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Resonance structures might correspond to excited states in polycyclic conjugated systems
Chemical Physics Letters 491 (2010) 20–22
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Resonance structures might correspond to excited states in polycyclic conjugated systems S.M. Azami a,*, R. Pooladi b, N. Setoudeh c a
Chemistry Department, College of Science, Yasouj University, Yasouj 75914-353, Iran Nanotechnology Research Institute, Shiraz University, Shiraz 71454, Iran c Material Engineering Department, School of Engineering, Yasouj University, Yasouj 75914-353, Iran b
a r t i c l e
i n f o
Article history: Received 18 November 2009 In final form 19 March 2010 Available online 24 March 2010
a b s t r a c t Resonance structures are presented for four small carbon nanotubes (CNTs) in the context of quantum chemical concepts. The results show that there might exist Kekulé structures, which are apparent excited electronic states and may mislead a chemist as being contributed to the CNT ground state. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction The Lewis resonance pattern of chemical systems is one of the cornerstone concepts in chemistry and interpretation of chemical reactions. Nowadays, Kekulé resonance structures are applied in a wide range of chemical systems including carbon nanotubes (CNTs) and fullerenes. In general, such analyses are based on the fact that resonance structures contribute to the molecular electronic stability and sometimes are responsible for molecular aromaticity. However, as will be shown, it is not so and there might exist certain structures which have absolutely no contribution to the system’s electronic structure. Development of molecular orbital (MO) theory in the past century allowed theoretical chemists to understand the nature of chemical systems and quantify chemical phenomena via their electronic wave functions. Although the electronic structures provided by the corresponding MOs were found to be extremely complicated for polyatomic systems, certain transformations of the MOs could translate quantum mechanical concepts to chemical ones [1]. The concept of localized orbitals is indeed the most sensible transformation in which the resultant orbitals are centered on atoms and pair of atoms. At the present time, many localization techniques are developed such as the localized orbitals by Foster and Boys [2], Edmiston and Ruedenberg [3], and natural bond orbitals (NBOs) by Weinhold [4,5]. NBOs are found to be highly transferable among analogous systems and resemble Lewis lone pair and bonds. One of the interesting concepts in quantum chemistry is chemical resonance and many researches are devoted to inspect this phenomenon via superposition principle. According to this principle, the electronic wave function can be rewritten as a summation over that of resonance structures. Eqs. (1) display the one-to-one * Corresponding author. E-mail address: [email protected] (S.M. Azami). 0009-2614/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2010.03.061
correspondence between vector algebra and quantum mechanical resonance superposition.
A ¼ ax i þ ay j þ az k ax ¼ i A
ð1aÞ
jwi ¼ c1 j/1 i þ c2 j/2 i þ c3 j/3 i þ c1 ¼ h/1 jwi
ð1bÞ
In the above equations, jwi stands for the molecular electronic wave function; while j/k i represents that of kth resonance electronic configuration. The coefficients ck are then determined by h/k jwi, the overlap between original electronic wave functions and kth resonance configuration, which can be computed efficiently by using the formula introduced by Löwdin [6]. It should be noted that the j/k i components must provide an orthonormal set to rely on h/k jwi as resonance weight. Otherwise, h/k jwi values can be considered as a criterion of how much a resonance structure resembles the corresponding quantum mechanical electronic wave function. In a work by the present author [7], the NBOs are used to construct electronic wave function of the resonance structures and their weights are determined as h/k jwi after some mathematical manipulations. Karafiloglou [8], has also taken advantage of Moffitt’s theorem [9] to expand the molecular electronic wave function into determinants composed of NBOs. Indirect superposition of the electronic structure has also been considered as natural resonance theory (NRT) [10–12] in which the density matrices are expanded instead of the wave functions. In the next section, computational details of the procedure is presented and Section 3 discusses resonance structures for four small CNTs. 2. Computational details The electronic wave functions are obtained at HF/6-31G level of theory utilizing GAUSSIAN 03 suite of programs [13], where the C–C and C–H bond lengths were fixed at 1.4 and 1.0 Å, respectively. The resonance wave function, jwres i, can be obtained as a finite
21
S.M. Azami et al. / Chemical Physics Letters 491 (2010) 20–22
sum over the wave functions of orthogonalized resonance structures, j/0k i, according to the following equation:
jwres i ¼ N
X
ci j/0i i
the values of h/k jwi are adopted to compare contributions of different resonance structures of four simple CNTs which are discussed in the next section.
