Dipole properties of PH3, PF3, PF5, PCl3, SiCl4, GeCl4, and SnCl4

ABSTRACT Experimental and theoretical photoabsorption cross sections combined with constraints provided by the Kuhn–Reiche–Thomas sum rule, the high-energy behaviour of the dipole oscillator strength density, static dipole polarisabilities, and molar refractivity data when available are used to construct dipole oscillator strength distributions for PH3, PF3, PF5, PCl3, SiCl4, GeCl4, and SnCl4. The distributions are used to predict dipole sum rules S(k), mean excitation energies I(k), and van der Waals C6 coefficients. GRAPHICAL ABSTRACT


Introduction
The dipole oscillator strength distribution (DOSD) of a molecule is specified by excitation energies E i and oscillator strengths f i for the discrete spectrum and by the differential dipole oscillator strength (DOS) function (df/dE) for the continuum of energies, E c ࣘ E < Ý [1]. The DOS is proportional to the photoabsorption cross section [2]. Many important physical properties are related to the DOSD moments defined by (1) for k < 5/2. Atomic units are used in the above equation and throughout this work except where explicitly stated otherwise. The moments with integer 2 ࣙ k ࣙ −1 can be expressed as ground-state expectation values [3][4][5][6]. The Kuhn-Reiche-Thomas (KRT) sum rule states that S(0) is the number of electrons N e in the molecule [3,4]. The S (2) moment is proportional to the sum of the electron density values at the nuclei [5,7], whereas S(−1) and S(1) are CONTACT Ajit J. Thakkar ajit@unb.ca Supplemental data for this article can be accessed at http://dx.doi.org/./... related to statistical electron correlation coefficients [8,9] in position and momentum space, respectively. The sums S(−1) and S(−3) are related, respectively, to the total and the total zero-angle differential cross sections for inelastic scattering in collisions of the molecule with charged particles [10,11]. The static dipole polarisability α is given by S(−2) [12]. The Cauchy moments, S(−4), S(− 6), … , are the coefficients of the Taylor expansion of the frequencydependent polarisability α(ω) at low frequencies, ω. The molar refractivity is given by R(ω) = (4πa 3 0 /3)N A α(ω) in which N A is Avogadro's constant. The DOSD can be used to obtain α(ω) via the sum-over-states expression Other important properties follow from the mean excitation energies in which L(k) are logarithmic moments of the DOSD defined by The average energy associated with the total inelastic scattering cross section for grazing collisions of fast charged particles with the target species is I(−1). Radiation damage theory requires the average energies I(0) and I(1) which are, respectively, related to the average energy loss (stopping power) and its mean fluctuation (straggling) in these collisions [10,11,13]. I(2) is related to the Lamb shift [14,15]. The quantity I(−2) is a nearly ideal [16,17] average energy to use in the London [18] formula for the coefficient of the spherically averaged induced-dipoleinduced-dipole C 6 R −6 term that dominates the interaction between freely rotating closed-shell molecules separated by a large distance R. With this choice of average energy, the London formula reduces to in which α X (0) and I X (−2) are the static dipole polarisability and mean excitation energy, respectively, for species X = A, B. An exact way to compute C 6 is to substitute the frequency-dependent polarisabilities for the two species into the Casimir-Polder expression [19] in which i = √ −1. The C 6 coefficients are essential ingredients in the construction of model, non-retarded, intermolecular potentials that are valid for all intermolecular distances [20][21][22][23][24]. If both C 6 and C 8 dispersion coefficients are available, then C 10 and higher order coefficients can be calculated to good accuracy from simple models [25,26].
This work reports the construction of DOSDs for PH 3 , PF 3 , PF 5 , PCl 3 , SiCl 4 , GeCl 4 , and SnCl 4 from experimental and theoretical photoabsorption cross sections combined with constraints provided by the KRT sum rule, molar refractivity data, static dipole polarisabilities, and the high-energy behaviour of the dipole oscillator strength density. The method and data used [27] are summarised in Sections 2 and 3, respectively. The coupledcluster polarisabilities calculated to serve as constraints, the distributions we obtain, the resulting dipole properties S(k) and I(k), and van der Waals C 6 coefficients are summarised and discussed in Section 4. Some final remarks are made in Section 5.

