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

Among the title species, a reliable and accurate equilibrium geometry ( re structure) is available only for PF3, which has been determined experimentally more than 20 years ago. Here, we report accurate re 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 re 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, re(PCl) = 203.94 pm and θe(ClPCl) = 100.18°; for PH3F2, re(PHeq) = 138.38 pm and re(PFax) = 164.15 pm; for PF5, re(PFeq) = 153.10 pm and re(PFax) = 157.14 pm; for PCl3F2, re(PCleq) = 200.21 pm and re(PFax) = 159.37 pm; and for PCl5, re(PCleq) = 201.29 pm and re(PClax) = 211.83 pm. The associated uncertainties are estimated to be ±0.10 pm and ±0.10°, respectively.

Introduction

The most meaningful representation of the geometry of a molecule is provided by its equilibrium structure (re 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 re 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 measured[1] constants B0 and C0 were converted[2] to the corresponding equilibrium values Be and Ce to determine a reliable and accurate re 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 re 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 reported[21] a composite procedure related to the so-called geometry scheme[20] 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 study[21] was benzene serving as a test molecule. We note that gradient schemes[18−20] 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 demanding[20] 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 re structure is available in the literature only for PF3.[2] For PCl3 and PF5, re structures were estimated from corresponding zero-point average (r) structures.[25,26] Gas electron diffraction data measured[27] for PF5 were reanalyzed[28] 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 (rg structure) were determined by gas-phase electron diffraction.[29] Finally, for PCl5 average structural parameters denoted rα, rg, and ra 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 re structure to check the respective theoretical best estimate provided by the current purely computational scheme.

Computational Methods

We adopt the composite computational approach outlined in ref (21) to determine theoretical best estimates for the equilibrium bond length re and bond angle θe 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 procedure[21] are repeated here (see eqs –4):As indicated by the first term on the right-hand side of eq , the structural parameter pe (re or θe) 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 optimized[23] for the explicitly correlated F12 methods.[22] At hydrogen, this basis reduces to the cc-pVQZ-F12 basis set.[24] The parameter pe evaluated at the CCSD(T)-F12b/CVQZ-F12 level of theory is expected to be reasonably close to the basis set limit[22] 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-F12[23] being equal to cc-pVTZ-F12[24] 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) determinant[38] 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 × 10–6Eh/a0. For further technical details concerning the (U)CCSD(T)-F12b calculations, the reader is referred to ref (21).

The term Δpe[T] (see eqs 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 pe 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 ). The acronym VTZ refers to the correlation-consistent polarized valence triple-ζ basis cc-pVTZ[45] for the H and F atoms and the tight d-augmented cc-pV(T+d)Z[46] 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), Δpe[T] was also calculated with the VQZ, V5Z, and up to V6Z basis sets, which are the n-tuple-ζ (n = 4, 5, 6) analogues[45−47] 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 CCSDT[48] 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 Δpe[(Q)] (see eqs and 3) is computed as the difference of pe 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), Δpe[(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 CFOUR[49] had to be used for CCSDT(Q), whereas analytic[54] 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 Δpe[(Q)] (and Δpe[Q]) term the associated CCSDT calculations were also based on a UHF determinant (UHF-CCSDT)[48] in these cases.

The final term Δpe[SR] (see eqs and 4) serves as a correction for scalar relativistic effects on the given structural parameter pe: This term was evaluated from the difference of pe 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 analogues[62−64] 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 recontracted[65] 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 Δpe[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 (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:

Test molecules: diatomic radicals PH, PF, and PCl (X 3Σ–; C∞), triatomic radicals PH2 and PF2 (X 2B1; C2), and PH3 (C3v).

Target molecules: PF3 (C3), PCl3 (C3), PH3F2 (D3), PF5 (D3), PCl3F2 (D3), and PCl5 (D3).

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 semiexperimental[13]re structure for these species. To be more specific, we utilized for the asymmetric top species PX2 (X = H, D, F) the measured[67−69] 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[7,8,11,12]A0 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] ΔXvib (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 procedure[75] 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 × 10–10Eh/a0. 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-constants[73] (using the CFOUR[49] 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-pVQZ[63] basis for H and F and aug-cc-pV(Q+d)Z[46] 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 ΔXvib, there is a small electronic (magnetic) contribution ΔXel, 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 CFOUR[49] 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 ΔXel values computed presently are contained in Tables S1 and S2 of the Supporting Information, together with the ΔXvib 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 PH3[81] 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 re(PH) in PH3F2 where a somewhat larger effect (0.012 pm) occurs.

