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Lowest-energy structures of cationic P2m+1+ (m = 1–12) clusters from first-principles simulated annealing
Chemical Physics Letters 485 (2010) 26–30
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Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Lowest-energy structures of cationic Pþ 2mþ1 (m = 1–12) clusters from first-principles simulated annealing Tao Xue a, Jing Luo a, Si Shen a, Fengyu Li a,b, Jijun Zhao a,b,* a b
School of Physics and Optoelectronic Technology, College of Advanced Science and Technology, Dalian University of Technology, Dalian 116024, China Laboratory of Materials Modification by Laser, Electron, and Ion Beams, Dalian University of Technology, Dalian 116024, China
a r t i c l e
i n f o
Article history: Received 14 August 2009 In final form 8 December 2009 Available online 11 December 2009
a b s t r a c t Lowest-energy structures of odd-sized cationic Pþ 2mþ1 (m = 1–12) clusters have been determined from first-principles simulated annealing followed by more accurate geometry optimization within the framework of density functional theory. For Pþ n with n > 11, the current global minimum structures are more favorable than those previously reported ones. A structural motif based on P7, P8, P9, and P10 building blocks and P2 bridge was revealed for the medium-sized Pþ n clusters. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Among elementary solids, phosphorus is quite unique since it exhibits a variety of structural phases [1] such as black (orthorhombic), violet (monoclinic), rhombohedral, metallic (cubic), red, white, and amorphous. Most structural phases of phosphorus solids can be viewed as superstructures built by small subunits like P4, P8, and P9 clusters. Thus, it is of fundamental interest to reveal the size-dependent growth mechanism of phosphorus clusters, which is also potentially useful for designing novel nanoscale materials and devices. Using laser vaporization technique, gas-phase phosphorus clusters have been generated in cluster beam and their size-dependent relative abundances have been recorded via time-of-flight (TOF) mass spectrometer [2–9]. In a pioneer work by Martin [2], cationic Pþ n clusters with up to n = 24 were obtained by quenching the vapour of red phosphorus in helium beam. Except at Pþ 11 , the relative abundances of Pþ n clusters oscillate with cluster size n and show the preference of the odd-sized cationic clusters (with even number of valence electrons). Later, Huang et al. [3–6] reported TOF mass spectra of cationic phosphorus clusters generated by direct laser vaporization of red phosphorus, and the maximum cluster size was extended to 89 atoms. In their mass spectra, except several þ þ þ even-numbered peaks of Pþ 4 ; P6 ; P10 and P14 , most odd-sized clusþ ters show strong densities. For larger Pn clusters (n P 25), the observed magic numbers can be written as n = 25 + 8k (k = 0, 1, 2, 3, 4, 5, 6, 7, 8) [3,6], suggesting that the most stable structures of larger phosphorus clusters may be built from P8 units. Bulgakov and co-workers [7–9] reported mass spectra of neutral, cationic, and
* Corresponding author. Address: School of Physics and Optoelectronic Technology, College of Advanced Science and Technology, Dalian University of Technology, Dalian 116024, China. Fax: +86 411 84706100. E-mail address: [email protected] (J. Zhao). 0009-2614/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2009.12.019
anionic phosphorus clusters in a wide size range, i.e., up to n = 40 for Pn, n = 91 for Pþ n , and n = 49 for Pn , respectively. In each case, the relative abundance show odd–even alternation and the clusters with even number of electrons are dominant in the mass spectra. For the cationic Pþ n clusters, the observed magic numbers are n = 7 and 21, while the most abundant clusters are P10 and P14 for the neutral species. So far, theoretical calculations on phosphorus clusters with first-principles methods mainly focused on the neutral species [10–21]. In a series of pioneering works, Jones and co-workers performed simulated annealing with density functional theory (DFT) to explore the ground state structures of phosphorus clusters up to 11 atoms [10–12]. According to their simulation, P8, which was previously assumed to be cubic [13,14], turns out to adopt a wedge-like configuration in analogy with the (CH)8 cuneane [10]. Häser et al. carried out ab initio MP2 calculations on even-sized Pn clusters with up to n = 28 and discussed the thermodynamic stabilities (with regard to P4) and electronic properties of these clusters [15,16]. Fullerene-like cage configurations have been considered for Pn cluster with n P 14, but these hollow-cage clusters were usually found to be unstable upon dissociation into small P4 clusters [17,18]. Recently, small phosphorus clusters containing two to eight atoms have been investigated by high-level quantum chemistry methods [19–21]. Compared to the extensive studies of the neutral phosphorus clusters, much less is known about the charged clusters, particularly, the cationic ones, which were most frequently produced experimentally. In Jones’ simulated annealing study [11], the structures of cationic Pþ n clusters up to n = 11 were determined. Feng et al. attempted to predict the most stable structures of cationic Pþ 2mþ1 clusters (m = 3, 4, 5, 6) by ab initio Hartree–Fock calculations [22]. Chen et al. [23,24] compared large number of structural isoþ þ mers for small Pþ 5 ; P7 ; and P9 clusters using DFT calculations. Later, Guo et al. investigated the geometries and electronic properties
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of neutral and charged Pn clusters up to n = 15 using DFT calculations with the B3LYP/6-311G(d) method [25]. For the larger cationic Pþ n with n = 25 + 8k (k = 1, 2, 3, 4, 5, 6, 7, 8), Chen et al. constructed a series of chain-like configurations by adding cuneane P8 units to a presumed Cs structure of Pþ 25 cluster [6]. Despite the previous efforts, our knowledge on the structural growth behavior of phosphorus clusters is still insufficient since previous theoretical studies were either focused on small clusters (i.e., less than 15 atoms) or considered only limited number of presumed configurations. For the medium-sized clusters with more than 10 atoms, the potential energy surfaces (PES) would become rather complicated. An unbiased global search is essential for locating the lowest-energy structures. Within the framework of DFT, in this Letter we performed simulated annealing study to determine the lowest-energy structures of odd-numbered cationic phosphorus clusters (up to 25 atoms). For the medium-sized Pþ n clusters with n > 11, we revealed new lowest-energy configurations that are more energetically favorable than the previously proposed structures. The size-dependent evolution of cluster geometry, binding energy, and HOMO–LUMO gap will be discussed.
