Dynamical electron correlation and the chemical bond. I. Covalent bonds in AH and AF (A = B–F)

Our ability to predict molecular properties and the energetics and outcomes of molecular processes using electronic structure theory is limited by our inability to fully account for the effects of electron correlation. In 1958, Löwdin¹ defined the total electron correlation energy (E_TEC) as the difference between the energy of the exact solution of the non-relativistic electronic Schrödinger equation (nrESE) and the exact solution of the Hartree–Fock (HF) equation,

While the HF equation can be accurately solved even for rather large molecules, the development of methods to accurately solve the non-relativistic electronic Schrödinger equation is still a work in progress, although, as noted below, significant advances have been made in the past few decades.

A few years later, Sinanoğlu² expanded on Löwdin’s definition of E_TEC, noting that there were two distinct contributions to the total electron correlation energy. First is a “near degeneracy” contribution that arises from the degeneracy of atomic configurations as the nuclear charge Z → ∞, e.g., the 2s² and 2p² configurations in the beryllium-like atoms, and the degeneracy of molecular configurations as the internuclear distance R → ∞, e.g., the degeneracy of the 1σ_g² and 1σ_u² configurations in H₂. These “near-degeneracy” contributions depend on the details of the electronic structure of the atom or molecule and, thus, give rise to irregularities in the total correlation energy. This contribution to the correlation energy is now referred to as non-dynamical (or static) electron correlation energy (E_NDEC). Because non-dynamical electron correlation arises from the degeneracy or near-degeneracy of configurations, it can be taken into account by simply including the associated configurations in a multiconfiguration wave function.

Second is a contribution to the total electron correlation energy as a result of the instantaneous correlations among the electrons. This type of correlation places significant constraints on the electronic wave function,^3,4 e.g., the requirement that the electronic wave function vanishes when the distance between any two electrons, r_ij, approaches zero. This contribution to the total correlation energy is referred to as dynamical electron correlation energy (E_DEC) and can only be properly described by including interelectronic terms, r_ij, in the electronic wave function. Unfortunately, the inclusion of such terms is computationally prohibitive except for the smallest atoms and molecules. On the other hand, if the wave function does not include r_ij terms, the solution of the non-relativistic Schrödinger equation converges very slowly, although the convergence of these solutions has been markedly improved by the development of basis sets that systematically approach the complete basis set limit (see, e.g., Refs. 5–11 and references therein). To address the problems associated with the inclusion of r_ij terms in the wave function, Kutzelnigg and co-workers^12,13 proposed efficient and effective approximate methods that reduce the computational requirements, although the resulting computations are still formidable. For recent reviews of these “explicitly correlated” electronic structure methods, see Refs. 14–16.

Given the above breakdown of the total electron correlation energy, we can rewrite E_TEC as the sum of the non-dynamical and dynamical correlation energies,

where we have noted the explicit dependence of these quantities on the molecular geometry, R. The benefit of separating these two types of electron correlation was recognized in the mid-1960s by Clementi and Veillard in their studies of the variation of the correlation energy for the first-row atoms.¹⁷ They reported calculations on a number of low-lying states of the first-row atoms with a two-configuration wave function that accounted for the atomic near-degeneracy electron correlation energy, i.e., a wave function containing the 2s²2pⁿ and 2s⁰2pⁿ⁺² configurations. They found that E_NDEC was strongly dependent on the electronic configuration of the atoms. For example, for the ground states of the first-row atoms, the magnitude of E_NDEC is zero for the lithium atom, very large for the beryllium atom (26.7 kcal/mol), steadily decreases from the beryllium to the boron and carbon atoms (19.5 and 10.9 kcal/mol, respectively), and then vanishes entirely thereafter. When Clementi and Veillard subtracted E_NDEC from an estimate of E_TEC for the ground states of the first-row atoms, the remainder, i.e., E_DEC, depended rather smoothly on the number of electrons in the atom (compare Figs. 1 and 2 in Ref. 17).

Over the years, there have been a number of articles that examined how these two types of correlation effects are described by various electronic wave functions (see Refs. 18–21) or how they affect molecular properties and processes (see Refs. 22–25). There have also been a few attempts to obtain a better understanding of the basic nature of non-dynamical and dynamical correlation in atoms and molecules. For example, Valderrama et al.²⁶ analyzed the non-dynamical and dynamical electron correlation effects in the beryllium atom isoelectronic series using electron pair densities. Handy and co-workers investigated the nature of non-dynamical and dynamical correlation in a number of molecules.^27,28 Sears and Sherrill²⁹ examined the use of different multiconfiguration wave functions to define the non-dynamical and dynamical correlation energy in a selected set of molecules. Hollett and Gill³⁰ argued that there were two types of non-dynamical (static) correlation—one that is captured in an unrestricted HF wave function and another that is not, and Crittenden³¹ proposed a hierarchy of static correlation models. Finally, Ramos-Cordoba and co-workers^32–34 have proposed a means of separating the effects of non-dynamical and dynamical correlation and developed local descriptors for these two types of electron correlation.

Despite this prior work, the impact of dynamical correlation on molecular properties and processes is still not well understood. This is, in part, due to the lack of an agreed upon zero-order wave function that can serve as a proper foundation for decomposing the total correlation energy of a molecule into its non-dynamical and dynamical components. Handy and co-workers^27,28 argued that non-dynamical correlation effects in molecules should be defined by a wave function that only includes the configurations needed to describe the dissociation of the molecule, the so-called “left-right” correlation. Others have argued that the Full Optimized Reaction Space^35–37 (FORS) and valence Complete Active Space Self-Consistent Field^38–40 (vCASSCF) wave functions should be used to define the contribution from non-dynamical correlation energy; see, e.g., the discussion in Ref. 29. However, FORS/CASSCF wave functions can include many configurations in addition to those needed to describe “left-right” correlation or atomic near-degeneracies. In determining a zero-order wave function to use in partitioning the total correlation energy into its non-dynamical and dynamical contributions, it is important to keep in mind both the arguments put forward by Sinanoğlu² and the findings of Clementi and Viellard.¹⁷ Although wave functions that describe both types of near-degeneracy effects discussed by Sinanoğlu contain more than “left-right” correlation, as we note later, atomic and molecular near-degeneracy effects often cannot be cleanly separated.

The Spin-Coupled Generalized Valence Bond (SCGVB) wave function⁴¹ is constructed to describe the dissociation of molecules, i.e., it explicitly includes left–right correlation, although in some instances symmetry considerations may require the use of a projected SCGVB wave function, e.g., an SCGVB(n_a,m_a) wave function.⁴² In addition, the SCGVB wave function accounts for near-degeneracy effects in the atoms, particularly the 2s-2p near-degeneracy effect in the first-row atoms. Thus, the SCGVB wave function accounts for both types of non-dynamical electron correlation effects as defined by Sinanoğlu.² In this paper, we use the SCGVB wave function to decompose the total correlation energy into its non-dynamical and dynamical components and examine how dynamical electron correlation impacts the formation of the covalent chemical bonds in two prototype molecular species: the AH and AF series of molecules with A = B–F. Of particular interest is the influence of dynamical electron correlation on the spectroscopic constants of the AH and AF molecules, (R_e, ω_e, D_e), as well as on the potential energy curves, ΔE(R), for forming the covalent bonds in these two series, ΔE_DEC(R). Our overall goal is to obtain a better understanding of the nature of dynamical electron correlation and how it influences molecular properties and processes.

