) do not show this outlier, although the variation of the SC-NEVPT2 energies with m is slightly larger than for the DMRG-SCF energies.