- Email: [email protected]
Nanoring structure and optical properties of Ga8As8
Chemical Physics Letters 381 (2003) 397–403 www.elsevier.com/locate/cplett
Nanoring structure and optical properties of Ga8As8 Yanlin Sun, Xiaoshuang Chen *, Lizhong Sun, Xuguang Guo, Wei Lu National Laboratory for Infrared Physics, Shanghai Institute of Technical Physics, Chinese Academy of Sciences, Shanghai 200083, PR China Received 10 July 2003 Published online: 23 October 2003
Abstract After several initial geometric configurations are relaxed, a ring structure of Ga8 As8 cluster is found by first-principles calculations. It is shown that GaAs bond lengths are somewhat longer or shorter than that in bulk GaAs. The calculated vibrational spectrum implies that the optimized geometry is located at the minimum point of the potential surface. There are only two strong peaks of IR absorption, one is at about 260 cm1 and another is at 349 cm1 . The vibrations of Ga–As bonds in plane of ring are the primary IR vibration at about 349 cm1 . Raman spectrum shows that the vibrations of two layers of the GaAs ring are the primary Raman vibration at about 222 cm1 . The dipole polarizability anisotropy invariant and hyperpolarizability also are discussed. 2003 Elsevier B.V. All rights reserved.
1. Introduction Clusters have been drawing a great deal of attention since clusters link the gap between isolated molecule and bulk material [1,2]. On the one hand, people are interested in how the structure and properties of clusters approach to that of bulk materials as their sizes (atomic number of clusters) increase. On the other hand, due to the existence of isomers for a certain size clusters, the studies of clusters become more complex. People naturally hope to explore the most stable structure of clusters. Therefore, it is very important to develop theoretical methods capable of resolving structure of clusters. At the same time, the structural and electronic *
Corresponding author. E-mail address: [email protected] (X. Chen).
properties of semiconductor clusters have been extensively studied because of their fundamental interest and potential application in nanoelectronics, such as small gallium arsenide, silicon and germanium clusters etc. Liao et al. have computed the optimized geometries and energy separations of several electronic states of Ga3 As2 and Ga2 As3 clusters by using the completed active space multiconfiguration selfconsistent-field (CASSCF) followed by multireference singles + doubles configuration interaction (MRSDCI) computations which included up to 1.9 million configurations [3]. The properties of the electronic states in distorted and undistorted trigonal bipyramid, and edge-capped tetrahedron structures of Ga2 As3 are computed. Deutsch et al. have Gaussian-2 (G2) theory for third-row non-transition elements to calculate energies of
0009-2614/$ - see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2003.09.115
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Y.L. Sun et al. / Chemical Physics Letters 381 (2003) 397–403
germanium clusters, Gen (n ¼ 2–5) [4]. The G2 energies are used to derive accurate binding energies for the clusters. The results for Ge2 and Ge3 are in agreement with experiment while there is some disagreement for Ge4 and Ge5 . The static polarizabilities ac of the isolated semiconductor clusters SiN , GaN AsM and GeN TeM have been investigated in dependence of cluster size and temperature [5]. The results for the SiN clusters are discussed within a two-band semiconductor model that includes a widening of the band gap due to quantum size effects. Additionally, the importance of defect like electronic states is discussed for SiN and GaN AsM clusters. The temperature dependence of the polarizability values for several GaN AsM and GeN TeM species gives evidence for vibronic (ionic) contributions to ac . The IR spectrum, Raman spectrum, polarization and hyperpolarization are the most important physical quantities of the clusters. These properties are directly related to the size and structure effect of clusters. As a kind of single moment clusters, silicon and germanium clusters have been extensively studied. In semiconductor materials, the composition semiconductors clusters occupy a big portion of these materials. In this Letter, we intend to investigate the optimized structure of Ga8 As8 , as well as the interesting optical properties. The studies are twofold: first, by calculating the observable properties from first principles, one can improve his or her understanding of the underlying physics behind the observed phenomena. Second, by comparing the discrepancies between experiment and theory, one can modify potential energy surfaces in order to improve their coincidence. Density function theory (DFT) is used to calculate the structures and energies of small clusters Ga8 As8 . We found a ring structure of Ga8 As8 cluster, with lowest total energy for a Ga8 As8 cluster, no any other structures are reported. This structure may be the most symmetrical structure.