The outline of this article is as follows. In Sec. II, we briefly review the theoretical and computational methods to be used in this study. In Sec. III, we report and discuss the impact of dynamical electron correlation on the spectroscopic constants and potential energy curves of the AH and AF molecules: first for the first-row diatomic hydrides (Sec. III A) and then for the first-row diatomic fluorides (Sec. III B). Finally, in Sec. IV, we discuss the conclusions to be drawn from the current studies.

The (un-normalized) Spin-Coupled Generalized Valence Bond (SCGVB) wave function^43–45 has the general form

where {ϕ_ci} and {ϕ_vi} refer to the doubly occupied core and valence orbitals of which there are n_c + n_v, {φ_ai} to the singly occupied active valence orbitals of which there are n_a, and $ΘS,MSna$ to an n_a-electron spin function for a state of total spin S and spin projection M_S. All orbitals in the SCGVB wave function, both doubly and singly occupied, as well as the spin function, $ΘS,MSna$⁠, are variationally optimized. The SCGVB wave function has the general form of a traditional covalent valence bond wave function, although, as noted by Coulson and Fischer⁴⁶ and discussed in more detail by Wilson,⁴⁷ optimization of the orbitals incorporates the effects of singly ionic configurations into the SCGVB wave function. The SCGVB wave function is a generalized form of the RHF wave function, which can be obtained by imposing constraints on the SCGVB wave function.

Although the orbitals are allowed complete variational freedom in the optimization of the SCGVB wave function, in most molecules, the resulting orbitals correspond to hybrid orbitals, bond pair orbitals, and lone pair orbitals in line with classical chemical concepts (it must be noted, however, that the hybrid orbitals are not the classical sp, sp², and sp³ hybrid orbitals^48,49). Because the atomic origins of the SCGVB orbitals are usually unmistakable, we often use the atomic orbital designations followed by a prime to identify them, e.g., the bond orbitals in the BH molecule are the B2p_z′ and H1s′ orbitals, which as R → ∞ become the B2p_z and H1s atomic orbitals.

The spin function in Eq. (3) is a linear combination of all of the linearly independent ways to couple the spins of the electrons to obtain a state of total spin S and spin projection M_S,⁵⁰ 

where $fSna$ is the number of linearly independent spin coupling modes. Optimization of the spin function, i.e., the {c_S,k} in Eq. (4), enables the SCGVB wave function to smoothly describe the transition from a spin function appropriate for the separated atoms to that for the molecule. Various spin bases, e.g., Kotani, Rumer, and Serber, can be used for the spin couplings, {$ΘS,MS;kna$}, with each spin basis offering insights into the electronic structure of the molecule as well as having advantages and disadvantages; see Ref. 50 for more details. However, the overall SCGVB wave function is independent of the spin basis used.

In its simplest form, the SCGVB wave function selects the doubly occupied valence orbitals and singly occupied active orbitals so that the wave function of the molecule dissociates to the RHF wave function for the atoms. There are, however, exceptions. In the molecules considered here, SCGVB theory describes the formally doubly occupied 2s orbital of the boron and carbon atoms with a pair of orbitals, (2s₋, 2s₊), that are spⁿ hybrid orbitals (n ≪ 1). For example, the SCGVB wave function for the carbon atom is

Thus, the SCGVB wave function for the carbon atom is a combination of the 2s²2p² and 2s⁰2p⁴ configurations, just the configurations needed to describe the 2s-2p near-degeneracy effect. Although Mok et al.²⁷ argued that the 2s-2p near degeneracy effect should be classified as a dynamical correlation effect, the 2pⁿ⁺² configuration must be retained in the SCGVB wave function to properly describe dissociation in states where the (2s₋, 2s₊) orbitals are not involved in bond formation.⁵¹ In other molecules, e.g., the ground states of BeH⁵² and C₂⁵³ and the lowest excited states of BH,⁵⁴ CH,⁵⁵ and CF,⁵⁶ the electrons in the (2s₋, 2s₊) orbitals are directly used to form bonds. Thus, the atomic near-degeneracy effect is an inherent part of “left-right” correlation.

For some of the molecules, a projected SCGVB wave function is needed to obtain a wave function with the proper symmetry. In this study, a projected SCGVB wave function must be used to obtain a wave function of ¹Σ⁺ symmetry for the ground states of BH and BF and, thereby, fully describe the 2s-2p near degeneracy effect in the boron atom (see Ref. 57). In the CH/CF and OH/OF molecules, a projected SCGVB wave function must be used to ensure that the ground states have ²Π symmetry. These types of projected SCGVB calculations can be performed using the n_a-electrons in m_a-orbitals SCGVB(n_a,m_a) method developed by Karadakov et al.⁴² 

Since the SCGVB or SCGVB(n_a,m_a) wave functions fully describe the effects of non-dynamical electron correlation for most molecules, including all the molecules considered in this study, the dynamical electron correlation energy is simply given by

where R refers to the internuclear distance. We used two different computational methods to estimate E_nrESE: (i) the valence CASSCF(SCGVB)+1+2^40,58,59 method, with the active space defined by the SCGVB wave function,^41,57 and (ii) the CCSD(T)/RCCSD(T)^60,61 method, which is based on an RHF/ROHF wave function. Although the vCAS(SCGVB)+1+2 method recovers less of the dynamical correlation energy than the CCSD(T)/RCCSD(T) method, the overall trends in E_DEC(R) from the two methods are very similar and the vCAS(SCGVB)+1+2 calculations describe E_TEC(R) even when the internuclear separations are very large.

In addition to E_DEC(R) [Eq. (6a)], we are interested in the differential dynamical correlation energy, i.e., the change in E_DEC(R) relative to its value at R = ∞,

as well as the equilibrium geometry-shifted differential dynamical correlation energy,

with ΔR = R − R_e. The latter two quantities require us to determine E_DEC(∞). For the SCGVB and vCAS(SCGVB)+1+2 calculations, the value of E_DEC(∞) was taken to be the energies at the largest value of R considered (R > 10 Å). The CCSD(T)/RCCSD(T) energies of the AH and AF molecules at R = ∞ are simply the sum of the CCSD(T)/RCCSD(T) energies of the two atoms.

All calculations used the aug-cc-pVQZ basis set for hydrogen and the first-row atoms.^5,6 This basis set is sufficiently close to a complete basis set that it yields accurate SCGVB energies as well as usefully accurate vCAS(SCGVB)+1+2 and CCSD(T)/RCCSD(T) energies (more so for the AH series than the AF series as we shall see below). All the calculations presented in this study were performed with the Molpro suite of quantum chemical programs (version 2010.1).^62,63 The CASVB module in Molpro was used to perform the SCGVB calculations^64,65 with Kotani spin functions.⁵⁰ 

As noted in Sec. I, our overall goal is to develop an understanding of the nature of dynamical electron correlation and how it influences the formation of chemical bonds and molecular properties and processes. The focus of the current study is on the covalent bonds in the ground states of the diatomic AH and AF molecules (A = B–F). This includes quantifying the effect of dynamical correlation on the spectroscopic constants, (R_e, ω_e, D_e), for each of these molecules as well as the dependence of E_DEC on the internuclear distance: E_DEC(R), ΔE_DEC(R), and ΔE_DEC(ΔR).