ð2Þ
i
3. Results and discussion
where ci ¼ h/0i jwres i and N is the normalization constant. For closedshell wave functions, h/k jwi can be computed by Löwdin general formula [6] as h/k jwi ¼ jhvki jvj ij2 , where vki and vj are ith and jth molecular orbitals of j/k i and jwi, respectively. In the present work,
Chemical structures of the CNTs, excluding the dangling hydrogen atoms, are shown in Fig. 1 and labeled by I–IV. System I is one ring of a small CNT in diameter which is claimed to be the smallest
Fig. 1. The chemical structure of CNTs together with their labels used in the text (the hydrogen atoms are not shown).
Fig. 2. A typical NBO in system II, representing a p bond.
Table 1 Resonance patterns for the CNTs.a IV
III
II
a
b
c
d
e
f
g
h
i
a
Only determining (double) bonds are shown to make the patterns clear, and the other double bonds can be uniquely depicted.
I
22
S.M. Azami et al. / Chemical Physics Letters 491 (2010) 20–22
Table 2 The values of CNT
I II III IV a
h/k jwi , h/a jwi
relative overlap between kth resonance wave function and individual resonance configurations with respect to h/a jwi.
Resonance configuration a
b
c
d
e
f
g
h
i
1.0000 1.0000 1.0000 1.0000
1.3499 0.0000a 2.0408 0.5500
0.0000 0.1235 1.5714 0.8000
0.0000 0.2881 0.0000 0.0004
0.0000 0.3786 0.0000 0.0005
0.0000 0.0006
0.0000 0.1178
0.0000 0.2678
0.0000 0.3417
The numerical error is ±1.0 108.
Table 3 Occupancies of p and p* orbitals for II and benzene according to NBO results. II
Benzene
a
b
1.59p 0.41p*
1.59p 0.41p*
1.67p 0.33p*
0.67p 1.33p*
diameter CNT [14]. The interaction of hydrogen molecule with such small CNT is recently analyzed by the present author, theoretically [15]. The resonance structures of the CNTs are displayed in Table 1. To keep clarity, several points are considered in this table: (i) equivalent resonance structures are excluded; (ii) the resonance schemes are unrolled and atoms located at the two ends are identical; (iii) only determining bonds are shown, i.e., the other double bonds can be uniquely identified. One of NBO p bonds corresponding to the double bonds in Table 1 is plotted in Fig. 2 to visualize its nodal shape and qualitative distribution. The other NBO p bonds have almost the same shape and distribution. As Fig. 2 shows, the NBO highly resemble a chemist’s imagination of p bond and can thus be safely considered as building blocks of resonance electronic configurations. The resonance structures depicted in Table 1 include both Kekulé and Dewar-type structures. Although there exist also many other resonance structures such as long-bonded and ionic configurations, we have limited the discussion to Kekulé and Dewar-type structures. However, a large number of structures are necessary to form a complete superposition of resonance structures. In order to compare resonance components, j/a i is considered as reference configuration as it has contributed to all of the CNTs MO wave function. Therefore,
h/k jwi h/a jwi
values, relative overlap of j/k i
configuration, are reported in Table 2. According to these values, the three Dewar-type resonance structures in I has no contribution to the CNT’s p system. For III, resonance structure b has significantly higher overlap with jwi than structure a. An interesting result is that resonance structure b has absolutely no contribution to the p system of II, as it possesses zero overlap with its quantum mechanical electronic wave function. Such failure of Lewis resonance structures has not yet been observed for Kekulé structures in polycyclic conjugated systems. Therefore, classical resonance structures may not be valid for all CNTs and may lead to unreal results. From quantum mechanical standpoint, such resonance structure corresponds to a mixture of excited states. To prove this, Moffitt’s theorem can be helpful. According to this theorem, any arbitrary Slater determinant can be rewritten as certain linear combination of Slater determinants composed of canonical molecular orbitals of the same system and basis set. Therefore, electronic state of resonance structure b in II can be decomposed as Eq. (3):
