Article pubs.acs.org/JPCA

Computational Tests of Quantum Chemical Models for Excited and Ionized States of Molecules with Phosphorus and Sulfur Atoms David K. Hahn, Krishans RaghuVeer, and J. V. Ortiz* Department of Chemistry and Biochemistry, Auburn University, Auburn, Alabama 36849-5312, United States ABSTRACT: Time-dependent density functional theory (TD-DFT) and electron propagator theory (EPT) are used to calculate the electronic transition energies and ionization energies, respectively, of species containing phosphorus or sulfur. The accuracy of TD-DFT and EPT, in conjunction with various basis sets, is assessed with data from gas-phase spectroscopy. TD-DFT is tested using 11 prominent exchangecorrelation functionals on a set of 37 vertical and 19 adiabatic transitions. For vertical transitions, TD-CAM-B3LYP calculations performed with the MG3S basis set are lowest in overall error, having a mean absolute deviation from experiment of 0.22 eV, or 0.23 eV over valence transitions and 0.21 eV over Rydberg transitions. Using a larger basis set, aug-pc3, improves accuracy over the valence transitions via hybrid functionals, but improved accuracy over the Rydberg transitions is only obtained via the BMK functional. For adiabatic transitions, all hybrid functionals paired with the MG3S basis set perform well, and B98 is best, with a mean absolute deviation from experiment of 0.09 eV. The testing of EPT used the Outer Valence Green’s Function (OVGF) approximation and the Partial Third Order (P3) approximation on 37 vertical first ionization energies. It is found that OVGF outperforms P3 when basis sets of at least triple-ζ quality in the polarization functions are used. The largest basis set used in this study, aug-pc3, obtained the best mean absolute error from both methods -0.08 eV for OVGF and 0.18 eV for P3. The OVGF/631+G(2df,p) level of theory is particularly cost-effective, yielding a mean absolute error of 0.11 eV.

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INTRODUCTION The toxicity of certain organophosphorus and organosulfur compounds can scarcely be more astonishing. The compound Oethyl S-[2-(diisopropylamino)ethyl] methylphosphonothioate, for example, is so lethal that skin contact with only 10 mg, or inhalation of only 30 mg per cubic meter over a period of 15 min, is potentially fatal to humans.1 Remote detection of this compound, better known as VX, other chemical warfare agents, and their thermal decomposition products (which may also be toxic) with spectroscopic sensors is therefore of interest to public safety, but this endeavor requires a knowledge of spectroscopic signatures that has yet to be developed. Theoretical calculation of such data is a welcome alternative to experimental observation when hazardous compounds like these are under consideration. The molecular size of some chemical warfare agents necessitates the use of the most cost-effective quantum chemical method for studying their excited electronic states, which presently would probably be linear-response time-dependent density functional theory (TD-DFT).2 This method computes excitation energies as the poles of a density response function for a time-dependent external potential, thereby providing information on numerous excited states in a single calculation. Its reliability often hinges on choosing an appropriate exchangecorrelation functional to describe excitation. Several TD-DFT studies of excitation energies of first row molecules have shown that functionals incorporating 20−25% Hartree−Fock exchange correlation typically have the best overall accuracy for low-energy valence transitions, whereas long-range corrected functionals are the best choice for Rydberg transitions.3−5 Unfortunately, TDDFT studies of molecules containing second row elements are less numerous, so the suitability of these functionals for investigating the electronic spectra of chemical warfare agents © 2014 American Chemical Society

is less certain. Only a few TD-DFT studies of phosphorus molecules are found in the literature, including those for P2,6 P4,7 PH2,8 and HCP.8 An application of TD-B3LYP to organosulfur molecules using the 6-31+G(d) basis set obtained a mean absolute deviation from experimental band maxima of 0.09 eV for n → π* transitions, and 0.24 eV for π → π* transitions,9 a level of accuracy surpassing that obtained for analogous sulfur-free compounds.10 As with TD-DFT, electron propagator theory (EPT) directly calculates the energy difference between initial and final states, and may provide numerous state energies in a single calculation. Both the partial third order (P3)11 and Outer Valence Green’s Function (OVGF)12,13 approximations to EPT compute ionization energies as eigenvalues of the sum of the Fock and self-energy operators in the canonical Hartree−Fock orbital basis. These methods use a truncated expansion of the selfenergy with respect to the electron−electron interaction, making them less expensive than ΔE(SCF) or ΔE(DFT) calculations and thus preferable for large systems. In P3, the expansion is to second order and neglects the off-diagonal elements of the selfenergy matrix. In OVGF, the expansion is to full third order and includes an estimate of higher order contributions. P3 is therefore more economical, but has been shown to outperform OVGF for certain first row molecules.11 For phosphorus and sulfur molecules, only a few EPT studies are found in the literature,14,15 so the performance quality of P3 and OVGF is not well-established. Received: March 11, 2014 Revised: April 29, 2014 Published: April 29, 2014 3514

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Table 1. Vertical Transition Energies of Phosphorus and Sulfur Molecules TD-DFT/APC3 − EXP EXP (eV)

species 1

SCH2(1 A2) SCl2(11B1) SCH3Cl(11A″) SCF2(11A2) SCl2(11A2) SCF3Cl(11A″) S(CH3)2(11A2) S(CH3)2(11B1) SO(CH3)2(11A″) P4 SCF3Cl(21A′)

n → π* nS → σ* nS → σ* n → π* nS → σ* nS → σ* n → σ* n → σ* n → π*

a

2.03 3.19b 3.49c 3.52d 3.57b 3.72e 5.20f 5.39f 5.53g,h 5.60i 5.79e,h

nCl → σ* SCH2(11B2) n → 4s 5.85j SCHCH3(21A′) π → π* 5.90j SO(CH3)2(21A′) n → σ* 6.03g 1 SCH3Cl(2 A′) nCl → 6.05c,h σ* SCH2(21A1) π → π* 6.13j P(CH3)3(21A′) n → 5s 6.15k 1 S(CH3)2(2 B1) n → 4s 6.15f 1 PH2CH3(2 A′) n → 4s 6.17k SH2(11B1) n → 4s 6.33l S(CH3)2(21A1) n → 4p 6.33f 1 PH2CH3(3 A′) n → 5s 6.33k 1 SCH2(2 B2) n → 4p 6.59j SO(CH3)2(1A′) π → π* 6.60g PH2CH3(11A″) n → π* 6.63k 1 SCH2(3 A1) n → 4p 6.84j 1 PH3(2 A1) n → 4s 6.89m SO(CH3)2(41A′) n → σ* 6.90g 1 SO(CH3)2(2 A″) π → σ* 7.12g 1 SCH2(4 B2) n → 5s 7.74j 1 SH2(2 A2) n → 4p 7.85l SO(CH3)2(31A″) π → σ* 7.92g 1 SH2(2 A1) n → 4p 8.03l 1 SCH2(5 B2) n → 5p 8.04j SCH2(41A1) n → 5p 8.18j PF3(11E) n → 4p 8.20n SH2(21B1) n → 4p 8.25l 1 SH2(5 B1) n → 5s 8.91l Mean Absolute Deviation (Val.) Mean Absolute Deviation (Ryd.) Mean Absolute Deviation Standard Deviation Maximum Deviation

