The phosphorus clusters Pn (n=1–6) and their anions: Structures and electron affinities

Journal of Molecular Structure: THEOCHEM 759 (2006) 225–238 www.elsevier.com/locate/theochem

The phosphorus clusters Pn (nZ1–6) and their anions: Structures and electron affinities Dan Wang, Cuiling Xiao, Wenguo Xu * Department of Chemistry, School of Science, Beijing Institute of Technology, Beijing 100081, People’s Republic of China Received 20 June 2005; accepted 31 October 2005 Available online 4 January 2006

Abstract The molecular structures and electron affinities of the Pn =PK n (nZ1–6) species have been examined using seven density functional theory (DFT) methods. The basis sets used in this work is of double-z plus polarization quality with additional diffuse s- and p-type functions, denoted DZPCC . The geometries are fully optimized with each DFT method independently. According to the total energies, the most stable isomers have been predicted. Then, the adiabatic electron affinity (EAad), the vertical electron affinity (EAvert), and the vertical detachment energy (VDE) were reported. The most reliable adiabatic electron affinities, obtained at the DZPCC BLYP level of theory, are 0.83 (P), 0.61 (P2), 1.57 (P3), 0.47 (P4), 4.03 eV (P5) and 2.05 (P6), respectively. These EAad values for P, P2 and P3 are in very good agreement with experimental values (average K K K absolute error 0.06 eV). The most reliable VDE, obtained at the DZPCC BLYP level of theory, are 0.73 ðPK 2 Þ, 2.18 ðP3 Þ, 1.70 ðP4 Þ, 4.10 eV ðP5 Þ K Þ, respectively. These values are in good agreement with experimental values (average absolute error 0.12 eV with P excluded). and 2.19 ðPK 3 6 q 2005 Published by Elsevier B.V. Keywords: Phosphorus cluster; Structure; Electron affinities; DFT

1. Introduction The electronic structure of the phosphorus atom is 1s22s22p63s23p3, with three unpaired electrons in 3p orbital that are available for bonding. Phosphorus is known to form a wide range of homoatomic clusters [1]. Experimental studies on phosphorus clusters have been performed for a long time. The clusters of P4 have been observed experimentally in the vapor phase of white phosphorus early in 1935 [2]. Martin [3] observed clusters of PC n (n up to 24) after quenching the vapor of red phosphorus in a helium beam. There have been some previous theoretical studies [4–16] of phosphorus clusters. In 1985, Raghavachari at al. [14] studied the geometries, binding energies and vibrational frequencies of P2, P4 and P8 using ab initio molecular orbital methods. They found that P4 is more stable than either P2 or P8. In 1990, Jones and Hohl theoretically investigated the structures of P2 o P8 using density functional method, combined with molecular dynamics and simulated annealing techniques [4]. They found that the most stable structure P8 with C2v symmetry is more * Corresponding author. Tel./fax: C86 10 68914780. E-mail address: [email protected] (W. Xu).

0166-1280/$ - see front matter q 2005 Published by Elsevier B.V. doi:10.1016/j.theochem.2005.10.038

stable than the 2P4 tetrahedra, but the energy difference is less than 0.5 eV. In 1991, Yilmaz [10] calculated the geometries of the low-lying isomers of phosphorus P2–P10 using the semiempirical modified neglect of diatomic overlap method. In 1992, Ha¨ser [12] investigated clusters of phosphorus with ab initio SCF and MP2 methods and concluded that no P8 cluster is stable with respect to 2P4. In 1993, Benjamin [13] took SCF MO calculations for P8, indicated that at the 4-31G* level the energy of P8 (cuneane) falls slightly below that of 2P4, while at the 6-31G* basis set of P8 (cuneane) lies above 2P4 only a few Kcal/mol. In 2000, Chen [15,16] acquired many isomers of P5, P7, P8, and P9 with the DFT method. There have been the important experimental studies of the electron affinities and other fundamental properties of the P atom and some Pn clusters. Early in 1961, Carette and Kervin C [17] observed the ions PK, PC 2 and P3 formed by dissociative resonance capture of P4 and obtained electron affinities for P, P2, and P3 of 0G0.5, 0.3G0.5, and 1G0.5 eV, respectively. In 1974, Bennett, Margrave, et al. [18] obtained an experimental value of EA(P)Z0.77G0.05 eV from the photodetachment cross section measurements. Then some of electron affinities for phosphorus clusters were reported [18–21]. In 1985, Snodgrass et al. [20] performed photoelectron spectroscopy on PK 2 , and gave the value of the adiabatic electron affinity of P2 of 0.589G0.025 eV and D0 PK 2 Z 4:88G0:06 eV. In 1995, Jones et al. [21] determined the electron affinities of several

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Pn (nZ1–9) clusters using the negative ion photoelectron spectra. They focused on the vertical detachment energies (VDE), and some adiabatic energies had been reported. Density functional theory (DFT)[22] has become a widely applicable computational tool, requiring much less computational effort than convergent quantum mechanical methods, such as coupled cluster theory. The application of gradientcorrected density functional theory has been shown to be effective for many inorganic species [23–30]. The theoretical prediction of electron affinities has historically been generally difficult due to the desired result being a small difference between two large energies; but recent work has shown that some DFT methods with carefully chosen basis sets are dependable for EA predictions with the average error of 0.15 eV. For a general discussion of the reliability of DFT studies of anions, the reader is referred to the 2002 review of Rienstra-Kiracofe, Tschumper, Schaefer, Nandi, and Ellison [31]. The object of the present study is to systematically apply several contemporary forms of density functional theory [22] to determine the electron affinities of the Pn (nZ1–6) series. Of specific interest is (a) the comparison of the theoretical electron affinities with the limited available experimental results; (b) the relationship between the neutral Pn molecules and their anions as reflected by the three types of energy separations, e.g. the adiabatic electron affinity (EAad), the vertical electron affinity (EAvert), and the vertical detachment energy of the anion (VDE). We would like to establish reliable theoretical predictions for these phosphorus clusters in the absence of experimental results and in some cases to challenge existing experiments.

