Electron correlation effects in cobalt fluorides CoF n

The molecular cobalt fluorides CoF 2 , CoF 3 and CoF 4 are studied and compared by employing different basis sets as well as Quantum Information Theory (QIT) to inves-tigate their correlation effects. These prototypical monomers may be systematically extended in size yielding a novel quasi 1-dimensional, strongly correlated model system consisting of cobalt atoms bridged by oxygen atoms and fluorine termination on both ends. Accurate correlation energies are obtained using Full Configuration Interaction (FCI) and Full Configuration Interaction Quantum Monte Carlo (FCIQMC) calculations and the results are compared to Coupled Cluster and Density Matrix Renormalization Group (DMRG) energies. The analysis indicates the cobalt atom requires a larger number of one-electron basis functions than fluorine and the use of localized molecular orbitals may facilitate calculations for the extended systems.

As the longer chains require the treatment of many (unpaired) electrons we here start by investigating the monomers CoF 2 and CoF 4 , as well as the intermediate CoF 3 . The ground state structure of CoF 4 has a tetrahedral (T d ) geometry and is therefore considered as well. The three compounds CoF 2 , CoF 3 , and CoF 4 (T d ) have been identified experimentally by matrix-isolated IR spectroscopy [15] and mass spectrometric data of the cationic species is available as well. [15,16] Theoretical studies on the CCSD(T)/aVTZ and B3LYP/aVTZ levels of theory [17] confirm the high-spin character and show that CoF 4 and CoF 3 are thermodynamically stable under the release of F or F 2. [17] The minimum structures are found to be linear (D ∞h ) for CoF 2, [17,18] trigonal planar (D 3h ) for CoF 3, [17,19,20] and tetrahedral (T d ) for CoF 4.
[ 17] A practical application of the compounds is the Fowler process, [21] the synthesis of fluorocarbons by reduction of CoF 3 to CoF 2 . The trifluoride is later regenerated by fluorination of CoF 2 with F 2 . Similar, molecular CoF 4 is of potential interest as an oxidation or fluorination reagent, as it constitutes the highest neutral cobalt fluoride. [16,22] The four model systems will be investigated by systematic variation of the one-electron basis set. Accurate correlation energies are obtained by conventional FCI or full configuration interaction quantum Monte Carlo (FCIQMC), depending on the basis set size. Additionally, density matrix renormalization group (DMRG) calculations and subsequent quantum information theory (QIT) analysis [23] are applied.

| Density matrix renormalization group and quantum information theory
Originally invented by White, [6] the DMRG method is especially suited to treat strong correlations in large active spaces and closely connected to QIT. [23] The latter allows for a quantitative analysis of the correlation (entanglement) effects of either individual or groups of orbitals. Both, DMRG and QIT, are described in detail in various reviews, [12,[24][25][26][27][28][29] therefore only a brief overview is given here.
In DMRG the electronic wave function jΨi is represented by the d-order tensor U containing the configuration interaction (CI) coefficients, where d is the number of orbitals. This tensor is then factorized into a matrix product state (MPS), where each (spatial) orbital i is represented by a matrix A i .
The indices α i label the four possible occupations of each spatial orbital, that is, the single-orbital basis states j ϕ i f g αi i. The orbitals are considered to be arranged on a chain and the corresponding matrices are connected to their neighbors by a virtual index. Contraction (multiplication) of two matrices along their shared virtual index creates a combined subsystem. Contraction of all matrices A i recovers the full wave function.
Numerical accuracy can be controlled by the number of blockstates M, which truncates each virtual index to its M largest contributions. The lower limit M = 1 recovers the HF solution and leads to small matrices, that is, low memory requirements. In the numeric exact limit M ! ∞ the FCI results are obtained. Alternatively, the accuracy can be adaptively controlled by the dynamic block state selection (DBSS) approach, [30,31] where the number of blockstates is chosen such that a user-defined truncation error for the density matrix is not exceeded. In the DMRG algorithm, an effective Hamiltonian for a subset of orbitals (typically two) is formed by contracting the MPS with the Hamiltonian over all except the selected orbitals. Diagonalization and factorization of the obtained eigenvector into the selected orbitals then leads to an updated set of matrices A i . This process is then iteratively repeated while sweeping forward and backward through the chain of orbitals until convergence is achieved. The CI problem is thus solved by diagonalizing many small, effective Hamiltonians instead of one large Hamiltonian.
QIT relies on the n-orbital reduced density matrices that are obtained by tracing over all but n orbital indices, for example, the 1-orbital reduced density matrix is given by and can be directly obtained from contraction of the MPS. The n-orbital reduced density matrix then encodes the entanglement of the subsystem of n orbitals with its environment. This may be quantified by calculating the von Neumann entropy [32] of all eigenvalues ω i, α . For example the 1-orbital entropy of orbital i is defined as Large values of S i indicate large entanglement, while values close to zero represent negligible entanglement of the corresponding orbital. Similar, the 2-orbital entropy S ij describes the amount of entanglement between both combined orbitals and the environment. But it does not include any entanglement between i and j, which may therefore be recovered as the difference between the 2-orbital entropy and both 1-orbital entropies. This is called the mutual information I ij .
Additionally, the so-called total correlation can be considered. On a side note, we would like to mention that the name total correlation might be misleading in the context of this work. A more intuitive name might be total entropy or total entanglement. However we will stick with the name established in previous literature.