j/b i ¼ c0 jw0 i þ
X i
ci jwi i
ð3Þ
where jw0 i and jwi i denote ground state and ith excited state, respectively. Since c0 = 0 for this case, electronic state of resonance structure b in II will be a mixture of excited electronic states. The mixing coefficients, {ci}, can be determined by either Moffitt’s theorem or Löwdin general formula. Another interesting result is the high overlap values of Dewar-type structures in II and IV. For IV, the relative overlap of structure i competes with b which is a Kekulé-type structure. The results for II and IV warn a chemist that Dewar-type structures might not be ignorable in interpreting of chemical structures including p system. It is interesting to compare IIb resonance structure with Dewar benzene in which an unconventional bond connects two carbon atoms. If this bond is of r-type, such resonance configuration has no contribution to the benzene’s ground state, i.e., h/Dewar jwi ¼ 0. Note that this result is evident as the unconventional bond is orthogonal to all MOs of benzene’s p system with respect to symmetry consideration of molecular orbitals. The other choice for the unconventional bond is a p-type bond. Such bond has been widely studied in the literature and its role is well explored in the benzene’s aromaticity [16]. Table 3 displays occupancies of p and p* orbitals for II (a, b), Kekulé and Dewar resonance structures in minimal basis set. As Table 3 shows, occupancy of p and p* orbitals for IIa and IIb are similar with those of Kekulé in benzene. However, the bond connecting carbons 1 and 4 in Dewar structure has occupancy of p* higher than the corresponding p by 0.66 which is in agreement with the fact Dewar structure has low weight in benzene’s p system. Also, unlike Dewar structure in benzene, the Dewar-type resonance structures in II contribute significantly to the ground state. Finally, it seems that there exists a symmetry-related phenomenon which is able to interpret both vanishing resonance IIb and large contribution of Dewar-type structure to the system’s resonance. Acknowledgements The authors would like to acknowledge the reviewers for their comments on the necessity of considering Dewar-type resonance structures and their contribution to the electronic ground states. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
H. Weinstein, R. Paunz, M. Cohen, Adv. At. Mol. Phys. 7 (1971) 97. J.M. Foster, S.F. Boys, Rev. Mod. Phys. 32 (1960) 300. C. Edmiston, K. Ruedenberg, Rev. Mod. Phys. 34 (1963) 457. A.E. Reed, L.A. Curtiss, F. Weinhold, Chem. Rev. 88 (1988) 899. J.P. Foster, F. Weinhold, J. Am. Chem. Soc. 102 (1980) 7211. P.-O. Löwdin, Phys. Rev. 97 (1955) 1474. A.H. Pakiari, S.M. Azami, Int. J. Quantum Chem. 108 (2008) 219. P. Karafiloglou, J. Comput. Chem. 22 (2001) 306. W. Moffitt, Proc. R. Soc. A218 (1953) 486. E.D. Glendening, F. Weinhold, J. Comput. Chem. 19 (1998) 593. E.D. Glendening, F. Weinhold, J. Comput. Chem. 19 (1998) 610. E.D. Glendening, J.K. Badenhoop, F. Weinhold, J. Comput. Chem. 19 (1998) 628. M.J. Frisch et al., GAUSSIAN 03, Gaussian, Pittsburgh, PA, 2003. H.Y. Peng et al., Appl. Phys. Lett. 77 (2000) 2831. S. Fakhraee, S.M. Azami, J. Chem. Phys. 130 (2009) 084113-1. L.J. Schaad, B. Andes Hess, Chem. Rev. 101 (2001) 1465.
