Ab initio analysis and harmonic force fields of silicon nanoclusters

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(1)Ab initio analysis and harmonic force fields of silicon nanoclusters Sonia M. Aguilera-Segura Department of Chemical Engineering, Universidad de los Andes, Bogotá D.C., Colombia. Jorge M. Seminario Department of Chemical Engineering, Materials Science and Engineering, Graduate Program, and Department of Electrical and Computer Engineering, Texas A&M University, College Station, Texas 77843, United States. ABSTRACT: Silicon clusters are analyzed at the nanoscale with the hope they become a substitute of present bulk silicon devices. We analyze electronic and charge states of very small Sin (n=1-96) clusters using sophisticated compound methods such as G1-G4, CBS, and W1, and a variety of DFT and traditional calculations with several sizes and qualities of basis sets. Results are compared with very precise experimental data when available. B3PW91 hybrid yield results comparable with all expensive compound procedures. All ionization potentials, electron affinities and dissociation energies are in good agreement with the experiment. We perform calculations of bigger clusters up to Si96 using B3PW91/6-31G(d). We calculate second derivative and thus the vibrational spectra for all clusters studied. Finally, we generate harmonic force fields that allow us to extend the vibrational behavior to millions of atoms in future MD calculations. INTRODUCTION The importance of interaction between theory and experiment has become increasingly appreciated during the past years. Computer simulation is a useful tool, booth for predictive purposes and to gain understanding of chemical problems. Ab initio molecular orbital theory or ‘first principles’ electronic structure methods are based upon quantum mechanics and therefore are capable of high accuracy predictions over a wide range of systems without needing of empirical information. However, ab initio molecular theory methods place a considerable demand on computer resources .1 In contrast, the density functional theory (DFT)2-4, which corresponds to a subgroup of ab initio molecular orbital methods, has become very popular in recent years to model molecular systems and chemical reactions. In a variety of problems DFT methods have proven to provide an effective tradeoff between computational cost and accuracy.5 The properties and structures of silicon clusters have been the subject of extensive experimental6-17 and theoretical18,19 (using ab initio electronic structure methods) studies in the lasts years due to the revolution in high-tech. It is used in a variety of products and applications as semiconductors, transistors, solar cells, photonics, optics, and spectroscopy among others.. In this work, we analyze electronic and charge states of very small clusters using sophisticated compound methods and a variety of DFT and traditional calculations with several sizes and qualities of basis sets. We calculate excitation energies, ionization potentials (IP), electron affinities, vibrational frequencies, excitation energies and current-voltage characteristics. Results are compared with very precise experimental data when available and used to determine possible features of new devices made using these clusters. We calculate second derivative and thus the vibrational spectra for all clusters studied up to around Si100. We generate harmonic force fields that allow us to extend the vibrational behavior to millions of atoms using MM/MD calculations. Our group found in the past that properties at the interfaces of electronic devices are due to features present within a distance of 1 nm. We can consider that such a distance is the minimum size of the material able to produce the required features at interfaces; the reminder large size still available in present devices is simply there because of limitations in the fabrication techniques. Thus this work has a goal to investigate the minimum size required to obtain the important properties of electronic devices and to determine if at those sizes we can get even better properties..

