Equilibrium Structures of the Phosphorus Trihalides PF₃ and PCl₃, and the Phosphoranes PH₃F₂, PF₅, PCl₃F₂, and PCl₅

Abstract

Among the title species, a reliable and accurate equilibrium geometry (r_e structure) is available only for PF₃, which has been determined experimentally more than 20 years ago. Here, we report accurate r_e structures for all title molecules, which were obtained using a composite computational approach based on explicitly correlated coupled-cluster theory (CCSD(T)-F12b) in conjunction with a large correlation-consistent basis set (cc-pCVQZ-F12) to take core–valence electron correlation into account. Additional terms were included to correct for the effects of iterative triple excitations (CCSDT), noniterative quadruple excitations (CCSDT(Q)), and scalar relativistic contributions (DKH2-CCSD(T)). The performance of this computational procedure was established through test calculations on selected small molecules (PH, PF, PCl, PH₂, PF₂, and PH₃). For PF₃, PCl₃, PH₃F₂, and PF₅ sufficiently accurate experimental ground-state rotational constants from the literature were used to determine semiexperimental r_e structures, which were found to be in excellent agreement with the corresponding best estimates from the current composite approach. The recommended equilibrium structural parameters are for PCl₃, r_e(PCl) = 203.94 pm and θ_e(ClPCl) = 100.18°; for PH₃F₂, r_e(PH_eq) = 138.38 pm and r_e(PF_ax) = 164.15 pm; for PF₅, r_e(PF_eq) = 153.10 pm and r_e(PF_ax) = 157.14 pm; for PCl₃F₂, r_e(PCl_eq) = 200.21 pm and r_e(PF_ax) = 159.37 pm; and for PCl₅, r_e(PCl_eq) = 201.29 pm and r_e(PCl_ax) = 211.83 pm. The associated uncertainties are estimated to be ±0.10 pm and ±0.10°, respectively.

1. Introduction

The most meaningful representation of the geometry of a molecule is provided by its equilibrium structure (r_e structure), mainly because it is independent of vibrational effects. The advantage of such a vibrationless structure becomes obvious when comparing geometries of related molecules: even subtle effects that may have some influence on the structural parameters can be discussed in terms of differences in the chemical bonding in these species. Such a discussion is less straightforward when nuclear-motion effects must be taken into account. Of course, such comparisons are most meaningful when the r_e structures in question are as reliable and accurate as possible. At least they should have been determined in a consistent way, for instance by high-level quantum-chemical calculations. Sometimes it is feasible experimentally to correct the measured ground-state rotational constants of a given polyatomic molecule (and its isotopologues if necessary) for the contributions due to zero-point vibrations. The resulting equilibrium rotational constants may then be used to derive the equilibrium geometry. Among the title species, such a purely experimental procedure was applied to PF₃ (phosphorus trifluoride): the measured¹ constants B₀ and C₀ were converted² to the corresponding equilibrium values B_e and C_e to determine a reliable and accurate r_e structure of PF₃.² However, such a purely experimental procedure has not yet been applied to the other title molecules. For PCl₃ (phosphorus trichloride),³⁻⁶ PH₃F₂ (difluorophosphorane),^7,8 and PF₅ (phosphorus pentafluoride)⁹⁻¹² ground-state rotational constants were measured, but there is not enough experimental information (vibration–rotation interaction constants) to transform them into equilibrium constants. For PCl₃F₂ (trichlorodifluorophosphorane) and PCl₅ (phosphorus pentachloride), spectroscopically derived rotational constants are not available at all.

Continuous advances in electronic structure methods and computational resources have made it possible to calculate the vibrational corrections to ground-state rotational constants of polyatomic molecules and their isotopologues with sufficient accuracy. Adopting these corrections, the resulting equilibrium rotational constants allow for the determination of the respective r_e structure, e.g., by a least-squares fitting procedure. Such a structure is called a semiexperimental equilibrium structure,¹³ due to the combination of experimental and theoretical data. This strategy has been pioneered by the work of Pulay, Meyer, and Boggs about 40 years ago.¹⁴ Its accuracy has been carefully established,¹⁵ and it can nowadays be applied almost routinely (e.g., see ref (16)).

Alternatively, equilibrium geometries of small and medium-sized molecules may also be obtained with an accuracy of about 0.1 pm for bond lengths and 0.1° for bond angles using state-of-the-art quantum-chemical methods. Typically, these methods are based on coupled-cluster (CC) theory including higher-order (at least quadruple) excitations and employing large basis sets, perhaps in combination with extrapolation techniques to estimate the complete basis set limit. Such rather expensive calculations have become feasible by the introduction of composite approaches.¹⁷⁻²⁰ Recently, we have reported²¹ a composite procedure related to the so-called geometry scheme²⁰ whose dominant term is based on explicitly correlated CC theory (CCSD(T)-F12b)²² and which takes core–valence electron correlation properly into account by employing an orbital basis set (cc-pCVQZ-F12) optimized for that purpose.²³ We have used this approach to calculate the equilibrium structural parameters of pyrazine (10 atoms), s-triazine (9 atoms), and s-tetrazine (8 atoms) with estimated uncertainties of ±0.10 pm and ±0.10° for bond distances and angles, respectively.²¹ The largest species treated in that study²¹ was benzene serving as a test molecule. We note that gradient schemes¹⁸⁻²⁰ have the advantage to provide stationary points corresponding to minima of the potential energy surface computed with a given approach. However, geometry schemes^20,21 are computationally less demanding²⁰ while they are capable of providing results that are very similar to those from gradient approaches.^20,21

In the present paper we apply the composite computational approach of ref (21) to compute the equilibrium geometries of the title molecules. As already mentioned, a reliable and accurate r_e structure is available in the literature only for PF₃.² For PCl₃ and PF₅, r_e structures were estimated from corresponding zero-point average (r_z) structures.^25,26 Gas electron diffraction data measured²⁷ for PF₅ were reanalyzed²⁸ to deduce equilibrium values for the bond distances, which are identical to those from ref (26) within the quoted error bars. For PH₃F₂, the effective ground-state (r₀) structure was inferred from the associated ground-state rotational constants.^7,8 For PCl₃F₂, mean internuclear distances (r_g structure) were determined by gas-phase electron diffraction.²⁹ Finally, for PCl₅ average structural parameters denoted r_α, r_g, and r_a are available.^30,31

Previously, we considered target and test molecules consisting only of first-row and hydrogen atoms.²¹ We start our current investigation by demonstrating that the chosen computational procedure is capable of accurately describing also the geometries of selected test molecules containing second-row atoms (P and Cl), namely, PH (X ³Σ^–), PF (X ³Σ^–), PCl (X ³Σ^–), PH₂ (X ²B₁), PF₂ (X ²B₁), and PH₃ (X ¹A₁). In addition, these small test species allow us to check convergence issues in our calculations. Unfortunately, the PCl₂ radical could not be included as a test species due to the lack of experimental data for rotational constants and equilibrium structural parameters. Whenever experimental ground-state rotational constants are known for the present target molecules, we determined a semiexperimental r_e structure to check the respective theoretical best estimate provided by the current purely computational scheme.

