PHYSICAL REVIEW B 71, 165433 共2005兲

Ab initio calculations of the photoelectron spectra of transition metal clusters Shen Li,1 M. M. G. Alemany,1,2 and James R. Chelikowsky1,3 1Department

of Chemical Engineering and Materials Science, Institute for the Theory of Advanced Materials in Information Technology Digital Technology Center, University of Minnesota, Minneapolis, Minnesota 55455, USA 2Departamento de Física de la Materia Condensada, Facultad de Física, Universidad de Santiago de Compostela, E-15782 Santiago de Compostela, Spain 3 Departments of Physics and Chemical Engineering, Institute of Computational Engineering and Sciences, University of Texas, Austin, Texas 78712, USA 共Received 13 December 2004; revised manuscript received 14 February 2005; published 27 April 2005兲 We report the results of an ab initio study of the photoelectron spectra of copper and vanadium anionic clusters. Our calculations are performed using a real-space pseudopotential approach based on the densityfunctional theory, and include final state effects. For each size of the cluster, we find the lowest energy structure. The calculated photoelectron spectra, using these ground state structures, reproduce the main features of those obtained by experiment. Our results demonstrate that real-space techniques are advantageous techniques for the study of the electronic and structural properties of transition metals clusters. DOI: 10.1103/PhysRevB.71.165433

PACS number共s兲: 33.80.Eh, 31.15.Ew, 36.40.⫺c

The study of the electronic and structural properties of transition metal 共TM兲 clusters is a very active field in computational science. The main reasons for this are two-fold: First, TM clusters hold great interest both from fundamental and technological points of view.1 For example, these clusters can possess magnetic moments very different from the bulk phase. As a consequence, TM clusters can serve as models for studying the transition from an atomiclike to solidlike regime as a function of the system size. Also, TM clusters can show catalytic activity when supported on solid substrates. Experiment shows that as small as four silver atoms can be catalytic centers in AgBr emulsions for latentimage formation in the phtographic process.2 Second, important breakthroughs in experimental techniques have allowed accurate data to be obtained for many properties of interest of TM clusters. Specifically, high resolution photoelectron spectra 共PES兲 are available for a wide range of photon energies and clusters sizes for TM elements. Computational methods that reproduce the PES would give not only the opportunity to interpret better the electronic structure information, which is directly obtained from the PES, but to also extract valuable information about the geometric structure of TM clusters. Standard quantum-chemistry methods have shown to be adequate for describing the electronic excitation energies of TM clusters. However, owing to their enormous computational load, their applicability is often limited to small clusters consisting of simple metals, in which the treatment of the d electrons is not necessary. To avoid such restrictions, workers often make use of techniques that employ pseudopotentials3 and density functional theory 共DFT兲.4 Pseudopotential theory allows one to focus on the chemically active valence electrons by replacing the strong all-electron atomic potential with a weak pseudopotential, which effectively reproduces the effects of the core electrons on the valence states. The pseudopotential approximation significantly reduces the number of eigenvalues to be handled. This is especially important for heavier elements. DFT replaces the many-electron wave function representation of the sys1098-0121/2005/71共16兲/165433共5兲/$23.00

tem, which is present in standard quantum-chemistry approaches, by a set of noninteracting one-electron wave functions with the same charge density as the original system. Even with these simplifications of the electronic structure problem, the range of sizes and complexity of the systems that can be investigated by quantum computations remains limited. DFT is formulated to be valid only for the electron ground state, so the study of excited state properties is outside of its scope. Another serious limitation comes from the basis sets that are normally used for expanding the oneelectron wave functions, e.g., localized basis sets and plane waves. Basis sets like Gaussians or atomic-like functions require an extensive testing of the basis. This entails an optimization of a multiple parameter space. Consequently, results obtained from these approaches can be sensitive to the basis choice. Plane-wave methods have the advantage that only one parameter 共the wavelength of the highest Fourier component used in the expansion兲 need be refined to control convergence, but they possess both “physical” and computational drawbacks. In a plane-wave representation, periodic boundary conditions are typically employed. If one wants to study a nonperiodic system such as a molecule or cluster, care must be taken to reproduce the vacuum accurately, since spurious interactions between replicated images of the system must be avoided. Also, a periodic representation is complicated by the study of charged systems: periodicity makes the system infinitely charged. To avoid this situation, an artificial uniform compensating charge is usually inserted in order to prevent a divergence of the total energy. Furthermore, plane-wave approaches make extensive use of fast Fourier transforms 共FFTs兲 for performing matrix-vector multiplications between the Hamiltonian matrix and the trial wave vectors. Since FFTs involve nonlocal operations, the efficiency of their implementation on parallel computers architectures is diminished by the need for global communications among processors. Here we illustrate that the main characteristics of the PES of TM clusters can be efficiently and accurately described using a real-space implementation of DFT and pseudopoten-

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PHYSICAL REVIEW B 71, 165433 共2005兲

