- Email: [email protected]
Model potential nonlocal density functional calculations of small cobalt clusters, Con,n⩽5
Computational Materials Science 22 (2001) 118±122
www.elsevier.com/locate/commatsci
Model potential nonlocal density functional calculations of small cobalt clusters, Con; n 6 5 M. Pereiro a,*, S. Man'kovsky b, D. Baldomir a, M. Iglesias a, P. Mlynarski a, M. Valladares a, D. Suarez a, M. Castro c, Juan E. Arias d b
a Departamento de Fõsica Aplicada, Facultad de Fõsica, Universidade de Santiago de Compostela, Santiago de Compostela, Spain Institute for Metal Physics of the National Academy of Sciences of Ukraine, 36 Acad. Vernadsky Blvd, UA-03680, Kiev 142, Ukraine c Departamento de Fõsica y Quõmica te orica, Facultad de Quõmica, Universidad Nacional Aut onoma de M exico, Del. Coyoacan, Ciudad Universitaria, C.P. 04510, M exico D.F., Mexico d Intituto de Investigaciones Technol ogicas, Universidade de Santiago de Compostela, Santiago de Compostela, Spain
Accepted 2 April 2001
Abstract The results of self-consistent nonlocal generalized gradient approximation (GGA) density functional calculations are reported for small cobalt clusters Con 2 6 n 6 5. An emphasis is made on a proper treatment of exchange and correlation eects. The enhancement of the magnetic moments as well as the bonding properties of these clusters are discussed in terms of the cluster size and symmetry. We compare some results from deMon±KS (D) module release 3.2 and the computational scheme of Sambe±Felton and Dunlap (SFD). Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction The rapid development of nanofabrication techniques followed by the discovery of some new and interesting eects, as the giant magnetoresistance to be mentioned as primary example, provides another impact for the theoretical study of transition metal clusters. Those systems present themselves a serious challenge for any ®rst principle calculation due to the heavy demand of a proper description of both exchange and correlation (XC) eects. Usually, the size of the cluster as well as the high level of correlation treatment needed causes a rapid failure of traditional ab
*
Corresponding author. E-mail address: [email protected] (M. Pereiro).
initio quantum chemistry methods based either on con®guration interaction (CI) expansion of Slater determinants or Moller±Plesset perturbation techniques. Fortunately, the density functional theory (DFT) [1,2] oers a computational alternative yet retaining the ab initio level. Among transition atom systems the cobalt clusters have been much less studied than others from the ®rst transition row of elements. The most recent paper devoted to this subject is the work of Li and Gu [3] based on local density approximation (LDA) in DFT and discrete diophantine technique used for evaluation of matrix elements with no geometry optimizations. Here we present more re®ned approach to the systematic study of both electronic and magnetic properties of Con clusters (2 6 n 6 5) using a fully self-consistent spin polarized nonlocal generalized gradient approximation (GGA)
0927-0256/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 1 ) 0 0 1 7 7 - X
M. Pereiro et al. / Computational Materials Science 22 (2001) 118±122
Gaussian density functional program [4] and deMon±KS (D) module release 3.2. The main purpose of the present work is to show how the electronic as well as the magnetic properties evolve with the increase of cluster size and compare some of the results between this two programs. The geometry of all clusters has been optimized. Also, for n 4 dierent less symmetrical structures have been also fully optimized for the sake of possible Jahn±Teller (JT) eects. The GGA level oers the state-of-the-art treatment of both XC eects and, at the same time, provides a computationally ecient and more precise way of evaluating molecular integrals by means of linear combinations of Gaussian-type orbitals (LCGTO).