2. Methods The IR spectrum, Raman spectrum, polarization and hyperpolarization calculations on Ga8 As8 ring have been performed after using the General Atomic and Molecular Electronic Structure System
(GAMESS version 6.0) [6] program to optimize first the structure of Ga8 As8 cluster. During the densityfunctional calculations, the Restricted Hartree– Fock (RHF) and 6-31G bases are chosen. The correlated wave functions are used, at the secondorder level of Moller–Plesset perturbation theory. RHF remains the most commonly calculated wave function. Thus, four decades after RoothaanÕs work new techniques for RHF calculations start to be proposed, particularly PulayÕs direct inversion in the iterative subspace (DIIS) interpolation and AlmlofÕs direct SCF (self-consistent-field). Both of these methods are implemented in GAMESS, which also provides a several direct energy minimization (DEM) schemes. We would like to briefly mention that the completed infrared and Raman spectra could be calculated by using these schemes. The integral absorption coefficient of an infrared transition is proportional to square of the first derivative of the dipole moment according to the normal coordinate. Such derivatives can be calculated by quantum chemistry and hence the infrared intensity can be estimated. In a similar manner, Raman spectral intensities can also be found by calculating the first derivative of the polarizability. Many ab initio programs automatically give IR and Raman intensities after a frequency calculation [7]. The molecular hyperpolarizabilities are evaluated as the sums of transition dipole moment matrix elements between the ground and excited states [8–11]. The first-order hyperpolarizability for the second-harmonic-generation (SHG) is given by all e3 X hgjrj jn0 ihn0 jri jnihnjrk jgi 2 42 n;n0 þ hgjrk jn0 ihn0 jri jnihnjrj jgi ! 1 1 þ 0 0 xn x ðxn xÞ xn þ x ðxn þ xÞ þ hgjrj jn0 ihn0 jri jnihnjrk jgi þ hgjri jn0 ihn0 jrk jnihnjrj jgi ! 1 1 þ 0 xn þ 2x ðxn þ xÞ x0n 2x ðxn xÞ þ hgjri jn0 ihn0 jrk jnihnjri jgi þ hgjrk jn0 ihn0 jrj jnihnjri jgi !# 1 1 þ ; ð1Þ 0 xn x ðxn 2xÞ x0n þ x ðxn þ 2xÞ
bijk þ bikj ¼
where jgi is the wave function of unperturbed ground state, jni and jn0 i are those for the excited
Y.L. Sun et al. / Chemical Physics Letters 381 (2003) 397–403
states with excitation energies of hxn , and hxn0 , respectively; and x the incident frequency. In the ordinary SOS (sum-over-states) calculations [8,12], the ground state wave function is obtained by a SCF calculation and those for the excited states are obtained by a single excitation configuration P interaction (SCI) calculation, i.e. jni ¼ i;a Ania ji ! ai. Here ji ! ai indicates a configuration obtained by one-electron excitation from an occupied MO (molecular orbit) ÔiÕ to an unoccupied MO ÔaÕ, with Ania is the CI coefficient for the state ÔnÕ. Then, by this assumption to the excited state wave functions, the dipole matrix elements in Eq. (1) can be expressed by one-electron dipole integrals over MOs. For instance, if the ground state is a closedshell singlet state, dipole matrix elements are: pffiffiffiX hgjxjni ¼ 2 Ania i;a
0
hnjxjn i ¼
XX i;a
0
Ania Anib ½hajxjbidij hijxjjidab :
ð2Þ
j;b
The actual formula for the SOS calculation can be obtained by inserting Eq. (2) into Eq. (1) [13]. An important factor when computing polarizabilities is that a sufficiently large basis set should be used with many polarization functions. The basis sets are adopted with a larger sets of [5s3p2d/3s2p]. The calculations are made at MP2/6-31G* optimized geometries. All computation are performed at temperature 298.150 K and pressure 1.00000 atm.
3. Results and discussion 3.1. The ring structure of Ga8 As8 cluster To search for the equilibrium structure of a Ga8 As8 cluster, some possible initial configurations must be defined as seeds. The search is started by defining some possible geometries by using some considerations of symmetry. Basing on the initial geometric configuration, we set up a limited step (200). In each step, the HF equation is solved exactly. The nuclei are then moved according to the forces. We then decompose the mixed density, move each partial density along with its atom, and re-overlap at the new geometry. After much itera-
399
tion, the maximum of the forces is less than 4.7031E)5 (hartrees/bohr), the kinetic energies of the nuclei approach to zero, and the electronic energy is nicely constant because the system is close to self-consistency. The process is stopped when selfconsistent condition is satisfied, and a stable (or substable) final calculated structure is obtained. Self-consistent field calculations are carried out with a convergence criterion of 10–5 a.u. on the total energy and electron density. Geometry optimizations are performed with the quadratic approximation. This is another version of an augmented Hessian technique where the shift parameter is chosen so that the steplength is equal to DXMAX (initial trust radius of the step). It is completely equivalent to the TRIM method. We use a convergence criterion of 0.0001 a.u. on the gradient and displacement, 0.5 for the maximum permissible value of the trust radius and 0.05 for minimum permissible value of the trust radius in the geometry optimization. After initial geometric configurations are relaxed, it is surprising that we found a ring structure for Ga8 As8 cluster. The structure is shown in Fig. 1, and its point groups and total energies are C4 and )33224.30 eV, respectively. The bond lengths and degrees for Ga8 As8 nanoring are shown in Table 1. There are two kinds bonds in the structure. Atom 1 (As) binds to three Ga atoms with the bond length , respectively. These GaAs bond 2.207 and 2.335 A lengths are somewhat longer or shorter than that in . For bulk GaAs, are the GaAs distance is 2.24 A example atoms 2 and 3 (Ga) bind to a As atom . The bond (atom 1), the bond length is 2.207 A length between Ga and As atom is shorter than that of bulk GaAs. Atom 9 (Ga) binds to atom 1 (As), the . The bond length between Ga bond length is 2.335 A and As atom is larger than that of bulk GaAs. From Fig. 1, we can know that Ga atoms are closer to the center of ring than As atoms. 3.2. IR and Raman spectra of Ga8 As8 nanoring Based on the optimized structure, the calculations of IR and Raman spectra on Ga8 As8 nanoring have been performed by using the GAMESS program. Fig. 1 shows the IR and Raman spectra of Ga8 As8 nanoring. The calculated
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Fig. 1. The strcture of Ga8 As8 nanoring predicted by DFT theory.
