3P138
Open-shell ELG 法の開発と新規磁性材料設計への応用（九大院・総理工 1，JST-CREST2）○Zhu Xun1，Liu Kai1，折本 裕一 1，青木 百合子 1, 2

Development of open-shell elongation method and its application toferromagnetic material design(Kyushu Univ.1，JST-CREST2) ○Zhu Xun1，Liu Kai1，Yuuichi Orimoto1，Yuriko Aoki1, 2

1. Introduction
Conventional (CONV) ab initio method has problem in treating huge systems as big timescale. Many linear scaling schemes including elongation (ELG) method1 have been proposed
during the last decades. Most linear scaling methods are still limited to closed-shell systems.
It is very desired to extend the initial closed-shell ELG method to be able to treat open-shell
systems such as organic molecule-based ferromagnets, which have many advantages andapplications.2, 3 We investigated the accuracy and efficiency of the developed open-shellelongation method for high-spin nonbonding butadiene radicals at the level of restricted
open-shell Hartree-Fock (ROHF). Furthermore, we investigated the organic ferromagnetism
using high-spin stability index Lijmin based on the nonbonding molecular orbitals (NBMOs)
obtained by ELG at the level of ROHF.
2. Method
2.1. Open-shell elongation method
A target system is divided into N units
and each unit is assumed to have an
unpaired electron with up spin (see Fig. 1).
The eigenvalue problem of a starting
cluster consisting of A and B parts issolved to initiate the elongation method.
Canonical MOs of the starting cluster islocalized into A1 part and B part. Next, an
attacking monomer M is added, and the eigenvalue problem of B and M is solved. After that,the B+M part is localized into A2 part and B part. The procedure is repeated until the desired
system size is reached. Cutoff technique can be included to speed up the calculations. As theeigenvalue problem is only solved for the B+M part, if the coupling between A1 and B + M
part is below threshold, A1 part can be ignored during the self-consistent-field process without
introducing big error in the next step. Finally, the total energy is calculated including thecutoff part.
2.2. High-spin stability index Lijmin
As it is still a challenging problem to obtain the exact energy difference between thelowest spin state and the highest spin state for a system with degenerate NBMOs, a high-spinstability index Lijmin was proposed for quick design of organic ferromagnets.4 Based on theNBMOs obtained by ELG method at the level of ROHF, we can obtain Lijmin as following:(1) Perform unitary transformation on NBMOs in iterative way as(1)
where ࣂ is the transformation parameter and the optimized ࣂ generates the Lijmin value that isthe minimum overlap of coefficients of NBMOs. As Lijmin varies with ࣂ, ࣂ should be
optimized in iterative way, and the detailed process of obtaining ࣂ can be seen in Ref. 4.
(2) Calculate the Lijmin value4, 5 as(2)where Lijmin corresponds to the minimum energy difference between the lowest spin (L) state
and the highest spin (H) state, that is, ΔE(L-H) = E(L) – E(H).
3. Results and discussion
We have performed calculations for model 1 in Fig. 2 at
the ROHF/6-31G level of theory. The distance between twounits is 6for the test. The number of units in the starting
cluster is 10, and 5 units are added each time. The first
cutoff calculation started from 25 units. The threshold ofN1N2Figure 2. Selected models. Black arrows
are unpaired electrons with up spin

density matrix is 10-8. The deviation of energy (per atom) from theconventional calculation is shown in Fig. 3a. The largest deviationis 1.90  10-8 a.u. per atom. As shown in Fig. 3b, the CPU time ofelongation method is faster than that of conventional method when
the system is big enough. We have also performed ELG
calculations at the level of ROHF/6-31G(d) for model 2 in Fig. 2
to obtain Lijmin, the number of units in the starting cluster is 6, and1 unit is added each time. ΔE(L-H) obtained by conventional
method at the ROHF/6-31G(d) level can indicate the level offerromagnetism. As shown in Fig. 4, ΔE(L-H) increases with Lijmin,which means Lijmin can indicate the degree of high-spin stability.
It indicates that the combination between open-shellelongation method and Lijmin index is promising for huge systems
and quick design of organic ferromagnet.
[1] Imamura, A.; Aoki, Y.; Maekawa, K. J. Chem. Phys. 1991, 95,

Figure 3. Comparison of energy
and CPU time between ELG andconventional methods at thelevel of ROHF/6-31G for model 1

5419-5431.
[2] Crayston, J. A.; Devine, J. N.; Walton, J. C. Tetrahedron 2000, 56,7829-7857.
[3] Barone, V.; Boilleau, C.; Cacelli, I.; Ferretti, A.; Monti, S.; Prampolini, G.
J. Chem. Theory Comput. 2013, 9, 300-307.
[4] Aoki, Y.; Imamura, A. Int. J. Quantum. Chem. 1999, 74, 491-502.
[5] Zhu, X.; Aoki, Y. Curr. Phys. Chem. 2013, 3, 99-112.

Figure 4. Variation ofΔE(L-H) with Lijmin
for model 2