Research: Science and Education
Getting the Weights of Lewis Structures out of Hückel Theory: Hückel–Lewis Configuration Interaction (HL-CI) Stéphane Humbel UMR 6180 - Chirotechnologies: Catalyse et Biocatalyse, CNRS/Université Paul Cézanne (Aix-Marseille III), Campus St Jérôme Case A 62, 13397 Marseille Cedex 20, France;
[email protected] One of the key chemical concepts relating to Lewis structures is resonance. This concept is made clear when several Lewis structures (resonance forms) have to be invoked to reach the correct description of a system (1). The resonance between the different Lewis structures is widely taught, for instance, through the case of the amide (Figure 1) with an electron donor (N) and an electron acceptor (O). Moreover, the planar geometry of the amide can only be explained by resonance structure II that has some weight in the total description. However, one needs appropriate ab initio methods and related codes (vide infra) to obtain the weight of ∼34%. Without the appropriate quantum chemistry codes, the chemist is left with qualitative rules that help him or her intuition when predicting the major structure (2). They can be summarized as Rule 1: In any case, the more the octet rule is fulfilled, the better (3). Rule 2: The best structure is consistent with the electronegativity of the atoms (e.g., a negative charge is best on the most electronegative atom), unless rule 1 is not obeyed. Rule 3: The less charge separation, the better, unless rule 1 is not obeyed.
The resonance in π systems is also a valuable tool to predict or explain a variety of behaviors of chemical systems, such as the location of the partial charges, specific geometrical pattern, selectivity in organic chemistry, and so forth. On the one hand, students possess an understanding of the electronic π delocalization through resonance between Lewis structures (4). On the other hand, this concept is widely taught through the Hückel theory, which gives access to resonance energy, aromaticity, partial charges, and fractional bond orders (5). Hückel theory is a cornerstone for the molecular orbital concept, and despite its relatively poor accuracy, it provides a fantastic playground for teaching quantum chemis-
A
B
Figure 1. Lewis structures for methanamide: (A) resonant hybrid and (B) the two resonance structures.
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try basics and the use of quantum chemistry for understanding chemistry. If we look at the resonance concepts, the Hückel theory gives the opportunity to evaluate the resonance energy by comparing the energy of a system to that of its major structure (determined by following the qualitative rules). This is shown for methanamide in Figure 2. Using Hückel theory, the delocalized system is found at Etot = 4α + 6.55β. The major Lewis structure, I, has two electrons in the lone pair of the nitrogen (at α + 1.37β) and the two others in the πC⫽O orbital, computed separately at α + 1.65β. The energy of the structure is thus easily evaluated at EI = 4α + 6.04β, and a resonance energy of 0.51β is found by the difference between Etot and EI. Other quantities, such as bonding energies, can also be computed and give students a sense of the electronic delocalization concept. However, it would be particularly appealing to be able to further link this energetic approach of the delocalization to the resonance between the Lewis structures, that is, to give the weights of each structure as a percentage. To do so, a variety of tools exist in standard quantum chemistry packages. One of these tools is the natural resonance theory (NRT) (6) procedure. Implemented in the natural bond orbital code NBO 5.0 (7), it is easily available in GAUSSIAN (8). After performing a geometry optimization to find the equilibrium geometry, the NRT program can be used to obtain the weights of the structures. For the amide, 66% is found for structure I and 34% for structure II (Figure 1).1 Despite its simplicity, this kind of computation cannot be done in a standard classroom. It requires computers and some knowledge of how to use the tools installed on them.
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Figure 2. Resonance energy of hybrid I versus total energy of methanamide calculated using Hückel theory.
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3). The diagonal terms of this HL-CI determinant are the energy difference between that of the corresponding Lewis structures and that of ψtot. They can be labeled ∆EI = EI − Etot and ∆EII. The energy Etot of the requested solution ψtot can be calculated independently with Hückel theory. Hence, in the HL-CI determinant every term is known but the offdiagonal term HI-II. The following expression is then used to obtain HI-II: H I− II = ± ∆E I ∆E II
(2)
Since ψI and ψII are orbitals in the Hückel theory, their coefficients CI and CII (eq 1) can be determined using one of the secular equations of the HL-CI determinant (eqs 3 or 4) and the normalization (eq 5): Figure 3. (Top) Hamiltonian matrix of the system written in the basis of the two Lewis structures. The corresponding overlap matrix is the identity matrix. (Bottom) Secular determinant of the HL-CI approach.