Table 1 The bond lengths and degrees for Ga8 As8 nanoring Lengths
Definition
Value ) (A
Degrees
Definition
Value
Degrees
Definition
Value
R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16 R17 R18 R19 R20 R21 R22 R23 R24
R(1,2) R(1,3) R(1,9) R(2,6) R(2,10) R(3,5) R(3,11) R(4,5) R(4,8) R(4,12) R(5,13) R(6,7) R(6,14) R(7,8) R(7,15) R(8,16) R(9,10) R(9,11) R(10,14) R(11,13) R(12,13) R(12,16) R(14,15) R(15,16)
2.2072 2.2073 2.3346 2.2069 2.3346 2.207 2.3346 2.207 2.2073 2.3346 2.3347 2.2071 2.3346 2.2073 2.3346 2.3347 2.2071 2.207 2.2073 2.2072 2.2072 2.2069 2.2072 2.2069
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17 A18 A19 A20 A21 A22 A23 A24
A(2,1,3) A(2,1,9) A(3,1,9) A(1,2,10) A(6,2,10) A(1,3,11) A(5,3,11) A(5,4,12) A(8,4,12) A(3,5,4) A(3,5,13) A(4,5,13) A(2,6,7) A(2,6,14) A(7,6,14) A(6,7,15) A(8,7,15) A(4,8,7) A(4,8,16) A(7,8,16) A(1,9,10) A(1,9,11) A(2,10,9) A(2,10,14)
105.3102 77.8493 77.8526 95.1048 95.0807 95.0884 95.0675 95.0681 95.0971 105.3384 77.8625 77.8713 105.3447 77.8633 77.8709 95.0591 95.0905 105.2949 77.8506 77.8465 95.1086 95.0995 77.8523 77.8574
A25 A26 A27 A28 A29 A30 A31 A32 A33 A34 A35 A36 A37 A38 A39 A40 A41 A42 A43 A44 A45 A46 A47 A48
A(9,10,14) A(3,11,9) A(3,11,13) A(9,11,13) A(4,12,13) A(4,12,16) A(13,12,16) A(5,13,11) A(5,13,12) A(6,14,10) A(6,14,15) A(7,15,14) A(7,15,16) A(14,15,16) A(8,16,12) A(8,16,15) L(1,2,6,10,)2) L(1,3,5,11,)2) L(5,4,8,12,)2) L(6,7,8,15,)2) L(10,9,11,1,)2) L(11,13,12,5,)2) L(10,14,15,6,-2) L(12,16,15,8,-2)
105.3299 77.8579 77.8602 105.3208 77.8697 77.8599 105.3398 95.0614 95.0595 95.0687 95.0557 77.8685 77.8564 105.3255 95.1038 95.1011 193.4 166.6216 166.6101 193.3739 193.4458 193.3369 166.6655 193.4526
vibrational spectrum has no imaginary frequency, implying that the optimized geometry is located at the minimum point of the potential surface. In total, there are 40 vibration modes (from 50.3613 to 349.8024 cm1 ) shown in Table 2, the strongest IR absorption modes are modes V41 and V42 . Modes V41 and V42 are the vibrations of Ga–As
bonds along X and Y axis (in plane of ring). Modes V41 and V42 are very similar, and their frequencies are almost same. In Fig. 2, modes V41 and V42 are the degenerated with same peak. There are only two strong peaks in the IR spectrum, one is at about 260 cm1 (mode V34 ) and another is at 349 cm1 (modes V41 and V42 ). Mode V34 is the
Y.L. Sun et al. / Chemical Physics Letters 381 (2003) 397–403 Table 2 All the vibration frequency and IR intensity (Raman activity), there are 39 mode from 50.3613 to 349.8024 cm1
V3 V4 V5 V6 V7 V8 V9 V10 V11 V12 V13 V14 V15 V16 V17 V18 V19 V20 V21 V22 V23 V24 V25 V26 V27 V28 V29 V30 V31 V32 V33 V34 V35 V36 V37 V38 V39 V40 V41 V42
Frequency (cm1 )
IR intensity (km/mol)
Raman activity 4 /amu) (A
50.3613 50.3617 82.3058 82.334 99.7225 99.7463 103.7396 110.3141 110.4054 116.6778 129.9056 129.9067 131.9878 132.6041 143.3803 143.4576 153.6146 184.3876 184.3904 194.629 194.6728 198.539 198.5406 207.3838 220.0171 220.0419 221.9139 225.243 225.2439 246.8662 246.9042 260.146 323.922 336.2424 336.2832 338.7645 338.7665 339.1144 349.7658 349.8024
0 0 0 0 0.5164 0.5143 3.3101 0 0 0 0 0 0 0 1.2833 1.2872 1.1145 0 0 0.3247 0.323 0 0 0 0.7535 0.759 0 0 0 0 0 77.8239 0 2E)4 3E)4 0 0 0 99.3356 99.459
0.318 0.3187 0.3633 0.3662 0 0 0 9.6522 9.654 2.1718 1.1096 1.1128 0 18.7676 0 0 0 0.1709 0.1699 0 0 0.0804 0.08 9.8438 1E)4 2E)4 84.973 3.0469 3.0464 1.4621 1.4659 0 0 3.2116 3.2072 1.867 1.8631 6E)4 0 0
vibration between the two layers of ring along Z axis. The two layers of ring are made of atom 1–8 and atom 9–16, respectively. From the IR spectrum, the vibrations of Ga–As bonds in plane of ring are the primary IR vibration at about 349 cm1 . The strongest Raman activity is mode V29 . Which is the vibration between the two layer of ring