We began our study of the impact of dynamical electron correlation on the covalent chemical bond by examining its impact on the covalent bonds in the AH molecules (A = B–F). The bond in BH is almost a pure covalent chemical bond, whereas the bond in FH is a very polar covalent bond. Thus, the AH series provides a progression of covalent bond types with increasing polarity from BH to FH. Furthermore, these molecules have a particularly simple electronic structure since the second atom in the molecule is a hydrogen atom. As shown by Clementi and Veillard,¹⁷ E_DEC has a smooth dependence on the number of electrons from BH to FH at R = ∞. However, how does E_DEC(R) depend on R? If E_DEC(R) has a smooth dependence on R, it would most likely be found in the AH series.

We began by examining the effect of dynamical correlation on the spectroscopic constants of the ground states of the AH molecules. The results of the SCGVB, vCAS(SCGVB)+1+2, and CCSD(T)/RCCSD(T) calculations on the first-row diatomic hydrides are summarized in Tables I–III. Included in Table I are the calculated equilibrium bond distances (R_e), vibrational frequencies (ω_e), and dissociation energies (D_e) for the AH series. The comparison of the results of the CCSD(T)/RCCSD(T) calculations with the available experimental data⁶⁶ indicates that these calculations provide an excellent description of the electronic structure of all AH molecules. Table I also lists the energies of the AH molecules at R_e, i.e., E_e, and Table II lists the SCGVB energies of the AH molecules (A = B–F) at the vCAS(SCGVB)+1+2 and CCSD(T)/RCCSD(T) values of R_e as well as the SCGVB, vCAS(SCGVB)+1+2, and CCSD(T)/RCCSD(T) energies at R = ∞. Finally, Table III contains the dynamical electron correlation energies, E_DEC, for the AH series (A = B–F) at the equilibrium internuclear distances, E_DEC(R_e), and at R = ∞, E_DEC(∞), plus the values of the differential dynamical electron correlation energy, ΔE_DEC(R_e) = E_DEC(R_e) − E_DEC(∞).

The dynamical correlation energies for the AH molecules at equilibrium, i.e., E_DEC(R_e), as well as at the separated atom limit, E_DEC(∞), are plotted in Fig. 1. The magnitude of E_DEC increases from −30.3884/−30.3808 kcal/mol in BH to −169.5986/−179.0195 kcal/mol in FH at R_e and from −23.5383/−23.3342 kcal/mol in BH and −146.2698/−152.8043 kcal/mol in FH at R = ∞, in line with the increasing number of electrons in the A atom [the first value is relative to the vCAS(SCGVB)+1+2 energies and the second to the CCSD(T)/RCCSD(T) energies, a notation that will be used throughout this report]. The increases in the magnitudes of E_DEC(R_e) and E_DEC(∞) from BH to FH are surprisingly close to linear. The R² values for a linear fit are 0.9969 and 0.9938 for the CCSD(T)/RCCSD(T) results at R_e and R = ∞, respectively, and 0.9969 and 0.9940 for the corresponding vCAS(SCGVB)+1+2 results. The associated root-mean-square (rms) errors are 2.9 and 3.6 kcal/mol at R_e and R = ∞, respectively, for the CCSD(T)/RCCSD(T) energies and 2.7 and 3.4 kcal/mol for the corresponding vCAS(SCGVB)+1+2 energies. In fact, the increase in E_DEC is almost exactly linear from NH to FH—the R² value for a linear fit of E_DEC for these three molecules is 0.9999 for the CCSD(T)/RCCSD(T) results at R_e and 0.9997 for the same results at R = ∞. The associated rms errors are now just 0.3 and 0.5 kcal/mol. The accuracies of the linear fits of E_DEC are essentially the same for the vCAS(SCGVB)+1+2 results at R_e and R = ∞. All in all, the simple dependence of both E_DEC(R_e) and E_DEC(∞) for the AH series is rather striking and suggests that E_DEC for the first-row molecules might indeed be a relatively smooth function of the number of electrons just as it was in the first-row atoms.¹⁷ 

Also plotted in Fig. 1 is the differential dynamical correlation energy, ΔE_DEC(R_e). The changes in ΔE_DEC(R_e) from BH to FH are far more modest than the changes in either E_DEC(R_e) or E_DEC(∞), i.e., most of the increase in the magnitude of E_DEC(R_e) is due to the increase in the dynamical electron correlation energy of the A atom. However, the magnitude of ΔE_DEC(R_e) in BH is significantly smaller than for the other AH molecules, just −6.8/−7.0 kcal/mol. This is to be compared to −15.3/−16.0 kcal/mol (CH), −17.2/−18.6 kcal/mol (NH), −20.1/−22.0 kcal/mol (OH), and −23.3/−26.2 kcal/mol (FH). Although ΔE_DEC(R_e) varies relatively smoothly for CH–FH, BH is a clear outlier. Prior SCGVB calculations on the BH molecule^51,67,68 offer some insights into the unusually small value of ΔE_DEC(R_e) in BH. There are only two pairs of valence orbitals in BH: a (B2s₋′, B2s₊′) lone pair and a (B2p_z′, H1s′) bond pair. The prior calculations found that the lone pair is polarized away from the bond pair as R decreases. As a result, the two valence pairs in BH are spatially separated from one another at R_e, which leads to the small value of ΔE_DEC(R_e). For the rest of the AH series, there is an increasing number of electrons in the A2pπ′ orbitals, which are essentially localized on the A atom. Interaction of the electrons in the A2pπ′ orbitals with those in the AH bond pair is largely responsible for the increase in ΔE_DEC(R_e) from CH–FH. This finding was the first hint that E_DEC may be more complicated than implied by the simple functional forms for E_DEC(R_e) and ΔE_DEC(∞), as shown in Fig. 1.

The change in D_e along the AH series as a result of dynamical correlation follows that for ΔE_DEC(R_e) discussed above, just with a change in sign. As such, the increases in D_e due to dynamical correlation, ΔD_e, range from just 6.8/7.0 kcal/mol in BH to 23.3/26.2 kcal/mol in FH. As a result of the change in ΔE_DEC(R_e) noted above, there is a large difference between the contribution of dynamical correlation to D_e(CH), 15.2/15.9 kcal/mol, vs that to D_e(BH). Thereafter, the contribution of dynamical correlation to the bond energy increases only modestly, to 17.2/18.6 kcal/mol in NH, 20.1/22.0 kcal/mol in OH, and 23.3/26.2 kcal/mol for FH. It is surprising that the systematic increase in the number of electrons in the A2pπ′ orbitals of the A atom from CH to FH has only a modest effect on the variation in D_e along the AH series. The contribution of ΔE_DEC(R_e) to D_e increases by only 1.9/2.6 kcal/mol from CH to NH, 2.9/3.4 kcal/mol from NH to OH, and 3.2/4.2 kcal/mol from OH to FH. The reason for such a modest increase in the contribution of ΔE_DEC(R_e) to D_e from CH to FH is unclear, since, e.g., FH has three more electrons in the A2pπ′ orbitals than CH. This is another hint that we may not fully understand the basic nature of dynamical electron correlation and how it correlates with other aspects of the electronic structure of the AH molecules.