2. Computational Methods

We adopt the composite computational approach outlined in ref (21) to determine theoretical best estimates for the equilibrium bond length r_e and bond angle θ_e of each molecule, with a slight modification in the scalar relativistic part (see below). For the sake of easy reference and clarity, the basic definitions of this procedure²¹ are repeated here (see eqs 1–4):

1

2

3

4

As indicated by the first term on the right-hand side of eq 1, the structural parameter p_e (r_e or θ_e) is initially optimized at the level of explicitly correlated CC theory employing the so-called F12b approximation^22,32,33 including all single and double excitations (CCSD)^34,35 and augmented by a perturbational estimate of the effects of connected triple excitations (CCSD(T)).³⁶ CVQZ-F12 denotes the correlation-consistent polarized core–valence quadruple-ζ basis cc-pCVQZ-F12 optimized²³ for the explicitly correlated F12 methods.²² At hydrogen, this basis reduces to the cc-pVQZ-F12 basis set.²⁴ The parameter p_e evaluated at the CCSD(T)-F12b/CVQZ-F12 level of theory is expected to be reasonably close to the basis set limit²² and therefore serves as the starting point in the present scheme. For the purpose of comparison, we also tested the smaller basis CVTZ-F12, which is the correlation-consistent polarized core–valence triple-ζ basis cc-pCVTZ-F12²³ being equal to cc-pVTZ-F12²⁴ at hydrogen. For the open-shell species serving as test molecules (PH, PF, PCl, PH₂, and PF₂) the unrestricted UCCSD(T)-F12b variant^33,37 was used, which is based on a high-spin restricted open-shell Hartree–Fock (ROHF) determinant³⁸ and the perturbative triples corrections are computed as defined in ref (39). Whenever using core–valence basis sets in this work, all electrons in the molecule under consideration were correlated, except for those occupying the 1s-like core molecular orbitals (MOs) of the second-row atoms. All (U)CCSD(T)-F12b geometry optimizations were done using the MOLPRO 2012 program.^40,41 The four-point formula implemented in this program was used for the numerical energy gradients, and the step sizes for distances and angles were set to 0.01 a₀ and 0.5°, respectively. Throughout this work, the largest internal gradient components at the stationary points were always less than 2 × 10^–6E_h/a₀. For further technical details concerning the (U)CCSD(T)-F12b calculations, the reader is referred to ref (21).

The term Δp_e[T] (see eqs 1 and 2) corrects the (U)CCSD(T)-F12b/CVQZ-F12 result for the effects of an iterative treatment of connected triple excitations. Accordingly, this term is obtained as the difference of p_e optimized at the level of CC theory with a full treatment of single, double, and triple excitations (CCSDT)^42,43 and at the CCSD(T)^34,36,44 level (see eq 2). The acronym VTZ refers to the correlation-consistent polarized valence triple-ζ basis cc-pVTZ⁴⁵ for the H and F atoms and the tight d-augmented cc-pV(T+d)Z⁴⁶ basis for the P and Cl atoms. To check basis set convergence in the case of the test molecules (PH, PF, PCl, PH₂, PF₂, and PH₃), Δp_e[T] was also calculated with the VQZ, V5Z, and up to V6Z basis sets, which are the n-tuple-ζ (n = 4, 5, 6) analogues⁴⁵⁻⁴⁷ of VTZ (n = 3). Whenever such valence-only basis sets were used, the frozen core approximation was applied (i.e., the 1s2s2p-like core MOs of P and Cl and the 1s-like core MO of F were constrained to be doubly occupied). For the open-shell test molecules (see above) the ROHF-based variants of CCSDT⁴⁸ and CCSD(T)³⁹ were used. These geometry optimizations were performed using analytic or numerical energy gradients as implemented in the CFOUR program.⁴⁹

The term Δp_e[(Q)] (see eqs 1 and 3) is computed as the difference of p_e optimized at the level of CCSDT augmented by a perturbative treatment of connected quadruple excitations (CCSDT(Q))^50,51 and at the CCSDT level. This approximate correction for the effects of quadruple excitations partly covers higher-order excitations in the cluster operator. The label VDZ stands for the double-ζ analogue of the VTZ basis.^45,46 To check basis set convergence in the case of the test molecules (see above), Δp_e[(Q)] was computed employing even larger basis sets (up to V6Z). The geometry optimizations at the CCSDT(Q) level were done with the use of the MRCC code^52,53 interfaced to CFOUR.⁴⁹ Numerical energy gradients as provided by CFOUR⁴⁹ had to be used for CCSDT(Q), whereas analytic⁵⁴ or numerical energy gradients were employed for CCSDT. For the test molecules, the CCSDT(Q) geometries were compared with those optimized at the level of CC theory with a full treatment of single, double, triple, and quadruple excitations (CCSDTQ),^53,55−57 employing basis sets up to VQZ. The CCSDTQ geometry optimizations were carried out in complete analogy to those at the CCSDT(Q) level. Due to program limitations, the CCSDT(Q) calculations had to be based on an unrestricted Hartree–Fock (UHF)⁵⁸ reference wave function for the open-shell test species. For the sake of compatibility, the corresponding CCSDTQ calculations were also carried out using a UHF reference function. Consequently, to be consistent within the Δp_e[(Q)] (and Δp_e[Q]) term the associated CCSDT calculations were also based on a UHF determinant (UHF-CCSDT)⁴⁸ in these cases.

The final term Δp_e[SR] (see eqs 1 and 4) serves as a correction for scalar relativistic effects on the given structural parameter p_e: This term was evaluated from the difference of p_e optimized at the CCSD(T) level using the Douglas–Kroll–Hess Hamiltonian of second order (DKH2)⁵⁹⁻⁶¹ and at the nonrelativistic (standard) CCSD(T) level. The acronym AWCVTZ refers to the augmented correlation-consistent polarized weighted core–valence triple-ζ basis (aug-cc-pwCVTZ).⁶²⁻⁶⁴ To check basis set convergence, we employed the larger basis sets AWCVQZ and AWCV5Z, which are the quadruple-ζ and quintuple-ζ analogues⁶²⁻⁶⁴ of AWCVTZ. At hydrogen, these basis sets reduce to the corresponding aug-cc-pVXZ (X = T, Q, 5) sets.⁶³ In conjunction with the DKH2 Hamiltonian, the correspondingly recontracted⁶⁵ versions of these basis sets were used (denoted as AWCVXZ-DK; X = T, Q, 5). We recall that the employment of a core–valence basis currently implies that all electrons in the molecule under study were correlated (see above for details). For the open-shell test species, these calculations were done using the unrestricted variant of CC theory based on ROHF orbitals (denoted UCCSD(T)).^39,66 The geometry optimizations concerning Δp_e[SR] were carried out using the MOLPRO 2012 program.⁴⁰ Numerical energy gradients were utilized in analogy to the (U)CCSD(T)-F12b calculations for the leading term in eq 1 (see above).

To summarize, the present composite approach was used to deduce theoretical best estimates of the geometry of the following molecules within the constraint of their point group symmetries:

• 1.Test molecules: diatomic radicals PH, PF, and PCl (X ³Σ^–; C_∞v), triatomic radicals PH₂ and PF₂ (X ²B₁; C_2v), and PH₃ (C_3v).
• 2.Target molecules: PF₃ (C_3v), PCl₃ (C_3v), PH₃F₂ (D_3h), PF₅ (D_3h), PCl₃F₂ (D_3h), and PCl₅ (D_3h).