401
along Z axis. Vibrations of the As atoms have small departure from Z axis, but these of the Ga atoms are accurated along Z axis. From the Raman spectrum, the vibration of two layers of the GaAs ring is the primary Raman vibration at about 222 cm1 . 3.3. Polarizabilities and hyperpolarizabilities of Ga8 As8 nanoring Polarizabilities and hyperpolarizabilities are important in chemical physics because they determine a number of molecular interactions, such as the magnitude of long-range intermolecular induction and dispersion forces, as well as the cross sections of different scattering and collision processes. We calculate the polarizabilities a and the first hyperpolarizabilities b. The individual tensor components (Ove Christiansen et al. have used this method in calculating CO and H2 O from coupled cluster [14]) of the isotropic polarizability are as follows a ¼ 13ðaXX þ aYY þ aZZ Þ the polarizability anisotropy invariant " # 2 2 2 1=2 ðaXX aYY Þ þ ðaYY aZZ Þ þ ðaZZ aXX Þ Da ¼ 2 and the hyperpolarizability average 1X bk ¼ ðbiiZ þ biZi þ bZii Þ 5 The Z axis passes though the center of Ga8 As8 nanoring and perpendicular to the ring. In Table 3, we report our computational results for the dipole, polarizabilities and first hyperpolarizabilities of Ga8 As8 nanoring. The dipole polarizability anisotropy invariant and hyperpolarizability are lZ ¼ 1:25243399E–05, Da ¼ 333 and bk ¼ 0:036883534246 (in a.u.), respectively.
4. Conclusions Based on the first-principles theory, we have studied the structure and optical properties of Ga8 As8 . The results are shown in the following. (a) After initial geometric configurations are relaxed, a ring structures is found for a Ga8 As8
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Fig. 2. The IR and Raman spectra of GaAs ring in term of DFT theory.
Table 3 The dipole, polarizability and hyperpolarizability of Ga8 As8 nanoring in DFT theory (in a.u.) with base field of 30.0010 Dipole
X 3.28260860D)05
Y 2.96625208D)05
Z )1.25243399D)05
Polarizability
XX 4.72020324D+02
YY 4.71780637D+02
ZZ 5.95095977D)03
XY )3.09718784D)04
XZ 67030513D)02
YZ 3.63335319D+02
XXX )1.17884705D)01
XXY )9.45207750D)02
XYY 1.01571666D)01
YYY )5.84887116D)02
XXZ 4.73034003D)02
XYZ )9.98570423D)03
YYZ )9.28255022D)02
XZZ 1.06053322D)02
YZZ 3.67827324D)03
HyperPolar
ZZZ 2.61178786D)02 All units are a.u., where a.u. ¼ 0.86566 · 1032 esu.
cluster. These Ga and As bond lengths are somewhat longer or shorter than that of bulk GaAs. (b) The calculated vibrational spectrums have no imaginary frequency, implying that the optimized geometry is located at the minimum point of the potential surface. (c) The Z axis passes through the center of Ga8 As8 nanoring and is perpendicular to the ring. The dipole polarizability anisotropy invariant and hyperpolarizability are lZ ¼
1:25243399E)05, Da ¼ 333 and bk ¼ 0:036883534246 (in a.u.), respectively.
Acknowledgements This work is supported by Chinese National Key Basic Research Special Fund, National Science Foundation, One-hundred-person project of Chinese Science Academy and Key Found of Shanghai Science and Technology Foundation.
Y.L. Sun et al. / Chemical Physics Letters 381 (2003) 397–403
The computational support is from Shanghai super computer center. I am very grateful to Professor Ronnie Kosloff and his secretary Eva Guez for their help.
References [1] P. Jena, B.K. Rao, S.N. Khanna, Physics and chemistry of small clusters, in: NATO ASI Series, vol. 158, Kluwer, Dordrecht, 1990. [2] E.R. Bernstein, Atomic and molecular clusters, in: Studies in Physical and Theoretical Chemistry, vol. 68, Elsevier, Amsterdam, 1990. [3] M.Z. Liao, Dingguo Dai, K. Balasubramanian, Chem. Phys. Lett. 239 (1995) 124. [4] P.W. Deutsch, L.A. Curtiss, J.P. Blaudeau, Chem. Phys. Lett. 270 (1997) 413.