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Resonance structures might correspond to excited states in polycyclic conjugated systems S.M. Azami a,*, R. Pooladi b, N. Setoudeh c a
Chemistry Department, College of Science, Yasouj University, Yasouj 75914-353, Iran Nanotechnology Research Institute, Shiraz University, Shiraz 71454, Iran c Material Engineering Department, School of Engineering, Yasouj University, Yasouj 75914-353, Iran b
a r t i c l e
i n f o
Article history: Received 18 November 2009 In final form 19 March 2010 Available online 24 March 2010
a b s t r a c t Resonance structures are presented for four small carbon nanotubes (CNTs) in the context of quantum chemical concepts. The results show that there might exist Kekulé structures, which are apparent excited electronic states and may mislead a chemist as being contributed to the CNT ground state. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction The Lewis resonance pattern of chemical systems is one of the cornerstone concepts in chemistry and interpretation of chemical reactions. Nowadays, Kekulé resonance structures are applied in a wide range of chemical systems including carbon nanotubes (CNTs) and fullerenes. In general, such analyses are based on the fact that resonance structures contribute to the molecular electronic stability and sometimes are responsible for molecular aromaticity. However, as will be shown, it is not so and there might exist certain structures which have absolutely no contribution to the system’s electronic structure. Development of molecular orbital (MO) theory in the past century allowed theoretical chemists to understand the nature of chemical systems and quantify chemical phenomena via their electronic wave functions. Although the electronic structures provided by the corresponding MOs were found to be extremely complicated for polyatomic systems, certain transformations of the MOs could translate quantum mechanical concepts to chemical ones [1]. The concept of localized orbitals is indeed the most sensible transformation in which the resultant orbitals are centered on atoms and pair of atoms. At the present time, many localization techniques are developed such as the localized orbitals by Foster and Boys [2], Edmiston and Ruedenberg [3], and natural bond orbitals (NBOs) by Weinhold [4,5]. NBOs are found to be highly transferable among analogous systems and resemble Lewis lone pair and bonds. One of the interesting concepts in quantum chemistry is chemical resonance and many researches are devoted to inspect this phenomenon via superposition principle. According to this principle, the electronic wave function can be rewritten as a summation over that of resonance structures. Eqs. (1) display the one-to-one * Corresponding author. E-mail address: [email protected] (S.M. Azami). 0009-2614/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2010.03.061
correspondence between vector algebra and quantum mechanical resonance superposition.
A ¼ ax i þ ay j þ az k ax ¼ i A
ð1aÞ
jwi ¼ c1 j/1 i þ c2 j/2 i þ c3 j/3 i þ c1 ¼ h/1 jwi
ð1bÞ
In the above equations, jwi stands for the molecular electronic wave function; while j/k i represents that of kth resonance electronic configuration. The coefficients ck are then determined by h/k jwi, the overlap between original electronic wave functions and kth resonance configuration, which can be computed efficiently by using the formula introduced by Löwdin [6]. It should be noted that the j/k i components must provide an orthonormal set to rely on h/k jwi as resonance weight. Otherwise, h/k jwi values can be considered as a criterion of how much a resonance structure resembles the corresponding quantum mechanical electronic wave function. In a work by the present author [7], the NBOs are used to construct electronic wave function of the resonance structures and their weights are determined as h/k jwi after some mathematical manipulations. Karafiloglou [8], has also taken advantage of Moffitt’s theorem [9] to expand the molecular electronic wave function into determinants composed of NBOs. Indirect superposition of the electronic structure has also been considered as natural resonance theory (NRT) [10–12] in which the density matrices are expanded instead of the wave functions. In the next section, computational details of the procedure is presented and Section 3 discusses resonance structures for four small CNTs. 2. Computational details The electronic wave functions are obtained at HF/6-31G level of theory utilizing GAUSSIAN 03 suite of programs [13], where the C–C and C–H bond lengths were fixed at 1.4 and 1.0 Å, respectively. The resonance wave function, jwres i, can be obtained as a finite
21
S.M. Azami et al. / Chemical Physics Letters 491 (2010) 20–22
sum over the wave functions of orthogonalized resonance structures, j/0k i, according to the following equation:
jwres i ¼ N
X
ci j/0i i
the values of h/k jwi are adopted to compare contributions of different resonance structures of four simple CNTs which are discussed in the next section.