HCTH

BLYP

PBE

B3-LYP

B98

PBE0

MPW1PW91

BMK

M062X

CAM-B3LYP

ωB97X-D

0.16 −0.34 −0.33 0.17 −0.49 −0.32 −0.46 −0.86 −0.35 −0.33 −0.66

0.11 −0.42 −0.40 0.06 −0.54 −0.39 −0.68 −1.07 −0.57 −0.50 −0.73

0.11 −0.38 −0.37 0.12 −0.53 −0.37 −0.52 −0.90 −0.43 −0.36 −0.71

0.18 −0.18 −0.24 0.15 −0.34 −0.23 −0.23 −0.47 0.09 −0.27 −0.41

0.20 −0.14 −0.22 0.21 −0.32 −0.20 −0.10 −0.28 0.23 −0.23 −0.38

0.20 −0.08 −0.17 0.24 −0.28 −0.16 −0.06 −0.23 0.30 −0.09 −0.31

0.21 −0.08 −0.17 0.23 −0.28 −0.16 −0.06 −0.25 0.28 −0.10 −0.31

0.17 −0.03 −0.16 0.21 −0.27 −0.16 0.18 0.25

−0.01 −0.08 −0.28 0.05 −0.40 −0.27 −0.10 −0.06

0.16 −0.01 −0.14 0.14 −0.23 −0.13 0.01 0.01

0.19 0.02 −0.12 0.21 −0.22 −0.11 0.13 0.17

−0.05 −0.20

−0.17 −0.30

−0.11 −0.20

−0.02 −0.20

−0.76 0.08 −1.27 −1.07

−1.05 −0.05 −1.43 −1.33

−0.84 0.07 −1.30 −1.14

−0.50 0.14 −0.69 −0.71

−0.31 0.24 −0.52 −0.61

−0.25 0.29 −0.45 −0.54

−0.28 0.28 −0.48 −0.54

0.14 0.22 0.17 −0.45

−0.16 0.09 −0.03 −0.56

−0.12 0.08 0.02 −0.45

0.01 0.13 0.12 −0.44

0.22 −0.83 −0.80 −0.86 −0.43 −0.76 −0.92 −0.79 −0.20 −0.12 −0.90 −0.82 −1.50 −1.25 −1.13 −0.96 −1.45 −0.79 −1.44 −1.23 −0.74 −1.16 −1.30 0.58 0.92 0.75 0.47 −1.50

0.11 −0.98 −1.16 −1.00 −1.00 −1.28 −0.66 −1.16 −0.44 −0.31 −1.33 −0.99 −1.70 −1.45 −1.61 −1.45 −1.63 −1.22 −1.92 −1.94 −0.88 −1.65 −1.74 0.68 1.27 0.98 0.58 −1.94

0.23 −0.84 −0.96 −0.86 −0.79 −1.08 −0.48 −0.94 −0.27 −0.18 −1.09 −0.82 −1.58 −1.29 −1.37 −1.18 −1.53 −0.99 −1.68 −1.51 −0.82 −1.39 −1.49 0.61 1.08 0.84 0.52 −1.68

0.15 −0.45 −0.60 −0.56 −0.59 −0.66 −0.21 −0.64 0.31 0.01 −0.72 −0.59 −0.92 −0.63 −1.06 −0.87 −0.73 −0.67 −1.37 −1.15 −0.49 −1.10 −1.22 0.37 0.73 0.55 0.41 −1.37

0.18 −0.29 −0.43 −0.39 −0.42 −0.47 −0.03 −0.45 0.47 0.10 −0.51 −0.40 −0.77 −0.41 −0.88 −0.67 −0.55 −0.48 −1.19 −0.94 −0.42 −0.89 −1.05 0.33 0.55 0.44 0.37 −1.19

0.22 −0.28 −0.37 −0.37 −0.35 −0.42 −0.01 −0.40 0.56 0.12 −0.46 −0.38 −0.71 −0.32 −0.82 −0.61 −0.45 −0.42 −1.13 −0.88 −0.41 −0.83 −0.98 0.29 0.51 0.40 0.37 −1.13

0.21 −0.30 −0.40 −0.38 −0.37 −0.44 −0.03 −0.42 0.55 0.12 −0.48 −0.39 −0.73 −0.34 −0.81 −0.63 −0.49 −0.47 −1.12 −0.88 −0.40 −0.87 −0.97 0.3 0.52 0.41 0.37 −1.12

0.27 0.18 0.05 0.08 0.03 −0.09 0.37 −0.10 1.02 0.38 −0.14 0.08 −0.26 0.31 −0.58 −0.33 0.04 −0.15 −0.89 −0.65 −0.16 −0.57 −0.71 0.25 0.29 0.27 0.36 1.02

0.17 −0.12 −0.28 −0.37 −0.35 −0.28 −0.02 −0.35 0.89 0.03 −0.33 −0.38 −0.39 0.10 −0.72 −0.59 −0.11 −0.33 −1.03 −0.65 −0.44 −0.86 −0.94 0.22 0.43 0.33 0.35 −1.03

0.13 0.02 −0.19 −0.26 −0.36 −0.15 0.08 −0.30 0.86 0.16 −0.25 −0.33 −0.38 0.09 −0.63 −0.48 −0.10 −0.26 −0.94 −0.59 −0.36 −0.73 −0.88 0.19 0.36 0.28 0.33 −0.94

0.18 0.12 −0.06 −0.09 −0.20 −0.08 0.18 −0.20 0.92 0.22 −0.19 −0.18 −0.31 0.27 −0.56 −0.45 −0.02 −0.28 −0.87 −0.59 −0.33 −0.71 −0.83 0.21 0.32 0.27 0.35 0.92

h Omitted from the statistical analysis. aref 54. bref 55. cref 56. dref 57. eref 58. fref 59. gref 60. iref 61. jref 41. kref 39. lref 40. mref 62. nref 63. (The reference value for the H2S(21B1) transition in Table 1 is the average for the range in the observed value, which is from 8.18 to 8.32 eV.40)

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This work assesses, mainly through comparison to experiment, the accuracy of TD-DFT in regard to the electronic transition energies, equilibrium geometries, and harmonic vibrational frequencies of a diverse collection of small phosphorus and sulfur molecules, using a selection of exchange correlation functionals and basis sets. In addition, OVGF and P3 are tested against the experimental ionization energies of such molecules with a selection of basis sets. From these comparisons, the best models for the study of phosphorus and sulfur molecule lowenergy excitation and photoionization spectroscopy are ascertained.