Several of the basis sets employed in calculations with the programs CFOUR[49] and MRCC[52] were downloaded from the EMSL basis set library using the Basis Set Exchange web portal.[83,84]

Results and Discussion

Test Molecules

The computed equilibrium geometries of the diatomic test molecules PH, PF, and PCl are collected in Table , and those of PH2, PF2, and PH3 are listed in Table . 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 re structures (see section ).

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)

See text.

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

Not calculated.

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

Value determined for PD.

Reference (87) for PH and ref (90) for PF.

Reference (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)

See text.

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

Not calculated.

Values evaluated for PD2 and PD3, respectively.

Reference (71).

Values 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 re 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 re 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 re 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 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 recontracted[65] basis sets, as before (see Tables 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 and 2 also show the best estimates of the re structures of the test molecules resulting from the current composite approach. They may be compared with the corresponding experimental re 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 re structures were determined for the PX2 (X = H, D, F) species (see section ). These structural data and their uncertainties were evaluated from the semiexperimental rotational constants Ae, Be, and Ce in complete analogy to the procedure used to infer the experimental r0 structure of PF2.[69] Using the experimental[67−69] ground-state rotational constants, the corresponding inertial defects Δ0 are presently calculated to be 0.0688 (PH2), 0.0958 (PD2), and 0.1906 u Å2 (PF2) whereas the respective semiexperimental equilibrium values Δe appear to be −0.000861, –0.000443, and +0.00202 u Å2. As expected, the Δe values are significantly closer to zero than their Δ0 counterparts. Additionally, we have determined semiexperimental re structures for PH3 and PD3 (see section ), 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 re structures of PH3 and PD3: in the former, the bonds are lengthened by 0.00017 pm and θe(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 experimental[71] determination of the re structures of PH3 and PD3 is fully justified by these results. The semiexperimental re structures of both species are in excellent agreement with their purely experimental[71] counterparts (see Table ).

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 experimental[71] values, respectively, to within 0.05° (see Table ). Applying a similar computational scheme as adopted presently, the re structure of PH2 was best estimated[92] to be re(PH) = 141.58 pm and θe(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 ). When excluding this correction, our current best estimate for the PH2 structure is re(PH) = 141.57 pm and θe(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 observation[21] 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 re 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)).

PF3 and PCl3

Table 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 re structures. The latter are also given in Table . 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 previously[72] (see Table , footnote a) and are included in Table 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)

From ref (72).

Not calculated.

See text.

Numerically identical to the best estimate in ref (72).

Reference[2] for PF3 and ref (25) for PCl3.

Estimate of uncertainty from ref (95).

Reference (94).

The current semiexperimental re structure of PF3 is identical with its experimental[2] counterpart (see Table ), both with regard to the estimated uncertainties of the former (±0.10 pm and ±0.10°, respectively) and the even smaller experimental[2] error bars (±0.014 pm and ±0.04°, respectively). The best estimated re structure is also in excellent agreement with experiment[2] (see Table ). For the sake of completeness, Table also contains another experimental equilibrium geometry of PF3 determined in an earlier investigation.[94] Both experimental[2,94]re 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 experimental[2]re structure appears to be the most reliable and accurate equilibrium geometry of PF3 presently available. Thus, it is the recommended re structure for PF3 (see Table ).

Turning to PCl3, the current semiexperimental re 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 ra structure at two different temperatures, an estimate of the differences ra – re, and hence a first estimate of the equilibrium geometry of PCl3. The most reliable re structure of PCl3 published[25] so far was derived from the r structure; it is identical with the semiexperimental re structure, within the corresponding uncertainties (see Table ). 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: re(PCl) = 203.94(10) pm and θe(ClPCl) = 100.18(10)°. The additional inclusion of the ground-state rotational constants[5] of the P35Cl237Cl species through an unweighted least-squares structural fit (using the STRFIT code of Kisiel[97]) 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: re(PCl) decreases by 0.01 pm and θe(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 re structure, the rotational constants of P35Cl3 (P37Cl3) are calculated to be Be = 2623.81 (2493.77) MHz and Ce = 1476.09 (1396.35) MHz (same unit as in the experimental work[3−6]). 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[5,6]B0 values of P35Cl3 and P37Cl3 as well as the experimental[5]X0 (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).

PH3F2 and PF5

Table contains the calculated bond lengths in PH3F2 and PF5, in particular those needed to arrive at the theoretical best estimates of the re 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 D3 point-group symmetry, just as in PF5. PH3F2 and PF5 are examples of hypervalent species. Although a discussion of hypervalency[100] is beyond the scope of this paper, we note that a decade ago a new model of hypervalent bonding has been introduced[101] termed recoupled pair bonding. This concept has also been applied to the series of PF (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)

Subscripts eq and ax refer to equatorial and axial ligands, respectively.

See text.

Reference (26).

Reference (28).