species. The default numerical integration grid (75,302) of Gaussian 98 was initially applied, but we also used the finer grid (99,590) to check suspicious results. The tight convergence of the SCF integrals was also used. The DZP basis set for phosphorus was constructed from the Scha¨fer–Horn–Ahlrichs [39] set of contracted Gaussian functions by adding a set of five pure d-type polarization functions ad(P) Z0.6 on each atom. Since, diffuse functions are important for the anions, the DZP basis was augmented with diffuse functions; each atom received one additional s-type and one set of p-type functions. The diffuse function orbital exponents were determined in an ‘even tempered sense’ as a mathematical extension of the primitive set, according to the prescription of Lee and Schaefer [40]. The diffuse function exponents were thus taken to be as(P)Z0.034480, and ap(P)Z 0.033460. The final basis was thus P(13s9p1d/7s5p1d). This extended basis is denoted as ‘DZPCC’. The total number of DZPCC basis functions ranged from 54 for P2 =PK 2 to 243 for P9 =PK 9. All Pn (nZ2–6) stationary point geometries were interrogated by the evaluation of their harmonic vibrational frequencies at each of the seven different levels of theory. Zero-point vibrational energies (ZPVE) evaluated at the seven levels are presented in Table 1. The ZPVE differences between Pn and PK n (nZ2–6) are quite small, in the range from 0.008 to 0.029 eV. These differences may be used as corrections to the adiabatic electron affinities. The electron affinities are evaluated as the difference of total energies in the following manner: the adiabatic electron affinity is determined as,

2. Theoretical methods

the vertical electron affinity by

EAad Z Eðoptimized neuralÞKEðoptimized anionÞ; EAvert Z Eðoptimized neuralÞ

The seven different density functional or hybrid Hartree– Fock/density functional forms used in our study are (a) Becke’s 1988 exchange functional [32] with Lee, Yang and Parr’s correlation functional [33] (BLYP); (b) The half and half exchange functional [34] with the LYP correlation functional (BHLYP); (c) Becke’s three-parameter hybrid functional [35] with the LYP correlation functional (B3LYP); (d) Becke’s 1988 exchange functional with Perdew’s correlation functional [36] (BP86); (e) Becke’s three-parameter hybrid functional [35] with Perdew’s correlation functional [36] (B3P86); (f) The exchange component of Perdew and Wang’s 1991 functional [37] (BPW91); and (g) This is Becke’s three-parameter functional [35] as above, with the non-local correlation provided by the Perdew and Wang 91 expression (B3PW91). All the electron affinities and molecular structures have been determined using the Gaussian 98 program suite [38]. Restricted methods were used for all closed shell systems, while unrestricted methods were employed for the open-shell

KEðanion at optimized neutral geometryÞ; and the vertical detachment energy of the anion by VDE Z Eðneural at optimized anion geometryÞ KEðoptimized anionÞ: 3. Results and discussion 3.1. P and PK The measured electron affinity of phosphorus was given to be 0.77G0.05 eV by Bennett and Margrave [18] early in 1974. Later, Hotop and Lineberger [20] gave a lower EA value of 0.74651G0.00030 eV for phosphorus in 1985. In 1995, Jones et al. [21] took photoelectron spectra of PK at hnZ2.33 and 3.49 eV photon energy, giving the experimental electron affinity of phosphorus is 0.75G0.05 eV. The electron affinity of the 4S state of the P atom was roughly estimated to be 0.85 eV with an empirical method by Jones et al. [21]. Our theoretical EA for P atom calculated at various methods of DFT with DZPCC basis sets, as well as the experimental data

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Table 1 Zero-point vibrational energies within the harmonic approximation for the most stable structures of Pn =PK n ðnZ 2–6Þ in eV Isomers

BHLYP

BLYP

B3LYP

BP86

B3P86

B3PW91

BPW91

P2 PK 2 DðP2 KPK 2Þ P3 PK 3 DðP3 KPK 3Þ P4 PK 4 DðP4 KPK 4Þ P5 PK 5 DðP5 KPK 5Þ P6 PK 6 DðP6 KPK 6Þ

0.053 0.044 0.009 0.084 0.099 K0.015 0.179 0.150 0.029 0.220 0.229 K0.009 0.260 0.248 0.011

0.046 0.039 0.008 0.074 0.090 K0.016 0.155 0.128 0.028 0.184 0.199 K0.015 0.239 0.226 0.013

0.049 0.041 0.008 0.076 0.095 K0.019 0.167 0.139 0.028 0.201 0.213 K0.012 0.284 0.273 0.011

0.047 0.040 0.008 0.077 0.093 K0.016 0.164 0.136 0.028 0.193 0.206 K0.013 0.249 0.238 0.011

0.050 0.042 0.008 0.080 0.098 K0.018 0.175 0.146 0.029 0.209 0.219 K0.010 0.269 0.259 0.010

0.050 0.042 0.008 0.080 0.097 K0.018 0.175 0.146 0.029 0.209 0.218 K0.010 0.268 0.258 0.010

0.047 0.040 0.008 0.078 0.094 K0.016 0.166 0.137 0.029 0.195 0.207 K0.012 0.252 0.241 0.011

All results obtained with the DZPCC basis set.