COMPUTATIONAL METHODS

Several prominent exchange-correlation functionals were used to compute TD-DFT transition energies: the pure DFT generalized gradient approximation (GGA) functionals HCTC,16 BLYP,17 and PBE;18 the hybrid GGA functionals B3LYP,19,20 B98,21 PBE0,22 and MPW1PW91,23 which have 20%, 22%, 25%, and 25% Hartree−Fock exchange correlation (HFX), respectively; the long-range corrected (LC) functional CAM-B3LYP,24 which has 19% HFX at short-range, a standard error function with parameter 0.33 Bohr−1 at intermediate range, and 65% HFX at 3515

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long-range; the LC functional ωB97X-D,25 which has 22% HFX at short-range, a standard error function with parameter 0.20 Bohr−1 at intermediate range, and 100% HFX at long-range; and the meta-hybrid GGA functionals BMK26 and M06-2X,27 which have 42% and 54% HFX, respectively. Vertical transition energies were computed at the MP3/MG3S optimized ground state geometry of each compound. For these calculations, each functional was paired with a 6-31+G(2df,p),28−31 MG3S,32 or aug-pc333−36 atomic basis set. The 631+G(2df,p) basis is a [5s4p2df] contracted set on third period elements that is of double-ζ quality in the valence space and triple-ζ quality in the polarization functions. MG3S is [7s6p3d2f] and aug-pc3 is [7s6p5d3f2g] on third period elements, and both of these basis sets are of triple-ζ valence quality. (aug-pc3 is herein denoted APC3.) Adiabatic transition energies were obtained from the total energy of the DFT/MG3S and TD-DFT/MG3S optimized structure of each molecule. For most species, the optimized structure could be found via an analytical gradient or eigenvaluefollowing procedure. For SCH3(12A′), however, neither of these procedures are successful at the TD-BMK/MG3S and TDCAM-B3LYP/MG3S levels of theory, so it was necessary to resort to single point energy calculations in order to determine the optimized structure. (This was accomplished by printing the geometries that are generated during a TD-DFT numerical frequency calculation, identifying the 11A′ lowest energy structure, and repeating until a converged geometry was reached.) The zero-point correction to the total energy was determined from a normal coordinate analysis of the optimized structure, using energy second derivatives computed analytically at DFT/MG3S and numerically at TD-DFT/MG3S. No scale factor was applied to the DFT or TD-DFT zero-point correction. The normal coordinate analyses also served to verify that each equilibrium structure corresponds to a minimum on the ground or excited state potential energy surface. As part of this study, an examination of phosphorus and sulfur molecule electronic transition energies and excited state properties was also conducted with the Configuration Interaction-Singles method.37 This method was found to be consistently less accurate than TD-DFT, but the results of this examination are not given further consideration here. P3 and OVGF ionization energies were computed within the frozen core orbital approximation at the MP3/MG3S optimized ground state geometry of each neutral compound, using the 631G(d), 6-31+G(2df,p), MG3S, and APC3 basis sets. All quantum chemical calculations were performed with Gaussian 0938 using the default settings in two-electron integral grid fineness, SCF convergence criterion, and structure optimization convergence criteria.

A statistical analysis of the residuals obtained for the valence and Rydberg transitions, and for the entire test set, is presented at the bottom of the table. The residuals pertaining to the 11A″ transition of dimethyl sulfoxide have been omitted from the statistical analysis because of ambiguity in the TD-DFT manifold of this molecule when meta-hybrid GGA or LC functionals are usedthe orbital transition identifying this state has nearly equal weight in two states of 1A″ symmetry at the MP3/MG3S equilibrium geometry. When hybrid functionals are used, TDDFT is within 0.3 eV of the observed SO(CH3)2(11A″) transition energy, as Table 1 shows. The assignments in Table 1 are taken from literature sources, except for those of trimethyl phosphine and methyl phosphine. The absorption spectrum of trimethyl phosphine features a very strong and broad band with a maximum at 6.15 eV,39 and in Table 1 this value is compared to the computed energy of the (10a′)11A′ → (11a′)21A′ transition. This transition appears third or fourth in the TD-DFT manifold, depending on functional choice, and is predicted to carry an oscillator strength of f ≈ 0.20, regardless of functional choice, which is at least two times larger than that of any other excitation. The computed energy of this transition is shifted over 1.0 eV by the addition of diffuse functions to the basis set, so it is considered a Rydberg transition in the statistical analysis. Table 1 shows that the deviation from experiment by the computed energy of the P(CH3)3(21A′) transition is steadily reduced by an increasing admixture of HFX, ranging from −0.98 eV for BLYP to 0.18 eV for BMK. The first two absorption bands in the electronic spectrum of methyl phosphine have a sharp maximum centered at 6.15 and 6.33 eV,39 and in Table 1 these values are compared to the first two transitions in the TD-DFT manifold, which are excitations from the (10a′)11A′ state to (11a′)21A′ and (12a′)31A′. The third absorption band, centered at 6.63 eV, is compared to the excitation energy of the (10a′)11A′ → (5a″)11A″ transition, which appears anywhere from third to fifth in the TD-DFT manifold, depending on functional choice. The 11A″ state has the greatest computed oscillator strength, just as the third absorption band has the greatest absorption band area. The computed excitation energy of the 11A″ transition is shifted less than 0.1 eV by inclusion of diffuse functions in the basis set, so this is considered a valence transition in Table 1, while the 21A′ and 31A′ transitions are shifted more than 1.0 eV and are considered Rydberg transitions. Thus, the error in the TD-DFT transition energy of 11A″ is reduced to a smaller extent than that of 21A′ and 31A′ by increasing the admixture of HFX, as Table 1 shows. Only the LC functionals yield transition energies within 0.3 eV of experiment for each of these transitions. Absorption spectra for hydrogen sulfide (H 2 S) and thioformaldehyde (CH2S) contain several assigned bands and therefore provide a test of the energetic ordering of the states in the computed manifold. The first H2S absorption band has a maximum at 6.33 eV attributed to the 11B1(n,4s) state.40 TDBMK overpredicts this value by only 0.03 eV, but the other functionals underpredict it by at least 0.20 eV. The dipole forbidden 11A2(n,Vp+Rd) state is 0.26 eV below 11B1 at the vertical geometry, according to TD-BMK. TD-M062X and the LC functionals also place 11A2 below 11B1, by 0.04−0.20 eV, while the hybrid and GGA functionals place 11A2 above 11B1 by 0.02−0.40 eV. The next three experimentally observed states, in increasing energy, are 21A2(n,4p), 21A1(n,4p), and 21B1(n,4p). TD-DFT/APC3 places 21A2 in correct order, but inverts 21B1 with 21A1, regardless of functional choice. Higher energy states in

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RESULTS Vertical Transition Energies. Table 1 shows the deviation from experiment by TD-DFT/APC3 vertical excitation energies for a set of 20 valence transitions and 18 Ry transitions of phosphorus and sulfur molecules. All experimental values are from the chemical literature and were obtained from absorption spectroscopy measurements taken in the gas-phase. Also shown is the symmetry species and orbital character of each excited state. The global hybrid functionals are listed from left to right in order of increasing percentage of HFX, and the smallest and largest absolute deviation obtained for each transition is listed in bold and italics, respectively. 3516

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Table 2. Mean Absolute Deviations from Experiment for TD-DFT Vertical Transition Energies of Phosphorus and Sulfur Molecules HCTH