experimental data within the best of our knowledge. As an alternative, we performed high-level ab initio calculations using the CCSD/ aug-cc-PVDZ method implemented in the GAUSSIAN 03 package [29], which is rather accurate in predicting the structures and bonding of small clusters. In Table 1, the geometry parameters, binding energies, vibrational frequencies, and ionization potentials from our theoretical calculations are compared with the experimental data for P2 and P4 [21,30] and the CCSD/aug-cc-PVDZ results for þ Pþ 3 and P5 . The theoretical P–P bond lengths of P2 and P4 are very close to the measured ones. Our PBE/TNP calculations overestimated the cluster binding energies by less than 5%. The computed vibrational frequencies agree well with the experimental data with an average deviation of 1.6%. The theoretical prediction of ionizaþ tion potentials is also satisfactory (see Table 1). For Pþ 3 and P5 clusters, the geometry parameters and vibrational frequencies from PBE/TNP calculations agree satisfactorily with the CCSD/aug-ccPVDZ results, whereas the theoretical binding energies by PBE/ TNP are systematically lower than the CCSD/aug-cc-PVDZ values. Overall speaking, the reasonable agreements between our PBE/ TNP calculations and the experimental data as well as the high-level CCSD/aug-cc-PVDZ results demonstrate that our computational method is reliable.
2. Theoretical methods
3. Results and discussion
In this work, the potential energy surfaces of the cationic phosphorus clusters were first explored by unbiased search using the simulated annealing (SA) technique incorporated with first principles molecular dynamics (FPMD), as implemented in the DMol3 code [26] (built in the Material Studio 4.3 package provided by Accelrys Inc.) The DMol3 code allows constant-temperature molecular dynamic simulations, in which the interatomic forces are obtained from first-principles calculations within the all-electron density functional theory. In the FPMD simulations, the local density approximation (LDA) was used to describe the exchange-correlation interaction, and the double numerical plus p and d polarization (DNP) basis set [26] was adopted. The Harris functional approximation [27] was employed to solve for the total energy and atomic forces in a non-self-consistent-field (non-SCF) way. The initial structures for simulated annealing were built randomly, and the total simulation time (1.3 ns) would be long enough to forget the original configurations. For each size, the cluster was independently annealed from a high starting temperature of 1500 K (about the melting point of phosphorus solids) down to 300 K via FPMD. The temperature was systematically reduced by increments of 100 K. At each temperature, MD simulation in a NVT ensemble lasts for 100 ps with a time step of 1 fs (i.e. 100 000 MD steps at each temperature). The structures of the cationic phosphorus clusters from simulated annealing were further optimized using the same DMol3 code. A triple numerical basis set including d and p polarization functions (TNP) was chosen. The exchange-correlation interaction was treated within the generalized gradient approximation (GGA) with the Perdew–Burke–Enzerhof (PBE) parameterization [28]. SCF calculations were done with convergence criterion of 106 Hartree on the total energy. All the structures were fully optimized without any symmetry constraint with a convergence criterion of 0.002 Hartree/Å for the forces and 0.005 Å for the displacement. Vibrational analysis of these Pþ n clusters in their equilibrium configurations has been carried out to ensure that there are no imaginary frequencies corresponding to the saddle points on the PES. The accuracy of the present PBE/TNP computational scheme has þ been assessed by benchmark calculations on small P2, Pþ 3 , P4 and P5 þ þ clusters. For the small cationic P3 and P5 clusters, there was no
For each cationic phosphorus cluster, there is no imaginary frequency in the equilibrium structures obtained from the combinational search of FPMD simulated annealing and local PBE/TNP minimization. Thus, the final structures of Pþ 2mþ1 (m = 1–12) clusters displayed in Fig. 1 are all true minima on the potential energy surface instead of the saddle points. þ þ þ þ 3.1. Pþ 3 , P 5 , P 7 , P 9 and P 11 þ þ þ þ For small cationic phosphorus clusters (Pþ 3 , P5 , P7 , P9 , and P11 ) their lowest-energy geometries have been determined by Guo et al. using density functional calculations with B3LYP method [25]. Our present calculations have successfully reproduced these most stable configurations for small Pþ 2mþ1 (m = 1–5) clusters, thus we will only discuss briefly the theoretical results. þ The equilibrium geometries of Pþ 3 and P5 are an equilateral triangle with D3h symmetry and a square pyramid with C4v symmetry, respectively. In analogy to the well-known cuneane P8 unit [6,10,24,25], the wedge-like configuration of Pþ 7 (C2v symmetry) can be viewed as a square-face-capped triangle prism with two broken bonds. The lowest-energy geometry of Pþ 9 (D2d symmetry) can be built from two P4 tetrahedra aligned vertically to each other
Table 1 P–P distance (r0), binding energy (Eb), vibrational frequencies (x0), and ionization þ potential (IP) for neutral P2, P4 and cationic Pþ 3 , P5 clusters from our PBE/TNP and CCSD/aug-cc-PVDZ calculations as well as experiments [21,29]. r0 (Å)
Eb (eV)
x0 (cm1)
IP (eV)
788.5 780.8
10.45 10.6 ± 0.1
P2 PBE/TNP Experiment
1.902 1.893
P4 PBE/TNP Experiment
2.207 2.21 ± 0.02
13.18 12.53
371.6, 459.7, 594.1 360.8, 466.3, 600.5
Pþ 3 PBE/TNP CCSD/aug-cc-PVDZ
2.11 2.14
11.74 13.31
474.0, 623.5 474.2, 650.9
– –
Pþ 5 PBE/TNP CCSD/aug-cc-PVDZ
2.26 2.30
19.15 22.40
457.9, 501.9, 509.6 471.4, 503.1, 523.5
– –
5.27 5.08
9.56 9.28 ± 0.1
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T. Xue et al. / Chemical Physics Letters 485 (2010) 26–30
Fig. 2. Isomer structures of Pþ n (n = 13, 15, 21, 25) reported in literature [6,18,22,25]. DE is the energy difference between this isomer and our lowest-energy structure. Fig. 1. Lowest-energy configurations of odd-sized Pþ n (n = 3–25) clusters.
