# Chemical Bonding: The Orthogonal Valence-Bond View

## Abstract

**:**

## 1. Introduction

#### 1.1. Bonding and Bonds

## 2. Covalent Bonding and Chemical Reactions

## 3. OVB Reading of FORS Wave Functions

_{MO}and symmetry of the partially occupied MOs (the active orbitals) and the number of electrons n

_{elec}that can be distributed among them (the active electrons). The information concerning the number of active electrons and active orbitals is abbreviated as CAS(n

_{elec}, n

_{MO}). The fragments are the reactants that are combined to give the product, while the definition of fragments is determined by the idea of how an elementary reaction proceeds and which MOs are needed to describe the reaction. If the choice of the active MOs and of the active electrons is correct, the FORS wave function will correctly describe static electron correlation during the reaction, although a quantitatively correct description needs to include dynamic correlation corrections. That means FORS wave functions do correctly describe the geometries of reactants, products and possible transition structures, but FORS energy differences between them are always too small [12]. The description of all reactions discussed in this paper is based on molecular structures where for fixed inter-fragment distances R, all fragment geometries are fully optimized.

_{MO}active MOs yields n

_{MO}FOs. When these FOs are used as active orbitals, the CSFs made with the FOs will describe the local processes during the reaction. Since the FOs are orthogonal, so are the CSFs made with them. The FO-CSFs describe which local states fragments are in, how these states are coupled and whether the fragments are neutral or ionic. A single diagonalization of the CI -matrix (vide infra) that was constructed with the FO-CSFs yields the energies and the weights of the FO-CSFs in the molecular state function. This analysis, which can be called an OVB reading of a FORS wave function, differs significantly in some aspects from a VB reading based on non-orthogonal orbitals.

_{2}→ C

_{2}H

_{4}

_{2}→ Si

_{2}H

_{4}

_{2}+ H

_{2}→ CH

_{4}

_{2h}symmetry, while the dimerization reaction of two silylenes was studied in D

_{2h}and C

_{2h}symmetry. Insertion of a carbene into the H–H single bond was studied in C

_{s}symmetry.

**Figure 1.**The types of fragment orbital (FO)-configuration state functions (CSFs) used for the insertion reaction. NB, the no-bond configuration, in which the lowest FO of each fragment is doubly occupied; X, a single excitation in one fragment; C, single-charge transfer; DX, double excitations in one fragment; DC, double-charge transfer; CX, single-charge transfer and single excitation; TT, singlet coupling of two local triplets; SS, singlet coupling of two local singlets; TX, double excitation in one fragment and a single excitation in the other; DXC, double excitation and a charge transfer; QX,quadruple excitation; σ and σ

^{∗}are the bonding and antibonding H

_{2}MOs, respectively.

#### 3.1. The Carbene Dimerization

_{2h}symmetry using the 6-31G* basis. The order of the active MOs is σ, π, π

^{∗}and σ

^{∗}. Of the 20 MO-CSFs, only 12 are of the A

_{g}symmetry, and the ethene ground state of A

_{g}symmetry was decomposed in these 12 MO-CSFs.

**Figure 2.**Carbene dimerization. The energy curve for the C–C bond stretching. Energy in Hartrees; bond length in Å.

^{2}π

^{2}|) has the highest weight; at equilibrium geometry, it is close to one. It decreases with increasing C–C distance, and other CSFs, like |2020| = |σ

^{2}π

^{∗2}|, then become important. At r(C–C) = 3.5 Å, the wave function is dominated by five CSFs with weights ≈ 0.2.

**Figure 3.**Carbene dimerization. Energy (

**left**) and weights (

**right**) of the six most important molecular orbital (MO)-CSFs. Energy in Hartrees; bond length in Å.

**Figure 4.**Carbene dimerization. HCH bond angle and CH bond length. Angles in degrees; bond lengths in Å.

**Figure 5.**Carbene dimerization. Energies (

**left**) and weights (

**right**) of the six most important FO-CSFs. Energy in Hartrees; bond length in Å.

#### 3.2. The Silylene Dimerization in D_{2h}

^{2}π

^{2}|) dominates as in the case of ethene; on stretching the Si–Si bond, the weight of the |2020| = |σ

^{2}π

^{∗2}| CSF increases, but at 3.5 Å, the weight of both CSFs drops to zero and the |2002| = |σ

^{2}σ

^{∗2}| CSF reaches a weight of more than 90% within few tens of an Ångstrom (Figure 7). Interpretation of the change of character of the wave function on the basis of MOs is impossible; after all, the change occurs at an Si–Si distance that is more than 50% larger than the equilibrium distance, so the meaning of bonding and antibonding σ and π orbitals is far from clear. The maximum in the total energy curve suggests that the MO switch π ↔ σ

^{∗}does not occur smoothly. Using a larger basis set is no remedy for this problem.

