The molecular electronic wave functions are expressed in quasi-atomic form. The corresponding global energy expression exhibits the unified resolution of the various modes of chemical bonding in terms of physical interactions. The bonding patterns in several molecules is elucidated.

## I. CHEMICAL MODEL AND PHYSICAL SYNERGISM

In 1889, Lavoisier proposed that all matter is composed of a limited number of “simple substances.”^{1} In the first decade of the 19th century, Dalton proposed that matter consists of molecules that are formed from the atoms of the elements, i.e., Lavoisier’s simple substances.^{2} Berzelius then hypothesized that atoms were held together electrostatically.^{2} However, in 1811, Avogadro showed that Gay-Lussac’s experiments on reactions in gases imply the existence of H_{2}, O_{2}, and N_{2}, i.e., bonding that is now called covalent.^{2,3} The subsequent developments in the emerging field of organic chemistry confirmed the widespread occurrence and importance of covalent bonding.^{2} They furthermore led to the conclusion that the cohesion of a molecule is the result of individual bonds between a few atoms.^{2} The vast successful development of chemistry since then has confirmed this “chemical atomic model” as an effective description of nature.

The main quantitative validation of the chemical model was the fact that the steady gravimetric advances led to the determination of the exact relative weights of all atoms, as was definitively discussed by Cannizzaro in 1860 at the famous Congress of Karlsruhe,^{2,4–6} and that, thereby, the compositions of all newly discovered molecules could be consistently accounted for. During the same time span, the developments and advances in electrochemistry led to the inference, notably by Faraday,^{7} that the “chemical affinities,” which create interatomic bonds, are in some way generated by electric forces. Helmholtz presented a detailed discussion of this subject in his Faraday lecture^{8} of 1881 and in an influential addendum to this lecture.^{9} In the latter, he hypothesizes that atoms have “electrical valence spots” and that bonds between the valence spots of different atoms are mediated by the weightless electrical quanta of an interatomic electric fluid that are attracted to the atomic valence spots.

Starting with Thompson’s discovery^{10} of the electron in 1897 and Bohr’s theoretical derivation^{11} of the electronic spectrum of hydrogen in 1913, the developments in experimental and theoretical physics established the *electronic* structure of the atoms. Starting with Heitler and London’s demonstration that quantum mechanics leads to the existence of the H_{2} molecule,^{12} it has furthermore been established that molecules consist of electrons moving in the field of nuclei. The aforementioned empirical validity of the chemical model implies that the interactions between electrons and nuclei combine to generate a *synergism of compound interactions* that provide the physical basis for the effective chemical model.

The electronic motions are governed by quantum mechanics, and, in the absence of external agents, such as electromagnetic fields, the electronic wavefunction of the molecule contains all intra-molecular electronic physics. Notably, the interactions that generate the bond-forming synergism are embedded in the molecular wave function.

The identification of the parts of this synergism requires the analysis of the *energy of molecule formation,*

where H = molecular Hamiltonian, Ψ = molecular electronic wave function, H_{A} = Hamiltonian of the free atom A, Ψ^{f}_{A} = electronic wave function of the free atom A. The present study expounds how, for valence space wave functions, a sequence of *internal transformations of Ψ* in fact resolves the energy of formation (1) in terms of components that exhibit the interactions of the bonding synergism. The underlying, detailed mathematical formulations have been given in a separate study.^{13}

Since this bonding resolution of the formation energy applies to a molecular system at any point of its potential energy surface, it elucidates not only the global bond pattern of a stable molecule. By the combination of several such analyses, energy changes along reaction paths, reaction energies, barriers etc., can be elucidated in terms of inter-atomic and intra-atomic bonding changes.

As is discussed in the Appendix, a full bonding analysis of the molecular energy of formation that is *exclusively* based on the *actual* molecular electronic wave function is not given by other energy decomposition analyses.

## II. INTRA-ATOMIC CHANGES AND INTER-ATOMIC INTERACTIONS IN BOND FORMATION

There exists an important difference between the chemical model and the physical synergism. In the chemical model, which was outlined in the first paragraphs of Sec. I, the possibility that bond formation can involve significant internal changes in the “indivisible” atoms was not considered. In the physical bonding synergism, by contrast, the molecule contains certain electronic substructures that are very similar to *but different from the electronic structures of the free atoms*. These substructures will be called *quasi-atoms*.