with a fourfold coordinated joint atom in between, which has been previously found by Guo et al. [25] and Chen et al. [23]. The most stable structure of Pþ 11 can be viewed as an eight-atom cuneane unit and a six-atom trigonal prism sharing three P atoms on a triangle face. It is noteworthy that the wedge-like P7 and P8 are the common building blocks for larger phosphorus clusters, as we will show in the following discussions. Starting from Pþ 5 , most atoms in the phosphorus clusters are threefold coordinated whereas there is one atom with fourfold coordination. The dominant threefold coordination can be understood by the electron configuration of phosphorus atoms, that is, each P atom contributes three electrons to form three covalent bonds with the neighboring P atoms, leaving two lone-pair electrons. þ 3.2. Pþ 13 and P 15 þ The structural evolution behavior is rather clear from Pþ 11 to P15 , can be obtained as shown in Fig. 1. The most stable structure of Pþ 13 by adding two atoms to the waist (the sharing triangle) of Pþ 11 . Inserting further two additional atoms between the waist and the triangle end of the Pþ 13 results in the most stable structure of Pþ 15 , which can be considered as a P8 cuneane and a P10 polyhedron sharing three atoms on a triangle. For the cationic phosphorus clusters with more than 11 atoms, the global minimum structures from unbiased FPMD annealing search have no point-group symmetry. Previously, symmetric configurations were proposed for Pþ 13 [22,25] and Pþ 15 [25] (both with Cs symmetry), which are depicted in Fig. 2. At the same PBE/TNP level of theory, the present lowest-energy structures (shown in Fig. 1) are more energetically þ favorable than the previous proposed ones for Pþ 13 and P15 by 1.4 and 1.2 eV, respectively. This indicates that an unbiased search is essential for revealing the lowest-energy structures of the medium-sized phosphorus clusters. The most stable structures of the odd-numbered cationic phosþ phorus clusters from Pþ 5 to P15 all satisfy the rule that there is only one four-coordinated atom as the key link point of the entire frame and each of the rest atoms connect with three other atoms. However, as cluster size further increases, we observed the emergence of two-coordinated atoms and even a five-coordinated atom (in Pþ 21 ). The descriptions for the lowest-energy structures of the larger þ þ þ þ Pþ 17 , P19 , P21 , P23 and P25 clusters are presented below. þ þ þ þ 3.3. Pþ 17 , P 19 , P 21 , P 23 and P 25
The equilibrium geometry of Pþ 17 has one pair of adjacent atoms, one with fourfold coordination and another with twofold coordina-
tion; the rest atoms are all threefold coordinated. This structure can be decomposed into two P8 subunits connected by the central two-coordinated atom. In the equilibrium configuration of Pþ 19 , there are three fourcoordinated atoms and one two-coordinated atom. One of the three four-coordinated atoms is adjacent to the two-coordinated atom, while the other two are connected with each other to form the waist of the tubular structure. The entire structure of Pþ 19 can be viewed as a wedge-like P7 subunit connected with a P10 subunit (twisted pentagonal prism) by a P2 bridge. Unlike the other medium-sized cationic Pþ n clusters (n = 11–25) with straight chain-like geometries, the lowest-energy structure of Pþ 21 is a curved tubular chain. Besides the regular threefold coordination, one twofold, two fourfold, and one fivefold coordinated atoms were found. The two-coordinated atom still pairs up with one four-coordinated atom. Both the two fourfold atoms are bonded with the fivefold atom. Previously, Bulgakov et al. proposed two possible structures for Pþ 21 [18], i.e., an edge-capped hollow cage with C2v symmetry and a chain-like tubular structure (P4 + P8 + P9) with Cs symmetry (see Fig. 2). Compared to the lowest-energy structure from our calculations, the C2v cage isomer is energetically less preferred by 1.75 eV, and the chain-like Cs isomer is 0.33 eV higher in energy. The most stable structure of Pþ 23 has one four-coordinated atom. The rest atoms are all with threefold coordination. Overall speaking, it has a quasi-linear tubular shape. Careful examination reveals that it consists of three subunits and one P2 bridge. The right side is a P6 unit formed by adding two P atoms on an open-edged P4 tetragon. On the left side, there are two subunits, a wedge-like P8 and a twisted pentagonal prism of P10, connected by sharing a P3 triangle. The left and right parts of the cluster are linked together by a P2 bridge. In the lowest-energy structure of Pþ 25 , there are two pairs of twofold coordinated atom and fourfold coordinated atom, and an additional four-coordinated atom. These two pairs of two-coordinated and four-coordinated atoms have very similar bonding environment and align symmetrically in the cluster. On each side of the pair, there are a cuneane P8 and a P10 subunit, respectively. Between them, there is a pentagonal P5 ring. In the previous study by Chen et al. [19], a chain-like Cs configuration constructed by P8 + P8 + P9 was proposed for Pþ 25 (see Fig. 2). Upon relaxation at the same PBE/TNP level, this structure is less favorable in energy than the present lowest-energy structure by 0.3 eV. From the above discussions, we can figure out several general rules for the structural motif of the cationic phosphorus clusters: (1) most medium-sized clusters exhibit tubular chain-like configurations with low symmetry or even without symmetry; (2) for a
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T. Xue et al. / Chemical Physics Letters 485 (2010) 26–30
Table 2 Symmetries, highest vibrational frequencies, total energies, average P–P bond lengths, HOMO–LUMO gaps, binding energies (Eb) of odd-numbered cationic phosphorus Pþ n clusters (up to n = 25) as well as the adiabatic ionization potentials (IP) of odd-sized neutral Pn clusters (up to n = 9). n
Symmetry
Highest frequency (cm1)
Total energy (Hartree)
Average bond length (Å)
Gap (eV)
Eb (eV/atom)
IP (eV)
3 5 7 9 11 13 15 17 19 21 23 25
D3h C4v C2v D2d Cs C1 C1 C1 C1 C1 C1 C1
632.51 509.57 562.82 537.37 492.62 493.38 482.03 511.32 541.83 526.29 473.55 518.31
1023.3475 1705.8490 2388.3375 3070.8263 3753.2958 4435.7767 5118.2631 5800.7274 6483.1696 7165.6547 7848.1822 8530.6252
