On the SmCo Dimer: A Detailed Density Functional Theory Analysis

J. Phys. Chem. A 2010, 114, 1897–1905

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On the SmCo Dimer: A Detailed Density Functional Theory Analysis Hu¨seyin Oymak*,† and S¸akir Erkoç‡ BK21 Physics Program and Department of Physics, Chungbuk National UniVersity, Cheongju 361-763, Korea, Electrical & Electronics Engineering Department, Atilim UniVersity, 06836 Ankara, Turkey, and Department of Physics, Middle East Technical UniVersity, 06531 Ankara, Turkey ReceiVed: September 11, 2009; ReVised Manuscript ReceiVed: December 21, 2009

Making use of 21 different exchange-correlation functionals, we performed density functional theory calculations, within the effective core potential level, to investigate some spectroscopic and electronic features of the SmCo dimer in its ground state. A particular emphasis was placed on the (spin) multiplicity of SmCo. Most of the functionals under discussion unanimously agreed that the multiplicity of SmCo should be 10. It was observed that the nature of interaction between Sm and Co atoms to form the SmCo dimer can be described, to a good approximation, by a Lennard-Jones curve. For the multiplicity value 10, the binding energy De was seen to be in the range 1.08-1.77 eV, while the equilibrium separation distance and the fundamental frequency were found to be re ) 2.975 ( 0.035 Å and ωe ) 120 ( 10 cm-1, respectively. Introduction Thanks to their current and potential uses in industrial and technological applications, transition metal-rare earth alloys and compounds have given a constant impetus to scientists and researchers for more than three decades. Out of them, the Sm-Co alloy system1-3 is perhaps the most celebrated one, for they possess an extraordinarily high magnetocrystalline anisotropy, leading to large coercivities, which renders them valuable especially in designing high-density magnetic recording devices, high energy-product permanent magnet materials with high Curie temperature, high-performance hard magnetic thin films, etc.4-14 What makes the Sm-Co system so special is that their properties are all settled by the 3d-electrons of Co atoms and the unpaired, highly correlated, and localized 4f-electrons of Sm atoms; the former are understood to be responsible for the magnetization and the Curie temperature, and the latter provide the strong magnetocrystalline anisotropy, which is essential for the large energy-product.6-9,15 The Sm-Co binary alloy system has several reported phases: SmCo2, SmCo3, SmCo5, Sm2Co7, Sm2Co17, Sm3Co, Sm5Co2, Sm5Co19, and Sm9Co4, which are stable under standard atmospheric conditions and exhibit magnetocrystalline anisotropy fields of different strengths.1-3,9,13 Out of these, SmCo5 and Sm2Co17 are the two most important permanent magnetic materials as they possess the highest magnetocrystalline anisotropy (up to 20 × 106 J · m-3) among all the other known hard magnetic materials.9-12 Moreover, these two alloys exhibit an exceptionally high Curie temperature (Tc ) 1020 K), which makes them unrivalled particularly in high-temperature applications.9,12 However, due to their high reactivity, making them subject to fast oxidation, the syntheses of nanostructured SmCo5 and Sm2Co17 have been extremely difficult until recently. The literature witnessed many attempts to synthesize such Sm-Co nanoparticles. In 2007, using a surfactant-assisted ball milling technique, Wang et al. succeeded in obtaining hard magnetic nanoparticles, of different sizes with narrow size distribution, * Corresponding author. Tel: +90 312 586 83 87. Fax: +90 312 586 80 91. E-mail: [email protected]. † Chungbuk National University and Atilim University. ‡ Middle East Technical University.

based on the SmCo5 and Sm2Co17 systems.11 In the same year, Hou et al. reported a way for the synthesis of SmCo5 magnets by the high-temperature reductive annealing of core/shellstructured Co/Sm2O3 nanoparticles.12 Finally, very recently, in 2008, Chinnasamy et al. announced the successful production of ferromagnetic air-stable Sm-Co nanoparticles using a onestep chemical synthesis method, which was said to be environmentally friendly and readily scalable to large volume synthesis to meet the needs for the advanced permanent magnet applications; they reported the presence of uniform, anisotropic bladelike nanoparticles approximately 10 nm in width and 100 nm in length.9 A theoretical study of the Sm-Co alloy system would be ultimately timely after these monumental experimental achievements. A thorough investigation directly on the bulk Sm-Co system6-8 poses, however, a big challenge to the quantum mechanical methods since their properties are the results of the interaction of 3d- and 4f-electrons of totally different natures. Beginning with the present work, we take another way: we start from the (hypothetical) SmCo dimer as the building blocks of the actual Sm-Co alloy system. After a detailed exploration of the structural and electronic properties of the SmCo dimer, we propose to study, in the subsequent works, the other dimers and trimers of Sm and Co atoms, like Sm2, Co2, SmCo2, Sm2Co, Sm3, Co3, and their higher derivatives, including SmCo5 up to the Sm2Co17 nanoparticle, covering all the small aggregates that imitate all the phases of the Sm-Co alloy system. Thus, from separate atoms to their dimers and trimers, and in turn, to their higher derivatives, we aim to scrutinize the evolution of structural, electronic, and magnetic properties of small Sm-Co nanoparticles. This work reports the results of density functional theory calculations, within the effective core potential level, carried out for the (hypothetical) SmCo dimer. All the calculations are performed by using the GAUSSIAN 03 package.16 Using 21 different exchange-correlation functionals, some spectroscopic (binding energy, equilibrium interatomic separation, and fundamental frequency) and electronic properties (highest occupied and lowest unoccupied molecular orbitals and their corresponding gap energy, dipole moment, and excess charges on the

10.1021/jp908792f  2010 American Chemical Society Published on Web 01/12/2010

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J. Phys. Chem. A, Vol. 114, No. 4, 2010