C I ∆E I + C II H I− II = 0
(3)
C I H I − II + C II ∆E E II = 0
(4)
CI
2
+ C II
2
= 1
(5)
As the weights are the square of the coefficients, wI = CI2 and wII = CII2, they can be obtained directly. In this article we propose a simple method, based on energies obtained with the Hückel theory, to compute the weights of the structures. We restrict the method to only two structures because this is straightforward to achieve. Larger interactions are beyond the scope of classroom use and will be published elsewhere (9). Hückel–Lewis Configuration Interaction Method (HL-CI)
Principle of HL-CI If a system, described formally with ψtot, is considered as a resonance between the Lewis structures I and II, each of them formally described with ψI and ψII, one should be able to write ψtot as a linear combination of ψI and ψII, (1)
ψ tot = C I ψ I + C II ψ II
where CI and CII are real coefficients and are related to the weights of the structures. Following the Coulson–Chirgwin definition (10), the weight of a structure i in a wave function is wi = Σj(Ci Cj Sij ), where j runs over all of the structures ψj, including ψi, and Sij is the overlap between the structures. If we consider that the Lewis structures, all the ψi, form an orthonormal basis (11), the above formula is reduced to wi = Ci2. Thus to obtain Ci2 is the final goal of the present method. To do so, the Hamiltonian matrix of the system is first written in the basis of our Lewis structures, and the corresponding overlap matrix is the identity matrix (Figure 3). The diagonal terms of the Hamiltonian matrices, HI-I and HII-II, are the energies of the Lewis structures EI and EII. They can be computed in the Hückel theory framework. (As some caution must be used here, these computations will be discussed in the next section.) Following standard quantum chemistry techniques, the two matrices H and S are used to build a secular determinant |H − E S| = 0. Moreover, this determinant can state that ψtot, of energy Etot, is a solution of the Schrödinger equation, hence |H − EtotS| = 0 (Figure www.JCE.DivCHED.org
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Obtaining the Energy of the Lewis Structures A Lewis structure has localized bonds and lone pairs. Its energy can be calculated easily by computing the energy of each concerned orbital, as we did in the introduction for EI. However, some caution is necessary because we are interested in energies that will be compared to Etot. Hence, we need to have a well-balanced treatment for all the structures and for the resonant hybrid. The same set of parameters for each atom must be used throughout.2 The C⫽N double bond of structure II, for instance, must be calculated with a two-electron nitrogen to be consistent with structure I and with the resonant hybrid. Another way of calculating the energy of each structure is to zero out unwanted interactions in the Hamiltonian matrix of the resonant hybrid in such a way that an appropriate localization of the electron pairs is obtained (Figure 4). The secular determinant, for structure I, for instance, is built from that of the complete system by zeroing out H23 (or HCN), in bold in Figure 4. Similarly, for structure II it is the H12 (or HC⫽O) term that is set to zero. This leads to two sub-determinants for each structure. It is a general rule for Lewis structures that only 2 × 2 (bonds) or 1 × 1 (lone pairs) sub-determinants have to be solved, which is easy for the students. The complete resolution for the methanamide is shown in Figure 4. From here we use the usual substitution x = (α − ε)兾β. The secular determinant of the resonant hybrid is first used to obtain the energy of the orbitals. With four electrons in the system, the first two orbitals are populated, and the value of Etot is obtained. This is a standard Hückel calculation. To get the energies of the Lewis structures (EI and EII), each secular sub-determinant is solved. For instance for structure I, we get two sets of solution, that of N alone, εN = α + 1.37β (which has two electrons in it), and that of C⫽O with the πC⫽O orbital at επC⫽O = α + 1.65β (which also has two electrons). This is strictly equivalent to the resolution made in the introduction of this article (Figure 2). With the ap-
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Figure 4. Complete calculation of the weights following the HL-CI approach on the methanamide test case.
propriate electron occupancy, we get the value for EI: 4α + 6.04β. We proceed in a similar manner for structure II, which must have two electrons in πC⫽N and two in the pO orbital.