| Full configuration interaction quantum Monte Carlo
The FCIQMC was recently introduced by Booth et al. [7] It combines the expansion of the electronic wave function in determinants from FCI with the propagation of random walkers from Quantum Monte Carlo. If the walker population is at equilibrium, then the number of walkers N i on each determinant jii is proportional to its CI coefficients C i The population dynamics is controlled by the coupling between two determinants jii and jji, given by the HamiltonianĤ.
Together with the imaginary-time Schrödinger equation where τ = it, this yields a set of coupled differential equations Here the parameter S is introduced to control the total number of walkers as described below.
After choosing an initial walker population, a three-step algorithm propagates the walker population. The first spawning step considers offdiagonal spawning events from a parent determinant i on a child determinant j. With a certain probability depending on the coupling matrix K ij , a new walker will be spawned. In the second step, diagonal death/cloning events will occur on each parent determinant, again with a certain probability. Finally all walkers from the previous two steps are summed on each determinant in the annihilation step, that is, walkers with opposite sign on the same determinant cancel out. In the long time limit, this algorithm will eventually equilibrate and oscillate about that equilibrium.
As the walkers on each determinant relate to the CI coefficients, the correlation energy may be obtained from the equilibrated population.
Different energy measures exist however. The first one is the energy shift S introduced in Equation (11), which is defined such that it reaches the correlation energy if the walker population remains constant. In practice, S will oscillate around the correlation energy, that is, may be averaged if one continues to sample the equilibrated population. Another measure is the projected energy which may be rearranged to The power of FCIQMC lies in the number of required walkers. Although a system-dependent number of critical walkers is required to reach a converged energy, this number is much smaller than the number of configurations a deterministic FCI calculation requires. The reason is that although not all determinants are populated by walkers in a certain population snapshot, they still remain accessible in the next iteration. Their effect on all other determinants (spawning of new walkers) can thus be sampled over time.
Typically, a FCIQMC calculation starts with a number of walkers on the HF configuration. The energy shift is then kept constant (eg, S = 0) and causes the total number of walkers to grow. Once a user-defined amount of walkers is reached, the calculation switches to constant N w mode and varies S which allows the walker population to equilibrate.
Two extensions to FCIQMC are the exact deterministic treatment of the most important determinants (semi-stochastic) to reduce the statistical error (noise) [33,34] and the initiator method, [35,36] where only determinants selected as an initiator may spawn walkers on previously unoccupied determinants. This makes the propagation more stable with respect to the number of walkers required. A further extension introduces superinitiators, [37] which marks all populations spawned from a superinitiator as a regular initiator.