Method
Our method is a refinement of the Meath method [28] and has been described in detail elsewhere [1]. Here, we provide only a terse summary. The available DOS data from [1,28] is used to obtain scaling factors for each interval. The constraints used are invariably the KRT sum rule and often molar refractivity values at one or two wavelengths. When refractivity data are unavailable, the static dipole polarisability α(0) = S(−2) can be used as a constraint. Moreover, we impose the known high-energy asymptotic behaviour [7] of the DOS density given [1] by in which the sum runs over the N n nuclei with charges Z K and the electron number density ρ( r) evaluated at the nuclear position vectors R K . Observe that μ 1 = S (2). An iterative procedure can be used to obtain S(2) and γ , provided that the unitless ratios μ j /μ 1 , j = 2, 3, can be estimated. Since quantities that depend on the electron density at the nucleus are quite insensitive to their molecular environment, a free-atom additive model [29] in which the μ j , j = 1, 2, 3, are obtained by replacing the electronic densities at the nuclei in Equation (8) by their free-atom counterparts is quite accurate [1]. Note that free-atom values of ρ(0) at the Hartree-Fock level are available for all the atoms up to Xe from Table IV of Koga et al. [30]. Moreover, observe that, at the Hartree-Fock level, Equation (5.49) of [31] reduces to ρ(0) = πb 8 Z 3 /8 in which b 8 Z 5 is the leading coefficient of the large momentum density expansion of the electronic momentum density [31,32]. Hence, Hartree-Fock values of ρ(0) for all the atoms up to lawrencium can be obtained from the values of b 8 in Table 2 of [33].
Therefore, a good way to utilise this Laurent expansion (Equation (7)) is to use the free-atom additive model values of μ 2 /μ 1 , μ 3 /μ 1 , and S(2) to determine an initial γ by requiring the three-term Laurent expansion to match the value of (df/dE) at E N . Then, subject the new distribution to the constraint procedure and obtain a new DOSD, a new S(2), and a new γ . This process is repeated until S(2) converges to six significant figures. In our experience, no more than 15 iterations are needed to reach convergence.
The above method is applied to a representative selection of the many distributions that can be constructed using different combinations of the available photoabsorption data and constraints. The distribution that leads to the smallest standard deviation s of the scaling parameters is chosen as the best one. If several distributions lead to very similar values of s, then, as of this work, the distribution leading to the smoothest DOS is selected. This small change goes some way to addressing the concern [34] that the method sometimes leads to DOSDs that are point-wise inaccurate, particularly at the energy interval boundaries.

Photoabsorption data
The photoabsorption data examined for each of the seven molecules is summarised below. Data are available over an extended energy range for five of the systems. The energy range over which photoabsorption data is available for GeCl 4 and SnCl 4 is somewhat limited, but fortunately it does not have gaps up to 30 eV.
There is a wealth of experimental photoabsorption data available for PH 3 over the range from 5 to 279 eV [35][36][37][38][39][40][41][42]. Moreover, there are random-phase approximation (RPA) calculations from 5 to 30 eV [43] and data from 12 to 75 eV computed by time-dependent (TD) density functional theory (DFT) [44]. There is a selection of experimental photoabsorption data over the ample range from 5 to 300 eV [39,45,46] for PF 3 and over the ample range from 4 to 350 eV [35,39,46,47] for PCl 3 . Au and Brion [48] reported low-resolution photoabsorption cross sections for PF 5 from 10 to 300 eV and highresolution cross sections from 10 to 50 eV and from 133 to 152.9 eV. We are not aware of any other experimental measurements for PF 5 .
For all the molecules, photoabsorption cross sections at higher energies are constructed from a free-atom additive model. Atomic data up to 30 keV can be taken from the semiempirical compilation of Henke et al. [62] and for higher energies up to 100 keV from the theoretical tabulations of Chantler [63,64]. Equation (7) is used for the asymptotic region from 100 keV to Ý, following the procedure described in Section 2 and elsewhere [1].