Dynamical electron correlation also has a significant effect on the calculated R_e’s and ω_e’s; see Fig. 2. The magnitude of the changes in R_e as a result of dynamical electron correlation, ΔR_e, decreases from −0.0153/−0.0156 Å in BH to −0.0027/−0.0017 Å in OH, before becoming slightly positive in FH (+0.0007/+0.0028 Å). The latter result is especially puzzling since one might have expected the magnitude of E_DEC(R) to steadily increase with decreasing R, thereby resulting in shorter equilibrium bond lengths for the entire AH series. The changes in ω_e, Δω_e, are equally puzzling. Based on the increased contribution of dynamical correlation to D_e from BH to FH, one might expect Δω_e to increase steadily along the series since dynamical correlation is deepening the potential energy curve. The equilibrium frequency does, in fact, increase steadily from BH to FH, but the contribution of dynamical electron correlation to ω_e does not. Dynamical electron correlation increases ω_e(BH) by +73.1/+69.8 cm⁻¹ and ω_e(CH) by +134.4/+142.3 cm⁻¹, as expected, but thereafter its contribution decreases from NH (+117.4/+126.3 cm⁻¹) to OH (+76.4/+76.4 cm⁻¹) and FH (+36.1/+11.4 cm⁻¹). Therefore, there is not a direct correlation between Δω_e and ΔD_e. All in all, the impact of dynamical correlation on both R_e and ω_e again suggests that dynamical electron correlation in molecules may be more complicated than implied by the atomic results of Veillard and Clementi.

To obtain insights into the cause of the changes in the spectroscopic constants noted above, we examined the dependence of E_DEC on the internuclear distance, R. Figure 3 presents two plots related to E_DEC(R). The first plot [Fig. 3(a)] is simply a plot of E_DEC(R). As expected, the curves for E_DEC(R) are shifted progressively downward from BH to FH, reflecting the increases in the magnitude of E_DEC(∞). Note that the shapes of the E_DEC(R) curves from the vCAS(SCGVB)+1+2 and CCSD(T)/RCCSD(T) calculations are nearly identical, although the difference in E_DEC(R) from the two calculations increases from BH to FH, in line with ΔE_DEC(R_e) above. Figure 4 shows a logarithmic plot of |ΔE_DEC(R)|. The value of |ΔE_DEC(R)| at R = 3.5 Å, the largest distance in the plot, decreases from BH to FH. This is a direct result of the decrease in the radial extent of the orbitals on the A atom from boron to fluorine, which, in turn, leads to a decrease in the interaction between the A and H atoms at large R. As R decreases from 3.5 Å, |ΔE_DEC(R)| increases for all the AH molecules and, except for BH, the increases are almost linear on the logarithmic plot. The curve for BH is clearly an anomaly, as it seems to have two nearly linear regions. As R continues to decrease the slopes of the E_DEC(R) curves eventually begin to decrease, bending downward until for the most polar molecule, FH, the slope becomes zero and then changes sign. There is a pronounced “kink” in many of the curves, i.e., a rapid change in slope, which is clearly seen in the BH curve. There is a distinct minimum in the FH curve.

The second plot in Fig. 3 [panel (b)] is a plot of the R_e-shifted, differential dynamical correlation energy, ΔE_DEC(ΔR); see Table I for the values of R_e from the vCAS(SCGVB)+1+2 calculations. The behavior seen in Fig. 3(b) is consistent with that in Figs. 3(a) and 4, although the variations in the curves are far more evident because of the expanded energy scale resulting from the subtraction of E_DEC(∞) from E_DEC(R). It is now clear that the ΔE_DEC(ΔR) curves do not smoothly depend on ΔR—with the new energy scale there are “kinks” in the curves for BH, CH, and NH and minima in the curves for NH, OH, and FH. The location of the “kink” shifts to smaller ΔR from BH to NH, which has an unusually shaped minimum, and then increases in OH, which has a broad flat minimum, and further increases in FH, which now has a distinct minimum at a positive value of ΔR. There may even be a minimum in the curve for CH, although, if so, it is very small.

The “kinks” and minima in the ΔE_DEC(ΔR) curves are intriguing—why are they there?—but, first, we must determine if they are of any consequence, i.e., will they affect the properties of the bound AH molecules? To address this question, we plotted the potential energy curves along with the ΔE_DEC(R) curves for each of the AH molecules, A = B–F, in Fig. 5. Any major changes in the ΔE_DEC(R) curves that fall within the bound portion of the potential energy curves will definitely affect the properties of the AH molecule. As can be seen, the “kink” in the ΔE_DEC(R) curve for BH falls just to the right of the repulsive wall of the potential energy curve, within the bound portion of the potential energy curve, while those in CH and NH fall just to the left of the repulsive wall, outside the bound portion of the curve. The minima in the OH and FH curves fall well inside the bound portion of the potential energy curve. Thus, the “kink” in the BH curve as well as the minima in the OH and FH curves will definitely influence the properties of the molecules, while the kinks for CH and NH probably will not. However, even if a “kink” lies outside the bound portion of the potential energy curve for an AH molecule, the properties of that molecule will be affected by any changes in the slopes and curvatures of the E_DEC(ΔR) curves inside the bound portion of the potential energy curve.

Based on the behavior of the curves of ΔE_DEC(ΔR), the unusual effect of dynamical electron correlation on the R_e’s and ω_e’s of the AH series (A = B–F) can now be understood. The magnitude as well as the sign of ΔR_e depends on the slope of the ΔE_DEC(ΔR) curve at ΔR = 0.0 Å (R_e). The slopes of the BH and CH curves are very similar at ΔR = 0.0 Å as are the changes in ΔR_e: −0.0153/−0.0156 Å in BH and −0.0155/−0.0161 Å in CH. The slope of the ΔE_DEC(ΔR) curve decreases noticeably for NH and, as a result, ΔR_e decreases significantly from CH to NH (−0.0087/−0.0088 Å). The slope further decreases for the OH curve (−0.0027/−0.0017 Å) and then for FH, the slope actually changes sign, resulting in a positive ΔR_e (+0.0007/+0.0028 Å). Thus, the decrease in the magnitude of ΔR_e from CH to OH as well as its change in the sign for FH is a direct result of the changes in the slopes of the ΔE_DEC(ΔR) curves at R_e, a change that is clearly evident in the curves in Fig. 3(b). Similar arguments apply to Δω_e, the change in the fundamental frequency induced by dynamical electron correlation, although, for ω_e, it is the changes in the curvatures of the ΔE_DEC(ΔR) curves at ΔR = 0.0 Å, which are critical.