are listed in Table 2. The EA values predicted by BLYP (0.83 eV), B3PW91 (0.82 eV), BPW91 (0.83 eV) and BHLYP (0.68 eV) are closer to the experimental value 0.75G0.05 eV given by Jones et al. [21] in 1995. The predictions of the other DFT methods are all higher than the experimental values. 3.2. P2 and PK 2 The optimized geometries of the ground states of P2 and its anion are given in Fig. 1. The P2 Molecule has a 1Sg ground state with DNh symmetry. The general trend for the bond lengths of P 2 is BLYPOBP86OBPW91OB3LYPO ˚ ) and B3PW91OB3P86OBHLYP. Our B3P86 (1.892 A ˚ B3PW91 (1.894 A) method’s values are consistent well with ˚ ) reported by N.J. Brassington the experimental value (1.893 A 2 K et al. [41]. Anionic P2 at Pg ground state had an experimental ˚ reported by Jones et al. [21]. Our P–P bond length of 1.969 A ˚ . The theoretical predictions are in range from 1.966 to 2.016 A ˚ DZPCC BHLYP (1.966 A) is closest to the experimental value, while the other DFT methods predict longer bond lengths. The P–P bond distance for the anionic PK 2 is roughly Table 2 The electron affinities of P in eV (kcal/mol in parentheses) Method

EA

BHLYP BLYP B3LYP BP86 B3P86 B3PW91 BPW91 Experiment

0.68(15.7) 0.83(19.1) 0.93(21.4) 1.00(23.1) 1.40(32.3) 0.82(18.9) 0.83(19.1) 0.75G0.05a 0.772G0.052b 0.74651G0.00030c

Values are not corrected for ZPVE and were obtained with the DZPCC basis set. a a Ref. [21]. b Ref. [18]. c Ref. [20].

˚ longer than that of the neutral, indicating a weaker P–P 0.092 A bond in the PK 2 anion than that in neutral P2, due to the excess electron occupying an antibonding molecular orbital 2pg which causes the bond strength decreased. There are several experimental studies of the electron affinity for the phosphorus dimer. In 1974, Bennett et al. [18] reported the experimental adiabatic electron affinity of the P2 molecule to be 0.24G0.23 eV from their electron impact studies. Later (1977) Feldmann et al. [19] reported the EA of P2 to be !0.64999 eV. In 1985, Snodgrass et al. [20] reported the value of 0.589G0.025 eV for P2. In 1995, Jones et al. [21] reported the most accurate electron affinities to date for P2, 0.63G0.05 eV and VDE 0.68G0.05 eV using photoelectron detachment measurements. Our theoretical neutral-anion energy calculations for P2, as well as experimental electron affinity data, are given in Table 3.1. The adiabatic electron affinity EAad is predicted to be 0.70 (BHLYP), 0.61 (BLYP), 0.78 (B3LYP), 0.85 (BP86), 1.34 (B3P86), 0.83 (B3PW91) and 0.75 eV (BPW91), respectively. The zero-point vibrational energy correction is so small (wC0.008 eV, Table 1) that it may be ignored. The theoretical values are high, compared with the experimental value (0.63G0.05 eV), except for the BLYP result (0.61 eV). The BLYP method provided a very reasonable prediction. The range for the theoretical vertical electron affinity EAvert is from 0.49 to 1.22 eV. There are no experimental or other theoretical data available. The vertical detachment energy

Fig. 1. Molecular geometries for neutral P2 and anionic PK 2 . All bond distances ˚. are in A

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Table 3.1 Adiabatic and vertical electron affinities of the neutral Pn (nZ2–3) clusters and vertical detachment energies of their anions in eV (kcal/mol in parentheses)

Table 3.3 Adiabatic and vertical electron affinities of the neutral Pn (nZ6) clusters and vertical detachment energies of their anions in eV (kJ/mol in parentheses)

Compound

Method

EAad

EAvert

VDE

Compound

Method

EAad

EAvert

VDE

P2

BHLYP BLYP B3LYP BP86 B3P86 B3PW91 BPW91 Expt.

0.70 0.61 0.78 0.85 1.34 0.83 0.75 0.630G0.050a 0.589G0.025b !0.64999c 0.24G0.23d 1.80 1.57 1.83 1.86 2.42 1.90 1.77 1.665G0.050a 0.92G0.37d

0.56 0.49 0.66 0.73 1.22 0.70 0.63

0.86 0.73 0.92 0.97 1.47 0.96 0.86 0.68G0.05a

P6

BHLYP BLYP B3LYP BP86 B3P86 B3PW91 BPW91 Expt.

2.24 2.05 2.28 2.37 2.91 2.39 2.28

2.09 1.91 2.13 2.24 2.77 2.25 2.15

2.40 2.19 2.42 2.50 3.05 2.53 2.41 2.220G0.050a

P3(3a/3bK)

BHLYP BLYP B3LYP BP86 B3P86 B3PW91 BPW91 Expt.

0.86 0.97 1.19 1.19 1.73 1.19 1.07

1.96 2.18 2.15 2.45 2.82 2.30 2.21 1.68G0.05a

Values are not corrected for ZPVE and were obtained with the DZPCC basis set. a Ref. [21]. b Ref. [20]. c Ref. [19]. d Ref. [18].

(VDE) of PK 2 ranges from 0.73 to 1.47 eV. The value of BLYP method is close to the experimental data [21]. The values of EAad, EAvert, and VDE are close to each other due to the small differences in geometry between the neutral and the anion (Tables 3.2 and 3.3).

Table 3.2 Adiabatic and vertical electron affinities of the neutral Pn (nZ4–5) clusters and vertical detachment energies of their anions in eV (kJ/mol in parentheses) Compound

Method

EAad

EAvert

VDE

P4(4a/4bK)

BHLYP BLYP B3LYP BP86 B3P86 B3PW91 BPW91 Expt. BHLYP BLYP B3LYP BP86 B3P86 B3PW91 BPW91 Expt.

