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Atomic Structure and Energetic Stability of Complex Chiral Silicon Nanowires Pavel V. Avramov,*,†,‡ Soma Minami,§ Stephan Irle,| Leonid A. Chernozatonskii,⊥ and Keiji Morokuma§ Siberian Federal UniVersity, 79 SVobodniy aV., Krasnoyarsk 660041, Russia, L. V. Kirensky Institute of Physics SB, Russian Academy of Sciences, Krasnoyarsk 660036, Russia, Fukui Institute for Fundamental Chemistry, Kyoto UniVersity, 34-3 Takano Nishihiraki, Sakyo, Kyoto 606-8103, Japan, Institute for AdVanced Research and Department of Chemistry, Nagoya UniVersity, Nagoya 464-8602, Japan, and Emanuel Institute of Biochemical Physics, Russian Academy of Sciences, 4 Kosigin st., Moscow, 119334, Russian Federation ReceiVed: February 23, 2010; ReVised Manuscript ReceiVed: July 16, 2010
Atomic and electronic structure and energetic stability of newly proposed pentagonal and hexagonal chiral complex silicon nanowires (NWs) composed of five or six 〈110〉 oriented crystalline fragments were studied using the ab initio DFT method. The chirality of the wires was caused by consecutive shifts of each fragment by 1/5 or 1/6 of the wire unit cell parameter and rotations of 4° and 3.3° for achiral pentagonal or hexagonal wires, respectively. Chirality causes the HOMO-LUMO gap to reduce by 0.1 eV. Chiral silicon nanowires are found to be metastable structures with a 4.5 (kcal/mol)/Si atom potential barrier for reversible chiral T achiral transformation. I. Introduction Chirality of nanoclusters is a challenging topic in modern nanoscience due to remarkable changes of electronic, optical, and mechanical properties of the species. Chiral1-4 (Si,1,2 SiO2,3 PbSe,4) and helical5-14 (ZnSe,5 SiO2 and Al2O3,6,7 GeTe and Sb2Te3,8 ZnO,9-11 GaN, ZnGa2O4 and ZnsnO4,12 SiC/SiO2,13 Au,14) structures of different compositions and shapes were discovered experimentally. Excluding chiral carbon nanotubes (see, for example, ref 15) and fullerenes,16 as well as icosahedral Au chiral clusters,17 the atomic structure and energetic stability of the key chiral and helical nanoclusters have not been studied theoretically. The atomic structure of chiral (n, m) carbon nanotubes was interpreted in terms of screw dislocation along the main tube axis.18 It was proposed that axial dissection of zigzag [(n + 1/2m, 0) for an even m or (n + 1/2m + 1/2, 0) for an odd m] nanotube wall and consequent sliding the sides by a vector b ) bγ + be [bγ ) m(-1/2,1) is screw dislocation Burger vector and be ) (-1/2, 0) or be ) 0 is a small edge component for odd or even m, respectively19] leads to formation of an ideal chiral (n, m) tube with bγ kink on the nanotube end. Silicon nanostructures with pentagonal symmetry were discovered experimentally.20-23 Pentagonal20 and truncated starshaped pentagonal21 wires demonstrate clear pentagonal symmetry and a diamond-type silicon lattice. The twinned nature of the nanoclusters consisting of at least five fragments of diamond-type silicon lattice was clearly seen in STM images.21-23 Previously, the atomic and electronic structures of 〈110〉 oriented complex pentagonal and hexagonal24 and star-shaped pentagonal and hexagonal25 silicon nanowires (Figure 1) were studied using DFT techniques. The wires have central pentagonal or hexagonal prisms as the basis surrounded by one or * Corresponding author. E.mail:
[email protected]. † Siberian Federal University. ‡ Kirensky Institute of Physics, Russian Academy of Sciences. § Kyoto University. | Nagoya University. ⊥ Emanuel Institute of Biochemical Physics, Russian Academy of Sciences.