2.112 2.260 2.248 2.298 2.255 2.250 2.246 2.240 2.244 2.259 2.250 2.243
0.10 2.60 2.41 2.96 1.75 1.98 2.33 1.20 0.91 1.01 1.81 1.41
0.032 1.500 2.079 2.402 2.559 2.692 2.800 2.846 2.852 2.911 3.011 2.991
7.62 7.20 7.19 6.66 – – – – – – – –
phosphorus cluster containing more than 15 atoms, there are at least one pair of twofold coordinated atom adjacent to a fourfold coordinated atom; (3) the common building units are P7, P8, P9, and P10 and the former three are all wedge-like; (4) P2 dimer can serve as bridge between those subunits. þ þ þ At those medium-sized Pþ 13 , P15 , P21 , P25 clusters, our combined search by FPMD annealing and PEB/TNP optimization have successfully located the low-energy cluster structures that are energetically preferred than the previous structures from hand-made construction. The simultaneous appearance of the P7, P8, P9, and P10 building units as well as the C1 symmetry of the entire cluster geometry indicate that it is almost impossible to construct the lowest-energy structure of a medium-sized phosphorus cluster by simple intuition. Hence, an unbiased search based on first-principles method is needed to reveal the structural evolution behavior of the phosphorus clusters. Table 2 summarizes the highest vibrational frequencies, total energies, average P–P bond lengths, HOMO–LUMO gaps, binding energies of the odd-numbered cationic phosphorus clusters up to Pþ 25 , as well as the adiabatic ionization potentials of odd-sized neutral Pn clusters (up to n = 9). One can see that binding energy generally rises with increasing cluster size. Thus, the clusters can continue to gain energy during the growth process. The binding energy rise rapidly for the small Pþ n clusters (n = 3–11), while the increasing trend becomes relatively slow for n P 13. This can be related to the appearance of chain-like tubular structural motif þ around Pþ 13 . Except for P3 , the average P–P bond lengths for all phosphorus clusters are all around 2.25 Å, insensitive to cluster size. As one of the characteristic properties, the HOMO–LUMO gap reflects relative chemical stability of a cluster. Relatively larger gaps (more than 2 eV) are found at n = 5, 7, 9, and 15. Among all cationic clusters studies, Pþ 9 possesses the maximum gap of 2.96 eV. In addition to the cationic Pþ 2mþ1 clusters up to m = 12, we have also searched for the most stable geometries of small neutral P2m+1 clusters with m = 1–4 and computed the adiabatic ionization potentials of these clusters using the same TNP/PBE method. The lowest-energy structures of P2m+1 clusters from our calculations are consistent with previous findings, that is, P3 is an equilateral triangle [10], P5 is an edge-capped tetrahedron [10,23,25], P7 is formed by adding a twofold atom to the boat-shape P6 [23,25], P9 is derived from the cuneane structure of P8 by face-capping one atom [11]. The computed IP values listed in Table 2 are in line with the previous theoretical and experimental data [25,29]. For example, the present ionization potentials of P5, P7, and P9 are 7.20, 7.19, and 6.66 eV, respectively, comparable to the previous theoretical values of 7.23, 7.43, and 6.79 eV [25], respectively. The neutral Pn clusters of larger size (n > 9) have not been considered here since it requires an extensive search of the most stable cluster structures.
4. Conclusions To summarize, the lowest-energy structures of the odd-numbered cationic phosphorus clusters (up to 25 atoms) are determined by the combination of first-principles simulated annealing and PBE/TNP optimization. Benchmark calculations on small P2 and P4 clusters show the accuracy of the present PBE/TNP method. For smaller Pþ n clusters with n 6 11, our simulations reproduced the lowest-energy configurations in previous study. However, for Pþ n clusters with more than 11 atoms, we revealed a new family of structural growth motif based on P7, P8, P9, and P10 building units and P2 bridge. These medium-sized Pþ n clusters usually possess tubular chain-like configurations with no point-group symmetry and there is at least one pair of twofold coordinated and fourfold coordinated atoms in each cluster. The most stable strucþ þ þ tures obtained for Pþ 13 , P15 , P21 and P25 are energetically more preferred than those previously proposed ones. The present results demonstrate that an unbiased search incorporated with first-principles method is crucial to find out the most stable structures and to understand the structural evolution mechanism of the phosphorus clusters. Acknowledgements This work was supported by the Undergraduate Innovative Research Training Program (081014115) and the Program for New Century Excellent Talents in University of China (NCET-060281) provided by the Ministry of Education of China. References [1] J. Donohue, The Structures of the Elements, Wiley, New York, 1974. [2] T.P. Martin, Z. Phys. D 3 (1986) 211. [3] R.B. Huang, Z.Y. Liu, P. Zhang, Y.B. Zhu, F.C. Lin, J.H. Zhao, L.S. Zheng, Chin. J. Struct. Chem. 12 (1993) 180. [4] R.B. Huang et al., Int. J. Mass. Spect. Ion. Process. 151 (1995) 55. [5] R.B. Huang, H.D. Li, Z.Y. Lin, S.H. Yang, J. Phys. Chem. 99 (1995) 1418. [6] M.D. Chen, J.T. Li, R.B. Huang, L.S. Zheng, C.T. Au, Chem. Phys. Lett. 305 (1999) 439. [7] A.V. Bulgakov, O.F. Bobrenok, V.I. Kosyakov, Chem. Phys. Lett. 320 (2000) 19. [8] A.V. Bulgakov et al., Phys. Solid State 44 (2002) 617. [9] A.V. Bulgakov, O.F. Bobrenok, I. Ozerov, W. Marine, S. Giorgio, A. Lassesson, E.E.B. Campbell, Appl. Phys. A 79 (2004) 1369. [10] R.O. Jones, D. Hohl, J. Chem. Phys. 92 (1990) 6710. [11] R.O. Jones, G. Seifert, J. Chem. Phys. 96 (1992) 7564. [12] P. Ballone, R.O. Jones, J. Chem. Phys. 100 (1994) 4941. [13] E. Fluck, C.M.E. Pavlidou, R. Janoschek, Phos. Sulph. 6 (1979) 469. [14] R. Ahlrichs, S. Brode, C. Ehrhardt, J. Am. Chem. Soc. 107 (1985) 7260. [15] M. Häser, U. Schneide, R. Ahlrichs, J. Am. Chem. Soc. 114 (1992) 9551. [16] M. Häser, O. Treutler, J. Chem. Phys. 102 (1995) 3703. [17] G. Seifert, T. Heine, P.W. Fowler, Eur. Phys. J. D 16 (2001) 341. [18] J.G. Han, J.A. Morales, Chem. Phys. Lett. 396 (2004) 27. [19] M.K. Denk, A. Hezarkhani, Heteroat. Chem. 16 (2005) 453. [20] D. Wang, C.L. Xiao, W.G. Xu, J. Mol. Struct. (Theochem) 759 (2006) 225.