The experimental ground-state rotational constants of PH₂,⁶⁷ PD₂,⁶⁸ PF₂,⁶⁹ PH₃,⁷⁰ PD₃,⁷¹ PF₃,⁷² PCl₃,^5,6 PH₃F₂,^7,8 and PF₅^11,12 were used to determine a semiexperimental¹³r_e structure for these species. To be more specific, we utilized for the asymmetric top species PX₂ (X = H, D, F) the measured⁶⁷⁻⁶⁹ constants A₀, B₀, and C₀, and for the oblate symmetric top molecules PH₃, PD₃, and PF₃ the B₀ and C₀ constants,⁷⁰⁻⁷² which refer to the so-called B-reduction of the rotational Hamiltonian for PH₃ and PD₃.^70,71 Furthermore, we used the B₀ constants^5,6 of the P³⁵Cl₃ and P³⁷Cl₃ isotopologues, as well as the published^7,8,11,12A₀ and B₀ constants for the prolate symmetric tops PH₃F₂ and PF₅. In order to deduce the equilibrium rotational constants, the corresponding zero-point vibrational corrections^13,73,74 ΔX_vib (X = A, B, C) are needed. For this purpose, we determined the required cubic normal coordinate force constants at the (UHF-)CCSD(T) level of theory with the use of the CFOUR code,⁴⁹ employing various basis sets (see below) and using a finite difference procedure⁷⁵ that involved displacements along reduced (dimensionless) normal coordinates (step size Δq = 0.05) and the calculation of analytic Hessians^76,77 at these displaced geometries. At the respective equilibrium geometries optimized at the same level of theory as adopted for the force field calculations, the internal gradient components were always less than 2 × 10^–10E_h/a_0. The vibration–rotation interaction constants (α-constants) were derived from the theoretical normal coordinate force constants by applying standard formulas based on second-order rovibrational perturbation theory.⁷³ In addition, the α-constants of PCl₃F₂ and PCl₅ were also calculated. For PCl₅, only those parts of the cubic force field were computed that are required for the calculation of the α-constants⁷³ (using the CFOUR⁴⁹ input option ANHARMONIC = VIBROT). The following basis sets were employed: AVQZ for PH₂, PD₂, PF₂, PH₃, PD₃, PF₃,⁷² P³⁵Cl₃, and P³⁷Cl₃; VQZ for PH₃F₂, PD₃F₂, PF₅, P³⁵Cl₃F₂, and P³⁷Cl₃F₂; VTZ for P³⁵Cl₅ and P³⁷Cl₅. The VQZ and VTZ basis sets have already been described (see above). The AVQZ basis is derived from VQZ by the addition of diffuse functions: it consists of the aug-cc-pVQZ⁶³ basis for H and F and aug-cc-pV(Q+d)Z⁴⁶ for P and Cl. Most of these results serve as predictions, in particular those for PD₃F₂, PCl₃F₂, and PCl₅ whose rotational constants have not yet been measured.

Besides the vibrational correction ΔX_vib, there is a small electronic (magnetic) contribution ΔX_el, which is related to the rotational g-tensor.^13,74,78 Such contributions were also included to investigate their effects on the semiexperimental structural parameters. Thus, the rotational g-tensors of the various species mentioned above were computed at the level of CCSD(T)/AWCVTZ theory at the associated best estimated equilibrium geometries from this study. The CFOUR⁴⁹ program was used for these computations to ensure gauge-origin independence and fast basis set convergence by employing rotational London atomic orbitals.^79,80 However, due to program limitations the rotational g-tensors were not calculated for the open-shell molecules. All g-factors and ΔX_el values computed presently are contained in Tables S1 and S2 of the Supporting Information, together with the ΔX_vib data (see above). While most of the g-factors stand as predictions, they may be compared with experiment in a few cases, e.g., for PH₃⁸¹ and PF₃.⁸² It turned out that neglecting these electronic contributions has no significant impact on the present equilibrium geometrical parameters: the bond lengths and angles change at most by 0.0033 pm and 0.0026°, respectively (these maximum values apply to PCl₃), the only exception being r_e(PH) in PH₃F₂ where a somewhat larger effect (0.012 pm) occurs.

Several of the basis sets employed in calculations with the programs CFOUR⁴⁹ and MRCC⁵² were downloaded from the EMSL basis set library using the Basis Set Exchange web portal.^83,84

3. Results and Discussion

3.1. Test Molecules

The computed equilibrium geometries of the diatomic test molecules PH, PF, and PCl are collected in Table 1, and those of PH₂, PF₂, and PH₃ are listed in Table 2. Considering the raw results, those calculated at the DKH2-(U)CCSD(T)/AWCV5Z-DK and (U)CCSD(T)-F12b/CVQZ-F12 levels of theory should be the most reliable and accurate ones, due to the size of the basis sets as well as the inclusion of core–valence electron correlation and scalar relativistic effects (in case of the DKH2 results). Moreover, structures optimized at the level of explicitly correlated CC theory are known to be reasonably close to the basis set limit.²² For this reason, they were selected as starting points to derive theoretical best estimates for the r_e structures (see section 2).

Table 1. Computed and Experimental Equilibrium Bond Lengths (pm) in PX (X = H, F, Cl).

		PH	PF	PCl
methoda	basis	re(PH)b	re(PF)b	re(PCl)b
CCSD(T)	VDZ	143.78	164.16	205.53
CCSD(T)	VTZ	142.58	159.69	203.33
CCSD(T)	VQZ	142.40	159.41	202.38
CCSD(T)	V5Z	142.36	159.28	202.08
CCSD(T)	V6Z	142.35	159.22	201.95
CCSD(T)	AWCVTZ	142.23	159.59	202.88
CCSD(T)	AWCVQZ	142.11	159.10	201.74
CCSD(T)	AWCV5Z	142.07	158.89	201.46
DKH2-CCSD(T)	AWCVTZ-DK	142.24	159.65	202.93
DKH2-CCSD(T)	AWCVQZ-DK	142.12	159.16	201.79
DKH2-CCSD(T)	AWCV5Z-DK	142.09	158.95	201.51
CCSDT	VDZ	143.84[.84]	164.26[.26]	205.69[.68]
CCSDT	VTZ	142.63[.63]	159.76[.75]	203.47[.47]
CCSDT	VQZ	142.46[.46]	159.46[.46]	202.51[.51]
CCSDT	V5Z	142.41[.41]	159.32[.32]	202.20[.20]
CCSDT	V6Z	142.40[.40]	159.26[.26]	202.07[.07]
CCSDT(Q)	VDZ	143.85	164.35	205.75
CCSDT(Q)	VTZ	142.65	159.84	203.55
CCSDT(Q)	VQZ	142.47	159.54	202.59
CCSDT(Q)	V5Z	142.43	159.41	202.28
CCSDT(Q)	V6Z	142.42	c	c
CCSDTQ	VDZ	143.85	164.32	205.73
CCSDTQ	VTZ	142.65	159.81	203.54
CCSDTQ	VQZ	142.47	159.52	c
CCSD(T)-F12b	CVTZ-F12	142.09	158.87	201.39
CCSD(T)-F12b	CVQZ-F12	142.07	158.81	201.31
best estimatea		142.14	159.03	201.57
experimentald		142.182772(89)e	158.9329(20)	201.4609(49)
experimentalf		142.2179(16)	158.96
experimentalg		142.140(22)

^aSee text.

^bDecimal places in square brackets refer to the corresponding UHF-CCSDT results (see text).

^cNot calculated.

^dReference (86) for PH, ref (89) for PF, and ref (91) for PCl.