| COMPUTATIONAL DETAILS
For the CoF 2 molecule, the electronic ground state is reported to be an 4 Δ g state and the first excited state 4 Σ − g is only about 5 mE h higher in energy. [18] Due to the reduction to the abelian group D 2h the two degenerate 4 Δ g fall into the A g and B 1g irreducible representations (IRREP), while 4 Σ − g is B 1g . The electronic states and the energetic order of the Co 3d orbitals are summarized in Figure 2. For simplicity we will focus on the A g component of the 4 Δ g ground state. However, it proved advantageous to optimize the molecular orbitals (MOs) on the CASSCF(7,5) level and include both electronic states in the state averaging procedure. On this level, the 4 Σ − g state is actually lower in energy than the 4 Δ g ground state. This, however, may be corrected using the higher levels of theory. The electronic, nondegenerate ground states for the other three models are in the total symmetric IRREP and orbitals are based on Restricted Open-Shell Hartree-Fock calculations. All FCI and DMRG calculations exclude core orbitals with eigenvalue below −3 E h (i.e. 1s, 2s, 2p, 3s, 3p on Co and 1s on F) but keep all virtual orbitals.
The four systems of interest are investigated by a systematic variation of the one-electron basis sets and FCI calculations, ranging from a minimal basis set up to cc-pVDZ [38] on all atoms. The MOs and FCI energies are obtained with the Molpro [39][40][41][42] software package. Where a conventional FCI is not possible due to the basis set size, FCIQMC [7] calculations are performed instead. FCIQMC calculations are first equilibrated using F I G U R E 2 The Co 3d orbitals in the CASSCF(7,5) calculation of CoF 2 using the cc-pVDZ basis set. The energetic order is plotted to the left. The occupation patterns and CI coefficients c i of all possible configurations and their corresponding electronic states are indicated to the right 1 × 10 6 walkers and are then stepwise increased to 1 × 10 7 and 1 × 10 8 walkers. Afterward the number of superinitiators is stepwise increased until the averaged correlation energy converges within a user-defined threshold (2 × 10 −3 E h ).
The chosen basis sets start with a minimal basis set based on cc-pVDZ on all atoms and then includes stepwise additional atomic orbital (AO) basis functions on the cobalt center, while the fluorine basis set is kept minimal. The largest basis set considers the full cc-pVDZ basis set on all atoms. Detailed information on all used basis sets is available in Tables S1 to S6. Furthermore, the model systems are investigated by DMRG calculations, using a density matrix cutoff for the DBSS approach of 1 × 10 −5 and an additional cutoff M max = 1024 for the maximum number of blockstates at each step. DMRG calculations and subsequent QIT analysis are performed using the DMRG Budapest code. [43] 4 | RESULTS AND DISCUSSION The correlation energies for the isolated ions at different basis sets are summarized in Table 1. Be aware that the fluoride F − at a minimal basis set does not have any correlation energy. The results are used to determine the amount of correlation energy that arises from combining the atoms in CoF n by defining the interatomic correlation energy The results for the cobalt fluorides are shown in Table 2. HF results for CoF 3 using the smaller basis sets (basis 4 and basis 5 in Table 2) are missing, as the calculations were unable to preserve the expected degeneracy in the Co 3d orbitals.

| Intra-vs inter-atomic correlation
While the minimal basis set results (basis 5) do not seem to follow any trend, one can observe the expected decrease, that is, more negative  . The effect of gaining more intra-atomic correlation energy when using the larger basis set on F (basis 1) may be confirmed by a QIT analysis of both basis sets (basis 1 vs basis 2), as shown in Tables S7 and S8. The mutual information of orbital pairs including one F 2p AO and one Co AO (eg, orbital pairs #2#5 and #2#6) is smaller using the larger basis set. In contrast, the mutual information between two F orbitals (without Co AO contributions) increases for the larger basis set (eg, #2#37 in the full cc-pVDZ basis, corresponding to #2#28 in the smaller basis set). For isosurface plots of all orbitals, refer to Supplementary Information 2. It should be noted, however, that a direct correspondence between the AO basis set effect and the QIT data is difficult, as the QIT data are based on delocalized MOs while the basis set effects are directly related to AO basis functions.

| AO mixing for bonding orbitals
Another trend that changes character depending on the AO basis set size is the interatomic correlation energy ΔE corr per F atom of the CoF 3 model (cf. Table 2). For the larger two basis sets (basis 1 and basis 2) it is very close to the CoF 2 value and much smaller than for CoF 4 . On the other hand, it is very close to CoF 4 if only the Co 4sp functions are added to the minimal basis set (basis 3). This might be connected to the character of the 3d orbitals, which are summarized in Table 3. The numbers represent MO indices and indicate whether the corresponding MO is mostly constituted by a single Co 3d AO (single number), or mixed with F 2p AOs (two numbers). For example in CoF 2 the MO with index 4 is almost identical with the 3d x 2 −y 2 AO, while the 3d z 2 AO contributes mostly to MO 3 and additionally to MO 5 with some smaller contribution. Thus MOs 3 and 5 may be considered as hybridized. Additionally, these two MOs form a pair of bonding and antibonding orbitals, that is, the latter has an additional nodal plane perpendicular to the Co─F bonding axes. Assignment of the MOs and their AO contributions is based on the isosurface plots (cf. Supplementary Information 2). In the CoF 2 model, the MOs are dominated by Co 3d AO contributions. The minor F 2p contributions in 3d z 2 may be related to the state-averaged CASSCF(5,7) including the 4 Σ − g as indicated in Figure 2. There the 3d z 2 orbital is singly occupied and thus gets mixed with other AOs. This needs to be corrected by a CI calculation targeting the 4 Δ g only. Both CoF 4 models show, in contrast to CoF 2 , F 2p contributions in all MOs related to the Co 3d orbitals, while CoF 3 shows a more mixed situation: The 3d z 2 orbital remains on its own, and the other four 3d orbitals are mixed with varying degrees of F 2p contributions. The mixed MOs would require more flexibility in the AO basis set, especially for the Co 3d orbitals that are highest in energy among the occupied orbitals. The F 2p AOs, however, which are energetically lower and thus are all doubly occupied, are sufficiently described by a smaller basis set. Therefore, the qualitative trend in interatomic correlation energy, which relates to the bonding description, does change when adding more AO basis functions on Co, but not on F. The much smaller value for CoF 2 however is already visible for smaller basis sets on Co as the 3d orbitals mix much less than for the other three cases.
The Co 3d orbitals and their mixing with the F 2p orbitals may also be interpreted in terms of the bonding situation. In case of AO mixing, the pairs of bonding and antibonding MO yield the largest mutual information of the systems (cf. Tables S7 to S11), which indicates a large and systematic problem of HF in describing these MOs. Correction on the CI level then requires large entanglement of these orbitals. As a result, increasing interatomic correlation energies are observed. In fact, such arguments have been made a long time ago [44][45][46] and similar observations have been made for N 2.
[47] There, the effect increased for large internuclear separations, where strong correlation becomes very important.

