On quadratic bondorder decomposition within molecular orbital space
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Abstract
A simple method of analysing and localization of canonical molecular orbitals for particular chemical bond using the MOresolved bondorder decomposition scheme is presented. An alternative definition of classical bond order orbitals is provided and links to communication theory of the chemical bond are outlined and briefly discussed. The introduced procedure of decomposition of quadratic bond orders allows one to analyse two as well as three center chemical bonds within the framework of the same theory.
Keywords
Wiberg’s index Bond order decomposition Localized orbital Molecular orbital Scattering operator Communication theory1 Introduction
The concept of chemical bond order [1] is deeply embedded in chemical intuition. Bond multiplicities are widely used in evaluating of electronic structure of molecules despite the fact, that they are not observables in quantummechanical sense. The main reason of their popularity is that they provide a very compact tool for rationalizing the results of theoretical calculations.
However, in typical ab initio calculations canonical MOs of poliatomics are delocalized over the whole molecule and thus one MO can simultanously reveal bonding character with respect to particular chemical bond and antibonding character with respect to another. There is a variety of computationaly efficient methods of localization and decomposition of the chemical bonds into \(\sigma ,\pi \) and \(\delta \) components: tranformation MO \(\rightarrow \) LMO by optimalization the expectation value of an appropriate operator [4, 5, 6], transformation MO \(\rightarrow \) NBO into the natural bond orbital (NBO) representation [7], transformation MO \(\rightarrow \) LOBO into the localized orbitals of bond order (LOBO) [8], symmetry based bondorder decomposition techniques [9], etc. These methods allow one to gain a very compact and intuitive picture of typical chemical bonds but sometimets (especially in the case of threecenter twoelectron chemical bonds, delocalized bond, very weak atomic interactions, etc.) they can lead to unreasonable results.
In this paper we introduce a simple method of analysing and localization of canonical molecular orbitals (CMO) for particular chemical bonds using MOresolved bond order decomposition scheme and the bond order orbitals (BOO) concept [10]. Also, the relation to the communication theory of the chemical bond (CTCB) [11, 12, 13, 14, 15, 16, 17, 18, 19, 20] is outlined and briefly discussed.
2 MOresolved decomposition
2.1 Electron population \(N_k\)
2.2 Covalency index \(w_{k,l}\)
3 Noiseless MOchannel: the eigenproblem of \(\hat{M}_{XY}\)

stochastic—typical for molecular communication systems with all MOs mutually communicated, i.e. with nonzeroed both, diagonal and offdiagonal elements of matrix of conditionalprobability amplitudes (21),

fully deterministic and noiseless—characteristic for information channels defined by unit matrix of the relevant conditional probabilities (offdiagonal communications zeroed).
Hence, diagonalization of matrix of the operator \(\hat{M}_{XY}\) within the space of all molecular orbitals (occupied and virtual), which is somehow similar to diagonalization of twocenter atomic blocks of the density matrix \({\gamma }\) [10], gives rise to alternative bond order orbitals (BOO) representation. As proved by Jug [10], the classical bond order orbitals appear in pairs with the opposite signs of its eigenvalues and vanishing eigenvalues one observes always when the number basis functions on each of the two atoms \(X\) and \(Y\) is different. The sum of all eigenvalues is equal to zero and the overall bond order can be obtained using the Mulliken overlap criterion [10] or—alternatively, the vector projection weighting procedure [31].
4 Numerical results
To examine how the proposed bondorder decomposition procedure manages with several representative types of chemical bonds we used wavefunctions calculated at RHF as well as DFT/B3LYP [34] theory levels within MINI [35] and ccpVDZ [36] basis sets, respectively, using the standard ab initio quantum chemistry package, Gamess [37]. The bond order orbitals were generated by special program written by authors and visualized (after deorthogonalization) using the molecular visualization program, Molekel [38].
Comparison of eigenvalues from diagonalization of twoatomic blocks of matrix \(\varvec{\gamma }\) and eigenvalues of operator \(\hat{M}_{XY}\) within CMO space for water molecule
Matrix  Atom/bond  Eigenvalues  