0.41 0.47 0.57 0.61 1.03 0.51 0.49

K0.52 K0.24 K0.23 K0.07 0.23 K0.28 K0.21

3.90 4.03 4.14 4.14 4.56 3.99 3.97

1.69 1.69 1.84 1.95 2.42 1.88 1.83

1.94 1.70 1.93 1.87 2.43 1.90 1.76 O1.350G0.050a 4.13 4.10 4.22 4.39 5.02 4.48 4.30 4.040G0.050a

P5(5a/5aK)

Values are not corrected for ZPVE and were obtained with the DZPCC basis set. a Ref. [21].

Values are not corrected for ZPVE and were obtained with the DZPCC basis set. a a Ref. [21].

3.3. P3 and PK 3 The phosphorus trimer has been identified in the gas phase for long time [42–44]. It was predicted by Murrell [42] to have a triangular (possible D3h) structure and subsequent calculations have confirmed that this form of P3 is subject to a small Jahn–Teller distortion. The Hartree–Fock calculations of Murrell et al. [44] showed that the final C2v structure having ˚ ) and a bond angle of 648. a bond length of 4.25 a.u. (2.249 A MNDO calculations [45] also produce a C2v ground state, with ˚ ) and two long bonds of one short bond of 3.543 a.u. (1.875 A ˚ 3.753 a.u. (1.986 A, a C2v bond angle of 58.38) The structures of P3 found in the present calculations are shown in Fig. 2. The relative energies (in eV) for the P3 systems are listed in Table 4. For the neutral P3 isomer, structure 3a with C2v-symmetry for the 2A2 state is the global minimum. It is an isosceles (nearly equilateral) triangle. Its theoretical apex bond angle ranges from 65.2 to 65.98, and the bond length for the two equal sides is in the range from 2.073 ˚ , predicted by the seven different functionals. This to 2.119 A prediction is similar to Jones et al. [4] results. We have optimized three other C2v symmetry structures with the 2B1, 2 B2 and 2A1 electronic states, respectively: a nearly equilateral ring structure (structure 3b), and two bent geometries (qO1008, structure 3d, and qZw928 structure 3e). Structure 3b is predicted to lie above structure 3a in energy by only 0.02–0.04 eV. Structure 3d is predicted to lie energetically above structure 3a by 1.03–1.28 eV. For structure 3e, four DFT methods (B3LYP, BHLYP, B3P86, and B3PW91) predict it to be a local minimum with much higher energy (1.81, 2.05, 1.93 and 1.93 eV) than structure 3a, but the other three DFT methods (BLYP, BP86, and BPW91) predict that it is not a stationary point and collapse to structure 3b. The ˚ linear geometry 3c with bond length from 1.948 to 1.995 A has higher energy, lying above structure 3a by 0.93–1.28 eV. This structure is a genuine minimum. It has all real vibrational frequencies, but there are the two components of the pu bending frequencies split. There were several theoretical reports for the trimer anion PK . 3 There are no experimental data available. Burdett and Marsden [46] and Hamilton and Schaefer [11] both found three low-lying minima: an equilateral triangle (D3h, 3 A), a linear

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˚. Fig. 2. Molecular geometries for neutral P3. All bond distances are in A

closed-shell singlet (DNh, 1Sg) and a bent (C2v) triplet. The former two minima were so close in energy that a definite prediction of the ground state was not possible. In Jones et al. [21] work with MD/DF calculations in 1995, they found that the linear structure has the lowest energy, but the D3h and C2v structures are only 0.06 and 0.27 eV less stable, respectively. In their study, CI calculations results show the D3h structure is 0.03 eV more stable. In our present calculations with DFT methods, we get some different results. The structures parameters for the anionic colorXredXPK 3 are displayed in Fig. 3. The relative energies (in eV) for the PK 3 isomers are 1 reported in Table 4. The linear structure 3aK of PK 3 ( Sg) has the lowest energy calculated by three DFT methods (BLYP, BP86 and BPW91), while the D3h ð3 AÞ\ structure 3bK has the lowest energy calculated by the other four DFT methods (BHLYP, B3LYP, B3P86 and B3PW91). The bent structure 3cK with C2v symmetry (1A1 electronic state) by all seven DFT methods has the highest energy (higher 0.27–0.38 eV than

structure 3aK, respectively). The bond distances in structure ˚ , which are w0.007 A ˚ 3aK are predicted to be 1.941–1.985 A shorter than their neutral counterpart structure 3c.The bond length of structure 3aK calculated by Jones at al. [21] is ˚ . Our BHLYP value is closest to it. In structure 3bK, 1.947 A ˚ . For the the bond distances are predicted to be 2.156–2.216 A K structure 3c , the bond distances are predicted to be 2.057– ˚ , and the bond angle is from 72.7 to 74.8 8, larger than 2.115 A that for the neutral 3a by w7–8 8. From the analysis above, the calculated geometries parameters by BHLYP method are in good agreement with the earlier work [11,21,46]. The theoretical EAad, EAvert, and VDE for structures 3a/3bK, as well as the experimental electronic affinity data, are listed in Table 3.1. The range of EAad is predicted from 1.57 to 2.42 eV with the seven DFT methods. The BLYP method predicts the smallest EAad (1.57 eV), and B3P86 method predicts the largest EAad (2.42 eV). There have been two different experimental EAs reported [18,21]. The 1974

Table 4 The relative energies (in eV) for the P3 =PK 3 isomers

3a 3b 3c 3d 3e 3aK 3bK 3cK

BHLYP

BLYP

B3LYP

BP86

B3P86

B3PW91

BPW91

0.00 0.04 1.28 1.36 2.05 0.00 K0.49 0.27

0.00 0.02 0.93 1.03 –a 0.00 0.14 0.38

0.00 0.02 1.09 1.18 1.81 0.00 K0.11 0.34

0.00 0.03 1.03 1.13 –a 0.00 0.05 0.36

0.00 0.02 1.19 1.28 1.93 0.00 K0.20 0.31

0.00 0.02 1.19 1.28 1.93 0.00 K0.23 0.30

0.00 0.02 5.03 1.15 –a 0.00 0.01 0.35

Values are not corrected for ZPVE. a a Not a stationary point, and it collapses to the structure 3b.