^eValue determined for PD.

^fReference (87) for PH and ref (90) for PF.

^gReference (88).

Table 2. Computed and Semiexperimental Equilibrium Geometries (pm, deg) of PX₂ (X = H, F) and PH₃ as Well as Experimental Equilibrium Geometries (pm, deg) of PH₃.

		PH2		PF2		PH3
methoda	basis	re(PH)b	θe(HPH)b	re(PF)b	θe(FPF)b	re(PH)	θe(HPH)
CCSD(T)	VDZ	143.06	91.714	162.57	98.894	142.54	93.358
CCSD(T)	VTZ	141.99	91.840	158.39	98.657	141.56	93.505
CCSD(T)	VQZ	141.86	91.871	158.12	98.427	141.47	93.554
CCSD(T)	V5Z	141.83	91.878	157.96	98.350	141.44	93.560
CCSD(T)	V6Z	141.82	91.878	157.90	98.300	141.44	93.561
CCSD(T)	AWCVTZ	141.67	91.761	158.18	98.239	141.27	93.426
CCSD(T)	AWCVQZ	141.57	91.812	157.76	98.320	141.17	93.487
CCSD(T)	AWCV5Z	141.53	91.813	157.57	98.321	141.14	93.492
DKH2-CCSD(T)	AWCVTZ-DK	141.67	91.695	158.23	98.258	141.27	93.339
DKH2-CCSD(T)	AWCVQZ-DK	141.57	91.745	157.81	98.338	141.17	93.399
DKH2-CCSD(T)	AWCV5Z-DK	141.54	91.746	157.62	98.338	141.14	93.404
CCSDT	VDZ	143.11[.11]	91.695[.695]	162.66[.66]	98.887[.888]	142.57	93.344
CCSDT	VTZ	142.03[.03]	91.825[.825]	158.43[.43]	98.647[.649]	141.59	93.496
CCSDT	VQZ	141.90[.90]	91.862[.862]	158.15[.14]	98.417[.419]	141.50	93.550
CCSDT	V5Z	141.87[.87]	91.871[.871]	157.99[.99]	98.339[.341]	141.47	93.558
CCSDT	V6Z	141.86[.86]	91.872[.872]	c	c	c	c
CCSDT(Q)	VDZ	143.11	91.685	162.74	98.911	142.58	93.332
CCSDT(Q)	VTZ	142.04	91.806	158.50	98.669	141.60	93.474
CCSDT(Q)	VQZ	141.92	91.839	c	c	141.51	93.524
CCSDTQ	VDZ	143.11	91.684	162.72	98.908	142.58	93.331
CCSDTQ	VTZ	142.04	91.804	c	c	141.60	93.473
CCSD(T)-F12b	CVTZ-F12	141.55	91.747	157.55	98.291	141.15	93.454
CCSD(T)-F12b	CVQZ-F12	141.53	91.807	157.50	98.304	141.13	93.489
best estimatea		141.57	91.72	157.67	98.34	141.17	93.38
semiexperimentala		141.636(8)	91.686(13)	157.612(2)	98.338(2)	141.195	93.378
semiexperimentala,d		141.612(2)	91.706(3)			141.185	93.383
experimentale						141.1607(83)	93.4184(95)
experimentale,f						141.1785(57)	93.4252(68)

^aSee text.

^bDecimal places in square brackets refer to the corresponding UHF-CCSDT results (see text).

^cNot calculated.

^dValues evaluated for PD₂ and PD₃, respectively.

^eReference (71).

^fValues determined for PD₃.

Upon comparison of the results from (ROHF-)CCSDT/VTZ and (ROHF-)CCSD(T)/VTZ, the full treatment of connected triple excitations appears to increase the bond lengths in the test molecules, by +0.03 pm in PH₃ up to +0.14 pm in PCl. The bond angles in PH₂, PF₂, and PH₃ decrease very slightly (0.009–0.015°). Upon enlargement of the basis from VTZ to VXZ (X = 6 for PH, PF, PCl, and PH₂; X = 5 for PF₂ and PH₃), the effects arising from the full treatment of connected triple excitations change the bond distances in PH, PH₂, PF₂, and PH₃ minutely (at most by 0.01 pm), and only slightly more in PF and PCl (decreasing by 0.02–0.03 pm). The corresponding effect on the bond angle is negligible in PF₂ (by 0.001°) and still tiny in PH₂ and PH₃ (decrease in absolute value by 0.009° and 0.007°, respectively). Thus, typically, the CCSDT versus CCSD(T) corrections are already converged reasonably well for the VTZ basis. This is in line with previous observations.¹⁷ Even in the two cases where the effects may not seem completely negligible (bond angles in PH₂ and PH₃), they are still so small in absolute value (on the order of 0.01°) that they do not significantly affect the final theoretical best estimate of the associated r_e structures.

Higher-order correlation effects beyond CCSDT are approximated by the differences between (UHF-)CCSDT(Q)/VDZ and (UHF-)CCSDT/VDZ results. Contributions from connected quadruple excitations to bond distances should converge faster with basis set size than those from connected triples.¹⁷ In the current test molecules, the perturbative treatment of connected quadruples slightly increases the bond lengths. In the case of PH bonds (PH, PH₂, PH₃) the elongations do not exceed 0.01 pm. They are somewhat more pronounced for the PF and PCl bonds (0.07–0.09 pm). The bond angles in PH₂ and PH₃ decrease by 0.010° and 0.012°, respectively, whereas the angle in PF₂ increases by 0.023°. In the present test molecules, these contributions agree in sign with the corresponding (ROHF-)CCSDT versus (ROHF-)CCSD(T) differences (see above) so that both effects enhance each other. The only exception from this finding is provided by the bond angle in PF₂ for which a partial cancellation of both effects occurs. When enlarging the basis from VDZ to VXZ (X = 6 for PH; X = 5 for PF and PCl; X = Q for PH₂ and PH₃; X = T for PF₂), the effects of the connected quadruple excitations on the bond distances in the test molecules change typically at most by 0.01 pm. The effects on the bond angles in PH₂ and PH₃ are slightly larger (increase in absolute value by 0.013° and 0.014°, respectively) but remain small enough to not significantly affect the final theoretical best estimate of the associated r_e structures. In view of these data, we evaluate the (UHF-)CCSDT(Q) versus (UHF-)CCSDT contributions with the VDZ basis, assuming that they are sufficiently converged to serve as increments when deriving theoretical best estimates for the geometrical parameters, as in our previous study.²¹ Generally, in order to achieve quantitative accuracy in the prediction of r_e structures quadruple excitations must be taken into account.^18,19 In this context, the question arises whether quadruple excitation effects on molecular geometries are sufficiently well described by the (UHF-)CCSDT(Q) approximation (i.e., by the noniterative treatment of quadruples). To check this issue, the geometries of the test molecules were additionally optimized at the (UHF-)CCSDTQ/VDZ level of theory. We find that the bond lengths in PH, PH₂, and PH₃ remain unaltered upon a full rather than a perturbative treatment of quadruple excitations. The bond distances in PF, PCl, and PF₂ are shortened by 0.02–0.03 pm. The bond angles in PH₂, PF₂, and PH₃ decrease by 0.001–0.003°. Essentially the same alterations occur in those cases where larger basis sets (VTZ, VQZ) could be employed (see Tables 1 and 2). Keeping in mind that the target accuracy of the present study is ±0.10 pm for bond lengths and ±0.10° for bond angles, these deviations are considered small enough to justify the use of the noniterative (UHF-)CCSDT(Q) approximation.