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[5] R. Schafer, S. Schlecht, J. Woenckhaus, J.A. Becker, Phys. Rev. Lett. 76 (3) (1996) 471. [6] M.W. Schmidt, K.K. Baldridge, et al., J. Comput. Chem. 14 (11) (1993) 1347. [7] T. Veszprer, Quantum Chemistry Fundamentals to Application, Kluwer Academic/Plenum Publishing, Dordrecht, New York, Norwell, 1998. [8] J. Zyss, D.S. Chemla, in: D.S. Chemla, J. Zyss (Eds.), Nonlinear Optical Properties of Organic Molecules and Crystals, vol. 1, Academic Press, New York, 1987. [9] B.J. Orr, J.F. Ward, Mol. Phys. 20 (1971) 513; B.M. Pierce, J. Chem. Phys. 91 (1989) 791. [10] S.J. Lalama, A.F. Garito, Phys. Rev. A 20 (1979) 1179. [11] W. Dirk, M.G. Kuzyk, Phys. Rev. A 39 (1989) 1219. [12] M. Tomonari, N. Gokubo, T. Takada, M.W. Feyereisen, J. Almlof, Chem. Phys. Lett. 203 (1993) 603. [13] M. Tomonari, N. Ookubo, T. Takada, H. Hirayama, Chem. Phys. Lett. 272 (1997) 199. [14] O. Christiansen, J. Gauss, J.F. Stanton, Chem. Phys. Lett. 305 (1999) 147.
Nanoring structure and optical properties of Ga8As8 Yanlin Sun, Xiaoshuang Chen *, Lizhong Sun, Xuguang Guo, Wei Lu National Laboratory for Infrared Physics, Shanghai Institute of Technical Physics, Chinese Academy of Sciences, Shanghai 200083, PR China Received 10 July 2003 Published online: 23 October 2003
Abstract After several initial geometric configurations are relaxed, a ring structure of Ga8 As8 cluster is found by first-principles calculations. It is shown that GaAs bond lengths are somewhat longer or shorter than that in bulk GaAs. The calculated vibrational spectrum implies that the optimized geometry is located at the minimum point of the potential surface. There are only two strong peaks of IR absorption, one is at about 260 cm1 and another is at 349 cm1 . The vibrations of Ga–As bonds in plane of ring are the primary IR vibration at about 349 cm1 . Raman spectrum shows that the vibrations of two layers of the GaAs ring are the primary Raman vibration at about 222 cm1 . The dipole polarizability anisotropy invariant and hyperpolarizability also are discussed. 2003 Elsevier B.V. All rights reserved.
1. Introduction Clusters have been drawing a great deal of attention since clusters link the gap between isolated molecule and bulk material [1,2]. On the one hand, people are interested in how the structure and properties of clusters approach to that of bulk materials as their sizes (atomic number of clusters) increase. On the other hand, due to the existence of isomers for a certain size clusters, the studies of clusters become more complex. People naturally hope to explore the most stable structure of clusters. Therefore, it is very important to develop theoretical methods capable of resolving structure of clusters. At the same time, the structural and electronic *
Corresponding author. E-mail address: [email protected] (X. Chen).
properties of semiconductor clusters have been extensively studied because of their fundamental interest and potential application in nanoelectronics, such as small gallium arsenide, silicon and germanium clusters etc. Liao et al. have computed the optimized geometries and energy separations of several electronic states of Ga3 As2 and Ga2 As3 clusters by using the completed active space multiconfiguration selfconsistent-field (CASSCF) followed by multireference singles + doubles configuration interaction (MRSDCI) computations which included up to 1.9 million configurations [3]. The properties of the electronic states in distorted and undistorted trigonal bipyramid, and edge-capped tetrahedron structures of Ga2 As3 are computed. Deutsch et al. have Gaussian-2 (G2) theory for third-row non-transition elements to calculate energies of
0009-2614/$ - see front matter 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2003.09.115
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Y.L. Sun et al. / Chemical Physics Letters 381 (2003) 397–403
germanium clusters, Gen (n ¼ 2–5) [4]. The G2 energies are used to derive accurate binding energies for the clusters. The results for Ge2 and Ge3 are in agreement with experiment while there is some disagreement for Ge4 and Ge5 . The static polarizabilities ac of the isolated semiconductor clusters SiN , GaN AsM and GeN TeM have been investigated in dependence of cluster size and temperature [5]. The results for the SiN clusters are discussed within a two-band semiconductor model that includes a widening of the band gap due to quantum size effects. Additionally, the importance of defect like electronic states is discussed for SiN and GaN AsM clusters. The temperature dependence of the polarizability values for several GaN AsM and GeN TeM species gives evidence for vibronic (ionic) contributions to ac . The IR spectrum, Raman spectrum, polarization and hyperpolarization are the most important physical quantities of the clusters. These properties are directly related to the size and structure effect of clusters. As a kind of single moment clusters, silicon and germanium clusters have been extensively studied. In semiconductor materials, the composition semiconductors clusters occupy a big portion of these materials. In this Letter, we intend to investigate the optimized structure of Ga8 As8 , as well as the interesting optical properties. The studies are twofold: first, by calculating the observable properties from first principles, one can improve his or her understanding of the underlying physics behind the observed phenomena. Second, by comparing the discrepancies between experiment and theory, one can modify potential energy surfaces in order to improve their coincidence. Density function theory (DFT) is used to calculate the structures and energies of small clusters Ga8 As8 . We found a ring structure of Ga8 As8 cluster, with lowest total energy for a Ga8 As8 cluster, no any other structures are reported. This structure may be the most symmetrical structure.