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˚ Fig. 3. Molecular geometries for anionic PK 3 . All bond distances are in A.

experimental result (0.92G0.37 eV) [18] seems too low, since our BHLYP (1.80 eV) BLYP (1.57 eV), B3LYP (1.83 eV) and BPW91 (1.77 eV) results are close to the recent (and more reliable) experimental value (1.665G0.050 eV) given by Jones et al. [21]. The range of EAvert is from 0.86 to 1.19 eV (except B3P86 1.73 eV), and the range of VDE is from 1.96 to 2.82 eV. Jones et al. [21] estimated VDE to be 1.68G0.05 eV, but their calculation results are more than 3.0 eV. Our predictions are all higher than the experimental data and lower than their calculation value. It is noted that the estimated VDE value by Jones et al. seems too low, which needs to be further experimental study. 3.4. P4 and PK 4 Bhaghavantam [47] performed the first measurements for P4 in 1930. Early electron diffraction measurements [2] were consistent with a tetrahedral structure with bond length of ˚ . This value has been refined by vibration– 2.13G0.04 A ˚ . The rotation Raman spectroscopy [41] to 2.2228G0.0009 A tetrahedral P4 structure is the dominant form of elemental phosphorus in the vapor phase. Under the assumption of rigid body librations of the P4 tetrahedra, PP distance of 2.209G ˚ has been inferred from a low-temperature X-ray 0.005 A structure analysis in crystalline white phosphorus [48]. Brundle et al. [49] performed a HF calculation at the experimental geometry and found excellent agreement with measured ionization potentials using a basis of s- and p-atomic orbitals. In 1990, Jones and Hohl [4] predicted three structures corresponding to local minima in the energy surface: the ˚ ; In 1994, most stable Td structure with bond length 2.207 A ˚ for the tetrahedral P–P Ballone and Jones [21] reported 2.212 A separation with the LSD method. In 1999, Boudon et al. [50] presented the first high-resolution infrared absorption study of P4 and found that the ground-state bond length of P4 is 219.58 pm. The value is significantly different from a value obtained from a Raman study of reference [41]. Five theoretical structures for neutral P4 species are displayed in Fig. 4. The relative energies (in eV) for the P4 isomers are reported in Table 5. According to the relative energies listed in Table 5, the most stable structure for the neutral P4 is 4a that displays Td symmetry in its 1A1 electronic ˚ )O state. The trend of the bond distances is BLYP (2.456 A ˚ )OBPW91 (2.220 A ˚ )OB3LYP (2.212 A ˚ )O BP86 (2.225 A

˚ )OB3P86 (2.194 A ˚ )OBHLYP (2.181 A ˚ ). B3PW91 (2.195 A The B3PW91 and B3P86 bond distances are consistent with the experimental value of Boudon et al. [50]. We found a ‘roof-shaped’ P4 structure 4b, with C2v symmetry for its 1A1 state, lying substantially (1.74– 2.16 kcal/mol) above the tetrahedral form. Structure 4c has planar C2v symmetry, lying energetically above structure 4a by 2.45–2.93 eV. We also located the D4h triplet structure 4d (3A1g state) that is higher than the ground state 4a in energy by 1.19–2.42 eV. The P–P bond distance is in the range of 2.107– ˚ . However, it is predicted to be a transition state using 2.221 A all seven DFT methods, and the magnitude of the imaginary vibrational frequency is larger than 200i cmK1. Getting rid of this imaginary frequency, 4d goes to a D2d local minimum 4e (3A1 ground state). The P–P bond distances in 4e are similar to those for the singlet structure 4b (see Fig. 4). It has almost the same energy as that of 4b. We found four stationary point structures for the anionic PK 4 displayed in Fig. 5. The corresponding relative energies are listed in Table 5. Structure 4aK is a planar rectangular (D2h) ˚ and 2.087– structure with bond length of 2.216–2.289 A ˚ . Structure 4bK is a ‘butterfly’(C2v symmetry) structure 2.150 A related to structure 4b for neutral P4. The bond distance of the common side is shorter than that of its neutral counterpart ˚ , while the distances of the other (structure 4b) by w0.18 A ˚ . The bonds are longer than those of structure 4b by w0.03 A K change of the dihedral angles shows that structure 4b is more puckered than the neutral structure 4b. In Jones et al.’s study [21], the ‘roof’ structure (4bK) was found to be the most stable structure. In our calculations, Structures 4aK is more stable than structure 4bK, though only 0.01–0.19 eV with seven DFT methods lower in energy. The chain structure 4cK is a local minimum with C2h symmetry, lying above structure 4bK by 0.01–0.79 eV. Structure 4dK is another local minimum that has higher energy than structure 4bK by 0.64–0.84 eV (Table 5). There is no experimental structure available for comparison. The theoretical EAad, EAvert, and VDE values, as well as the experimental results, are listed in Table 3.2. Since, the very few difference between structure 4aK and 4bK for anionic PK 4 in total energies, according to the previous research [21], we choose structure 4bK to calculate EAs for P4. Our predicted the EAad value for P4 ranges from 0.41 to 0.61 eV except B3P86 (1.03 eV). The EAvert value is predicted to have negative values, ranging from K0.52 to K0.21 eV except B3P86