(2) Table 1. Compound procedures used in this work, their highest level methods and largest basis sets (In All Cases the highest methods are not used with the largest basis set), methods used for the geometry optimization, specific basis set, and number of functions for Si highest basis sets Procedure. Method. Opt method. Basis set. Si basis. nSi. G1. QCISD(T,E4T). MP2. 6-311G(2df,p). 6s,5p,2d,1f. 38. G2. QCISD(T,E4T). MP2. 6-311+G(3df,2p). 7s,6p,3d,1f. 47. G3. QCISD(T,E4T). MP2. GTLarge. 7s,6p,4d,3f. 66. G4. CCSD(T, E4T, FrzG4). B3LYP. GFHFB2. 8s,7p,4d,3f,2g,1h. 99. CBS-4M. MP4SDQ. HF. CBSB1. CBS-QB3. CCSD(T). B3LYP. CBSB3. 7s,6p,3d,2f. 54. 7s,6p,3d,2f. 54. W1U. CCSD(T). B3LYP. augh-cc-pVQZ+2df. 7s,6p,4d,3f,2g. 84. W1BD. BD(T). B3LYP. augh-cc-pVQZ+2df. 7s,6p,4d,3f,2g. 84. W1RO. CCSD(T). B3LYP. augh-cc-pVQZ+2df. 7s,6p,4d,3f,2g. 84. METHODOLOGY Ab initio theory is based on total energy calculations of the electronic system as a function of certain atomic displacements. Most ab initio calculations use the Bohr-Oppenheimer approximation to solve the Schrödinger equation, H=E, where E is the total energy of the system, the wave function, and H the Hamiltonian operator. There are generally one of two types: Hartree Fock, which considers each electron to experience effects of all of the other electrons combined, and several correlated methods, which considers individual electron interactions, with some extent. Accuracy of results depends on the degree of electron correlation and the size of the basis set used. In this work, GAUSSIAN-0920 program is used to calculate several Si clusters using complete basis set (CBS) methods (CBS-4M,21,22 and CBSQB322,23), Gaussian-1 to Gaussian-4 methods (G1,24,25 G2,26 G3,27 and G428), W129 methods (W1U, W1BD, W1RO), and the B3PW91 functional with several sizes of basis sets. The DFT B3PW9130 hybrid functional includes the Becke-331,32 exchange and the Perdew-Wang-9116,33,34 correlation functional. The basis sets used with B3PW91 are the 6-31G(d) and 6-31G+(d), the 6-311G(d) and 6311+(d), as well as the correlation consistent (cc) including diffuse functions (aug): aug-cc-pVDZ, aug-cc-pVTZ, aug-cc-pVQZ, and aug-cc-pV5Z. Table 1 shows the main characteristics of the compound methods used in this work. Under method, the most advanced method is used by each procedure indicated, and under basis set, the largest basis set used is shown; however, these two, method an basis set, are not necessarily used simultaneously in the calculations. Also indicated in Table 1 are the methods used for the geometry optimizations. In. addition, the type and size of the largest basis sets for Si are also indicated. Geometry optimizations are performed with the MP2/6-31G(d) level for G1-G3 and B3LYP/6-31G(d,2f) for G4 methods; HF for CBS-4M using 3-21G(d); and B3LYP/cc/pVTZ+d for the W1 methods, and B3LYP/6-311G(2d,d,p) level of theory is used for the CBS-QB3. The W1 methods use the largest basis set. RESULTS AND DISCUSION When predictive calculations are going to be used, it is a good idea to perform tests to small systems that have been studied experimentally before by others and compare existing data with the results obtained. It can give us a good idea about the accuracy and convenience of the methods used and also let us knows how far we can go with each procedure. Even if we know that there are very accurate procedures, they are generally very expensive computationally and so worthless if we can get very similar results with another that is less expensive. The first comparison that can be done with experiments using the highest levels of theory corresponds to the excitation energies to the lowestenergy states of the Si atom. These excited states have different multiplicity from the ground state of the Si atom in our case. For Si (Table 2) the ground state is a triplet, thus we calculate the excitation energies to the lowest state of the singlet and the quartet and compare them to precise experimental energies tabulated on the NIST Computational Chemistry Comparison and Benchmark DataBase (CCCBD)17 which contains a variety of experimental information used in this work. All compound methods, except CBS-4M, yield better results than the DFT ones for the excitation energy from the ground state triplet of Si.