| Comparison of different correlation measures
Largest overall mutual information is observed for CoF 4 D 4h (I max 77,78 = 0:4066) and CoF 4 T d (I max 5,7 = 0:2096), while CoF 3 and CoF 2 show smaller values (cf. Tables S7 to S11). This agrees with the trend of strong correlation as measured by the total correlation I tot (QIT based) and the coupled cluster-based T 1 and D 1 diagnostics, summarized in Table 4. In general, the FCIQMC energies are much smaller than the DMRG results, as the latter is primarily developed for static correlation and thus struggles with the dynamic correlation contribution. FCIQMC on the other hand is able to consider both. The table also includes DMRG results using Pipek-Mezey [48] localized orbitals. As observed earlier, additional basis functions on fluorine reduce interatomic correlations between Co and F, that is, intra-atomic correlations are preferred. This may be exploited by using localized MOs. In turn this requires more mixing of orbitals participating in the chemical bonds. The localized orbitals result in a much lower DMRG energy although it used much looser numerical parameters (DBSS DM cutoff 1 × 10 −4 vs 1 × 10 −5 ). The obtained MPS for the localized orbitals is thus much more compact and exact. Additionally, the total correlation I tot increased, that is, overall more entanglement is included.
The CoF 4 DMRG calculations for the T d structure are closer to the FCIQMC energies than for D 4h (column ΔE FCI -DMRG in Table 4) (note that both systems have very similar FCIQMC correlation energies, cf. E corr for basis 1 in Table 2). At the same time T d shows slightly less total correlation I tot . Together with the T 1 and D 1 diagnostics that indicate D 4h to be more strongly correlated, one would expect DMRG to perform better for D 4h than for T d . Assuming DMRG is indeed able to capture all strong correlation effects, this means the D 4h result is missing some important dynamic correlation effects. A possible explanation could be the stronger multireference character in connection with the number of blockstates used. In the less strongly correlated case T d , only a few configurations directly related to strong correlation generate excited configurations related to dynamic correlation. In D 4h on the other hand, more configurations related to strong correlation are required, which in turn generate a larger number of configurations for dynamic correlation. Using the same numerical cutoff parameter for both systems will therefore lead to a different quality and accuracy of the calculation.
The RCCSD(T) energies in Table 4 are in rough agreement with FCIQMC results. While the correlation energy for CoF 2 is underestimated, the CoF 4 energies are overestimated. Following this trend the CoF 3 energies are in very well agreement (less than 1 mE h ), which may therefore be the result of error cancelation.

| SUMMARY
In summary, the results suggest a more complete AO basis set on cobalt than for fluorine is required, and intra-atomic are more important than interatomic correlations. Thus the use of localized MOs seems beneficial. The treatment of the extended CoO chains should therefore consider localized orbitals and one may try to reduce the number of basis functions on oxygen/fluorine. The latter suggestion however assumes the results may directly be transferred from fluorine to oxygen and therefore requires careful validation.

ACKNOWLEDGMENTS
Financial support by the International Max Planck Research School "Functional Interfaces in Physics and Chemistry" as well as computing time provided by the high performing computing facilities (ZEDAT) at Freie Universität Berlin is gratefully acknowledged.