\(\lambda _1\)  \(\lambda _2\)  \(\lambda _3\)  \(\lambda _4\)  \(\lambda _5\)  
\(\varvec{\gamma }\)  O  2.00  2.00  2.00  1.23  1.20 
\(\varvec{\gamma }\)  H  0.79  0.00  0.00  0.00  0.00 
\(\varvec{\gamma }\)  O–H  0.98  0.98  0.00  0.00  0.00 
\(\mathbf O _X\)  O  2.00  2.00  2.00  0.38  0.36 
\(\mathbf O _X\)  H  0.31  0.00  0.00  0.00  0.00 
\(\mathbf O _{XY}\)  O–H  0.96  0.00  0.00  0.00  0.00 
\(\mathbf V _{XY}\)  O–H  0.96  0.00  0.00  0.00  0.00 

Wibergtype bond orders calculated within Löwdinorthogonalized atomic orbital representation [22],

Wibergtype bond orders calculated within “physically”orthogonalized atomic orbital representation, recently proposed by the authors [39],
 Mayer’s bond orders [3, 25] calculated within nonorthogonal atomic orbital representation. In this particular case the appropriate elements of matrix \(\mathbf O _{XY}\) are defined as follows:where \(\mathbf c _n\) and \(\mathbf c _n\) are the relevant columns of the LCAO matrix.$$\begin{aligned} \left( \mathbf O _{XY}\right) _{m,n} = \sum ^X_k\sum ^Y_l \left( \mathbf C ^o\mathbf C ^{o\dagger }\mathbf S \right) _{k,l}\left( \mathbf c ^o_n\mathbf c ^{o\dagger }_m\mathbf S \right) _{l,k}, \end{aligned}$$(36)
Comparison of Wibergtype bond orders \(W_{XY}\) and its components \(W_{XY}^\sigma , W_{XY}^\pi \) and \(W_{XY}^r\) from BOOresolved decomposition scheme for several molecules at two different computational levels: RHF/MINI and B3LYP/ccpVDZ
Chemical bond  RHF / MINI  B3LYP / ccpVDZ  

