Why edge inversion? Theoretical characterization of the bonding in the transition states for inversion in F _n NH_(3−n) and F_nPH_(3−n) (n = 0–3)

Abstract

As first noted by Dixon et al. (J Am Chem Soc 108:2461–2462, 1986), heavily fluorinated pyramidal phosphorus compounds, e.g., F _n PH_(3−n) with n > 1, invert through a T-shaped transition state (edge inversion) rather than the D_3h-like transition states (vertex inversion) found in the corresponding nitrogen compounds and less fluorinated phosphorus compounds. Subsequent studies by Dixon and coworkers established that this is a general phenomenon and has important chemical consequences. But what is the reason for the change in the structure of the transition state? Recent theoretical investigations have resulted in the discovery of a new type of chemical bond, the recoupled pair bond. In particular, it was found that recoupled pair bond dyads account for the hypervalency of the elements beyond the first row. In this paper, we show that recoupled pair bond dyads also account for the existence of the edge inversion pathway in heavily fluorinated phosphorus compounds and likely account for the presence of the lower energy inversion pathways in pyramidal compounds of other elements beyond the first row.

1 Introduction

The structures, energetics and properties of molecules formed from elements in the first row of the periodic table, Li to Ne, can be dramatically different from those formed from elements in the subsequent rows—the so-called first-row anomaly. The anomaly manifests itself in a number of ways, such as the inability of the first row p-block elements to form hypervalent species. N and P are an example of this anomaly: P is able to form hypervalent molecules such as PF₅ and PCl₅, while N only forms NF₃ and NCl₃. Another manifestation of the difference between N and P has drawn quite a bit of attention. The ground states of NH₃, NF₃, PH₃ and PF₃ are all pyramidal, as expected. However, NH₃, NF₃ and PH₃ invert through a transition state with D_3h symmetry, while the transition state for inversion in PF₃ is T-shaped with C_2v symmetry.

Investigations of the T-shaped pnictogen transition states for inversion were inspired by the experimental synthesis of the first molecule containing a T-shaped, tricoordinated hypervalent phosphorous structure, 5-aza-2,8-dioxa-3,7-di-tert-butyl-l-phosphabicyclo[3.3.0]octa-3,6-diene (ADPO) [1, 2]. Another hypervalent 10-P-3 compound, an intermediate, was discovered by Lochschmidt and Schmidpeter at approximately the same time [3]. Following up on this discovery, Dixon and co-workers [4–8] explored the structures of the transition states for inversion in phosphorus compounds. They performed calculations on PH₃, FPH₂, F₂PH and PF₃ and found that F₂PH and PF₃ have T-shaped transition states while FPH₂ and PH₃ have D_3h–like transition states. They pointed out that when inversion occurs through a T-shaped structure, the lone pair orbital is in the molecular plane, not perpendicular to the plane as in the D_3h-like transition states. Inversion through a T-shaped transition state is referred to as edge inversion, while inversion through a D_3h-like transition state is referred to as vertex inversion.

Arduengo, Dixon and Roe experimentally verified the edge inversion mechanism in a bi-cyclo[3.3.0] octane ring system that is a saturated analog of ADPO by measuring the barrier to inversion at the tricoordinated phosphorous atom [5]. Although ADPO contains a T-shaped phosphorus structure in its ground state, its saturated analog contains a pyramidal phosphorus structure in its ground state and a T-shaped phosphorus structure in its transition state. They argued that the conjugated π system of ADPO stabilizes the T-shaped structure while the saturated analog of ADPO is not so stabilized. The electronic structure of all of these molecules can be easily understood once one recognizes the ability of phosphorous to form both covalent and recoupled pair bonds in simple tricoordinated molecules such as PF₃ [9], as will be shown in this paper.

Although edge inversion has been well established for pyramidal molecules with central atoms beyond the first row and with electronegative ligands, the reason for the difference in inversion pathway upon fluorination has been the subject of some debate. Both perturbation molecular orbital arguments involving the HOMO–LUMO gap [10] as well as pseudo-Jahn–Teller effects [11] have been used to rationalize the T-shaped transition state structures. Woon and Dunning, on the other hand, noted that the two axial bonds in the transition state for inversion in PF₃ closely resemble those in the PF₂(A²Π) state, which has a recoupled pair bond dyad, and concluded that the ability of the P atom to form a very stable recoupled pair bond dyad is the source of this anomaly [9]. In fact, the formation of recoupled pair bonds is the basis for the bonding in hypervalent molecules, as shown in studies on SF _n [12], PF _n [9], ClF _n [13] and a number of related compounds. The T-shaped transition state for inversion of PF₃ should, in fact, be considered hypervalent, even though it is tricoordinated, because it possesses one of the hallmarks of hypervalent compounds—a recoupled pair bond dyad.

In the present study, we systematically investigated the ground and transition states for inversion of NH₃, PH₃ and their F substituents, F _n NH_(3−n) and F _n PH_(3−n) (n = 1–3), and show that other than PF₃, F₂PH is the only species with a T-shaped transition state. Accurate predictions of the structures and energies of the F _n NH_(3−n) and F _n PH_(3−n) (n = 0–3) species were obtained by using coupled cluster methods [14–17] with large correlation consistent basis sets [18–21]. We then used generalized valence bond (GVB) theory [22, 23] to obtain insights into the nature of the bonding in these molecules and explain the similarities and differences in the structural and energetic trends of the N and P species.

The layout of the paper is as follows: Sect. 2 describes the computational methods we used for this study, including a brief overview of GVB theory; Sect. 3 presents the optimized geometries and energetics for the ground and transition states of F _n NH_(3−n) and F _n PH_(3−n), proposes the three major questions we need to answer, analyzes the GVB wave functions of related atoms and molecules, discusses the role recoupled pair bonding plays in the transition states and answers the questions raised; and finally, we summarize our findings in Sect. 4.