From the curves of the dynamical correlation energy in Figs. 3 and 4, it is clear that the magnitude of the correlation energy does not simply increase as the atoms are drawn closer and closer together. The decrease in the slope of |ΔE_DEC(R)| with decreasing R is likely a result of the increasing interpenetration of the charge clouds on the two atoms that results in a saturation of |ΔE_DEC(R)|. However, what causes the unusual changes in the ΔE_DEC(ΔR) curves for the AH molecules as ΔR decreases further? In particular, why does ΔE_DEC(ΔR) have a “kink” in the BH curve and a minimum as well as a maximum in the FH curve? To answer these questions, we examined this region for both of these molecules in more detail; the resulting ΔE_DEC(ΔR) curves are plotted in Fig. 6. As shown in Fig. 6(a), there is a clear change in the slope of the BH curve at ΔR ≈ −0.35 Å from both the vCAS(SCGVB)+1+2 and CCSD(T) calculations. This change in slope is similar to, although less dramatic than the minimum–maximum (min–max) seen in the ΔE_DEC(ΔR) curve for FH in Fig. 6(b), but it has the same general character; it is just that, for BH, the minimum and maximum are energetically very close. For the FH molecule, the ΔE_DEC(ΔR) curve reaches a minimum around ΔR ≈ 0.1 Å and rises from this point to a maximum at ΔR ≈ −0.6 Å, at which point it begins to drop again, appearing to head toward the ΔE_DEC(ΔR) value at ΔR = −R_e, i.e., R = 0.0 Å, for the neon atom, the values of which are also plotted in the figure.⁶⁹ 

Since the min–max region is most pronounced in FH, we analyzed the SCGVB wave function for this molecule. For FH, the SCGVB configuration is (F1s′)²(F2s′)²(F2p_x′)²(F2p_y′)²(F2p_z′, H1s′) with the spins of the electrons in the (F2p_z′, H1s′) orbital pair singlet coupled. In Fig. 7, we plot the SCGVB orbitals of FH at selected points from ΔR = +0.5 Å to ΔR = −0.8 Å. At ΔR = +0.5 Å, the FH orbitals are those expected for the SCGVB description of an FH molecule with a polar covalent bond: a (F2p_z′, H1s′) bond pair with the H1s′ orbital having significant F2p_z atomic character, an F2s′ orbital that is polarized away from the FH bond pair, and (F2p_x′, F2p_y′) orbitals that are largely unchanged from those in the fluorine atom. At ΔR = +0.1 Å, we see a noticeable change in the H1s′ orbital—it has spatially contracted and delocalized even more onto the fluorine atom. This change in the size of the orbital continues as ΔR decreases. The F2p_z′ orbital also appears to have contracted slightly. At ΔR = −0.8 Å, even the F2s′ and F2p_x′ orbitals appear to be slightly smaller relative to their shapes at ΔR = +0.5 Å. Clearly, the SCGVB orbitals of FH are changing in not-so-subtle ways from ΔR = +0.5 to −0.8 Å.

To help quantify the changes in the FH orbitals, we performed an atomic orbital analysis of the SCGVB orbitals.^48,49 This analysis projects the atomic F2s, F2p and H1s orbitals onto the SCGVB orbitals, providing a quantitative measure, namely, $PX2(φai)$⁠, of the atomic X orbital composition of the SCGVB φ_ai orbital—the closer the value of $PX2(φai)$ is to unity, the closer the φ_ai orbital resembles the atomic X orbital.⁷⁰ As shown in Fig. 7, the F2p_z′ and H1s′ bond orbitals are the orbitals most affected by the decrease in ΔR. In Fig. 8, we plot the atomic orbital compositions of these two orbitals as a function of ΔR. The evolution of the composition of these two bond orbitals from ΔR = +0.6 Å to ΔR = −0.8 Å is summarized below.

• F2p_z′ orbital. At ΔR = +0.6 Å, the F2p_z′ orbital is essentially an atomic F2p_z orbital with $PF2pz2(F2pz′)$ = 0.995. As ΔR decreases, the F2p_z component decreases and the H1s component increases. The latter reaches a maximum at ΔR ≈ −0.35 Å, after which the H1s component of the F2p_z′ orbital decreases. In this region, the F2p_z component of the F2p_z′ orbital continues to decrease until ΔR ≈ −0.7 Å, at which point it starts to increase again. By ΔR = −0.8 Å, the F2p_z′ orbital has acquired a substantial amount of F2s + F2p_z atomic character, $PF2s +2pz2(F2pz′)$ = 0.836, although its character is dominated by the F2p_z atomic orbital, $PF2pz2(F2pz′)$ = 0.702.

• H1s′ orbital. The change in the H1s′ orbital is far more dramatic. At ΔR = +0.6 Å, the H1s′ orbital is largely an H1s atomic orbital, $PH1s2(H1s′)$ = 0.855, but with a significant contribution from the F2p_z atomic orbital, $PF2p2(H1s′)$ = 0.274, as expected for a polar covalent bond. As ΔR decreases, the (F2s, F2p) character of the orbital increases rapidly and by ΔR = −0.8 Å, the H1s′ orbital has largely become a F2p_z orbital, $PF2pz2(H1s′)$ = 0.843. At this distance, $PF2s2(H1s′)$ is just 0.151.

Note that the crossing between the $PF2s + F2p2(H1s′)$ and $PH1s2(H1s′)$ curves occurs at ΔR ≈ +0.1 Å, essentially at the same point that the ΔE_DEC(ΔR) curve for FH has a minimum.

In summary, at ΔR = +0.6 Å, the F2p_z′ and H1s′ orbitals resemble those expected for a polar covalent FH bond, but, by ΔR = −0.8 Å, both of these orbitals resemble F2p_z orbitals.

Both the visual inspection of the SCGVB orbitals and the atomic orbital analysis, especially the latter, indicate that there are substantial changes in nature of the orbitals in the region around R_e. From ΔR = ∞ to ΔR = +0.6 Å, the electronic structure of the FH molecule has an (F2p_z′, H1s′) bond pair and the magnitude of ΔE_DEC(ΔR) is continuously increasing. As ΔR decreases from +0.6 Å, the (nominal) H1s′ orbital increasingly resembles an F2p_z atomic orbital and at ΔR ≈ +0.1 Å, it has become an equal mixture of an H1s atomic orbital and an F2p_z atomic orbital (the atomic F2s component is still small at this distance). From the minimum in the ΔE_DEC(ΔR) curve at ΔR ≈ +0.1 Å to the maximum in the curve at ΔR ≈ −0.6 Å, the electronic structure of the FH molecule undergoes a transition to one that resembles an F⁻-like anion plus a nearby proton with a (F2p_z′, F2p_z″)-like orbital pair, not a (F2p_z′, H1s′) bond pair. The decrease in the magnitude of ΔE_DEC(ΔR) in this region directly correlates with this change in the electronic structure of the FH molecule. As ΔR decreases below −0.6 Å, the electronic structure of the FH molecule begins to resemble that of a neon atom with all orbitals doubly occupied except for the (Ne2p_z′, Ne2p_z″) pair and ΔE_DEC(ΔR) approaches that of the corresponding neon atom.

Given the above analysis for the BH and FH molecules, the overall behavior of the ΔE_DEC(ΔR) curves for the AH molecules seen in Fig. 3(b) can now be understood. From large ΔR to an inflection point, say ΔR^†, the magnitude of ΔE_DEC(ΔR) increases nearly exponentially (except for BH, which seems to have two nearly exponential regions). For ΔR < ΔR^†, the slopes of the ΔE_DEC(ΔR) curves decrease. As ΔR continues to decrease, there is a clear “kink” in the ΔE_DEC(ΔR) curve for BH and CH at a value of ΔR, ΔR*, which decreases from BH to CH. For the more polar molecules, NH–FH, the change in the ΔE_DEC(ΔR) curves leads to minima at ΔR*, the depth and location of which are strongly influenced by the polarity of the AH bond—the more polar the bond, the larger the depth and the larger ΔR*. By ΔR*, the electronic structure of the molecule has begun transitioning from a (A2p_z′, H1s′) bond pair to a polarized (A2p_z′, A2p_z″)-like anion pair with the nearby proton resulting in a noticeable contraction of the orbitals related to those of the A⁻ anion. This change in the electronic structure of the molecule as ΔR decreases leads to the observed decrease in the magnitude of the dynamical electron correlation energy. The decrease in the magnitude of ΔE_DEC(ΔR) continues until ΔR^‡, at which point there is the maximum in the curve. For ΔR < ΔR^‡, the magnitude of ΔE_DEC(ΔR) increases again, this time tending toward the value of ΔE_DEC(ΔR) appropriate for the corresponding (A+1) atom with an [(A+1)2p_z′, (A+1)2p_z″] pair instead of a (A2p_z′, H1s′) bond pair. All of the curves for ΔE_DEC(ΔR) shown in Fig. 3(b) appear to follow this pattern with the behavior becoming more and more pronounced with the increasing polarity of the AH bond.