BLYP

PBE

B3-LYP

Basis Set 6-31+G(2df,p) MG3S APC3

0.56 0.57 0.58

0.56 0.63 0.68

0.56 0.59 0.61

0.39 0.39 0.34

6-31+G(2df,p) MG3S APC3

0.54 0.52 0.92

0.67 0.65 1.27

0.54 0.50 1.06

0.32 0.24 0.73

6-31+G(2df,p) MG3S APC3

0.55 0.55 0.75

0.64 0.64 0.98

0.55 0.54 0.84

0.35 0.32 0.55

B98

PBE0

MPW1-PW91

BMK

Mean Absolute Deviation (Valence) 0.36 0.34 0.33 0.44 0.36 0.32 0.33 0.4 0.33 0.29 0.3 0.25 Mean Absolute Deviation (Rydberg) 0.25 0.25 0.25 0.54 0.18 0.18 0.18 0.47 0.55 0.51 0.52 0.29 Mean Absolute Deviation 0.31 0.29 0.29 0.49 0.27 0.25 0.25 0.44 0.44 0.40 0.41 0.27

M06-2X

CAM-B3LYP

ωB97X-D

0.3 0.26 0.22

0.27 0.23 0.19

0.37 0.29 0.21

0.29 0.21 0.43

0.28 0.21 0.36

0.33 0.26 0.32

0.3 0.24 0.33

0.27 0.22 0.28

0.35 0.27 0.27

Table 3. Adiabatic Transition Energies of Phosphorus and Sulfur Molecules TD−DFT/MG3S − EXP species SPF2(22A′) π → π* HSO(12A′) σ → π* SCH2(11A2) n → π* SCHCH3(21A) n → π* PH2(12A1) n→σ SC(CH3)2(11A″) n → π* HPO(11A″) n → π* HPCl(12A′) σ→π SCH2CH3(32A″) σ → π SCH3(12A′) σ→π PH(13Π+) σ→π PO2(12B1) σ→π PO(12Σ+) π→σ SO2(11B1) σ → π* P2(11Πg) σ → π* PCH(11A″) π → σ* SO(23Σ−) π → π* S(CH3)2(11B1) n → σ* SCH2(21A′) π → π* Mean Absolute Error Standard Deviation Maximum Error a

EXP (eV) 1.63a 1.78b 2.03c 2.22c 2.27d 2.33c 2.42e 2.71f 2.82g 3.27h 3.63i 3.74j 3.81k 3.87l 4.28m 4.31n 5.16o 5.44p 5.54c

HCTH

BLYP

PBE

B3-LYP

B98

PBE0

MPW1PW91

BMK

M062X

CAM-B3LYP

0.02 0.14 0.01 0.00 −0.02 −0.08 −0.15 0.01 −0.03 −0.06 −0.11 −0.24 −0.01 −0.22 −0.09 −0.28 0.06 −0.83 −0.05 0.13 0.20 −0.83

−0.17 0.06 −0.09 −0.10 0.10 −0.17 −0.22 −0.02 −0.29 −0.05 0.00 −0.19 −0.16 −0.82 −0.21 −0.38 −0.07 −0.87 −0.32 0.23 0.25 −0.87

−0.06 0.12 −0.07 −0.07 0.13 −0.14 −0.24 0.01 −0.11 −0.01 0.08 −0.24 −0.10 −0.74 −0.16 −0.32 0.11 −0.76 −0.14 0.19 0.24 −0.76

−0.06 0.07 0.04 0.05 0.09 0.03 −0.07 0.07 −0.03 0.10 0.06 −0.05 −0.05 −0.14 −0.12 −0.46 0.01 −0.34 −0.20 0.11 0.14 −0.46

−0.07 0.05 0.06 0.07 0.00 0.05 −0.05 0.03 −0.07 0.12 0.00 −0.06 −0.12 −0.06 −0.08 −0.45 0.03 −0.20 −0.21 0.09 0.13 −0.45

0.04 0.13 0.08 0.11 0.15 0.10 −0.06 0.13 0.21 0.32 0.16 −0.09 −0.16 −0.01 −0.04 −0.39 0.12 −0.13 −0.05 0.12 0.16 −0.39

0.04 0.13 0.09 0.11 0.17 0.10 −0.04 0.14 0.22 0.39 0.16 −0.08 −0.17 −0.02 −0.03 −0.40 0.10 −0.14 −0.07 0.13 0.17 −0.40

−0.19 −0.17 0.03 0.07 −0.04 0.09 −0.05 0.02 0.00 0.44 0.05 −0.04 −0.26 0.02 −0.33 −0.53 0.17 0.54 −0.33 0.17 0.25 0.54

−0.24 −0.32 −0.11 −0.06 −0.23 −0.03 −0.22 −0.14 −0.21 −0.13 −0.27 −0.16 0.08 −0.14 −0.32 −0.94 0.05 0.10 −0.42 0.22 0.22 −0.94

−0.06 0.09 0.06 0.10 0.02 0.12 −0.01 0.06 0.13 0.04 0.03 0.03 −0.11 0.07 −0.06 −0.53 0.10

ref 64. bref 65. cref 41. dref 66. eref 67. fref 68. gref 69. href 70. iref 71. jref 72. kref 73. lref 74.

the TD-DFT/APC3 H2S manifold have ambiguous orbital character, so a comparison to experiment is not possible. For CH2S, the first five observed excited states are 11A2(n,π*), 1 1 B2(n,4s), 21A1(π,π*), 21B2(n,4p), and 31A1(n,4p),41 and TDDFT/APC3 correctly orders the first two of these states regardless of functional choice. However, only TD-BMK/ APC3 and TD-LC/APC3 correctly order the first four excited states, and only TD-LC/APC3 correctly orders all five of these states. The sixth excited state is 21A2(n,4p), according to multireference configuration interaction calculations,42 and according to TD-LC/APC3, but this state has not been observed in CH2S spectra. Higher energy CH2S states could not be assigned to experiment due to ambiguous orbital character. The statistical analysis of the residuals in Table 1 shows a general improvement in overall accuracy as the percentage of HFX incorporated in the functional is increased. The pure DFT GGA functionals yield a mean absolute error (MAE) of more

m

−0.15 0.11 0.16 −0.53

ωB97X-D −0.03 0.05 0.08 0.11 −0.04 0.14 −0.07 0.01 0.09 0.27 −0.03 −0.02 0.20 0.05 −0.05 −0.43 0.12 0.33 −0.12 0.12 0.16 −0.43

ref 6. nref 47. oref 75. pref 76.

than 0.70 eV over the entire test set, but this is reduced to 0.55 eV for B3LYP, 0.41 eV for MPW1PW91, and 0.27 eV for BMK. A further increase in HFX, as implemented in M06-2X, increases the MAE to 0.33 eV, and no improvement in MAE is obtained on going from BMK to either LC functional. The breakdown of the MAE according to orbital character shows the percentage of HFX having a much greater effect on the computed accuracy of Rydberg transitions; between HCTH and BMK, the MAE improves by 0.33 and 0.63 eV for valence and Rydberg transitions, respectively. This is due of course to the need for the asymptotically correct −1/r form of the potential to describe long-range electronic interactions. CAM-B3LYP is best in overall accuracy for the valence transitions, with MAE = 0.19 eV, while BMK is best for the Rydberg transitions, with MAE = 0.29 eV. Table 2 compares the MAE for calculations performed with three basis sets, 6-31+G(2df,p), MG3S, and APC3, on the excitation energies in Table 1. Over the valence transitions, going 3517

