Density-Functional Theory Studies on Beryllium Metal Fragments of 81

Chapter 29

Density-Functional Theory Studies on Beryllium Metal Fragments of 81, 87, and 93 Atoms Richard B. Ross , C. William Kern , Shaoping Tang , and Arthur J. Freeman 1

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Downloaded by COLUMBIA UNIV on March 17, 2013 | http://pubs.acs.org Publication Date: May 5, 1996 | doi: 10.1021/bk-1996-0629.ch029

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PPG Industries, P.O. Box 9, Allison Park, PA 15101 Department of Chemistry and Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208 1

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Density functional molecular orbital theory has been applied tofragmentsof bulk beryllium consisting of 81, 87, and 93 atoms. Mulliken net charges for all atoms are found to be close to the bulk value of zero. ΔSCF ionization potentials are found to be 1.7 to 1.9 eV larger than the bulk workfunction (3.92 eV). Valence electron density plots provide greater detail than electron density plots from previous Hartree-Fock-Roothan studies. Valence electron deformation plots indicate increased electron density along the principal axis of symmetry (z) (perpendicular to the xy basal plane) in comparison to the x and y axial directions. In addition to comparisons with experimental bulk properties, the results are compared to previous Hartree-Fock-Roothan studies and density functional models of other beryllium clusters. It is possible to conclude that the calculated binding energies converge to near the bulk experimental value at ~70-80 atoms. The accuracy of the current model to determining binding energies and atomic populations forfragmentsof bulk beryllium as small as 81 atoms provides additional evidence that density functional methods are a useful tool to characterize the electronic structure of bulk metal systems. The structure and dynamics of metal clusters are being characterized in increasing detail with the advent of modern instrumentation and sophisticated spectroscopic techniques. The experimental studies are providing increased understanding into the nature of intermetallic binding in clusters. For theoretical studies on clusters composed of more than a few atoms, two possible approaches to characterize the many-body interactions within the clusters are the Hartree-Fock-Roothan (HFR) and density functional theories (DFT). The methodologies reduce the N dependence of highly correlated N-orbital schemes to a more tractable dependence of N -N . The Hartree-Fock-Roothan method leads to the best single determinant 7

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Current address: Materials Science Laboratory, Texas Instruments, Inc., Mail Stop 147, Dallas, T X 75243 0097-6156/96/0629-0435$15.00/0 © 19% American Chemical Society In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

Figure 1. Three-dimensional perspective views of the 81, 87, and 93 atom clusters. The atoms in the clusters occupy positions as in the bulk hep lattice.



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uncorrelated wave function while the density functional theory method replaces the full Hamiltonian with an approximate correlated potential. Previously, HFR studies have been carried out by Ross, Kern, Ermler, Pitzer and coworkers on clusters of 13 (7), 19 (2), 21 (2), 33 (2), 39 (2), 51 (3), 57 (3), 69 (4), 81 (5), 87 (5), 93 (6), 105 (d), 111 (6), 123 (6), and 135 (7)beryllium atoms. In these studies, evidence is presented for the convergence of bulk properties such as binding energy and net charges with cluster size. In addition, the size of the cluster required for convergence has been seen to depend on the property of interest. A HFR study has also been carried out by Pettersson and Bauschlicher on a cluster of 55 beryllium atoms (8). A Hartree-Fock band structure calculation (9), density functional band theory studies (10,11), and orthogonalized plane wave studies (72,13) have also been performed in an examination of the electronic properties of bulk beryllium. Recently, a density functional molecular orbital theory study has been performed by Tang and coworkers on a cluster of 135 beryllium atoms (14). In the present paper, density functional theory studies are reported for clusters of 81, 87, and 93 beryllium atoms. Calculated binding energies, ionization potentials, net charges, and electron densities are given and compared to previous work and experimental data where available. Calculations

Three dimensional views of the metal fragments can be seen in Figure 1. The atoms occupy the positions as they would in the bulk hep beryllium lattice. As discussed in detail previously (2), thefragmentsare derived by adding sets of atoms as they are found on successive coordination spheres about a central beryllium atom. The calculations employ the local density functional approach (LDA) for molecules as implemented (75a) in the program DMol. The exchange and correlation potential employed is the explicit form of V * ^ given by Hedin and coworkers (16). A detailed discussion of the formalism of the method has been presented previously by Delley (15b). The basis functions in DMOL are generated numerically from the local density functional solutions for free atoms and for positively charged atoms. Five basis functions are used for beryllium which contain neutral Be-2s, Be -2s, and hydrogenic C-2p orbitals. The C-2p orbitals are used as polarization functions. Ground state neutral self-consistent field (SCF) studies have been carried out for all fragments. SCF studies have also been performed on the +1 ion. ASCF ionization potentials have been computed then as the difference of converged neutral and first ionized states. The studies for the closed shell neutral and first ionized states have been carried out in spin-restricted and spin-unrestricted manners, respectively. The degree of convergence of the self-consistent iterations, measured by root mean square (rms) changes in the charge density, is set at 10" which allows the total energy to converge to 10" Ry. +2

