Sains Malaysiana 41(4)(2012): 445-452

Effects of Spin Contamination on the Stability and Spin Density of Wavefunction of  Graphene: Comparison between First Principle and Density Functional Methods

(Kesan Pelumusan Spin Terhadap Kestabilan dan Ketumpatan Spin bagi Fungsi  Gelombang Grafen pelumusan: Perbandingan antara Kaedah Prinsip Pertama  dan Kaedah Kefungsian Ketumpatan)

Lee Sin Ang* & Shukri Sulaiman

Physical Sciences Programme, School of Distance Education

Universiti Sains Malaysia, 11800 Penang, Malaysia

Mohamed Ismail Mohamed-Ibrahim

Chemical Sciences Programme, School of Distance Education

Universiti Sains Malaysia, 11800 Penang, Malaysia

Diserahkan: 13 Mei 2011 / Diterima: 7 Oktober 2011

ABSTRACT

The effects of spin contamination on the stability and the spin densities of a model of graphene in the Hartree-Fock wavefunction (HF), Møller-Plesset perturbation theory (second order, MP2 and fourth order, MP4) and density functional theory (B3LYP and PBEPBE) are reported. It was found that the Hartree-Fock and MP2 wavefunctions of graphene suffer from the contamination from higher spin states and spin projection method failed to project out the spin contaminants. The spin density from HF was overestimated, while for MP2 it has the wrong trend. B3LYP and PBEPBE wave functions however have negligible contamination for higher spin states. Comparison with reported results showed that the spin

densities at the center of the molecule from the pure functionals of PBEPBE were underestimated. Based on the comparison made, it was concluded that among the methods considered, the suitable one for use in the calculations of pristine graphene was B3LYP.

Keywords: Density functional theory; grapheme; Hartree-Fock; Møller-Plesset perturbation theory; spin

contamination

ABSTRAK

Kesan pelumusan spin ke atas kestabilan dan ketumpatan spin di dalam fungsi gelombang Hartree-Fock (HF), teori gangguan Møller-Plesset (tertib kedua, MP2 dan tertib keempat, MP4) dan teori fungsi ketumpatan (B3LYP dan PBEPBE) bagi model grafen dilaporkan. Didapati bahawa fungsi gelombang HF dan MP2 bagi grafen mengalami pelumusan daripada keadaan spin lebih tinggi dan kaedah projeksi gagal mengeluarkan pelumusan putaran. Nilai ketumpatan spin daripada HF adalah terlebih anggar, manakala untuk MP2 ia mempunyai corak yang salah. B3LYP dan PBEPBE mempunyai pelumusan spin yang boleh diabaikan. Perbandingan dengan keputusan yang telah dilaporkan menunjukkan bahawa ketumpatan spin pada bahagian tengah molekul daripada kefungsian tulen PBEPBE adalah terkurang anggar. Berdasarkan perbandingan yang dilakukan, disimpulkan bahawa antara kaedah-kaedah yang dipertimbangkan, B3LYP merupakan kaedah yang sesuai untuk pengiraan melibatkan grafen tulen.

Kata kunci: Grafin; Hartree-Fock; pencemaran putaran; teori gangguan Møller-Plesset; teori fungsi ketumpatan

RUJUKAN

Amos, T. & Snyder, L.C. 1964. Unrestricted Hartree-Fock Calculations. I. An improved method of computing spin properties. Journal of Chemical Physics 41: 1773-1783.

Andrews, J.S., Jayatilaka, D., Bone, R.G.A., Handy, N.C. & Amos, R.D. 1991. Spin contamination in single-determinant wave functions. Chemical Physics Letters 183: 423-431.

Baker, J., Scheiner, A. & Andzelm, J. 1993. Spin contamination in density functional theory. Chemical Physics Letters 216: 380-388.

Burnham, D.R. 1969. Spin contamination in PPP unrestricted Hartree-Fock wave functions. Theoretical Chemistry Accounts: Theory, Computation and Modeling (Theoretica Chimica Acta) 13: 428-432.

Chen, W. & Schlegel, H.B. 1994. Evaluation of S2 for correlated wave functions and spin projection of unrestricted Møller-Plesset perturbation theory. Journal of Chemical Physics 101: 5957-5968.

Chuang, Y.-Y., Coitino, E.L. & Truhlar, D.G. 1999. How should we calculate transition state geometries for radical reactions? the effect of spin contamination on the prediction of geometries for open-shell saddle points. Journal of Physical Chemistry A 104: 446-450.

