Heisenberg coupling constant predicted for molecular magnets with pairwise spin-contamination correction
Graphical abstract
Introduction
Single molecule magnets (SMMs) are promising candidates for the implementation of quantum computing [1], [2], [3], data storage, and spintronics [4], [5], [6], [7], [8], [9], [10]. Numerous new SMMs have been reported with a wide variety of topologies and nuclearities, incorporating a variety of transition metal atoms. A majority of molecules demonstrating SMM behavior have been synthesized using manganese, iron or nickel. Examples include Mn12 magnetic wheels like [Mn12O12(O2CCH2But)16(H2O)4] with S=8 [11] and [Mn12O12(O2CCHCl2)8(O2CCH2But)8(H2O)3], with S=10 [12] ground state. Another SMM complex with Jahn-Teller isomerism is [Mn12O12(O2CC6H4–2–CH3)16(H2O)4]·CH2Cl2·2H2O that was reported by Rumberger et al. [13]. Other topologies seen in SMMs include Mn4 dicubane [14] and the Mn4 cubane [15] complexes, and rod-shaped Mn6 clusters. One-dimensional chains of weakly interacting SMMs are also known, such as the complex [Mn4(hmp)6Cl2]n(ClO4)2n, reported by Yoo et al. [16]. Smaller molecular wheels include [Mn4(anca)4(Htea)2(dbm)2]·2.5Et2O [17], while larger wheels include Mn24 one, which consists of eighteen Mn(III) ions and six Mn(IV) ions [18].
Due to the size of these systems, spin-polarized or unrestricted Kohn–Sham formalism, known as broken symmetry Density Functional Theory (BS DFT) is the only first-principle method capable to describe their electronic structure. DFT is widely used to make accurate prediction of structure and properties for the molecules and solids, including aggregation and crystallization,[19], [20], [21], [22] and their emission fingerprints [23], reaction mechanisms and reaction rates [24], [25], [26], linear and non-linear optical properties [27], [28], [29], [30]. However, conceptual disadvantage of BS DFT approach is that spin-polarized Slater determinant no longer is an eigenfunction of the spin operator. The average value of 〈Ŝ2〉 is not equal to the correct value of Sz(Sz+1) (here Sz is ½ of the difference in total numbers of α and β electrons) [31]. This situation is known as spin contamination, and 〈Ŝ2〉 is often used as its measure. The common rule [32] is to neglect spin contamination if 〈Ŝ2〉 differs from Sz(Sz+1) by less than 10%. Due to of spin contamination, spin density becomes incorrect, electron energy deviates from that of the pure spin state, and molecular geometry may be distorted toward the diradical one. Some researchers argue that despite this incorrect spin-density, the total density and electron energy in BS DFT are predicted correctly [33]. Hence, the phenomenon of spin-contamination should be ignored. In fact, spin energy gap, predicted with BS DFT, can be found in a reasonable agreement with experimental values [34], [35]. Other researchers, however, recognize spin contamination as a problem affecting the energy. Possible solutions to spin contamination problem include constrained DFT [36], [37] and spin purification schemes [38], [39], discussed below.
Heisenberg exchange coupling constant J is often used to describe the difference in energy between the low and the high spin state. Positive value of J corresponds to ferromagnetic, and negative value corresponds to anti-ferromagnetic coupling. Since BS DFT does not produce the energies of the pure spin states, the expression for J must account for spin contamination. The following expressions had been suggested for this purpose [40], [41], [42], [43], [44]:
Of these three, J3 suggested by Yamaguchi [45], [46] can be reduced to J1 and J2 in the weak and strong coupling limits, respectively. Here EBS is the energy of the low-spin unrestricted Kohn–Sham (KS) determinant, ET is the energy of the high-spin KS determinant, 〈Ŝ2〉BS and 〈Ŝ2〉T are average values of the respective operators, and Smax is high-spin value for the operator Sz.
A more complicated expressions for variable spin-correction, including dependence of J on the overlap between corresponding spin polarized orbitals p and q were also derived recently [47], [48]. Taking the orbital overlap into account resulted in more accurate J values for Cu2+ binuclear complexes [48], [49]. However, this variable spin-correction approach had not been applied to systems with two or more correlated electron pairs. In this contribution we derive the new approach to spin contamination correction, and apply it to study two examples of binuclear molecular magnets.
Section snippets
Theoretical derivation
Here we propose an alternative approach to variable spin-correction, based on canonical Natural Orbitals (NO) [50]. First, let us consider a diatomic system AB with one correlated electron pair, such as stretched H2 molecule. We assume that restricted Kohn–Sham formalism yields higher energy for this system than unrestricted one, as the case of H2 molecule far from equilibrium. Unrestricted KS description produces the natural orbitals a, b as eigenvectors of the total density matrix with the
Implementation and numerical results
All calculations were done with Gaussian03 [52] program using B3LYP exchange-corrlation functional and TZV basis set. Spin-correction described above in theory section is implemented as a combination of unix shell script and FORTRAN code. It reads Natural Orbitals (NO) printout from Gaussian03 job (keyword used was Punch=NO) and converts them into spin-polarized molecular orbitals. Script uses a threshold parameter to identify the correlated pair. The spin polarization of the electron core was
Conclusions
We derived an expression to extract the energy of the pure singlet state expressed in terms of energy of BS DFT solution, the occupation number of the bonding NO, and the energies of the high and broken symmetry low and intermediate spin states built on these bonding and antibonding NOs (as opposed to self-consistent KS orbitals). Thus, unlike spin-contamination correction schemes by Noodleman, Yamaguchi, and Ruiz, spin-correction is introduced for each correlated electron pair individually.
Acknowledgments
This work was supported by the Russian Science Foundation, Contract no. 14-43-00052. The authors acknowledge the University of Central Florida Advanced Research Computing Center (https://arcc.ist.ucf.edu) for providing computational resources and support. SG acknowledges DOE Stockpile Stewardship Academic Alliance Program under Grant # DE-FG03-03NA00071.
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