(3) to its singlet state. G2 yields the exact value, and G1-G4 yield also great agreement with errors of 2.3, 2.9 and 3.9%, respectively. W1U, W1BD and W1RO procedures yield errors of 9.6, 9.2 and 9.5%, respectively. CBS-BQ3 method yields an error of 10.3%. On the other hand, the B3PW91 and the CBS-4M errors are in the range of 51.6 to 65.8%. All results using B3PW91 functional are above the experimental value and show errors of 0.46 to 0.51 eV, compared to the experimental value of 0.781 eV. Table 2. Total energies at 0 K in Ha and relative energies (ΔE) in eV for the lowest singlet state of the neutral Si atom for several levels of theory*. Method. basis set. energy (Ha). B3PW91. 6-31G(d). -289.27126. E (eV) 1.30. B3PW91. 6-31+G(d). -289.27301. 1.27. B3PW91. 6-311G(d). -289.29337. 1.27. B3PW91. 6-311+G(d). -289.29379. 1.26. B3PW91. aug-cc-pVDZ. -289.29000. 1.27. B3PW91. aug-cc-pVTZ. -289.29760. 1.24. B3PW91. aug-cc-pVQZ. -289.29969. 1.25. B3PW91. aug-cc-pV5Z. -289.30217. 1.25. G1. -288.90573. 0.76. G2. -288.90460. 0.78. G3. -289.19441. 0.76. G4. -289.20947. 0.75. CBS-4M. -288.88868. 1.18. CBS-QB3. -288.89965. 0.86. W1U. -289.80150. 0.86. W1BD. -289.80155. 0.85. W1RO. -289.80150. 0.86. expt.. 0.78117. * The excitation energies (E) are from the ground state triplet of the neutral Si. The experimental energy corresponds to the J=2, and the error in eV corresponds to the difference between the calculated value and the experimental data.. Results in Table 2 show that the B3PW91 functional cannot achieve the best results for the singlet excited state of the Si atom. The use of highest basis sets does not improve the performance of the B3PW91 functional when calculating the first excitation energy. It is possible that the calculated value of the singlet is located in the range between the lowest-energy and the second lowest-energy singlet state for the atom. It would appear that B3PW91 has difficulties to reach the singlet state when the ground state is a triplet. CBS-APNO and. the aug-cc-pV6Z cannot be used because they are not available for Si. Table 3. Total energies at 0 K in Ha and relative energies (ΔE) in eV for the first excited quintet of the neutral Si atom*. Method. basis set. energy (Ha). B3PW91. 6-31G(d). -289.17356. E (eV) 3.95. B3PW91. 6-31+G(d). -289.17416. 3.96. B3PW91. 6-311G(d). -289.19426. 3.97. B3PW91. 6-311+G(d). -289.19445. 3.97. B3PW91. aug-cc-pVDZ. -289.18997. 4.00. B3PW91. aug-cc-pVTZ. -289.19656. 3.99. B3PW91. aug-cc-pVQZ. -289.19941. 3.98. B3PW91. aug-cc-pV5Z. -289.20169. 3.98. G1. -288.78613. 4.02. G2. -288.78634. 4.00. G3. -289.06959. 4.15. G4. -289.08358. 4.13. CBS-4M. -288.79195. 3.82. CBS-QB3. -288.78335. 4.03. W1U. -289.68232. 4.10. expt.. 4.13217. *The ground state in all levels of theory is a triplet, and it is used as a reference. The experimental energy corresponds to the J=2, and error in eV corresponds to the difference between the calculated value and the experimental energy.. We also calculate the excitation energies to the quintet. Results in Table 3 show that the B3PW91 functional using relatively small basis sets competes in quality with the most sophisticated compound methods. The best results are obtained for B3PW91/aug-cc-pVDZ. Except for G3, G4 and W1 methods, which yield an excellent agreement with the experiment, the B3PW91 with basis sets 631G(d) and 6-311G(d) yields similar results to those from the expensive G1, G2 and CBS-QB3. The worst results are obtained with CBS-4M. G4 yields the exact value, and G1-G3 are also in excellent agreement with errors of 2.8, 3.3 and 0.5%, respectively. W1U procedure yields an error of 0.8%. CBS-BQ3 method yields an error of 8.3%. The B3PW91 errors are in the range of 3.3 to 4.3%, which is better that for the singlet state. The size of the basis set slightly improves the results. CBSAPNO and the aug-cc-pV6Z cannot be used because they are not available for Si. W1BD and W1RO had convergence problems to calculate the quintet excited state..