Claxton, T.A. & McWilliams, D. 1970. The restriction of spin contamination in unrestricted Hartree Fock wave functions. Theoretical Chemistry Accounts: Theory, Computation, and Modeling (Theoretica Chimica Acta) 16: 346-350.

Cohen, A.J., Tozer, D.J. & Handy, N.C. 2007. Evaluation of <S2> in density functional theory. Journal of Chemical Physics 126: 214104.

Cramer, C.J., Dulles, F.J., Giesen, D.J. & Almlöf, J. 1995. Density functional theory: excited states and spin annihilation. Chemical Physics Letters 245: 165-170.

Cremaschi, P., Gamba, A., Morosi, G. & Simonetta, M. 1976. Influence of spin contamination and basis set on electrostatic potential and Hfs coupling constants of organic radicals. Theoretical Chemistry Accounts: Theory, Computation, and Modeling (Theoretica Chimica Acta) 41: 177-182.

Davidson, E.R. & Clark, A.E. 2005. Spin polarization and annihilation for radicals and diradicals. International Journal of Quantum Chemistry 103: 1-9.

Dias, J.R. 2008. Resonance-theoretic calculation of the ground state spin density of the π-system of edge atoms on grapheme nanodots and nanoribbons. Chemical Physics Letters 467: 200-203

Frisch, M.J., Trucks, G.W., Schlegel, H.B., Scuseria, G.E., Robb, M.A., Cheeseman, J. R., Montgomery, J.J.A., Vreven, T., Kudin, K.N., Burant, J.C., Millam, J.M., Iyengar, S.S., Tomasi, J., Barone, V., Mennucci, B., Cossi, M., Scalmani, G., Rega, N., Petersson, G.A., Nakatsuji, H., Hada, M., Ehara, M., Toyota, K., Fukuda, R., Hasegawa, J., Ishida, M., Nakajima, T., Honda, Y., Kitao, O., Nakai, H., Klene,

M., Li, X., Knox, J. E., Hratchian, H.P., Cross, J.B., Bakken,V., Adamo, C., Jaramillo, J., Gomperts, R., Stratmann, R.E., Yazyev, O., Austin, A. J., Cammi, R., Pomelli, C., Ochterski, J.W., Ayala, P.Y., Morokuma, K., Voth, G.A., Salvador, P., Dannenberg, J.J., Zakrzewski, V.G., Dapprich, S., Daniels, A.D., Strain, M. C., Farkas, O., Malick, D.K., Rabuck, A.D., Raghavachari, K., Foresman, J.B., Ortiz, J.V., Cui, Q., Baboul, A.G., Clifford, S., Cioslowski, J., Stefanov, B.B., Liu, G., Liashenko, A., Piskorz, P., Komaromi, I., Martin, R. L., Fox, D.J., Keith, T., Al-Laham, M. A., Peng, C.Y., Nanayakkara, A., Challacombe, M., Gill, P. M. W., Johnson, B., Chen, W., Wong, M.W., Gonzalez, C. & Pople, J. A. 2004. Gaussian 03, Revision E.01, Wallingford CT, Gaussian, Inc. Gao, X., Zhou, Z., Zhao, Y., Nagase, S., Zhang, S.B. & Chen, Z. 2008. Comparative Study of Carbon and BN Nanographenes: Ground Electronic States and Energy Gap Engineering. Journal of Physical Chemistry C 112: 12677-12682.

Geim, A.K. 2009. Graphene: Status and Prospects. Science 324: 1530-1534.

Geim, A.K. & Novoselov, K.S. 2007. The rise of graphene. Nature Materials 6: 183-191.

Grafenstein, J., Kraka, E., Filatov, M. & Cremer, D. 2002. Can unrestricted density-functional theory describe open shell singlet biradicals? International Journal of Molecular Sciences 3: 360-394.

Handy, N. C., Knowles, P.J. & Somasundram, K. 1985. On the convergence of the Møller-Plesset perturbation series. Theoretical Chemistry Accounts: Theory, Computation and Modeling (Theoretica Chimica Acta) 68: 87-100.

Hod, O., Barone, V. & Scuseria, G.E. 2008. Half-metallic graphene nanodots: A comprehensive first-principles theoretical study. Physical Review B 77: 035-411.

Jiang, D.-e., Sumpter, B.G. & Dai, S. 2007. First principles study of magnetism in nanographenes. Journal of Chemical Physics 127: 124703-5.