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tials. The procedure we employ for calculating the PES is as follows. We first obtain the electronic ground state of the clusters. Then, we calculate the excitation energies of the clusters by accounting for final states effects 共that is effects derived from the relaxation of eigenstates upon the removal of one electron in the cluster兲. We use the method proposed by Massobrio, Pasquarello and Car 共MPC兲.5 In order to assess the validity of this procedure, we compute the excitations energies of clusters of two TM elements with a very different electronic structure, Cu共关Ar兴3d104s1兲 and V共关Ar兴3d34s2兲, for which extensive PES data is available. Real-space methods6 not only allow for the exact representation of the electronic problem of a system within DFT, but they also avoid the limitations that are present in “traditional” DFT-approaches. In such methods, there is no “formal” basis. The calculations are performed directly on a realspace grid. Convergence is assured by controlling only one parameter: the grid spacing. There is no need to introduce artificial periodicity in dealing with localized systems. Our real-space method is based on the self-consistent solution of the Kohn-Sham equations on a uniform orthogonal threedimensional grid.7 In order to implement the Laplacian operator, we employ a higher-order finite-difference method; for the exchange and correlation potential we use the local spin density approximation given by Ceperley and Alder,8 and the electron-ion term is determined using nonlocal normconserving ionic pseudopotentials cast in the KleinmanBylander form.9,10 In doing so we obtain a matrix representation that is local. Consequently, highly efficient diagonalization procedures can be employed to extract the required eigenvalue/eigenstate pairs.11 Also, the code is easily parallelized by a domain decomposition approach with only limited communication between processors.12 We placed negatively charged Cu and V clusters at the center of a spherical domain chosen so that the wave functions smoothly vanish at its boundary 共normally, the outer atom of the clusters was ⬃9 a.u. 共1 a.u.= 0.529 Å兲 removed from the boundary兲. The Cu 共V兲 core electrons are represented by pseudopotentials generated for the reference atomic configuration 3d104s14p0 共3s23p63d3兲 using the Troullier-Martins prescription,9 with radial cutoffs of 2.35/ 2.50/ 2.35 a.u. 共1.55/ 2.55/ 2.50 a.u.兲 for the s, p, and d channels, respectively. The Kohn-Sham equations were iterated self-consistently until the difference between the input and output potentials was less than 0.001 eV. Convergence was attained for a grid spacing of 0.35 a.u. 共0.30 a.u.兲 for the Cu 共V兲 clusters. For each cluster size, we typically constructed four or five topologically distinct structures, and determined the ground-state geometry by direct comparison of the total energies of the structures after optimization. The clusters were relaxed until the force component for each Cartesian direction 共as derived from the Hellmann-Feynman theorem13兲 was less than 0.001 a.u. 共⬃0.05 eV/ Å兲. Once the ground-state electronic structure of the anionic clusters is obtained, one can analyze the PES by direct comparison with the density of states 共DOS兲 calculated from the single-particle eigenvalues.14 Such attempts have been tried with relative success for TM elements using different DFT

approaches.5,15 As noted above, one has to examine the foundations of DFT in order to find an explanation for discrepancies between the theoretical and experimental data. MPC proposed a simple method for calculating single-particle excitation energies.5 They obtained promising results when applied to the study of the PES of Cu clusters. We adopt their approach here, albeit with a different implementation. Within their method, the excitation energy corresponding to the kth eigenstate, ⌬SCF共k兲, is calculated as 0 − 共k兲 − E initial , ⌬SCF共k兲 = Efinal

共1兲

0 共k兲 is the total energy of the neutral cluster obwhere Efinal tained after the removal of the electron in the original an− . When determining ionic cluster whose total energy is E initial the final excited state for the neutral cluster, the electronic cloud surrounding the hole is allowed to relax, but the geometric structure of the cluster is held fixed 共a vertical excitation process is considered兲. The relaxation of the electronic degrees of freedom entails a self-consistent process. In each step of the process, a tenfinal , of the neutral cluster is astative set of eigenstates, m signed, and the Hamiltonian of the system constructed. A new set of eigenstates is obtained by operating the Hamiltonian on the eigenstates. In order to maintain the wave function assignment of the hole to the kth eigenstate during the self-consistent process, the eigenstates are expanded in a basis of the eigenstates for the original cluster,

mfinal = 兺 Cmlinitial , l

共2兲

l

where the coefficients Cml are obtained by a diagonalization final procedure. The process is initiated by approximating m initial with the eigenvectors m , after zeroing the occupation number of the kth eigenstate. The excitation energy spectrum is obtained by calculating ⌬SCF共k兲 for each eigenstate of the original cluster. In our calculations, both hole and valence states were allowed to relax. However, if the final states are close in energy, we often found difficulties in achieving convergence. In these cases, we did a constrained relaxation where the hole state was kept frozen while the remaining states were allowed to relax. We estimate the error introduced by using the constrained scheme to be less than 0.2 eV for the clusters considered. The study of the excitation spectra of Cu and V clusters constitutes a stringent test of the method, as these clusters exhibit important differences in their PES. Vanadium can be considered a “typical” TM element, with a partially occupied 共and chemically active兲 d band. In contrast, the valency in copper is determined by a d band that is filled with a single electron located in the s band. Such an electronic configuration, which is characteristic of all “noble” metals, makes Cu clusters resemble alkali metal clusters in their electronic shell structure.16 However, it has been shown that the inclusion of the d electrons 共e.g., s − d hybridization兲 is necessary to properly describe the electronic properties of Cu clusters.5,17

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FIG. 1. Ground-state structures for Cu−n 共n = 3, 5, 8, and 9兲 and V−n 共n = 3 and 4兲 as obtained from our real-space pseudopotential approach.