Scalar relativistic effects on the equilibrium molecular geometries are presently evaluated from the differences between the DKH2-(U)CCSD(T)/AWCVTZ-DK and (U)CCSD(T)/AWCVTZ results. On an absolute scale, these effects are quite small for molecules containing first- and second-row atoms only:⁸⁵ the bonds in PF, PF₂, and PCl are lengthened by 0.05–0.06 pm, whereas the bond distances in PH, PH₂, and PH₃ remain virtually unchanged. The bond angles in PH₂ and PH₃ decrease by 0.066° and 0.087°, respectively, whereas the angle in PF₂ increases by 0.019°. To examine basis set convergence, we performed analogous geometry optimizations at the (U)CCSD(T) level of theory with and without the DKH2 Hamiltonian employing the AWCVQZ and AWCV5Z basis sets. The DKH2 calculations were carried out using the corresponding recontracted⁶⁵ basis sets, as before (see Tables 1 and 2). It turns out that the scalar relativistic effects calculated with the AWCVTZ basis vary in absolute value at most by 0.01 pm and 0.002°, respectively, when enlarging the basis as stated above. Thus, basis set convergence of the scalar relativistic effects appears to be reached for the AWCVTZ basis.

Tables 1 and 2 also show the best estimates of the r_e structures of the test molecules resulting from the current composite approach. They may be compared with the corresponding experimental r_e structures for PH,⁸⁶⁻⁸⁸ PF,^89,90 PCl,⁹¹ and PH₃.⁷¹ However, for PH₂ and PF₂ there are no such experimental data available in the literature. For this reason semiexperimental r_e structures were determined for the PX₂ (X = H, D, F) species (see section 2). These structural data and their uncertainties were evaluated from the semiexperimental rotational constants A_e, B_e, and C_e in complete analogy to the procedure used to infer the experimental r₀ structure of PF₂.⁶⁹ Using the experimental⁶⁷⁻⁶⁹ ground-state rotational constants, the corresponding inertial defects Δ₀ are presently calculated to be 0.0688 (PH₂), 0.0958 (PD₂), and 0.1906 u Ų (PF₂) whereas the respective semiexperimental equilibrium values Δ_e appear to be −0.000861, –0.000443, and +0.00202 u Ų. As expected, the Δ_e values are significantly closer to zero than their Δ₀ counterparts. Additionally, we have determined semiexperimental r_e structures for PH₃ and PD₃ (see section 2), the uncertainties of which should be roughly similar to those from experiment.⁷¹ We note that the calculated electronic contributions to the rotational constants (see Table S2 of the Supporting Information) have a particularly small effect on the semiexperimental r_e structures of PH₃ and PD₃: in the former, the bonds are lengthened by 0.00017 pm and θ_e(HPH) decreases by 0.0013° due to the magnetic corrections, while in the latter the corresponding effects are smaller by about a factor of 2 (0.000081 pm and 0.00063°, respectively). Thus, the neglect of these corrections in the experimental⁷¹ determination of the r_e structures of PH₃ and PD₃ is fully justified by these results. The semiexperimental r_e structures of both species are in excellent agreement with their purely experimental⁷¹ counterparts (see Table 2).

The deviations between the best estimated bond lengths and their experimental or semiexperimental analogues usually do not exceed 0.10 pm. A slightly larger deviation (0.11 pm) occurs for the internuclear distance in the PCl radical. The best estimated bond angles in PH₂, PF₂, and PH₃ agree with the semiexperimental and experimental⁷¹ values, respectively, to within 0.05° (see Table 2). Applying a similar computational scheme as adopted presently, the r_e structure of PH₂ was best estimated⁹² to be r_e(PH) = 141.58 pm and θ_e(HPH) = 91.78°. Scalar relativistic effects on the molecular geometry were neglected.⁹² The corresponding correction indeed vanishes for the bond distance in PH₂ whereas relativity appears to narrow the bond angle by 0.066° (see Table 2). When excluding this correction, our current best estimate for the PH₂ structure is r_e(PH) = 141.57 pm and θ_e(HPH) = 91.78°, which is in almost perfect agreement with the result from ref (92) (see above).

In our previous investigation,²¹ we noted that structural data computed at the CCSD(T)-F12b/CVQZ-F12 level of theory differ from those obtained at the CCSD(T)-F12b/CVTZ-F12 level at most by 0.02 pm and 0.04°, respectively. However, this observation²¹ refers to molecules consisting only of first-row atoms and hydrogen. Considering the test molecules of this paper, the deviations may be somewhat larger: 0.02–0.08 pm and 0.01–0.06°, respectively. Similar deviations are seen for the title molecules (see below), with the notable exception of the axial bond length in PCl₅ (0.13 pm). Thus, in the presence of second-row atoms the significantly smaller CVTZ-F12 basis may be employed for the calculation of molecular geometries whenever the larger basis set is no longer feasible due to the size of the molecule under study or when the target accuracy is somewhat relaxed.

The results obtained for the best estimated r_e structures of the test molecules suggest that the current composite approach is suited to predict equilibrium bond lengths and angles with an accuracy of about ±0.10 pm and ±0.10°, respectively, even if second-row atoms are present. Therefore, the same accuracy is expected to be achievable for the actual target molecules (see also ref (93)).

3.2. PF₃ and PCl₃

Table 3 collects the computed equilibrium structural parameters of PF₃ and PCl₃ which are needed in the present composite approach to derive the theoretical best estimates of the corresponding r_e structures. The latter are also given in Table 3. We include some further theoretical results, which are not involved in the evaluation of the best estimates but may be useful to check convergence issues. We point out that many results concerning PF₃ have already been published previously⁷² (see Table 3, footnote a) and are included in Table 3 for the sake of easy reference. They are discussed in more detail in ref (72).

Table 3. Computed, Semiexperimental, and Experimental Equilibrium Geometries (pm, deg) of PF₃ and PCl₃.

		PF3		PCl3
method	basis	re(PF)	θe(FPF)	re(PCl)	θe(ClPCl)
CCSD(T)	VDZ	160.84a	97.500a	207.32	100.517
CCSD(T)	VTZ	156.84a	97.662a	205.64	100.384
CCSD(T)	VQZ	156.61a	97.593a	204.74	100.249
CCSD(T)	AVQZ	156.71a	97.525a	204.79	100.171
CCSD(T)	AWCVTZ	156.61	97.494	205.16	100.189
CCSD(T)	AWCVQZ	156.24a	97.565a	204.14	100.198
DKH2-CCSD(T)	AWCVTZ-DK	156.65	97.496	205.21	100.186
DKH2-CCSD(T)	AWCVQZ-DK	156.28a	97.566a	204.19	100.194
CCSDT	VDZ	160.90a	97.492a	207.43	100.526
CCSDT	VTZ	156.86a	97.656a	205.72	100.397
CCSDT	VQZ	156.62	97.589	b	b
CCSDT(Q)	VDZ	160.98a	97.503a	207.50	100.533
CCSDT(Q)	VTZ	156.93	97.665	b	b
CCSD(T)-F12b	CVTZ-F12	156.05	97.556	203.80	100.191
CCSD(T)-F12b	CVQZ-F12	156.00a	97.556a	203.75	100.171
best estimatec		156.14d	97.56d	203.95	100.19
semiexperimentalc		156.10(10)	97.57(10)	203.94(10)	100.18(10)
experimentale		156.099(14)	97.57(4)	203.9(3)	100.28(10)f
experimentalg		156.1(1)	97.7(2)

^aFrom ref (72).