**Figure 6.**Planar silylene dimerization. The energy curve for the Si–Si bond stretching. Energy in Hartrees; bond length in Å.

**Figure 7.**Planar silylene dimerization. Weights of the six most important MO-CSFs. Bond length in Å.

**Figure 9.**Planar silylene dimerization. Energies

**(left)**and weight

**(right)**of the six most important FO-CSFs. Energies in Hartrees; bond lengths in Å.

#### 3.3. The Silylene Dimerization in C_{2h}

**Figure 10.**Non-planar silylene dimerization. The energy curve for the Si–Si bond stretching. Energy in Hartrees; bond length in Å.

**Figure 11.**Non-planar silylene dimerization. Weights of the six most important MO-CSFs. Bond lengths in Å.

**Figure 12.**Non-planar silylene dimerization. HSiH bond angle, SiH bond length and pucker angle. Angle in degrees; bond length in Å.

**Figure 13.**Non-planar silylene dimerization. Energies (

**left**) and weights (

**right**) of the six most important FO-CSFs. Energy in Hartrees; bond length in Å.

#### 3.4. The Insertion of Carbene into H_{2}

_{s}symmetry. For all 20 CSFs, the coefficient is different from zero for symmetry reasons. The parameter R used as the reaction coordinate (Figure 14) is the normal distance of the carbon atom from the molecular axis in H

_{2}. The energy curve of the ground state (Figure 15) is as unspectacular as the energy curves for the carbene dimerization. According to the energy curve, bonding starts at about R = 1.5 Å.

^{1}A

_{1}state; 1.107 Å and 102.4 deg [26]. At R = 1.5 Å, both parameters change in a discontinuous way to values typical for carbene in the

^{3}B

_{1}state. At the same distance, the bond length of the hydrogen molecule doubles, which is impossible when the molecule is bonded. The distances between the carbon atom and the two hydrogen atoms are different in the initial phase of the insertion reaction, i.e., when R ≥ 1.5 Å, the carbon is not pointing to the H

_{2}midpoint; rather, the carbene and the hydrogen molecules approach each other in a parallel fashion, with the carbon atom closer to one hydrogen atom in H

_{2}than to the other. At R = 1.5 Å the two C–H distances are equal, and the carbene has rotated from a parallel to a perpendicular position with respect to the hydrogen molecule, so the symmetry changes from C

_{s}to C

_{2v}.

_{2}. At short distances, where the lowest excited carbene triplet state

^{3}B

_{1}is coupled with the lowest excited H

_{2}triplet state to an overall singlet, TT dominates. Additionally, in between these two regions, around R = 1.5 Å, CSF X is important, where the hydrogen molecule is in its ground state, σ

^{2}, and the carbene is in the excited singlet state

^{1}B

_{1}. This CSF has zero weight at large, as well as at small R values; it appears when bonding starts and disappears again when bonding is finished. Its role is to prepare the carbene in the

^{1}A

_{1}state for bonding. According to basic chemical principles, covalent bonding between fragments is only possible if unpaired electrons are available. Accordingly, the two singlet coupled electrons in the s lone pair orbital cannot contribute to covalent bonding, whereas the electrons in the triplet state can. However, this needs an excitation of one electron from the s to the p orbitals and a spin flip of one electron. The excitation without spin flip is described by X. Spin flip in one fragment needs spin flip in the second to guarantee that the ground-state multiplicity of the molecular wave function is not changed. As soon as this occurs, TT dominates, and X no longer describes a physical process in the system and disappears. The lowest triplet state of H

_{2}is dissociative, which is in accord with the sudden increase in bond length when TT becomes dominant.

**Figure 16.**Carbene insertion into H

_{2}. Dependence of the geometry parameters on R. Energy in Hartrees; R in Å.