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˚ . Italics fonts for structure 4d indicate transition state. Fig. 4. Molecular geometries for neutral P4. All bond distances are in A

Table 5 The relative energies (in eV) for the P4 =PK 4 isomers

4a 4b 4c 4d 4e 4aK 4bK 4cK 4dK

BHLYP

BLYP

B3LYP

BP86

B3P86

B3PW91

BPW91

0.00 2.16 2.90 1.19 1.94 K0.15 0.00 0.79 0.78

0.00 1.74 2.45 2.03 1.79 K0.19 0.00 0.01 0.64

0.00 1.94 2.68 2.14 1.91 K0.16 0.00 0.23 0.72

0.00 1.93 2.71 2.33 2.05 K0.05 0.00 0.59 0.77

0.00 2.12 2.92 2.41 2.16 K0.03 0.00 0.62 0.84

0.00 2.13 2.93 2.42 2.16 K0.01 0.00 0.63 0.84

0.00 1.97 2.76 2.37 2.10 K0.02 0.00 0.65 0.78

Italics for structure 4d indicate that it is a transition state. Values are not corrected for ZPVE.

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˚ Fig. 5. Molecular geometries for anionic PK 4 . All bond distances are in A.

(0.23 eV), while the VDE values vary from 1.70 to 2.43 eV calculated with different DFT methods respectively. Jones et al. [21] reported the VDE to be 1.35G0.05 eV using the photoelectron spectroscopy, and our BLYP method predicts the VDE value (1.70 eV) closest to the experiment. The EAad, EAvert and VDE values are significantly different for P4 =PK 4 due to the striking difference in geometries between the global minima of neutral P4 (tetrahedral) and the anionic PK 4 (butterflyshaped). 3.5. P5 and PK 5 The five isomers for P5 and five isomers for PK 5 optimized by seven DFT methods are shown in Figs. 6 and 7, respectively. The corresponding relative total energies of the

structures are listed in Table 6. All structures listed here are real local minima. As indicated by the relative total energies in Table 6, the structure 5a is the global minima of neutral P5. It is derived from 4a by breaking one bond and adding one twofold atom with C2v symmetry. No experimental studies have been reported for the structures of neutral P5. In 1990, Jones and Dohl [4] predicted the same structure to be the most stable structure for neutral P5. At the same time, they characterized other structures: the structure pentagon with a Jahn–Teller distortion (C2v symmetry) similar to our structure 5b; a D3h propeller like structure, with three atoms symmetrically placed with respect to a central bond of length 4.52 a.u.; a C4v structure, with a P atom placed centrally above a P4 square. In 1994, Ballone and Jones [6] reported the C2v structure 5a to be the global minimum too. In 2000, Chen

Table 6 The relative energies (in eV) for the P5 =PK 5 isomers

5a 5b 5c 5d 5e 5aK 5bK 5cK 5dK 5eK

BHLYP

BLYP

B3LYP

BP86

B3P86

B3PW91

BPW91

0.00 0.29 0.49 1.58 1.55 0.00 1.89 2.11 2.29 3.61

0.00 K0.33 0.32 1.00 0.89 0.00 1.67 2.24 2.09 2.91

0.00 K0.10 0.40 1.23 1.15 0.00 1.78 2.19 2.20 3.19

0.00 K0.15 0.43 1.23 0.97 0.00 1.63 2.07 2.13 2.83

0.00 0.10 0.51 1.46 1.24 0.00 1.73 2.03 2.23 3.12

0.00 0.13 0.52 1.53 1.24 0.00 1.70 1.99 2.21 3.09

0.00 K0.08 0.46 1.28 0.99 0.00 1.61 2.01 2.12 2.79

Italics for structures 5c and 5d indicate levels of theory at which a transition state was found. Values are not corrected for ZPVE.

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˚ . Italics fonts for structures 5c and 5d indicate transition state. Fig. 6. Molecular geometries for neutral P5. All bond distances are in A

et al. [15] also mentioned the C2v most stable structure for P5. In the present research, similar to the previous studies, we predict that the global minimum is the structure 5a (C2v symmetry at the 2B1 state). It is more stable than structure 5b by 0.10–0.29 eV with three DFT methods (BHLYP, B3P86 and B3PW91) while less stable than structure 5b by 0.10– 0.33 eV with other DFT methods (BLYP, B3LYP, BP86 and

BPW91). For structure 5b, the BHLYP, BLYP, BP86 and BPW91 methods predict a planar structure with C2v symmetry, while the other three methods predict a Cs structure slightly twisted from the plane with a dihedral angle (6.7–8.3 8). A Cs structure 5c, lies above structure 5a by 0.32–0.53 eV (Table 6). Structures 5d and 5e (Fig. 6) have not been

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˚ Fig. 7. Molecular geometries for anionic PK 5 . All bond distances are in A.

reported before. They are characterized as local minima but with high energies (Table 6). Structure 5d, containing a fourmembered ring, with a Cs symmetry lies above the global minimum 5a by 1.00–1.58 eV, while the D2d structure 5e by 0.89–1.55 eV. Baudler et al. [51] have identified the anionic PK 5 as a ring of unsubstituted twofold coordinated P atoms. This planar anion was prepared by reaction between white phosphorus and tetrahydrofuran in solution [52]. Baird [53] performed MNDO calculations for PK 5 , and found a planar pentagonal ˚ ). In 1995, (D5h) structure with bond length 3.58 a.u. (1.895 A Jones et al. [21] predicted the planar D5h structure with bond