In Fig. 1 we plot the ground state structures calculated for several Cu−n 共n = 3, 3, 5, 8, and 9兲 and V−n 共n = 3 , 4兲 clusters. We previously discussed the structures of V clusters elsewhere.22 All these structures possess C2v symmetry, and are largely consistent with previous DFT-based calculations.5,18,19 The most stable structure for Cu−3 is a linear chain, which is lower by ⬃0.2 eV/ atom than an isosceles triangle. A planar trapezoidal structure represents the ground state geometry for Cu−5 ; this structure is more stable than a trigonal bipyramid by ⬃0.06 eV/ atom, which is the second most stable structure for n = 5. The lowest energy structures for Cu−8 are a bicapped octahedron and a capped pentagonal bipyramid 共separated in energy by about 0.02 eV/ atom兲. By adding an atom capped at the bottom of the bicapped octahedron we obtained the most favorable structure for Cu−9 . A similar structure was found for n = 9 by Jug et al.18 共the atom is added at the top instead than at the bottom兲. However, MPC obtained as ground state structure a bicapped pentagonal bipyramid;5 in our calculations such a structure is only ⬃0.02 eV/ atom higher in energy than that plotted in Fig. 1. The ground state structures obtained for n = 3, 5 and 9 have a zero magnetic moment; the structure obtained for n = 8 has magnetic moment one B. In Fig. 2 we compare the DOS and excitation energy obtained for the Cu anionic structures plotted in Fig. 1 with available experimental data.17 The excitation energy calculated by MPC using a plane-wave approach for clusters with n = 3, 5, and 9 is also shown.5 Our theoretical values were broadened using a Gaussian distribution function with width at a half-maximum of 0.08 eV. The energy scale of the theoretical results was shifted by a constant so the highest occupied molecular orbital 共HOMO兲 of each cluster was aligned to the corresponding peak in the PES. The shift in

FIG. 2. DOS and excitation energies 共⌬SCF兲 as obtained from our real-space pseudopotential approach for the Cu−n ground-state structures represented in Fig. 1. Results for the excitation energies as obtained from a plane-wave pseudopotential approach for n = 3, 5, and 9 共Ref. 5兲 are also shown 共“other theory;” upper lines兲. The theoretical results are compared with available experimental data 共Ref. 17; dashed lines兲.

energy is estimated to be ⬃2 eV for the DOS, and as small as ⬃0.2 eV for the excitation energy. We attribute the small offset in the excitation energy curves to relaxations in the geometry of the clusters during the electronic excitation process that were not taken into account in our theory. In the comparison between theory and experiment, one must focus on the positions of the peaks in the spectra and not in their relative intensities. The intensities are affected by angularmomentum dependent photodetachment cross sections which are not included in theory. The different nature of the 3d and 4s bands in Cu is reflected in the experimental spectra of the Cu−n clusters. The corelike character of the 3d electrons give rise to a “band” in the PES that is quite localized. The position of this band shifts upwards monotonically in energy when the size of the cluster increases,20 and its onset is determined to be roughly 2 eV below the HOMO, at a binding energy of ⬃4 eV for the clusters studied in this work. The features that occur between the HOMO and the onset of the 3d band are mostly s-like. Owing to the delocalized character of the 4s valence electrons, these features strongly depend on the surface size and geometry of the clusters, and do not show any evident pattern with increasing cluster size. As one can see from Fig. 2, the DOS calculated from the single particle eigenvalues give a very poor description of the PES for the smaller clusters studied, Cu−3 and Cu−5 ; even a qualitative assignment of most of the theoretical peaks to

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corresponding peaks in the PES is not possible 共e.g., for Cu−3 , experiment clearly shows the existence of two distinct features before the onset of the d band, while exhibits five closely spaced peaks with binding energies less than 4 eV兲. For Cu−8 and Cu−9 , the two first major peaks discernible in the PES are identifiable in the DOS, although their relative position is not perfectly matched, but beyond the second peak the experimental features are not properly reproduced, i.e., theory tends to put peaks very close in energy. The limitations of the single particle eigenvalues in reproducing the PES are overcome when the final state effects are taken into account. As we see in Fig. 2, one can identify all the main features of the PES through the theoretical excitation energies. Also, the relative position of the experimental features is accurately reproduced. The most important mismatch between theory and experiment occurs for Cu−3 , where theory yields doublet peaks in contrast to the experimental feature at ⬃3.5 eV. The experimental feature is divided in several subpeaks. Our results for the excitation energy agree with experiment better than the results obtained by MPC,5 especially for Cu−5 and Cu−9 . The PES of these two clusters clearly show the existence of three distinguishable peaks before the onset of the 3d band at ⬃4 eV. MPC’s spectra for Cu−5 only gives two peaks, with a separation between them that is off by ⬃0.5 eV when compared with experiment. For Cu−9 , three distinct peaks also are discernible from the results obtained by MPC, but their distribution along the 0 – 4 eV binding energy range differs greatly from that observed in experiment. The differences between our results and MPC’s for Cu−9 can be attributed, at least in part, to geometry differences between the ground state structures for which the excitation spectra was calculated. The PES of vanadium clusters posses characteristics that are rather different than that observed in copper. In vanadium, the 3d shell produces features that are not as “well localized” as in copper, and that contribute significantly to regions of the spectra that are close to the Fermi level.21 In recent work performed with our real-space approach, we have shown that a generally good description of the PES of V−n clusters is attainable through the DOS, particularly for n ⬎ 4 共Ref. 22兲. Here we wish to estimate how the inclusion of the final state effects affect the previously calculated spectra. In Fig. 3, we plot the DOS and excitation energies calculated for the ground state structures obtained for V−3 and V−4 , an isosceles triangle and a planar rhombohedral structure, respectively 共see Fig. 1兲. The excitation energies obtained for similar geometric structures by Grönbeck and Rosén using a DFT method that employ a linear combination of atomic orbitals as a basis are also shown.23 Theory is compared with the available experimental data.24 共The DOS and excitation energy curves were shifted by approximately 3 and 0.5 eV, respectively, to align the HOMO to the corresponding peak in the PES.兲 As we see in Fig. 3, the PES curve for V−3 is characterized by four distinct peaks below ⬃2.5 eV. Four features are also distinguishable from the DOS, but their positions do not match experiment as well as do the peaks of the excitation energy curve. As regards V−4 , the PES shows five features in the 0 – 4 eV binding energy range 共at approximately 1.8, 2.3, 2.6, 3.2, and 3.9 eV兲. These features are already identifiable from the DOS, but the agree-