Obtaining the Weight of a Lewis Structure The remainder of Figure 4 details the actual HL-CI part. First we want to determine the off-diagonal term of the HLCI determinant, HI-II. Following the general principle exposed earlier, this term is the square root of the product ∆EI∆EII (eq 2). Of course two roots can be envisaged, a positive (᎑0.71β) and a negative one (+0.71β). Like the Hückel parameter β, the off-diagonal term must be negative, and only that solution is to be retained, +0.71β. This value is then used in the secular equations of the HL-CI determinant. The two equations are redundant (as is the case in Hückel as well) and the normalization constraint must be used to obtain the weights of the structures, CI2 = 66% and CII2 = 34%. These values are coincidentally equal to the ab initio results (Figure 1). In general, one should not expect those two methods to match so well. 1058
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Typical Examples Here we applied HL-CI to a few simple examples and tested the results against that of the ab initio NRT method. Because of the underlying Hückel approximation, one should not expect great accuracy; however, the aim of this part is to show that no large errors are encountered that would spoil the student’s chemical knowledge. If any significant error occurs, some understanding of it should be given to the student. As for the ab initio calculations, each molecular structure has been optimized with Cs symmetry constraint and characterized as a minimum by second derivative analysis. The level of calculation is B3LYP and the basis set 6−31+G(d) ensures ionic structures to be appropriately handled. With these geometries, two NRT calculations were done, one at the B3LYP兾6−31+G(d) level, and one at the Hartree–Fock (HF) level with the same basis set. These two series of calculation should help establish the variation of the weights within the NRT method. The values obtained from these two cal-
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Research: Science and Education Table 1. Tested Examples Type
Structures
Energies
Details
HL-CI
NRT-a
NRT-b
Amide
Etot = 4α + 6.55β
HI-II = +0.71β
(hN:=1.37, kCN:=0.89
EI = 4α + 6.04β
∆EI = –0.51β
66
66
69c
EII = 4α + 5.56β
∆EII = –0.99β
34
34
31c
hO.=0.97, kCO.=1.06)
d
Butadiene
Etot = 4α + 4.47β
HI-II = +1.08β
EI = 4α + 4.00β
∆EI = –0.47β
84
78
82
EII = 4α + 2.00β
∆EII = –2.47β
16
22
18
Acrolein
Etot = 4α + 5.81β
HI-II = +0.97β
(hO.=0.97, kCO.=1.06)
EI = 4α + 5.30β
∆EI = –0.50β
79
85
90
EII = 4α + 3.94β
∆EII = –1.87β
21
15
10
Enamine
Etot = 4α + 5.08β
HI-II = +0.70β
(hN:=1.37, kCN:=0.89)
EI = 4α + 4.74β
∆EI = –0.34β
81
79
83
EII = 4α + 3.62β
∆EII = –1.46β
19
21
17
α-Imino carbonium
Etot = 2α + 3.18β
HI-II = +0.82β
(hN.=0.51, kCN.=1.02)
EI = 2α + 2.61β
∆EI = –0.57β
68
68
73
EII = 2α + 2.00β
∆EII = –1.18β
32
32
27
Formic acid
Etot = 4α + 7.70β
HI-II = +0.51β
(hO.=0.97, kCO.=1.06
EI = 4α + 7.48β
∆EI = –0.22β
85
73
78
hO:=2.09, kCO:=0.66)
EII = 4α + 6.50β
∆EII = –1.20β
15
26
22
Enolate
Etot = 4α + 6.32β
HI-II = +0.50β
(hO:=2.09, kCO:=0.66)
EI = 4α + 6.18β
∆EI = –0.14β
93
60
63
EII = 4α + 4.56β
∆EII = –1.76β
7
40
37
Enol
Etot = 4α + 6.32β
HI-II = +0.50β
(hO:=2.09, kCO:=0.66)
EI = 4α + 6.18β
∆EI = –0.14β
93
86
90
EII = 4α + 4.56β
∆EII = –1.76β
07
14
10
a-
b-
NRT calculations with B3LYP/6-31+G(d) geometries and wave functions. NRT calculations with B3LYP/6-31+G(d) geometries and HF/6-31+G(d) c d wave functions. Data for resonance structure I is given in the top row and data for structure II in the bottom row. Heteroatoms, X, are incorporated as αX = α + hXβ and βXY = kXYβ, where the default values of the parameters, hX and kXY, are taken from Van–Catledge’s list.