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Table 4. TD-B3LYP/MG3S Excited State Equilibrium Properties species 2

SPF2(2 A′) HSO(12A′) SCH2(11A2) SCH(CH3)(21A) PH2(12A2) SC(CH3)2(11A″) HPO(11A″) HPCl(12A′) SCH2CH3(32A″) SCH3(12A′) PH(13Π+) PO2(12B1) PO(12Σ+) SO2(11B1) P2(11Π g) PCH(11A″) SO(23Σ−) S(CH3)2(11B1) SCH2(21A′)

excited state

f

Δ⟨S2⟩

coefficients

geometry

2 1 1 1 1 1 1 1 2 2 1 1 2 1 4 1 4 1 2

0.0095 0.0001 0.0000 0.0000 0.0020 0.0000 0.0019 0.0029 0.0006 0.0006 0.0051 0.0103 0.0034 0.0024 0.0000 0.0012 0.0484 0.0012 0.0524

0.011 0.005 0.0 0.0 0.004 0.0 0.0 0.005 0.005 0.005 0.009 0.002 0.005 0.0 0.0 0.0 0.015 0.0 0.0

0.990 H − 1→ L (β) 0.995 H → L 0.708 H → L 0.708 H → L 0.996 H → L (β) 0.707 H → L 0.705 H → L 0.992 H → L 0.996 H − 1 → L 0.996 H − 1 → L 0.714 H → L + 1 (β), 0.691 H → L (β) 0.990 H → L 0.985 H → L +1 (α) 0.706 H → L 0.500 H → L, −0.500 H − 1 → L + 1 0.695 H → L, −0.129 H − 1 → L + 1 0.665 H − 1 → L, 0.665 H → L + 1 0.700 H → L 0.614 H − 1 → L, −0.347 H − 2 → L

RPS = 2.0921, RPF = 1.5798, ∠SPF = 103.0, ∠FPF = 96.6 RSH = 1.3518, RSO = 1.6397, ∠HSO = 94.3 RSH = 1.0811, RSC = 1.6741, ∠SCH = 120.4 RSC = 1.6900, RCC = 1.4918, ∠SCC = 122.8 RPH = 1.3958, ∠HPH = 121.5 RSC = 1.7062, RCC = 1.4925, ∠SCC = 118.8 RPH = 1.4611, RPO = 1.5440, ∠HPO = 97.5 RPH = 1.4151, RPCl = 2.0299, ∠HPCl = 113.7 RSC = 2.1164 RSC = 2.1094, ∠SCH = 98.9, ∠SCH = 99.0 RPH = 1.4848 RPO = 1.5069, ∠OPO = 177.9 RPO = 1.4571 RSO = 1.5271, RSO = 1.5279, ∠OSO = 94.7 RPP = 1.9926 RPC = 1.6945, RCH = 1.0913, ∠PCH = 130.2 RSO = 1.8099 RSC = 1.7789, RSC = 1.7790, ∠CSC = 105.7 RSC = 1.9160, ∠SCH = 111.4

For the 11B1 state of dimethyl sulfide, no minimum energy structure could be located at the CAM-B3LYP/MG3S level of theory. In C2v, C2, or Cs symmetry, the S(CH3)2(11B1) equilibrium geometry is a saddle point on the CAM-B3LYP/ MG3S potential energy surface, and in C1 symmetry, the optimization algorithms converge on dissociation toward SCH3 + CH3. Apparently, 11B1 is unbound at this level of theory due to nonadiabatic interaction with 11A2 at the 11B1 minimum. (This conical intersection is actually located along a C−S bond stretching coordinate near the 11B1 minimum, according to highlevel ab initio calculations.44) Table 3 shows the residuals pertaining to S(CH3)2(11B1) that are obtained with the other functionals, but these are omitted from the statistical analysis since the data in this set are incomplete. These residuals range in size from 0.10 eV at TD-M06-2X/MG3S to −0.87 eV at TDBLYP/MG3S. TD-B3LYP/MG3S equilibrium properties of each excited state are shown in Table 4. Most transitions in this test set are to low energy (ΔE < 4 eV) valence excited states with single excitation character, and therefore fall within the regime of linear response TD-DFT under the adiabatic approximation. Excited state spin contamination is nearly absent from each open shell system, as demonstrated by the difference between excited and ground state expectation values for the total spin-squared operator, Δ⟨S2⟩, so the results for these states are physically meaningful.45 While most transition energies in Table 3 are reasonably wellreproduced at TD-DFT/MG3S, those for PCH(11A″) considerably underestimate experiment, regardless of functional choice. The size of the error increases with the percentage of HFX, ranging from −0.28 eV for TD-HCTH/MG3S to −0.94 eV for TD-M06-2X/MG3S, and it is not decreased when LC functionals are used, indicating that PCH(11A″) has little or no Rydberg or charge transfer character. Thus, the self-interaction error is likely to be small in these calculations. Calculations with the equation-of-motion coupled cluster method yield To = 4.31 eV for PCH(11A″),46 in perfect agreement with the observed value.47 The MAEs in Table 3 range from 0.09 eV for TD-B98 to 0.23 eV for TD-BLYP. Both HCTH and the hybrid functionals

from 6 to 31+G(2df,p) to APC3 slightly worsens the MAE for GGA functionals, slightly improves the MAE for hybrid functionals, and significantly improves the MAE for meta-hybrid and LC functionals. TD-BMK, in particular, undergoes an unusually large improvement in MAE, by 0.15 eV, between MG3S and APC3. Over the Rydberg transitions, going from 6 to 31+G(2df,p) to MG3S improves all MAEs, but going from MG3S to APC3 worsens all MAEs except that of TD-BMK, which improves by 0.18 eV. This result may perhaps stem from the manner in which the BMK functional was parametrized. BMK is a reparameterization of the τ-HCTH functional, and τHCTH is a hybrid version of HCTH that includes the kinetic energy density.26 The parametrization of BMK used the TZ3Pf +diffuse basis set (triple-ζ, triple polarization, plus diffuse and “tight” d functions), whereas the HCTH functional was parametrized using the smaller TZ2P basis set (triple-ζ, double polarization). Consequently, the parameters of the BMK functional are less affected by basis set superposition error than those of HCTH, making BMK more transferable to a “large” basis set such as APC3.43 As Table 2 shows, the TD-CAM-B3LYP/MG3S level of theory is noteworthy in that it predicts transition energies of valence states (MAE = 0.23 eV) and Rydberg states (MAE = 0.21 eV) with roughly equal and relatively high overall accuracy. Among the present models, it provides the most cost-effective means of computing the low-energy vertical electronic manifold of phosphorus and sulfur compounds. Adiabatic Transition Energies. Adiabatic transition energies are given sole consideration in Table 3, since these have been computed with only the MG3S basis set in the present study, and since their experimental values are more accurate than those of vertical transitions. All experimental values in Table 3 were obtained from gas-phase absorption or emission spectroscopy. Experimental and theoretical Te values are compared for P2(11Πg), SO(23Σ−), PO(12Σ−), and HPO(11A″), while the remaining transition energies are To values, which include the zero-point correction. The smallest and largest residual pertaining to each test subject is listed in bold and italics, respectively, and a statistical analysis of the residuals is presented at the bottom the table. 3518

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Table 5. Excited State Equilibrium Structural Parameters of Phosphorus and Sulfur Molecules TD−DFT/MG3S − REF species PO (1 Σ ) P2 (11Πg) HPO (11A″) HPO (11A″) HPCl (12A′) HPCl (12A′) PH2 (12A1) SO (23Σ−) HSO (12A′) HSO (12A′) F2PS (22A′) F2PS (22A′) PCH(11A″) SCH2 (11A2) SCH3 (12A′) PH2 (12A1) HPO (11A″) HPCl (12A′) F2PS (22A′) 2 +

param.