Because of these electronic structure differences, the energies of the quasi-atoms lie above the energies of the free atoms by virtue of the atomic variation principle. As a consequence, the energy lowering that causes bond formation is actually the result of *a non-trivial* c*ompetition between the anti-bonding energy increase due to the change of the free atoms into quasi-atoms and the bonding energy lowering due to the interactions between the quasi-atoms*. This competition is expressed by resolving the energy of molecule formation in Eq. (1) as

where E(adapt) contains the change from free atoms to quasi-atoms, and E(interact) contains the interactions between quasi-atoms.

The quantification of the competition inherent in Eq. (2) requires the knowledge of the energy of the non-interacting quasi-atoms in the molecule, viz.,

so that

In the present analysis, the values of the quasi-atomic energies H_{A}(qa) are determined by the *internal transformations* of Ψ that generate the energy resolution, as mentioned after Eq. (1). In order to stay as close as possible to the chemical model, a major consideration guiding the formulation of the internal transformations is *to reduce the antibonding energy increases caused by the modification of the free atoms into the quasi-atoms as much as possible.* This criterion leaves no room for the independent choice of an arbitrary model wave function to represent a non-interacting modified atom in a molecule.

## III. QUASI-ATOMIC STRUCTURE OF THE MOLECULAR VALENCE SPACE WAVE FUNCTION

The present bonding analysis is formulated for MCSCF wave functions generated by the *full molecular orbital valence space*. The reasons are as follows: The basic structure of the periodic table is determined by the *valence space* parts of the *atomic* wave functions, which are generated by the orbital spaces that are spanned by the optimized atomic minimal basis sets. The lowest atomic states are determined by the configuration interaction in the atomic valence space. Furthermore, chemical bonding is dominated by the interactions in the optimized *valence space* part of the *molecular* wave function.^{14} The molecular valence space is generated by the optimized molecular orbital valence space, whose *dimension* is the union of the dimensions of the constituent atomic valence spaces. Hence, any bonding analysis has to begin with an analysis of the full molecular valence space. This analysis is also a prerequisite for subsequent bonding analyses of dynamic correlations. Specific concrete options within this framework are discussed in detail in Ref. 13.

In all examined molecules, calculations have shown^{15–22} that the optimized *molecular* valence space wave function can, in fact, be generated *by a set of molecular orbitals that are very close to the corresponding optimized minimal basis set orbitals of the constituent free atoms*.

Accordingly, the resolution of the molecular valence space wave function Ψ in terms of quasi-atomic substructures given in Eqs. (2)–(4) is exhibited most clearly when Ψ is expressed as a linear combination of determinants that are formed from the set of those orthogonal molecular orbitals that are *closest* to free-atom orbitals. These *“quasi-atomic molecular orbitals”* (QUAOs) are determined by projecting the optimized orbital valence space *of each free atom* onto the *full molecular* orbital valence space and then symmetrically orthogonalizing the full orbital set. Further context regarding the QUAOs will be discussed in Sec. V. Some illustrative examples will be discussed in Sec. VIII.

In this QUAO-based representation, the molecular valence space wave function can then be organized as the orthogonal quasi-atomic configurational expansion,

where Ψ^{d}_{Aν} are the spin–orbital products of QUAOs on atom A, similarly for Ψ^{d}_{Bν}, Ψ^{d}_{Cν}, … with the restriction that, for a given Ψ^{d}, *all* component determinants Ψ^{d}_{ν} have the same electron population on atom A, the same population on atom B, and similarly for atoms C, D, …. Thus, each Ψ^{d}*individually* represents a different specific *distribution of the electrons over the quasi-atoms,* with an integer number of electrons on each atom, some neutral, some ions, each in some atomic state. In neutral molecules, the strongly dominant distribution in Ψ typically is the distribution Ψ^{0}, in which each atom is neutral, i.e., the number of electrons is equal to the nuclear charge.

