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Calculation of Magnetic Couplings with Double-Hybrid Density Functionals Tobias Schwabe and Stefan Grimme* Organic Chemistry Institute, University of M€ unster, M€ unster, Germany
ABSTRACT The performance of double-hybrid density functionals for the computation of molecular magnetic coupling constants in the framework of a standard spin Hamiltonian is assessed. A variety of different ferromagnetically and antiferromagnetically spin-coupled test systems is considered. A significant and systematic improvement with respect to common and more recent functionals is found. SECTION Molecular Structure, Quantum Chemistry, General Theory
olecular (and in particular organic) magnets are promising new materials for optical and electronic devices.1-3 Therefore a good description of these systems on a quantum chemical level is desireable, which would support a more efficient rational design of new magnets. As a byproduct, one gains insight into the physics of the magnetic exchange (coupling) mechanisms. The key quantity for the problem is the magnetic (exchange) coupling constant (J) between magnetic sites. It is defined by the phenomenological Heisenberg-Dirac-van Vleck spin Hamiltonian X ^ HDVV ¼ Jij S^i 3 S^j ð1Þ H Æi, jæ
Table 1. Magnetic Couplings ( J) from B3LYP with Various Basis Sets
M
J in cm-1 system
lit.a,b
def2-SVP def2-TZVPP
H-He-H, 1.25 Å
-4860
-8393
-9139
-8894
H-He-H, 1.625 Å
-544
-994
-1069
-1054
H-He-H, 2.0 Å
-50
-109
-117
-116
1924
2218
2118
-1224 -91
-1185 -199
-1297 -167
R4-Dehydrotoluene 1050 - 1749 biverdazyl [Cu2Cl6]2-
-769 0 - (-40)
a Taken from ref 9 b He-systems: 6-31þþG(d,p) R-4-dehydrotoluene: 6-31þG(d) biverdazyl: 6-31G(d) (C, N, O) and 6-31þþG(d,p) (H) [Cu2Cl6]2-: 6-3111þG (Cu) and 6-31G(d) (Cl).
where S^i is the spin operator on center i and Æi, jæ indicates a summation over nearest neighbors. In routine computations of J, nowadays mainly density functional theory (DFT)-based methods are used, which provide a very good compromise between accuracy and computational effort. Accurate multireference wave function approaches are computationally too demanding for interesting systems, although CASPT2 can be used when high accuracy is needed and the applicable resource requirements can be fullfilled. Although Kohn-Sham DFT is inherently only a single-reference method, it can be applied to the problem with some success by using the broken symmetry (BS) approach. However, results are heavily influenced by the quality of the applied approximations to the exchange-correlation functional;a fundamental key in all DFT computations. We here demonstrate how more accurate and systematically improved magnetic couplings can be computed with the new family of double-hybrid density functionals (DHDFs).4-6 As shown below, they even outperform already reliable meta-hybrid7 and range-separated functionals.8 To assess the quality of magnetic couplings derived from DHDF computations, we investigated a standard database for this problem.9 It consists of three model systems, two organic biradicals, and seven bicopper organometallic complexes (see Table 2). Labels correspond to the respective CSDB reference codes.
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ref.a
Within the BS approach, several alternative formulations for J have been proposed,10-16 and they mostly differ in their dependence on the S^2 expectation value (which is related to an ongoing debate of the extent to which this value has a physical meaning in a Kohn-Sham framework; see, e.g., ref 17.). Recently, Moreira and Illas derived a formulation based on the Ising model for which no S^2 values are needed.16 For two magnetic centers with sz = 1/2, it reads J ¼ 2ðjBSæ - jTæÞ
ð2Þ
where |BSæ is the energy of the BS state, and |Tæ is that of the triplet state. We do not want to contribute further to the discussion of which formulation is more valid, but basically make a comparison to results with other functionals from the literature. Therefore all our computations are based on eq 2. First we looked at the basis set dependence of J using B3LYP18-20 for some of the smaller systems. Results are given in Received Date: February 17, 2010 Accepted Date: March 18, 2010 Published on Web Date: March 25, 2010
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DOI: 10.1021/jz100218y |J. Phys. Chem. Lett. 2010, 1, 1201–1204
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Table 2. Magnetic Couplings (J) for the Open-Shell Database from Experiment and with Different Density Functionals J in cm-1 system
ref.