(4) Table 4. First, second, third and fourth Ionization Potentials (IP1, IP2, IP3, IP4, respectively) and their errors (Δ) from the experimental values of Si (all units are in eV)*. Method. basis set. B3PW91. 6-31G(d). 8.24. IP1 0.09. 16.50. IP2 0.16. 33.32. IP3 -0.17. 45.29. IP4 0.14. B3PW91. 6-31+G(d). 8.25. 0.10. 16.51. 0.16. 33.31. -0.18. 45.31. 0.16. B3PW91. 6-311G(d). B3PW91. 6-311+G(d). 8.25. 0.09. 8.24. 0.09. 16.47. 0.12. 16.47. 0.12. 33.27. -0.22. 45.39. 0.24. 33.28. -0.22. 45.39. 0.24. B3PW91. aug-cc-pVDZ. 8.26. 0.11. 16.49. 0.14. 33.32. -0.17. 45.30. 0.16. B3PW91. aug-cc-pVTZ. 8.25. 0.09. 16.47. 0.12. 33.27. -0.23. 45.40. 0.26. B3PW91 B3PW91. aug-cc-pVQZ. 8.25. aug-cc-pV5Z. 8.25. 0.09. 16.49. 0.15. 33.27. -0.22. 45.42. 0.28. 0.09. 16.49. 0.14. 33.26. -0.23. 45.41. 0.26. G1. 8.10. -0.06. 16.22. -0.12. 33.33. -0.16. 44.64. -0.50. G2. 8.10. -0.05. 16.22. -0.12. 33.32. -0.17. 44.64. -0.50. G3. 8.13. -0.03. 16.30. -0.04. 33.59. 0.10. 44.97. -0.17. G4. 8.14. -0.01. 16.33. -0.02. 33.62. 0.13. 44.99. -0.15. CBS-4M. 7.97. -0.18. 16.09. -0.25. 33.43. -0.07. 44.76. -0.39. CBS-QB3. 8.07. -0.08. 16.23. -0.11. 33.29. -0.20. 44.63. -0.51. W1U. 8.13. -0.02. 16.28. -0.06. 33.46. -0.03. 45.11. -0.03. W1BD. 8.13. -0.02. 16.28. -0.06. expt.. 17. IP1. IP2. 8.152. *The ground states of singlet, respectively.. The following tests correspond to the ionization potentials (IPs) of Si. Table 4 shows the first four ionization potentials of Si. For the first IP (IP1) all methods yield excellent results with errors smaller than 1.3% except for CBS-4M with an error of 2.2%. The best fit to the experiment correspond to G4 followed by all W1 methods with errors of only 0.01 and 0.02 eV. Errors are practically independent of the size of the basis set. All DFT methods yield excellent agreement with experiment. The errors for the second (IP2), third (IP3) and fourth ionization potentials (IP4) are larger than those for the IP1. G4, G3, W1 methods (W1U and W1BD) yields the best values for the IP1 with errors of 0.1, 0.3, and 0.4%; all B3PW91 levels yield better results than CBS-4M, and B3PW91 competes with G1 and G2. For the IP3, W1U and, amazingly, CBS4M, yield the best results with errors of 0.03 and 0.07 eV; all levels of B3PW91 competes with expensive G and CBS-QB3 methods with errors around 0.3 and 0.7%. For W1U yields the best result with an error of 0.07%. However, for the IP4, all B3PW91 levels are better than CBS, G1, and G2 compound methods; G3 and G4 yield the same errors. Amazingly, except for W1, the best results are obtained with the B3PW91 using 6-31G(d). All B3P1W1 levels yield. 16.346. IP3. 33.493. -. IP4. -. -. 45.142. are triplet, doublet, singlet, doublet, and. results above the experimental value. The opposite happen with compound methods for the IP4. The W1RO compound method does not yield results for any IP and W1BD does not yield results for IP3 and IP4 due perhaps the incompatibility of the wave functions at the B3LYP and Hartee-Fock levels. Our next quantity to check is the electron affinity (EA) of Si. In Table 5, as excepted the best results are obtained with diffuse functions. Values calculated using diffuse functions yields results above the experimental value. All results, except for CBS-4M and B3P191 with 6-31G(d) basis, are in good agreement with the experimental value. All W1 and G3 compound methods practically reproduce the experiment; all B3PW091 levels (except 6-31G(d)) compete with all expensive G (except G3) and CBSQB3 compound methods with errors of 0.08 to 0.1 eV and 0.3 to 0.7 eV, respectively. In addition, there are not converge problems with W1RO and W1BD procedures for the triplet of the anion Si-1 as a difference with the IPs. The next analysis corresponds to the dissociation energies of small Sin clusters. Table 6 shows that best results for the dimer are obtained with G3 and B3PW91/aug-cc-pVQZ. Except for B3PW91 with small basis sets, all results are in good agreement with the experiment with errors around.