atoms) of SmCo are presented. The main emphasis is placed on the multiplicities of separate Sm and Co atoms, and the SmCo dimer. At the end we attempt to describe the interaction between Sm and Co atoms via a Lennard-Jones curve. Method of Calculations In the present work density functional theory (DFT) is utilized to investigate the ground state of the SmCo dimer. It proved its usefulness especially when the standard quantum mechanical methods are computationally expensive or incapable for a particular atomic, molecular, or solid-state system. With its results agreeing sufficiently well with the observed data, DFT is today one of the most preferred methods in computational chemistry and physics.17-19 Here use three only-exchange functionals for the SmCo dimer calculations: HFS, XAlpha, and HFB, as they appear in GAUSSIAN 03. The HFS (Hartree-Fock-Slater) functional contains Slater’s theoretical proportionality constant R ) 2/3 to the density F4/3 to reproduce the HF (Hartree-Fock) exchange energy.17,18,20 It is commonly known as the “local spin density exchange” (LSDA) and regarded as an approximation to the Kohn-Sham procedure. The second functional XAlpha contains an empirical proportionality constant R ) 0.7 to F4/3.17,18,20 The last one, HFB (Hartree-Fock-Becke), is the exchange functional of Becke in which the exchange functional of Slater as well as the density gradient corrections21 are incorporated. The four standalone functionals, VSXC, HCTH93, HCTH147, and HCTH407, are tried in the present work. As their name implies, they are self-contained and are not used with any other functional. The VSXC (van Voorhis-Scuseria) functional is a gradient-corrected exchange-correlation functional that includes the improvement from the kinetic energy density τ.22 The remaining HCTH (Hamprecht-Cohen-Tozer-Handy) functionals are also gradient-corrected exchange-correlation functionals, which incorporate numerical exchange-correlation potentials, experimental energetics, and nuclear gradients.23-25 We also used 14 different, well-known, hybrid functionals, which merge the HF exchange with other DFT based exchangecorrelation. The B3LYP functional26 is Becke’s celebrated threeparameter hybrid functional, with the LYP (Lee-Yang-Parr) correlation functional, which includes both local and nonlocal correlation terms.27,28 The B3LYP functional is perhaps, to date, the most widely used exchange-correlation functional. In B3P86 and B3PW91 functionals, again Becke’s three-parameter functional26 is used, but with the respective gradient-corrected correlation functionals P86 (Perdew)29 and PW91 (PerdewWang).30-33 The functionals B1B95 and B1LYP contain Becke’s one-parameter hybrid functionals;34 the former includes the gradient-corrected, τ-dependent B95 correlation functional. The B1LYP functional was implemented by Adamo and Barone.35 Another hybrid functional mPW1PW91 combines the PW91 correlation functional with the mPW (modified Perdew-Wang) exchange functional, due again to Adamo and Barone.36 There are three hybrid functionals that are revisions or modifications to Becke’s and Schmider’s B97 hybrid functional:37,38 B98,38 B71,23 and B972.39 The PBE1PBE functional,40,41 also known as PBE0, contains the gradient-corrected correlation functional PBE (Perdew-Burke-Ernzerhof) and was hybridized by Adamo;42 it imposes an exchange to correlation ratio 1/3. O3LYP is another three-parameter hybrid functional, like B3LYP, by Cohen and Handy.43 We also use two half-and-half functionals, BHandH and BHandHLYP; the former contains the HF and LSDA exchange functionals with the LYP correlation functional and, in addition to these, also involves Becke’s density gradient

Oymak and Erkoç corrections.21 And the final one, BMK (Boese-Martin), is a τ-dependent hybrid functional.44 An integral part of this work is the use of the compact effective potential (CEP) basis functions with ECP triple-split basis, namely, CEP-121G basis functions, which are now being widely exploited in quantum chemistry, especially for calculations in the study of compounds containing heavy elements.45-47 They are successfully used to calculate the equilibrium structures and spectroscopic properties of several small molecules.45 In the effectiVe core potential (ECP) or pseudopotential methods, only the valence electrons of an atom are explicitly treated rather than considering all of its electrons; the effects of the remaining nonvalence electrons and the nucleus (i.e., the “core”) of the atom, which are almost always quite involved, are supposed to modify the “effective” (pseudo)potential in which the valence electrons move. These methods are therefore an indispensable remedy when treating systems with atoms having too many electrons. Because of their accuracy and reliability, the ECP methods have been highly popular among chemists for many years; they give quite satisfactory results with errors, relative to the experiment, comparable to corresponding ones obtained from all-electron ab initio methods.48 (The interested reader is referred to two important reviews for more detailed information on the ECP methods: ref 49 and, the recent one, ref 50. The latter, with its references to another reviews, is especially valuable.) We should mention that although GAUSSIAN 03 offers a wide choice of basis functions to use, most of them are unavailable to the 62-electron Sm atom of this work. Only three classes of basis functions happen to be suitable for the SmCo dimer; one of them is the CEP-121G basis functions used in this work, and the others are the UGBS (universal Gaussian basis set)51 and the SDD (Stuttgart-Dresden) basis functions (the latter are also ECP-based).52 We made an attempt to use SDD for SmCo, but it was computationally far more expensive than CEP-121G. We did not try UGBS. Results and Discussion This work starts with a thorough investigation on the ground states of separate Co and Sm atoms. We aim to determine which methods, among the 21 DFT methods under discussion, are capable of giving the proper ground state, with the correct multiplicity, which possesses the minimum total energy. We attach importance, to some extent, to this preliminary study, for if any particular method is to result in valuable insight into the properties of SmCo dimer, it is intuitively reasonable to expect that method to lead to the standard ground-state multiplicities for the separate atoms. As a byproduct, we shall later use the minimum energy values obtained in this part in constructing the Lennard-Jones potential curves for the SmCo dimer. To this end, we performed self-consistent-field (SCF) calculations using the 21 DFT methods, with CEP-121G basis, for the isolated Co and Sm atoms. As we shall see shortly, some methods for some multiplicity values did not yield any definite result, due to sometimes a nonconvergent- or occasionally a confused-SCF process. When we were confronted with such cases, we did not resort to any further way (e.g., removing symmetry condition by using “nosymm” keyword or using the quadratic convergence method, etc.) to force the calculation to converge or yield a definite result. Throughout this work we followed this precept, thus providing consistency in the obtained results. (A SCF calculation with or without symmetry condition, for example, was seen to end up usually with a significant discrepancy in the self-consistent energy values.)

DFT Analysis of the SmCo Dimer

J. Phys. Chem. A, Vol. 114, No. 4, 2010 1899

TABLE 1: Total Energies, Em (Hartrees), for a Co Atom for Different Multiplicities, m (2S + 1)a method

-E2

-E4

-E6

-E8

-E10

-E12

HFS XAlpha HFB VSXC HCTH93 HCTH147 HCTH407 B3LYP B3P86 B3PW91 B1B95 B1LYP mPW1PW91 B98 B971 B972 PBE1PBE O3LYP BHandH BHandHLYP BMK

143.6035 144.2851 144.4087 145.1235 145.5236 145.5663* 145.6075* 145.1134 145.4932 145.1410* 145.0915 145.0030 145.1156* 145.0683 145.0281 145.2484* 145.0443* 145.2852 144.5443* 144.9438 144.4685*

nc 144.3068* 144.4319* nc nc 145.5525 145.5650 145.1243* 145.5072* 145.1409 nc 145.0132* 145.1153 145.1060* 145.0657* 145.2441 145.0438 145.2966* 144.5432 144.9525* 144.4560

143.5406 144.2059 144.3398 145.0044 145.3905 145.4327 145.4760 144.9957 145.3901 145.0530 144.9889 144.8901 145.0333 144.9789 144.9362 145.1342 144.9608 145.1619 144.4166 144.8507 144.3925

143.3099 143.9668 144.0880 144.7307 145.1026 145.1401 145.1838 144.7171 145.1071 144.7806 144.7087 144.6153 144.7511 144.6945 144.6509 144.8514 144.6797 144.8794 144.2029 144.5939 144.0752

142.9205 143.5685 143.6643 144.2865 144.6422 144.6765 144.7243 144.2683 144.6658 144.3313 144.2683 144.1686 144.3189 144.2311 144.1864 144.3931 144.2493 144.4298 143.7920 144.1651 143.5844

139.9752 140.6099 140.6998 141.2186 141.5871 141.6262 141.6709 141.2179 141.6178 141.2912 141.2229 141.1180 141.2794 141.1827 141.1359 141.3251 141.2115 141.3670 140.7446 141.1081 140.5684

∆E2-4 +0.5905 +0.6313 -0.3755 -1.1565 +0.2966 +0.3810 -0.0027 +0.2776 -0.0082 +1.0259 +1.0231 -0.1170 -0.0136 +0.3102 -0.0299 +0.2367 -0.3401

a The starred ones show the lowest energy. “nc” indicates a “not completed” calculation due to a nonconvergent-SCF or confused-SCF process. The energy difference ∆E2-4 ) E2 - E4 is in eV.