FIG. 3. DOS and excitation energies 共⌬SCF兲 as obtained from our real-space pseudopotential approach for the V−3 and V−4 groundstate structures represented in Fig. 1. Results for the excitation energies as obtained from a method that employs a linear combination of atomic orbitals as a basis 共Ref. 23兲 are also shown 共“other theory;” upper lines兲. The theoretical results are compared with available experimental data 共Ref. 24; dashed lines兲.

ment with experiment is improved when final state effects are included in the calculations. In particular, the “shape” of the first and second peaks are better reproduced, as well as the position of the remaining peaks. Our results for the excitation energy for V−4 are similar to those obtained by Grönbeck and Rosén.23 However, their results for V−3 only show two distinct peaks with binding energies lower than 2.5 eV. In summary, we have presented ab initio results for the excitation energies of copper and vanadium clusters. The calculations were performed using a method that is based on the self-consistent solution of the Kohn-Sham equations on a uniform real-space grid. A higher-order finite-difference method is combined with ab initio pseudopotentials. The good agreement found between the calculated excitation energy curves and the experimental photoelectron spectra establishes the adequacy of such techniques for studying the electronic and structural properties of transition metal clusters. The successful application of real-space approaches to the study of these complex materials is very attractive, owing to their advantages over other techniques. In marked contrast to Cu and V clusters, we found that we could accurately reproduce the photoemission spectra of Si clusters without the inclusion of final state relaxations.14 We attribute this difference to d-state correlations which are present in Cu and V clusters, and absent in the Si clusters. This difference is more pronounced in the smaller clusters, i.e., less than a half dozen atoms.

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This work was in part by the National Science Foundation under Grants No. DMR-0130395 and No. DMR-0325218 and the U.S. Department of Energy under Grants No. DEFG02-89ER45391 and No. DE-FG02-03ER15491. The cal-

for example, Metal Clusters, edited by W. Ekardt 共Wiley, New York, 1999兲. 2 P. Fayet, F. Granzer, G. Hegenbart, E. Moisar, B. Pischel, and L. Wöste, Phys. Rev. Lett. 55, 3002 共1985兲. 3 J. C. Phillips, Phys. Rev. 112, 685 共1958兲. 4 P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 共1964兲; W. Kohn and L. J. Sham, ibid. 140, A1133 共1965兲. 5 C. Massobrio, A. Pasquarello, and R. Car, Phys. Rev. Lett. 75, 2104 共1995兲; C. Massobrio, A. Pasquarello, and R. Car, Phys. Rev. B 54, 8913 共1996兲. 6 J. R. Chelikowsky, N. Troullier, K. Wu, and Y. Sadd, Phys. Rev. B 50, 11355 共1994兲; J. R. Chelikowsky, N. Troullier, and Y. Saad, Phys. Rev. Lett. 72, 1240 共1994兲; M. M. G. Alemany, M. Jain, L. Kronik, and J. R. Chelikowsky, Phys. Rev. B 69, 075101 共2004兲; T. L. Beck, Rev. Mod. Phys. 74, 1041 共2000兲; J.-L. Fattebert and J. Bernholc, Phys. Rev. B 62, 1713 共2000兲; G. Zumbach, N. A. Modine, and E. Kaxiras, Solid State Commun. 99, 57 共1996兲, F. Gygi and G. Galli, Phys. Rev. B 52, R2229 共1995兲; and references therein. 7 J. R. Chelikowsky, M. M. G. Alemany, M. Jain, L. Kronik, and Y. Saad 共unpublished兲. 8 D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 556 共1980兲. 9 N. Troullier and J. L. Martins, Phys. Rev. B 43, 1993 共1991兲. 10 L. Kleinman and D. M. Bylander, Phys. Rev. Lett. 48, 1425 共1982兲. 11 R. Lehoucq, D. C. Sorensen, and C. Yang, Arpack User’s Guide: Solution of Large-Scale Eigen-value Problems With Implicitly 1 See,

culations were performed at the Minnesota Supercomputing Institute and at the National Energy Research Scientific Computing Center 共NERSC兲. M.M.G.A. also acknowledges support from the “Ministerio de Educación y Ciencia” of Spain under the program “Ramón y Cajal.”