˚ as the most stable structure of PK length 2.096 A 5 . Chen et al. [15] reported the same prediction in 2000. In the present work, as indicated in Table 6, it was reproduced that structure 5aK with D5h symmetry at 1 A electronic state is the most stable isomer of PK 5 . The bond length is in the range of 2.097– ˚ optimized with seven DFT methods respectively. Our 2.149 A BHLYP bond length prediction is in good agreement with the ˚ ) [21]. Structure 5bK with Cs Jones’s work (2.096 A symmetry has a higher energy by 1.61–1.89 eV. The threedimensional C2v structure (5cK)(look like a butterfly that similar to 5a) has a higher energy by 1.93–2.24 eV. These two structures (5bK, 5cK) were also given by Chen et al.

D. Wang et al. / Journal of Molecular Structure: THEOCHEM 759 (2006) 225–238

235

˚. Fig. 8. Molecular geometries for P6. All bond distances are in A

[15]. In the present work, we also found two other structures of PK 5 that have not reported before, i.e. the structure containing a four-member ring (5dK) with C2v symmetry comparative with structure 5cK in energy, and the D2d structure 5eK (Fig. 7) which has the highest energy in all stable geometries calculated on PK 5 (Table 6). The electron affinities of P5 calculated by seven DFT methods and the experimental value are given in Table 3.2. The EAad ranges from 3.90 to 4.56 eV. The BHLYP method gives the lowest EAad (3.90 eV). No experimental EAad was reported. The BLYP (4.03 eV) value may be the most reliable prediction, according to the predictions of P, P2, P3 and P4. The EAvert ranges from 1.69 to 2.42 eV. The VDE ranges from 4.10 to 5.02 eV, the values are so large, indicating that the anion is quite stable with respect to electron detachment. The experimental VDE was reported to be 4.040G0.050 eV by Jones et al. [21]. Our BLYP (4.10 eV) and BHLYP (4.13 eV) methods give the nearest values. The striking differences between EAad, EAvert, and VDE are due to the large changes in geometry between the global minima for P5 (a butterfly-shaped) and PK 5 (a planar pentagon).

3.6. P6 and PK 6 Baird [53] performed MNDO calculations on three forms of P6. The most stable is the prismane (D3h) analog. The benzene analog structure lies by 0.22 eV higher and a C2v structure lies by a similar amount above that. The same three isomers were treated in the HF approximation with 6-31G basis sets by Nagase and Ito [54]. In their study, the C2v structure was predicted to be the most stable structure of P6. Jones and Hohl [4] found several structures for P6. They also predicted that the C2v structure is the most stable structure. We performed seven DFT methods calculations on P6, have the similar results with the previous researches [4,54]. The four P6 structures displayed in Fig. 8 and the relative energy was listed in Table 7. The most stable structure with C2v symmetry derived from the P5 (Fig. 6, 5a) by replacing the single atom outside the C2v roof by a dimmer (Fig. 8, 6a). The structure 6b with D3h symmetry lies by 0.52–0.70 eV above 6a. The structure 6c with D2 symmetry converted from D6h lies by 0.34–0.96 eV above the global minima. Another structure 6d has C2h symmetry lies 0.60–1.03 eV higher than 6a.

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Table 7 The relative energies (in eV) for the P6 =PK 6 isomers

6a 6b 6c 6d 6aK 6bK 6cK 6dK 6eK

BHLYP

BLYP

B3LYP

BP86

B3P86

B3PW91

BPW91

0.0 0.52 0.96 1.03 0.0 0.45 1.82 2.10 2.46

0.0 0.70 0.34 0.60 0.0 0.24 0.60 1.35 1.71

0.0 0.63 0.63 0.77 0.0 0.33 0.67 1.76 2.05

0.0 0.61 0.58 0.62 0.0 0.27 0.53 1.69 1.93

0.0 0.54 0.86 0.80 0.0 0.36 0.58 2.03 2.26

0.0 0.52 0.88 0.78 0.0 0.34 0.57 2.02 2.25

0.0 0.58 0.63 0.62 0.0 0.27 0.51 1.73 1.96

Values are not corrected for ZPVE.

˚ Fig. 9. Molecular geometries for anionic PK 6 . All bond distances are in A.