TABLE 2: Total Energies, Em (Hartrees), for a Sm Atom for Different Multiplicities, m (2S + 1)a method

-E1

-E3

-E5

-E7

-E9

-E11

∆E5-7


79.9584 80.3755 80.9181 nc 81.5766 81.5997 81.6132 76.8073 81.4798 81.1398 80.9244 80.8706 81.0401 81.1156 81.1120 81.1721 80.9677 nc 80.1091 80.4275 80.6300

80.1926 80.6164 80.8771 80.5602* 81.5506 81.5526 81.6708 81.2668 81.6100 81.1674 81.1521 81.0906 81.0845 81.0910 81.0753 81.0925 nc 81.3007 80.2103 80.6351 80.6493

80.5960* nc 81.1737 80.4904 81.8689* 81.9173 81.9213* 81.2887 81.6808 81.3620 nc 81.1463 81.2624 81.2672 81.2473 81.3405 81.2102 81.5345 80.3652 80.7377 80.7412

80.5705 nc 81.2635* 70.7348 81.8171 81.9231* nc 81.4192* 81.7788* 81.4632* 81.4755* 81.2367* 81.3640* 81.3809* 81.3613* 81.4632* 81.3172* 81.6235* 80.5406* 80.8198* 80.8522*

69.2956 69.4772 69.4024 70.9952 nc nc 81.8241 69.9502 70.2344 81.3845 81.4269 69.8524 81.2962 81.2884 81.2667 81.3677 81.2437 70.0162 80.4637 80.7398 80.7596

79.0995 80.5983 nc 78.3544 81.3485 81.3935 81.4414 80.6688 81.0605 80.7492 80.8285 80.4743 80.6567 80.6471 80.6228 80.7262 80.6039 80.9413 79.8222 80.0053 80.1079

-0.6939

∆E9-7

+2.4436 -1.4095 +0.1578 +3.5511 +2.6667 +2.7538 +2.4599 +2.7647 +3.0939 +3.1021 +3.3388 +2.9116 +2.4218 +4.7729 +2.2341 +3.0205

+2.1415 +1.3225 +1.8449 +2.5171 +2.5742 +2.5987 +2.0000 +2.0926 +2.1769 +2.5198

a

The starred ones show the lowest energy. “nc” indicates a “not completed” calculation due to a nonconvergent-SCF or confused-SCF process. The energy differences ∆Ei-k ) Ei - Ek are in eV.

The results are tabulated in Tables 1 and 2, which give selfconsistent energy values for a series of possible multiplicity values for each atom. The ground-state electron configuration of the Co atom is [Ar] 3d74s2. Because of its three unpaired 3d-electrons, the expected multiplicity, 2S + 1 (hereafter we shall use m for 2S + 1 and Em for the corresponding total energy), for Co is m ) 4 (its ground-state level is given as 4 F9/2). We note that, since a Co atom has totally 27 electrons, the possible multiplicity values that we should test are like 2, 4, 6, .... Of course, the methods used are different and so are the corresponding energy values. The first feature to be observed in Table 1 is that not all the methods lead to the expected m ) 4 result; only nine methods, namely XAlpha, HFB, B3LYP, B3P86, B1LYP, B98, B971, O3LYP, and BHandHLYP, succeed in identifying the Co atom with multiplicity 4 as possessing the lowest energy value, E4.

We observe that the general trend among the different multiplicities is that the self-consistent energy values from E6 to E12 are always increasing, sometimes drastically, especially in passing from E10 to E12. It is also to be noted that the E6 value, for a specific method, is always more than the E2 value; the difference between them varies from 1.7116 to 3.6354 eV and is thus significant. Therefore, we are left with the conclusion that either the m ) 4 or m ) 2 case gives the lowest energy, as is clear from the Table 1. To what extent is a method successful or not in picking the m ) 4 case to possess the lowest energy? To answer this question, we tabulated in the last column of Table 1 the energy difference between E4 and E2, where the plus sign means success, and the minus sign failure. We see that all the standalone functionals failed to give the desired result even for only one Co atom. This is an early indication that we should

1900


not give much credence to these methods in dealing with the subsequent SmCo dimer calculations, because the ∆E2-4 values, among the successfully converged calculations, are -0.3755 eV for HCTH147 and -1.1565 eV for HCTH407, not insignificant discrepancies. A similar conclusion might be drawn for BMK and B972 methods, for they also give ∆E2-4 ) -0.3401 and -0.1170 eV values, respectively, though the latter is less severe. The remaining four unsuccessful hybrid functionals, B3PW91, mPW1PW91, PBE1PBE, and BHandH, with their respective ∆E2-4 values as -0.0027, -0.0082, -0.0136, and -0.0299 eV, seem to have insignificant, or tolerable, discrepancies in spotting E2, instead of the expected E4 result, as being the lowest energy. On the other hand, it is clearly plain from Table 1 that all the hybrid functionals with an -LYP correlation part are always thriving in predicting the lowest E4 result; with their ∆E2-4 values 0.2966 eV for B3LYP, +0.2776 eV for B1LYP, +0.3102 eV for O3LYP, and +0.2367 eV for BHandHLYP, all being significant, sharp, and clear. In this respect two only-exchange functionals XAlpha (∆E2-4 ) +0.5905 eV) and HFB (∆E2-4 ) +0.6313 eV), as well as other well-known hybrid functionals B3P86 (∆E2-4 ) +0.3810 eV), B98 (∆E2-4 ) +1.0259 eV), and B971 (∆E2-4 ) +1.0231 eV) manage to acclaim the desired lowest E4 result as being the champion among the other possible multiplicities. We here do not make any comment on the B1B95 method for the time being; there will be some later comments in the discussion of the SmCo dimer. Before passing to the Sm atom, it might be convenient to mark XAlpha, HFB, B3LYP, B3P86, B1LYP, B98, B971, O3LYP, and BHandHLYP as “reliable” candidate methods to perform the subsequent SmCo dimer calculations, and B3PW91, mPW1PW91, PBE1PBE, BHandH, and B972 as “suspicious” candidates. The remaining ones already proved unavailing. With the similar outcomes from the Sm atom, this list will be tougher, as we shall see shortly. What about the isolated Sm atom? It has totally 62 electrons and its ground-state electron configuration is [Xe] 4f66s2. With its 6 unpaired 4f-electrons, a Sm atom has the expected multiplicity m ) 7, along with its ground-state level 7F0. Because of an even number of total electrons, the potential multiplicity values to be tested are 1, 3, 5, .... Calculations for a Sm atom seem more fruitful than those for a Co atom, for, as seen from Table 2, this time only five methods, namely HFS, XAlpha, VSXC, HCTH93, and HCTH407 did not spot the expected E7 result as being the lowest energy; all the remaining methods are definitely successful. Contrary to the Co case, there appears no definite pattern among different energies for the Sm atom. To measure quantitatively the success of a method in picking the m ) 7 case to possess the lowest energy, we also tabulate in Table 2 the energy difference between E7 and E5 and between E7 and E9 (wherever it is meaningful), with the same ( sign convention used in Table 1. Except for HFB, which has ∆E5-7 ) +2.4436 eV, the only-exchange functionals are insufficient to predict E7 as the lowest energy; HFS picks E5 as the lowest one, with a significant discrepancy of ∆E5-7 ) -0.6939 eV, while XAlpha fails to converge even for E7 and E5. Apart from HCTH147 for which ∆E5-7 ) +0.1578 eV, the standalone functionals are seen as weak for Sm, just as they for Co; HCTH93 and HCTH407 say E5 is the minimum energy, with a large discrepancy of ∆E5-7 ) -1.4095 eV for the former, whereas VSXC chooses the unexpected E3 as the minimum one. It is striking to observe from Table 2 that the desired m ) 7 result is decisively picked as having the lowest energy by all the 13 hybrid functionals (this automatically includes the hybrid functionals with an -LYP correlation part), though there happens