Figure 1. Three projections of (a) 〈110〉 oriented fragment cut out from silicon crystal and star-shaped (b) pentagonal and (c) hexagonal silicon nanowires. The structures of the central Si15- or Si18-based cores are presented in red. The main 〈110〉 wire axis is presented as red arrows.
several layers of hexagonal prisms and display a pronounced segment structure. The pentagonal silicon wires are the most energetically stable structures among several NWs with small
10.1021/jp1016399 2010 American Chemical Society Published on Web 08/16/2010
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diameter (e10 nm), designed by connection of triangular prisms cut out along the 〈110〉 direction.24 In this work, we designed and studied the atomic and electronic structure and energetic stability of hypothetical complex silicon chiral nanowires of pentagonal and hexagonal symmetry, built up by a combination of five or six uniform 〈110〉 oriented fragments cut from crystalline silicon and connected through equivalent {111} facets with a k/5 or m/6 (where k and m are integers and equal to 0-4 or 0-5 for pentagonal and hexagonal wires, respectively) shift of the unit cell vector along the 〈110〉 direction, respectively. All fragments of the pentagonal and hexagonal wires were rotated 4.0° or 3.3° along the 〈110〉 direction, respectively. On the basis of the DFT calculations, we discovered that the wires are metastable structures with a potential energy barrier of 4.5 (kcal/mol)/Si atom for the reversible chiral T achiral transformation. Chirality decreases semiconducting band gaps and may lead to rotation of optical polarization of the species. II. Methods of Electronic Structure Calculations The Gaussian 0326 code was used to calculate the electronic structure of complex silicon nanowires. Geometry optimization was performed using the analytic energy gradient at the B3LYP/ 3-21G* level of theory.27 All relative energies were calculated by taking into account the counterpoise correction for the basis set superposition error (BSSE). For the systems under study the BSSE corrections were found to be smaller than 3 kcal/ mol. At present, the B3LYP method is probably the most widely applied among all DFT approaches. It was designed especially for the accurate description of dissociation limits for a wide variety of molecules. The accuracy of the B3LYP method in describing the dissociation energies estimated for the G2 benchmark molecular set is equal to 3.5 kcal/mol.27-31 The method predicts much higher accuracy (1 kcal/mol32) in description of relative energetic stabilities of isomers with the same nature of chemical bonding. III. Results and Discussion A. Construction of Chiral Nanowires. Achiral pentagonal or hexagonal24,25,33 complex wires, particularly the star-shaped ones25 (Figure 1b,c), display a pronounced segment structure perpendicular to the main 〈110〉 wire axis. The atomic structure of the wires is determined by the symmetry of the central core. The central cores of pentagonal and hexagonal achiral wires (Figure 1b,c, marked by red) are formed by Si15 or Si18 segments, respectively. The segments are followed by each other along the 〈110〉 direction and have common pentagons/hexagons rigorously perpendicular to the main axis. Infinite pentagonal or hexagonal cores can be considered themselves as 1D superfine nanowires with 1D unit cells consisting of Si10 or Si12 fragments (since the Si15 and Si18 fragments of infinite nanowires have two shared pentagons or hehagons, the 1D unit cell of the wire cores consists of 10 or 12 silicon atoms, respectively, and translation vector bw is equal to 3.622 Å).24,25,33 Addition of external silicon layers (Figure 1b,c, marked by blue) leads to formation of more complex pentagonal/hexagonal wires with more complex structure, and unit cells consist of 40 or 48 silicon atoms for pentagonal and hexagonal star-shaped nanowires (Figure 1b,c), respectively.24,25,33 For all wires, hydrogen atoms were used to saturate surface dangling bonds. The schematic of the central pentagonal Si15-based core of achiral pentagonal wire is presented in Figure 2a (the core is also marked by red in Figure 1a). The core consists of a
Figure 2. ( a) Schematic of 1D unit cell of the central fragment of acharal pentagonal wire. The main axis is perpendicular to the basic pentagons and is shown as a red arrow. Hexagons a, b, ..., e form a unit cell fragment by sharing common bonds (1 and 3, 4 and 6, etc.) of the a and e and a and b hexagons. (b) Atomic structure of the chiral pentagonal star-shaped silicon nanowire. Silicon atoms are in blue; hydrogen atoms are in red. (c) Atomic structure of the 1D unit cell of the chiral pentagonal central core. The dashed arrows denote the way of bond sharing.