\(W_{XY}\)  \(W_{XY}^{\sigma }\)  \(W_{XY}^{\pi }\)  \(W_{XY}^r\)  \(W_{XY}\)  \(W_{XY}^{\sigma }\)  \(W_{XY}^{\pi }\)  \(W_{XY}^r\)  
Wiberg’s bond orders calculated within “geometrically” orthogonalized AO representation  
\(\text{ N }_2\)  3.00  1.00  2.00  0.00  3.42  1.00  2.00  0.42 
HF  0.92  0.92  0.00  0.00  1.21  0.91  0.00  0.30 
CO  2.49  0.95  1.54  0.00  3.06  0.96  1.72  0.38 
NaCl  0.46  0.46  0.00  0.00  1.37  0.60  0.00  0.77 
\(\text{ C }_2\text{ H }_6\) ( C–C )  1.14  1.00  0.00  0.14  1.22  0.92  0.00  0.30 
\(\text{ C }_2\text{ H }_4\) ( C–C )  2.13  1.00  1.00  0.13  2.17  0.93  0.95  0.29 
\(\text{ C }_2\text{ H }_2\) ( C–C )  3.03  1.00  2.00  0.03  3.09  0.97  1.92  0.20 
\(\text{ C }_6\text{ H }_6\) ( C–C )  1.64  0.98  0.56  0.10  1.63  0.90  0.52  0.21 
Wiberg’s bond orders calculated within “physically” orthogonalized AO representation  
\(\text{ N }_2\)  3.00  1.00  2.00  0.00  3.00  1.00  2.00  0.00 
HF  0.79  0.79  0.00  0.00  0.89  0.88  0.00  0.01 
CO  2.03  0.75  1.28  0.00  2.29  0.87  1.42  0.00 
NaCl  0.25  0.25  0.00  0.00  0.31  0.18  0.00  0.13 
\(\text{ C }_2\text{ H }_6\) ( C–C )  0.96  0.96  0.00  0.00  1.00  0.95  0.00  0.05 
\(\text{ C }_2\text{ H }_4\) ( C–C )  1.98  0.98  1.00  0.00  2.03  0.95  1.00  0.08 
\(\text{ C }_2\text{ H }_2\) ( C–C )  3.00  1.00  2.00  0.00  2.71  0.73  2.00  \(\)0.02 
\(\text{ C }_6\text{ H }_6\) ( C–C )  1.41  0.96  0.56  0.11  1.33  0.85  0.55  \(\)0.07 
Mayer’s bond orders calculated within nonorthogonal AO representation  
\(\text{ N }_2\)  3.00  1.42  2.00  0.42  2.93  1.32  2.00  \(\)0.39 
HF  0.86  1.13  0.00  0.27  1.04  1.17  0.00  \(\)0.13 
CO  2.38  1.29  1.50  0.41  2.62  1.22  1.64  \(\)0.24 
NaCl  0.38  0.50  0.00  0.12  0.86  0.52  0.00  0.34 
\(\text{ C }_2\text{ H }_6\) ( C–C )  1.02  1.27  0.00  0.25  1.12  1.29  0.00  \(\)0.17 
\(\text{ C }_2\text{ H }_4\) ( C–C )  2.03  1.28  1.00  0.25  2.13  1.26  0.98  \(\)0.11 
\(\text{ C }_2\text{ H }_2\) ( C–C )  3.01  1.14  2.00  0.13  2.71  1.24  1.98  \(\)0.51 
\(\text{ C }_6\text{ H }_6\) ( C–C )  1.45  1.32  0.60  0.47  1.46  1.26  0.60  \(\)0.40 
The component \(W_{XY}^r\) stands for the sum of all eigenvalues that represent neither \(\sigma \) nor \(\pi \) chemical bonds and usually have quite small values. Thus, \(W^r_{XY}\ne 0\) may indicate a correction from the outer hybrids (e.g. in the case of conjugated \(\pi \)bonds), antibonding orbitals, orthogonalization artemaths, etc.. At the first glance we can observe in Table 2 that decomposition of Mayer’s bond orders leads to very large values of \(W^r_{XY}\) as well as \(\sigma \)eigenvalues that remarkably exceed 1 which practically disqualify this variant of BOdecomposition. Using the orthogonalized basis sets ensures the appropriate eigenvalues of \(\sigma \)components, however \(W^r_{XY}\) can still assume significant nonzero values. Particularly, the overestimated values of Wibergtype bond orders defined within Löwdinorthogonalized extended (but also minimal) basis set are determined in the greater part by large \(W^r_{XY}\) values, even for diatomics. A comparison with the relevant values calculated using “physical” orthogonalization procedure allows one to draw a conclusion that the “geometrical”(involving all canonical molecular orbitals regardless of its occupations [39]) orthogonalization originally proposed by Löwdin significantly affects the quadratic bondorder indices with contributions that do not represent any of pure bond components. Thus, using the recently proposed by the authors orthogonalization procedure seems to be the most suitable for bond order analysis and all further calculations presented in this paper involve the “physically”orthogonalized AOrepresentation.
Comparison of BOcomponents: \(W_{XY}^\sigma , W_{XY}^\pi \) and \(W_{XY}^r\) from BOOresolved decomposition scheme for several cycloalkenes and its aromatic equivalents
Molecule  \(W_{\text{ C=C }}^{\sigma }\)  \(W_{\text{ C=C }}^{\pi }\)  \(W_{\text{ C=C }}^r\)  Molecule  \(W_{\text{ C=C }}^{\sigma }\)  \(W_{\text{ C=C }}^{\pi }\)  \(W_{\text{ C=C }}^r\) 