Oymak and Erkoç to be some nonconvergent results for B1B95, PBE1PBE, and O3LYP. This unanimity among the hybrid functionals is marked by significantly large ∆E5-7 and ∆E9-7 values. We here also draw attention to sometimes large and sometimes drastically lower E9 values, which is possible only with eight unpaired electrons and this, in turn, requires one of the two 6f-electrons be transferred to 4f energy level. Except B3LYP, B3P86, B1LYP, and O3LYP, the remaining hybrid functionals do not exclude this likelihood. We also finally notice that, for Sm, functionals with a B- exchange part (including HFB) all have not failed in leading to the lowest E7 result. According to the above discussion, we here mark HFB, HCTH147, B3LYP, B3P86, B3PW91, B1LYP, mPW1PW91, B98, B971, B972, PBE1PBE, O3LYP, BHandH, BHandHLYP, and BMK as “reliable” candidate methods to perform the subsequent SmCo dimer calculations, and B1B95 as a “suspicious” candidate. Comparing this list with the previous one for Co, we mark HFB, B3LYP, B3P86, B1LYP, B98, B971, O3LYP, and BHandHLYP as the “toughest” candidates for the SmCo dimer calculations; it should not be unexpected that they will give the most reliable information about the SmCo dimer. Now we come to the SmCo dimer for which we carried out two sets of large calculations. In the first set, surmising somehow that the interaction between Sm and Co atoms is Lennard-Jones type and that the interatomic separation re in equilibrium is in the range 1.0-5.0 Å, we set the initial interatomic distance r to a value between this interval and asked GAUSSIAN 03 to geometrically optimize the SmCo dimer by performing SCF calculations, with the CEP-121G basis. We did this for each one of the 21 DFT methods under discussion, for seven different possible multiplicities from m ) 2 to 14 (note the odd total electron number in SmCo), and 41 different initial interatomic distances, from r ) 1.0 Å until 5.0 Å, increasing by 0.1 Å in each subsequent calculation. Our aim is to determine, for a specific method, the multiplicity value at which the total energy assumes its minimum and, in turn, to determine, for the soobtained multiplicity value, the spectroscopic constants (binding energy De, equilibrium interatomic separation re, and fundamental frequency ωe) and some electronic properties [highest occupied molecular orbital (HOMO), lowest unoccupied molecular orbital (LUMO), HOMO-LUMO gap energy, dipole moment, and excess charges on the atoms] of the SmCo dimer. Because especially of the 62-electron Sm, a SmCo dimer calculation, even with CEP-121G basis, is a demanding task. Not all of the 41 calculations for a given multiplicity and method manage to converge properly, leaving the calculation uncomplete. For example, only two for VSXC (for the m ) 8 case), eight for HCTH147 (m ) 2), 12 for B1B95 (m ) 8), and 20 calculations for BMK (m ) 6) are seen to complete appropriately. Sometimes we could not obtain even a single complete calculation, e.g., VSXC (m ) 10), HCTH93 (m ) 12), and HCTH407 (m ) 4). In this respect the only-exchange functionals (HFS, XAlpha, and HFB) give a fairly moderate and the standalone functionals (VSXC and HCTH’s) a usually poor performance. As may be expected, the remaining hybrid functionals’ accomplishments are usually great. For instance, 41 for B3LYP (m ) 8) and 36 calculations for PBE1PBE (m ) 10) are complete. Given a particular method and multiplicity, a SmCo dimer optimization calculation usually leads to some small number of specific total energy and corresponding optimized interatomic separation values. For example, for all of the 39 converged calculations (among totally 41 trials) of B3LYP (m ) 10), total energy is between -226.5801 and -226.5884 hartrees, with



TABLE 3: Total Energies, Em (Hartrees), for the SmCo Dimer for Different Multiplicities, m (2S + 1)a method

-E2

-E4

-E6

-E8

-E10

-E12

-E14

∆E6-10


224.3362 225.4645 225.6692 225.9035 227.4711 227.3297 227.3926 226.5069 227.1656 226.5329 226.5598 226.1952 226.4112 226.3477 226.2896 226.5141 226.2841 226.7678 224.8767 225.6872 225.2484

224.4053 225.5358 225.7302 225.9322 227.5626 227.6602 nc 226.5832 227.3348 226.6633 226.6332 226.2942 226.5399 226.5219 226.4629 226.7610 226.4188 226.9734 225.1361 225.8080 225.3593

224.4167* 225.5547* 225.7322 225.9410* 227.5823* 227.6859* 227.7645* 226.5856 227.3369 226.6652 226.6367 226.2909 226.5414 226.5177 226.4655 226.7642 226.4204 226.9750 225.1378 225.8054 225.3802*

224.3992 225.5346 225.7233 225.7757 227.5706 227.6728 227.7426 226.5788 227.3296 226.6579 226.6071 226.2898 226.5343 226.5183 226.4592 226.7570 226.4132 226.9677 225.1303 225.7999 225.3653