2. Methods The IR spectrum, Raman spectrum, polarization and hyperpolarization calculations on Ga8 As8 ring have been performed after using the General Atomic and Molecular Electronic Structure System
(GAMESS version 6.0) [6] program to optimize first the structure of Ga8 As8 cluster. During the densityfunctional calculations, the Restricted Hartree– Fock (RHF) and 6-31G bases are chosen. The correlated wave functions are used, at the secondorder level of Moller–Plesset perturbation theory. RHF remains the most commonly calculated wave function. Thus, four decades after RoothaanÕs work new techniques for RHF calculations start to be proposed, particularly PulayÕs direct inversion in the iterative subspace (DIIS) interpolation and AlmlofÕs direct SCF (self-consistent-field). Both of these methods are implemented in GAMESS, which also provides a several direct energy minimization (DEM) schemes. We would like to briefly mention that the completed infrared and Raman spectra could be calculated by using these schemes. The integral absorption coefficient of an infrared transition is proportional to square of the first derivative of the dipole moment according to the normal coordinate. Such derivatives can be calculated by quantum chemistry and hence the infrared intensity can be estimated. In a similar manner, Raman spectral intensities can also be found by calculating the first derivative of the polarizability. Many ab initio programs automatically give IR and Raman intensities after a frequency calculation [7]. The molecular hyperpolarizabilities are evaluated as the sums of transition dipole moment matrix elements between the ground and excited states [8–11]. The first-order hyperpolarizability for the second-harmonic-generation (SHG) is given by all e3 X hgjrj jn0 ihn0 jri jnihnjrk jgi 2 42 n;n0 þ hgjrk jn0 ihn0 jri jnihnjrj jgi ! 1 1 þ 0 0 xn x ðxn xÞ xn þ x ðxn þ xÞ þ hgjrj jn0 ihn0 jri jnihnjrk jgi þ hgjri jn0 ihn0 jrk jnihnjrj jgi ! 1 1 þ 0 xn þ 2x ðxn þ xÞ x0n 2x ðxn xÞ þ hgjri jn0 ihn0 jrk jnihnjri jgi þ hgjrk jn0 ihn0 jrj jnihnjri jgi !# 1 1 þ ; ð1Þ 0 xn x ðxn 2xÞ x0n þ x ðxn þ 2xÞ
bijk þ bikj ¼
where jgi is the wave function of unperturbed ground state, jni and jn0 i are those for the excited
Y.L. Sun et al. / Chemical Physics Letters 381 (2003) 397–403
states with excitation energies of hxn , and hxn0 , respectively; and x the incident frequency. In the ordinary SOS (sum-over-states) calculations [8,12], the ground state wave function is obtained by a SCF calculation and those for the excited states are obtained by a single excitation configuration P interaction (SCI) calculation, i.e. jni ¼ i;a Ania ji ! ai. Here ji ! ai indicates a configuration obtained by one-electron excitation from an occupied MO (molecular orbit) ÔiÕ to an unoccupied MO ÔaÕ, with Ania is the CI coefficient for the state ÔnÕ. Then, by this assumption to the excited state wave functions, the dipole matrix elements in Eq. (1) can be expressed by one-electron dipole integrals over MOs. For instance, if the ground state is a closedshell singlet state, dipole matrix elements are: pffiffiffiX hgjxjni ¼ 2 Ania i;a
0
hnjxjn i ¼
XX i;a
0
Ania Anib ½hajxjbidij hijxjjidab :
ð2Þ
j;b
The actual formula for the SOS calculation can be obtained by inserting Eq. (2) into Eq. (1) [13]. An important factor when computing polarizabilities is that a sufficiently large basis set should be used with many polarization functions. The basis sets are adopted with a larger sets of [5s3p2d/3s2p]. The calculations are made at MP2/6-31G* optimized geometries. All computation are performed at temperature 298.150 K and pressure 1.00000 atm.