D. Wang et al. / Journal of Molecular Structure: THEOCHEM 759 (2006) 225–238

For the anionic PK 6 , we found five stable structures displayed in Fig. 9. And the relative energies are listed in Table 7. From Table 7, it is clear that the most stable isomer in all the present calculations was the C2v form (Fig. 9, 6aK) that is relative to the structure of 6a. Structures 6bK (C2h) and 6cK (D3h) lie about 1 .0eV higher than 6aK. Structures 6dK (C2v) and 6eK (D2h) are both planar structures and have higher energies. Jones et al. [21] took MD/DF and LSD method calculation to PK n clusters and found three isomers for PK 6 (same with our structure 6aK, 6bK and 6cK). No experimental data for comparison. The electron affinities of P6 =PK 6 calculated by seven DFT methods and the experimental value are given in Table 3.3. The EAad ranges from 2.05 to 2.91 eV calculated with seven DFT methods respectively. The BLYP method gives the lowest EAad (2.05 eV). No experimental EAad was reported. The EAvert ranges from 2.09 to 2.77 eV. The VDE ranges from 2.19 to 3.05 eV. The experimental VDE was reported to be 2.220G 0.050 eV by Jones et al. [21]. Our BLYP method gives the nearest value. The small differences between EAad, EAvert, and VDE are due to the same geometry of the global minima for P6 and PK 6. 4. Conclusions We have performed a theoretical study of density functional theory calculations with the DZPCC basis set on the phosphorus clusters, focusing on the geometries and electronic affinities. 4.1. The structures trends We have performed calculation of the structures of isomers for Pn and PK n (nZ1–6), i.e. locating the minima in the energy surfaces for each molecules and determining the energy differences between the anions and the neutral clusters at the ground states of geometry, and the most stable isomer have been predicted. With a valence configuration of 3s2 3p1x 3p1y 3p1z for P atom, they like to form threefold chemical bond and three-dimensional structures. In the present predictions, for the neutral Pn clusters, only the structures of P2 and P3 are planar. For the anionic Pn clusters, the structures of Pn (nZ2, 3, 4, 5) are planar. 4.2. Electron affinities of Pn clusters The BLYP method is the most reliable method for predicting the electron affinities for these molecular systems. The adiabatic EAs are predicted to be 0.83 (P), 0.61 (P2), 1.57 (P3), 0.47 (P4), 4.03 (P5) and 2.05 (P6) eV. The EAad values for Pn increase in a zig-zag pattern as n increases from 1 to 6. The larger EAs are associated with the close-shell anionic systems K ðPK 3 ; P5 Þ, which are clearly more stable. For the seven DFT methods, the average errors for the neutral-anion energy separations (compared with the most reliable available experimental data, namely the EAs for P, P2, P3 are 0.09 (BHLYP), 0.06 (BLYP), 0.17 (B3LYP), 0.22 (BP86), 0.71

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(B3P86), 0.17 (B3PW91), and 0.10 eV (BPW91), respectively. The average errors of VDE for PK n (nZ2–6) are 0.26 (BHLYP), 0.20 (BLYP), 0.33 (B3LYP), 0.43 (BP86), 0.96 (B3P86), 0.44 (B3PW91) and 0.31 eV (BPW91), respectively. It can also be found from the tables that the VDEs for PK 3 by different methods differ considerably from the experimental data. If the VDE of PK 3 was excluded, better results were obtained. The average errors of VDE for PK n (2–6, except 3) are 0.26 (BHLYP), 0.12 (BLYP), 0.30 (B3LYP), 0.35 (BP86), 0.92 (B3P86), 0.40 (B3PW91) and 0.26 eV (BPW91), respectively. Therefore, the experimental data for PK 3 maybe suspectable and further experimental studies are suggested. Our theoretical predictions will hopefully provide strong motivation for further experimental studies of these important phosphorus clusters and their anions. Acknowledgements The research was supported by Key Laboratory of Theoretical and Computational Chemistry of Jilin University of China. We are grateful to Professor Henry F. Schaefer III of University of Georgia, who supplied us the DZPCC basis sets for P element. References [1] N.N. Greenwood, A. Earnshaw, Chemistry of the Elements, Pergamon Press, Oxford, 1985. [2] L.R. Maxwell, S.B. Hendricks, V.M. Mosley, J. Chem. Phys. 3 (1935) 699. [3] T.P. Martin, Z. phys. D3 (1986) 211. [4] R.O. Jones, D. Hohl, J. Chem. Phys. 92 (1990) 6710. [5] R.O. Jones, G. Seifert, J. Chem. Phys. 96 (1992) 7564. [6] P. Ballone, R.O. Jones, J. Chem. Phys. 100 (1993) 4941. [7] G. Seifert, R.O. Jones, J. Chem. Phys. 96 (1992) 2951. [8] M. Ha¨ser, O. Treutler, J. Chem. Phys. 102 (1995) 3703. [9] R.B. Huang, H.D. Li, Z.Y. Lin, S.H. Yang, J. Phys. Chem. 99 (1995) 1418. [10] H. Yilmaz, J. Mol. Struct. (Theochem) 257 (1992) 285. [11] T.P. Hamilton, H.F. Schaefer III, Chem. Phys. Lett. 166 (1990) 303. [12] M. Ha¨ser, U. Schneider, R. Ahlrichs, J. Am. Chem. Soc. 114 (1992) 9551. [13] B.M. Gimarc, D.S. Warren, Inorg. Chem. 32 (1993) 1850. [14] Krishnan Raghavachari, R.C. Haddon, J.S. Binkley, Chem. Phys. Lett. 122 (1985) 219. [15] M.D. Chen, R.B. Huang, L.S. Zheng, Q.E. Zhang, C.T. Au, Chem. Phys. Lett. 325 (2000) 22. [16] M.D. Chen, R.B. Huang, L.S. Zheng, C.T. Au, J. Mol. Struct. (Theochem) 499 (2000) 195. [17] J.D. Carette, L. Kerwin, Can. J. Phys. 39 (1961) 1300. [18] S.L. Bennett, J.L. Margrave, J.L. Franklin, J. Chem. Phys. 61 (1974) 1647. [19] D. Feldman, T. Rackwitz, H.J. Kaiser, E. Heincke, Z. Naturforsch. A 32 (1977) 600. [20] J.T. Snodgrass, J.V. Coe, C.B. Freidhoff, K.M. Mchugh, K.H. Bowen, Chem. Phys. Lett. 122 (1985) 352. [21] R.O. Jones, G. Gantefor, Hunsicker, P. Pieperhoff, J. Chem. Phys. 103 (1995) 9549. [22] W. Kohn, A.D. Becke, R.G. Parr, J. Phys. Chem. 100 (1996) 12974. [23] R.A. King, V.S. Mastryukov, H.F. Schaefer, J. Chem. Phys. 105 (1996) 6880. [24] G.S. Tschumper, J.T. Fermann, H.F. Schaefer, J. Chem. Phys. 104 (1996) 3676.

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