224.4076 225.5413 225.7352* nc nc nc nc 226.5884* 227.3397* 226.6680* 226.6405* 226.2992* 226.5445* 226.5283* 226.4690* 226.7667* 226.4236* 226.9779* 225.1415* 225.8129* 225.3668

224.3688 225.5034 225.6820 225.8220 nc nc nc 226.5253 227.2784 226.6104 226.5733 226.2407 226.4886 226.4577 226.4028 226.6944 226.3712 226.9093 225.1053 225.7792 225.2818

224.2934 225.4250 225.6161 225.7019 227.3933 nc nc 226.4283 227.1936 226.5298 226.4988 226.1451 226.4144 226.3590 226.2949 226.5820 226.2909 226.8163 225.0175 225.6883 225.2431

-0.2476 -0.3646 +0.0816

+0.0762 +0.0762 +0.0762 +0.1034 +0.2259 +0.0844 +0.2884 +0.0952 +0.0680 +0.0871 +0.0789 +0.1007 +0.2041 -0.3646

a The starred ones show the lowest energy. “nc” indicates a “not completed” calculation due to a nonconvergent-SCF or confused-SCF process. The energy difference ∆E6-10 ) E6 - E10 is in eV.

corresponding optimized interatomic separation values between 2.9573 and 3.0221 Å, most of them being in the middle of this interval. Another example is HCTH93 (m ) 8), for which there are nine results with -227.5705 hartrees and r ∼ 2.33 Å, one result with -227.5658 hartrees and r ) 2.3436 Å, two results with -227.5588 hartrees and r ∼ 2.50 Å, and seven results with -227.5563 hartrees and r ∼ 2.98 Å. A final example is HFS (m ) 6), for which all the 16 converged results are the same: -224.4167 hartrees and r ) 2.3522 Å. Table 3 shows the so-obtained self-consistent energy values for a series of possible multiplicity values for SmCo. The most prominent feature seen from this table is that all the hybrid functionals but BMK indicate that the ground state of SmCo should be the one with multiplicity 10, while the remaining onlyexchange and standalone functionals, including also BMK, indicate it is the one with multiplicity 6. A ground state for SmCo with multiplicity 10 means that the ingredient atoms Sm and Co do not change their individual electronic nature drastically in coming together to form SmCo so that they somehow keep their original respective multiplicities 7 and 4. This should be plausible, because, put in another way, the 4f orbitals of Sm and the 3d orbitals of Co are somewhat different in nature and are in different energy levels, the electronic interaction between the six 4f-electrons of Sm and seven 3delectrons of Co is only fairly strong to form SmCo; the change in electronic configuration of the individual atoms is barely discernible. It is under these circumstances that we may expect to describe the interaction between Sm and Co atoms as being Lennard-Jones type, yet only to a good approximation. If this happens to be the case, we may further reckon the binding energy De to be necessarily small and the equilibrium separation value to be a moderate one. We will later say more about these points. We see from Table 3 that energies are arranged in some particular orders. For B3P86, mPW1PW91, B971, B972, PBE1PBE, O3LYP, and BHandH, the order is E10 < E6 < E4 < E8 < E12 < E14 < E2. For HFB, B3LYP, B3PW91, and B1B95, only E14 and E2 are interchanged in this order. That is to say, most of the methods, especially the hybrid functionals, including the only-exchange functional HFB, do agree on their results.

Even all 21 methods result in the same four lowest energies as (E10 or E6) < (E8 or E4); only the ordering among them is different at times. Agreeing that the ground state of SmCo should be linked with the multiplicity m ) 10, it is clear from Table 3 that all the methods in our “toughest” list, i.e., HFB, B3LYP, B3P86, B1LYP, B98, B971, O3LYP, and BHandHLYP (along with B3PW91, B1B95, mPW1PW91, B972, PBE1PBE, and BHandH) are flawlessly successful, as is clearly seen from their sometimes small but usually significant ∆E6-10 values. Our predictions from Tables 1 and 2 about the SmCo calculations and the methods to be used for them hold. According to our previous discussions, it is not unexpected to see that two of the only-exchange functionals, HFS and XAlpha, fail to lead to mark E10 as being the lowest energy; their respective ∆E6-10 values are not insignificant at all: -0.2476 and -0.3646 eV. That all the standalone functionals, with their several incomplete calculations, produce E6 as being the lowest energy, instead of E10, is unsurprising, too, for they already proved futile for the SmCo calculations. Finally, the hybrid functional BMK gave an initial sign in Table 1 that it might not be powerful for SmCo; it led to the m ) 6 result, with a not insignificant ∆E6-10 ) -0.3646 eV value. The spectroscopic constants of the SmCo dimer calculated for the multiplicity value at which the total energy is minimum are presented in Table 4, which is now arranged according to the multiplicity, m, values. To our best knowledge, there exist no experimental data nor theoretical outcomes in the literature for the SmCo dimer. Consequently, Table 4 contains only the results of this work, which are seemingly the first to appear in the literature. We begin with the binding energy De, which is SmCo - E7Sm - E4Co), calculated, e.g., for HFB as De ) -C(E10 where C is the hartree-to-electronvolt conversion factor, C ) SmCo , E7Sm, and E4Co are 27.211383 eV/Hartree; the values for E10 taken from Tables 3, 2, and 1, respectively. Similarly, for Co - ESm mPW1PW91, it is De ) -C(ESmCo 10 7 - E2 ), and for BMK, De ) -C(E6SmCo - E7Sm - E2Co). The scheme is obvious: always the minimum energies, for a particular method, in Tables 1-3 are involved in this calculation. Whenever the data from Tables 1 and 2 seem unhealthy or not convincing, we prefer not to

1902


Oymak and Erkoç

TABLE 4: Spectroscopic Constants of the SmCo Dimer Calculated for the Multiplicity, m (2S + 1), at Which the Total Energy Is Minimuma method

m

De

re

ωe

q(Sm)

µ

HFB B3LYP B3P86 B3PW91 B1LYP mPW1PW91 B98 B971 B972 PBE1PBE O3LYP BHandH BHandHLYP B1B95 HFS XAlpha VSXC HCTH93 HCTH147 HCTH407 BMK

10 10 10 10 10 10 10 10 10 10 10 10 10 10 6 6 6 6 6 6 6

1.0841 1.2206 1.4622 1.7378 1.3408 1.7658 1.1283 1.1409 1.4975 1.6872 1.5732 1.5410 1.1051

3.0719 2.9671 2.9465 2.9686 2.9892 2.9819 3.0076 3.0062 3.0055 2.9818 3.0098 2.9705 3.0870 2.9828 2.3523 2.3437 2.9026 2.5145 2.4993 2.5322 4.1571

118.6 126.3 129.7 126.9 122.7 124.5 117.6 117.8 119.5 124.1 120.2 122.0 111.3 123.3 189.4

0.6309 0.6291 0.6223 0.6187 0.6271 0.6196 0.5870 0.5845 0.5760 0.6196 0.6300 0.6283 0.6095 0.6161 1.0098 0.9989 0.6016 0.9777 0.9753 0.9648 0.1692