sequence of closed Si15 fragments connected to each other by shared silicon pentagons oriented normal to the main 〈110〉 wire direction. A Si15 fragment consists of a belt of hexagons a-e, parallel to the wire 〈110〉 direction, connected by shared bonds numbered 1, 3, 4, and 6. This belt is “unrolled” in the schematic at the bottom of Figure 2a. For achiral wires the bonds 2 and 5 of the a-e hexagons form the basic pentagons that are exactly perpendicular to the main 〈110〉 axis and serve as boundaries of the 1D unit cell (Figures 1b and 2a). The shared bonds (n, 1)/(l, 3), (n, 6)/(l, 4), (n, 3)/(r, 1), and (n, 4)/(r, 6) form the specific structure of the achiral Si15 central fragment, where the hexagon index n is equal to {a, b, ..., e} and l and r are the indexes of the left (l) and right (r) neighbors of the nth hexagon. The second integer index is a bond number (Figure 2a). Again, top and bottom bonds numbered 2 and 5 of all hexagons form pentagons perpendicular to the main 〈110〉 direction. Shifting all 〈110〉 fragments on k*1/5 or m*1/6 of the 1D unit cell vector along the 〈110〉 direction and rotation by 4.0° or 3.3° leads to formation of the chiral pentagonal or hexagonal wire, respectively (Figure 2b,c). The wire chirality is caused by a different way of bond sharing of the central cores. In contrast to the achiral structures, shifting and rotation of the fragments leads to additional sharing of the bonds numbered 2 and 5 of the hexagons separated by additional five hexagons (a and f, for example) of the core, and the unrolled belt of pentagons is no longer exactly perpendicular to the 〈110〉 direction. The 1D unit cells of the chiral wire cores consist of 15 (pentagonal wire) or 24 (hexagonal wire) hexagons and are
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TABLE 1: Structural Parameters, Average Binding Energies per Silicon Atom, and Band Gaps for Different Nanowire Types cluster
length, Å
effective perpendicular size, Å Parent 〈110〉 Oriented Wires 16.739 × 10.858b
Si atom binding energy,a kcal/mol/atom
band gap, eV
-87.17 -88.42 -90.30 -90.93
2.802 2.693 2.485 2.478
Si132H88 Si164H104 Si388H216 Si420H232
16.588 22.715 45.332 48.941
Si140H90 Si380H210
14.503 38.321
Complex Pentagonal Achiral Star-Shaped Wires 16.628 -87.79 -90.30
2.869 2.595
Si140H94 Si380H214
17.193 42.762
Complex Pentagonal Chiral Star-Shaped Wires 16.628 -86.54 -89.68
2.767 2.509
Si168H108
14.140
Complex Hexagonal Achiral Star-Shaped Wire 18.069 -87.17
2.654
16.921
Complex Hexagonal Chiral Star-Shaped Wire 18.069 -85.91
2.631
Si168H112 a
The energy of chemical binding of hydrogen atom with silicon was estimated on the basis of B3LYP/3-21G* calculations of the SiH4 molecule and is equal to -82.28 kcal/mol. b Since a pristine 〈110〉 oriented nanowire has a rhombus cross section (Figure 1a), we present the perpendicular effective sizes of the wire in both directions.
3 or 4 times longer than the unit cells of the parent achiral wire cores, respectively. Actually, the formation of chiral wires is equivalent to introduction of a screw dislocation defect in complex achiral pentagonal or hexagonal wires. Axial dissection of the pentagonal/ hexagonal wire core and consequent sliding of the sides by the bw vector along the 〈110〉 direction leads to formation of an ideal chiral wire with a bw kink on the wire end. Complex silicon nanoclusters can be designed by addition of external silicon layers of a tetrahedral nature to the core structures and termination by hydrogens on the wire surface.34 In achiral wires the external silicon layers form segments that are connected to each other along to the main 〈110〉 axis.24,25 In contrast to achiral wires, the external silicon layers of the chiral wires form ribbons that coil around the chiral cores. In such a case, the silicon-silicon bonds are strained and squeeze the coiled external ribbons with decreasing the chiral angle and elongation of the units cells up to 20 and 30 hexagons for pentagonal and hexagonal wires with one silicon external layer, respectively. B. Structure and Stability of the Achiral and Chiral Wires. Average binding energies per silicon atom as well as structural parameters and band gap widths of the studied clusters are presented in Table 1. For the sake of comparison, four 〈110〉 parent fragments of the silicon crystal of rhombus cross sections (Figure 1a) with saturated surface dangling bonds by hydrogen atoms (Si132H88, Si164H104, Si388H216, and Si420H232 clusters) were calculated as well as complex pentagonal achiral (Si140H90 (Figure 3a) and Si380H210) and chiral (Si140H94 (Figure 3a) and Si380H214) and hexagonal achiral (Si168H108) and chiral (Si168H112 (Figure 3b)) clusters. In calculating the binding energy of silicon, one needs the H binding energy of a hydrogen atom with a silicon atom (Ebind ), which was estimated from a calculation of the SiH4 molecule H SiH4 as follows: Ebind ) (Etot - ESi - 4EH)/4 ) -82.28 kcal/mol, 4 is the total energy of the SiH molecule (-182248.61 where ESiH tot 4 kcal/mol) at the optimized geometry, and ESi (-180674.00 kcal/ mol) and EH (-311.37 kcal/mol) are the B3LYP energies of free Si (3P) and H (2S) atoms. The average silicon binding Si energies per silicon atom (Ebind ) of the wires were calculated