2. Computational details The present results have been obtained using a computational scheme of Sambe and Felton [5] later improved by Dunlap et al. (SFD) [6] and program deMon±KS (D) module release 3.2. Despite the use of an orbital basis sets SFD introduces two auxiliary basis sets which are used in the self-consistent scheme to ®t the charge density and XC potential. However, after the SCF convergency is reached the XC energy is recalculated by numerical integration over the grid points. The core electrons of cobalt atoms are described by the use of a model potential (MP) in concert with a 15 electrons orbital basis set. This MP has been optimized for a speci®c use in DFT scheme [7]. The inner 3p shell has been moved into a basis set, thus allowing the explicit treatment of important 3p±3d correlations in metal systems. Also, the use of MP minimizes the basis set superposition errors
119
(BSSE) which should be taken into account when analysing close lying states on an all electron basis set level; and on the other, it allows to keep the same computational treatment of cobalt clusters for all nuclearities studied. Having in mind the latter purpose, the grid used in the ®tting procedures consisted of 12 angular grid along with a 32 point radial mesh per atom. The GGA treatment of XC was done through Perdew and Wang (PW86) [8] nonlocal functional for exchange and Perdew (P86) [9,10] nonlocal functional for correlation along with the VWN [11] parametrization of the correlation energy of a homogeneous electron gas. Details of the implementation of these density gradient nonlocal functionals as well as the corresponding nonlocal XC potentials are described in [4]. The orbital valence basis set with contraction pattern (2111=211 =311 ) with (5,3,3; 5,3,3) auxiliary set was chosen to be appropriate for all clusters. In atomic calculations the electron density has been spherically and nonspherical averaged and the total energy of its d7 s2 state has been used as an reference to calculate the binding energies of the clusters.
3. Results 3.1. Cobalt atom The experimental ground state of the cobalt atom has the con®guration d7 s2 placing its d8 s1 state 0.418 eV (D) higher and 1.06 eV in (SFD) too. In Table 1 total energies are reported for both spherical and nonspherical cobalt atoms at MCP level of treatment. Having a look to this Table, we can come to the conclusion that nonspherical cal-
Table 1 Total energy of the ground and ®rst excited state computed with MCP calculations (LSDA and GGA level) MCP
Spherical Nonspherical
LSDA PW91 LSDA PW86 PW91
E 4s2 3d7 (a.u.)
E 4s1 3d8 (a.u.)
DE (eV)a
)107.04054 )107.41956 )107.08156 )107.56046 )107.48976
)107.05686 )107.43929 )107.06772 )107.52798 )107.47126
)0.444 )0.537 +0.376 +0.883 +0.503
Spherical and nonspherical approximations are used in cobalt atom. Results corresponds to D. We de®ne DE E 4s1 3d8 E 4s2 3d7 . The experimental separation is equal to 0.418 eV.
a
120
M. Pereiro et al. / Computational Materials Science 22 (2001) 118±122
culations at MCP level improve all theoretical results, of the all-electron type, so far obtained [14]. Indeed, our present MCP results for the energy separation between the ground, d7 s2 , and excited, d8 s1 , states of the Co atom (equal to + 0.883 eV (D) at the NS-MCP-PW86) is in very good agreement with the experimental value. 3.2. Co2 We start the molecular calculations with cobalt dimer. Very few studies have been performed for this system. An excellent review for the Co2 as well as for other transition metal dimers has been given by Salahub [12]. From that reference a limited CI calculation gives unbound state with respect to (SFD) d7 s2 Co atom with bond length of 2.56 A (D) while LSD calculations give 2.07 A and 2.01 A (SFD) and 1.96 (D). In the present study we use a spin polarized version of DFT. That means that for every possible dierence between the number of spin up and spin down electrons (from now on denoted as N) we look for an electronic con®guration giving the lowest energy. 1 We also use, in all molecular systems, symmetry rules for reducing the number of molecular calculations. In this case a D4h symmetry has been adopted though we assign the electronic con®guration in terms of the in®nite symmetry group. The ground state has been found for N 4 with the following electron assignment: 1r2g 2r2g 1p4u 1d3u 1p2g 1r1u ; (SFD) and giving the bond distance of 1.96 A binding energy of 0.865 eV (SFD) (or 0.433 eV per atom ± in latter calculations the binding energy per atom will be given as being more appropriate for viewing its dependence on cluster nuclearity). By making the following promotion: 1d#u ! 1r#u ; we optimized also this excited state which gave the In that same optimized bond distance of 1.96 A. case the adiabatical and vertical energy dierence 1
In following, all results corresponds to SFD.