Restarted Arnoldi Methods 共SIAM, Philadelphia, 1998兲. M. M. G. Alemany, M. Jain, and J. R. Chelikowsky 共unpublished兲. 13 H. Hellmann, Einführung in die Quantumchemie 共Deuticke, Leipzig, 1937兲; R. P. Feynman, Phys. Rev. 56, 340 共1939兲. 14 N. Binggeli, J. L. Martins, and J. R. Chelikowsky, Phys. Rev. Lett. 68, 2956 共1992兲; X. Jing, N. Troullier, D. Dean, N. Binggeli, and J. R. Chelikowsky, Phys. Rev. B 50, 12234 共1994兲. 15 H. Häkkinen, B. Yoon, U. Landman, X. Li, H.-J. Zhai, and L.-S. Wang, J. Phys. Chem. A 107, 6168 共2003兲. 16 W. A. de Heer, Rev. Mod. Phys. 65, 611 共1993兲. 17 C.-Y. Cha, G. Ganteför, and W. Eberhardt, J. Chem. Phys. 99, 6308 共1993兲. 18 K. Jug and B. Zimmermann, J. Chem. Phys. 116, 4497 共2002兲. 19 P. Calaminici, A. M. Köster, N. Russo, and D. R. Salahub, J. Chem. Phys. 105, 9546 共1996兲. 20 O. Cheshnovsky, K. J. Taylor, J. Conceicao, and R. E. Smalley, Phys. Rev. Lett. 64, 1785 共1990兲. 21 M. Iseda, T. Nishio, S. Y. Han, H. Yoshida, A. Terasaki, and T. Kondow, J. Chem. Phys. 106, 2182 共1997兲. 22 S. Li, M. M. G. Alemany, and J. R. Chelikowsky, J. Chem. Phys. 121, 5893 共2004兲. 23 H. Grönbeck and A. Rosén, J. Chem. Phys. 107, 10620 共1997兲. 24 H. Wu, S. R. Desai, and L. S. Wang, Phys. Rev. Lett. 77, 2436 共1996兲. 12

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Ab initio calculations of the photoelectron spectra of transition metal clusters Shen Li,1 M. M. G. Alemany,1,2 and James R. Chelikowsky1,3 1Department

of Chemical Engineering and Materials Science, Institute for the Theory of Advanced Materials in Information Technology Digital Technology Center, University of Minnesota, Minneapolis, Minnesota 55455, USA 2Departamento de Física de la Materia Condensada, Facultad de Física, Universidad de Santiago de Compostela, E-15782 Santiago de Compostela, Spain 3 Departments of Physics and Chemical Engineering, Institute of Computational Engineering and Sciences, University of Texas, Austin, Texas 78712, USA 共Received 13 December 2004; revised manuscript received 14 February 2005; published 27 April 2005兲 We report the results of an ab initio study of the photoelectron spectra of copper and vanadium anionic clusters. Our calculations are performed using a real-space pseudopotential approach based on the densityfunctional theory, and include final state effects. For each size of the cluster, we find the lowest energy structure. The calculated photoelectron spectra, using these ground state structures, reproduce the main features of those obtained by experiment. Our results demonstrate that real-space techniques are advantageous techniques for the study of the electronic and structural properties of transition metals clusters. DOI: 10.1103/PhysRevB.71.165433

PACS number共s兲: 33.80.Eh, 31.15.Ew, 36.40.⫺c

The study of the electronic and structural properties of transition metal 共TM兲 clusters is a very active field in computational science. The main reasons for this are two-fold: First, TM clusters hold great interest both from fundamental and technological points of view.1 For example, these clusters can possess magnetic moments very different from the bulk phase. As a consequence, TM clusters can serve as models for studying the transition from an atomiclike to solidlike regime as a function of the system size. Also, TM clusters can show catalytic activity when supported on solid substrates. Experiment shows that as small as four silver atoms can be catalytic centers in AgBr emulsions for latentimage formation in the phtographic process.2 Second, important breakthroughs in experimental techniques have allowed accurate data to be obtained for many properties of interest of TM clusters. Specifically, high resolution photoelectron spectra 共PES兲 are available for a wide range of photon energies and clusters sizes for TM elements. Computational methods that reproduce the PES would give not only the opportunity to interpret better the electronic structure information, which is directly obtained from the PES, but to also extract valuable information about the geometric structure of TM clusters. Standard quantum-chemistry methods have shown to be adequate for describing the electronic excitation energies of TM clusters. However, owing to their enormous computational load, their applicability is often limited to small clusters consisting of simple metals, in which the treatment of the d electrons is not necessary. To avoid such restrictions, workers often make use of techniques that employ pseudopotentials3 and density functional theory 共DFT兲.4 Pseudopotential theory allows one to focus on the chemically active valence electrons by replacing the strong all-electron atomic potential with a weak pseudopotential, which effectively reproduces the effects of the core electrons on the valence states. The pseudopotential approximation significantly reduces the number of eigenvalues to be handled. This is especially important for heavier elements. DFT replaces the many-electron wave function representation of the sys1098-0121/2005/71共16兲/165433共5兲/$23.00