ð2Þ
i
3. Results and discussion
where ci ¼ h/0i jwres i and N is the normalization constant. For closedshell wave functions, h/k jwi can be computed by Löwdin general formula [6] as h/k jwi ¼ jhvki jvj ij2 , where vki and vj are ith and jth molecular orbitals of j/k i and jwi, respectively. In the present work,
Chemical structures of the CNTs, excluding the dangling hydrogen atoms, are shown in Fig. 1 and labeled by I–IV. System I is one ring of a small CNT in diameter which is claimed to be the smallest
Fig. 1. The chemical structure of CNTs together with their labels used in the text (the hydrogen atoms are not shown).
Fig. 2. A typical NBO in system II, representing a p bond.
Table 1 Resonance patterns for the CNTs.a IV
III
II
a
b
c
d
e
f
g
h
i
a
Only determining (double) bonds are shown to make the patterns clear, and the other double bonds can be uniquely depicted.
I
22
S.M. Azami et al. / Chemical Physics Letters 491 (2010) 20–22
Table 2 The values of CNT
I II III IV a
h/k jwi , h/a jwi
relative overlap between kth resonance wave function and individual resonance configurations with respect to h/a jwi.
Resonance configuration a
b
c
d
e
f
g
h
i
1.0000 1.0000 1.0000 1.0000
1.3499 0.0000a 2.0408 0.5500
0.0000 0.1235 1.5714 0.8000
0.0000 0.2881 0.0000 0.0004
0.0000 0.3786 0.0000 0.0005
0.0000 0.0006
0.0000 0.1178
0.0000 0.2678
0.0000 0.3417
The numerical error is ±1.0 108.
Table 3 Occupancies of p and p* orbitals for II and benzene according to NBO results. II
Benzene
a
b
1.59p 0.41p*
1.59p 0.41p*
1.67p 0.33p*
0.67p 1.33p*
diameter CNT [14]. The interaction of hydrogen molecule with such small CNT is recently analyzed by the present author, theoretically [15]. The resonance structures of the CNTs are displayed in Table 1. To keep clarity, several points are considered in this table: (i) equivalent resonance structures are excluded; (ii) the resonance schemes are unrolled and atoms located at the two ends are identical; (iii) only determining bonds are shown, i.e., the other double bonds can be uniquely identified. One of NBO p bonds corresponding to the double bonds in Table 1 is plotted in Fig. 2 to visualize its nodal shape and qualitative distribution. The other NBO p bonds have almost the same shape and distribution. As Fig. 2 shows, the NBO highly resemble a chemist’s imagination of p bond and can thus be safely considered as building blocks of resonance electronic configurations. The resonance structures depicted in Table 1 include both Kekulé and Dewar-type structures. Although there exist also many other resonance structures such as long-bonded and ionic configurations, we have limited the discussion to Kekulé and Dewar-type structures. However, a large number of structures are necessary to form a complete superposition of resonance structures. In order to compare resonance components, j/a i is considered as reference configuration as it has contributed to all of the CNTs MO wave function. Therefore,
h/k jwi h/a jwi
values, relative overlap of j/k i
configuration, are reported in Table 2. According to these values, the three Dewar-type resonance structures in I has no contribution to the CNT’s p system. For III, resonance structure b has significantly higher overlap with jwi than structure a. An interesting result is that resonance structure b has absolutely no contribution to the p system of II, as it possesses zero overlap with its quantum mechanical electronic wave function. Such failure of Lewis resonance structures has not yet been observed for Kekulé structures in polycyclic conjugated systems. Therefore, classical resonance structures may not be valid for all CNTs and may lead to unreal results. From quantum mechanical standpoint, such resonance structure corresponds to a mixture of excited states. To prove this, Moffitt’s theorem can be helpful. According to this theorem, any arbitrary Slater determinant can be rewritten as certain linear combination of Slater determinants composed of canonical molecular orbitals of the same system and basis set. Therefore, electronic state of resonance structure b in II can be decomposed as Eq. (3):