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CHEMICAL APPLICATIONS OF DENSITY-FUNCTIONAL THEORY

The ground state neutral calculations on Be i, Be , and Be^ have been carried out on a Cray Y-MP supercomputer requiring 66, 85, and 64 minutes of CPU time, respectively. The +1 ion SCF studies have been carried out on a Silicon Graphics R8000 workstation and required 351, 820, and 681 minutes for the same respective atom clusters. 8

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Discussion for Selected Properties

Binding Energy. Calculated binding energies for the fragments are shown in Table I and plotted in Figure 2. Also shown for comparison are binding energies calculated from LDA studies on a fragment of 135 atoms (14) as well as on a series of smaller fragments (77). The experimental binding energy (18) and binding energies calculated for a series of HFR clusters (7-7) are also included for comparison.

Table L Calculated Binding Energies (kcai/mon for Beryllium Ousters

Fragment Bel3 Bel9 Be21 Be33 Be39 Be51 Be57 Be69 Be81 Be87 Be93 Bel05 Belli Bel23 Bel35 Bulk Exp

HFR 12.0 17.2 17.9 16.5 21.7 25.8 25.4 28.2 28.3 28.5 28.0 29.9 30.5 29.6 31.7

DFT* 50.6 59.8 59.4 59.4 65.0 70.1 70.8 73.0 73.2 73.4 73.2

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*DFT studies for clusters of 13 to 69 from Ref. 17. Binding energy for Be^s from Ref. 14. HFR binding energies from Ref 1-7. 'Tlef. 18.

b



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As can be seen in the figure and table, the DFT binding energies converge to near the bulk value of 75.3 kcal/mol at -70-80 atoms. Specifically, the binding energies for 69, 81, 87, and 93 atomfragmentsare found to be 73.0, 73.2, 73.4, and 73.2 kcal/mol, respectively. As the cluster size is increased to 135 atoms, the binding energy is overestimated slightly (77.5 kcal/mol) compared to the experimental value (75.3 kcal/mol (18)). It should be noted, however, that the binding energy will likely increase somewhat more if d-polarization functions are included in the basis set. As can also be seen in Figure 2, the HFR binding energies begin to converge with cluster size at a point similar to the DFT results (-70-80 atoms). However, the calculated values of the binding energies are 28-32 kcal/mol for the largest clusters which is well below the bulk value of 75.3 (18). The underestimation of binding energy for the HFR clusters is likely due in largest part to the omission of electron correlation effects which are approximated in the DFT methodology through the exchange-correlation potential. As can also be seen in Figure 2, the trends for increasing binding energy with cluster size for both the DFT and HFR methodologies are similar. For example, there is an increase between 13 and 19 atoms followed by a leveling off to 33 atoms. The binding energy then increases between 33 and 51 atoms. The binding energy then increases sharply to 57 atoms followed by a more gradual increase to 135 atoms in both cases. The similarity of trends between the two methodologies also increases confidence that the lowest states have been found in the HFR clusters which require a search amongst low-lying electron configurations. An estimate of the contribution of electron correlation to the binding energy can be obtained by subtracting the Ben HFR and DFT binding energies which yields a difference of 38.6 kcal/mol. Calculation of this difference for clusters from 19 through 135 atoms produces a rather narrow range of values from 42.6 to 45.8 kcal/mol which are only about 15% greater than the Ben difference. Adding the Bei difference of 42.6 kcal/mol to the HFR Bens binding energy yields a binding energy of 74.3 kcal/mol which is to within 1 kcal/mol of the bulk binding energy (18). This analysis suggests that well over 95% of the electron correlation contribution to the binding energy of bulk beryllium metal is localized in the Be^ cluster. 9

Ionization Potential. Calculated ASCF ionization potentials from the studies are shown in Table II. As with binding energies, calculated values from previous HFR studies (7-7) and a DFT study on Bei 5 (14) are included for comparison. In this case, for lack of any other measurements, the calculated ionization potentials are compared to the experimental workfunction of the bulk metal (3.92 eV) (79). As can be seen from the calculated values in the table, the DFT ASCF ionization potentials range from 1.7 to 1.9 eV greater than the experimental workfunction. In contrast, the HFR ASCF ionization potentials agree to within 0.4 eV. 3

Atomic Charge. Net atomic charges calculated from Mulliken population analyses (20) are summarized in Table HI. They are also shown in Figure 3 as a function of radius R for successive coordination shells of the symmetry-distinct