converges to a single value of 0.213 eV. For the equilibrium separation of the ground state another excited state has been calculated by promoting a 1d#u electron into 1p#g . The obtained (vertical) energy dierence was 0.531 eV. The potential energy curve of the ground state near equilibrium has been ®tted using a third-degree polynomial in order to obtain the vibrational frequency as well as the inharmonic term. The calculated values are 452:9 cm 1 for xe and 1.90 for xe xe . That value of vibrational frequency seems to be too large when considered along with the calculated binding strength but this can be accounted for a `cooperative' eect of moderate grid size and MP use. The value of the binding energy can be compared to the estimated experimental value [13] <1.2 eV. The highest occupied orbital for the Co2 ground state is the 1b1u of minority spin with its eigenvalue )5.97 eV which, in the view of DFT [1,2], has its physical interpretation as being the ionization potential (IP) of a system. It is worthwhile to mention that although the highest occupied Kohn±Sham orbital is the only one having a direct physical meaning while the all remaining ones should be viewed, in principle, as just a pure mathematical construction which helps solving the eective Kohn±Sham one-particle equations, their numerical similarity to real orbitals is used to make estimations of the degree of (sp; sd) hybridizations by means of Mulliken population analysis as well as plots of orbitals which, in turn, provide help in the analysis of the bonding mechanisms in molecular structures. The atomic state of a Co atom within the dimer has been evaluated as s0:874 p0:07 d8:06 thus closely resembling the d8 s1 state. 3.3. Co3 In the Co3 case only one structure has been studied: the equilateral triangle. Using the C3v symmetry the ground state has been found for an electron spin up±spin down dierence N 7 with The a Co±Co equilibrium distance of 2.04 A. binding energy per atom is 0.56 eV and the Co atomic state is estimated to be s0:97 p0:16 d7:87 . The bonding is mainly due to the s electrons though there is some admixture of d-type electrons in the bonding molecular orbitals.
M. Pereiro et al. / Computational Materials Science 22 (2001) 118±122
3.4. Co4 The cobalt tetramer has been inspected in a more detailed way than other clusters where only the most symmetric structures have been extensively examined. At ®rst we investigated the tetrahedral structure. Even with the imposed Td symmetry constraint there is a huge number of closely lying states and care has to be taken in order to properly assign the ground state. The one we have found corresponds to N 10 with an and binding optimized Co±Co distance of 2.99 A energy per atom equal to 0.98 eV. The highest occupied Kohn±Sham orbital is of T2 symmetry, in the majority spin `band' and having two electrons, thus being a candidate for the JT distorsion. The atomic electron con®guration in Co4 is s1:12 p0:29 d7:58 . Next, we have assumed a square geometry for cobalt tetramer imposing the D4h constraint. The optimized Co±Co distance was also equal to with N 10. The adjacent spin up/down 2.29 A dierences N 8 and N 12 produced less stable states being placed above the ground state by 0.56 and 1.85 eV, respectively (vertical values). The optimized Td structure is more stable than the optimized D4h one by 0.96 eV. The analysis of the eigenvalue spectra of Co4 in the square geometry shows that the highest occupied orbital containing one electron has e symmetry. As the JT eect is possible here, the symmetry has been lowered from D4h to D2h and the rhombus structure has been optimized. The ground state has been found for a along with a 72° Co±Co distance equal to 2.19 A angle for N 10. As expected, the optimized D2h structure is more stable than the square geometry by 0.29 eV. Clearly, by choosing the square
121
geometry, other distortions are possible, though, due to the rather small energy dierences between the JT structures, the quality of both the grid and basis sets must be enhanced ± possibly with BSSE corrections. On the other hand, the change of geometry may imply the change in N, therefore changing the magnetic moments. That, in turn, is directly connected with a rather big rearrangement of the XC energies so the JT eects in transition metal clusters must be discussed along with the level of XC treatment. 3.5. Co5 The geometry of Co5 has been assumed as equilateral trigonal bipyramid so the C3v symmetry has been adopted in order to reduce the time of computations during geometry optimizations. The lowest state has been obtained for N 13 and although another state Co±Co distance 2.35 A with N 11 has also been optimized leading to the same bond distance as in the former case, lying only 0.34 eV higher than the ground state. The Mulliken population analysis gives an average atomic hybridization of cobalt as s1:039 p0:462 d7:562 .
4. Conclusions The electronic, structural and magnetic properties of small cobalt clusters have been studied within the frame of the density functional method. The nonlocal GGA treatment of XC eects within DFT scheme oers the most accurate description of `many-body' problem in a present day ab initio calculations upon transition metal
Table 2 Electronic and magnetic properties of cobalt clusters as a function of cluster size Property
Co2
Co3
Co4
Co5
Binding energy per atom (eV) Magnetic moment per atom (lB ) `Kohn±Shan' IP (eV)a Optimized Co±Co distance (A) Population of d orbital
0.433 2.0 5.97# 1.96 8.06
0.596 2.33 5.65" 2.04 7.87
0.980 2.50 5.91" 2.29 7.58
0.856 2.60 5.99" 2.35 7.56
SFD results. Eigenvalue of highest occupied Kohn±Sham orbital with opposite sign (" = # denote whether it is of majority/minority spin).