The summation ∑_{d} in Eq. (6) generates *migrations* of electrons between the quasi-atoms. Thereby, fractional charges can be (though do not have to be) created on the quasi-atoms. Thus, *the configuration mixing of the Ψ*^{d} *in* *Eq. (6)* *defines and quantifies the concept of interatomic electron sharing*.

## IV. QUASI-ATOMIC RESOLUTION OF THE MOLECULAR VALENCE SPACE ENERGY

The quasi-atomic resolution of Ψ in terms of Ψ^{d} according to Eq. (6) leads to the following, quasi-atomic resolution of the molecular energy for a neutral molecule

where the normalization C_{0}^{2} + ∑_{d≠0}C_{d}^{2} = 1 has been used.

The physical origins of the various terms in this equation are embodied in the constituent energy integrals with respect to the QUAOs. The *intra-atomic energies* are given by the one- and two-electron integrals over QUAOs on the same atom. Two kinds of *interatomic energies* occur, which are exemplified by the following integrals [in which (Aa), (Bb), … denote orbital a on atom A, orbital b on atom B, …]:

and

where A ≠ B, C, and D are arbitrary, and a, b, … denote QUAOs on atoms A, B, …, respectively. The interference energies are consequences of the wave nature of Ψ. Interference densities and interference energies have been analyzed by Nascimento for many molecules.^{23} The physical meanings of the integrals in Eqs. (8) and (9) are manifest and provide the direct basis for the physical interpretations that are attributed to various terms in the energy analysis of Eq. (8).

The first term on the right-hand side of Eq. (8) contains the energies that exist independently of electron sharing, viz., the intra-atomic energies of the orthogonal quasi-atoms, H^{0}_{A}, and the “quasi-classical” Coulombic interactions, H_{AB}(quasi-classical Coulombic) = H^{0}_{AB} between them,

The second term in Eq. (8) contains the changes that electron sharing generates in the intra-atomic energies and in the interatomic Coulombic interactions, viz.,

The energies H_{AB}(sharing, intra-atomic and Coulombic) are sums of three kinds of interactions:

Coulombic energy changes due to the

*increase in the mutual penetration*of electrons from different atoms when the electrons are shared between the atoms. These repulsions generate the generally recognized “tendency of electrons to avoid each other,” which is the cause of the “strong correlation” terms in the wave function. This role of the penetration energy has been graphically exhibited for H_{2}.^{24}The energies associated with

*charge transfers*between the quasi-atoms, which consist of positive electron detachment energies, negative electron attachment energies, ionic attractions, and ionic repulsions. Together, they yield the total effect of charge-transfer on the energy.Non-classical Coulombic interactions, such as the interactions that generate, e.g., long-range dispersion forces.

The third term in Eq. (8) contains the interference energies between the QUAOs of different atoms, viz.,

(The three- and four-center interferences have been folded into the two-center interferences.) In all molecules examined so far, these terms have been found to be the origin of the energy lowering that causes covalent bonding. Since the existence of these terms distinguishes the wave mechanical description of electron sharing from any possible classical description of electron sharing, it is apparent why quantum mechanics can yield covalent bonding whereas classical mechanics cannot. In certain simple situations, the interference energy becomes identical with Mulliken’s classic MO resonance integral β = γ − αS.^{25} Explicit formulas for Eqs. (11)–(13) are given in Ref. 13.

## V. QUASI-ATOMIC RESOLUTION OF THE MOLECULAR ENERGY OF FORMATION

As discussed in Sec. II, critical quantities for the fundamental resolution of Eq. (2) for the energy of formation of the molecule are the energies of the non-interacting quasi-atoms, H_{A}(qa), of Eq. (3). The H_{A}(qa) are intra-quasi-atomic energies that are calculated with those orbitals in the molecular valence space that are *closest to corresponding free-atom orbitals*. These molecular orbitals are determined for each atom by projecting the optimized orbital valence space of that free atom onto the full molecular orbital valence space^{26} by means of a singular value decomposition of the respective overlap matrix.^{13} They have been called molecule-adapted atomic orbitals, MAAOs.^{13} By virtue of their separate constructions, the MAAOS of different atoms are mutually nonorthogonal.