Following the study of the AH series, we extended our study of the effects of dynamical correlation on covalent chemical bonds to the AF series (A = B–F). This series differs from the AH series in two fundamental ways. First, bond polarity runs in the opposite direction—the BF bond is the most polar covalent bond in the series with the polarity decreasing from BF to F₂. In fact, the bond in F₂ is a pure covalent bond. Second, lone pairs are present on both atoms with the repulsive interaction between the lone pairs, i.e., Pauli repulsion, increasing along the series from BF to F₂. Thus, the AF series provides a progression of covalent bond types that are distinctly different than, although still related to, those for the simpler AH series.

Again, we first examined the role of dynamical electron correlation on the AF molecules at their respective equilibrium geometries, focusing on E_DEC(R_e) and ΔE_DEC(R_e) along with the spectroscopic constants R_e, ω_e, and D_e. The results of the SCGVB, vCAS(SCGVB)+1+2, and CCSD(T)/RCCSD(T) calculations on the first-row diatomic fluorides are summarized in Tables IV–VI. Table IV includes the calculated equilibrium total energies (E_e), bond distances (R_e), vibrational frequencies (ω_e), and dissociation energies (D_e). The comparison of the results of the CCSD(T)/RCCSD(T) calculations with the available experimental data⁶⁶ indicates that these calculations provide a very good description of the molecules, although the agreement is, as expected, somewhat worse than for the AH series. Table V contains the SCGVB energies of the AF molecules at the vCAS(SCGVB)+1+2 and CCSD(T)/RCCSD(T) values of the equilibrium internuclear distances (R_e) as well as the SCGVB, vCAS(SCGVB)+1+2, and CCSD(T)/RCCSD(T) energies at R = ∞. Finally, Table VI contains the dynamical correction energies (E_DEC) for the AF molecules at the equilibrium internuclear distances (R_e) and at R = ∞ plus ΔE_DEC(R_e) = E_DEC(R_e) − E_DEC(∞).

The dynamical correlation energies for the AF molecules, E_DEC(R_e), E_DEC(∞), and ΔE_DEC(R_e), are plotted in Fig. 9. For the AF series, the vCAS(SCGVB)+1+2 calculations recoup less of E_DEC at R_e relative to the CCSD(T)/RCCSD(T) calculations, with the percentages ranging from 93.3% in BF (Δ = −13.7 kcal/mol) to 91.8% in F₂ (Δ = −26.9 kcal/mol). The percentages at R = ∞ are only slightly better, 94.1% for BF (Δ = −10.4) to 92.6% in FH (Δ = −22.7 kcal/mol). Although these results could possibly be improved by using a full vCASSCF+1+2 calculation to calculate the energy of the AF molecules, as we have noted before,⁵⁷ such calculations often lead to incorrect orbitals and inconsistencies in the energy, so we chose not to do so. We do not believe this leads to a significant problem since the trends in the vCAS(SCGVB)+1+2 and CCSD(T)/RCCSD(T) results, especially ΔE_DEC(R_e), are very similar.

As we saw before, the magnitudes of E_DEC(R_e) and E_DEC(∞) increase from BF to F₂, and the increases are again nearly linear, especially so from NF to F₂. What is most interesting in the AF series are the changes in ΔE_DEC(R_e) from BF to F₂. The magnitude of ΔE_DEC(R_e) is significantly less for BF (−23.6/−27.0 kcal/mol) than for CF (−30.9/−36.1 kcal/mol), undoubtedly for the same reason as in the AH series, but then it essentially plateaus from CF to NF (−29.8/−36.6 kcal/mol) before decreasing for OF (−24.8/−31.4 kcal/mol) and F₂ (−17.1/−21.2 kcal/mol) [again, the first value is from vCAS(SCGVB)+1+2 calculations and the second from CCSD(T)/RCCSD(T) calculations]. This behavior is quite different than that observed in the AH series [compare with the plot of ΔE_DEC(R_e) in Fig. 1]. The increase and then decrease in the magnitude of ΔE_DEC(R_e) from BF to F₂ is at least partially a result of the large increase in the R_e’s in the AF series from CF to F₂ illustrated in Fig. 9 (see also the discussion below) combined with the decreasing size of the A2s and A2p orbitals from boron to fluorine. The combination of these two factors keeps the electrons in the orbitals centered on the two atoms at R_e increasingly separated along the progression from CF to F₂, leading to a decrease in the magnitude of ΔE_DEC(R_e) from NF to F₂.

As noted in Sec. III A, the change in D_e along the AF series as a result of dynamical electron correlation is simply the negative of that for ΔE_DEC(R_e). This means that the contribution of dynamical electron correlation to D_e also increases from BF to CF, essentially plateaus from CF to NF, and then decreases from NF to OF and F₂. We will not discuss the changes in D_e further; the impact of dynamical correlation on D_e can be inferred from the changes in ΔE_DEC(R_e) above.

We must, however, discuss the changes in R_e. In the AH series, the bond lengths become shorter and shorter from BH to FH as expected from the decreasing size of the A2p orbitals involved in the AH bond; see Fig. 10. In the AF series, on the other hand, the calculated R_e’s increase from BF to F₂, although only slightly from BF to CF but then nearly linearly from CF to F₂. This increase in R_e is predicted by the SCGVB calculations and is due to the increase in Pauli repulsion between the lone pairs on the two atoms from CF to F₂. In CF, the 2pπ′ orbitals first become occupied, with the number of electrons in the 2pπ′ orbitals increasing linearly from CF to F₂. As a result, from CF to F₂, SCGVB calculations predict that R_e increases nearly linearly. The inclusion of dynamic electron correlation reduces the Pauli repulsion between the electrons in the orbitals on the two atoms, shortening the AF bonds, i.e., ΔR_e is negative with the BF molecule being the exception, ΔR_e = −0.0007/+0.0027 Å, for the reasons discussed earlier. ΔR_e is also relatively small for CF (−0.0064/−0.0019 Å) but then increases from NF (−0.0273/−0.0233 Å) to OF (−0.0557/−0.0577 Å) before decreasing slightly for F₂ (−0.0527/−0.0545 Å); see Fig. 11.

The changes in Δω_e are also quite significant for the later molecules in the series. Although it is just +5.4/−10.3 cm⁻¹ and +43.3/+25.7 cm⁻¹ for BF and CF, respectively, it increases to +129.7/+132.0 cm⁻¹, +212.8/+259.7 cm⁻¹, and +192.5/+223.6 cm⁻¹ for NF, OF, and F₂, respectively. For F₂, the difference is 24% of ω_e, hardly a negligible amount. All in all, comparing the AH and AF series, we find that the impact of dynamical correlation on R_e and ω_e in the AF series is the opposite of that in the AH series; compare Fig. 11 with Fig. 2. This can be largely attributed to the important role played by the Pauli repulsion between the lone pairs of the two atoms in the AF series.