_{2}fragment. In X, only the hydrogen σ orbital is doubly occupied, but the two active electrons at the carbon atom occupy the orthogonal 2s and 2p

_{π}AOs (angular correlation), which helps them to avoid each other, and therefore, a contraction of these two AOs is much less unfavorable than when the two electrons are in the same AO. For all other CSFs, contraction of the active FOs is favorable as a result of the two local spin flips.

## 4. What We Can Learn from the OVB Analysis

_{2h}, where both reactants are already in their corresponding lowest high-spin state and where NB therefore has zero weight throughout the whole reaction. This finding is in accord with Lewis’ idea that covalent bonding between reactants is only possible if both fragments have unpaired electrons that can be singlet coupled to an electron pair. In the cases of carbene and silylene, the two electron spins must be coupled to a local high-spin state, which is only possible when they occupy the s and the p lone pair orbitals and are thus angularly correlated. Furthermore, the carbene insertion reaction shows that the high-spin state of the hydrogen molecule is best represented by a triplet excitation of the doubly-occupied σ MO, in which the two unpaired electrons are left-right correlated. The

^{1}A

_{1}state of the carbene reactant in this reaction is an excited fragment state, so the change from the singlet to triplet state is indeed a de-excitation, a process that seems to occur in two steps: First, the fragment goes from the low-lying

^{1}A

_{1}state to the higher lying

^{1}B

_{1}state; the second step is a spin flip in both fragments. The

^{1}B

_{1}state only helps to prepare the singlet carbene for bonding; at the equilibrium geometry, its weight is already zero. That such a CSF is important can only be seen when the whole reaction is investigated, not when an OVB analysis is only made at the equilibrium geometry.

**Figure 17.**Carbene insertion into H

_{2}.

**(Left)**Energies and weights of all FO-CSFs with weight larger than 0.1.

**(Right)**Energy and weights of the three most important FO-CSFs. Energy in Hartrees; R in Å.

## 5. The Differences between Conventional VB and OVB

#### 5.1. The Basis of Conventional VB

_{2}is based on the minimization of the expectation value of the molecular Hamiltonian:

_{HL}has ${}^{1}\text{\Sigma}_{g}^{+}$ symmetry.

_{HL}is given by:

_{I}is the spatial part of the so-called ionic wave function of ${}^{1}\text{\Sigma}_{g}^{+}$ symmetry; Ψ

_{T}is the spatial part of the ${}^{3}\text{\Sigma}_{u}^{+}$ wave action; and Ψ

_{S}is the spatial part of the wave function describing the ionic ${}^{1}\text{\Sigma}_{u}^{+}$ state.

_{HL}describes the stable H

_{2}ground state qualitatively correctly. The energy curve of the ionic wave function does have a local minimum, but at a too-large equilibrium distance, and the stabilization energy with respect to two isolated hydrogen atoms is close to zero. The triplet-energy curve is completely repulsive, and the energy curve of the second ionic wave function lies very high. Nevertheless, the quantitative agreement between the experimental results and the theoretical results obtained with the HL wave function is poor.

_{e}= 0.8679 Å and a dissociation energy of D

_{e}= 304.5 kJ/mol; the best experimental values are r

_{e}= 0.74117 Å and D

_{e}= 456.8 kJ/mol, so the predicted equilibrium distance is 17% too long and the dissociation energy is 33% too small.

_{HL}Ψ

_{HL}+ c

_{I}Ψ

_{I}, the so-called Weinbaum function, is dominated by the HL wave function for all interatomic distances R from very large values to distances smaller than the equilibrium distance, as the absolute value of the CI coefficient of the HL wave functions is always much larger than that of the ionic wave function, |c

_{HL}| >> |c

_{I}|. Using the Weinbaum function, the equilibrium distance becomes even worse, R

_{e}= 0.884 Å, while the dissociation energy improves slightly, D

_{e}= 311.6 kJ/mol. These results are obtained with 1s AOs for the free hydrogen atom; when a basis function χ(r) = Ne

^{−ζr}with a variational parameter ζ is used instead, the results are considerably improved: using a simple HL wave functions and ζ

_{opt}= 1.17 gives R

_{e}= 0.7356 Å and D

_{e}= 364.7 kJ/mol; when ζ is optimized with the energy of the Weinbaum function, the equilibrium distance is R

_{e}= 0.757 Å and D

_{e}= 388.0 kJ/mol. By adding a p polarization function to the 1s AO with the optimized ζ, the results can again be improved: R

_{e}= 0.746 Å and D

_{e}= 397.5 kJ/mol with errors of 0.7% for the bond distance and 13% for the dissociation energy. These results seem to justify the view that it is the HL wave function that describes the major part of the stabilizing processes in the hydrogen molecule; all other contributions give just minor improvements.