a
122
M. Pereiro et al. / Computational Materials Science 22 (2001) 118±122
systems. Also, the use of the optimized model potential with the 3p±3d correlations explicitly treated by shifting the 3p shell into a valence basis set ensures a uni®ed and accurate description of all clusters within a computational method. All clusters show an abundance of close-lying states. Therefore, for each cluster nuclearity, its ground state has to be assigned with care. The main purpose of this paper was to show how the basic properties of Co clusters evolve with the cluster size and compare some results from SFD and D. We imposed some constrains in order to achieve a reasonably short computing time. The key among them was the assumed high symmetry of each cluster studied. The highly symmetric structures exhibit degeneracy of the highest occupied Kohn± Sham orbitals leading therefore to JT distortions. This was investigated partially in the case of a planar tetramer, obtaining results that throw an armative answer. Nevertheless, its energetics, as well as the changes of cluster magnetic moments, are small and our general conclusions are still valid. As it can be seen, the magnetic moment of Co clusters is enhanced when compared to its bulk value of 1:72 lB . That enhancement can be accounted for an increased exchange splitting due to the decrease of the coordination number. See Table 2 for details.
Acknowledgements One of the authors (M. Pereiro) would like to thank Universidad de Santiago de Compostela (Departamento de Fõsica Aplicada) and the organizing committee of the X Workshop on Computational Materials Science (CMS2000), for the ®nancial support.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
P. Hohenberg, W. Kohn, Phys. Rev. 136 (1964) B864. W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1133. Z. Li, B. Gu, Phys. Rev. B 47 (1993) 13611. P. Møynarski, D.R. Salahub, Phys. Rev. B 43 (1991) 1399. H. Sambe, R.H. Felton, J. Chem. Phys. 62 (1975) 1122. B.I. Dunlap, J.W.D. Connolly, J.R. Sabin, J. Chem. Phys. 71 (1979) 3396. J. Andzelm, E. Radzio, D.R. Salahub, J. Chem. Phys. 83 (1985) 4573. J.P. Perdew, Y. Wang, Phys. Rev. B 33 (1986) 8800. J.P. Perdew, Phys. Rev. B 33 (1986) 8822. J.P. Perdew, Phys. Rev. B 34 (1986) 7406E. S.H. Vosko, L. Wilk, M. Nusair, Can. J. Phys. 58 (1980) 1200. D.R. Salahub, in: K.P. Lawley (Ed.), Ab initio Methods in Quantum Chemistry ± II, Wiley, New York, 1987. D.A. Hales, C.X. Su, L. Lian, P.B. Armentrout, J. Chem. Phys. 100 (1994) 1049. C. Jamorski, A. Martõnez, M. Castro, D.R. Salahub, Phys. Rev. B 55 (1997) 10905.
www.elsevier.com/locate/commatsci
Model potential nonlocal density functional calculations of small cobalt clusters, Con; n 6 5 M. Pereiro a,*, S. Man'kovsky b, D. Baldomir a, M. Iglesias a, P. Mlynarski a, M. Valladares a, D. Suarez a, M. Castro c, Juan E. Arias d b
a Departamento de Fõsica Aplicada, Facultad de Fõsica, Universidade de Santiago de Compostela, Santiago de Compostela, Spain Institute for Metal Physics of the National Academy of Sciences of Ukraine, 36 Acad. Vernadsky Blvd, UA-03680, Kiev 142, Ukraine c Departamento de Fõsica y Quõmica te orica, Facultad de Quõmica, Universidad Nacional Aut onoma de M exico, Del. Coyoacan, Ciudad Universitaria, C.P. 04510, M exico D.F., Mexico d Intituto de Investigaciones Technol ogicas, Universidade de Santiago de Compostela, Santiago de Compostela, Spain
Accepted 2 April 2001
Abstract The results of self-consistent nonlocal generalized gradient approximation (GGA) density functional calculations are reported for small cobalt clusters Con 2 6 n 6 5. An emphasis is made on a proper treatment of exchange and correlation eects. The enhancement of the magnetic moments as well as the bonding properties of these clusters are discussed in terms of the cluster size and symmetry. We compare some results from deMon±KS (D) module release 3.2 and the computational scheme of Sambe±Felton and Dunlap (SFD). Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction The rapid development of nanofabrication techniques followed by the discovery of some new and interesting eects, as the giant magnetoresistance to be mentioned as primary example, provides another impact for the theoretical study of transition metal clusters. Those systems present themselves a serious challenge for any ®rst principle calculation due to the heavy demand of a proper description of both exchange and correlation (XC) eects. Usually, the size of the cluster as well as the high level of correlation treatment needed causes a rapid failure of traditional ab
*
Corresponding author. E-mail address: [email protected] (M. Pereiro).