culations differ by about 5%. In acrolein (Table 1) the major structure is found at 85% with the B3LYP wave function, while it is as large as 90% using HF. In all the cases (Table 1), we used the standard Hückel parameters (12) that best describe the expected major Lewis structure, according to the qualitative rules given in the introduction. Except for one case, the enolate, an unexpectedly good accuracy is encountered. In most cases, the difference between the HL-CI and the NRT results at the B3LYP level is not much larger than that between the ab initio levels themselves. Except for the enolate, the biggest difference is about 10%: for formic acid. The major structure is found at 85% in HLCI versus 73% (or 78%) with NRT at the B3LYP level (or HF). Such a difference is satisfactory, especially for teaching purposes. The case of enolate is somewhat surprising with respect to the other results. Structure I is too favored, 93%, compared to the ab initio results (60%). Insight to this error can be gained by examining the enol case. With the current Hückel parameterization both the enol and the enolate have a two-electron oxygen. They have the same Hamiltonian matrix and the same number of π electrons. The results of the Hückel theory are thus identical for both systems. The HLCI method being only an additional layer on top of Hückel theory, all our results are also identical for enol and enolate www.JCE.DivCHED.org
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(93% for structure I). Now, if we look at the results for enol with the NRT calculation at the B3LYP level (or HF), we find they are close to the HL-CI results. The major structure I is indeed found at 86% (or 90%) versus 93% for HL-CI. Thus one shall conclude that the parameters for a two-electron oxygen atom in Hückel (αO: = α + 2.09β, βCO: = 0.66β) are appropriate to describe the neutral O of the enol but not the anionic O− of the enolate. Such an anionic two-electron O− should be re-parameterized as a better electron-donating atom, which is beyond the scope of this article (13).
Teaching HL-CI The mathematical skills required for HL-CI are not different from those used to handle the Hückel theory. Because the additional computations of the energies of the structures only concern small 2 × 2 and 1 × 1 sub-determinants, the HL-CI method is even restricted to the simplest mathematical cases. The technique of cleaning up the Hamiltonian matrix of the resonant hybrid is frequently taught to get the resonance energy. This cleaning of a unique Hamiltonian matrix is also convenient to make sure a unique set of parameters is used throughout. To show the requirement of having the same parameterization in the Lewis structures and in the resonant hybrid, one can use the second structure of the amide (II). Because
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Conclusions
Figure 5. Energy results obtained using different hybridization (compare with Figure 4).