REF (pm & °) a

re 146.3 re 198.87b re(P−H) 146.71c re(P−O) 155.79c re(P−H) 140.63d re(P−Cl) 200.35d re 139.39e re 177.5f re(S−H) 134.2g re(S−O) 166.1g re(P−F) 155.6h re(P−S) 204.2h ro(P−C) 169.0i ro(S−C) 168.0i ro(S−C) 205.7j θe 121.68e θe 97.4c θe 116.08d θe(F−P− 96.8h F) F2PS (22A′) θe(F−P− 105.1h S) HSO (12A′) θe 95.7g Mean Absolute Deviation (Bonds) Mean Absolute Deviation (Angles) a

HCTH

B-LYP

PBE

B3-LYP

B98

PBE0

MPW1PW91

BMK

M062X

CAM-B3LYP

ωB97X-D

0.53 −1.30 0.17 −0.42 1.16 1.21 0.28 2.19 1.11 −3.84 3.80 0.96 0.93 −0.31 4.85 0.48 −1.91 −2.22 −0.25

1.58 2.53 0.90 1.75 2.55 7.71 1.16 2.26 4.99 2.25 −0.31 5.78 9.07 2.17 11.29 0.33 −2.97 −8.18 −0.57

1.31 0.71 1.11 0.84 2.40 3.11 1.30 1.62 2.19 2.28 −2.12 4.95 4.74 0.92 4.01 0.01 −3.05 −5.01 −0.70

−0.59 0.39 −0.60 −1.39 0.88 2.64 0.19 3.49 0.98 −2.13 2.38 5.01 0.45 −0.59 5.24 −0.17 0.09 −2.34 −0.18

−0.73 1.12 −0.67 −1.92 0.63 2.11 0.08 2.47 1.08 −2.88 1.71 4.50 0.43 −0.48 3.31 −0.07 0.71 −2.11 −0.02

−1.11 −1.17 −0.49 −2.46 0.70 −0.13 0.29 0.47 0.87 −3.41 1.27 1.28 −0.14 −1.82 −1.04 0.17 1.12 −1.37 −0.13

−1.26 −1.14 −0.77 −2.51 0.45 −0.03 0.09 0.78 0.72 −3.30 1.27 1.46 −0.17 −1.92 −0.46 0.09 1.14 −1.38 −0.10

−1.05 3.94 −2.01 −2.54 −0.04 0.09 −0.61 1.66 1.16 −2.24 0.52 2.53 −0.63 −0.88 −1.76 −0.09 1.02 −1.72 −0.30

−1.71 −0.60 −1.61 −3.14 0.06 0.53 −0.61 0.74 −0.27 10.34 0.55 −1.34 −0.86 −2.95 −0.81 −0.67 2.68 0.16 −0.63

−1.60 −1.32 −1.15 −3.47 0.30 0.73 −0.09 1.43 0.56 −3.31 0.94 1.62 −0.71 −2.70 −0.13 −0.21 3.06 −1.00 0.06

−1.29 1.71 −0.98 −3.11 0.52 0.57 0.03 −0.73 0.62 −3.87 1.12 0.34 −0.47 −2.37 −1.11 0.37 2.47 −0.95 0.11

−2.12

−4.95

−4.22

−2.06

−1.55

−1.10

−1.05

1.29

1.96

−0.46

0.25

−1.13 1.49 1.35

−1.72 3.86 3.12

−1.65 2.29 2.44

−1.40 1.68 1.04

−1.28 1.55 0.96

−1.41 1.16 0.88

−1.47 1.11 0.87

−2.40 1.43 1.14

−6.91 1.81 2.17

−1.27 1.33 1.01

−1.11 1.29 0.88

ref 77. bref 6. cref 67. dref 78. eref 79. fref 75. gref 80. href 64. iref 8. jref 81.

length for HSO(12A′) that overestimates experiment by only 0.9 and 1.4 pm, respectively. The computed value of ⟨S2⟩ in both of these unrestricted calculations is very close to the exact value of 0.75. The TD-DFT bond length for the P−O bond in HPO(11A″) is also sensitive to functional choice. Its deviation from experiment generally increases with the percentage of HFX, ranging from −0.4 pm for HCTH to −3.1 pm for M06-2X, and both LC functionals also deviate by over −3 pm. In comparison, an ab initio calculation on HPO(12A″) at a high theoretical level, QCISD/aug-cc-pVTZ, overestimates Re(P−O) by 1.9 pm.49 Contributions from core correlation, and extrapolation of the QCISD result to the complete basis set limit, are needed in order to get within 1 pm of experiment, so the accuracy of TD-HCTH in this instance obviously is fortuitous. The relative accuracy of the 11 functionals considered in this study over the set of geometric parameters in Table 5 is similar to what is obtained for the adiabatic transition energies in Table 3, despite the difference in size between these two test sets. HCTH and the hybrid functionals improve on BLYP and PBE in terms of the MAE in both bond lengths and bond angles, but no further improvement is obtained by using meta-hybrid or LC functionals. TD-MPW1PW91 has the smallest MAE for bond lengths (1.1 pm) and bond angles (0.9°). Similar accuracy is found in the TD-B3LYP/aug-TZVPP geometric parameters of first row molecules.8 Ground state B3LYP/cc-pVTZ+d bond lengths between phosphorus or sulfur and elements O, C, and H consistently fall within 1.0 pm of experiment,50 while the present TD-B3LYP results obtained with the larger MG3S basis set meet this standard in 7 of 11 cases in Table 5. Both TD-BLYP and TD-M06-2X geometries are prone to very large errors. In addition to the above-mentioned failure with