References (7) and (8) for PH3F2 and ref (12) for PF5.

Estimate of uncertainties from ref (95).

Reference (27).

Turning to the equilibrium geometry of PH3F2, the best estimated re structure and its semiexperimental counterpart are in excellent agreement, with deviations of only 0.04 and 0.03 pm for re(PHeq) and re(PFax), respectively (see Table ). As before, the error bars of the semiexperimental re distances are assumed to equal those typically observed for the best estimates in our test calculations (±0.10 pm; see subsection ). 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 ).

The theoretical best estimate for the re structure implies the following values for the rotational constants of PH3F2 (PD3F2): Ae = 2.909996 (1.456117) cm–1 and Be = 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[7,8]A0 value was estimated[8] 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 re 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 ). The most recent experimental[26]re structure of PF5 was derived from the r structure. Prior to this experimental work,[26] the gas electron diffraction data[27] were reanalyzed[28] to deduce not only a barrier height (3.4 kcal/mol) for the pseudorotation[103] in PF5 but also the equilibrium internuclear distances (see Table , no uncertainties quoted in ref (28)). The two experimental[26,28]re structures agree within the error limits of ±0.3 pm estimated in the more recent work.[26] Comparing the semiexperimental re structure with experiment,[26] the equatorial bond lengths differ by 0.2 pm, which is still covered by the experimental[26] uncertainty, whereas the difference between the axial bond lengths is somewhat more pronounced (0.46 pm). Similar to PH3F2, there is also an experimental[12]r0 structure for PF5 which was determined from the measured[11,12]A0 and B0 constants. As expected, the r0 bond lengths are somewhat larger (by about 0.3 pm) than their semiexperimental re counterparts (see Table ). While the equatorial r0 bond lengths are also larger (0.4–0.5 pm) than their experimental[26,28]re 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 re distance in PF5 is shorter than the equilibrium bond length in PF3 by about 3 pm (see Tables and 4 for the best estimated, semiexperimental, and experimental re structures). Similarly, the PH re distance in PH3F2 is shorter than the equilibrium bond length in PH3 by about 2.8 pm (see Tables and 4). The axial re distance in PF5 turns out to be much smaller than in PH3F2, by about 7 pm (see Table ). It is well-known that the axial bonds in PF5 are distinctly longer than the equatorial ones.[27] Considering the best estimated and semiexperimental re 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 re structure, the rotational constants of PF5 are calculated to be Ae = 3781.45 MHz and Be = 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 re 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 re structures of PH3F2 and PF5 (see Table ) as the most reliable and accurate equilibrium geometries available at present for these two species.

PCl3F2 and PCl5

The computed equilibrium bond lengths including the best estimates for PCl3F2 and PCl5 are reported in Table . The rotational constants of both species have not yet been measured. Therefore, the semiexperimental strategy[13] could not be applied in these two cases. However, the rg structures of both phosphoranes as well as the rα and ra structures of PCl5 are known from gas electron diffraction studies.[29−31] The experimental[29,30]rg and rα internuclear distances are listed in Table . Like PH3F2 and PF5 (see subsection ), PCl3F2 and PCl5 form trigonal bipyramidal molecules of D3 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)

Subscripts eq and ax refer to equatorial and axial ligands, respectively.

See text.

Reference (29) for PCl3F2 and ref (30) for PCl5.

Reference (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 re structures available to date for PCl3F2 and PCl5 (see Table ).

In PCl3F2, the best estimated PCl and PF re distances are smaller than the corresponding rg values[29] 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 experimental[29] error limits (see Table ). The PCl bonds in PCl3F2 may be compared with those in PCl3, showing that the best estimated PCl re distance is smaller in the phosphorane species by 3.7 pm (see Tables 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 and 5). In the corresponding rg structures,[27,29] this elongation is slightly smaller (1.9 pm).

Using the theoretical best estimate of the re structure of PCl3F2, the rotational constants of P35Cl3F2 (P37Cl3F2) are calculated to be Be = 1647.47 (1585.41) MHz and Ce = 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 ), PH3F2 and PF5 (see subsection ), 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 rg 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 experimental[30] uncertainties (see Table ), which appears to be quite substantial. This is in contrast to the situation in PCl3F2 (see above). The associated rα – re 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 rg (10.4(4) pm) and rα (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 rg distances in PCl5 are compared with those in PCl3F2 (see Table ).

The best estimated re structure of PCl5 implies that the rotational constants of P35Cl5 (P37Cl5) are Ae = 1188.97 (1124.74) MHz and Be = 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).

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 ).

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 re 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 re 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 re structures (essentially identical with the best estimates; see Tables and 4), whereas for PF3 the purely experimental[2]re 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 re structures of PCl3F2 and PCl5 available at present (see Table ).