tem, which is present in standard quantum-chemistry approaches, by a set of noninteracting one-electron wave functions with the same charge density as the original system. Even with these simplifications of the electronic structure problem, the range of sizes and complexity of the systems that can be investigated by quantum computations remains limited. DFT is formulated to be valid only for the electron ground state, so the study of excited state properties is outside of its scope. Another serious limitation comes from the basis sets that are normally used for expanding the oneelectron wave functions, e.g., localized basis sets and plane waves. Basis sets like Gaussians or atomic-like functions require an extensive testing of the basis. This entails an optimization of a multiple parameter space. Consequently, results obtained from these approaches can be sensitive to the basis choice. Plane-wave methods have the advantage that only one parameter 共the wavelength of the highest Fourier component used in the expansion兲 need be refined to control convergence, but they possess both “physical” and computational drawbacks. In a plane-wave representation, periodic boundary conditions are typically employed. If one wants to study a nonperiodic system such as a molecule or cluster, care must be taken to reproduce the vacuum accurately, since spurious interactions between replicated images of the system must be avoided. Also, a periodic representation is complicated by the study of charged systems: periodicity makes the system infinitely charged. To avoid this situation, an artificial uniform compensating charge is usually inserted in order to prevent a divergence of the total energy. Furthermore, plane-wave approaches make extensive use of fast Fourier transforms 共FFTs兲 for performing matrix-vector multiplications between the Hamiltonian matrix and the trial wave vectors. Since FFTs involve nonlocal operations, the efficiency of their implementation on parallel computers architectures is diminished by the need for global communications among processors. Here we illustrate that the main characteristics of the PES of TM clusters can be efficiently and accurately described using a real-space implementation of DFT and pseudopoten-

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©2005 The American Physical Society

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tials. The procedure we employ for calculating the PES is as follows. We first obtain the electronic ground state of the clusters. Then, we calculate the excitation energies of the clusters by accounting for final states effects 共that is effects derived from the relaxation of eigenstates upon the removal of one electron in the cluster兲. We use the method proposed by Massobrio, Pasquarello and Car 共MPC兲.5 In order to assess the validity of this procedure, we compute the excitations energies of clusters of two TM elements with a very different electronic structure, Cu共关Ar兴3d104s1兲 and V共关Ar兴3d34s2兲, for which extensive PES data is available. Real-space methods6 not only allow for the exact representation of the electronic problem of a system within DFT, but they also avoid the limitations that are present in “traditional” DFT-approaches. In such methods, there is no “formal” basis. The calculations are performed directly on a realspace grid. Convergence is assured by controlling only one parameter: the grid spacing. There is no need to introduce artificial periodicity in dealing with localized systems. Our real-space method is based on the self-consistent solution of the Kohn-Sham equations on a uniform orthogonal threedimensional grid.7 In order to implement the Laplacian operator, we employ a higher-order finite-difference method; for the exchange and correlation potential we use the local spin density approximation given by Ceperley and Alder,8 and the electron-ion term is determined using nonlocal normconserving ionic pseudopotentials cast in the KleinmanBylander form.9,10 In doing so we obtain a matrix representation that is local. Consequently, highly efficient diagonalization procedures can be employed to extract the required eigenvalue/eigenstate pairs.11 Also, the code is easily parallelized by a domain decomposition approach with only limited communication between processors.12 We placed negatively charged Cu and V clusters at the center of a spherical domain chosen so that the wave functions smoothly vanish at its boundary 共normally, the outer atom of the clusters was ⬃9 a.u. 共1 a.u.= 0.529 Å兲 removed from the boundary兲. The Cu 共V兲 core electrons are represented by pseudopotentials generated for the reference atomic configuration 3d104s14p0 共3s23p63d3兲 using the Troullier-Martins prescription,9 with radial cutoffs of 2.35/ 2.50/ 2.35 a.u. 共1.55/ 2.55/ 2.50 a.u.兲 for the s, p, and d channels, respectively. The Kohn-Sham equations were iterated self-consistently until the difference between the input and output potentials was less than 0.001 eV. Convergence was attained for a grid spacing of 0.35 a.u. 共0.30 a.u.兲 for the Cu 共V兲 clusters. For each cluster size, we typically constructed four or five topologically distinct structures, and determined the ground-state geometry by direct comparison of the total energies of the structures after optimization. The clusters were relaxed until the force component for each Cartesian direction 共as derived from the Hellmann-Feynman theorem13兲 was less than 0.001 a.u. 共⬃0.05 eV/ Å兲. Once the ground-state electronic structure of the anionic clusters is obtained, one can analyze the PES by direct comparison with the density of states 共DOS兲 calculated from the single-particle eigenvalues.14 Such attempts have been tried with relative success for TM elements using different DFT

approaches.5,15 As noted above, one has to examine the foundations of DFT in order to find an explanation for discrepancies between the theoretical and experimental data. MPC proposed a simple method for calculating single-particle excitation energies.5 They obtained promising results when applied to the study of the PES of Cu clusters. We adopt their approach here, albeit with a different implementation. Within their method, the excitation energy corresponding to the kth eigenstate, ⌬SCF共k兲, is calculated as 0 − 共k兲 − E initial , ⌬SCF共k兲 = Efinal