As we saw in the AH series, the changes in E_DEC(R) for the AF series (A = B–F) are critical to understanding how dynamical electron correlation affects the properties of the AF molecules at their equilibrium geometries. Figure 12 presents two plots for the AF series, one for E_DEC(R) and one for ΔE_DEC(ΔR). Although the magnitudes of E_DEC(R) for the AF series noted in Fig. 12(a) are much larger than for the AH series, the shapes of the E_DEC(R) curves are similar. As expected, the increase in the magnitude of ΔE_DEC(R) as R decreases is somewhat larger in the AF series than in the AH series. In both cases, the magnitude of ΔE_DEC(R) at large R increases nearly exponentially, although in the AF series the curves are bent slightly more upward at large R; see Fig. 13. Again, as in the AH series, the slope of the curve decreases as R decreases beyond some critical point. That point is the largest for BF with a clear shift to smaller R along the series BF–F₂.

The second plot [Fig. 12(b)] is a plot of ΔE_DEC(ΔR); see Table IV for the values of R_e from the vCAS(SCGVB)+1+2 calculations. The following three trends are observed in this figure:

• The increase in the magnitude of ΔE_DEC(ΔR) begins at the largest ΔR in BF, with the curves being shifted progressively inward from BF to F₂. This behavior can largely be attributed to the decreasing size of the A2s and A2p orbitals from boron to fluorine.

• The magnitude of ΔE_DEC(ΔR) at ΔR = 0.0 Å (R_e) increases from BF to NF but then decreases from NF to F₂. This is, of course, the same trend that we observed in ΔE_DEC(R_e), the reasons for which were discussed above.

• The ΔE_DEC(ΔR) curves for the very polar BF, CF, and NF molecules show clear signs of having minima just as occurred in the very polar FH molecule. For BF, there is an unusually broad minimum at slightly less than ΔR = 0.0 Å (R_e), and for CF and NF, the minima occur around ΔR ≈ −0.35 Å (CF) and ΔR ≈ −0.4 Å (NF). Although there is no sign of minima in the curves for OF and F₂, there may well be small minima or “kinks” for ΔR < −0.5 Å, just as there was in the AH series for the least polar molecules. However, these points will lie at very high energies on the repulsive wall of the relevant potential energy curves, not in the bound region of the potential energy curve.

Although the behavior seen in Fig. 12(b) is again unusual, given the results for the AH series, it is no longer surprising. The ΔE_DEC(ΔR) curves for the AF series reflect much of what was observed in the AH series when the differences in the two series are taken into account.

Based on the behavior of the ΔE_DEC(ΔR) curves in the AF series, we can again understand the unusual effect of dynamical electron correlation on the differences between the R_e’s and ω_e’s, predicted by the SCGVB and fully correlated, vCAS(SCGVB)+1+2 and CCSD(T)/RCCSD(T), calculations. ΔR_e is approximately the same in BF and CF, −0.0007/+0.0027 Å and −0.0064/−0.0019 Å, respectively, which is consistent with the smaller slopes of their ΔE_DEC(ΔR) curves around ΔR = 0.0 Å (R_e). The magnitude of ΔR_e increases rather dramatically from NF to OF, −0.0273/−0.0233 Å (NF) and −0.0557/−0.577 Å (OF), before nearly plateauing from OF to F₂, −0.0527/−0.0545 Å. These changes in ΔR_e are, once again, a direct result of the differing slopes of the ΔE_DEC(ΔR) curves at ΔR = 0.0 Å. Similar arguments apply to the changes in Δω_e, although, once again, it is the curvatures of the ΔE_DEC(ΔR) curves at ΔR = 0.0 Å, which are relevant for predicting Δω_e.

In Fig. 14, the potential energy curves along with the curves of ΔE_DEC(R) of the AF series are plotted. None of the “kinks” or minima in these curves occurs in the bound regions of the potential energy curves. Thus, they are not expected to have a significant effect on the properties of these molecules. However, changes in the slope and curvature of the ΔE_DEC(R) curves are clearly evident for the more polar molecules, especially BF and CF, which will lead to changes in the properties of these molecules just as was observed for the spectroscopic constants.

In Fig. 15, the ΔE_DEC(ΔR) curves for the BF molecule are plotted over an extended range of ΔR. As can be seen, these curves resemble those for the FH curve, although the effect of the more polar BF bond is clearly evident in the very broad minimum in the region around ΔR = 0.0 Å. In contrast to the curves for the FH molecule, the ΔE_DEC(ΔR) curves for the BF molecule from the vCAS(SCGVB)+1+2 and CCSD(T) calculations are significantly different. The minimum in the curve from the CCSD(T) calculations is at significantly larger ΔR and more sharply curved. On the other hand, the maxima in the two curves are nearly at the same value of ΔR, between −0.6 and −0.7 Å. Overall, however, the general shape of the ΔE_DEC(ΔR) curves from the two calculations are remarkably similar.

The changes in the atomic orbital composition of the bond orbitals for the BF molecule, (B2p_z′, F2p_z′), in the region around ΔR = 0.0 Å, as revealed by the atomic orbital projection analysis, are illustrated in Fig. 16. At ΔR = 1.5 Å, which is a point off the plot to the right, the B2p_z′ orbital is largely a B2p_z hybrid atomic orbital with $PB2s + B2p2(B2pz′)$ = 0.877 and h_2p/2s = 7.03, which has delocalized slightly on the fluorine atom, $PF2p2(B2pz′)$ = 0.170, and the F2p_z′ orbital is essentially an F2p_z atomic orbital with $PF2p2(F2pz′)$ = 0.996. As ΔR decreases, these orbitals change as follows:

• B2p_z′ orbital. By ΔR = +0.6 Å, the B2p_z′ orbital has delocalized onto the fluorine atom to such an extent that $PB2s + B2p2(B2pz′)$ ≈ $PF2s + F2p2(B2pz′)$⁠. From ΔR = +0.6 Å to ΔR = −0.3 Å, the B2p_z atomic component of the B2p_z′ orbital decreases and the F2p_z atomic component increases, becoming 0.379 and 0.866, respectively, at ΔR = −0.3 Å. Below ΔR = −0.3 Å, the F2p_z component of the B2p_z′ orbital decreases and the B2p_z component continues to decrease. By ΔR = −0.73 Å, the leftmost point on the curve, the nominally B2p_z′ orbital has no definitive atomic character, with its major components, the F2s + F2p_z atomic orbitals equal to 0.622 and its B2s + B2p_z atomic orbitals just equal to 0.011.

• F2p_z′ orbital. At ΔR = +0.6 Å, the F2p_z′ orbital is still largely a F2p_z atomic orbital, $PF2p2(F2pz′)$ = 0.954. By ΔR = 0.0 Å, the F2p_z′ orbital has changed little, adding a small measure of B2p_z atomic character. As ΔR continues to decrease, however, the F2p_z character of this orbital continues to decrease dropping to just 0.567 at ΔR = −0.73 Å. Thus, at this point, the nominal F2p_z′ orbital is also not well defined.