_{HL}is frequently related to the form of the wave function in which each AO is occupied by exactly one valence electron, which is thought to exactly represent covalency: Each atom contributing one electron to the bonding electron pair. Ψ

_{HL}is therefore also called a covalent or a neutral wave function, and the same holds true for the triplet wave function Ψ

_{T}. The ionic wave functions, on the other hand, describe a cation/anion pair, they differ in their relative phases and in their symmetry. Ψ

_{HL}and Ψ

_{I}describe mutually exclusive electron distributions at large distances; the two wave functions are orthogonal to each other at large distances. At intermediate and short distances, the eigenfunctions of the (2,2)-CI problem are linear combinations of Ψ

_{HL}and Ψ

_{I}.

#### 5.2. The Non-Orthogonality of VB-CSFs

_{GS}and for the excited state E

_{ES}are shown together with the energies of the two CSFs, E

_{HL}and E

_{I}.

**Figure 19.**The energies of the ground state and the excited state and for the HL and the ionic CSF. Energy in Hartrees; bond length in Å.

_{GS}is nearly identical to the E

_{HL}curve, whereas the energy curves E

_{ES}and E

_{I}only get close for large distances. This means, whereas Ψ

_{HL}describes the ground state very well, the excited state is not dominated by Ψ

_{I}, except at large distances. One can also see that, for short distances, the energy curves of E

_{HL}and E

_{I}get very close and become identical for very small distances. This is due to the fact that Ψ

_{HL}and Ψ

_{I}are not orthogonal to each other; the overlap between the two wave functions is:

_{ion}and Ψ

_{HL}are nearly zero and nearly one, respectively, confirming that the ground state is well represented by the HL CSF alone. At distances smaller than the equilibrium distance when both matrices become singular, the CSFs are becoming linearly dependent and the CI coefficients of the CSFs are approaching infinity. Since the CI vectors of the generalized eigenvalue problem are orthogonal with respect to the metric S, the squares of the CI coefficients are not proper weights of the CSFs; instead, the Chirgwin–Coulson weights are mostly used to describe molecular electronic structures by their fractional ionic character [28]. However, since Ψ

_{HL}describes two neutral hydrogen atoms only at large distances, but the same cation/anion pair as Ψ

_{I}at small distances, such a characterization does not have unique physical relevance. Instead, the question arises: What does it mean calling Ψ

_{HL}a covalent wave function, when it describes a neutral situation only at large distances, but an ionic one at small distances? One might as well claim that at small distances, Ψ

_{I}is covalent. One immediate consequence is: From the form of a wave function, one cannot infer what kind of electron distribution it describes. This is in contrast with common belief: In the valence bond (VB) view. . . , the electrons are viewed to interact so strongly that there is negligible probability of finding two electrons in the same orbital. The wave function is thus considered to be dominated by purely covalent contributions in which each electron is spin paired to another electron [29]. Another consequence is: The non-orthogonality of VB CSFs poses difficulties for the interpretation of wave functions that are more severe than the numerical problems of the VB method frequently mentioned.

#### 5.3. The Role of Interference in Conventional VB

_{HL}so well suited to describe bonding in H

_{2}? To answer this question, we reorder the contributions in the energy expression. The separation of the energy into classical and interference contributions shows (Figure 22) that only the interference of the non-orthogonal AOs causes bonding; the energy curve for the classical contributions without interference contributions is purely repulsive.

**Figure 22.**Partitioning of the Heitler–London energy into classical contributions and contributions caused by interference.

_{2}; the repulsive character of the triplet state is due to destructive interference. This can be seen from the one-particle densities normalized to the number of particles:

_{I}= ρ

_{HL}and ρ

_{S}= ρ

_{T}. Therefore, both ${\text{\Sigma}}_{g}^{+}$ states show constructive interference, whereas both ${\text{\Sigma}}_{u}^{+}$ states show destructive interference.