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Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Lowest-energy structures of cationic Pþ 2mþ1 (m = 1–12) clusters from first-principles simulated annealing Tao Xue a, Jing Luo a, Si Shen a, Fengyu Li a,b, Jijun Zhao a,b,* a b
School of Physics and Optoelectronic Technology, College of Advanced Science and Technology, Dalian University of Technology, Dalian 116024, China Laboratory of Materials Modification by Laser, Electron, and Ion Beams, Dalian University of Technology, Dalian 116024, China
a r t i c l e
i n f o
Article history: Received 14 August 2009 In final form 8 December 2009 Available online 11 December 2009
a b s t r a c t Lowest-energy structures of odd-sized cationic Pþ 2mþ1 (m = 1–12) clusters have been determined from first-principles simulated annealing followed by more accurate geometry optimization within the framework of density functional theory. For Pþ n with n > 11, the current global minimum structures are more favorable than those previously reported ones. A structural motif based on P7, P8, P9, and P10 building blocks and P2 bridge was revealed for the medium-sized Pþ n clusters. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Among elementary solids, phosphorus is quite unique since it exhibits a variety of structural phases [1] such as black (orthorhombic), violet (monoclinic), rhombohedral, metallic (cubic), red, white, and amorphous. Most structural phases of phosphorus solids can be viewed as superstructures built by small subunits like P4, P8, and P9 clusters. Thus, it is of fundamental interest to reveal the size-dependent growth mechanism of phosphorus clusters, which is also potentially useful for designing novel nanoscale materials and devices. Using laser vaporization technique, gas-phase phosphorus clusters have been generated in cluster beam and their size-dependent relative abundances have been recorded via time-of-flight (TOF) mass spectrometer [2–9]. In a pioneer work by Martin [2], cationic Pþ n clusters with up to n = 24 were obtained by quenching the vapour of red phosphorus in helium beam. Except at Pþ 11 , the relative abundances of Pþ n clusters oscillate with cluster size n and show the preference of the odd-sized cationic clusters (with even number of valence electrons). Later, Huang et al. [3–6] reported TOF mass spectra of cationic phosphorus clusters generated by direct laser vaporization of red phosphorus, and the maximum cluster size was extended to 89 atoms. In their mass spectra, except several þ þ þ even-numbered peaks of Pþ 4 ; P6 ; P10 and P14 , most odd-sized clusþ ters show strong densities. For larger Pn clusters (n P 25), the observed magic numbers can be written as n = 25 + 8k (k = 0, 1, 2, 3, 4, 5, 6, 7, 8) [3,6], suggesting that the most stable structures of larger phosphorus clusters may be built from P8 units. Bulgakov and co-workers [7–9] reported mass spectra of neutral, cationic, and
* Corresponding author. Address: School of Physics and Optoelectronic Technology, College of Advanced Science and Technology, Dalian University of Technology, Dalian 116024, China. Fax: +86 411 84706100. E-mail address: [email protected] (J. Zhao). 0009-2614/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2009.12.019
anionic phosphorus clusters in a wide size range, i.e., up to n = 40 for Pn, n = 91 for Pþ n , and n = 49 for Pn , respectively. In each case, the relative abundance show odd–even alternation and the clusters with even number of electrons are dominant in the mass spectra. For the cationic Pþ n clusters, the observed magic numbers are n = 7 and 21, while the most abundant clusters are P10 and P14 for the neutral species. So far, theoretical calculations on phosphorus clusters with first-principles methods mainly focused on the neutral species [10–21]. In a series of pioneering works, Jones and co-workers performed simulated annealing with density functional theory (DFT) to explore the ground state structures of phosphorus clusters up to 11 atoms [10–12]. According to their simulation, P8, which was previously assumed to be cubic [13,14], turns out to adopt a wedge-like configuration in analogy with the (CH)8 cuneane [10]. Häser et al. carried out ab initio MP2 calculations on even-sized Pn clusters with up to n = 28 and discussed the thermodynamic stabilities (with regard to P4) and electronic properties of these clusters [15,16]. Fullerene-like cage configurations have been considered for Pn cluster with n P 14, but these hollow-cage clusters were usually found to be unstable upon dissociation into small P4 clusters [17,18]. Recently, small phosphorus clusters containing two to eight atoms have been investigated by high-level quantum chemistry methods [19–21]. Compared to the extensive studies of the neutral phosphorus clusters, much less is known about the charged clusters, particularly, the cationic ones, which were most frequently produced experimentally. In Jones’ simulated annealing study [11], the structures of cationic Pþ n clusters up to n = 11 were determined. Feng et al. attempted to predict the most stable structures of cationic Pþ 2mþ1 clusters (m = 3, 4, 5, 6) by ab initio Hartree–Fock calculations [22]. Chen et al. [23,24] compared large number of structural isoþ þ mers for small Pþ 5 ; P7 ; and P9 clusters using DFT calculations. Later, Guo et al. investigated the geometries and electronic properties
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of neutral and charged Pn clusters up to n = 15 using DFT calculations with the B3LYP/6-311G(d) method [25]. For the larger cationic Pþ n with n = 25 + 8k (k = 1, 2, 3, 4, 5, 6, 7, 8), Chen et al. constructed a series of chain-like configurations by adding cuneane P8 units to a presumed Cs structure of Pþ 25 cluster [6]. Despite the previous efforts, our knowledge on the structural growth behavior of phosphorus clusters is still insufficient since previous theoretical studies were either focused on small clusters (i.e., less than 15 atoms) or considered only limited number of presumed configurations. For the medium-sized clusters with more than 10 atoms, the potential energy surfaces (PES) would become rather complicated. An unbiased global search is essential for locating the lowest-energy structures. Within the framework of DFT, in this Letter we performed simulated annealing study to determine the lowest-energy structures of odd-numbered cationic phosphorus clusters (up to 25 atoms). For the medium-sized Pþ n clusters with n > 11, we revealed new lowest-energy configurations that are more energetically favorable than the previously proposed structures. The size-dependent evolution of cluster geometry, binding energy, and HOMO–LUMO gap will be discussed.