2 Computational methods

The calculations presented in this study were performed with the Molpro suite of quantum chemical programs (version 2008.1 and 2010.1) [24]. In order to provide accurate geometries and energetics, the structures, energies and frequencies of the F _n NH_(3−n) and F _n PH_(3−n) molecules were determined with single-reference restricted singles and doubles coupled cluster theory with perturbative triples [CCSD(T)] [14–17]. For geometry optimizations and energies, augmented correlation consistent basis sets of quadruple zeta quality (aug-cc-pVQZ) were used for the first row atoms (H, N and F), and the corresponding d-function augmented set [aug-cc-pV(Q + d)Z] was used for the second row P atom. For frequency calculations, aug-cc-pVTZ basis sets were used for the first row atoms and an aug-cc-pV(T + d)Z basis set was used for the P atom [18–21]. The frequency calculations were performed at geometries optimized using the same basis sets. The shorthand notation AVXZ (X = T, Q) will be used to represent the sets of a specific quality (including the extra d-function on the P atom). The frequency calculations enabled us to conclusively identify the ground states (all real frequencies) and transition states (one imaginary frequency) for all of the species.

The inversion barrier for each molecule is calculated as the difference between the electronic energy of the transition state and the ground state, i.e., it does not include the zero point energy correction. We did this to focus on the effect of the changes in the electronic structure of the molecules upon inversion; the reader can easily correct these numbers using the vibrational frequencies given in the tables. Energies are quoted with two significant figures after the decimal place for comparison with other theoretical calculations.

Multireference configuration interaction (MRCI) calculations based on valence complete active space self-consistent field (CASSCF) wave functions with the quadruples corrections (+Q) [25–30] were used to determine the excitation energies of the N and P atoms using the AVQZ basis set.

To characterize the nature of the bonding in the F _n NH_(3−n) and F _n PH_(3−n) species, we used the generalized valence bond (GVB) method [22, 23]. The GVB wave function is well suited to analysis of the bonding in molecules as it describes bond-breaking processes properly. The GVB wave function is also inherently more accurate than the Hartree–Fock (HF) wave function, including the most important non-dynamical correlation effects represented in a valence CASSCF wave function. At the same time, the GVB wave function is concise, offering a clear physical picture of the electronic structure of a molecule that is readily connected with those of the atoms or fragments of which it is composed.

In the GVB framework, a covalent bond is formed by singlet coupling two electrons in a pair of overlapping, singly occupied GVB orbitals concentrated on the two atoms involved in the bond. A recoupled pair bond, on the other hand, results from a two-center, three-electron interaction. Nevertheless, it is also easily described in GVB theory because each electron has its own orbital [31]. The recoupled pair bond dyad, which is of particular interest here, is simply two singlet-coupled bonding pairs—one from the original recoupled pair bond and the other from a covalent bond formed with the electron left over from forming the recoupled pair bond. The remarkable stability of the recoupled pair bond dyad is a direct result of the ionicity of these two bonds—a dyad is only found when the two ligands are very ionic [32, 33]. The GVB orbitals, orbital overlaps and spin coupling functions provide a concise picture of the electronic structure of the molecule. The fully variational GVB method is equivalent to the spin-coupled VB method [34], and the CASVB [35–37] program implemented in Molpro was used to perform the calculations with the AVQZ basis sets.

The GVB/spin-coupled VB wave function for a molecular system of n _a active electrons with total spin S and projection M is:

$$\Psi_{\text{GVB}} = \hat{a}\phi_{d1} \phi_{d1} \phi_{d2} \phi_{d2} \ldots \phi_{{dn_{d} }} \phi_{{dn_{d} }} \varphi_{a1} \varphi_{a2} \ldots \varphi_{{an_{a} }} \alpha \beta \alpha \beta \ldots \alpha \beta \Theta_{S,M}^{{n_{a} }} .$$

(1)

In the above equation, $\hat{a}$ is the antisymmetrizer; the set of orbitals, {ϕ _di }, are the set of n _d doubly occupied core and valence orbitals, and the set of orbitals, {φ _ai }, are the set of n _a singly occupied active valence orbitals. The total number of electrons is N _e  = 2n _d  + n _a . The doubly occupied valence orbitals do not directly participate in bonding, although, as we shall see, they can affect which type of bonds are formed. The active orbitals are distinct, singly occupied and non-orthogonal orbitals. The spatial product of orbitals in Eq. (1) is multiplied by a product of αβ spin functions associated with the doubly occupied orbitals times a spin function, $\varTheta_{{S,{M}}}^{{n_{a} }}$, for the electrons in the active orbitals. This spin function is a linear combination of spin eigenfunctions, also known as spin basis functions, which represent the unique ways in which the spins of the n _a electrons in the active orbitals can be coupled to give a total spin of S. Kotani spin functions [38] were used in our study. The Kotani functions are orthogonal to each other, and, therefore, the contribution of each spin function to the total GVB wave function, the weight w _k , is simply the square of its coefficient.

We found that one of the spin eigenfunctions was dominant for all of the molecular systems that we studied here, and it is the perfect pairing (PP) function:

$$\varTheta_{{{S},{M},{\text{PP}}}}^{{n_{a} }} = \frac{(\alpha \beta - \beta \alpha )}{\sqrt 2 }\frac{(\alpha \beta - \beta \alpha )}{\sqrt 2 } \cdots \frac{(\alpha \beta - \beta \alpha )}{\sqrt 2 }$$

(2)

As shown in Eq. (2), the perfect pairing spin function singlet couples all of the active electrons into electron pairs. A pair of singlet-coupled orbitals can describe a lone pair, in which case the orbitals are highly overlapping and concentrated on one atom. Or the orbital pair can describe a bond if the orbitals are overlapping and concentrated on two atoms. The GVB calculations for the states reported in this paper are all 6-in-6 calculations, i.e., there are six active electrons in six GVB orbitals. All spin functions were included in the calculations, although, as noted, the PP spin function was always dominant (w _PP = 0.91–0.99+). This means that there are three pairs of singlet-coupled orbitals, or, since we kept the lone pair orbitals doubly occupied in all calculations, three bonds in both the ground and transition states of the F _n NH_(3−n) and F _n PH_(3−n) molecules.