^bNot calculated.

^cSee text.

^dNumerically identical to the best estimate in ref (72).

^eReference² for PF₃ and ref (25) for PCl₃.

^fEstimate of uncertainty from ref (95).

^gReference (94).

The current semiexperimental r_e structure of PF₃ is identical with its experimental² counterpart (see Table 3), both with regard to the estimated uncertainties of the former (±0.10 pm and ±0.10°, respectively) and the even smaller experimental² error bars (±0.014 pm and ±0.04°, respectively). The best estimated r_e structure is also in excellent agreement with experiment² (see Table 3). For the sake of completeness, Table 3 also contains another experimental equilibrium geometry of PF₃ determined in an earlier investigation.⁹⁴ Both experimental^2,94r_e structures are in complete accordance when considering the uncertainties (±0.1 pm and ±0.2°, respectively) quoted in ref (94). In view of the agreement with the current theoretical best estimate and the semiexperimental result, the purely experimental²r_e structure appears to be the most reliable and accurate equilibrium geometry of PF₃ presently available. Thus, it is the recommended r_e structure for PF₃ (see Table 3).

Turning to PCl₃, the current semiexperimental r_e structure is almost identical with the theoretical best estimate, the differences (0.01 pm and 0.01°, respectively) being 1 order of magnitude smaller than the expected uncertainties (±0.10 pm and ±0.10°, respectively) of both of them. This excellent agreement validates both results. The molecular structure of PCl₃ was investigated by gas electron diffraction;⁹⁶ this study provided an r_a structure at two different temperatures, an estimate of the differences r_a – r_e, and hence a first estimate of the equilibrium geometry of PCl₃. The most reliable r_e structure of PCl₃ published²⁵ so far was derived from the r_z structure; it is identical with the semiexperimental r_e structure, within the corresponding uncertainties (see Table 3). However, we expect the present semiexperimental structural parameters to be superior in terms of reliability and accuracy, and we thus recommend the following equilibrium geometry for PCl₃: r_e(PCl) = 203.94(10) pm and θ_e(ClPCl) = 100.18(10)°. The additional inclusion of the ground-state rotational constants⁵ of the P³⁵Cl₂³⁷Cl species through an unweighted least-squares structural fit (using the STRFIT code of Kisiel⁹⁷) after having corrected them for the effects of zero-point vibrations and electronic contributions (see Table S1 of the Supporting Information) does not alter the recommended geometry significantly: r_e(PCl) decreases by 0.01 pm and θ_e(ClPCl) increases by 0.01°. These changes are 1 order of magnitude smaller than the estimated error limits and are therefore currently neglected.

In contrast to PF₃ whose ground-state rotational constants are well determined experimentally (see refs (1) and (72) as well as references therein), the axial ground-state rotational constants C₀ of P³⁵Cl₃ and P³⁷Cl₃ have not yet been measured. Making use of the theoretical best estimate of the r_e structure, the rotational constants of P³⁵Cl₃ (P³⁷Cl₃) are calculated to be B_e = 2623.81 (2493.77) MHz and C_e = 1476.09 (1396.35) MHz (same unit as in the experimental work³⁻⁶). Adopting theoretical (CCSD(T)/AVQZ) vibrational and theoretical (CCSD(T)/AWCVTZ) electronic corrections, the ground-state rotational constants amount to B₀ = 2616.83 (2487.24) MHz and C₀ = 1470.38 (1391.04) MHz. Analogously, we obtain for P³⁵Cl₂³⁷Cl: A₀ = 2615.64 MHz, B₀ = 2531.29 MHz, and C₀ = 1443.32 MHz. It comes as no surprise that the electronic contributions are smaller in absolute value than the corresponding vibrational corrections (see Table S1 of the Supporting Information). The experimental^5,6B₀ values of P³⁵Cl₃ and P³⁷Cl₃ as well as the experimental⁵X₀ (X = A, B, C) constants of P³⁵Cl₂³⁷Cl are underestimated theoretically by 0.02%. The errors of the C₀ values predicted for the two totally symmetric species of PCl₃ are expected to be very similar to those of the other constants. Using the r₀ structure of PCl₃ from ref (5)C₀ is tentatively estimated to be 1474 (1394) MHz, thus being about 0.2% larger than our best C₀ values predicted above for P³⁵Cl₃ (P³⁷Cl₃).

3.3. PH₃F₂ and PF₅

Table 4 contains the calculated bond lengths in PH₃F₂ and PF₅, in particular those needed to arrive at the theoretical best estimates of the r_e structures. Both molecules assume a trigonal bipyramidal geometry.^7,27 According to Muetterties’ rule,^98,99 the two axial positions will be occupied by the most electronegative ligands. The structure of PH₃F₂ conforms to this rule, i.e., the H (F) atoms take up the equatorial (axial) sites resulting in D_3h point-group symmetry, just as in PF₅. PH₃F₂ and PF₅ are examples of hypervalent species. Although a discussion of hypervalency¹⁰⁰ is beyond the scope of this paper, we note that a decade ago a new model of hypervalent bonding has been introduced¹⁰¹ termed recoupled pair bonding. This concept has also been applied to the series of PF_n (n = 1–5) species.¹⁰²

Table 4. Computed and Semiexperimental Equilibrium as Well as Experimental Effective Ground-State Structures (pm) of PH₃F₂ and PF₅ and Experimental Equilibrium and Mean Internuclear Distances (pm) in PF₅.

		PH3F2		PF5
method	basis	re(PHeq)a	re(PFax)a	re(PFeq)a	re(PFax)a
CCSD(T)	VDZ	139.89	167.02	157.29	160.82
CCSD(T)	VTZ	138.82	164.14	153.89	157.73
CCSD(T)	VQZ	138.78	164.33	153.68	157.64
CCSD(T)	AVQZ	138.80	164.72	153.74	157.72
CCSD(T)	AWCVTZ	138.49	164.68	153.57	157.64
CCSD(T)	AWCVQZ	138.46	164.27	153.25	157.32
DKH2-CCSD(T)	AWCVTZ-DK	138.43	164.73	153.56	157.64
DKH2-CCSD(T)	AWCVQZ-DK	138.40	164.32	153.24	157.31
CCSDT	VDZ	139.92	167.06	157.33	160.86
CCSDT	VTZ	138.84	164.16	153.90	157.74
CCSDT(Q)	VDZ	139.93	167.11	157.40	160.92
CCSD(T)-F12b	CVTZ-F12	138.46	164.03	153.10	157.12
CCSD(T)-F12b	CVQZ-F12	138.45	164.00	153.06	157.08
best estimateb		138.42	164.12	153.13	157.15
semiexperimentalb		138.38(10)	164.15(10)	153.10(10)	157.14(10)
experimentalc				152.9(3)	157.6(3)
experimentald				153.0	157.6
r0 structuree		139.4(2)	164.68(2)	153.43(30)f	157.46(30)f
rg structureg				153.4(4)	157.7(5)

^aSubscripts eq and ax refer to equatorial and axial ligands, respectively.

^bSee text.

^cReference (26).

^dReference (28).

^eReferences (7) and (8) for PH₃F₂ and ref (12) for PF₅.