experimental data within the best of our knowledge. As an alternative, we performed high-level ab initio calculations using the CCSD/ aug-cc-PVDZ method implemented in the GAUSSIAN 03 package [29], which is rather accurate in predicting the structures and bonding of small clusters. In Table 1, the geometry parameters, binding energies, vibrational frequencies, and ionization potentials from our theoretical calculations are compared with the experimental data for P2 and P4 [21,30] and the CCSD/aug-cc-PVDZ results for þ Pþ 3 and P5 . The theoretical P–P bond lengths of P2 and P4 are very close to the measured ones. Our PBE/TNP calculations overestimated the cluster binding energies by less than 5%. The computed vibrational frequencies agree well with the experimental data with an average deviation of 1.6%. The theoretical prediction of ionizaþ tion potentials is also satisfactory (see Table 1). For Pþ 3 and P5 clusters, the geometry parameters and vibrational frequencies from PBE/TNP calculations agree satisfactorily with the CCSD/aug-ccPVDZ results, whereas the theoretical binding energies by PBE/ TNP are systematically lower than the CCSD/aug-cc-PVDZ values. Overall speaking, the reasonable agreements between our PBE/ TNP calculations and the experimental data as well as the high-level CCSD/aug-cc-PVDZ results demonstrate that our computational method is reliable.
2. Theoretical methods
3. Results and discussion
In this work, the potential energy surfaces of the cationic phosphorus clusters were first explored by unbiased search using the simulated annealing (SA) technique incorporated with first principles molecular dynamics (FPMD), as implemented in the DMol3 code [26] (built in the Material Studio 4.3 package provided by Accelrys Inc.) The DMol3 code allows constant-temperature molecular dynamic simulations, in which the interatomic forces are obtained from first-principles calculations within the all-electron density functional theory. In the FPMD simulations, the local density approximation (LDA) was used to describe the exchange-correlation interaction, and the double numerical plus p and d polarization (DNP) basis set [26] was adopted. The Harris functional approximation [27] was employed to solve for the total energy and atomic forces in a non-self-consistent-field (non-SCF) way. The initial structures for simulated annealing were built randomly, and the total simulation time (1.3 ns) would be long enough to forget the original configurations. For each size, the cluster was independently annealed from a high starting temperature of 1500 K (about the melting point of phosphorus solids) down to 300 K via FPMD. The temperature was systematically reduced by increments of 100 K. At each temperature, MD simulation in a NVT ensemble lasts for 100 ps with a time step of 1 fs (i.e. 100 000 MD steps at each temperature). The structures of the cationic phosphorus clusters from simulated annealing were further optimized using the same DMol3 code. A triple numerical basis set including d and p polarization functions (TNP) was chosen. The exchange-correlation interaction was treated within the generalized gradient approximation (GGA) with the Perdew–Burke–Enzerhof (PBE) parameterization [28]. SCF calculations were done with convergence criterion of 106 Hartree on the total energy. All the structures were fully optimized without any symmetry constraint with a convergence criterion of 0.002 Hartree/Å for the forces and 0.005 Å for the displacement. Vibrational analysis of these Pþ n clusters in their equilibrium configurations has been carried out to ensure that there are no imaginary frequencies corresponding to the saddle points on the PES. The accuracy of the present PBE/TNP computational scheme has þ been assessed by benchmark calculations on small P2, Pþ 3 , P4 and P5 þ þ clusters. For the small cationic P3 and P5 clusters, there was no
For each cationic phosphorus cluster, there is no imaginary frequency in the equilibrium structures obtained from the combinational search of FPMD simulated annealing and local PBE/TNP minimization. Thus, the final structures of Pþ 2mþ1 (m = 1–12) clusters displayed in Fig. 1 are all true minima on the potential energy surface instead of the saddle points. þ þ þ þ 3.1. Pþ 3 , P 5 , P 7 , P 9 and P 11 þ þ þ þ For small cationic phosphorus clusters (Pþ 3 , P5 , P7 , P9 , and P11 ) their lowest-energy geometries have been determined by Guo et al. using density functional calculations with B3LYP method [25]. Our present calculations have successfully reproduced these most stable configurations for small Pþ 2mþ1 (m = 1–5) clusters, thus we will only discuss briefly the theoretical results. þ The equilibrium geometries of Pþ 3 and P5 are an equilateral triangle with D3h symmetry and a square pyramid with C4v symmetry, respectively. In analogy to the well-known cuneane P8 unit [6,10,24,25], the wedge-like configuration of Pþ 7 (C2v symmetry) can be viewed as a square-face-capped triangle prism with two broken bonds. The lowest-energy geometry of Pþ 9 (D2d symmetry) can be built from two P4 tetrahedra aligned vertically to each other
Table 1 P–P distance (r0), binding energy (Eb), vibrational frequencies (x0), and ionization þ potential (IP) for neutral P2, P4 and cationic Pþ 3 , P5 clusters from our PBE/TNP and CCSD/aug-cc-PVDZ calculations as well as experiments [21,29]. r0 (Å)
Eb (eV)
x0 (cm1)
IP (eV)
788.5 780.8
10.45 10.6 ± 0.1
P2 PBE/TNP Experiment
1.902 1.893
P4 PBE/TNP Experiment
2.207 2.21 ± 0.02
13.18 12.53
371.6, 459.7, 594.1 360.8, 466.3, 600.5
Pþ 3 PBE/TNP CCSD/aug-cc-PVDZ
2.11 2.14
11.74 13.31
474.0, 623.5 474.2, 650.9
– –
Pþ 5 PBE/TNP CCSD/aug-cc-PVDZ
2.26 2.30
19.15 22.40
457.9, 501.9, 509.6 471.4, 503.1, 523.5
– –
5.27 5.08
9.56 9.28 ± 0.1
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Fig. 2. Isomer structures of Pþ n (n = 13, 15, 21, 25) reported in literature [6,18,22,25]. DE is the energy difference between this isomer and our lowest-energy structure. Fig. 1. Lowest-energy configurations of odd-sized Pþ n (n = 3–25) clusters.