4.7749 4.4700 4.2422 4.2144 4.4911 4.1847 3.8193 3.7878 3.6391 4.1240 4.0206 4.2536 4.4178 4.2479 5.3683 5.3979 3.7701 6.1217 6.0765 5.9727 2.1352

5.3470 1.6186

134.8 137.5 131.5 54.4

a

Binding energy De is in eV, equilibrium interatomic separation re is in Å, and the fundamental frequency ωe is in cm-1. Also given are the calculated excess charge q(Sm) on the Sm atom (in units of electron charge |e|) and the dipole moment µ (Debyes). Note that for the excess charge on the Co atom, we have q(Co) ) -q(Sm).

calculate De, as in the cases of B1B95, HFS, XAlpha, VSXC, HCTH93, and HCTH407. We see that the methods considered in this work reach no consensus about the binding energy De of SmCo; the De values in Table 4 are in a somewhat bad range of 1.08-1.77 eV for m ) 10. The situation goes from bad to worse if we look at the standalone HCTH147 method, which indicates an abnormal 5.35 eV De value for m ) 6. We surprisingly notice that even similar methods lead to no close results at all for De, for example, B971 (1.1409 eV) and B972 (1.4975 eV), and BHandH (1.5410 eV) and BHandHLYP (1.1051 eV). Although it is unsafe here to give an average De value for SmCo, since the just mentioned range is undesirably large, we should nonetheless point out that the binding energies for m ) 10 are indeed small values, as we mentioned before. As seen from Table 4, contrary to the binding energy case, the values for the Sm-Co bond, re, obtained for the multiplicity m ) 10 are all in a good range of 2.94-3.09 Å. It seems safe to give a more compact figure as re ) 2.975 ( 0.035 Å for the Sm-Co bond length. We note that the re values are really not so small or large values, as we argued before. This time similar methods give close results, for example, B3LYP (2.9671 Å) and B1LYP (2.9892 Å), and B98 (3.0076 Å), B971 (3.0062 Å), and B972 (3.0055 Å), maybe except BHandH (2.9705 Å) and BHandHLYP (3.0870 Å). On the other hand, the re values calculated for the m ) 6 case are not in agreement at all; they are scattered in the interval from 2.3437 Å for alpha to 4.1571 Å for BMK. Nevertheless, we note that all the three HCTH methods give an re value around 2.5 Å. The situation for the fundamental frequency ωe is the same as that for the bond length: while the ωe values for m ) 10 are all in close proximity to each other, those for m ) 6 are again spread over a large interval, which is 54-190 cm-1. It is to be noted that, except for De but including q and µ values, the results of the hybrid functional BMK are persistently far more different than those of the other methods, no matter what the multiplicity is. For the multiplicity m ) 10, the fundamental frequency

values in Table 4 can be given compactly as ωe ) 120 ( 10 cm-1. We note in passing how the three Becke methods, B98 (117.6 cm-1), B971 (117.8 cm-1), and B972 (119.5 cm-1) give very close results. In Table 4, although the binding energy De values are spread over a little wide range, all the bond length, re, and fundamental frequency, ωe, values (as well as q and µ values) for the multiplicity m ) 10 are consistently close to each other or are spread over a reasonably narrow range. This implies, apart from their consistency, the reliability of the methods (HFB, B3LYP, B3P86, B3PW91, B1LYP, mPW1PW91, B98, B971, B972, PBE1PBE, O3LYP, BHandH, and BHandHLYP; the bold ones are much more recommended as being the “toughest” ones) to scrutinize the features of SmCo. Just for an example, a skillfully designed CASSCF study with, for example, the hybrid functional B3LYP is highly likely to lead to far more improved results for SmCo. Table 4 summarizes also the calculated excess charge on Sm, q(Sm), and the closely related quantity dipole moment, µ, for SmCo. [The excess charge on the Co atom is q(Co) ) -q(Sm).] The appeared values for m ) 10 are q(Sm) ) 0.6 ( 0.03|e| and µ ) 4.2 ( 0.6 D; the latter has more uncertainty than the former. A net charge separation and the resultant dipole moment are due largely to the unpaired electrons that reside in not much spoiled Sm and Co ingredients of the SmCo dimer. We note again that B3LYP (0.6291|e| and 4.4700 D) and B1LYP (0.6271|e| and 4.4911 D), and B98 (0.5870|e| and 3.8193 D), B971 (0.5845|e| and 3.7878 D), and B972 (0.5760|e| and 3.6391 D), give close outcomes for the respective q(Sm) and µ values. This time, except for VSXC and BMK, the figures of q(Sm) for m ) 6 display a noticeable unanimity around 0.98|e|. The same situation holds, though not that much pronounced, for the dipole moment, µ, figures, which are centered around 5.8 D. We calculated HOMO-LUMO gaps of the SmCo dimer for the multiplicities at which the total energy is minimum, which are given in Table 5. The HOMO-LUMO gap of SmCo quantifies the measure of its excitability; the bigger the gap, the more it can be excited, and the more, in turn, stable it is, and vice versa. In this respect, the HOMO, LUMO (the “frontier” orbitals), and the resultant HOMO-LUMO gap regulate the manner in which the SmCo dimer interacts with any other atom(s) or species. Also, the HOMO-LUMO gap, which is a measurable quantity by means of some spectroscopic techniques, is especially helpful in connecting the optimized geometry of any species to its higher derivatives; in the present work, it paves the way for a more detailed exploration from the SmCo dimer to the industrially and scientifically important SmCo5 and Sm2 Co17, which will be the subjects of our next studies. Table 5 reports just our preliminary results for SmCo, though not necessarily informative for the time being. We immediately notice that the only-exchange functional HFB, among the m ) 10 results, leads to Eg(R) and Eg(β) values significantly lower than the remaining methods. The reason for this may be attributed to its lack of correlation energy part. On the other extreme, the same quantities from the hybrid functionals BHandH and BHandHLYP are much larger than the others. For the all remaining methods for m ) 10, the figures are unanimous to some moderate degree: Eg(R) ) 2.55 ( 0.3 eV and Eg(β) ) 2.55 ( 0.7 eV, with a bigger uncertainty for the latter. The situation for the m ) 6 results is again the same as that in Table 4: the figures are badly scattered, seemingly obeying no principle or regulation. It is worth noting once more that the figures



TABLE 5: HOMO and LUMO Energies (Hartrees), and HOMO-LUMO Gap (Eg) Energies (eV) of the SmCo Dimer, Calculated for the Multiplicity, m (2S + 1), at Which the Total Energy Is Minimum method

m

-HOMO (R)

-LUMO (R)

Eg(R)

-HOMO (β)

-LUMO (β)

Eg(β)

HFB B3LYP B3P86 B3PW91 B1LYP mPW1PW91 B98 B971 B972 PBE1PBE O3LYP BHandH BHandHLYP B1B95 HFS XAlpha VSXC HCTH93 HCTH147 HCTH407 BMK