significantly improve on BLYP and PBE in terms of the MAE, but the LC functionals provide no further improvement, and use of meta-hybrids increases the MAE in relation to the hybrids. These results suggest that by incorporating approximately 22% HFX into HCTH, one would create a functional having exceptional accuracy within this test set, at least for calculations using the MG3S basis set. Excited State Equilibrium Geometries. Table 5 shows the percentage deviation from reference bond lengths and bond angles by TD-DFT/MG3S equilibrium structural parameters. The TD-DFT geometries of PO(12Σ+), SPF2(22A′), and PH2(12A1) are compared to results from high-level ab initio calculations, while the remaining TD-DFT geometries are compared to results from high resolution gas phase spectra. All reference values are equilibrium values except for those of PCH(11A″), SCH2(11A2), and SCH3(21A′), which contain the effect of vibrational averaging. This effect is assumed to be small and the residuals obtained for these test subjects are included in the statistical analysis. Most TD-DFT parameters in Table 5 are within 1.5 pm of the reference value, but the deviation in the computed S−O bond length of HSO(12A′) ranges from −2.1 pm at TD-B3LYP/ MG3S to 10.3 pm at TD-M06-2X/MG3S. When the TD-B3LYP optimization is repeated with a larger basis set, APC3, a deviation of −2.4 pm is obtained. The inclusion of multiple-electron excitations in the response formalism may be needed to accurately compute this bond length, as these excitations are neglected in linear response TD-DFT.48 Alternatively, one may repeat the optimization with unrestricted DFT by promoting a β electron from an occupied a′ molecular orbital of the ground state to a virtual a″ orbital. Such calculations at the UB3LYP/ MG3S and UM06-2X/MG3S level of theory yield a S−O bond 3519

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Table 6. Excited State Harmonic Vibrational Frequencies of Phosphorus and Sulfur Molecules (TD−DFT/MG3S − EXP)/EXP × 100% species

mode

PO (1 Σ ) ωe(P−O) P2 (11Πg) ωe(P−P) HPO (11A″) ωe(P−H) HPO (11A″) ωe(P−O) HPCl (12A′) ωe(P−Cl) SO (23Σ−) ωe(S−O) HPO (12A′) ωe(HPO) HPCl (12A′) ωe(HPCl) PCH (12A″) ν(P−C) SCH2 (11A2) ν(S−C) SCH3 (12A′) ν(S−C) HSO (12A′) ν(S−H) HSO (12A′) ν(S−O) HSO (12A′) ν(HSO) Mean Absolute % Deviation 2 +

a

EXP (cm−1) a

1164 619b 1848c 871c 529d 630e 564c 622d 951f 820f 410g 2769h 702h 828h

HCTH

BLYP

PBE

B3-LYP

B98

PBE0

MPW1-PW91

BMK

M06-2X

CAM-B3LYP

ωB97X-D

0 3 7 7 −2 −13 0 −5 0 9 −1 −6 6 −1 4

−2 −6 4 2 −32 −16 7 −19 −3 3 −9 −10 2 −3 8

−1 −2 5 6 −10 −10 1 −12 −1 6 2 −9 4 −3 5

5 0 −2 15 −1 −9 1 −2 2 11 5 −6 7 2 5

6 −2 −3 18 1 −7 −1 −1 3 10 13 −7 8 2 6

8 5 −1 19 7 −3 −5 3 4 16 16 −5 9 2 8

8 4 −1 19 7 −4 −4 4 4 16 15 −5 9 3 8

11 0 2 25 8 5 −6 10 8 12 29 −7 5 6 10

12 −1 2 24 7 −4 −14 17 1 18 16 0 18 73 15

10 4 0 23 5 −5 −5 6 5 18 18 −4 8 4 9

8 −2 −1 22 7 −5 −4 10 5 18 24 −3 11 4 9

ref 73. bref 6. cref 67. dref 82. eref 75. fref 8. gref 81. href 65.

HSO(12A′), TD-M06-2X predicts a linear geometry for PCH(11A″), as opposed to the ∼130° bent configuration obtained with CASSCF,46 or with TD-DFT when any of the other functionals in Table 5 are used. These poor geometries are reflected in large errors in TD-M06-2X adiabatic transition energies (−0.32 eV and −0.94 eV errors, respectively, for HSO(12A′) and PCH(11A″)) in Table 3. Excited State Vibrational Frequencies. Percentage deviations from experiment in unscaled TD-DFT harmonic vibrational frequencies are shown in Table 6. All reference values come from gas-phase laser-induced fluorescence spectroscopy, and all reference values except for those of PCH(11A″), SCH 2 (1 1 A 2 ), and HSO(1 2 A′) have been corrected for anharmonicity. Note that Table 6 is consistent with Table 5 in that errors in stretch mode frequencies correlate with errors in equilibrium bond lengths. The frequency of the P−O stretch mode of HPO(11A″) is overestimated by up to 23%, for example, reflecting a 3% underestimation of the P−O bond length. The mean absolute percentage error (MA%E) shown in Table 6 tends to increase with the percentage of HFX and ranges from 4% for HCTH to 15% for M06-2X. Better accuracy is obtained from TD-DFT/aug-TZVPP vibrational frequencies of first row molecules, which typically fall within 4% of the (mostly harmonic) experimental values.8 Vertical Ionization Energies. Errors in computed vertical ionization energies of phosphorus- and sulfur-containing species are shown in Table 7. The test set consists of 31 HOMO, 4 HOMO−1, and 2 HOMO−2 first ionization energies, and almost all reference values come directly from gas-phase photoelectron spectroscopy. In the case of PH2 and PO, the reference value was estimated by adding the experimental gasphase adiabatic ionization energy to the difference between calculated vertical and adiabatic ionization energies. For PH2 this difference is 0.0 eV at CISD/TZ3P(2f,2d)+2diff,51 and for PO it is 0.15 eV at ROHF/6-31G(d).52 The residuals pertaining to the ionization energy of the phenyl sulfide radical, S(C6H5), are omitted from the statistical analysis shown at the bottom of Table 7, as the SCF calculation on this molecule would not converge when the APC3 basis set was used, so the data in this set are incomplete. The pole strength of the final state of each species in Table 7 exceeded 0.90 in most cases, and 0.85 in all cases. The molecular

orbital approximation upon which P3 and OVGF are based is valid for pole strengths exceeding 0.80.53 The computed values of each ionization energy (IE) in Table 7 vary significantly within the present range of basis sets, and for nearly all species the IE increases with each augmentation of the basis set. Comparison of IEs computed using the MG3S and APC3 basis sets suggests convergence to the basis set limit of OVGF for a few species, i.e., POF3, PF3, P4, and P, while a basis set effect as large as ∼0.1 eV is apparent for S, SH, SCH2, and SCH3. P3 appears to converge more slowly than OVGFon going from P3/MG3S to P3/APC3, most IEs in Table 7 increase by approximately 0.1 eV and only that of SC2H5 remains unchanged. The statistical analysis of the residuals in Table 7 shows that, for OVGF, use of the 6-31G(d) basis set is sufficient to obtain an MAE of 0.44 eV, and expanding to 6-31+G(2df,p) improves all IEs and decreases the MAE to 0.11 eV. Further expansion to MG3S improves almost all OVGF IEs and decreases the MAE to 0.08 eV, or 1.8 kcal/mol. The OVGF/MG3S IEs overshoot experiment in some cases, so going from MG3S and APC3 has no effect on the MAE. P3 is similar in overall accuracy to OVGF, within the present test set, when the 6-31G(d) basis set is used, yielding a 0.41 eV MAE. However, expanding the basis to 631+G(2df,p) results in a smaller improvement in MAE, to 0.21 eV, and further expansion to MG3S and APC3 has only a small effect on the MAE. Thus, OVGF is about 0.1 eV better than P3 in MAE within the present test set when the three larger basis sets are considered. However, some of this difference in performance can be traced to a few high energy ionization processes, specifically those of PF2NCO, SO2, PCH, and PF2CN. For each of these cases OVGF/6-31+G(2df,p) is within 0.10 eV of experiment while P3/APC3 deviates from experiment by at least 0.20 eV. In general, P3 tends to overestimate the larger IEs in Table 7, and underestimate the smaller IEs, when the larger basis sets are used. The higher and lower energy ionization processes in Table 7 therefore warrant separate consideration. Over the 24 IEs in Table 7 above 9.8 eV, OVGF/6-31+G(2df,p) and P3/APC3 have a MAE of 0.09 and 0.21 eV, respectively. Over the 13 IEs below 9.8 eV, OVGF/6-31+G(2df,p) and P3/APC3 have a MAE of 0.07 and 0.13 eV, respectively. Among the present models, OVGF/6-31+G(2df,p) appears to provide the best compromise 3520