3 Results and discussion

3.1 Similarities and differences in the F _n NH_(n–3) and F _n PH_(3−n) (n = 0–3) ground states and transition states for inversion

3.1.1 Molecular structures

The optimized structures and geometrical parameters of the ground state (GS, X¹A₁) and transition state (TS, ¹A₁) of all of the F _n NH_(3−n) and F _n PH_(3−n) (n = 0–3) molecules are tabulated in Tables 1, 2, 3 and 4 and illustrated in Fig. 1. The ground states of F _n NH_(3−n) and F _n PH_(3−n) are pyramidal with the singly occupied 1s orbitals of H(²S) and the singly occupied 2p orbitals of F(²P) forming normal 2-electron, 2-center covalent bonds with the three singly occupied 2p and 3p orbitals on N(⁴S) and P(⁴S), respectively. However, the inversion transition states fall into two different structural categories. NH₃, NF₃ and PH₃ have planar transition states with D_3h symmetry (θ _XYX = 120°). The transition states of FNH₂ and FPH₂ are planar with C_2v symmetry and similar bond angles: θ _HNF = 114.0° and θ _HPF = 114.9°. The transition state of F₂NH is also of C_2v symmetry with θ _HNF at 123.7°. These latter three C_2v transition states are, in fact, D_3h-like and fall into the same structural category as NH₃, NF₃ and PH₃.

Table 1 The geometries, frequencies and total energies for the ground states (GS) of the F _n NH_(3−n) (n = 0–3) molecules from CCSD(T) calculations with the indicated basis sets

Table 2 The geometries, frequencies and total energies for the inversion transition states (TS) of the F _n NH_(3−n) (n = 0–3) molecules from CCSD(T) calculations with the indicated basis sets

Table 3 The geometries, frequencies and total energies for the ground states (GS) of the F _n PH_(3−n) (n = 0–3) molecules from CCSD(T) calculations with the indicated basis sets

Table 4 The geometries, frequencies and total energies for the inversion transition states (TS) of the F _n PH_(3−n) (n = 0–3) molecules from CCSD(T) calculations with the indicated basis sets

Fig. 1

Optimized geometries of the F _n NH_(3−n) and F _n PH_(3−n) (n = 0–3) ground states (GS, X¹A₁) and the transition states for inversion (TS, ¹A₁) obtained from CCSD(T)/AVQZ calculations. Bond distances are in Å and bond angles in degree. θ corresponds to bond angle and τ corresponds to dihedral angle. N is color coded in blue, P in rusty orange, F in cyan and H in gray

The transition states for F₂PH and PF₃, on the other hand, are outliers. In the transition state for inversion of F₂PH, the central P atom and two of the F atoms, the axial F atoms, lie almost on a straight line with θ _FPF = 169.8° and θ _HPF = 84.9°. This quasi-linear structure is also present in PF₃, where the FPF axial angle is 176.0° and the angle between the axial and equatorial F atom is 88.0°. These two transition states are referred to as T-shaped transition states as first characterized by Dixon and coworkers [4–8] and are referred to as edge transition states in contrast to the transition states for the other six F _n NH_(3−n) and F _n PH_(3−n) molecules, which are referred to as vertex transition states. The above observations lead to Question #1: What is the cause of the differences in the structures of the transition states of (F₂NH, F₂PH) and (NF₃, PF₃), as well as those of (PH₃, FPH₂) and (F₂PH, PF₃)?

Besides the obvious structural differences, a closer examination of the bond distances shows that for all of the molecules that have D_3h or D_3h-like transition states, i.e., NH₃, FNH₂, F₂NH₃, NF₃, PH₃ and FPH₂, the bond lengths in the transition states are all shorter than the ground state bond lengths. However, the differences in the bond lengths in F₂PH and PF₃ do not follow the same simple pattern. For F₂PH, the PH bond distance is slightly shorter (Δ = −0.010 Å) and the PF bond distances are significantly longer (Δ = +0.064 Å) in the transition state than in the ground state. For PF₃, the transition state has two types of PF bonds. The two axial PF bonds are much longer (Δ = +0.063 Å) than the ground state PF bond, and the equatorial PF bond is slightly shorter (Δ = −0.006 Å) than the ground state PF bond.

Table 5 summarizes the bond distance changes between the ground state and transition state of all eight molecules. The percentage changes with respect to the bond lengths in the ground state are also listed (in parentheses). The percentage change for molecules in the D_3h-like category ranges from −1.4 % to −3.1 %. On the other hand, the percentage change for the PH bond in F₂PH is only −0.7 %, which is very close to the −0.4 % change for the equatorial PF bond in F₂PF. The percentage change for the axial PF bonds when n = 2 and n = 3 is 4.0 %, which is larger than any of the differences in the other molecules. These data lead to Question #2: Why do the bond distances of the D_3h-like transition states decrease and those of the T-shaped transition states behave very differently?

Table 5 The changes in the bond lengths, ΔR, in F _n NH_(3−n) and F _n PH_(3−n) (n = 0–3), between the ground states (GS) and the inversion transition states (TS): ΔR = R(TS) − R(GS), in Å

3.1.2 Inversion barrier

Here, as noted in Sect. 2, the inversion barrier is defined as the difference in the electronic energies of the transition state and the ground state, i.e., we ignore differences in the zero-point energies even though vibrational frequencies are reported, because we want to focus on the variations in the barriers caused by the changes in the electronic structure of the molecules. Figure 2 shows the plot of the inversion barrier with respect to n, the number of F atoms in the molecule. When N is the central atom, the barrier height increases monotonically from 5.34 to 82.79 kcal/mol as n increases from 0 to 3. The rate of increase increases with n as well, essentially doubling with each additional F atom: 10.31 kcal/mol from n = 0–1, 22.88 kcal/mol from n = 1–2 and 44.26 kcal/mol from n = 2–3. When P is the central atom, the barrier height increases from 33.25 kcal/mol to just 54.00 kcal/mol as n increases from 0 to 3. The rate of increase is 17.29 kcal/mol from n = 0–1, but only 1.50 kcal/mol from n = 1–2 and 1.96 kcal/mol from n = 2–3. Thus, from n = 1–3, the change in the height of the inversion barrier is vastly different in the N series than in the P series: The total increase in the barrier height for F _n NH_(3−n) is 77.45 kcal/mol while the barrier height increase for F _n PH_(3−n) is less than a third of that, 20.75 kcal/mol. So Question #3 is: Why does the barrier change very little from n = 1–3 in the F _n PH_(3−n) series, instead of increasing dramatically like the F _n NH_(3−n) series does?