^fEstimate of uncertainties from ref (95).

^gReference (27).

Turning to the equilibrium geometry of PH₃F₂, the best estimated r_e structure and its semiexperimental counterpart are in excellent agreement, with deviations of only 0.04 and 0.03 pm for r_e(PH_eq) and r_e(PF_ax), respectively (see Table 4). As before, the error bars of the semiexperimental r_e distances are assumed to equal those typically observed for the best estimates in our test calculations (±0.10 pm; see subsection 3.1). Experimentally,^7,8 only the effective ground-state structure of PH₃F₂ is available. As expected, the semiexperimental and the best estimated PH and PF equilibrium bond lengths are smaller than the associated r₀ distances, by about 1.0 and 0.5 pm, respectively (see Table 4).

The theoretical best estimate for the r_e structure implies the following values for the rotational constants of PH₃F₂ (PD₃F₂): A_e = 2.909996 (1.456117) cm^–1 and B_e = 0.160179 (0.155895) cm^–1 (same unit as in the experimental work^7,8). Applying both theoretical (CCSD(T)/VQZ) vibrational and theoretical (CCSD(T)/AWCVTZ) electronic corrections, we obtain A₀ = 2.866446 (1.440804) cm^–1 and B₀ = 0.159112 (0.154924) cm^–1. Compared to experiment,^7,8 the purely theoretical A₀ and B₀ constants of PH₃F₂ are too low by 0.054% and too large by 0.031%, respectively. In this context we note that the uncertainty of the experimental^7,8A₀ value was estimated⁸ to be 0.003 cm^–1, which corresponds to 0.105%, about twice as much as the error of the theoretical A₀ value. The rotational constants given above for PD₃F₂ serve as predictions. The associated errors should be about the same as those in PH₃F₂.

Turning to PF₅, the situation is very similar to that found for PH₃F₂ as well as for PF₃ and PCl₃: the best estimated and the semiexperimental r_e structures are essentially the same, the deviations being as small as 0.03 and 0.01 pm, respectively, for the equatorial and axial bond distances (see Table 4). The most recent experimental²⁶r_e structure of PF₅ was derived from the r_z structure. Prior to this experimental work,²⁶ the gas electron diffraction data²⁷ were reanalyzed²⁸ to deduce not only a barrier height (3.4 kcal/mol) for the pseudorotation¹⁰³ in PF₅ but also the equilibrium internuclear distances (see Table 4, no uncertainties quoted in ref (28)). The two experimental^26,28r_e structures agree within the error limits of ±0.3 pm estimated in the more recent work.²⁶ Comparing the semiexperimental r_e structure with experiment,²⁶ the equatorial bond lengths differ by 0.2 pm, which is still covered by the experimental²⁶ uncertainty, whereas the difference between the axial bond lengths is somewhat more pronounced (0.46 pm). Similar to PH₃F₂, there is also an experimental¹²r₀ structure for PF₅ which was determined from the measured^11,12A₀ and B₀ constants. As expected, the r₀ bond lengths are somewhat larger (by about 0.3 pm) than their semiexperimental r_e counterparts (see Table 4). While the equatorial r₀ bond lengths are also larger (0.4–0.5 pm) than their experimental^26,28r_e analogues, in contrast to expectation this does not hold for the axial bonds. However, a definitive assessment is difficult due to the relatively large uncertainties (±0.3 pm)^26,95 of these structures.^12,26

The equatorial r_e distance in PF₅ is shorter than the equilibrium bond length in PF₃ by about 3 pm (see Tables 3 and 4 for the best estimated, semiexperimental, and experimental r_e structures). Similarly, the PH r_e distance in PH₃F₂ is shorter than the equilibrium bond length in PH₃ by about 2.8 pm (see Tables 2 and 4). The axial r_e distance in PF₅ turns out to be much smaller than in PH₃F₂, by about 7 pm (see Table 4). It is well-known that the axial bonds in PF₅ are distinctly longer than the equatorial ones.²⁷ Considering the best estimated and semiexperimental r_e as well as the r₀ structure,¹² this difference appears to be 4.0 pm, in good agreement with the result from gas electron diffraction.²⁷

Using the theoretical best estimate of the r_e structure, the rotational constants of PF₅ are calculated to be A_e = 3781.45 MHz and B_e = 3145.63 MHz (same unit as in the experimental work^11,12). With the use of theoretical (CCSD(T)/VQZ) vibrational and theoretical (CCSD(T)/AWCVTZ) magnetic corrections we obtain A₀ = 3765.39 MHz and B₀ = 3132.61 MHz. The experimental^11,12 values for A₀ and B₀ in PF₅ are underestimated theoretically by 0.025–0.035%.

We complete this discussion for PH₃F₂ and PF₅ by emphasizing that the semiexperimental r_e structures of both species are in excellent agreement with the corresponding theoretical best estimates. Thus, both types of results validate each other. Therefore, we recommend the semiexperimental r_e structures of PH₃F₂ and PF₅ (see Table 4) as the most reliable and accurate equilibrium geometries available at present for these two species.

3.4. PCl₃F₂ and PCl₅

The computed equilibrium bond lengths including the best estimates for PCl₃F₂ and PCl₅ are reported in Table 5. The rotational constants of both species have not yet been measured. Therefore, the semiexperimental strategy¹³ could not be applied in these two cases. However, the r_g structures of both phosphoranes as well as the r_α and r_a structures of PCl₅ are known from gas electron diffraction studies.²⁹⁻³¹ The experimental^29,30r_g and r_α internuclear distances are listed in Table 5. Like PH₃F₂ and PF₅ (see subsection 3.3), PCl₃F₂ and PCl₅ form trigonal bipyramidal molecules of D_3h point-group symmetry in the gas phase.²⁹⁻³¹ Appropriate to Muetterties’ rule,^98,99 in PCl₃F₂ the most electronegative ligands (F atoms) occupy the two axial positions such that the equatorial sites are left over for the less electronegative ligands (Cl atoms).

Table 5. Computed Equilibrium and Experimental Average Structural Parameters (pm) of PCl₃F₂ and PCl₅.

		PCl3F2		PCl5
method	basis	re(PCleq)a	re(PFax)a	re(PCleq)a	re(PClax)a
CCSD(T)	VDZ	202.85	163.57	204.97	214.94
CCSD(T)	VTZ	202.02	159.73	203.14	213.22
CCSD(T)	VQZ	201.11	159.74	202.14	212.59
CCSD(T)	AVQZ	201.06	159.95	202.14	212.73
CCSD(T)	AWCVTZ	201.23	159.84	202.40	213.03
CCSD(T)	AWCVQZ	200.42	159.52	201.46	212.08
DKH2-CCSD(T)	AWCVTZ-DK	201.22	159.89	202.43	213.03
DKH2-CCSD(T)	AWCVQZ-DK	200.40	159.56	201.49	212.08
CCSDT	VDZ	202.93	163.62	205.08	215.04
CCSDT	VTZ	202.07	159.75	203.21	213.30
CCSDT(Q)	VDZ	203.00	163.68	205.17	215.09
CCSD(T)-F12b	CVTZ-F12	200.11	159.31	201.14	211.83
CCSD(T)-F12b	CVQZ-F12	200.10	159.24	201.10	211.70
best estimateb		200.21	159.37	201.29	211.83
rg structurec		200.5(3)	159.6(2)	202.3(3)	212.7(3)
rα structured				201.7(3)	212.4(3)

^aSubscripts eq and ax refer to equatorial and axial ligands, respectively.