H-He-H, 1.25 Å H-He-H, 1.625 Å
a
b
a
a
LC-ωPBE
M06
B3-LYP
-4860
-6176
-7834
-544
-718
-741
-50 1050 to 1749
-76 2963
biverdazylc
-769
[Cu2Cl6]2-
0 to -40 111
H-He-H, 2.0 Å dehydrotoluene
YAFZOU
BH-LYP
B2-PLYP
B2GP-PLYP
B2GP-LYP
-8393
-9139
-7091
-6892
-6317
-5710
-994
-1069
-827
-777
-705
-651
-41 2632
-109 1924
-117 2218
-89 3025
-83 1769
-75 1627
-69 4463
-1636
-1184
-1224
-1185
-2181
-214
32
-3216
-15
5
-91
-199
-45
-121
-61
1
154
294
194
223
101
164
123
69
2
1.5
3
4
2
0
-15
-11
0
-11
-39
-15
-61
36
17
15
19
18
XAMBUI PATFIA
B3-LYP
CAVXUS
-19
-14
-6
-21
-31
-8
-17
-14
-4
CUAQAC02 BISDOW
-286 -382
-273 -371
-436 -632
-429 -634
-462 -695
-144 -170
-262 -336
-177 -222
-87 -94
a
Taken from ref 9. b Taken from ref 26. c 1,10 ,5,50 -Tetramethyl-6,60 -dioxo-3,30 -biverdazyl.
slightly. This is in contrast to thermochemical5,27,28 and excited state29 applications. Note that the DHDFs are less sensitve to the amount of Fock-exchange than standard hybrid functionals. This can be deduced when differences between BH-LYP/B3-LYP and B2-PLYP/B2GP-PLYP are compared. The results also demonstrate that the perturbative correlation when based on Kohn-Sham orbitals is also rather robust for open-shell systems as opposed to standard MP2, which often fails in such cases.30 In conclusion we have shown that significant and systematic improvements for the theoretical determination of magnetic couplings can be obtained with double-hybrid density functionals. For a standard open-shell database, a better performance compared to two common hybrid functionals and even some improvements compared to two more recent functionals is documented. Once more, the general robustness of the DHDF approach and its applicability to general quantum chemical problems is underlined. Most computations for this work have been carried out with the TURBOMOLE program package,31-35 version 6.0. The B2GP-PLYP functional was implemented in TURBOMOLE, version 5.9. For the B2GP-PLYP BS computations, start densities from previous B2-PLYP runs were employed. For the Coulomb and exchange integrals as well as the correlation contribution, we made use of the resolution-of-the-identity36-38 integral approximation. All basis sets, including the auxiliary basis sets, were taken from the program libraries.21-25 All geometries are model or crystal structures and are the same for the |BSæ and the |Tæ state.
Table 1 together with values from the literature. It can be seen that the difference between the results for split valence basis set (def2-SVP)21-24 and the larger triple-ζ basis set (def2TZVPP)22-25 is smaller than that between def2-SVP and the (mixed) basis sets used previously. We think that def2-SVP allows a fair comparison with the previous data and it is used for all further computations. We shortly describe and motivate the DFT methods employed. We chose B3-LYP, the most common hybrid functional, to establish comparability to previous investigations. BH-LYP18 is applied to investigate the effect of different amounts of Fock-exchange (i.e., 50% vs 20% in B3-LYP) on the performance of hybrid functionals. We used the two double-hybrids B2-PLYP4 and B2GP-PLYP5 that differ by the amount of Fock-exchange and perturbative correlation. We also considered a method that only employs the hybrid density functional part of B2GP-PLYP without the additional perturbation correction (denoted as B2GP-LYP). All values are given in Table 2. Also for the complete set, a fair agreement between published and our computed values is found for B3-LYP. This supports our above-mentioned view that a comparison of all methods is feasible on a qualitative basis. As can be seen, the DHDFs significantly improve on the performance of common hybrid functionals. All J's are corrected into the right direction when the perturbative correlation contribution is added. For example, the value of J (dehydrotolouene) is 4463 cm-1 for B2GP-LYP, but 1627 cm-1 for B2GP-PLYP. On the other hand, the too low value of 69 cm-1 for J(YAFZOU) with B2GP-LYP is increased to 123 cm-1 with the respective DHDF. Similar trends can be found for B2-PLYP (53% Fock-exchange) in comparison to BH-LYP. B2-PLYP performs slightly better than B2GP-PLYP. Also with regard to the more recent functionals, LC-ωPBE and M06, we notice a systematic improvement, which is, however, most predominant for the organic systems. Obviously, the higher amount of Fock-exchange in B2GP-PLYP (65% in comparison to 53% in B2-PLYP) deteriorates the outcome
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AUTHOR INFORMATION Corresponding Author: *To whom correspondence should be addressed. E-mail: grimmes@ uni-muenster.de.
ACKNOWLEDGMENT We kindly acknowledge the help of Prof. Illas who provided us the geometries for all systems.
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DOI: 10.1021/jz100218y |J. Phys. Chem. Lett. 2010, 1, 1201–1204
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