共1兲

0 共k兲 is the total energy of the neutral cluster obwhere Efinal tained after the removal of the electron in the original an− . When determining ionic cluster whose total energy is E initial the final excited state for the neutral cluster, the electronic cloud surrounding the hole is allowed to relax, but the geometric structure of the cluster is held fixed 共a vertical excitation process is considered兲. The relaxation of the electronic degrees of freedom entails a self-consistent process. In each step of the process, a tenfinal , of the neutral cluster is astative set of eigenstates, m signed, and the Hamiltonian of the system constructed. A new set of eigenstates is obtained by operating the Hamiltonian on the eigenstates. In order to maintain the wave function assignment of the hole to the kth eigenstate during the self-consistent process, the eigenstates are expanded in a basis of the eigenstates for the original cluster,

mfinal = 兺 Cmlinitial , l

共2兲

l

where the coefficients Cml are obtained by a diagonalization final procedure. The process is initiated by approximating m initial with the eigenvectors m , after zeroing the occupation number of the kth eigenstate. The excitation energy spectrum is obtained by calculating ⌬SCF共k兲 for each eigenstate of the original cluster. In our calculations, both hole and valence states were allowed to relax. However, if the final states are close in energy, we often found difficulties in achieving convergence. In these cases, we did a constrained relaxation where the hole state was kept frozen while the remaining states were allowed to relax. We estimate the error introduced by using the constrained scheme to be less than 0.2 eV for the clusters considered. The study of the excitation spectra of Cu and V clusters constitutes a stringent test of the method, as these clusters exhibit important differences in their PES. Vanadium can be considered a “typical” TM element, with a partially occupied 共and chemically active兲 d band. In contrast, the valency in copper is determined by a d band that is filled with a single electron located in the s band. Such an electronic configuration, which is characteristic of all “noble” metals, makes Cu clusters resemble alkali metal clusters in their electronic shell structure.16 However, it has been shown that the inclusion of the d electrons 共e.g., s − d hybridization兲 is necessary to properly describe the electronic properties of Cu clusters.5,17

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FIG. 1. Ground-state structures for Cu−n 共n = 3, 5, 8, and 9兲 and V−n 共n = 3 and 4兲 as obtained from our real-space pseudopotential approach.

In Fig. 1 we plot the ground state structures calculated for several Cu−n 共n = 3, 3, 5, 8, and 9兲 and V−n 共n = 3 , 4兲 clusters. We previously discussed the structures of V clusters elsewhere.22 All these structures possess C2v symmetry, and are largely consistent with previous DFT-based calculations.5,18,19 The most stable structure for Cu−3 is a linear chain, which is lower by ⬃0.2 eV/ atom than an isosceles triangle. A planar trapezoidal structure represents the ground state geometry for Cu−5 ; this structure is more stable than a trigonal bipyramid by ⬃0.06 eV/ atom, which is the second most stable structure for n = 5. The lowest energy structures for Cu−8 are a bicapped octahedron and a capped pentagonal bipyramid 共separated in energy by about 0.02 eV/ atom兲. By adding an atom capped at the bottom of the bicapped octahedron we obtained the most favorable structure for Cu−9 . A similar structure was found for n = 9 by Jug et al.18 共the atom is added at the top instead than at the bottom兲. However, MPC obtained as ground state structure a bicapped pentagonal bipyramid;5 in our calculations such a structure is only ⬃0.02 eV/ atom higher in energy than that plotted in Fig. 1. The ground state structures obtained for n = 3, 5 and 9 have a zero magnetic moment; the structure obtained for n = 8 has magnetic moment one B. In Fig. 2 we compare the DOS and excitation energy obtained for the Cu anionic structures plotted in Fig. 1 with available experimental data.17 The excitation energy calculated by MPC using a plane-wave approach for clusters with n = 3, 5, and 9 is also shown.5 Our theoretical values were broadened using a Gaussian distribution function with width at a half-maximum of 0.08 eV. The energy scale of the theoretical results was shifted by a constant so the highest occupied molecular orbital 共HOMO兲 of each cluster was aligned to the corresponding peak in the PES. The shift in

FIG. 2. DOS and excitation energies 共⌬SCF兲 as obtained from our real-space pseudopotential approach for the Cu−n ground-state structures represented in Fig. 1. Results for the excitation energies as obtained from a plane-wave pseudopotential approach for n = 3, 5, and 9 共Ref. 5兲 are also shown 共“other theory;” upper lines兲. The theoretical results are compared with available experimental data 共Ref. 17; dashed lines兲.

energy is estimated to be ⬃2 eV for the DOS, and as small as ⬃0.2 eV for the excitation energy. We attribute the small offset in the excitation energy curves to relaxations in the geometry of the clusters during the electronic excitation process that were not taken into account in our theory. In the comparison between theory and experiment, one must focus on the positions of the peaks in the spectra and not in their relative intensities. The intensities are affected by angularmomentum dependent photodetachment cross sections which are not included in theory. The different nature of the 3d and 4s bands in Cu is reflected in the experimental spectra of the Cu−n clusters. The corelike character of the 3d electrons give rise to a “band” in the PES that is quite localized. The position of this band shifts upwards monotonically in energy when the size of the cluster increases,20 and its onset is determined to be roughly 2 eV below the HOMO, at a binding energy of ⬃4 eV for the clusters studied in this work. The features that occur between the HOMO and the onset of the 3d band are mostly s-like. Owing to the delocalized character of the 4s valence electrons, these features strongly depend on the surface size and geometry of the clusters, and do not show any evident pattern with increasing cluster size. As one can see from Fig. 2, the DOS calculated from the single particle eigenvalues give a very poor description of the PES for the smaller clusters studied, Cu−3 and Cu−5 ; even a qualitative assignment of most of the theoretical peaks to