^{−ζr}is used, contraction of the electron density can be accounted for by a variable exponent ζ, and polarization may be represented by a non-spherical AO by adding a p-type basis function to the 1s AO. Such an AO is similar to hybrid AOs. Calculations with such modified AOs were done in the early days of quantum chemistry, but they were always just seen as a way to improve the quantitative agreement between calculated and experimental data. To explain chemical bonding, the spherical 1s AOs of free hydrogen atoms were considered to be sufficient.

_{a}= N (a + ϵb) and b is replaced by the linear combination ϕ

_{b}= N (b + ϵa) with a small positive ϵ depending on the interatomic distance and a normalization coefficient. These so-called semi-localized AOs are nodeless and non-orthogonal; they are the basis of the generalized VB method (GVB) by Goddard [31].

#### 5.4. Orthogonal VB

_{2}[40,41].

_{2}molecule as correctly as do linear combinations of ${\text{\Psi}}_{\mathrm{HL}}^{n}$ and ${\text{\Psi}}_{\text{I}}^{n}$, regardless of the fact that the one-particle density of the former CSFs shows destructive interference and that of the latter shows constructive interference. The correct description of bonding does not depend on a certain choice of AOs.

_{2}ground state, the linear combination of OVB CSFs is also a very good description of the ground state. Because the OVB CSFs are orthogonal to each other, the neutral CSF can never describe an ionic electron distribution and vice versa. However, since the overlap integral S increases with decreasing distance R, the ionic contribution increases, as well. At large distances R, where S = 0, ${\text{\Psi}}_{\mathrm{HL}}^{n}$ is identical to ${\text{\Psi}}_{\mathrm{HL}}^{o}$, and the electron distribution in the ground state is strictly neutral or covalent. With increasing S, the ionic contribution increases, as well, but the contribution of ${\text{\Psi}}_{\mathrm{HL}}^{o}$ is always larger than that of ${\text{\Psi}}_{\mathrm{ion}}^{o}$. For R approaching zero, both coefficients of the linear combination approach 1/$\sqrt{2}$, which means the weight of covalent and ionic CSF is 1/2 for R = 0 (Figure 23). Pilar [44] in his Elementary Quantum Chemistry summarized the findings by McWeeny as follows: This means that the concepts of covalent and ionic character are not unique. In using the Slater–Pauling method for polyatomic molecules, it has been standard practice among chemists to speak of the relative importance of ionic structures in a molecule in terms of the coefficients of the corresponding wave function in the total wave function. The above analysis shows that such an interpretation does not have a unique physical significance. Of course, this is not at all surprising in the light of a previous discussion. . . , where it was shown that the so-called covalent and ionic functions used to describe H

_{2}have an overlap of 0.95 and thus have no unique interpretation in terms of fractional ionic character. One must then conclude that any Slater–Pauling covalent wave function which predicts stable chemical bonding does so only because the wave function contains ionic wave functions in terms of OAO’s. In conclusion, the use of OAO’s in the VB method leads to a clearer electrostatic picture of chemical bonding but destroys the chemist’s simple concepts of covalent and ionic character. In light of the relation of VB and OVB CSFs and the implications for their interpretation (known for half a century), statements of the following kind are surprising: . . . we may say that the symmetric orthonormalization gives very close to the poorest possible linear combination for determining the lowest energy. This results from the added kinetic energy of the orbitals produced by a node that is not needed.. . . We have here a good example of how unnatural orthogonality between orbitals on different centers can have serious consequences for obtaining good energies and wave functions [45].

#### 5.5. OVB and Chemical Bonding

- (1)
- Covalent bonding is the result of the lowering of kinetic energy through inter-atomic electron delocalization, called electron-sharing. Delocalization is caused by constructive interference during the superposition of hydrogen AOs. The electrostatic interactions due to charge accumulation in the internuclear region are not bonding, as is frequently claimed, but debonding.
- (2)
- Electron-sharing is accompanied by intra-atomic contraction and polarization. Contraction causes a decrease in the intra-atomic electrostatic energy and an increase in the intra-atomic kinetic energy in the deformed atoms in the molecule.
- (3)
- Intra-atomic contraction enhances the inter-atomic lowering of the kinetic energy and, thus, contributes to energy minimization.
- (4)
- The antagonistic changes of intra-atomic and inter-atomic energy contributions cause a variational competition between electrostatic and kinetic energy; the wave function that achieves the optimal total energy is obtained by variational optimization.
- (5)
- The atom-centered orbitals describing the deformed atoms are quasi-AOs; their shape depending on the distance between the interacting atoms. Near equilibrium distance, they are more contracted than the free AOs, causing the lowering of electrostatic energy; at larger distances, they may be even more expanded than in the free atom, because then the electron can better expand into spatial regions not available for the electron in the free atom when the AOs are superimposed.