initio quantum chemistry methods based either on con®guration interaction (CI) expansion of Slater determinants or Moller±Plesset perturbation techniques. Fortunately, the density functional theory (DFT) [1,2] oers a computational alternative yet retaining the ab initio level. Among transition atom systems the cobalt clusters have been much less studied than others from the ®rst transition row of elements. The most recent paper devoted to this subject is the work of Li and Gu [3] based on local density approximation (LDA) in DFT and discrete diophantine technique used for evaluation of matrix elements with no geometry optimizations. Here we present more re®ned approach to the systematic study of both electronic and magnetic properties of Con clusters (2 6 n 6 5) using a fully self-consistent spin polarized nonlocal generalized gradient approximation (GGA)
0927-0256/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 1 ) 0 0 1 7 7 - X
M. Pereiro et al. / Computational Materials Science 22 (2001) 118±122
Gaussian density functional program [4] and deMon±KS (D) module release 3.2. The main purpose of the present work is to show how the electronic as well as the magnetic properties evolve with the increase of cluster size and compare some of the results between this two programs. The geometry of all clusters has been optimized. Also, for n 4 dierent less symmetrical structures have been also fully optimized for the sake of possible Jahn±Teller (JT) eects. The GGA level oers the state-of-the-art treatment of both XC eects and, at the same time, provides a computationally ecient and more precise way of evaluating molecular integrals by means of linear combinations of Gaussian-type orbitals (LCGTO).
2. Computational details The present results have been obtained using a computational scheme of Sambe and Felton [5] later improved by Dunlap et al. (SFD) [6] and program deMon±KS (D) module release 3.2. Despite the use of an orbital basis sets SFD introduces two auxiliary basis sets which are used in the self-consistent scheme to ®t the charge density and XC potential. However, after the SCF convergency is reached the XC energy is recalculated by numerical integration over the grid points. The core electrons of cobalt atoms are described by the use of a model potential (MP) in concert with a 15 electrons orbital basis set. This MP has been optimized for a speci®c use in DFT scheme [7]. The inner 3p shell has been moved into a basis set, thus allowing the explicit treatment of important 3p±3d correlations in metal systems. Also, the use of MP minimizes the basis set superposition errors
119
(BSSE) which should be taken into account when analysing close lying states on an all electron basis set level; and on the other, it allows to keep the same computational treatment of cobalt clusters for all nuclearities studied. Having in mind the latter purpose, the grid used in the ®tting procedures consisted of 12 angular grid along with a 32 point radial mesh per atom. The GGA treatment of XC was done through Perdew and Wang (PW86) [8] nonlocal functional for exchange and Perdew (P86) [9,10] nonlocal functional for correlation along with the VWN [11] parametrization of the correlation energy of a homogeneous electron gas. Details of the implementation of these density gradient nonlocal functionals as well as the corresponding nonlocal XC potentials are described in [4]. The orbital valence basis set with contraction pattern (2111=211 =311 ) with (5,3,3; 5,3,3) auxiliary set was chosen to be appropriate for all clusters. In atomic calculations the electron density has been spherically and nonspherical averaged and the total energy of its d7 s2 state has been used as an reference to calculate the binding energies of the clusters.
3. Results 3.1. Cobalt atom The experimental ground state of the cobalt atom has the con®guration d7 s2 placing its d8 s1 state 0.418 eV (D) higher and 1.06 eV in (SFD) too. In Table 1 total energies are reported for both spherical and nonspherical cobalt atoms at MCP level of treatment. Having a look to this Table, we can come to the conclusion that nonspherical cal-
Table 1 Total energy of the ground and ®rst excited state computed with MCP calculations (LSDA and GGA level) MCP
Spherical Nonspherical
LSDA PW91 LSDA PW86 PW91
E 4s2 3d7 (a.u.)
E 4s1 3d8 (a.u.)