By contrast, the energy resolution that has been exhibited in the preceding Sec. IV is contingent on the orthogonality of the QUAOs. The QUAOs were therefore obtained from the MAAOs by symmetric orthogonalization, as noted in Sec. III. Consequently, the energies H_{A}(qa) of the non-orthogonal non-interacting quasi-atoms in the molecule in Eq. (3) differ from the intra-atomic energies of the orthogonal quasi-atoms, H^{0}_{A} of Eq. (11), in the energy resolution of Sec. IV, by the orthogonalization energy.

The use of orthogonal quasi-atomic orbitals is in accordance with the quantum mechanical rule that unambiguous interpretations of a system are based on orthogonal expansions. Thus, when a molecule is considered by itself, without reference to any other systems (e.g., the free atoms), its orthogonal QUAOs are the appropriate interpretative basis. This is also manifest by the clear resolutions obtained for the wave function in Sec. III and for the energy in Sec. IV. In terms of the non-orthogonal MAAOs, such resolutions in terms of quasi-atomic substructures would require numerous arbitrary and convoluted definitions, if at all possible. Since, in many contexts, a relation exists between antisymmetrization and orthogonalization energies, the latter are commonly perceived as energy increases caused by the exclusion principle forcing atoms into smaller spaces in a molecule. This type of energy has therefore been called Pauli repulsion.

Although the interatomic orthogonalizations are interatomic operations, they are not due to interatomic electron sharing or physical interactions. Being atomic orbital changes, they can be considered as atomic adaptations. Therefore, it seems reasonable to include them in the energy E(adapt) of Eq. (4) as follows:

where H_{A}(adapt-intra) is the increase in free atom energy due to intra-atomic configurational promotions and orbital modifications, viz.,

and H_{A}(adapt-inter) is the energy increase of the quasi-atom due to its orthogonalization to the other atoms in the molecular environment.

The quasi-atomic resolution of the energy E(interact) of Eq. (5) is provided by Eqs. (8)–(13) of the preceding Sec. IV.

Thus, the energy of formation of Eq. (2) is the result of the following synergism:

It is possible to absorb the orthogonalization energies proportionally in the Coulombic and the interference energies and, thereby, shift them from E(adapt) to E(interact).^{13} This may be useful in the context of the transferability of E(adapt).

All components of the synergism given by Eqs. (16)–(18) are directly expressed in terms of the intelligible fundamental physical interactions between the quasi-atomic orbitals that were identified in Eqs. (9) and (10). The explicit mathematical formulations are given in Ref. 13. Since the formulas involve only the density matrices, they are independent of the explicit construction of the wave function Ψ and can be used to identify differences in different wave functions.

## VI. COMPETITION BETWEEN E(interact) AND E(adapt)

Since all parts of the synergism embodied in Eqs. (16)–(18) derive from the same wave function, this resolution of E(formation) quantifies the competition between the antibonding atomic modifications E(adapt) and the bonding interactions E(interact), which was discussed in Sec. II. A case for which this competition has been found to be consequential is the explanation of the long-standing puzzle that Si_{2}H_{2} has a dibridged geometry while C_{2}H_{2} is linear. Guidez *et al.* have shown^{27} that, *in both molecules*, E(interact) is more bonding in the linear structure than in the dibridged structure, thus favoring the linear structure, while E(adapt) is more anti-bonding in the linear structure than in the dibridged structure, thus favoring the dibridged structure. However, *in the competitive sum* *(16)* *of the two*, E(interact) prevails in C_{2}H_{2}, while E(adapt) prevails in Si_{2}H_{2}, so that different geometries result in the two molecules. Table I shows the relevant quantitative values^{27} for the two isomerization reactions.

Δ = (linear − dibridged) . | ΔE(adapt) . | ΔE(interact) . | ΔE(formation) . |
---|---|---|---|

HCCH | 616 | −705 | −88 |

HSiSiH | 442 | −414 | 28 |

Δ = (linear − dibridged) . | ΔE(adapt) . | ΔE(interact) . | ΔE(formation) . |
---|---|---|---|

HCCH | 616 | −705 | −88 |

HSiSiH | 442 | −414 | 28 |

^{a}

Data from Guidez (Ref. 27). MCSCF calculation of the full valence space of ten electrons in ten orbitals with an uncontracted cc-pVQZ basis.