Refractivity data
A recent exhaustive compilation [65] of gas-phase polarisability measurements reveals that there are data for six of the molecules under consideration but not for PF 3 . Of these, the values for PH 3 , PF 5 , PCl 3 , and GeCl 4 are derived from dielectric permittivity experiments and have relatively large error bars [65]. Thus, only the experimental molar refractivity measurements for SiCl 4 [66,67] and SnCl 4 [67] were considered as constraints. Instead, we turn to theoretical static electronic dipole polarisabilities as constraints for the remaining five molecules. These are described in the next section.
Finally, we also calculated polarisabilities for all seven molecules using the CCSD(T) method [69,82], a coupledcluster (CC) method based on single (S) and double (D) substitutions [69,83] plus a non-iterative correction for triple (T) substitutions. The CCSD(T) polarisability calculations use the def2-TZVPD basis set [84] optimised for molecular response calculations. Then, improved values were obtained from CCSD(T)/q = CCSD(T)/def2-TZVPD + in which = MP2/def2-QZVPD − MP2/def2-TZVPD is a correction for residual basis set effects. All these computed polarisabilities and the experimental values recommended by Hohm [65] are summarised in Table 1. Other calculations reported in the literature for PH 3 include values of 30.6 au at the MP2 level [85] and 30.04 au at the CCSD(T)/CBS level [86]. There is also a published [87] CCSD(T)/aug-cc-pVTZ It is interesting to note that the static polarisabilities of PF 3 and PF 5 are almost identical. This means that they cannot be accounted for by either a free-atom (level 1) or a dressed-atom (level 2) additive model using the terminology of a hierarchy of additive models [29]. The lowest level additive model required is level 3 where the polarisability depends not only on the atomic number of an atom but also its coordination number [29]. The polarisabilities of PF 3 and PF 5 make it evident that a pentavalent phosphorus atom has a smaller polarisability than a trivalent phosphorus atom. Apparently, this is due to the lone pair electrons on the trivalent phosphorus being used for bonding in the pentavalent phosphorus atom.

Dipole sum rules and mean excitation energies
The computer program of [1] implementing the method described there and summarised in Section 2 was used to construct many constrained DOSDs using various combinations of the photoabsorption data described in Section 3.1, the KRT sum rule, the high-energy asymptotic behaviour, and one or two low-energy constraints chosen as follows. Coupled-cluster calculations are considered the 'gold standard' [69] and therefore we use only CCSD(T) polarisabilities, as opposed to MP2 or DFT values, as constraints. The CCSD(T)/CBS α from Muller and Woon [86] was tried as a constraint for PH 3 . The CCSD(T)/q polarisabilities presented in Section 4.1 were used as constraints for the DOSDs of PH 3 , PF 3 , PF 5 , PCl 3 , and GeCl 4 , whereas the constraints for SiCl 4 and SnCl 4 were based on the refractivity data mentioned in Section 3.2. In this manner, the number of constrained DOSDs tried ranged from 30 for GeCl 4 to 182 for SiCl 4 . In six cases, the DOSD with the smallest standard deviation s of the scaling factors was selected as the recommended DOSD. In the case of PF 5 , the DOSD with the second lowest s was selected because it was smoother at the energy interval boundaries and its s was only 1% higher than the lowest one. Further details including the energy ranges and sources of the photoabsorption data are gathered in the supplementary information.
The recommended dipole sum rules S(k) and the mean excitation energies for each of the seven molecules are listed in Tables 2 and 3, respectively. The only previously published values of S(k) other than S(−2) = α(0) for these systems that we are aware of are those of Olney et al. [88] for PH 3 , PF 3 , PF 5 , and PCl 3 . Their values were obtained from the DOSDs measured by Brion's group [41,45,47,48] normalised to the valence-shell KRT sum rule for PH 3 and PF 3 , and normalised to static polarisabilities for PF 5 and PCl 3 . The mean absolute percent deviations (MAPDs) of their S(k) relative to ours are 8.40%, 6.45%, 5.13%, 7.31%, 9.68%, and 11.9%, respectively, for k = −1, −2, …, −6.
The S(k) can be compared with values obtained from a free atom additive model [29] based on values of the atomic moments. In particular, using S(−2) from coupled-cluster computations of atomic polarisabilities [89][90][91] [92,93] with S a (−2) > S(−2) as do more than 97% of systems [93]. We estimate the uncertainties in our S(k) to be no more than ±3% for 2 ࣙ k ࣙ − 2 and ±4% for −3 ࣙ k ࣙ − 6.

Dipole-dipole dispersion C 6 coefficients
The dipole-dipole dispersion C 6 coefficients listed in Table 4 for the rotationally averaged long-range interaction between the seven species studied in this work and 56 other species were calculated using the pseudospectra from the supplementary information and ones reported earlier for He [95], H [26], Ne, Ar, Kr, and Xe [27], C 60 [96], O 2 , O 3 [97], pyridine [1], and 46 other species The C 6 coefficients are listed in Table 4. The tripledipole coefficients C 9 can easily be obtained from the pseudospectra.

Concluding remarks
The latest procedure, described in [1] and summarised in Section 2, for constructing constrained DOSDs makes it possible to impose the high-energy asymptotic behaviour on a DOSD for any molecule. It was used for seven previously unexamined species in this work and is being applied to 11 halomethanes and haloethanes; the results will be reported soon.