3. Results and discussion 3.1. The ring structure of Ga8 As8 cluster To search for the equilibrium structure of a Ga8 As8 cluster, some possible initial configurations must be defined as seeds. The search is started by defining some possible geometries by using some considerations of symmetry. Basing on the initial geometric configuration, we set up a limited step (200). In each step, the HF equation is solved exactly. The nuclei are then moved according to the forces. We then decompose the mixed density, move each partial density along with its atom, and re-overlap at the new geometry. After much itera-
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tion, the maximum of the forces is less than 4.7031E)5 (hartrees/bohr), the kinetic energies of the nuclei approach to zero, and the electronic energy is nicely constant because the system is close to self-consistency. The process is stopped when selfconsistent condition is satisfied, and a stable (or substable) final calculated structure is obtained. Self-consistent field calculations are carried out with a convergence criterion of 10–5 a.u. on the total energy and electron density. Geometry optimizations are performed with the quadratic approximation. This is another version of an augmented Hessian technique where the shift parameter is chosen so that the steplength is equal to DXMAX (initial trust radius of the step). It is completely equivalent to the TRIM method. We use a convergence criterion of 0.0001 a.u. on the gradient and displacement, 0.5 for the maximum permissible value of the trust radius and 0.05 for minimum permissible value of the trust radius in the geometry optimization. After initial geometric configurations are relaxed, it is surprising that we found a ring structure for Ga8 As8 cluster. The structure is shown in Fig. 1, and its point groups and total energies are C4 and )33224.30 eV, respectively. The bond lengths and degrees for Ga8 As8 nanoring are shown in Table 1. There are two kinds bonds in the structure. Atom 1 (As) binds to three Ga atoms with the bond length , respectively. These GaAs bond 2.207 and 2.335 A lengths are somewhat longer or shorter than that in . For bulk GaAs, are the GaAs distance is 2.24 A example atoms 2 and 3 (Ga) bind to a As atom . The bond (atom 1), the bond length is 2.207 A length between Ga and As atom is shorter than that of bulk GaAs. Atom 9 (Ga) binds to atom 1 (As), the . The bond length between Ga bond length is 2.335 A and As atom is larger than that of bulk GaAs. From Fig. 1, we can know that Ga atoms are closer to the center of ring than As atoms. 3.2. IR and Raman spectra of Ga8 As8 nanoring Based on the optimized structure, the calculations of IR and Raman spectra on Ga8 As8 nanoring have been performed by using the GAMESS program. Fig. 1 shows the IR and Raman spectra of Ga8 As8 nanoring. The calculated
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Fig. 1. The strcture of Ga8 As8 nanoring predicted by DFT theory.
Table 1 The bond lengths and degrees for Ga8 As8 nanoring Lengths
Definition
Value ) (A
Degrees
Definition
Value
Degrees
Definition
Value
R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16 R17 R18 R19 R20 R21 R22 R23 R24
R(1,2) R(1,3) R(1,9) R(2,6) R(2,10) R(3,5) R(3,11) R(4,5) R(4,8) R(4,12) R(5,13) R(6,7) R(6,14) R(7,8) R(7,15) R(8,16) R(9,10) R(9,11) R(10,14) R(11,13) R(12,13) R(12,16) R(14,15) R(15,16)
2.2072 2.2073 2.3346 2.2069 2.3346 2.207 2.3346 2.207 2.2073 2.3346 2.3347 2.2071 2.3346 2.2073 2.3346 2.3347 2.2071 2.207 2.2073 2.2072 2.2072 2.2069 2.2072 2.2069
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17 A18 A19 A20 A21 A22 A23 A24
A(2,1,3) A(2,1,9) A(3,1,9) A(1,2,10) A(6,2,10) A(1,3,11) A(5,3,11) A(5,4,12) A(8,4,12) A(3,5,4) A(3,5,13) A(4,5,13) A(2,6,7) A(2,6,14) A(7,6,14) A(6,7,15) A(8,7,15) A(4,8,7) A(4,8,16) A(7,8,16) A(1,9,10) A(1,9,11) A(2,10,9) A(2,10,14)
105.3102 77.8493 77.8526 95.1048 95.0807 95.0884 95.0675 95.0681 95.0971 105.3384 77.8625 77.8713 105.3447 77.8633 77.8709 95.0591 95.0905 105.2949 77.8506 77.8465 95.1086 95.0995 77.8523 77.8574
A25 A26 A27 A28 A29 A30 A31 A32 A33 A34 A35 A36 A37 A38 A39 A40 A41 A42 A43 A44 A45 A46 A47 A48
A(9,10,14) A(3,11,9) A(3,11,13) A(9,11,13) A(4,12,13) A(4,12,16) A(13,12,16) A(5,13,11) A(5,13,12) A(6,14,10) A(6,14,15) A(7,15,14) A(7,15,16) A(14,15,16) A(8,16,12) A(8,16,15) L(1,2,6,10,)2) L(1,3,5,11,)2) L(5,4,8,12,)2) L(6,7,8,15,)2) L(10,9,11,1,)2) L(11,13,12,5,)2) L(10,14,15,6,-2) L(12,16,15,8,-2)
105.3299 77.8579 77.8602 105.3208 77.8697 77.8599 105.3398 95.0614 95.0595 95.0687 95.0557 77.8685 77.8564 105.3255 95.1038 95.1011 193.4 166.6216 166.6101 193.3739 193.4458 193.3369 166.6655 193.4526
vibrational spectrum has no imaginary frequency, implying that the optimized geometry is located at the minimum point of the potential surface. In total, there are 40 vibration modes (from 50.3613 to 349.8024 cm1 ) shown in Table 2, the strongest IR absorption modes are modes V41 and V42 . Modes V41 and V42 are the vibrations of Ga–As