Fig. 2

Computed inversion barriers for F _n NH_(3−n) and F _n PH_(3−n) (n = 0–3) from CCSD(T)/AVQZ calculations. The inversion barrier is the electronic energy difference between the transition state and the ground state for each molecule without zero-point energy correction

As an aside, we did locate higher lying “transition states” in both F₂PH and PF₃. The geometry of the D_3h-like transition state in F₂PH is R _PH = 1.375 Å, R _PF = 1.571 Å and θ _HPF = 124.8°. This structure lies 32.72 kcal/mol above the lower transition state, at 84.76 kcal/mol. It is a true transition state with only one imaginary frequency: 1,158.2i cm⁻¹ (because of its high energy, we did not follow the reaction path from this transition state to see where it led). When PF₃ is constrained to have a D_3h structure, R _PF = 1.633 Å. The D_3h structure lies 33.12 kcal/mol higher than the T-shaped transition state and 87.12 kcal/mol above the ground state PF₃. However, this configuration has three imaginary frequencies (2 × 214i, 596i cm⁻¹) and therefore is not a true transition state.

3.2 p-Recoupled pair bonding in F₂PH and PF₃ and s-recoupled pair bonding in the other molecules

In the last section, we posed three questions. In this section, we show that the answers to all of these questions center on the ability of the P atom to form recoupled pair bonds with F and, more specifically, p-recoupled pair bond dyads with two F atoms. In molecules other than F₂PH and PF₃, the inversion transition states involve formation of s-recoupled pair bond dyads.

3.2.1 GVB description of the N and P atoms

In order to understand the differences between F _n NH_(3−n) and F _n PH_(3−n) (n = 0–3), we need to understand the differences between the central atoms in these molecules. Figure 3 presents the GVB orbital diagrams for the ground and low-lying excited states of the N and P atoms. The diagrams represent the valence electrons and orbitals. The ground states of the N and P atoms are ⁴S states with three singly occupied valence p orbitals in each atom. In the N/P(⁴S) diagram, the big circle represents the valence s orbital, the small circle represents the out-of-plane p _x orbital, and the two dumbbell shapes represent the two in-plane (p _y , p _z ) orbitals. The dots represent electron occupations. The ground states of F _n NH_(3−n) and F _n PH_(3−n) (n = 0–3) have three normal covalent bonds formed with the electrons in these three singly occupied p orbitals. The pyramidal structures of these molecules are a natural result of the orientations of the three p orbitals in the atom.

Fig. 3

HF and GVB diagrams for N and P atoms; n = 2 for N and n = 3 for P atom. The subscripts “L”, “R”, “I” and “O” represent left, right, inner and outer. The electrons in orbital pairs connected with a red line are singlet coupled

Although the connection of the ground states of the N and P atoms with the ground states of the F _n NH_(3−n) and F _n PH_(3−n) molecules is straightforward, the transition states do not correlate with the ground state atoms. Rather, they correlate with the first excited (²D) states of the atoms. In this state, one electron from a p orbital is excited into one of the other p orbitals with the original p orbital no longer occupied in the configuration (this is schematically represented by the configuration s²p ² _x p ¹ _y or s²p ² _z p ¹ _y in Fig. 3). Now, the atoms can form both covalent bonds (with the singly occupied p _y orbital) as well as recoupled pair bonds and recoupled pair bond dyads (with the s² or p² lone pairs). The calculated excitation energy from the ⁴S state to the ²D state is 55.33 kcal/mol for the N atom and 32.23 kcal/mol for the P atom (MRCI + Q/AVQZ); these numbers are to be compared to 54.97 and 32.48 kcal/mol from the NIST Atomic Spectra Tables [39]. Because of this energetic difference, recoupled pair bonds and recoupled pair bond dyads in P will lie at much lower energies than in N.

As noted above, there are two doubly occupied orbitals in the ²D state, i.e., 2s² and 2p² for N and 3s² and 3p² for P atom. Although these orbitals are doubly occupied in the HF wave function, in the GVB wave function s² and p² lone pairs are each described by two singly occupied, non-orthogonal lobe orbitals. Figure 4 shows the s and p GVB lobe orbitals for the N(²D) and P(²D) states. The s lobe orbitals of both atoms are very similar in shape and orientation, although the P orbitals are more diffuse and span a larger spatial region. The s lobe orbitals are well separated spatially, residing on opposite sides of the central atom, and have overlaps of only 0.768 (N) and 0.776 (P); we refer to them as the s_L and s_R lobe orbitals. In the GVB orbital diagrams 1a and 2a in Fig. 3, the s_L and s_R lobe orbitals are connected with a red line, representing the fact that they are singlet coupled.

Fig. 4

GVB orbitals and overlaps for the N(²D) and P(²D) states. A red line means that the electrons in these singly occupied orbitals are singlet coupled; the values of the orbital overlaps are given above the line. Contours are ±0.10, ±0.15, ±0.20 and ±0.25. Red contour represents positive, and blue contour represents negative orbital phase

On the other hand, the 2p lobe orbitals of N(²D) and the 3p lobe orbitals of P(²D) are very different. Both 2p lobe orbitals on the N(²D) state look like p orbitals, one tighter than the original 2p_z orbital and the other one more diffuse. They lie in the same spatial region, and we refer to them as inner and outer lobe orbitals, 2p_I and 2p_O. In the N(²D) GVB orbital diagram 1b in Fig. 3, these orbitals are represented with two dumbbell shapes, one inside of the other with a red line connecting them. The 3p lobe orbitals of P(²D), on the other hand, are distorted 3p_z orbitals, with one more concentrated on the left side of the P atom and the other more concentrated on the right side; they have an overlap of 0.839 and are referred to as 3p_L and 3p_R lobe orbitals. In diagram 2b in Fig. 3, they are represented with two half-dumbbell shapes connected with a red line.