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corresponding peaks in the PES is not possible 共e.g., for Cu−3 , experiment clearly shows the existence of two distinct features before the onset of the d band, while exhibits five closely spaced peaks with binding energies less than 4 eV兲. For Cu−8 and Cu−9 , the two first major peaks discernible in the PES are identifiable in the DOS, although their relative position is not perfectly matched, but beyond the second peak the experimental features are not properly reproduced, i.e., theory tends to put peaks very close in energy. The limitations of the single particle eigenvalues in reproducing the PES are overcome when the final state effects are taken into account. As we see in Fig. 2, one can identify all the main features of the PES through the theoretical excitation energies. Also, the relative position of the experimental features is accurately reproduced. The most important mismatch between theory and experiment occurs for Cu−3 , where theory yields doublet peaks in contrast to the experimental feature at ⬃3.5 eV. The experimental feature is divided in several subpeaks. Our results for the excitation energy agree with experiment better than the results obtained by MPC,5 especially for Cu−5 and Cu−9 . The PES of these two clusters clearly show the existence of three distinguishable peaks before the onset of the 3d band at ⬃4 eV. MPC’s spectra for Cu−5 only gives two peaks, with a separation between them that is off by ⬃0.5 eV when compared with experiment. For Cu−9 , three distinct peaks also are discernible from the results obtained by MPC, but their distribution along the 0 – 4 eV binding energy range differs greatly from that observed in experiment. The differences between our results and MPC’s for Cu−9 can be attributed, at least in part, to geometry differences between the ground state structures for which the excitation spectra was calculated. The PES of vanadium clusters posses characteristics that are rather different than that observed in copper. In vanadium, the 3d shell produces features that are not as “well localized” as in copper, and that contribute significantly to regions of the spectra that are close to the Fermi level.21 In recent work performed with our real-space approach, we have shown that a generally good description of the PES of V−n clusters is attainable through the DOS, particularly for n ⬎ 4 共Ref. 22兲. Here we wish to estimate how the inclusion of the final state effects affect the previously calculated spectra. In Fig. 3, we plot the DOS and excitation energies calculated for the ground state structures obtained for V−3 and V−4 , an isosceles triangle and a planar rhombohedral structure, respectively 共see Fig. 1兲. The excitation energies obtained for similar geometric structures by Grönbeck and Rosén using a DFT method that employ a linear combination of atomic orbitals as a basis are also shown.23 Theory is compared with the available experimental data.24 共The DOS and excitation energy curves were shifted by approximately 3 and 0.5 eV, respectively, to align the HOMO to the corresponding peak in the PES.兲 As we see in Fig. 3, the PES curve for V−3 is characterized by four distinct peaks below ⬃2.5 eV. Four features are also distinguishable from the DOS, but their positions do not match experiment as well as do the peaks of the excitation energy curve. As regards V−4 , the PES shows five features in the 0 – 4 eV binding energy range 共at approximately 1.8, 2.3, 2.6, 3.2, and 3.9 eV兲. These features are already identifiable from the DOS, but the agree-

FIG. 3. DOS and excitation energies 共⌬SCF兲 as obtained from our real-space pseudopotential approach for the V−3 and V−4 groundstate structures represented in Fig. 1. Results for the excitation energies as obtained from a method that employs a linear combination of atomic orbitals as a basis 共Ref. 23兲 are also shown 共“other theory;” upper lines兲. The theoretical results are compared with available experimental data 共Ref. 24; dashed lines兲.

ment with experiment is improved when final state effects are included in the calculations. In particular, the “shape” of the first and second peaks are better reproduced, as well as the position of the remaining peaks. Our results for the excitation energy for V−4 are similar to those obtained by Grönbeck and Rosén.23 However, their results for V−3 only show two distinct peaks with binding energies lower than 2.5 eV. In summary, we have presented ab initio results for the excitation energies of copper and vanadium clusters. The calculations were performed using a method that is based on the self-consistent solution of the Kohn-Sham equations on a uniform real-space grid. A higher-order finite-difference method is combined with ab initio pseudopotentials. The good agreement found between the calculated excitation energy curves and the experimental photoelectron spectra establishes the adequacy of such techniques for studying the electronic and structural properties of transition metal clusters. The successful application of real-space approaches to the study of these complex materials is very attractive, owing to their advantages over other techniques. In marked contrast to Cu and V clusters, we found that we could accurately reproduce the photoemission spectra of Si clusters without the inclusion of final state relaxations.14 We attribute this difference to d-state correlations which are present in Cu and V clusters, and absent in the Si clusters. This difference is more pronounced in the smaller clusters, i.e., less than a half dozen atoms.

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This work was in part by the National Science Foundation under Grants No. DMR-0130395 and No. DMR-0325218 and the U.S. Department of Energy under Grants No. DEFG02-89ER45391 and No. DE-FG02-03ER15491. The cal-

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