(5) 0.6% for B3PW91/aug-cc-pVQZ to 3.7%. Small basis sets (6-31G(d), 6-31G+(d), 6-311G(d), 6311G+(d)) yield errors around 6.1 to 7.6%, which is still in good agreement with the experiment. Roughly at ~12 kcal/mol is located the first excited state singlet of the dimer, and at ~33 kcal/mol is located the excited state quintet. The effect of the zero-point vibrational energy is between 0.70 and 0.76 kcal/mol for all levels of theory, and it is considered in all calculations. Table 5. Electron Affinities and Their Errors with Respect to the Experimental Value of Si*. EA (eV). error (eV). For the EA, G methods yield the best results. G3 reproduces the experiment and the others have error of only 0.1 eV. CBS-QB3 and W1 methods have an error of 0.4 eV respect to the experiment and CBS-4M and B3PW91 with all basis sets yield error around 0.1 to 0.29 eV. As expected better results are obtain when diffuse function are used. The anion ground state of the dimer found in all levels of theory is a doublet and roughly at 30 kcal/mol and 37 kcal/mol is located the first excited state quartet for when calculated with B3P191 and all compound methods, respectively. Table 6. Total energies in Ha and ground state neutral molecule*. for the. Method. basis set. B3PW91. 6-31G(d). 1.17. -0.22. B3PW91. 6-31+G(d). 1.47. 0.08. Method. basis set. energy (Ha). De (eV). 6-31G(d). -578.75203. 3.11. B3PW91. 6-311G(d). 1.30. -0.09. B3PW91. B3PW91. 6-311+G(d). 1.47. 0.08. B3PW91. 6-31+G(d). -578.75351. 3.11. B3PW91. aug-cc-pVDZ. 1.49. 0.10. B3PW91. 6-311G(d). -578.79245. 3.06. B3PW91. aug-cc-pVTZ. 1.48. 0.09. B3PW91. 6-311+G(d). -578.79447. 3.10. aug-cc-pVDZ. -578.79130. 3.20. B3PW91. aug-cc-pVQZ. 1.47. 0.08. B3PW91. B3PW91. aug-cc-pV5Z. 1.47. 0.08. B3PW91. aug-cc-pVTZ. -578.80676. 3.27. G1. 1.32. -0.07. B3PW91. aug-cc-pVQZ. -578.81177. 3.28. G2. 1.35. -0.04. B3PW91. aug-cc-pV5Z. -578.81691. 3.29. -577.98591. 3.22. G3. 1.38. -0.01. G1. G4. 1.35. -0.04. G2. -577.98375. 3.19. CBS-4M. 1.22. -0.17. G3. -578.56549. 3.29. CBS-QB3. 1.36. -0.03. G4. -578.59384. 3.26. -577.98277. 3.22. W1U. 1.39. 0.00. CBS-4M. W1BD. 1.40. 0.01. CBS-QB3. -577.98077. 3.22. W1RO. 1.40. 0.01. W1U. -579.78501. 3.24. W1BD. -579.78493. 3.24. W1RO. -579.78501. 3.24. expt.. 1.390. *The ground state of Si and Si quartet, respectively.. -1. 17. are triplet and. It is also calculated the IP and EA for the dimer. Results are shown in Table 7 and Table 8, respectively. All IP results yield excellent agreement with experiment. B3PW91 method with all basis set yield equal or better results than all expensive compound procedures. Best results are obtain with B3PW91 method and 6-311G(d), augcc-pVTZ, and aug-cc-pVTZ basis sets with an error of 0.01 eV. All compound methods yield errors of 0.02 or 0.03 eV except for CBS-4M and CBS-QB3 that yield error of 0.33 and 0.10 eV, respectively. The cation ground state of the dimer found in all levels of theory is a quartet and roughly at 13kcal/mol is located the first excited state doublet.. expt.**. 3.313. *The ground state in all levels of theory is a triplet **Experimental value of 319.7 kJ/mol35. We calculate the IP, EA and De for the trimer. Table 9 shows the IP for the Si3. Best results are obtained with all W1 and CBS-QB3 procedures and B3PW91 with aug-cc-pVTZ and aug-cc-pVQZ basis sets with errors of 0.1 to 0.3 eV. All B3P191 levels of theory yield better results than expensive G2-G4 procedures with error around 0.01 to 0.15 eV for the former and 0.18 to 0.47 for the G methods. It is found differences in the ground state of the cation: for all levels of B3PW91 except 6-31G(d), G3 and CBS-4M ground state is a quartet with a first excited state doublet roughly at 2 kcal/mol. For the rest of the G method, all W1 and CBS-QB3 procedures the.