It would seem to be challenging to describe this competition accurately if E(adapt) is obtained from a model wave function that is different from the wave function that yields E(interact).

## VII. KINETIC INTERFERENCE BETWEEN QUAOs AS DRIVER OF COVALENT BINDING

Table II exhibits the resolution formulated in Eqs. (16)–(18) for the energy of the formation of the C_{2} molecule.^{17} It lists the global values of the molecule for the essential energy components of the synergism and, moreover, their kinetic and potential components (see Sec. 6 of Ref. 17). The most important implications of these data are the following. The binding energy of −232 mh is the result of the competition between the anti-bonding +1824 mh of the total (intra + inter) atomic adaptation energy and the bonding −2056 mh of the total interactions. The bonding of the interactions is essentially due to the interference energy of −2067 mh, and the interference energy is bonding due to the kinetic interference energy of −3601 mh, while the potential interference energy is antibonding (+1534 mh). *Notably, the sum-total of all parts of E(interact) other than the kinetic interference is antibonding.*

. | E^{b}
. | T^{b}
. | V^{b}
. |
---|---|---|---|

Energy of formation^{c} | −232 | 232 | −464 |

Adapt-intra-atom | 990 | 1143 | −153 |

Adapt-inter-atom^{d} | 834 | 2709 | −1875 |

Interact^{e} | −2056 | −3620 | 1564 |

Coulombic | −195 | 0 | −195 |

Penetration | 206 | −19 | 225 |

Interference | −2067 | −3601 | 1534 |

. | E^{b}
. | T^{b}
. | V^{b}
. |
---|---|---|---|

Energy of formation^{c} | −232 | 232 | −464 |

Adapt-intra-atom | 990 | 1143 | −153 |

Adapt-inter-atom^{d} | 834 | 2709 | −1875 |

Interact^{e} | −2056 | −3620 | 1564 |

Coulombic | −195 | 0 | −195 |

Penetration | 206 | −19 | 225 |

Interference | −2067 | −3601 | 1534 |

^{a}

MCSCF calculation of the full valence space of eight electrons in eight orbitals with cc-pVQZ basis orbitals. Data from Ref. 17.

^{b}

Energies in millihartree (1 mh ≈ 0.627 kcal/mol).

^{c}

Energy of formation = Adapt-intra-atom + Adapt-inter-atom + Interact.

^{d}

Interatomic orthogonalization energy (Pauli repulsion).

^{e}

Interact = Coulombic + Penetration + Interference.

The observation that binding is essentially *driven* by the kinetic energy lowering associated with electron sharing/interference has been made in all systems examined so far, diatomic and polyatomic.^{28–30} It was first noted for H_{2} in 1962^{31} and analyzed for H_{2}^{+} in 1970.^{32} In 2014, an exhaustive in-depth analysis^{29,33} elucidated why covalent bonds are created by the *drive of electron waves to lower their kinetic energy through expansion from one atom to several atoms*. Recently, it has furthermore been demonstrated^{34} that the kinetic energy-driven formation of bonds is not limited to particles with Coulombic interactions but is also operative for systems of particles with non-Coulombic attractions and repulsions, for which the virial theorem in the form 2T = −V is not valid. Fundamentally, the role of the kinetic energy is a consequence of the quantum mechanical equation of motion being determined by the kinetic energy part of the Hamiltonian.

The kinetic interference energy is given by the sum

over all QUAO pairs between all different atoms in a molecule. The p_{Aa,Bb} are the respective elements of the first-order density matrix in the QUAO representation, which have traditionally been called (density) bond orders.^{35} The expression (19) has been found to be negative in all examined molecules.

Furthermore, without diminishing their quasi-atomic character, the QUAOs *on any atom* can be orthogonally mixed *among each other* to generate, *by an unbiased maximization*,^{36} “hybrid” QUAOs that are oriented in such a way that only very few of the bond orders in Eq. (19) have significant magnitudes, as will be apparent from the molecules discussed in Sec. VIII. Thus, *only a very few QUAO pairs contribute significantly to the global kinetic interference energy lowering* of *Eq. (19)**. Each of these significant contributions has been found to be individually negative for all (∼100) bonds in all examined molecules*.^{28} The energy of the formation of a molecule can therefore be understood as the sum of the drives to form the individual significant bonds between atom pairs, in confirmation of the chemical model.