10 10 10 10 10 10 10 10 10 10 10 10 10 10 6 6 6 6 6 6 6

0.0922 0.1208 0.1430 0.1238 0.1193 0.1268 0.1186 0.1156 0.1144 0.1253 0.1072 0.1353 0.1385 0.1212 0.0537 0.0658 0.0977 0.0665 0.0689 0.0666 0.1481

0.0552 0.0268 0.0471 0.0271 0.0167 0.0221 0.0235 0.0231 0.0216 0.0222 0.0244 0.0040 0.0060 0.0381 0.0391 0.0488 0.0441 0.0544 0.0611 0.0635 0.0265

1.0068 2.5584 2.6090 2.6338 2.7919 2.8485 2.5873 2.5165 2.5241 2.8041 2.2528 3.7914 3.9312 2.2602 0.3978 0.4618 1.4569 0.3290 0.2147 0.0844 3.3084

0.0704 0.1566 0.1793 0.1566 0.1619 0.1661 0.1648 0.1609 0.1563 0.1663 0.1254 0.1817 0.1792 0.1627 0.1072 0.1129 0.1124 0.1184 0.1245 0.1259 0.1351

0.0402 0.0606 0.0748 0.0523 0.0518 0.0468 0.0717 0.0733 0.0735 0.0498 0.0576 0.0286 0.0321 0.0465 0.0261 0.0288 0.1056 0.0485 0.0546 0.0601 0.0242

0.8234 2.6115 2.8444 2.8398 2.9938 3.2474 2.5315 2.3853 2.2528 3.1696 1.8463 4.1685 4.0017 3.1609 2.2066 2.2896 0.1842 1.9023 1.9007 1.7905 3.0186

resulting from the hybrid functional BMK are much more different than those for the remaining m ) 6 cases. In the second set of SmCo calculations we carried out “singlepoint” SCF computations, again with the CEP-121G basis, for only those 14 methods in Table 4 or 5 in which the total energy values are minima for the multiplicity m ) 10. With the aim of exploring the nature of the interaction between Sm and Co atoms, so checking whether it is really of Lennard-Jones type or not, as forecasted in above discussions, we scanned the interatomic distance r from 1.0 to 7.0 Å, increasing by 0.001 Å

in each subsequent calculation. The graphs in Figure 1 present the associated outcomes for the most informative interval, 1.8-5.0 Å. The calculation of the formation energy Ef is made in the same manner as that of the binding energy De. Then, for SmCo - E7Sm - E4Co), where HFB, for example, it is Ef ) -C(E10 C is the usual hartree-to-electronvolt conversion factor, the values for E7Sm and E4Co are taken respectively from Tables 2 SmCo is this time the present scan result. and 1, and E10 Except for B1B95, all the results are just like that shown in Figure 1d: most of the data from the 6000 calculations are

Figure 1. Nominal Lennard-Jones curves, describing the nature of the interaction between Sm and Co atoms, for the 14 methods in Table 4 or 5 in which the total energy values are minimum for the multiplicity m ) 10.

1904


arranged in a clearly distinguished order, resulting in a not-sosmooth and commonly broken curve. Nevertheless, we indeed manage to observe the Lennard-Jones nature, which we were after, for all the methods (even for B1B95). Of course we do not see, for a particular method, a perfect, flawless LennardJones curve, for there are many data that appear off the main curve; a feature that should be seen to be normal because in a “single-point” calculation, GAUSSIAN 03 forces the system to converge just as it is given at the outset, without allowing any change in the separation distance r or in any other parameter. To give a more compact and succinct presentation of the outcomes, we first remove such obviously “bad” off-data in all the graphs, then carry out a spline interpolation for the missing data. Figure 1a presents the so-obtained, nominally LennardJones, curves for the “toughest” methods, except O3LYP. For the ease of the comparison of previous and present results, we include vertical and horizontal lines (with the same line pattern as the corresponding main curve) for spotting visually the equilibrium separation distance re and the binding energy De values. Impressively enough, all seven methods in Figure 1a lead almost to the same re value around 3.0 Å, which is the aforementioned value in Table 4. Specifically, the re values from Figure 1a (and the corresponding ones from Table 4) are 3.02 Å (3.07 Å) for HFB, 2.96 Å (2.97 Å) for B3LYP, 2.92 Å (2.95 Å) for B3P86, 2.99 Å (2.99 Å) for B1LYP, 3.03 Å (3.01 Å) for B98, 3.03 Å (3.01 Å) for B971, and 3.08 Å (3.09 Å) for BHandHLYP; agreement is sufficiently satisfactory to a high degree. What about the corresponding binding energy De values? They are 1.08 eV (1.08 eV) for HFB, 1.07 eV (1.22 eV) for B3LYP, 1.30 eV (1.46 eV) for B3P86, 1.18 eV (1.34 eV) for B1LYP, 0.92 eV (1.13 eV) for B98, 0.93 eV (1.14 eV) for B971, and 0.99 eV (1.11 eV) for BHandHLYP, a fair agreement, with not so big discrepancy. We note that the De interval in Figure 1a is plainly narrower (centered around 1.1 eV) than that in Table 4; perhaps the figures from the latter are more reliable since they are the ones obtained as results of geometrical optimizations. It is also noted in passing how two Becke functionals, B98 and B971, result in almost the same LennardJones curves, with identical De and re values for SmCo, but at the same time, these two curves cross the Ef ) 0 line, a feature they should not have exhibited. This is the first, and the last, peculiarity to observe in this work, if we are to speak of a proper Lennard-Jones nature for the SmCo dimer. Figure 1b shows the likewise nominal Lennard-Jones curves for the remaining four methods in Table 4. The De and re values read from these graphs are 3.05 Å and 1.65 eV for mPW1PW91, 3.00 Å and 1.14 eV for B972, 2.94 Å and 1.48 eV for PBE1PBE, and 2.95 Å and 1.49 eV for BHandH. These are to be compared with the corresponding values 2.98 Å and 1.77 eV for mPW1PW91, 3.01 Å and 1.50 eV for B972, 2.98 Å and 1.69 eV for PBE1PBE, and 2.97 Å and 1.54 eV for BHandH. Here again the re values agree fairly well with each other; the same holds also for the De values, except for the B972 functional, which exhibits a grave discrepancy. Finally, we are left with two hybrid functionals, B1B95 and O3LYP, which display more than one curve in their r versus Ef graphs. We clearly distinguish five distinct curves, all in the Lennard-Jones nature, in Figure 1c for B1B95. All of them share the same re value around 3.0 Å, a good estimate comparing with the 2.98 Å value in Table 4; one of them, however, indicates a De value around 2.0 eV, one around 1.2 eV, and three around 1.5 eV. Note that we did prefer not to calculate the De value for B1B95 in Table 4 because of the insufficient data from Tables 1 and 2, which seemed to us odd at that time.