In summary, at ΔR = −0.73 Å, neither of the two bond orbitals in BF are well described by their B2p_z and F2p_z atomic character. The complicated nature of the two bond orbitals is likely a result of the magnitude of the combined nuclear charge of the boron and fluorine nuclei at R = 0.5354 Å (ΔR = −0.73 Å). The presence of these two highly charged nuclei would be expected to cause major changes in the electronic structure of the molecule as the united atom is approached. This is in distinct contrast to the AH series where a smooth transition from the molecule to the united atom was observed.

Our ability to predict molecular properties and the outcomes of molecular processes using electronic structure theory is limited by our ability to properly describe the effects of electron correlation. There are two distinct types of electron correlation: non-dynamical correlation, which is a result of near-degeneracies among electronic configurations that are specific to the atomic or molecular state, and dynamical correlation, which is a result of the instantaneous correlation of the electrons. Non-dynamical correlation can be taken into account by using the SCGVB wave function, but the dynamical correlation is far more difficult to describe and, as such, is the limiting factor in accurate predictions of molecular properties and the outcomes of molecular processes.

Clementi and Veillard¹⁷ showed that, for the ground states of the first-row atoms, the dynamical correlation energy has a relatively smooth dependence on the number of electrons in the atom. For any diatomic molecules formed from these atoms, this also is clearly the case at R = ∞. In this study, we determined the impact of the dynamical correlation energy, E_DEC(R), on the formation of the covalent bonds in the AH and AF molecules with A = B–F. These two series represent two distinct types of covalent bonds. In the AH series, the polarity of the bond increases from BH to FH, while in the AF series, it decreases from BF to F₂. Furthermore, there are lone pairs on both atoms in the AF series and these would be expected to have a significant effect on E_DEC(R). Our study examined the effect of dynamical electron correlation on the spectroscopic constants, (R_e, ω_e, D_e), of the AH and AF molecules as well as the dependence of the dynamical correlation energy on the internuclear separation: E_DEC(R), ΔE_DEC(R) = E_DEC(R) − E_DEC(∞), and ΔE_DEC(ΔR) = E_DEC(ΔR) − E_DEC(∞) with ΔR = R − R_e.

Although E_DEC varies relatively smoothly for the ground states of the first-row atoms from beryllium to neon, this does not carry over to the AF and AF molecules at their equilibrium geometries. For example, in the AH series, the changes in D_e caused by dynamical correlation, ΔD_e, increases substantially from BH to CH and then increases only slowly from CH to FH, despite the increasing number of electrons. The corresponding changes in R_e, ΔR_e, were equally puzzling—ΔR_e was approximately the same in BH and CH, decreased in magnitude from CH to OH, and then, actually became positive for FH. In the AF series, ΔD_e first increased and then decreased and the changes in ΔR_e were essentially the reverse of those in the AH series. Similar findings were obtained for ω_e. These changes in (R_e, ω_e, D_e) could only be understood by examining the change in E_DEC(R).

Changes in the behavior of the E_DEC(R) curves in the region of the equilibrium geometries of the AH and AF molecules resulted in irregular changes in the spectroscopic constants. Determination of the R dependence of E_DEC, specifically as ΔE_DEC(ΔR), showed that the variation of ΔE_DEC(ΔR) is complicated as ΔR approaches zero (R_e) and, as a result, has an erratic impact on the spectroscopic constants. There are the following four major regions in ΔR to consider:

1. At large ΔR, the increase in the magnitude of ΔE_DEC(R) is nearly exponential from R = ∞ to some smaller value of ΔR, ΔR^†. As ΔR^† is passed, the rate of increase slows with decreasing ΔR, an indication that the interpenetration of the charge clouds associated with the two atoms, leads to saturation of the dynamical correlation energy. The region over which the slope of the ΔE_DEC(R) curve decreases largely depends on the size of the A2s and A2p orbitals with some contribution from the polarity of the bond. The region from ΔR = ∞ to ΔR = ΔR^† is important but relatively uninteresting since the behavior of ΔE_DEC(R) is easily described.

2. At distances less that ΔR^†, the slope continues to decrease and, in the most polar molecules, the ΔE_DEC(ΔR) curve has a minimum at a value of ΔR, ΔR*. In the less polar molecules, there can be a “kink” in ΔE_DEC(ΔR), i.e., a rapid change in the slope. The location of ΔR* largely depends on the polarity of the AH or AF bond as well as the size of the A2s and A2p orbitals. For this reason, the variation in the value of ΔR* is complicated, e.g., in the AH series, the ΔR* associated with the “kink” decreases from BH to NH with the ΔR* associated with the minimum increasing from NH to FH. The changes in the magnitude, slope, and curvature of the ΔE_DEC(ΔR) curves around R_e (ΔR = 0.0 Å) directly correlate with the observed changes in the calculated spectroscopic constants, (ΔD_e, ΔR_e, Δω_e), of the AH and AF molecules.

3. In the most polar molecules, the minimum in ΔE_DEC(ΔR) is followed by a maximum at a much shorter ΔR, ΔR^‡. In the region, from ΔR* to ΔR^‡, the electronic structure of the molecules changes from one resembling a polar A–H or A–F bond to one resembling a polarized A⁻H⁺ or A⁺F⁻ ion pair with an adjacent positively charged nucleus. The most significant changes are in the bond pairs as they transition to a (A2p_z′, A2p_z″)-like lone pair in the AH series or a (F2p_z′, F2p_z″)-like lone pair in the AF series. In the transition region, the magnitude of ΔE_DEC(ΔR) decreases as ΔR decreases.

4. At ΔR < ΔR^‡, the orbitals for the AH molecules slowly approach those appropriate for the (A+1) atom with a [(A+1)2p_z′, (A+1)2p_z″] pair (the state of the united atom in these calculations is not a pure state). The changes in the orbitals are much more tortuous for the AF molecules as the combined nuclear charges of the two nuclei are very large and the changes in the electronic structures of the AF molecules as they approach the united atom are very complicated.

In summary, this study found that the nature of the ΔE_DEC(R) curves for both the AH and AF series, A = B–F, is far more complicated than initially anticipated. Although the behavior of the curves at large R was as expected with the magnitude of ΔE_DEC(R) increasing nearly exponentially with decreasing R, their behavior as R → R_e was not. The unusual behavior of the ΔE_DEC(R) curves at short R is driven by the changes in the electronic structure of the molecules, most notably by the changes in the orbitals involved in the bond. These changes are strongly dependent on the polarities of the AH or AF bonds. For the AH series, there was a transition from a (A2p_z′, H1s′) bond pair to a (A2p_z′, A2p_z″)-like lone pair, which results in a decrease in the magnitude of ΔE_DEC(R). A similar result was found for the AF series, although the changes in the electronic structure of the AF molecules as the united atoms are approached, which is far more complex for these molecules. The unusual behavior of the ΔE_DEC(R) curves was more pronounced when the covalent bonds were very polar although it does not necessarily disappear for less polar bonds. The results reported here clearly indicate that we have much to learn about the nature of dynamical electron correlation and its effect on the chemical bond and other molecular properties and processes.

This research was supported, in part, by the Center for Scalable Predictive Methods for Excitations and Correlated phenomena (SPEC), which is funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Chemical Sciences, Geosciences and Biosciences Division, as part of the Computational Chemical Sciences Program at Pacific Northwest National Laboratory.

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.