Both of the lone pairs on the N and P atoms are potentially available for recoupling to form bonds. However, the ease with which a lone pair can be recoupled is dependent on two factors: (1) the spatial orientation of the lobe orbitals and (2) the overlap of the orbitals. To form a strong recoupled pair bond or recoupled pair bond dyad, the lobe orbitals must be localized in different spatial regions and the overlap of the lone pair orbitals must be significantly less than one (the smaller, the better). It also helps if the ligand is very electronegative, because this will reduce the Pauli exchange-repulsion between the bonds (or between the bond and the electron in the left over orbital) formed by recoupling the lone pair [31].

As noted above, the N and P s_L and s_R orbitals are spatially well separated and their overlaps, 0.768 (N) and 0.776 (P), are similar. So, s-recoupled pair bonds can be formed in both N and P. The overlap between the N 2p_I and 2p_O orbitals (0.858) is slightly larger than that between the P 3p_L and 3p_R orbitals (0.839), and both overlaps are much larger than the overlaps between the (2s_L, 2s_R) and (3s_L, 3s_R) lobe orbitals. But, more importantly, the 2p_I and 2p_O orbitals occupy essentially the same spatial region, while the 3p_L and 3p_R orbitals are spatially separated. Therefore, it is far more favorable for P to participate in p-recoupled pair bonding than N. However, the (3p_L, 3p_R) orbitals of the P atom are not as spatially separated as the (3s_L, 3s_R) orbitals, so it will be more difficult to form recoupled bonds with the 3p lone pair than with the 3s lone pair.

From the GVB orbital diagram of the P(²D) state in Fig. 3, one can easily see that three strong bonds can be formed: with the singly occupied (3s_L, 3s_R, 3p _y ) orbitals in 2a and the singly occupied (3p_L, 3p_R, 3p _y ) orbitals in 2b. In 2a, the remaining lone pair is an out-of-plane 3p _x lone pair, and in 2b, it is an in-plane 3s pair. As will be demonstrated and discussed in Sects. 3.2.2 and 3.2.3, with 2a, the resulting structures of the transition states correspond to that of the D_3h-like structures, which have an out-of-plane lone pair. With 2b, the resulting structures will be those of the T-shaped transition states for inversion with an in-plane lone pair. Since the lobe orbitals in 1a, 2a and 2b in the N(²D) and P(²D) states have different properties than the 2p and 3p orbitals in the N(⁴S) and P(⁴S) states, the bonds formed with these orbitals in the transition states will have different lengths, strengths and spatial orientations than those in the ground states.

Which lobe orbitals will be used to form bonds depends on the strength of the resulting bonds as well as the resulting interactions with the electrons in the other orbitals in the molecule. In particular, we note that the formation of bonds with the (3s_L, 3s_R, 3p _y ) orbitals has a lone pair perpendicular to the yz plane that will have repulsive interactions with the lone pair orbitals in F that are also perpendicular to the yz plane. This is not the case when bonds are formed with the (3p_L, 3p_R, 3p_y) orbitals where the lone pair lies in the molecular plane.

3.2.2 GVB orbitals of the transition states

As discussed above, the ground states of the eight molecules in our study have three normal covalent bonds formed by singlet coupling the singly occupied ligand orbitals with the three singly occupied p orbitals of N(⁴S) and P(⁴S). Figure 5 shows the GVB bonding orbitals of the eight transition states, along with the doubly occupied lone pair orbital. When there are equivalent bonds, the GVB orbitals of only one bond are shown. The overlap between the two orbitals that form a bond is also shown. The GVB wave functions of all of the eight transition state molecules are predominantly PP spin coupled with w _PP ranging from 0.91 to more than 0.99.

Fig. 5

Selected GVB orbitals and overlaps for the inversion transition states of F _n NH_(3−n) and F _n PH_(3−n) (n = 0–3). The GVB calculations are 6-in-6 calculations. When two or three bonds are equivalent, the two GVB orbitals constituting one bond are shown. The doubly occupied orbital on the central atom for each molecule is shown following the GVB orbitals. Contours are ±0.10, ±0.15, ±0.20 and ±0.25

NH₃ and PH₃ each have three equivalent bonds. Upon bond formation, the orbitals on the two atoms polarize, hybridize, expand or contract, delocalize, etc. in response to the presence of the other atoms. However, it is clear that, for instance, one of the orbitals participating in the NH bond closely resembles the 2s lobe orbital of N shown in Fig. 4, although with somewhat more 2p character than in the N atom, and the other orbital resembles the 1 s orbital of the H atom. The lone pair orbital is essentially the out-of-plane 2p_x orbital in the N(²D) state. This is consistent with the orbital diagram 1a in Fig. 3. The three equivalent NH bonds result from resonance between the 2s-recoupled pair bond dyad and the normal covalent bond with the 2p_y orbital. The resulting structure has D_3h symmetry because this arrangement reduces the Pauli repulsion between the three bonds, and so, the GVB orbitals on the N atom in the NH₃ transition state are resonance averages (hybrids) of the two 2s lobe orbitals and the 2p_y orbital. The situation is similar in PH₃. The reason for the shorter bond distances in the transition states is that the orbitals on N and P have large s orbital components in the transition states, and s orbitals are closer to the nucleus than p orbitals, especially for the P atom. Therefore, the bonds formed with these orbitals are shorter than those formed with p orbitals in the ground states (Question #2).

The N atom is more electronegative than the P atom, so NH bonds are more polarized toward the N atom than PH bonds. A result of this is that the three orbitals centered on N have much higher overlaps (0.782) with each other than do the three orbitals centered on P (0.442). This leads to larger Pauli repulsion between the bond pairs in NH₃. One impact of this repulsive interaction is that the H 1s-like bond orbital develops a node in the region of the N atom. As a result, the overlap of the bond pair is smaller in NH₃ (0.534) than in PH₃ (0.813).