DE (eV)a
)107.04054 )107.41956 )107.08156 )107.56046 )107.48976
)107.05686 )107.43929 )107.06772 )107.52798 )107.47126
)0.444 )0.537 +0.376 +0.883 +0.503
Spherical and nonspherical approximations are used in cobalt atom. Results corresponds to D. We de®ne DE E 4s1 3d8 E 4s2 3d7 . The experimental separation is equal to 0.418 eV.
a
120
M. Pereiro et al. / Computational Materials Science 22 (2001) 118±122
culations at MCP level improve all theoretical results, of the all-electron type, so far obtained [14]. Indeed, our present MCP results for the energy separation between the ground, d7 s2 , and excited, d8 s1 , states of the Co atom (equal to + 0.883 eV (D) at the NS-MCP-PW86) is in very good agreement with the experimental value. 3.2. Co2 We start the molecular calculations with cobalt dimer. Very few studies have been performed for this system. An excellent review for the Co2 as well as for other transition metal dimers has been given by Salahub [12]. From that reference a limited CI calculation gives unbound state with respect to (SFD) d7 s2 Co atom with bond length of 2.56 A (D) while LSD calculations give 2.07 A and 2.01 A (SFD) and 1.96 (D). In the present study we use a spin polarized version of DFT. That means that for every possible dierence between the number of spin up and spin down electrons (from now on denoted as N) we look for an electronic con®guration giving the lowest energy. 1 We also use, in all molecular systems, symmetry rules for reducing the number of molecular calculations. In this case a D4h symmetry has been adopted though we assign the electronic con®guration in terms of the in®nite symmetry group. The ground state has been found for N 4 with the following electron assignment: 1r2g 2r2g 1p4u 1d3u 1p2g 1r1u ; (SFD) and giving the bond distance of 1.96 A binding energy of 0.865 eV (SFD) (or 0.433 eV per atom ± in latter calculations the binding energy per atom will be given as being more appropriate for viewing its dependence on cluster nuclearity). By making the following promotion: 1d#u ! 1r#u ; we optimized also this excited state which gave the In that same optimized bond distance of 1.96 A. case the adiabatical and vertical energy dierence 1
In following, all results corresponds to SFD.
converges to a single value of 0.213 eV. For the equilibrium separation of the ground state another excited state has been calculated by promoting a 1d#u electron into 1p#g . The obtained (vertical) energy dierence was 0.531 eV. The potential energy curve of the ground state near equilibrium has been ®tted using a third-degree polynomial in order to obtain the vibrational frequency as well as the inharmonic term. The calculated values are 452:9 cm 1 for xe and 1.90 for xe xe . That value of vibrational frequency seems to be too large when considered along with the calculated binding strength but this can be accounted for a `cooperative' eect of moderate grid size and MP use. The value of the binding energy can be compared to the estimated experimental value [13] <1.2 eV. The highest occupied orbital for the Co2 ground state is the 1b1u of minority spin with its eigenvalue )5.97 eV which, in the view of DFT [1,2], has its physical interpretation as being the ionization potential (IP) of a system. It is worthwhile to mention that although the highest occupied Kohn±Sham orbital is the only one having a direct physical meaning while the all remaining ones should be viewed, in principle, as just a pure mathematical construction which helps solving the eective Kohn±Sham one-particle equations, their numerical similarity to real orbitals is used to make estimations of the degree of (sp; sd) hybridizations by means of Mulliken population analysis as well as plots of orbitals which, in turn, provide help in the analysis of the bonding mechanisms in molecular structures. The atomic state of a Co atom within the dimer has been evaluated as s0:874 p0:07 d8:06 thus closely resembling the d8 s1 state. 3.3. Co3 In the Co3 case only one structure has been studied: the equilateral triangle. Using the C3v symmetry the ground state has been found for an electron spin up±spin down dierence N 7 with The a Co±Co equilibrium distance of 2.04 A. binding energy per atom is 0.56 eV and the Co atomic state is estimated to be s0:97 p0:16 d7:87 . The bonding is mainly due to the s electrons though there is some admixture of d-type electrons in the bonding molecular orbitals.