_{2}is completely analogous; because of the additional electron-electron repulsion, the total bonding energy is not twice the bonding energy of ${\text{H}}_{2}^{+}$, but only 85% of it [53]. For the many-electron molecules B

_{2}, C

_{2}, N

_{2}, O

_{2}and F

_{2}, the basic conclusions remain valid: Because of the larger number of interacting electrons, the deformation of the atoms in the molecule (due to electrostatic interactions or due to the Pauli exclusion principle) becomes more important, and the wave function adjustment may become a subtle problem.

_{2}molecule, the outer diagonal element of the CI matrix, the coupling matrix element, describes the transfer of an electron from one atom to the other, thus changing the neutral electron distribution into an ionic one. The same role is played by the “hopping integral” in the Hubbard model (the author is grateful to Prof. Ruedenberg for having pointed out this fact).

**Figure 24.**(

**Left**) Shape of the 1s AOs and squared AOs for different atom distances; (

**right**) shape of the 1s orthonormalized AOs (OAOs) and squared OAOs for different atom distances.

#### 5.6. Diabaticity of OVB CSFs

_{2}molecule by Angeli et al. [61] are in full agreement with the electronic structure approach: When the DMOs are OAOs, the adiabatic states are linear combinations of the neutral and ionic OVB CSFs, which show electronic uniformity along the whole reaction coordinate.

_{2}molecules, Nakamura and Truhlar showed that the second and third excited singlet states can be characterized by the σ → σ

^{∗}excitation in either of the two H

_{2}molecules. Therefore, the DMOs must be FOs localized on each molecule. When the delocalized MOs, using the input data for this example, are Procrustes localized on the hydrogen molecules and these localized FOs are used as DMOs, the diabatic states are identical with those constructed with Nakamura’s DMOs. This show, that, at least for this example, the FOs are indeed DMOs. More investigations are necessary to find out whether or not OVB CSFs are always diabatic; investigations using the coupling matrix method will be made in Angeli’s group, and those using the electronic structure approach will be made in the group of Sax.

## 6. Discussion

^{2}hybrid orbitals are needed to correctly describe the equilibrium geometry of planar disilene, but this post festum argument completely ignores the information that we have for the spin reorganization in the reacting fragments during the dimerization reaction.

^{3}hybrids.

^{1}A

_{1}state into the local

^{3}B

_{1}state, the carbene is prepared for bonding, but as one can see, first, there is a local excitation into the

^{1}B

_{a}state, by which the singlet-coupled valence electrons residing in the s-type lone pair AO become distributed among the s- and p-AOs and (only) then does the spin flip occurs simultaneously in both the carbene and hydrogen fragments. In the carbene fragment, this process is a de-excitation into the triplet ground state, whereas in the hydrogen molecule, it is an excitation into the unbound triplet state. The Fermi correlation in the carbene triplet state again causes a strong contraction of the electron densities in the fragments and a strong energetic stabilization. Due to the low overall symmetry, more than two ionic CSFs describe the electron sharing and, thus, the major contribution to chemical bonding.

## 7. Computational Methods

## 8. Conclusions

## Acknowledgments

## Conflicts of Interest

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Sax, A.F.
Chemical Bonding: The Orthogonal Valence-Bond View. *Int. J. Mol. Sci.* **2015**, *16*, 8896-8933.
https://doi.org/10.3390/ijms16048896

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Chemical Bonding: The Orthogonal Valence-Bond View. *International Journal of Molecular Sciences*. 2015; 16(4):8896-8933.
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2015. "Chemical Bonding: The Orthogonal Valence-Bond View" *International Journal of Molecular Sciences* 16, no. 4: 8896-8933.
https://doi.org/10.3390/ijms16048896