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Table 7. Vertical Ionization Potentials of Phosphorus and Sulfur Species theory − experiment OVGF

P3

species

EXP (eV)

6-31G*

6-31+G (2df,p)

MG3S

APC3

6-31G*

6-31+G (2df,p)

MG3S

APC3

PO P(CH3)3 SC(CH3)2 S−Ph S(CH3)2 SCHCH3 C2H5SCl SCH2CH3 SCH3 SCH2 P4 PH2CH3 SCl2 PH2 PCCH3 HSO PHCH2 S SH2 P SH PH3 P2 PCH PHCH2 PHF2 SCH2 PF2CN PF2NCS PCCH3 PF2NCO PF3 SO2 PCH PHCH2 POF3 SCH2 MAE SD

8.54a,b 8.60c 8.60d 8.63e,f 8.67g 8.98d 8.99h 9.08i 9.26j 9.38k 9.60l 9.62m 9.70n 9.82o,a 9.89p 9.92q 10.30p 10.36r 10.43g 10.49r 10.50e 10.59s 10.60t 10.64u 10.70p 11.00v 11.76k 11.90v 11.90v 12.19m 12.20v 12.23v 12.50k 12.86u 13.20p 13.52w 13.85k

−0.67 −0.46 −0.43 −0.89 −0.39 −0.5 −0.28 −0.51 −0.55 −0.53 −0.23 −0.37 −0.28 −0.29 −0.28 −0.31 −0.47 −0.86 −0.56 −0.6 −0.82 −0.55 −0.38 −0.26 −0.61 −0.54 −0.35 −0.44 −0.29 −0.36 −0.5 −0.57 −0.48 −0.45 −0.13 −0.17 −0.22 0.44 0.17

−0.14 −0.12 −0.05 −0.61 −0.04 −0.11 0.04 −0.07 −0.1 −0.14 −0.02 −0.07 −0.13 −0.11 −0.01 −0.14 −0.18 −0.3 −0.17 −0.03 −0.34 −0.21 −0.18 0.01 −0.25 −0.12 −0.07 −0.07 0.04 −0.03 −0.05 −0.14 −0.01 −0.09 0.13 0.23 0.08 0.11 0.12

−0.11 −0.05 0.01 −0.39 0.02 −0.05 0.09 −0.01 −0.06 −0.08 0.04 0.02 −0.09 −0.04 0.05 0.19 −0.12 −0.24 −0.09 0 −0.26 −0.11 −0.14 0.07 −0.17 −0.05 −0.03 −0.02 0.07 0.04 −0.03 −0.09 0.03 −0.03 0.2 0.25 0.14 0.08 0.11

−0.06 0.00 0.09

−0.57 −0.5 −0.61 −0.77 −0.51 −0.67 −0.35 −0.63 −0.68 −0.7 −0.03 −0.47 −0.31 −0.39 0.01 −0.44 −0.37 −0.9 −0.65 −0.37 −0.91 −0.62 −0.11 0.03 −0.73 −0.61 −0.23 −0.25 −0.2 −0.21 −0.13 −0.57 −0.24 −0.31 −0.03 −0.35 −0.1 0.41 0.25

−0.09 −0.25 −0.31 −0.52 −0.24 −0.36 −0.12 −0.29 −0.35 −0.39 0.1 −0.24 −0.24 −0.19 0.2 −0.01 −0.15 −0.49 −0.33 −0.16 −0.54 −0.37 −0.06 0.22 −0.45 −0.27 −0.02 0.07 0.04 0.03 0.25 −0.21 0.18 −0.05 0.14 0.02 0.08 0.21 0.21

−0.01 −0.18 −0.23 −0.31 −0.17 −0.29 −0.05 −0.22 −0.26 −0.31 0.15 −0.16 −0.16 −0.13 0.27 0.1 −0.08 −0.4 −0.23 −0.12 −0.44 −0.29 −0.01 0.29 −0.37 −0.15 0.04 0.18 0.14 0.1 0.37 −0.07 0.33 0.02 0.2 0.18 0.16 0.19 0.22

0.06 −0.09 −0.12

0.09 0.02 0.16 0.00 0.04 0.01 0.05 0.07 −0.02 −0.01 0.09 0.25 −0.08 −0.14 0.00 0.01 −0.17 −0.06 −0.12 0.1 −0.13 −0.03 0.02 0.00 0.09 0.08 0.01 −0.08 0.09 0.01 0.24 0.25 0.18 0.08 0.10

−0.05 −0.18 0.06 −0.22 −0.15 −0.2 0.23 −0.07 −0.05 −0.06 0.35 0.2 0.01 −0.28 −0.12 −0.08 −0.33 −0.2 0.05 0.36 −0.28 −0.1 0.13 0.23 0.19 0.18 0.48 −0.04 0.44 0.1 0.3 0.26 0.26 0.18 0.21

a t

Estimate. bref 52. cref 83. dref 84. eref 85. gref 86. href 87. iref 88. jref 89. kref 90. lref 91. mref 92. nref 93. oref 51. pref 92. qref 94. rref 95. sref 96. ref 97. uref 98. vref 99. wref 100. fOmitted from the statistical analysis.

■

between accuracy and computational cost for studying the ionization processes of sulfur and phosphorus compounds.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected], Phone: 334-844-4043.

CONCLUSION

Notes

The authors declare no competing financial interest.

■

TD-DFT and EPT may provide useful information regarding the spectroscopy of molecules containing phosphorus or sulfur. Comparison to experiment indicates that CAM-B3LYP/MG3S calculations are suitable for predicting the low-energy vertical electronic transitions of these compounds, while B98/MG3S is excellent for low-energy adiabatic transitions. The OVGF method is more accurate than the P3 method in predicting the ionization energies of these compounds if basis sets of triple-ζ quality in the polarization functions are used.

ACKNOWLEDGMENTS Computational resources were provided by the Alabama Supercomputer Center and the Arctic Region Supercomputing Center. This work was supported by the Defense Threat Reduction Agency.

■

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