(6) cation ground state is a doublet with a first excited state quartet roughly at 4 kcal/mol. Table 7. Ionization potential (IP) and errors from the experimental value of molecule for several levels of theory*. Method. basis set. B3PW91. 6-31G(d). 7.94. IP (eV) 0.02. B3PW91. 6-31+G(d). 7.94. 0.02. B3PW91. 6-311G(d). 7.91. -0.01. B3PW91. 6-311+G(d). 7.94. 0.02. B3PW91. aug-cc-pVDZ. 7.94. 0.02. B3PW91. aug-cc-pVTZ. 7.91. -0.01. B3PW91. aug-cc-pVQZ. 7.91. -0.01. B3PW91. aug-cc-pV5Z. 7.90. -0.02. G1. 7.95. 0.03. G2. 7.94. 0.02. G3. 7.89. -0.03. G4. 7.92. 0.00. CBS-4M. 7.59. -0.33. CBS-QB3. 7.82. -0.10. W1U. 7.90. -0.02. W1BD. 7.91. -0.02. 7.90. -0.02. W1RO. IP (eV). expt.. 7.92136. expt.. 7.9215. *The cation ground state found for all levels of theory is a quartet.. Results in Table 10 show excellent agreement with experimental EA for the trimer with errors around 0.1 for B3PW91/aug-cc-pVDZ to 1.02 for G4. All levels of B3PW91 compete with all expensive compound methods. The use of diffuse functions practically does not improve the results. Increasing the basis improves slightly the results. The anion ground state for all levels of theory is a doublet with the first excited quartet state at roughly 33 kcal/mol. Table 11 shows that best results for the dissociation energy of the trimer are obtained CBS4M, which reproduces the experimental value; B3PW91 with aug-cc-pVQZ and aug-cc-pV5Z basis sets with an error of 0.2 eV and all W1 procedures with error of 0.09 eV. All results are in good agreement with experiment and all level of B3PW91 competes with all expensive compound G procedures with errors around 3.5 and 5.6%.. Table 8. Electron affinity (EA) and errors from the experimental value of molecule for several levels of theory*. Method. basis set. B3PW91. 6-31G(d). 1.91. EA (eV) -0.29. B3PW91. 6-31+G(d). 2.10. -0.10. B3PW91. 6-311G(d). 1.97. -0.23. B3PW91. 6-311+G(d). 2.10. -0.10. B3PW91. aug-cc-pVDZ. 2.04. -0.16. B3PW91. aug-cc-pVTZ. 2.05. -0.15. B3PW91. aug-cc-pVQZ. 2.04. -0.16. B3PW91. aug-cc-pV5Z. 1.97. -0.23. G1. 2.21. 0.01. G2. 2.21. 0.01. G3. 2.20. 0.00. G4. 2.21. 0.01. CBS-4M. 2.03. -0.17. CBS-QB3. 2.16. -0.04. W1U. 2.16. -0.04. W1BD. 2.16. -0.04. W1RO. 2.16. -0.04. expt.. EA (eV). 2.199. 37. 2.2026. expt.. *The anion ground state found for all levels of theory is a doublet. Table 9. Ionization Potential (IP) and errors from the experimental value of molecule for several levels of theory Method. basis set. IP (eV). error (eV). B3PW91. 6-31G(d). 8.22. 0.10. B3PW91. 6-31+G(d). 8.27. 0.15. B3PW91. 6-311G(d). 8.24. 0.12. B3PW91. 6-311+G(d). 8.27. 0.15. B3PW91. aug-cc-pVDZ. 8.22. 0.10. B3PW91. aug-cc-pVTZ. 8.14. 0.02. B3PW91. aug-cc-pVQZ. 8.13. 0.01. B3PW91. aug-cc-pV5Z. 8.21. 0.09. G1. 8.21. 0.09. G2. 8.30. 0.18. G3. 8.40. 0.28. G4. 8.59. 0.47. CBS-4M. 7.90. -0.22. CBS-QB3. 8.11. -0.01. W1U. 8.14. 0.02. W1BD. 8.15. 0.03. W1RO. 8.14. 0.02. expt.. 8.12.

(7) Table 10. Electron affinity (EA) and errors from the experimental value of molecule for several levels of theory Method. basis set. EA (eV). error (eV). B3PW91. 6-31G(d). 2.23. -0.06. B3PW91. 6-31+G(d). 2.34. 0.05. B3PW91. 6-311G(d). 2.25. -0.04. B3PW91. 6-311+G(d). 2.34. 0.05. B3PW91. aug-cc-pVDZ. 2.30. 0.01. B3PW91. aug-cc-pVTZ. 2.35. 0.06. B3PW91. aug-cc-pVQZ. 2.33. 0.04. B3PW91. aug-cc-pV5Z. 2.26. -0.03. 2.24. -0.05. G1 G2. 2.25. -0.04. G3. 2.32. 0.03. G4. 1.27. -1.02. CBS-4M. 2.45. 0.16. CBS-QB3. 2.28. -0.01. W1U. 2.31. 0.02. W1BD. 2.31. 0.02. W1RO. 2.31. 0.02. expt.. 2.29. It is found differences in the ground state of the neutral trimer: for all levels of B3PW91, W1U, and W1RO methods it is a triplet with a first excited state singlet roughly at 11 kcal/mol. For all G, CBS, and W1BD procedures the ground state is a singlet with a first excited state singlet roughly at 39 kcal/mol. The multiplicity of the ground state of Si3 is predicted to be singlet38-41 but there is also a lowlying triplet38,42-46 state which has associated a combination transition between a symmetric stretching vibration and a bending vibration43,44 that produces difficulties when determining the correct state. The effect of the zero-point vibrational energy is 1.7 kcal/mol for all levels of theory, and it is considered