## VIII. QUASI-ATOMIC KINETIC BONDING ANALYSIS

Serendipitously, the interatomic kinetic interference energies, which drive the formation of the covalent bonds, are much easier to calculate than the interatomic potential energy terms. Therefore, the analysis of the structure of the kinetic interference energy offers a simple route to gaining relevant insights regarding the origin of the bonding patterns in molecules. To this end, kinetic bond orders (KBOs) have been defined^{37} for each QUAO pair by

where the scale factor 0.1 roughly compensates for the omission of the antibonding energy terms noted in the last sentence of the first paragraph of Sec. VII. As a result, the KBO values lie in the range of conventional bond energies.

The combination of the information contained in the KBOs, in the QUAO occupations,^{38} and in the QUAO hybridizations^{39} has provided an effective basis for identifying and elucidating the interactions that drive unusual bonding motifs that challenge conventional intuition as well as conventional bonding. Notably, KBOs of different bonds in a molecule are additive.

As illustrative examples, Figs. 1 and 2 exhibit the KBO analyses for the molecule diborane, H_{2}BH_{2}BH_{2}, and for the noble gas compound HXeCCH. In these figures, the QUAOs are displayed by their contours of 0.1 (electron/bohr^{3})^{1/2}. The QUAO electron occupations are listed in blue below each QUAO. Bond orders are shown between any two QUAOs for which |KBO| > 0.6 kcal/mol. The density bond orders p_{Aa,Bb} are in black font; the kinetic bond orders KBO_{AaBb} in kcal/mole are in black font on a yellow background.

In diborane^{40} (Fig. 1), only the two symmetry-unique bonds are shown. It is seen that an outside hydrogen QUAO and one oriented boron QUAO form a two-center CH bond, containing two electrons that are equally shared between the two atoms, with a KBO of −29.9 kcal/mol. The boron QUAO has 74% 2p-character. The central hydrogen QUAO is bonded to another oriented hybrid QUAO of each boron, yielding a three-center bonding motif, which contains two electrons. The hydrogen has abandoned 0.06 electrons to each boron QUAO. Each BH part of the central three-center two-electron bond has a KBO of −14.3 kcal/mol, very close to half the KBO of the two-center BH bond on the outside. In addition, there is a weak bond (KBO = −4 kcal/mol) between the two boron atoms. Remarkably, no significant KBO interactions exist between the shown BHB bond and the other, not-shown, symmetry-related (BHB)′ three-center two-electron bond. These conclusions are implied by the density matrices without any reference to the orbital construction of the molecular wave function.

For HXeCCH, as shown by Duchimaza-Heredia *et al.*^{19} (Fig. 2), *all* bonds and lone pairs are shown, exhibiting the global bonding pattern. There are three lone pairs on Xenon of essentially 5s, 5px, 5py, character. The two-center, two-electron CH bond is polarized toward the carbon QUAO, which is a 2s-2pσ hybrid pointing toward the hydrogen QUAO. The two carbon atoms are linked by three bonds: one between 2s-2pσ QUAOs pointing toward each other, one between the carbon 2px QUAOs, and one between the 2py QUAOs.

A three-center four-electron bond is formed by the hydrogen QUAO on the left, the 5pσ QUAO on xenon, and the 2s-2pσ QUAO on carbon pointing toward Xe. This bond is made possible by xenon abandoning 0.84 of its two electrons, donating 0.195*e* to the hydrogen QUAO and 0.645*e* to the carbon QUAO. Thereby, interferences between the QUAOs are made possible, viz., −30.7 kcal/mol between xenon and hydrogen, −23.5 kcal/mol between xenon and carbon, and −4.1 kcal/mol between hydrogen and carbon. A doubly occupied Xe-5pσ orbital could not have interferences.^{41}