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N u m b e r of A t o m s Figure 2. Calculated binding energies (kcal/mol) as a function of cluster size. Previously calculated HFR values (1-7) are included for comparison as are values from experiment (18) and DFT studies on a series of smaller clusters (7 7) and on Be

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groups of atoms. Positive numbers represent a gain of charge and negative numbers a loss of charge. As can be seen in the table and figure, calculated net charges for the central atom, BeO, are -0.038, -0.009, and +0.007 e for the 81, 87, and 93 atom clusters, respectively. These values are close to the bulk beryllium metal value of zero. The net charges for the remaining symmetry groups of atoms oscillate around the zero line with similar but not completely identical patterns. The absolute values of net charge range from 0.002 to 0.14 e indicating again the approximately neutral character of the atoms in the fragments. Lack of atomic relaxation may be partially responsible for the charges not being calculated to be exactly 0.0 e. Similar observations have been seen for net atomic charges in a DFT study by Tang et al. on Be^s (14). The net charge on the central beryllium atom was found to be 0.03 electrons. The net charges for the other symmetry-distinct groups of atoms also oscillate around the zero line without showing a clear trend. The absolute value of net charges are found to be small and range from 0.0-0.08 e. The similarity of the net charges for the 81, 87, 93, and 135 atom clusters indicates that they are either independent of cluster-size or converge to bulk limits in the DFT methodology at clusters of 81 atoms or less. In contrast, HFR net charges calculated previously (5,6) for 81, 87, and 93 atomfragmentsrange in absolute value from 0.02-1.0, 0.01-1.19, and 0.02-1.12 e, respectively. The charges on the central atom for the lowest states in the 81, 87, and 93 atom clusters were found to be 0.17, 0.18, and -0.06 e, respectively. The HFR average net charge for the central atom for the lowest states of 105 (6), 111 (d), 123 (d), and 135 (7) atom clusters have been found to be -0.09, -0.06, -0.06,


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Table Π. ASCF Ionization Potentials (eV) from the DFT studies and Lowest States of HFR Studies


Fragment Bel3 Be81 Be87 Be93 Bel 05 Belli Bel23 Bel35

Δ SCF Ionization Potential (eV) DFT HFR 4.85 4.28 5.90 3.78 5.70 4.20 5.81 4.31 4.48 4.03 4.29 5.85 a

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HFR ionization potentials from Réf. 1 (Bei ), Ref. 5 (Be i, Be ), Ref. 6 through Bem), and Ref. 7 (Be^s). W . 14. a

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(Ββ93

and 0.02 e. The net charge on the central atom trends towards the bulk value of zero with increasing cluster size and hence is cluster-size dependent in the HFR methodology. This is in contrast to the DFT studies for which all net charges are close to zero for 81, 87, 93, and 135 atom clusters. This suggests that the DFT valence electron density is distributed more evenly over the atom cores compared to the HFR density which omits electron correlation. Charge Density. Contour plots of the valence and deformation charge densities for the 81, 87, and 93 atom clusters are shown in Figures 4, 5, and 6 for the XY, XZ, and YZ planes, respectively. The deformation charge density is computed by subtracting the sum of atomic charges from the total charge for each respective cluster so that the charge redistributions resulting from DFT binding are seen more clearly. The dark and light lines represent charge gained or lost, respectively. In the XY plane (Figure 4), there are two distinct triangular charge environments. The distinct environments are due to the presence of atoms in planes above and below one of the triangular regions. Examination of both the valence and deformation charge distributions for all three clusters display this characteristic pattern. The two different types of triangular regions alternate in six triangular segments around the central atom. In all three clusters, one region consists of a circular contour. In Be i, the second region consists of 2 contours which are beginning to form a distorted hexagon shape. In Be 7 and Be93, in the second region the three contours have transformed even more to a distorted hexagon. Examination of additional atoms in the next shell around the central atom also shows two different triangular regions of similar shapes. The increased electron density in the hexagon region, in comparison to the circular region, is indicative of the presence of atoms in planes above and below this region. 8

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Cluster BeO BeA BeB -0.038 0.014 0.036 Besi -0.009 -0.002 0.04 Be 0.007 -0.006 0.02 Bees

Τ

BeF BeG BeH BeC BeD BeE 0.042 0.008 -0.022 -0.014 0.07 -0.031 0.017 0.02 -0.024 -0.003 0.063 -0.05 0.009 0.009 0.027 -0.14 0.074 -0.067

Tflhl ΤΠ Net Atomic Charges (tl in the Density Functional Theory Model

BeJ BeK Bel -0.065 -0.01 -0.004 -0.064 -0.008 0.004 -0.057 0.011 0.013


BeL BeM BeN 0.081 0.117 0.015 0.017 0.022 0.07

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Density-Functional Theory Studies on Beryllium Metal Fragments of 81

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