The Chemical Bond Fundamental Aspects Of Chemical Bonding
Cherie Trojak
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where \(\hat\mathcalA\) is the antisymmetrizer. For 1-electron chemical bonds, the constructive interference on initial bond formation arises from resonating a single electron between the fixed fragment orbitals of the two centers:
Physically, ΔECov allows the wavefunctions of the two prepared fragments to interact, delocalizing the electron or electrons that will form a bond from one fragment to both, and spin coupling them if 2 (or more) are involved. Since the resonance character of (2) and (3) enables an electron (or two) in fixed fragment orbitals to delocalize, ΔECov is the energy change where KE lowering is anticipated, prior to orbital contraction. ΔECov is so named because it will be significant for covalent bonds (and conversely less or not at all significant for more ionic bonds and charge-shift bonds).
The old physical picture does not generalize because it conflated resonance (the true origin of the bond in H2) with delocalization (the physical picture of lowered KE via spreading out the electrons)34. The interaction of different electron configurations need not be associated directly with either electron delocalization in space or its anticipated effects on kinetic energy, based on a particle-in-a-box picture. The perspective that wavefunction interference is the origin of the covalent bond has been found by not only this work, but also Ruedenberg, Nascimento, Bacskay, and many others6,9,10,11,12,13,14,15,20,21,23,24,35,36,37,38,39,40,41,42,43,44. The fact that this picture is recovered by many models speaks to its primacy as the fundamental origin of bonding, while systematic kinetic energy lowering appears in only certain models and in certain systems.
All calculations were carried out with a development version of the Q-Chem program48. The aug-cc-pCVTZ basis set was used for all EDA and KE calculations. The geometries used for dissociation curves were those obtained from a constrained HF optimization with the aug-cc-pVTZ basis set. Equilibrium geometries were also obtained from HF/aug-cc-pVTZ calculations.
Step 1: The two halves of the bond are computed as isolated restricted open-shell systems at the geometry that they will adopt in the bonded system. The energy difference (due to geometric distortion) between the infinite separation geometry fragments and the geometry they adopt in the complex is termed ΔEGEOM. In the case of 1-electron bonds, one half of the bond is treated as a radical and the other as a cation (in whichever configuration is lower in energy, all cases studied here are symmetrical).
Step 4: This wavefunction is then optimized with respect to a set of on-fragment virtual orbitals that describe contraction, ΔECon. These on-fragment virtual orbitals are those which are necessary to exactly describe the response of the electron density to a perturbation in the nuclear charge18. One such contraction orbital is required for each occupied orbital. Depending on how it mixes with its parent orbital, the response orbital may describe either contraction or expansion. In practice, only contraction occurs in the bonding regime.
Step 5: Further on-fragment relaxation is permitted that corresponds to electronic polarization, ΔEPol. ΔEPol is computed by ALMO-constrained optimization of each fragment with a set of on-fragment virtual orbitals. These virtuals are fragment electric response functions (FERFs)22 that exactly describe the response of the occupied orbitals to uniform electric fields (3 dipolar functions per occupied orbital) and their gradients (5 quadrupolar functions per occupied orbital). The ALMO constraint is a Hilbert space constraint which forbids CT between fragments and the use of the FERF virtual functions provides a well-defined basis set limit for polarization and also ensures that the asymptotic behavior of this term matches the theoretical expectation.
For 1-electron bonded systems (all radical cations in this work), the calculation is done as above (with neutral atoms), except that when the ALMO-constrained singlet CSF is formed, the resulting beta electron is removed. This ensures orbital symmetry between the two halves of the 1-electron bonded system.
All intermediates at all steps are valid, spin-pure, properly anti-symmetrized wavefunctions, allowing us to extract kinetic energy, even though this was not the initial goal. The kinetic energy was evaluated at various points during this constrained variational optimization to understand how these different physical processes affect the kinetic energy of the system. We focus on four kinetic energy changes during the bond-forming procedure: the change from relaxed fragments to prepared fragments (ΔTPrep), the change from prepared fragments at infinite separation (which is typically extremely close to that of the electronically relaxed fragments) to the spin-coupled wavefunction (ΔTCov), the change due to orbital contraction (ΔTCon), and the final change due to polarization and CT (ΔTPCT).