It should be noted that this charge transfer creates a substantial collateral *energy increase*^{42} because the large ionization potential for prying an electron away from xenon (+ 279.74 kcal/mol) is nowhere near counterbalanced by the other effects of this charge transfer, i.e., the electron affinities of hydrogen and carbon (which add up to about −13 kcal/mol) and the generated ionic interactions.^{42} Thus, the antibonding charge transfer is *forced* by the covalent bond formation. (The anti-bonding effect is much larger in this molecule than in the three-center bond of XeF_{2}, where the electron affinity of 2F is −157 kcL/mol). The conjecture^{43} that the three-center bond of Xe has an electrostatic origin is a misguided interpretation of ionic VB structures and exhibits the hazards of interpretations that stray too far from the basic physical content of the actual molecular wave function.

Among the other bonding motifs that have been elucidated by the KBO analysis^{15–22} are some transition metal bonds. In rhodium boride, the rhodium atom has been shown by Schoendorff *et al.*^{22} to bond to boron not only with the 4dσ and 4dπx and 4dπy type QUAOs but also using a (5s-5p) QUAO to form a fourth bond in the ground state (X^{1}Δ) and in the lowest excited state (a^{3}Δ) state. The participation of the 4d, 5s, and 5p orbitals in forming the bonding QUAOs on rhodium is shown in detail in Fig. 3 for both states. The next five states use the same QUAOs but have more complex bonding patterns.^{22}

In the {(C_{5}H_{5})_{2}Zr[N(SiHMe_{2})_{2}]}^{+} cation, the zirconium atom has been shown to use all 5p orbitals, which are empty in the free atom ground state, to form nine (5s-5p) QUAOs with about equal occupations. Of these, three are bonded to each C_{5}H_{5} ring, one bonds to the N atom, and each of the remaining two forms a three-center, two-electron, so-called pyragostic, Zr–H–Si bond.^{20}

## IX. SUMMARY

A system of orbital and configurational transformations casts the molecular electronic wave function into a form that exhibits the quasi-atoms and their interactions that are embedded in the wave function. The transformations resolve the molecular energy of formation into a quasi-atomic synergism that yields a unified analysis of the various modes of chemical bonding. The following observations have been made on the molecules that have been examined:

Covalent bonding is determined by the valence-space part of the molecular electronic wave function. Appropriate internal transformations cast its density matrices and energy into a sum of terms that represents a resolution in terms of constituent quasi-atoms and interactions between them. Specifically, the orbitals that span the valence space wave function can be and are transformed into a set of quasi-atomic molecular orbitals (QUAOs) that are very close to the orbitals of the free atoms. The representation of the valence space wave function in terms of the QUAOs yields quasi-atomic resolutions of the molecular density matrices and the energy. These resolutions reveal the quasi-atomic substructures (quasi-atoms) that are embedded in the wave function and identify the synergisms that generate the global bonding pattern. Thereby, the energy of the formation of the molecule is quantitatively expressed as the result of interactions with fundamental physical meanings.

Molecules with unconventional as well as conventional bonding motifs have been successfully elucidated by this approach.

All information for the analysis is extracted from the actual valence space wave function. No additional auxiliary model wave functions are involved. Therefore, the resolution yields the relative weights of the various contributions to the energy of formation as they are inherent in the actual wave function. This is particularly relevant for quantifying the competition between bonding and antibonding component interactions that are parts of the overall bonding synergism, notably the interatomic bonding forces vs the antibonding atomic modifications and the bonding covalent interferences vs the non-interference effects of electron sharing. The results of this intrinsic energy resolution represent benchmarks, notably with regard to the mentioned energy competitions.

## ACKNOWLEDGMENTS

The author expresses his warm appreciation to Professor W. H. Eugen Schwarz for his substantial interactions and sound counseling. The present work was supported, in part, by the National Science Foundation under Grant No. CHE-1565888 to Iowa State University and, in part, by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences, through the Ames Laboratory at Iowa State University under Contract No. DE-AC02-07CH11358.

## AUTHOR DECLARATIONS

### Conflict of Interest

The author has no conflicts to disclose.

### Author Contributions

**Klaus Ruedenberg**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Software (supporting); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal).