Figure 3. (a) Optimized structure of (a) achiral (Si140H90) and chiral (Si140H94) pentagonal and (b) achiral (Si168H108) and chiral (Si168H112) hexagonal star-shaped clusters. Si SiH H using the expression Ebind ) (Etot - nESi - m(EH + Ebind ))/n, SiH where Etot is the total energy of the SinHm cluster at optimized geometry. Since the pentagonal and hexagonal star-shaped complex wires display less energetic stability in comparison with parent 〈110〉 fragments of comparable lengths and cross sections,25 Table 1 indicates the same metastable nature of all chiral clusters. Despite small systematic energy differences (0.62-1.26 kcal/mol/Si atom, Table 1) between achiral and chiral species of comparable lengths for both pentagonal and hexagonal wires, considering the reliability is at best 1 kcal/mol at the B3LYP level, strictly speaking, we have to consider chiral and achiral wires of the same type (pentagonal or hexagonal) and comparable lengths as energetically nearly equal. The parent wires (Si132H88, Si164H104, Si338H216, Si420H232) with the same cross sections display a typical34 increase of the average energy of the Si atom chemical binding with the increasing the length of the clusters (Table 1). In comparison with achiral starshaped wires, the chirality of the chiral ones causes a slight decrease in band gaps of the wires and may lead to optical polarization. Increasing the length of silicon nanowires leads to a gradual increasing and decreasing of energetic stability and band gap width, respectively, approaching saturation limits.33 Ab initio DFT calculations of the set of pristine 〈110〉 oriented nanowires display the same behavior of the physical properties (Table 1).
Complex Chiral Silicon Nanowires The atomic structures of achiral and chiral nanowires are more complex and require larger models to be calculated with the same length of clusters. For this reason, we limited our study to calculations of several complex structures with the number of silicon atoms from 140 (pentagonal achiral and chiral nanowires) to 380 (also pentagonal achiral and chiral nanowires). C. Energy Barriers for Chiral T Achiral Transformation. A minimal of chiral star-shaped cluster Si140H94 (Figure 3b) was used to calculate the potential energy barrier of structural chiral T achiral transformation. For the sake of comparison, a short achiral pentagonal four-sectioned star-shaped wire (Si140H90, Figure 3a) was used as a reference structure. The formation of the chiral wires from achiral ones (in this case, the transition achiral Si140H90 f chiral Si140H94) due to the sliding of the 〈110〉 oriented segments leads to formation of two dangling bonds at both wire tips due to the different number of surface hydrogen atoms. To keep saturated all surface bonds, four additional hydrogen atoms were incorporated into the chiral structure. Because of the different number of atoms and chemical bonds, a direct calculation of the potential energy barrier at the transition state corresponding to the chiral T achiral transformation is impossible, and we have to resort to an approximate estimate, as described below. To avoid calculations of artificial open-shell multiradical systems (silicon wires with surface dangling bonds and nonbonded hydrogen atoms), we used a complex approach to compute the potential energy curve of the reversible chiral T achiral transformation: first, we performed a set (31 points) of single point B3LYP calculations of the achiral wire shifting the 〈110〉 fragments toward formation of the chiral structure with effective synchronized gradual movement (31 points, 30 intervals, each interval was equal to 0.06 Å) of four “extra” hydrogen atoms from 3.23 Å distance to equilibrium Si-H distance (1.43 Å) to saturate new forming dangling bonds. The hydrogen atom approach was effectively calculated by addition of the SiH3-H potential energy curve multiplied by 4 to the potential energy curve in the achiral f chiral direction. To study transformation of the Si140H90 cluster to Si140H94 (achiral f chiral