Kudin, K.N. 2008. Zigzag Graphene Nanoribbons with Saturated Edges. ACS Nano 2: 516-522.

Li, X. & Paldus, J. 2000. Effect of spin contamination on the prediction of barrier heights by coupled-cluster theory: F+H2>HF+H reaction. International Journal of Quantum Chemistry 77: 281-290.

Liu, B., Jia, D., Meng, Q. & Rao, J. 2007. A novel method for preparation of hollow carbon spheres under a gas pressure atmosphere. Carbon 45: 668-670.

Löwdin, P.-O. 1955. Quantum theory of many-particle systems. III. Extension of the Hartree-Fock scheme to include degenerate systems and correlation effects. Physical review 97: 1509.

Menon, A.S. & Radom, L. 2008. Consequences of spin contamination in unrestricted calculations on open shell species: effect of hartree-fock and møller-plesset contributions in hybrid and double-hybrid density functional theory approaches Journal of Physical Chemistry A 112: 13225-13230.

Montoya, A., Truong, T.N. & Sarofim, A.F. 2000. Spin contamination in hartree fock and density functional theorywave functions in modeling of adsorption on graphite. Journal of Physical Chemistry A 104: 6108-6110.

Nandi, P. K., Kar, T. & Sannigrahi, A. B. 1996. Effect of spin contamination in UHF wavefunctions on charge density based local quantities. Journal of Molecular Structure: THEOCHEM 362: 69-75.

Novoselov, K.S., Geim, A.K., Morozov, S.V., Jiang, D., Zhang, Y., Dubonos, S. V., Grigorieva, I.V. & Firsov, A.A. 2004. Electric field effect in atomically thin carbon films. Science 306: 666-669.

Novoselov, K.S., Jiang, D., Schedin, F., Booth, T.J., Khotkevich, V.V., Morozov, S. V. & Geim, A.K. 2005. Two-dimensional atomic crystals. Proceedings of the National Academy of Sciences of the United States of America 102: 10451-10453.

Peralta-Inga, Z., Murray, J.S., Edward Grice, M., Boyd, S., O’Connor, C. J. & Politzer, P. 2001. Computational characterization of surfaces of model graphene systems. Journal of Molecular Structure: THEOCHEM 549: 147-158.

Plakhutin, B., Gorelik, E., Breslavskaya, N., Milov, M., Fokeyev, A. A., Novikov, A., Prokhorov, T., Polygalova, N., Dolin, S. & Trakhtenberg, L. 2005. Anomalous values of Ŝ2 before and after annihilation of the first spin contaminant in UHF wave function. Journal of Structural Chemistry 46: 195-203.

Schlegel, H.B. 1986. Potential energy curves using unrestricted Møller-Plesset perturbation theory with spin annihilation. Journal of Chemical Physics 84: 4530-4534.

Schlegel, H.B. 1988. Møller-Plesset perturbation theory with spin projection. Journal of Physical Chemistry 92: 3075-3078.

Sendt, K. & Haynes, B.S. 2005. Density functional study of the chemisorption of O2 on the zig-zag surface of graphite. Combustion and Flame 143: 629-643.

Sendt, K. & Haynes, B.S. 2007. Density Functional Study of the Chemisorption of O2 Across Two Rings of the Armchair Surface of Graphite. Journal of Physical Chemistry C 111: 5465-5473.

Shrivastava, K.N. 2011. Laughlin’s wave function and angular momentum. International Journal of Modern Physics B 25: 1301-1357.

Snyder, L.C.K Amos, T. 1965. Unrestricted Hartree-Fock Calculations. II. Spin Properties of Pi-Electron Radicals. Journal of Chemical Physics 42: 3670-3683.

Son, Y.-W., Cohen, M.L. & Louie, S. G. 2006a. Energy Gaps in Graphene Nanoribbons. Physical Review Letters 97: 216803.

Son, Y.-W., Cohen, M.L. & Louie, S. G. 2006b. Half-metallic graphene nanoribbons. Nature 444: 347-349.

Wittbrodt, J.M. & Schlegel, H. B. 1996. Some reasons not to use spin projected density functional theory. Journal of Chemical Physics 105: 6574-6577.

Xu, Y.-J. & Li, J.-Q. 2005. The interaction of N2 with active sites of graphite: A theoretical study. Chemical Physics Letters 406: 249-253.

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