## DATA AVAILABILITY

### APPENDIX: COMPARISON WITH OTHER BONDING ANALYSES

A reviewer has requested comparisons to existing energy decomposition analyses. Most generally applied are the analyses of Morokuma–Ziegler–Baerends–Michalak^{44–47} and Morokuma–Streitweiser–Glendening–Weinhold.^{44,48,49} Many other bonding analyses have been proposed,^{50} including a recent one by Levine *et al.*,^{51} which are not widely used. A discussion that would do justice to all analyses would go beyond the limits of the present narrative. The following comments are confined to clarifying the essential elements that distringuish the present approach from existing energy decomposition analyses.

Whereas the mentioned generally used analyses in Refs. 45–48 are designed for density functional or Hartree–Fock calculations, the present analysis is formulated for multi-configurational wave functions in the molecular orbital valence space. Such MCSCF wave functions, of which there exist a variety, are effective in describing changes, including unusual ones, in covalent bonding patterns along reaction paths and thus the processes of bond forming and breaking.

The present analysis is based on the following premises:

All bond-forming interactions are embedded in the

*actual*electronic wave function of a molecule.Atoms are preserved in a molecule in a modified form. This implies that the

*actua*l molecular wave function contains substructures that have the character of*quasi-atoms.*The energy-of-formation of a molecule is the result of a

*competition*between the antibonding energy increase due to the modifications that create the quasi-atoms from free atoms and the bonding energy decrease due to the interactions between the quasi-atoms.

Within this context, the bonding synergism is formulated subject to the following guiding tenets:

All interaction energies are identified in their entirety in the energy of the

*actual*wave function of the molecule and the actual energies of the separated free atoms. No energies from other arbitrarily postulated model wave functions are employed.Accordingly, the wave functions and energies of the quasi-atoms in a molecule are identified by

*internal*orbital and configurational transformations of the*actual*molecular wave function, which automatically break the energy down into quasi-atom energies and interactions between quasi-atoms. The quasi-atoms in a molecule are*not*defined by separate independent wave functions.Among the criteria that determine the internal transformations, an essential requirement is that the quasi-atoms differ as little as possible from the respective free atoms so as to reduce the antibonding energy increases caused by the modification of the free atoms into the quasi-atoms as much as possible. This criterion, which conforms to the spirit of the chemical perception that an atom in a molecule is similar to a free atom, eliminates any arbitrariness in the quasi-atoms.

The observance of these tenets entails, among others, the following results:

In the actual molecular wave function, the quasi-atoms are given by their first and second order density matrices. Contrary to generally made assumptions, these density matrices are

*not deducible from a single atomic wave function*but from ensembles of entangled atomic wave functions.The density matrix resolution yields the interatomic interactions directly as coulombic and interference energies between quasi-atomic orbitals and not as complex quantities involving molecular orbitals or differences of intermediate wave functions. Examinations of the simple bonds driving interatomic kinetic interference energies directly exhibit the different global bonding patterns in complex molecules.

Simply put, in the present analysis, the quasi-atom in a molecule is the modified atom that is closest to the free atom under the constraint that the actual molecular electronic wave function is taken as the *only* source for identifying the embedded bond creating synergism. The fact that the quasi-atoms obtained by this unbiased route (as given by their density matrices) indeed turn out to be close to free atoms represents a theoretical-physical validation of the chemical model. The rigorous insights into this synergism should shed light on bonding analysis based on intuitive premises.

## REFERENCES

Avogadro should be given credit for discovering the covalent bond.

This is emphasized by Helmholtz in Ref. 8.

See Fig. 5 in Sec. 6.4 of Ref. 31.

_{2}H

_{2}not linear? An intrinsic quasi-atomic bonding analysis

_{2}BH

_{2}BH

_{2}, H

_{2}CO and the isomerization HNO → NOH

The occupation of the QUAO (Aa) is given by the diagonal element p_{AaAa} of the first order density matrix.

The orientation-hybridization is discussed in Sec. 2.3 of Ref. 18.

Section 3.7 in Ref. 18.

This is a special case of a general limitation regarding occupation-bond order combinations due to the antisymmetry of the wave function. See Sec. 2.2 of Ref. 18.

See Sec. 3.4 in Ref. 19.

See the citations 10–16 in the introduction of Ref. 19.