Upon a single substitution of H by F, the geometries do not change much. But there are significant changes in some of the orbitals. The N or P orbital involved in the bond to the F atom has delocalized onto the F atom as would be expected for a polar covalent bond. This delocalization builds N^δ+F^δ− (P^δ+F^δ−) character into the GVB wave function. The orbitals for the other two bond pairs resemble those in NH₃ and PH₃. Due to the interaction with the out-of-plane, doubly occupied 2p_x orbitals on F, the doubly occupied, out-of-plane 2p_x and 3p_x lone pair orbitals on N and P acquire antibonding character. Also, since F is very electronegative, the overlaps between the orbitals on the central atoms are reduced and the node in the H 1s-like orbital in NH₃ disappears upon F substitution. Regardless of these small changes, the nature of the bonding orbitals on the central atoms in FNH₂ and FPH₂ is similar to that of NH₃ and PH₃: hybrids of an s-recoupled pair bond dyad and a p-covalent bond.

Once another H atom is substituted with an F atom, there are marked differences in the geometrical and orbital structure of F₂NH and F₂PH. F₂NH is very similar to FNH₂, although two F atoms pull more electron density away from the central N atom than one F atom and the orbitals are less dense around the N atom. The overlaps between the bonding orbitals increase for both the NH and NF bonds. The out-of-plane, doubly occupied 2p_x orbital on N has acquired antibonding character on both F atoms, evidence of increased repulsive interactions between the electrons in the N lone pair and those in the doubly occupied 2p_x orbitals on F. On the other hand, F₂PH does not resemble FPH₂. The geometry of the transition state is now T-shaped, and the lone pair orbital is no longer a 3p_x-like orbital perpendicular to the molecular plane but a 3s lobe-like orbital in the molecular plane. This latter orbital has acquired a measure of antibonding character because of the interaction of the electrons in this orbital with the doubly occupied in-plane 2p orbitals on the F atom. In addition, the orbitals participating in the axial FPF bond in the F₂PH transition state are 3p lobe orbitals instead of 3s lobe orbitals.

The two axial PF bonds in F₂PH are, in fact, a p-recoupled pair bond dyad (Question #1). The P–F bond distance is very similar to that found in the A²Π state of PF₂, 1.649 Å (F₂PH) versus 1.639 Å (PF₂), a molecule known to have a p-recoupled pair bond dyad [9]. This is the reason for the unusual increase in the length of the PF axial bonds between the ground and transition state: p-recoupled pair bond dyads have bond distances that are significantly longer than the corresponding covalent bonds [9, 12, 13, 31, 40]. In contrast, the P orbital participating in the equatorial PH bond is the singly occupied 3p orbital (See 2b in Fig. 3). This bond is a normal polar covalent bond, which leads to only a slight decrease in the bond length (−0.7 % in Table 3) going from the ground state to the transition state (Question #2). Note that the p-recoupled pair bond dyad in F₂PH is almost linear, θ _FPF = 169.2°, as suggested in the orbital diagram 2b in Fig. 3, and as the geometry of PF₂(A²Π) suggests (θ _FPF = 180°). This is due to the dominant 3p character in the 3p lobe orbitals and the repulsive forces between the two PF bonds, the PH bond and the doubly occupied distorted 3s-like orbital. The quasi-linear structure is consistent with our finding that p-recoupled pair bond dyads are structurally rigid and prefer nearly collinear arrangements [9, 12, 13, 31, 40].

When all of the H atoms are substituted with F atoms, the NF₃ transition state has three equivalent NF bonds and D_3h symmetry. The doubly occupied orbital is an out-of-plane N 2p_x-like orbital with antibonding character on all three F atoms. The overlap between the two bonding orbitals constituting a NF bond increases from 0.697 in F₂NH to 0.721 in NF₃ as more F atoms pull more electron density into the bonding region. PF₃ is similar to F₂PH, with the only difference being that the PH covalent bond in F₂PH becomes a PF polar covalent bond. The two axial bonds are, again, a p-recoupled pair bond dyad formed with the P 3p lobe orbitals. The axial PF bonds in PF₃ are similar in length to those in F₂PH, 1.630 Å (PF₃) versus 1.649 Å (F₂PH) and are much longer than the PF bond in the PF₃ ground state (1.567 Å). For the equatorial bond, on the other hand, the percentage change in the bond length is just −0.4 % (0.006 Å), consistent with the equatorial PF bond being a covalent bond formed with a 3p orbital. Therefore, the seemingly random bond length changes in F₂PH and PF₃ (Question #2) are, in fact, not random: The p-recoupled pair bond dyads of the transition states have much longer bond distances than the covalent bonds, and the equatorial covalent bonds of the transition states are only slightly shorter than the covalent bonds of the ground states. The doubly occupied P orbital in PF₃ is similar to that in F₂PH, a distorted 3 orbital with some antibonding character associated with the F in-plane lone pairs.

3.2.3 Further comparison of s- and p-recoupled pair bond dyads

The analysis of the GVB orbitals of (PH₃, FPH₂, F₂PH, PF₃) shows that the dramatic change in the structures of the transition states for F₂PH and PF₃ is a result of the formation of a p-recoupled pair bond dyad in these species (Question #1). In this section, we take a step back to examine the triatomic molecules, because recoupled pair bond dyads also exist in NH₂, NF₂, PH₂ and PF₂, although in their excited states not in their ground states. Here, we focus on PH₂ and PF₂ as two examples of s- and p-recoupled pair bond dyads.