bonds along X and Y axis (in plane of ring). Modes V41 and V42 are very similar, and their frequencies are almost same. In Fig. 2, modes V41 and V42 are the degenerated with same peak. There are only two strong peaks in the IR spectrum, one is at about 260 cm1 (mode V34 ) and another is at 349 cm1 (modes V41 and V42 ). Mode V34 is the
Y.L. Sun et al. / Chemical Physics Letters 381 (2003) 397–403 Table 2 All the vibration frequency and IR intensity (Raman activity), there are 39 mode from 50.3613 to 349.8024 cm1
V3 V4 V5 V6 V7 V8 V9 V10 V11 V12 V13 V14 V15 V16 V17 V18 V19 V20 V21 V22 V23 V24 V25 V26 V27 V28 V29 V30 V31 V32 V33 V34 V35 V36 V37 V38 V39 V40 V41 V42
Frequency (cm1 )
IR intensity (km/mol)
Raman activity 4 /amu) (A
50.3613 50.3617 82.3058 82.334 99.7225 99.7463 103.7396 110.3141 110.4054 116.6778 129.9056 129.9067 131.9878 132.6041 143.3803 143.4576 153.6146 184.3876 184.3904 194.629 194.6728 198.539 198.5406 207.3838 220.0171 220.0419 221.9139 225.243 225.2439 246.8662 246.9042 260.146 323.922 336.2424 336.2832 338.7645 338.7665 339.1144 349.7658 349.8024
0 0 0 0 0.5164 0.5143 3.3101 0 0 0 0 0 0 0 1.2833 1.2872 1.1145 0 0 0.3247 0.323 0 0 0 0.7535 0.759 0 0 0 0 0 77.8239 0 2E)4 3E)4 0 0 0 99.3356 99.459
0.318 0.3187 0.3633 0.3662 0 0 0 9.6522 9.654 2.1718 1.1096 1.1128 0 18.7676 0 0 0 0.1709 0.1699 0 0 0.0804 0.08 9.8438 1E)4 2E)4 84.973 3.0469 3.0464 1.4621 1.4659 0 0 3.2116 3.2072 1.867 1.8631 6E)4 0 0
vibration between the two layers of ring along Z axis. The two layers of ring are made of atom 1–8 and atom 9–16, respectively. From the IR spectrum, the vibrations of Ga–As bonds in plane of ring are the primary IR vibration at about 349 cm1 . The strongest Raman activity is mode V29 . Which is the vibration between the two layer of ring
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along Z axis. Vibrations of the As atoms have small departure from Z axis, but these of the Ga atoms are accurated along Z axis. From the Raman spectrum, the vibration of two layers of the GaAs ring is the primary Raman vibration at about 222 cm1 . 3.3. Polarizabilities and hyperpolarizabilities of Ga8 As8 nanoring Polarizabilities and hyperpolarizabilities are important in chemical physics because they determine a number of molecular interactions, such as the magnitude of long-range intermolecular induction and dispersion forces, as well as the cross sections of different scattering and collision processes. We calculate the polarizabilities a and the first hyperpolarizabilities b. The individual tensor components (Ove Christiansen et al. have used this method in calculating CO and H2 O from coupled cluster [14]) of the isotropic polarizability are as follows a ¼ 13ðaXX þ aYY þ aZZ Þ the polarizability anisotropy invariant " # 2 2 2 1=2 ðaXX aYY Þ þ ðaYY aZZ Þ þ ðaZZ aXX Þ Da ¼ 2 and the hyperpolarizability average 1X bk ¼ ðbiiZ þ biZi þ bZii Þ 5 The Z axis passes though the center of Ga8 As8 nanoring and perpendicular to the ring. In Table 3, we report our computational results for the dipole, polarizabilities and first hyperpolarizabilities of Ga8 As8 nanoring. The dipole polarizability anisotropy invariant and hyperpolarizability are lZ ¼ 1:25243399E–05, Da ¼ 333 and bk ¼ 0:036883534246 (in a.u.), respectively.
4. Conclusions Based on the first-principles theory, we have studied the structure and optical properties of Ga8 As8 . The results are shown in the following. (a) After initial geometric configurations are relaxed, a ring structures is found for a Ga8 As8
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Fig. 2. The IR and Raman spectra of GaAs ring in term of DFT theory.
Table 3 The dipole, polarizability and hyperpolarizability of Ga8 As8 nanoring in DFT theory (in a.u.) with base field of 30.0010 Dipole
X 3.28260860D)05
Y 2.96625208D)05
Z )1.25243399D)05
Polarizability
XX 4.72020324D+02
YY 4.71780637D+02
ZZ 5.95095977D)03
XY )3.09718784D)04
XZ 67030513D)02
YZ 3.63335319D+02
XXX )1.17884705D)01
XXY )9.45207750D)02
XYY 1.01571666D)01
YYY )5.84887116D)02
XXZ 4.73034003D)02
XYZ )9.98570423D)03
YYZ )9.28255022D)02
XZZ 1.06053322D)02
YZZ 3.67827324D)03
HyperPolar
ZZZ 2.61178786D)02 All units are a.u., where a.u. ¼ 0.86566 · 1032 esu.
cluster. These Ga and As bond lengths are somewhat longer or shorter than that of bulk GaAs. (b) The calculated vibrational spectrums have no imaginary frequency, implying that the optimized geometry is located at the minimum point of the potential surface. (c) The Z axis passes through the center of Ga8 As8 nanoring and is perpendicular to the ring. The dipole polarizability anisotropy invariant and hyperpolarizability are lZ ¼
1:25243399E)05, Da ¼ 333 and bk ¼ 0:036883534246 (in a.u.), respectively.
Acknowledgements This work is supported by Chinese National Key Basic Research Special Fund, National Science Foundation, One-hundred-person project of Chinese Science Academy and Key Found of Shanghai Science and Technology Foundation.
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The computational support is from Shanghai super computer center. I am very grateful to Professor Ronnie Kosloff and his secretary Eva Guez for their help.
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