^bSee text.

^cReference (29) for PCl₃F₂ and ref (30) for PCl₅.

^dReference (30).

As suggested by our current test calculations and by the results obtained for PF₃, PCl₃, PH₃F₂, and PF₅ (see above), the best estimated bond lengths in PCl₃F₂ and PCl₅ are expected to be accurate to within ±0.10 pm. In the absence of any semiexperimental or experimental equilibrium geometries for these two species, it is obvious that our best estimates are the most reliable and accurate r_e structures available to date for PCl₃F₂ and PCl₅ (see Table 5).

In PCl₃F₂, the best estimated PCl and PF r_e distances are smaller than the corresponding r_g values²⁹ by 0.3 and 0.2 pm, respectively. Qualitatively, this is in line with expectation. However, we note that these differences are still covered by the experimental²⁹ error limits (see Table 5). The PCl bonds in PCl₃F₂ may be compared with those in PCl₃, showing that the best estimated PCl r_e distance is smaller in the phosphorane species by 3.7 pm (see Tables 3 and 5). The comparison of the axial bonds in PCl₃F₂ with those in PF₅ reveals that substitution of the equatorial F by Cl atoms elongates these bonds by 2.2 pm (best estimate; see Tables 4 and 5). In the corresponding r_g structures,^27,29 this elongation is slightly smaller (1.9 pm).

Using the theoretical best estimate of the r_e structure of PCl₃F₂, the rotational constants of P³⁵Cl₃F₂ (P³⁷Cl₃F₂) are calculated to be B_e = 1647.47 (1585.41) MHz and C_e = 1201.83 (1136.90) MHz. Taking into account theoretical (CCSD(T)/VQZ) vibrational and theoretical (CCSD(T)/AWCVTZ) electronic corrections yields B₀ = 1642.23 (1580.47) MHz and C₀ = 1197.92 (1133.27) MHz. On the basis of our experience with analogously computed rotational constants of PF₃ (see ref (72)), PCl₃ (see subsection 3.2), PH₃F₂ and PF₅ (see subsection 3.3), the B₀ and C₀ values predicted for PCl₃F₂ should be accurate to within 0.05%.

In PCl₅, the bond lengths at equilibrium (best estimates) are distinctly shorter than the corresponding r_g distances,³⁰ by 1.0 and 0.9 pm, respectively, for the equatorial and axial bonds. These differences correspond to at least three times the experimental³⁰ uncertainties (see Table 5), which appears to be quite substantial. This is in contrast to the situation in PCl₃F₂ (see above). The associated r_α – r_e differences are somewhat less pronounced (0.4 and 0.6 pm, respectively). The axial bonds in PCl₅ are much longer than the equatorial ones, according to the best estimates by 10.5 pm. Similar such differences are found in the r_g (10.4(4) pm) and r_α (10.7(4) pm) structures.³⁰ In PF₅, the corresponding difference is considerably smaller (close to 4 pm; see above). Replacing the axial F atoms in PCl₃F₂ by Cl atoms lengthens the equatorial PCl bonds by 1.1 pm (best estimate). This effect seems to be more pronounced (1.8(4) pm) when the equatorial r_g distances in PCl₅ are compared with those in PCl₃F₂ (see Table 5).

The best estimated r_e structure of PCl₅ implies that the rotational constants of P³⁵Cl₅ (P³⁷Cl₅) are A_e = 1188.97 (1124.74) MHz and B_e = 960.15 (908.28) MHz. Correcting them for the effects of zero-point nuclear motions (CCSD(T)/VTZ) and for electronic contributions (CCSD(T)/AWCVTZ) results in A₀ = 1184.94 (1120.99) MHz and B₀ = 956.82 (905.18) MHz. Just as for PCl₃F₂, the values of the rotational constants of PCl₅ serve as predictions whose maximum errors should not exceed 0.05% (see above).

4. Conclusions

Accurate molecular equilibrium geometries of selected trivalent (PF₃ and PCl₃) and pentavalent (PH₃F₂, PF₅, PCl₃F₂, and PCl₅) phosphorus compounds are reported. These geometries were calculated by means of a composite ab initio approach, which is based on explicitly correlated coupled-cluster theory (CCSD(T)-F12b) employing a large correlation-consistent orbital basis set (CVQZ-F12) to include core–valence electron correlation. The equilibrium structures optimized at this level of theory are corrected for the effects of an iterative treatment of triple excitations (CCSDT/VTZ versus CCSD(T)/VTZ) and of a noniterative treatment of quadruple excitations (CCSDT(Q)/VDZ versus CCSDT/VDZ). Scalar relativistic effects (DKH2-CCSD(T)/AWCVTZ-DK versus CCSD(T)/AWCVTZ) are also included.

Extensive test calculations on diatomic (PH, PF, PCl) and triatomic (PH₂, PF₂) radicals as well as on phosphane (PH₃) establish the accuracy of the composite procedure. The errors of the best estimated bond lengths usually do not exceed ±0.10 pm, and the best estimated bond angles turn out to be accurate to within ±0.05°. We find that basis set convergence of the leading term of the composite approach is not as fast as observed previously in molecules consisting only of first-row atoms and hydrogen.²¹ Generally speaking, the CVQZ-F12 basis may often be replaced by the smaller CVTZ-F12 basis without significant loss of accuracy, but doing so seems to compromise the present target accuracy of ±0.10 pm in some cases, for example in the case of the axial bond lengths in PCl₅ (see Table 5).

For PF₃, PCl₃, PH₃F₂, and PF₅ sufficiently accurate ground-state rotational constants are available from high-resolution rotational and vibrational spectroscopy. We corrected these constants for the effects of zero-point vibrations (CCSD(T)/AVQZ and CCSD(T)/VQZ, respectively) and for electronic contributions related to the rotational g-tensor (CCSD(T)/AWCVTZ). The resulting empirical equilibrium rotational constants were used to determine semiexperimental r_e structures of these molecules. Throughout we find that these structures are in excellent agreement with the corresponding best theoretical estimates. This cross-validation confirms the accuracy of both approaches and suggests that best theoretical estimates can be used confidently instead of the semiexperimental r_e structures for species whose rotational constants have not yet been measured. The magnetic corrections of the experimental ground-state rotational constants are generally found to be negligible for structural purposes. In the case of PCl₃, PH₃F₂, and PF₅ we recommend the current semiexperimental r_e structures (essentially identical with the best estimates; see Tables 3 and 4), whereas for PF₃ the purely experimental²r_e structure continues to be the most reliable and accurate equilibrium geometry published to date.

The best estimated equilibrium geometries of PCl₃F₂ and PCl₅ are expected to be of the same accuracy (±0.10 pm for bond lengths) as those for PF₃, PCl₃, PH₃F₂, and PF₅. Therefore, we recommend these best estimates as the most reliable and accurate r_e structures of PCl₃F₂ and PCl₅ available at present (see Table 5).

Supporting Information Available

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.9b04406.

• Two tables containing additional numerical results: computed vibrational corrections to the ground-state rotational constants of all title and selected test (PH₂, PD₂, PF₂, PH₃, PD₃) molecules as well as computed rotational g-tensors and related electronic contributions to the rotational constants of all title and closed-shell test (PH₃, PD₃) molecules (PDF)

The authors declare no competing financial interest.