The ground state of PH₂ is a ²B₁ state, bound by two normal covalent bonds. The first excited state is a ²A₁ state. The ²A₁ state derives from a linear ²Π state that contains a 3s-recoupled pair bond dyad, and relaxes to a bent geometry upon geometry optimization, incorporating additional 3p character into the P bonding orbitals. So let us compare the recoupled pair bond dyad in the ²Π and ²A₁ states of PH₂ with the bonds in the PH₃ and FPH₂ transition states to understand how the s-recoupled pair bond dyad evolves in these species. Figure 6 shows the GVB orbitals in PH₂, PH₃ and FPH₂ that are centered on the P atom along with the doubly occupied, out-of-plane 3p-like lone pair orbitals. The PH₂(²Π) state has two 3s-like lobe orbitals as well as singly and doubly occupied 3p(π)-like orbital on the P atom. The 3s lobe orbitals are slightly delocalized onto the H atoms compared to the 3s lobe orbitals in the P atom (Fig. 4). In order to strengthen the bond and reduce the repulsion between the electrons in these orbitals and because mixing between (hybridization of) s and p orbitals is facile, the ²Π state rearranges to become a strongly bent ²A₁ state with a bond angle of 121.8°, reducing the energy by 19.66 kcal/mol. The 3p orbital is pushed away from the two PH bonds, picking up 3s character. So the three singly occupied P orbitals in the ²A₁ state of PH₂ are not pure 3s lobe orbitals or 3p orbitals, but a mixture of the two. However, their origins are clear.

Fig. 6

GVB orbitals centered on the P atom for the linear PH₂(²Π) configuration, the bent PH₂(²A₁) excited state, and for the transition states for inversion in FPH₂(¹A₁) and PH₃(¹A₁)

The singly occupied P orbitals in the PH₃ transition state look very much like the corresponding orbitals in the PH₂(²A₁) state. The doubly occupied out-of-plane orbital is largely unchanged. The orbitals of the FPH₂ transition state are similar except that the P bonding orbital delocalizes onto the F atom (consistent with the fact that the F atom is more electronegative than the P atom), and the out-of-plane lone pair orbital acquires antibonding character due to repulsive interactions of the electrons in this orbital with those in the F 2p orbital.

Similar comparisons for PF₂, F₂PH and PF₃ are given in Fig. 7. The ground state of PF₂ is also a ²B₁ state, and the lowest-lying excited state is a ²Π state. In PF₂, in contrast to PH₂, the first excited state is linear. The PF₂(²Π) state contains a 3p-recoupled pair bond dyad, a singly occupied 3p-like orbital and a doubly occupied polarized 3s lone pair orbital (8a₁) [9]. As noted previously, p-recoupled pair bonds prefer linear or quasi-linear geometries [9, 12, 13, 31, 40]. Comparing the three molecules containing the p-recoupled pair bond dyad, one can see that the 3p-like lobe orbitals remain largely unchanged. The biggest changes occur for the 3p-like orbital and the doubly occupied 3s-like orbital as the third bond forms in F₂PH and F₂PF. The 3p-like P orbital localizes in the bonding region as the PH bond forms and then delocalizes onto the more electronegative F atom when the PF bond forms. The 3s-like orbitals gain antibonding character and are pushed away from the covalent bonds. Comparing Figs. 6 and 7, one can see the s-recoupled pair bond dyad is indeed much more structurally flexible than the p-recoupled pair bond dyad.

Fig. 7

GVB orbitals centered on the P atom for the linear PF₂(²Π) excited state and the transition states for F₂PH(¹A₁) and PF₃(¹A₁)

As shown above, it is possible for the same central atom to form different types of recoupled pair bonds and recoupled pair bond dyads. As shown in Fig. 4, the overlap between the 3s_L and 3s_R lobe orbitals of the P atom is 0.776, while the overlap between 3p_L and 3p_R is 0.839. Therefore, it is much easier to form an s-recoupled pair bond than a p-recoupled pair bond. In fact, forming a p-recoupled pair bond dyad requires electronegative ligands, which separates the two highly overlapping p lobe orbitals and reduces the Pauli repulsion between the resulting bond pairs [31]. The H atom is not sufficiently electronegative to recouple the P 3p² lone pair. Therefore, the P(²D) state forms an s-recoupled pair bond dyad with two H atoms and a p-recoupled pair bond dyad with two F atoms. Although the F atom is also able to recouple a 3s² lone pair, an s-recoupled pair bond dyad does not form in F₂PH and PF₃ because the P lone pair orbital would then be perpendicular to the molecular plane, resulting in strong Pauli repulsive interactions with the electrons in the doubly occupied orbitals on the F atoms. These repulsions are minimized if the lone pair on the P atom is in a 3s-like orbital, which is polarized away from the F atoms as is the case for the p-recoupled pair bond dyad. Therefore, the p-recoupled pair bond dyad is preferred, which gives rise to the low-barrier inversion pathways in F₂PH and PF₃ (Question #3). Even though the lone pair on the P atom is not involved in bond formation, it clearly influences which types of bonds are formed.

3.3 Transition state formation pathways

To summarize the above GVB analysis of the transition states, GVB diagrams of the pathways for forming the inversion transition states of two representative molecules, NF₃ and PF₃, are shown in Fig. 8. The NF₃ transition state goes through a pathway involving the formation of an s-recoupled pair bond dyad. The two F atoms recouple the 2s² electrons of N(²D) and form the NF₂(²Π) configuration, which rearranges to the bent ²A₁ state. The third F atom forms a normal (polar) covalent bond with the singly occupied orbital in the NF₂(²A₁) state, leading to the NF₃ D_3h transition state. The doubly occupied orbital on N is perpendicular to the molecular plane. The three NF bonds are equivalent as a result of the ease with which s and p orbitals hybridize (mix). The very high barrier to inversion in NF₃ is, in large part, due to the strong repulsions between the electrons in the lone pair orbital on the N atom and those in the lone pair orbitals on the F atom.