j/b i ¼ c0 jw0 i þ
X i
ci jwi i
ð3Þ
where jw0 i and jwi i denote ground state and ith excited state, respectively. Since c0 = 0 for this case, electronic state of resonance structure b in II will be a mixture of excited electronic states. The mixing coefficients, {ci}, can be determined by either Moffitt’s theorem or Löwdin general formula. Another interesting result is the high overlap values of Dewar-type structures in II and IV. For IV, the relative overlap of structure i competes with b which is a Kekulé-type structure. The results for II and IV warn a chemist that Dewar-type structures might not be ignorable in interpreting of chemical structures including p system. It is interesting to compare IIb resonance structure with Dewar benzene in which an unconventional bond connects two carbon atoms. If this bond is of r-type, such resonance configuration has no contribution to the benzene’s ground state, i.e., h/Dewar jwi ¼ 0. Note that this result is evident as the unconventional bond is orthogonal to all MOs of benzene’s p system with respect to symmetry consideration of molecular orbitals. The other choice for the unconventional bond is a p-type bond. Such bond has been widely studied in the literature and its role is well explored in the benzene’s aromaticity [16]. Table 3 displays occupancies of p and p* orbitals for II (a, b), Kekulé and Dewar resonance structures in minimal basis set. As Table 3 shows, occupancy of p and p* orbitals for IIa and IIb are similar with those of Kekulé in benzene. However, the bond connecting carbons 1 and 4 in Dewar structure has occupancy of p* higher than the corresponding p by 0.66 which is in agreement with the fact Dewar structure has low weight in benzene’s p system. Also, unlike Dewar structure in benzene, the Dewar-type resonance structures in II contribute significantly to the ground state. Finally, it seems that there exists a symmetry-related phenomenon which is able to interpret both vanishing resonance IIb and large contribution of Dewar-type structure to the system’s resonance. Acknowledgements The authors would like to acknowledge the reviewers for their comments on the necessity of considering Dewar-type resonance structures and their contribution to the electronic ground states. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
H. Weinstein, R. Paunz, M. Cohen, Adv. At. Mol. Phys. 7 (1971) 97. J.M. Foster, S.F. Boys, Rev. Mod. Phys. 32 (1960) 300. C. Edmiston, K. Ruedenberg, Rev. Mod. Phys. 34 (1963) 457. A.E. Reed, L.A. Curtiss, F. Weinhold, Chem. Rev. 88 (1988) 899. J.P. Foster, F. Weinhold, J. Am. Chem. Soc. 102 (1980) 7211. P.-O. Löwdin, Phys. Rev. 97 (1955) 1474. A.H. Pakiari, S.M. Azami, Int. J. Quantum Chem. 108 (2008) 219. P. Karafiloglou, J. Comput. Chem. 22 (2001) 306. W. Moffitt, Proc. R. Soc. A218 (1953) 486. E.D. Glendening, F. Weinhold, J. Comput. Chem. 19 (1998) 593. E.D. Glendening, F. Weinhold, J. Comput. Chem. 19 (1998) 610. E.D. Glendening, J.K. Badenhoop, F. Weinhold, J. Comput. Chem. 19 (1998) 628. M.J. Frisch et al., GAUSSIAN 03, Gaussian, Pittsburgh, PA, 2003. H.Y. Peng et al., Appl. Phys. Lett. 77 (2000) 2831. S. Fakhraee, S.M. Azami, J. Chem. Phys. 130 (2009) 084113-1. L.J. Schaad, B. Andes Hess, Chem. Rev. 101 (2001) 1465.