\(\text{ C }_3\text{ H }_4\)  0.98  0.96  0.03  \(\text{ C }_3\text{ H }_3^{+}\)  0.96  0.45  0.00 
\(\text{ C }_4\text{ H }_4\)  0.95  1.00  0.00  \(\text{ C }_4\text{ H }_4^{2}\)  0.94  0.50  0.26 
\(\text{ C }_5\text{ H }_6\)  0.96  0.93  0.01  \(\text{ C }_5\text{ H }_5^{}\)  0.96  0.60  0.19 
\(\text{ C }_6\text{ H }_8\)  0.97  0.94  0.01  \(\text{ C }_6\text{ H }_6\)  0.96  0.56  0.11 
\(\text{ C }_7\text{ H }_8\)  0.97  0.93  0.02  \(\text{ C }_7\text{ H }_7^{+}\)  0.96  0.48  0.07 
\(\text{ C }_8\text{ H }_8\)  0.97  0.95  0.01  \(\text{ C }_8\text{ H }_8^{2}\)  0.96  0.56  0.20 
\(\text{ C }_9\text{ H }_{10}\)  0.97  0.95  0.01  \(\text{ C }_9\text{ H }_{10}^{}\)  0.96  0.56  0.15 
\(\text{ C }_{10}\text{ H }_{10}\)  0.97  0.94  0.01  \(\text{ C }_{10}\text{ H }_8\)  0.96  0.51  0.12 
Finally, it was of our interest to study bond order orbitals of 3center 2electron chemical bonds. BOO and \(\text{ BOO }^*\) were calculated at RHF/MINI theory level for selected X–Y–Z configurations of atoms in the following molecules: \(\text{ HF }_2^{}\) (F–H–F), \(\text{ B }_2\text{ H }_6\) (B–H–B), \(\text{ C }_4\text{ H }_6\) (\(\text{ C }_1=\text{ C }_2\)–\(\text{ C }_3\)) and \(\text{ C }_6\text{ H }_6\) (\(\text{ C }_1\)–\(\text{ C }_2\)–\(\text{ C }_3\)) and results are presented in Figure 3. As we can see, eigenvectors of matrices \(\mathbf O _{XYZ}\) and \(\mathbf V _{XYZ}\) reliably identify the 3center bond order orbitals in bifluoride anion and diborane molecule. The associated eigenvalues of BOO and \(\text{ BOO }^{*}\) (\(\lambda _{1\sigma _3}\) and \(\lambda _{1\sigma ^{*}_3}\), respectively) in both species indicate a single, covalent bonds which is in agreement with chemical intuition.
5 Conclusions
In this paper we have proposed a scheme of decomposing chemical bond into anew defined the bond order orbitals which constitute an alternative for those originally defined by Jug. Solving the eigenproblem of the scattering operator, originated from the orbital communication theory, allows one to tackle the problem of two and three center bond analysis within the same framework. The new definitions give rise to alternative interpretation of molecular orbital of the valence shell in the case of \(n\)atomic molecule (closedshell electron configuration) as a bond order orbital of \(n\)center2electron chemical bond. The calculations we have carried out clearly show that the new method of bond order decomposition properly copes with identifying the relevant bond components (\(\sigma , \pi \), etc..). An analysis of numerical results brings to light the fact that the most transparent and reliable picture of the chemical bond structure can be obtained within the minimal basis set of “physically” orthogonalized atomic orbitals.
The chemical bond decomposition involving the eigenproblem of the appropriate scattering operator within the orthogonalized MO space is still need of thorough examination and this study is currently in progress.
Notes
Acknowledgments
This work was supported by the TDonation for Young Scientists and PhD Students, Grant No. PSP: K/DSC/000133 (from Department of Chemistry, Jagiellonian University).
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