we have there an sp2 hybridized nitrogen atom and an sp3like oxygen (Figure 5), one might use the parameters of a one-electron nitrogen and of a two-electron oxygen to build the Hamiltonian matrix and the secular determinant of this structure. Doing so, an energy of 4α + 6.79β is obtained (Figure 5) that is about 0.25β lower than Etot. Of course one can not consider any of the resonant contributors to be more stable than the resonant hybrid, and such a parameterization must be rejected to the benefit of a common parameterization for all the structures, identical to that of the resonant hybrid, as explained above. From a deeper quantum chemistry point of view, the technique uses a difficult concept for the beginning students; that is, One can write the Hamiltonian matrix in the basis of something else than orbitals: that of electronic configurations. This refers to the configuration interaction (CI) technique, which is traditionally taught in specialized courses. Moreover, we use the CI technique “upside down”: We use the result (Etot) to evaluate the missing matrix element (HI-II), while the CI technique normally aims at finding Etot from the matrix. However, we believe that restricted to such a simple twoconfiguration interaction, with meaningful configurations such as Lewis structures, much understanding can be gained without a tremendous effort. The method can also be an excellent introduction to the CI technique used in advance courses on quantum chemistry, provided the students have some knowledge of resonance in organic chemistry. If a student were to ask for a mathematical expression of the ψi’s, which is not necessary in the present method, a good enough answer would be to consider the Hartree products of the electron-pairs rather than the more accurate Slater determinants. For instance the ψi’s of the amide are given in eqs 6 and 7, with bars to indicate a spin-down electron: ΨI(1, 2, 3, 4) = π C
O(1) π C O(2 ) p N(3 ) p N( 4 )
(6)
ΨII(1, 2, 3, 4) = π C
N(1) π C N(2 ) p O(3) p O( 4 )
(7)
and
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The Hückel–Lewis CI technique extends to the Hückel theory the field of the resonance between Lewis structures. Of course, as in usual Hückel calculations, resonance energies can be obtained, but HL-CI gives an easy access to the weight of each structure. Although we obtained good accuracy, one should not forget that the technique uses the Hückel parameterized method, which is somewhat inaccurate. An accuracy of ∼10% compared to the ab initio methods (with the NRT method) makes the method fully acceptable for teaching. It appears that some caution is required when considering charged systems such as the enolate. This inaccuracy is inherent to the (simple) set of parameters we are using and is not specific to the HL-CI method. It could be solved with appropriate parameters for the two-electron anionic O−, that does not exist in the parameterization we are using (12a). From the quantum chemistry point of view, the technique uses additional knowledge, which is writing the Hamiltonian matrix on the basis of Lewis configurations. Although this might seem difficult for beginning students, the fact that each configuration is meaningful should help teaching such a technique in most universities. The HL-CI method does not require any mathematical skills beyond those necessary to handle the Hückel theory. It is even restricted to the simplest mathematical cases (2 × 2 sub-determinants at most). This permits use of the HL-CI method for constructing quite interesting new problems, which can be easy to solve. Finally, we believe HL-CI opens a new playground for teaching quantum chemistry and its use to understand organic chemistry. Acknowledgments The development of this method owes much to several discussions with M. Linares over the past two years. The author is also particularly grateful to his colleagues from Marseille (V. Ledentu, D. Hagebaum-Reignier, N. Ferré, and F. Fotiadu) and Paris (B. Braïda, P. C. Hiberty, and F. Volatron) for their judicious remarks and friendly encouragement. A. S. Olive is gratefully acknowledged for her friendly reading of the manuscript. Notes 1. The NRT code can find numerous resonance structures, but it can also be constrained to specific bonding schemes. If the interaction between I and II is explicitly requested, these weights are found (B3LYP/6-31+G(d)). 2. Different parameters for the different structures would lead to an unbalanced treatment, as will be seen later.
Literature Cited 1. A fantastic contribution on the subject is Wheland’s book: Wheland, G. W. Resonance in Organic Chemistry; Wiley: New York, 1955. 2. (a) Vollhardt K. P.; Schore, N. E. Organic Chemistry, 2nd ed.; W. H. Freeman and Cy: New York, 1994. (b) Marsh J. Advanced Organic Chemistry, 4th ed.; Wiley: New York, 1992. 3. The predominance of the octet rule will be valid all over this article, which deals only with situations where the octet rule
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4.
5. 6.
7.
8.
