DFT calculations of effective exchange integrals at the complete basis set limit on oxo-vanadium ring complex
Graphical abstract
Effective exchange integral (J) values at the complete basis set (CBS) limit are calculated.
Introduction
Since the discovery of the Ferric wheel by Lippard and co-workers [1], many polynuclear metal ring complexes that show interesting magnetic behaviors have been reported [2], [3], [4], [5], [6], [7]. Among a great deal of ring complexes, ones consist of the odd-membered ring are especially expected to show a spin frustration and a characteristic magnetic properties. The first example was {Cr8Ni} and {Cr7(VO)2} odd-membered ring complexes that are known as “the magnetic möbius strip” [8], [9]. These odd-membered ring complexes indeed showed unique magnetic properties, however there were hetero ions that caused different magnetic interactions in the ring. Therefore a pure homometallic ring has been expected to explain the magnetic behavior of the odd-membered systems. Recently, Hoshino et al. has realized the homometallic odd-membered ring complex that consists of seven oxo-vanadium units sandwiched by two β-cyclodextrins i.e. Na7[(VO)7Na7(H2O)7(CD)2]·nH2O (1) as illustrated in Fig. 1(A). A magnetic susceptibility suggested that the exchange interactions between oxo-vanadium units in the complex were anti-ferromagnetic however their intensity were very small about |Jexp| = 0.1 ∼ 0.2 cm−1. Because such weak exchange is often composed of some complicated but well-balanced interactions, quantitative theoretical calculations of the exchange couplings must be necessary to clarify an origin of the magnetic properties of the complex.
On the other hand, a recent progress of computers and density functional theory (DFT) realizes the first principal calculations of electronic structures of real large molecules without any simplifications of the functional groups, so that a direct estimation of the exchange interactions becomes feasible. In addition, there have been advances in a computational scheme to obtain the J values, especially by Ginsberg, Noodleman, Bencini, Ruitz and Yamaguchi et al. [11], [12], [13], [14], [15], [16], [17]. As a consequence, the J values of many diradical or polyradical species have been calculated to explain their magnetic behaviors. Those results have indicated that such DFT calculations and the above schemes are quite effective for the J values however other problems have appeared. In case of the DFT calculations of the J values, one faces, at least, two controversial points, i.e. a selection of functional sets [18], [19], [20] and basis sets. Both points strongly affects to the calculated results of the J values, however there have been no systematic examinations. The reason seems to be caused by a difficulty in separating the functional set error from the basis set error, because those two issues are often correlated each other. One solution in order to eliminate the basis set error is an estimation of the J values at the complete basis set (CBS) limit by the extrapolation method. If the J values at the CBS limit are estimated, one can focus on errors of the functional sets. In this way, it is important to estimate the J values at the CBS limit not only for discussions of the weak magnetic interactions in the magnetic compounds but also for the examination of the applicability of functional sets to the calculations of the J values. Nevertheless the way to compute the J values at the CBS limit has not been reported yet.
In this paper, we calculate the J values at the CBS limit with the use of the oxo-vanium dimer-unit models constructed by the experimental geometry of the real (VO)7 ring complex (1) [10]. For the purpose, we first propose a new scheme for the calculations of the J values at the CBS limit and apply it to the dimer models. We next examine many basis sets to elucidate which basis set can reproduce the J values at the CBS limit. Finally, we compare the calculated J values at the CBS limit among several DFT functional sets to examine which functional set is suitable for the (VO)7 ring complex.
Section snippets
Theoretical background
The electronic structures of the degenerated systems are usually described by multi-reference methods such as the complete active space (CAS) method [21], [22]. And the BS method, which expresses the degeneracy by splitting alpha and beta electrons into different orbitals are also often used as an alternative way [23]. The BS method can approximately involve the static correlation effect at a lower cost of computations, however the spin contamination is often appeared as a non-negligible error
Functional sets and basis sets
In order to examine the dependency of the functional sets, three types of hybrid DFT functional sets i.e. BHandHLYP [31], B3LYP [32] and CAM-B3LYP [33] are examined. All those functional sets consist of Becke88 exchange [34] and LYP correlation functions [35] but have different Hartree–Fock ratios and mixing procedures. As mentioned above, Dunning’s correlation-consistent polarized valence basis sets [36] are utilized to obtain total energies of the CBS limit. In addition, Pople’s split valence
Results and discussions
First, in order to examine the accuracy of the approximation of Eq. (6) that assumes , JAP values of Eq. (4) are compared with that corresponds to Eq. (6),where in Eq. (4) is substituted by in Eq. (7). Total energies, values of BS and HS states on each model are calculated with cc-pVXZ (X = D, T and Q) basis sets as summarized in Table S1 in Supplementary information. And JAP and values are obtained
Summary
In this paper, we examined dependencies of the calculated J values on basis sets and functional sets in comparison with J values at the CBS limit. For the purpose, the new scheme is proposed to obtain the J value at the CBS limit. This is the first example to obtain the J values at the CBS limit. Calculated results reproduce small J values of the (VO)7 complex suggesting an applicability of the hybrid DFT to a series of the oxo-vanadium ring complexes. Among the examined three hybrid functional
Acknowledgement
This work has been supported by Grant-in-Aid for Scientific Research on Innovative Areas “Coordination program” (No. 22108515) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) and by Grants-in-Aid for Scientific Research (KAKENHI) (No. 23350064) from Japan Society for the Promotion of Science (JSPS). And this work also has been supported by JST, CREST.
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