Oymak and Erkoç The reason for this intriguing, yet amusing, feature demands a higher level investigation (a good candidate is again a CASSCF study) and remains elusive for the time being. A very similar situation occurs in the O3LYP (one of the “toughest”) case, too, though not as severe as in the B1B95 case, as is seen in Figure 1d. We distinguish in its graph three distinct curves, one of them is barely discernible. All of them share the common re ≈ 3.0 Å value, which is identical to the one in Table 4. We see two well separated De values around 1.2 and 1.5 eV, the latter being the closest one to the 1.57 eV value in Table 4. It is interesting that, unlike the hybrid B1B95 functional, O3LYP did not give any hint in Tables 1-3 of this multicurve graph. Before closing, it seems appropriate to mention a subtle point. Due to its inherent 4f electrons, which cause all the major difficulties in obtaining the properties of the Sm-Co system, the heavy element Sm occupies the central part in the present work. Therefore, it is not surprising that the relativistic corrections will make surely an impact, which might be significant, on the isolated Sm atom and, in turn, on the spectroscopic and electronic properties of the SmCo dimer. Since the ECP methods partially compensate for them, as we pointed out before, our study does not exclude completely the relativistic corrections, and this was an important reason, among others, why we chose the CEP-121G basis functions to employ in this work. As a matter of fact, there exists a more sophisticated way to take into account the relativistic corrections. It is the DouglasKroll-Hess (DKH) method53-56 in which a scalar relativistic Hamiltonian is exploited for all-electron calculations of heavy atoms when the ECP functionals are not precise enough to lead to reliable results. Unfortunately, the GAUSSIAN 03 package, which is the main tool of this work, allows for DKH calculations only for the transition metals in the first and second rows of the periodic table, not for the Sm element in particular. Of course, we could have utilized the DKH method for the Co atom, but to little avail, because the comparison of the energies from different methods would then be meaningless. If a similar study for the SmCo dimer, using a DKH method implemented in another chemical calculation package, happens to appear in the near future, we expect that its results will not be drastically different, just discernible instead, from those presented in this work. In any case, our aim here is to give a stimulus to the researchers for further studies about the SmCo dimer; it is highly likely then that the outcomes of the present work might be used as markers for probing comparatively the relativistic corrections of the ECP and DKH methods. Conclusions In this work the SmCo dimer was studied in detail by carrying out DFT calculations within the effective core potential level (CEP-121G). Making use of 21 different exchange-correlation functionals that exist in GAUSSIAN 03, we reported some spectroscopic and electronic properties of SmCo, most being the first to appear in the literature to this day, to the best of our knowledge. To find out how reliable and effective they were in exploring the properties of SmCo, we first put the 21 functionals to the test by looking at whether they led to the ground state, which had the minimum total energy, with the conventional multiplicity value, for separate Co and Sm atoms. We saw that only seven among the 21 functionals were capable of identifying the Co atom with m ) 4 and the Sm atom with m ) 7 as possessing the lowest energy value. These functionals were HFB, B3LYP, B3P86, B1LYP, B98, B971, O3LYP, and BHandHLYP, which we marked as the “toughest” candidates for the SmCo calculations. All of the standalone functionals

DFT Analysis of the SmCo Dimer (VSXC, HCTH93, HCTH147, and HCTH407) and two onlyexchange functionals (HFS and XAlpha) were so unsuccessful in this multiplicity test that they gravely impaired their credibility for the subsequent SmCo calculations. The same multiplicity calculations were then performed for SmCo; all the hybrid functionals but BMK predicted that the ground state of SmCo should be the one with multiplicity m ) 10, while the others predicted it should be the one with m ) 6. By virtue of the previous test results, we plausibly surmised that the ground state of SmCo should be associated with multiplicity m ) 10. We attributed this value to a possible fact that the electronic configurations of Sm and Co do not change drastically in forming SmCo and concluded that the interaction between Sm and Co atoms might then be described by a Lennard-Jones type curve. We then calculated the spectroscopic constants of SmCo for the multiplicity value at which the total energy was a minimum. For m ) 10, the binding energy De was found in the range 1.08-1.77 eV, the equilibrium separation distance re was 2.975 ( 0.035 Å, and the fundamental frequency ωe was 120 ( 10 cm-1. The excess charge on Sm q(Sm) and the dipole moment µ for SmCo was also calculated; for m ) 10 they were q(Sm) ) 0.6 ( 0.03|e| and µ ) 4.2 ( 0.6 D. We saw that all the re, ωe, q, and µ values for m ) 10 were consistently close to each other, while, on the other hand, the variation in De values was fairly large. We next presented HOMO-LUMO gaps for SmCo; the figures for m ) 10 were the same: Eg(R) ) 2.55 ( 0.3 eV and Eg(β) ) 2.55 ( 0.7 eV. We then performed “singlepoint” SCF calculations for m ) 10 to explore the nature of the Sm-Co interaction. We indeed observed the Lennard-Jones nature for most of the functionals under question. The De and re values obtained from the Lennard-Jones curves were compared with those previously obtained; they agreed well with each other. Finally, we discussed two hybrid functionals, B1B95 and O3LYP, which intriguingly displayed more than one curve in their formation energy graphs. On this occasion, we propose to continue this research toward the larger derivatives of Sm-Co clusters. They are highly likely to give rise to valuable insights into the evolution from separated Sm and Co atoms to SmpCoq microclusters. Our target systems will be SmCo5 and Sm2Co17, with the emphasis on their electromagnetic properties. Acknowledgment. This work is supported by BK21 Physics Program via Department of Physics, Chungbuk National University. H.O. expresses his deepest gratitude to Jong-Bae Hong from Seoul National University and Suh-Kun Oh, SeongCho Yu, Seung-Kee Han, and Cheong-Ho Han from Chungbuk National University for their overwhelming kindness and hospitality during his stay in Korea the Beautiful. References and Notes (1) Magnetism: Fundamentals; du Tre´molet de Lacheisserie, E´., Gignoux, D., Schlenker, M., Eds.; Springer Science+Business Media, Inc.: New York, 2005. (2) Magnetism: Materials & Applications; du Tre´molet de Lacheisserie, E´., Gignoux, D., Schlenker, M., Eds.; Springer Science+Business Media, Inc.: New York, 2005. (3) CRC Handbook of Chemistry and Physics, 84th ed.; Lide, D. R., Ed.; CRC Press LLC: Boca Raton, FL, 2003. (4) Sun, S.; Murray, C. B.; Weller, D.; Folks, L.; Moser, A. Science 2000, 287, 1989. (5) Zeng, H.; Li, J.; Liu, J. P.; Wang, Z. L.; Sun, S. Nature (London) 2002, 420, 395. (6) Larson, P.; Mazin, I. I. J. Appl. Phys. 2003, 93, 6888. (7) Larson, P.; Mazin, I. I.; Papaconstantopoulos, D. A. Phys. ReV. B 2003, 67, 214405.

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JP908792F

On the SmCo Dimer: A Detailed Density Functional Theory Analysis

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