in all calculations. A consequence in the differences of the multiplicies of the trimer is the predicted angle. For those cluster whose ground state is a triplet, the angle predicted is 60.0°. The average angle predicted for those cluster with ground state singlet is 79.0 ± 0.2° which is in good agreement with previous MP4 calculations by Rohlfing and Raghavachari.47 Optimized Sin (n=3-10) neutral clusters using sophisticated G4 method are shown in Figure 1. We also calculate these clusters with B3PW91 functional and 6-31G(d) basis set and it can be observed that the geometries are practically the. same: except for Si3 and Si6, all geometries are the same with small differences in some bond lengths. For the trimer, as previously discussed, the differences on the multiplicities of the ground state change radically the angle obtained. For the Si6 cluster, both bond lengths and angles have significant differences. Apparently the structure obtained with the DFT method is transition geometry of the G4 method. Table 11. Total energies in Ha and the ground state neutral molecule. and eV for. Method. basis set. Energy (Ha). De (kcal/mol). De (eV). B3PW91. 6-31G(d). -868.21345. 161.2. 6.99. B3PW91. 6-31+G(d). -868.21521. 160.8. 6.97. B3PW91. 6-311G(d). -868.27367. 159.2. 6.90. B3PW91. 6-311+G(d). -868.27612. 160.3. 6.95. B3PW91. aug-cc-pVDZ. -868.26948. 162.5. 7.05. B3PW91. aug-cc-pVTZ. -868.29670. 162.5. 7.05. B3PW91. aug-cc-pVQZ. -868.30485. 168.1. 7.29. B3PW91. aug-cc-pV5Z. -868.31312. 169.0. 7.33. G1. -867.07973. 174.7. 7.58. G2. -867.07214. 177.0. 7.68. G3. -867.94245. 173.0. 7.50. G4. -867.98684. 173.0. 7.50. CBS-4M. -867.06539. 168.7. 7.31. CBS-QB3. -867.06814. 172.1. 7.46. W1U. -869.77056. 170.5. 7.39. W1BD. -869.77061. 170.6. 7.40. W1RO. -869.77061. 170.5. 7.40. 168.6. 7.31. expt.. Using the structures of Figure 1 we calculate IPs and EAs for Sin (n=4-10) using only B3PW91/6-31G(d) and G4. All results for the IP and EA shown in Figure 2 and Figure 3, respectively, are in excellent agreement with experiment. For the IP, B3PW91 hybrid with 6-31G(d) basis set yields errors around 0.1 to 10.0 % in comparison with G4 that yields errors around 0.1 to 5.9%. In general G4 yield more accurate results than DFT for the IP, but the errors are small enough if we consider that G4 is much more expensive than B3PW91/6-31G(d). For the EA, DFT yield better results than G4 with errors around 0.5 to 10.7%. In addition to the great agreement of the results calculated with DFT, it is also important to say that it is more difficult to obtain the geometries and results with the computationally expensive G4..

(8) Figure 1. Sin(n=3-10) clusters optimized with G4 method. All geometries correspond to the neutral ground state singlet and correspond to local minima.. 4,00. 9. 7 3,00. 6 5. Expt. G4. De (eV). IP (eV). 8. DFT. 4 0. 5 10 Size of cluster Sin. 15. 2,00 DFT. Figure 2. Ionization potential of Sin (n=1-15) clusters using G4 and DFT B3PW91/6-31G(d). Calculated values for Sin (n=18,15, 17,36 10) are compared with experimental data.. Expt. 1,00 1. 10 Size of cluster (n). 100. Figure 4. Dissociation energies per atom of Sin clusters from TABLE and TABLE show a tendency (FIGURe) to a bulk value of for silicon crystal. Calculated values with G4 and B3PW91 hibryd and 6-31G(d) are compared with experimental values.9,10,11,35,50. 3 EA (eV). G4. 2 1 Expt. G4. DFT. 0 0. 5 10 Size of cluster Sin. 15. Figure 3. Electron affinity of Sin (n=1-15) clusters using G4 and DFT B3PW91/6-31G(d). Calculated values for Sin(n=1-10) 6, 13, 14,17, 37, 45-47,51,52 are compared with experimental data.. For the construction of larger clusters (Si20 and up), we take a crystal cubic unit cell with experimental bulk properties of a=5.43094 Å48 and cell volume of 160.186 Å3 to build the inputs of the Sin clusters of Figure 5. We calculate dissociation energies for all neutral clusters as shown in Figure 4. G4 and B3PW91/6-31G(d) methods were used for the optimizations. All results are in great agreement with experiment. As previously discussed, for the.