with a fourfold coordinated joint atom in between, which has been previously found by Guo et al. [25] and Chen et al. [23]. The most stable structure of Pþ 11 can be viewed as an eight-atom cuneane unit and a six-atom trigonal prism sharing three P atoms on a triangle face. It is noteworthy that the wedge-like P7 and P8 are the common building blocks for larger phosphorus clusters, as we will show in the following discussions. Starting from Pþ 5 , most atoms in the phosphorus clusters are threefold coordinated whereas there is one atom with fourfold coordination. The dominant threefold coordination can be understood by the electron configuration of phosphorus atoms, that is, each P atom contributes three electrons to form three covalent bonds with the neighboring P atoms, leaving two lone-pair electrons. þ 3.2. Pþ 13 and P 15 þ The structural evolution behavior is rather clear from Pþ 11 to P15 , can be obtained as shown in Fig. 1. The most stable structure of Pþ 13 by adding two atoms to the waist (the sharing triangle) of Pþ 11 . Inserting further two additional atoms between the waist and the triangle end of the Pþ 13 results in the most stable structure of Pþ 15 , which can be considered as a P8 cuneane and a P10 polyhedron sharing three atoms on a triangle. For the cationic phosphorus clusters with more than 11 atoms, the global minimum structures from unbiased FPMD annealing search have no point-group symmetry. Previously, symmetric configurations were proposed for Pþ 13 [22,25] and Pþ 15 [25] (both with Cs symmetry), which are depicted in Fig. 2. At the same PBE/TNP level of theory, the present lowest-energy structures (shown in Fig. 1) are more energetically þ favorable than the previous proposed ones for Pþ 13 and P15 by 1.4 and 1.2 eV, respectively. This indicates that an unbiased search is essential for revealing the lowest-energy structures of the medium-sized phosphorus clusters. The most stable structures of the odd-numbered cationic phosþ phorus clusters from Pþ 5 to P15 all satisfy the rule that there is only one four-coordinated atom as the key link point of the entire frame and each of the rest atoms connect with three other atoms. However, as cluster size further increases, we observed the emergence of two-coordinated atoms and even a five-coordinated atom (in Pþ 21 ). The descriptions for the lowest-energy structures of the larger þ þ þ þ Pþ 17 , P19 , P21 , P23 and P25 clusters are presented below. þ þ þ þ 3.3. Pþ 17 , P 19 , P 21 , P 23 and P 25
The equilibrium geometry of Pþ 17 has one pair of adjacent atoms, one with fourfold coordination and another with twofold coordina-
tion; the rest atoms are all threefold coordinated. This structure can be decomposed into two P8 subunits connected by the central two-coordinated atom. In the equilibrium configuration of Pþ 19 , there are three fourcoordinated atoms and one two-coordinated atom. One of the three four-coordinated atoms is adjacent to the two-coordinated atom, while the other two are connected with each other to form the waist of the tubular structure. The entire structure of Pþ 19 can be viewed as a wedge-like P7 subunit connected with a P10 subunit (twisted pentagonal prism) by a P2 bridge. Unlike the other medium-sized cationic Pþ n clusters (n = 11–25) with straight chain-like geometries, the lowest-energy structure of Pþ 21 is a curved tubular chain. Besides the regular threefold coordination, one twofold, two fourfold, and one fivefold coordinated atoms were found. The two-coordinated atom still pairs up with one four-coordinated atom. Both the two fourfold atoms are bonded with the fivefold atom. Previously, Bulgakov et al. proposed two possible structures for Pþ 21 [18], i.e., an edge-capped hollow cage with C2v symmetry and a chain-like tubular structure (P4 + P8 + P9) with Cs symmetry (see Fig. 2). Compared to the lowest-energy structure from our calculations, the C2v cage isomer is energetically less preferred by 1.75 eV, and the chain-like Cs isomer is 0.33 eV higher in energy. The most stable structure of Pþ 23 has one four-coordinated atom. The rest atoms are all with threefold coordination. Overall speaking, it has a quasi-linear tubular shape. Careful examination reveals that it consists of three subunits and one P2 bridge. The right side is a P6 unit formed by adding two P atoms on an open-edged P4 tetragon. On the left side, there are two subunits, a wedge-like P8 and a twisted pentagonal prism of P10, connected by sharing a P3 triangle. The left and right parts of the cluster are linked together by a P2 bridge. In the lowest-energy structure of Pþ 25 , there are two pairs of twofold coordinated atom and fourfold coordinated atom, and an additional four-coordinated atom. These two pairs of two-coordinated and four-coordinated atoms have very similar bonding environment and align symmetrically in the cluster. On each side of the pair, there are a cuneane P8 and a P10 subunit, respectively. Between them, there is a pentagonal P5 ring. In the previous study by Chen et al. [19], a chain-like Cs configuration constructed by P8 + P8 + P9 was proposed for Pþ 25 (see Fig. 2). Upon relaxation at the same PBE/TNP level, this structure is less favorable in energy than the present lowest-energy structure by 0.3 eV. From the above discussions, we can figure out several general rules for the structural motif of the cationic phosphorus clusters: (1) most medium-sized clusters exhibit tubular chain-like configurations with low symmetry or even without symmetry; (2) for a
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Table 2 Symmetries, highest vibrational frequencies, total energies, average P–P bond lengths, HOMO–LUMO gaps, binding energies (Eb) of odd-numbered cationic phosphorus Pþ n clusters (up to n = 25) as well as the adiabatic ionization potentials (IP) of odd-sized neutral Pn clusters (up to n = 9). n
Symmetry
Highest frequency (cm1)
Total energy (Hartree)
Average bond length (Å)
Gap (eV)
Eb (eV/atom)
IP (eV)
3 5 7 9 11 13 15 17 19 21 23 25
D3h C4v C2v D2d Cs C1 C1 C1 C1 C1 C1 C1
632.51 509.57 562.82 537.37 492.62 493.38 482.03 511.32 541.83 526.29 473.55 518.31
1023.3475 1705.8490 2388.3375 3070.8263 3753.2958 4435.7767 5118.2631 5800.7274 6483.1696 7165.6547 7848.1822 8530.6252