direction), a set of single point calculations was performed for 31 points of the potential energy curve along the 〈110〉 direction. For all 30 intervals of the potential energy curve, the first 〈110〉 oriented fragment of the complex wire was motionless and the second fragment was shifted A/30 ) 0.024 Å (where A is equal to one-fifth of the unit cell length of the achiral wire (3.622 Å) and was equal to 0.724 Å). Each step all fragments were rotated respective to the 〈110〉 wire axis by 0.07°. Due to small distortions of the wire silicon core and large distances of four “extra” hydrogen atoms, the initial stages of the achiral f chiral transformation seem to be correct. Since the adding of extra hydrogen atoms was taken into account effectively by adding of the energetic correction to the potential energy curve, at the final stages of the transformation a chiral structure with four new dangling bonds (chiral Si140H90) was formed. For such a structure, the effective correction of the curve using the Si-H potential energy curve for the SiH4 molecule does not seem to be appropriate due to the presence of the dangling bonds. To study the potential energy curve in the chiral f achiral direction (transforming the Si140H94 cluster into the Si140H90 one), the same mesh grid for the reaction coordinate was chosen. At the initial stages, the sliding and rotation of the fragments along the 〈110〉 direction caused small structural distortions that could be treated as a small perturbation of the chiral structure. At the final stages, an artificial achiral Si140H94 wire with four “extra”
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Figure 4. Potential curves of the chiral f achiral (filled triangles), achiral f chiral (crosses), and final reversible chiral T achiral (empty circles on the solid line) transformations of pentagonal star-shaped complex wire Si140H94. The curves were calculated by effectively taking into account removal/addition of four additional hydrogen atoms.
hydrogen atoms bonded to the silicon core was formed. Again, the chiral f achiral potential curve was corrected by subtracting the SiH3-H potential energy curve multiplied by 4 to take into account the removal of 4 “extra” hydrogen atoms from the system. There should be only one transition state for the chiral T achiral transformation in both directions, which we expect to be located in the vicinity of point number 16 (middle of the reaction coordinate) since the energy difference between achiral and chiral wires is small. To obtain the potential energy curve of transformation, the potential energy curves obtained for both directions were combined for correct description of both chiral f achiral and achiral f chiral pathways (Figure 4). The height of the potential energy barrier [5.0 (kcal/mol)/Si atom] was estimated as the energy of the crossing of the curves in both directions at point number 18. The final potential energy curve displays different potential barriers from left to right [4.1 (kcal/ mol)/Si atom] and from right to left [5.0 (kcal/mol)/Si atom]. Since the difference between the barriers [0.9 (kcal/mol)/Si atom] is smaller than the accuracy of the B3LYP method [1 (kcal/mol)/Si atom, see above], we have to consider the barriers in both directions the same and equal to 4.5 (kcal/mol)/Si atom, which is much higher than a typical mistake (1 kcal/mol for isomer stability) of the B3LYP method. IV. Conclusions The atomic structures and average Si binding energies of complex silicon chiral wires of pentagonal and hexagonal symmetry were calculated with the DFT method. The structural chirality of the wires was caused by a consequent shift of parent 〈110〉 oriented fragments along the main 〈110〉 axis. It was shown that the chiral wires are energetically metastable with a potential energy barrier of structural chiral T achiral transformation equal to 4.5 (kcal/mol)/Si atom. The chirality causes the decrease of the semiconducting bandgap and may lead to rotation of the optical polarization of the wires. Acknowledgment. This work was supported by a CREST (Core Research for Evolutional Science and Technology) grant in the Area of High Performance Computing for Multiscale and Multiphysics Phenomena from the Japan Science and Technology Agency (JST) and a collaborative RFBR-JSPS grant No. 09-02-92107-ЯΦ. S.I. also acknowledges support by the Program for Improvement of Research Environment for Young Researchers from Special Coordination Funds for Promoting Science and Technology (SCF) commissioned by the Ministry of Education, Culture, Sports, Science, and Technology (MEXT)
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