(9) Figure 5. Clusters from Table which corresponds to local minima. The hybrid B3PW91 and 6-31G(d) basis set were used for the optimization. All clusters have charge neutral and multiplicity 1.0. dimer and the trimer both methods yield excellent agreement. For Sin (n=2-8) can be said that G4 yield the best results practically reproducing the experiments with errors of 0.4 to 6.1%. In all cases B3PW91/6-31G(d) underestimate the experimental value with errors of 4.4 to 10.3%. Errors are from 0.10 to 0.37 eV and increase slightly with size of cluster. Second derivatives are calculated with the procedure Fuerza49 for all cluster with the goal to create harmonic force constants for further use in MD simulations clusters. The clusters vibrational spectra show a systematic behavior thus we expect the force constants to follow a similar trend. The relationship between their force constants and bond lengths shows an inverse potentials tendency (Fig 6). The regression to the fourth order inverse potential yields a correlation factor higher than 0.95, which is appropriate for the present discussion. Higher orders could present better accuracy but the order of the equation is unnecessary at this level. According to the bond distribution shown in Figure 6 the average. bond force constant is 183.02 kcal mol-1 Å-2 with a bond length average of 2.36 Å. As a reference the bulk bond length is 2.35248. The fourth order inverse potential equation for the force constants versus the bond length is shown in Equation 1. (. ). Equation 1. Relation between force constants and bond lengths. FUTURE WORK A MD simulation can be performed for a Si crystal and determine, e.g., its molar heat capacity (Cp) to prove the bond force constants we obtained for silicon bonds. The LJ parameters used for this.

(10) Figure 6. Primary y axis: Relationship between Si-Si bond force constants (k12) in kcal mol-1 Å-2 and their bond lengths (b12) in Å of all clusters calculated with B3PW91 hybrid and 6-31G(d) basis set. Secondary y axis: Bond length frequency distribution.. simulation would be  equal to 2.102 according to the bond distribution and  of 0.0703 according with the calculated force constant of 183.02 kcal mol-1 Å-2 CONCLUSIONS The hybrid B3PW91 functional with several basis sets (6-31G(d), 6-31G+(d), 6-311G(d), 6-311G+(d), aug-cc-pVDZ, aug-cc-pVTZ, aug-cc-pVQZ, aug-ccpV5Z) yields great results with respect to experimental data and is comparable with high quality compound methods (G1-G4, CBS-4M, CBSQB3, and all W1 methods). All IPs, EAs, excitation energies, and dissociation energies are in great agreement with the experiment. Strong differences between B3PW91 hybrid and compound procedures are found on the ground states neutral and cation of the trimer. It is calculated second derivative for all cluster studied and it is obtained a fourth order inverse potential regression with a correlation factor higher than 0.95 for the relationship between bond force constants and bond lengths. The LJ parameters found for the future MD simulation are be  equal to 2.102 according to the bond distribution and  of 0.0703 according with the calculated force constant of 183.02 kcal mol-1 Å-2. REFERENCES (1) Dorsett, H.; White, A. Overview of molecular modelling and Ab initio molecular orbital methods suitable for use with energetics materials,. DSTO Aeronautical and maritime research laboratory, 2000. (2) Kohn, W. Physical Review Letters 1996, 76, 3168. (3) Kohn, W.; Sham, L. J. Physical Review 1965, 140, A1133. (4) Hohenberg, P.; Kohn, W. Physical Review 1964, 136, B864. (5) Seminario, J. M. Modern Density Functional theory: A tool for chemestry; Elsevier: Amsterdam, 1995. (6) Arnold, C. C.; Kitsopoulos, T. N.; Newmark, D. M. J. Chem. Phys. 1339, 99. (7) Fuke, K.; Tsukamoto, K.; Misaizu, F. Zeitschrift für Physik D 1992, 26, 204. (8) Fuke, K.; Tsukamoto, K.; Misaizu, F.; Sanekata, M. J. Chem. Phys. 1993, 99. (9) Gingerich, K. A.; Ran, Q.; Schmude Jr, R. W. Chemical Physics Letters 1996, 256, 274. (10) Ran, Q.; Schmude Jr, R. W.; Miller, M.; Gingerich, K. A. Chemical Physics Letters 1994, 230, 337. (11) Schmude, R. W.; Ran, Q.; Gingerich, K. A. J. Chem. Phys. 1993, 99. (12) Jarrold, M. F.; Honea, E. C. J. Phys. Chem. 1991, 95, 9181. (13) Kitsopoulos, T. N.; Chick, C. J.; Zhao, Y.; Newmark, D. M. J. Chem. Phys. 1991, 95, 1441. (14) Kitsopoulos, T. N.; Chick, C. J.; Weaver, A.; Newmark, D. M. J. Chem. Phys. 1990, 93. (15) Kostko, O.; Leone, S. R.; Duncan, M. A.; Ahmed, M. The Journal of Physical Chemistry A 2009, 114, 3176..

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