2.112 2.260 2.248 2.298 2.255 2.250 2.246 2.240 2.244 2.259 2.250 2.243
0.10 2.60 2.41 2.96 1.75 1.98 2.33 1.20 0.91 1.01 1.81 1.41
0.032 1.500 2.079 2.402 2.559 2.692 2.800 2.846 2.852 2.911 3.011 2.991
7.62 7.20 7.19 6.66 – – – – – – – –
phosphorus cluster containing more than 15 atoms, there are at least one pair of twofold coordinated atom adjacent to a fourfold coordinated atom; (3) the common building units are P7, P8, P9, and P10 and the former three are all wedge-like; (4) P2 dimer can serve as bridge between those subunits. þ þ þ At those medium-sized Pþ 13 , P15 , P21 , P25 clusters, our combined search by FPMD annealing and PEB/TNP optimization have successfully located the low-energy cluster structures that are energetically preferred than the previous structures from hand-made construction. The simultaneous appearance of the P7, P8, P9, and P10 building units as well as the C1 symmetry of the entire cluster geometry indicate that it is almost impossible to construct the lowest-energy structure of a medium-sized phosphorus cluster by simple intuition. Hence, an unbiased search based on first-principles method is needed to reveal the structural evolution behavior of the phosphorus clusters. Table 2 summarizes the highest vibrational frequencies, total energies, average P–P bond lengths, HOMO–LUMO gaps, binding energies of the odd-numbered cationic phosphorus clusters up to Pþ 25 , as well as the adiabatic ionization potentials of odd-sized neutral Pn clusters (up to n = 9). One can see that binding energy generally rises with increasing cluster size. Thus, the clusters can continue to gain energy during the growth process. The binding energy rise rapidly for the small Pþ n clusters (n = 3–11), while the increasing trend becomes relatively slow for n P 13. This can be related to the appearance of chain-like tubular structural motif þ around Pþ 13 . Except for P3 , the average P–P bond lengths for all phosphorus clusters are all around 2.25 Å, insensitive to cluster size. As one of the characteristic properties, the HOMO–LUMO gap reflects relative chemical stability of a cluster. Relatively larger gaps (more than 2 eV) are found at n = 5, 7, 9, and 15. Among all cationic clusters studies, Pþ 9 possesses the maximum gap of 2.96 eV. In addition to the cationic Pþ 2mþ1 clusters up to m = 12, we have also searched for the most stable geometries of small neutral P2m+1 clusters with m = 1–4 and computed the adiabatic ionization potentials of these clusters using the same TNP/PBE method. The lowest-energy structures of P2m+1 clusters from our calculations are consistent with previous findings, that is, P3 is an equilateral triangle [10], P5 is an edge-capped tetrahedron [10,23,25], P7 is formed by adding a twofold atom to the boat-shape P6 [23,25], P9 is derived from the cuneane structure of P8 by face-capping one atom [11]. The computed IP values listed in Table 2 are in line with the previous theoretical and experimental data [25,29]. For example, the present ionization potentials of P5, P7, and P9 are 7.20, 7.19, and 6.66 eV, respectively, comparable to the previous theoretical values of 7.23, 7.43, and 6.79 eV [25], respectively. The neutral Pn clusters of larger size (n > 9) have not been considered here since it requires an extensive search of the most stable cluster structures.
4. Conclusions To summarize, the lowest-energy structures of the odd-numbered cationic phosphorus clusters (up to 25 atoms) are determined by the combination of first-principles simulated annealing and PBE/TNP optimization. Benchmark calculations on small P2 and P4 clusters show the accuracy of the present PBE/TNP method. For smaller Pþ n clusters with n 6 11, our simulations reproduced the lowest-energy configurations in previous study. However, for Pþ n clusters with more than 11 atoms, we revealed a new family of structural growth motif based on P7, P8, P9, and P10 building units and P2 bridge. These medium-sized Pþ n clusters usually possess tubular chain-like configurations with no point-group symmetry and there is at least one pair of twofold coordinated and fourfold coordinated atoms in each cluster. The most stable strucþ þ þ tures obtained for Pþ 13 , P15 , P21 and P25 are energetically more preferred than those previously proposed ones. The present results demonstrate that an unbiased search incorporated with first-principles method is crucial to find out the most stable structures and to understand the structural evolution mechanism of the phosphorus clusters. Acknowledgements This work was supported by the Undergraduate Innovative Research Training Program (081014115) and the Program for New Century Excellent Talents in University of China (NCET-060281) provided by the Ministry of Education of China. References [1] J. Donohue, The Structures of the Elements, Wiley, New York, 1974. [2] T.P. Martin, Z. Phys. D 3 (1986) 211. [3] R.B. Huang, Z.Y. Liu, P. Zhang, Y.B. Zhu, F.C. Lin, J.H. Zhao, L.S. Zheng, Chin. J. Struct. Chem. 12 (1993) 180. [4] R.B. Huang et al., Int. J. Mass. Spect. Ion. Process. 151 (1995) 55. [5] R.B. Huang, H.D. Li, Z.Y. Lin, S.H. Yang, J. Phys. Chem. 99 (1995) 1418. [6] M.D. Chen, J.T. Li, R.B. Huang, L.S. Zheng, C.T. Au, Chem. Phys. Lett. 305 (1999) 439. [7] A.V. Bulgakov, O.F. Bobrenok, V.I. Kosyakov, Chem. Phys. Lett. 320 (2000) 19. [8] A.V. Bulgakov et al., Phys. Solid State 44 (2002) 617. [9] A.V. Bulgakov, O.F. Bobrenok, I. Ozerov, W. Marine, S. Giorgio, A. Lassesson, E.E.B. Campbell, Appl. Phys. A 79 (2004) 1369. [10] R.O. Jones, D. Hohl, J. Chem. Phys. 92 (1990) 6710. [11] R.O. Jones, G. Seifert, J. Chem. Phys. 96 (1992) 7564. [12] P. Ballone, R.O. Jones, J. Chem. Phys. 100 (1994) 4941. [13] E. Fluck, C.M.E. Pavlidou, R. Janoschek, Phos. Sulph. 6 (1979) 469. [14] R. Ahlrichs, S. Brode, C. Ehrhardt, J. Am. Chem. Soc. 107 (1985) 7260. [15] M. Häser, U. Schneide, R. Ahlrichs, J. Am. Chem. Soc. 114 (1992) 9551. [16] M. Häser, O. Treutler, J. Chem. Phys. 102 (1995) 3703. [17] G. Seifert, T. Heine, P.W. Fowler, Eur. Phys. J. D 16 (2001) 341. [18] J.G. Han, J.A. Morales, Chem. Phys. Lett. 396 (2004) 27. [19] M.K. Denk, A. Hezarkhani, Heteroat. Chem. 16 (2005) 453. [20] D. Wang, C.L. Xiao, W.G. Xu, J. Mol. Struct. (Theochem) 759 (2006) 225.
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