Fig. 8

Formation pathway diagrams of the ¹A₁ inversion transition states for NF₃ and PF₃

The PF₃ transition state goes through a p-recoupled pair bond dyad pathway. The two F atoms recouple the 3p² electrons of P(²D) to form the PF₂(²Π) state. The third F atom then forms a normal covalent bond with the remaining singly occupied P 3p orbital, resulting in a T-shape transition state structure. The lone pair orbital on P in PF₃ is a distorted 3s-like orbital polarized away from the F atoms in order to reduce the repulsions between the electrons in this orbital and those in the F lone pair orbitals. The collinear structure of the p-recoupled pair bond dyad remains almost unchanged in the process, with the dyad bonds bending slightly away from the lone pair. Not only are the axial bond lengths nearly the same in PF₂(A²Π) and PF₃(¹A₁, TS), but the F₂P–F ground and transition state bond energies are nearly the same, which means that the magnitude of the inversion barrier is nearly the same as the energy difference between the PF₂(X²B₁) and PF₂(A²Π) states, 54.00 versus 52.94 kcal/mol. This process is illustrated in Fig. 9. The bond energy of the equatorial PF bond in the transition state is 133.07 kcal/mol (the bond energy reported in ref. [9] is incorrect), similar to the 134.13 kcal/mol bond energy of the covalent PF bond in the ground state. Also, the energy lowering for forming a p-recoupled pair bond dyad from the atoms in PF₂(A²Π) is 206.45 kcal/mol, which is only 20.71 kcal/mol less stable than forming the two covalent bonds in PF₂(X²B₁), 227.16 kcal/mol. In fact, as noted earlier, the strength of recoupled pair bond dyads is the reason for the existence of hypervalent species such as PF₅ [9] and SF₆ [12].

Fig. 9

Energy and bond length changes during the formation of the ground state and inversion transition state of PF₃, beginning with the atoms. The energies are in kcal/mol, and bond distances are in Å

4 Conclusions

In this article, we report accurate CCSD(T) calculations on the ground and transition states for inversion of the F _n NH_(3−n) and F _n PH_(3−n) (n = 0–3) molecules along with a detailed analysis of the GVB wave functions for these molecules at the calculated stationary points. All of the molecules go through D_3h-like transition states, except for PF₃ and F₂PH, which have T-shaped transition states, a fact first reported by Dixon and coworkers [4–8]. For all of the molecules with D_3h-like transition states, the bond distances are shorter than those in their ground states, a result of the increased s character in the N and P bond orbitals in the transition states. However, for F₂PH and F₂PF, the transition state is T-shaped and the axial PF bonds are much longer than those in the ground state, while the equatorial PH and PF bonds are slightly shorter than those in the F₂PH and PF₃ ground states.

Likewise, there is a dramatic difference in the dependence of the barrier height for inversion as the number of F atoms (n) increases. The height of the inversion barrier height increases dramatically for the F _n NH_(3−n) series as n increases, from 5.34 (NH₃) to 82.79 (NF₃) kcal/mol. However, for the F _n PH_(3−n) series, the barrier height increases substantially from n = 0 to n = 1 (33.25–50.54 kcal/mol), but thereafter it increases only modestly (from 50.54 to 54.00 kcal/mol).

The explanation for the anomalous behavior of the F₂PH and PF₃ molecules is simple. In the transition states of both F₂PH and PF₃, the nearly collinear PF bonds are a result of the formation of p-recoupled pair bond dyads. A hydrogen atom is not sufficiently electronegative to recouple the lone pair in the P(²D) excited state, and therefore, these bonds are only found in F₂PH and PF₃. Formation of the p-recoupled pair bond dyads in the transition states of these two molecules is favored because it minimizes the repulsion between the electrons in the lone pair on P and those in the lone pairs on the F atoms. Since a p-recoupled pair bond dyad prefers a nearly collinear arrangement of the F–P-F atoms, the transition states in F₂PH and PF₃ are T-shaped. The lengthening of the axial PF bonds is a result of the formation of p-recoupled pair bond dyads in F₂PH and PF₃, while the third bond in each species is a covalent bond similar to that in the ground state.

The ground state of F₂PH and PF₃ arises from the addition of an H or F atom to the ground state of PF₂, the X²B₁ state, while the T-shaped transition arises from the corresponding additions to the first excited state of PF₂, the A²Π state. The energy of the PF₂(X²B₁) and PF₂(A²Π) states differs by 52.94 kcal/mol [9]. If the strengths of the PH and PF bonds in the F₂PH and PF₃ transitions states are similar to those in the ground states, as would be expected, one would predict barrier heights close to this value, which, indeed, is the case: 52.04 kcal/mol (F₂PH) and 54.00 kcal/mol (PF₃). In addition, it should be noted that the lengths of the covalent PH and PF bonds are similar in the ground and transition states.

In summary, the transition states for NH₃, FNH₂, F₂NH, NF₃, PH₃ and FPH₂ involve the formation of s-recoupled pair bond dyads. Neither the H or F atoms are able to recouple the 2p² lone pair in the N(²D) state, so all of the transition states in the F _n NH_(3−n) series possess s-recoupled pair bond dyads (or hybrids thereof). H is not able to recouple the 3p² lone pair of P atom, and thus, the PH₃ and FPH₂ transition states also contain s-recoupled pair bond dyads. Extensive mixing between the s-recoupled pair bond dyad and the remaining covalent bond results in the D_3h-like transition state structures and vertex inversion pathways. The F atom is sufficiently electronegative to recouple both the 3s² and the 3p² electrons of the P atom, but the p-recoupled pair bond dyad is preferred in F₂PH and PF₃ over the s-recoupled pair bond dyad because the repulsion between the out-of-plane lone pair on P and the lone pairs on F is smaller in the former case. This leads to much lower inversion barriers than expected for F₂PH and PF₃ based on the trends in the F _n NH_(3−n) and the first two members of the F _n PH_(3−n) series. The p-recoupled pair bond dyad is rigid and stays almost collinear when other bonds are formed, which results in the T-shaped transition states and edge inversion in F₂PH and PF₃.

The T-shaped pnictogen structures in the ground state and transition states of compounds such as ADPO and its saturated analog also result from the formation of p-recoupled pair bond dyads by the pnictogen elements from the second row and beyond. Factors that stabilize the T-shaped structures of these compounds include electronegative ligands (facilitating the formation of the recoupled pair bond dyad) and conjugated π system (allowing delocalization into the space usually occupied by the lone pair). Such compounds are expected to be widespread; their chemistry has been reviewed in detail by Arduengo and Stewart [2].