is obeyed. For specific comments on the subject see for instance (a) Suidan, L.; Badenhoop, J. K.; Glendening, E. D.; Weinhold, F. J. Chem. Educ. 1995, 72, 583–586. (b) Weinhold, F. J. Chem. Educ. 1999, 76, 1141–1146. (c) Weinhold, F. J. Chem. Educ. 2005, 82, 526–527 and the reply from Purser, G. H. J. Chem. Educ. 2005, 82, 528–529. (d) Purser, G. H. J. Chem. Educ. 1999, 76, 1013–1018. (e) Purser, G. H. J. Chem. Educ. 2001, 78, 981–983. For modern valence bond computations of similar resonance see, for instance, (a) the bond distorted orbitals (BDO) method: Mo, Y.; Lin, Z.; Wu, W.; Zhang, Q. J. Phys. Chem. 1996, 100, 11569–11572. (b) the valence bond B.O.N.D. (VBB) method: Linares, M.; Braïda, B.; Humbel, S. J. Phys. Chem. A 2006, 110, 2505–2509. See, for instance, (a) LoBue, J. M. J. Chem. Educ. 2002, 79, 1378. (b) Hanson, R. M. J. Chem. Educ. 2002, 79, 1379. (a) Glendening, E. D.; Weinhold, F. J. Comput. Chem. 1998, 19, 593–609. (b) Glendening, E. D.; Weinhold, F. J. Comput. Chem. 1998, 19, 610–627. (c) Glendening, E. D.; Badenhoop, J. K.; Weinhold, F. J. Comput. Chem. 1998, 19, 628–646. NBO 5.0. Glendening, E. D.; Badenhoop, J. K.; Reed, A. E.; Carpenter, J. E.; Bohmann, J. A.; Morales, C. M.; Weinhold, F. Theoretical Chemistry Institute, University of Wisconsin, Madison, WI, 2001; http://www.chem.wisc.edu/~nbo5 (accessed Mar 2007). Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Zakrzewski, V. G.; Montgomery, J. A., Jr.; Stratmann, R. E.; Burant, J. C.; Dapprich, S.; Millam, J. M.; Daniels, A. D.; Kudin, K. N.; Strain, M. C.; Farkas, O.; Tomasi, J.; Barone, V.; Cossi, M.; Cammi, R.; Mennucci, B.; Pomelli, C.; Adamo, C.; Clifford, S.; Ochterski, J.; Petersson, G. A.; Ayala, P. Y.; Cui, Q.; Morokuma, K.; Malick, D. K.; Rabuck, A. D.; Raghavachari, K.; Foresman,
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9. 10. 11.
12.
13.
J. B.; Cioslowski, J.; Ortiz, J. V.; Stefanov, B. B.; Liu, G.; Liashenko, A.; Piskorz, P.; Komaromi, I.; Gomperts, R.; 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.; Andres, J. L.; Gonzalez, C.; Head-Gordon, M.; Replogle, E. S.; Pople, J. A. Gaussian 98, Revision A.11; Gaussian, Inc.: Pittsburgh, PA, 1998. Hagebaum-Reignier, D.; Girardi, R.; Carissan, Y.; Humbel, S. J. Mol. Struct. (THEOCHEM), accepted for publication. Chirgwin, B. H.; Coulson, C. A. Proc. Roy. Soc. Lond. A. 1950, 201, 196–209. In ab initio valence bond calculations such as the bond distorted orbital (BDO) (ref 4a) or our valence bond B.O.N.D. (VBB) (ref 4b), the structures of course do overlap. However, this kind of approximation is frequently used in so-called “effective Hamiltonian” techniques. It is used as well for orbitals in Hückel Theory. The β off-diagonal terms then compensate the error introduced by such a drastic approximation. Similarly here, the off-diagonal term of the Hamiltonian will have to compensate the error we are introducing with this approximation. A similar approach has been used for instance for describing short strong hydrogen bonds in S. Humbel J. Phys. Chem. A 2002, 106, 5517–5520. (a) Van-Catledge F. A. J. Org. Chem. 1980, 45, 4801–4802. (b) These parameters are used in several codes such as HMO plus, Wissner, A. Tetrahedron Computer Methodology 1990, 3, 63–71 and in SHMO, Rauk, A. and Canning, R. HMO plus (Macintosh): http://www.stolaf.edu/depts/chemistry/courses/ toolkits/247/hmo-plus-203.hqx SHMO (java): http:// www.chem.ucalgary.ca/SHMO/ (accessed Mar 2007). Such a problem with Hückel parameters for charged systems is discussed for instance in Trong Anh, N. Orbitales Frontières; InterEditions/CNRS Editions: Paris, 1995; in French.
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