All calculations in this work were carried out with a development version of the Q-Chem 5.2 software package. All systems utilized code now available to general users and described in the user manual.
Chemical bonding theory is of utmost importance to chemistry, and a standard paradigm in which quantum mechanical interference drives the kinetic energy lowering of two approaching fragments has emerged. Here we report that both internal and external reference biases remain in this model, leaving plenty of unexplored territory. We show how the former biases affect the notion of wavefunction interference, which is purportedly recognized as the most basic bonding mechanism. The latter influence how bonding models are chosen. We demonstrate that the use of real space analyses are as reference-less as possible, advocating for their use. Delocalisation emerges as the reference-less equivalent to interference and the ultimate root of bonding. Atoms (or fragments) in molecules should be understood as a statistical mixture of components differing in electron number, spin, etc.
Here we show that even these accepted points necessarily imply the choice of both internal (state) or external (energetic) references which bias interpretations. We examine the nature and consequences of those biases and show how the consideration of interacting atoms or molecules as objects in real space minimizes the reference bias as much as possible. By introducing this real space picture we show that it is electron delocalization that underpins chemical bonding.
Ruedenberg and coworkers7,17,18 have pointed out that building a theory of chemical bonding requires as a first, absolutely necessary prerequisite to postulate that atoms, or larger entities if necessary, are somehow preserved in molecules. Chemical bonds occur among interacting moieties that we must single out from the final stable or metastable molecular arrangements. Since quantum mechanics is an intrinsically non-separable theory, how these atoms are introduced and manipulated provides a very first source of bias.
Further chemical analysis requires establishing a reference, which is standardly taken as the H atom and its exact one-electron states (Supplementary Note 1), and the molecular wavefunction is recast in terms of a set of so-called quasi-atomic orbitals (QUAOs, ϕa,b), which are then compared to the isolated ϕ1s function and used to provide an exact energy and density decomposition. Squaring the Ψ amplitude leads to a sum of the squares of the quasi-atomic densities and to an interference term: \(\rho =\phi _a^2+\phi _b^2+\rho _I\). It is found that the QUAOs pass from slight expansion at large R to significant contraction at equilibrium, and that it is only the interference TI lowering which drives the system to an equilibrium geometry, the accumulation of density in the internuclear region being also entirely caused by interference. In this standard model, it is the constructive interference of the atomic functions that allow the electron to dilute or delocalize. Many researchers have contributed to the details of this image over the years20,21,22,23,24,25.
Summarizing: (i) Associating interference as the driving force behind chemical bonding is, to a large extent, the result of assuming a set of internal references. Since it needs not be imposed, it cannot be taken as the final root of the chemical bond, which we ascribe to delocalization in its most general sense; (ii) Even for the simplest dihydrogen case, the consideration of a neutral reference disregards the fact that at equilibrium about half the time one of the H atoms bears two electrons.
Ubiquitous reference biases permeate the theory of chemical bonding, one of the pillars of modern chemistry. On the one hand, the internal references needed to interpret molecular wavefunctions in terms of atomic (i.e., chemical) components, lead to the interference terms that lie at the root of the standard model of the chemical bond. On the other, most energy decomposition analyses, which become essential to discern among conflicting or alternative bonding images, rest on choosing external (energetic) references for the set of fragments that are meant to interact with each other. However, interacting fragments in-a-molecule are entities in pesudo-mixed quantum states, with fluctuating (i.e., not fixed) electron populations, electronic, and spin states. Failing to consider this fact leads to a variety of electron count or electron spin biases. The historical use of energy-promoted and/or spin-excited fragments to rationalize the nature of chemical links finds a rigorous conceptual root in considering fragments as open quantum systems. We have argued that real space analyses are as reference-less as possible, needing only the specification of a chemically sensible atomic partition of the space. If that ingredient is provided, no biases remain, and all the electron counts, electronic and spin states of the interacting fragments are part of the output, not of the input of the procedure. In this way, the best bonding model for a system is automatically read from the results of the analysis.
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