M. Pereiro et al. / Computational Materials Science 22 (2001) 118±122
3.4. Co4 The cobalt tetramer has been inspected in a more detailed way than other clusters where only the most symmetric structures have been extensively examined. At ®rst we investigated the tetrahedral structure. Even with the imposed Td symmetry constraint there is a huge number of closely lying states and care has to be taken in order to properly assign the ground state. The one we have found corresponds to N 10 with an and binding optimized Co±Co distance of 2.99 A energy per atom equal to 0.98 eV. The highest occupied Kohn±Sham orbital is of T2 symmetry, in the majority spin `band' and having two electrons, thus being a candidate for the JT distorsion. The atomic electron con®guration in Co4 is s1:12 p0:29 d7:58 . Next, we have assumed a square geometry for cobalt tetramer imposing the D4h constraint. The optimized Co±Co distance was also equal to with N 10. The adjacent spin up/down 2.29 A dierences N 8 and N 12 produced less stable states being placed above the ground state by 0.56 and 1.85 eV, respectively (vertical values). The optimized Td structure is more stable than the optimized D4h one by 0.96 eV. The analysis of the eigenvalue spectra of Co4 in the square geometry shows that the highest occupied orbital containing one electron has e symmetry. As the JT eect is possible here, the symmetry has been lowered from D4h to D2h and the rhombus structure has been optimized. The ground state has been found for a along with a 72° Co±Co distance equal to 2.19 A angle for N 10. As expected, the optimized D2h structure is more stable than the square geometry by 0.29 eV. Clearly, by choosing the square
121
geometry, other distortions are possible, though, due to the rather small energy dierences between the JT structures, the quality of both the grid and basis sets must be enhanced ± possibly with BSSE corrections. On the other hand, the change of geometry may imply the change in N, therefore changing the magnetic moments. That, in turn, is directly connected with a rather big rearrangement of the XC energies so the JT eects in transition metal clusters must be discussed along with the level of XC treatment. 3.5. Co5 The geometry of Co5 has been assumed as equilateral trigonal bipyramid so the C3v symmetry has been adopted in order to reduce the time of computations during geometry optimizations. The lowest state has been obtained for N 13 and although another state Co±Co distance 2.35 A with N 11 has also been optimized leading to the same bond distance as in the former case, lying only 0.34 eV higher than the ground state. The Mulliken population analysis gives an average atomic hybridization of cobalt as s1:039 p0:462 d7:562 .
4. Conclusions The electronic, structural and magnetic properties of small cobalt clusters have been studied within the frame of the density functional method. The nonlocal GGA treatment of XC eects within DFT scheme oers the most accurate description of `many-body' problem in a present day ab initio calculations upon transition metal
Table 2 Electronic and magnetic properties of cobalt clusters as a function of cluster size Property
Co2
Co3
Co4
Co5
Binding energy per atom (eV) Magnetic moment per atom (lB ) `Kohn±Shan' IP (eV)a Optimized Co±Co distance (A) Population of d orbital
0.433 2.0 5.97# 1.96 8.06
0.596 2.33 5.65" 2.04 7.87
0.980 2.50 5.91" 2.29 7.58
0.856 2.60 5.99" 2.35 7.56
SFD results. Eigenvalue of highest occupied Kohn±Sham orbital with opposite sign (" = # denote whether it is of majority/minority spin).
a
122
M. Pereiro et al. / Computational Materials Science 22 (2001) 118±122
systems. Also, the use of the optimized model potential with the 3p±3d correlations explicitly treated by shifting the 3p shell into a valence basis set ensures a uni®ed and accurate description of all clusters within a computational method. All clusters show an abundance of close-lying states. Therefore, for each cluster nuclearity, its ground state has to be assigned with care. The main purpose of this paper was to show how the basic properties of Co clusters evolve with the cluster size and compare some results from SFD and D. We imposed some constrains in order to achieve a reasonably short computing time. The key among them was the assumed high symmetry of each cluster studied. The highly symmetric structures exhibit degeneracy of the highest occupied Kohn± Sham orbitals leading therefore to JT distortions. This was investigated partially in the case of a planar tetramer, obtaining results that throw an armative answer. Nevertheless, its energetics, as well as the changes of cluster magnetic moments, are small and our general conclusions are still valid. As it can be seen, the magnetic moment of Co clusters is enhanced when compared to its bulk value of 1:72 lB . That enhancement can be accounted for an increased exchange splitting due to the decrease of the coordination number. See Table 2 for details.
Acknowledgements One of the authors (M. Pereiro) would like to thank Universidad de Santiago de Compostela (Departamento de Fõsica Aplicada) and the organizing committee of the X Workshop on Computational Materials Science (CMS2000), for the ®nancial support.
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