Charge transfer mechanism of SERS for metal–molecule–metal junction supported by graphene and boron-doped graphene

Abstract

The Raman and absorption spectra of a Ag₂–PATP–Au₂ junction adsorbed on graphene and boron-doped graphene were investigated by using density functional theory (DFT) and time-dependent DFT methods. The interactions between the graphene and junction result in charge transfer (CT) from the graphene to the junction due to their different work functions. This CT leads to charge redistribution on the junction, and then the changes of static polarizabilities, which directly influence the enhancement of normal Raman spectra. The absorption spectra show that the graphene and boron-doped graphene induce some CT excited states in the visible and infrared regions. When the energy of incident light is close to the energy of these CT excited states, these electronic transitions will be excited, which leads to the enhancement of pre-resonant Raman scattering (pre-RRS) spectra. In pre-RRS spectra, the B-doped model has stronger Raman intensities, since it produces more CT excited states with intense oscillator strength near the incident light than the graphene model. The non-totally symmetric modes (b₂) are strongly enhanced as well as the totally symmetric modes (a₁), indicating the contribution of Herzberg–Teller (HT) scattering. The charge difference densities (CDDs) method was employed to directly visualize the CT from the graphene sheet to the molecule.

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Introduction

Since it was discovered in 1974, surface enhanced Raman scattering (SERS) has attracted much attention, due to its high sensitivity in detecting the structure of materials.^1–5 Compared to the normal Raman scattering, SERS can enhance normally weak Raman signals by several orders of magnitude, and has evolved into a powerful and reliable analytical tool for the ultra-sensitive detection of analytes even at the single-molecule level.^6–9 Numerous experimental and theoretical works have focused on the enhancement mechanism.^10,11 In general, two mechanisms are widely accepted, which are electromagnetic mechanism (EM) and chemical mechanism (CM), respectively. The former arises from the surface plasmon resonance of the metal substrates² and the latter is caused by the change in the electronic structure of molecule adsorbed on metal surface, companying by charge transfer from metal to molecule or vice versa.^12,13

Conventional SERS substrates were based on the roughness surface of a noble metal such as Ag, Au, Cu, and so on, which take advantage of the enhancement of EM. Usually, the relatively weak CM and EM coexist with EM dominant. Therefore, it is difficult to distinguish CM from EM to investigate the precise enhancement mechanism. To figure out this issue, several substrates have been used to separate CM from EM for further study of CM, such as metal substrate with flat surface,³ gold nanoparticle-coated substrate,¹⁴ and graphene substrate.¹⁵ Graphene, a single sheet of carbon atoms in a 2D honeycomb crystal structure, has attracted large interests from scientists due to its fascinating electronic properties.^16,17 Besides, graphene also exhibits amazing Raman scattering properties. Xie et al.¹⁸ firstly observed a fluorescence quenching effect of fluorescent dyes adsorbed on graphene, and obtained their Raman signals. Recently, Ling et al.¹⁵ demonstrated that graphene can be used as a substrate for Raman enhancement. Moreover, the surface plasmon on graphene is in the range of terahertz rather than in the visible range, therefore graphene does not support EM in SERS. The absence of EM simplifies the investigation of the CM in SERS, but simultaneously, it causes a considerable loss of sensitivity. Huang and coworkers¹⁹ combined graphene with conventional metallic SERS substrates, achieving a higher sensitivity of SERS detection. The combination of Au or Ag nanoparticles (NPs) and graphene sheets becomes a rising star in SERS especially for the detection of low concentration molecules.^20–22 By decorating Au or Ag NPs on graphene sheets, the aggregation of the NPs on graphene sheet and the strong electronic interactions between Au or Ag NPs and the graphene cause a coupled EM, which is considered to be responsible for the significantly enhanced Raman signal of the analytes. Kong et al.^23,24 investigated the Raman enhancement of pyridine adsorbed on graphene, by using density functional theory (DFT), to explore the influence of defects in SERS. Their results revealed that the graphene effect on enhancement of SERS could be improved by doping boron and nitrogen atoms. Nevertheless, the picture about the enhancement mechanism in this system is still incomplete. Further theoretical researches are necessary.

In this work, we systematically investigated the CM in SERS with graphene and boron-doped (B-doped) graphene as substrate that supports the metal–molecule–metal junction using DFT method. Concerning CM, Sun et al.'s work²⁵ had visualized the charge transfer in molecule–metal complex and metal–molecule–metal junction by employing the charge difference densities (CDDs) method and given the direct evidence of CM in SERS. Herein, we combined the metal–molecule–metal junction and graphene to detect the SERS properties of junctions supported by graphene. The PATP molecule was taken as the probe molecule to study the Raman properties. CDDs method was used to observe the charge transfer from graphene to junction under incident light.

Method

All the quantum chemical calculations were performed by the Gaussian 09 suite.²⁶ The model of graphene discussed in this work consists of a 4 × 4 graphene supercell (32C atoms), terminated with hydrogen atoms at the edges. The ground state geometries of Ag₂–PATP–Au₂ junction adsorbed on graphene and B-doped graphene were optimized using DFT method²⁷ with BP86 functional.^28,29 The basis sets for carbon, nitrogen, sulphur, boron and hydrogen atoms were 6-311 g++(d, p). For the Au atoms, the valence electron and the internal shells were described by the basis functions of LANL2DZ, which is the corresponding relativity effective core potentials.³⁰ The frequency analysis was carried out with the same functional and basis sets. There were no imaginary frequencies through the frequency analysis, ensuring that all of the structures were stable. Their normal Raman scattering (NRS) spectra and pre-resonant Raman scattering (pre-RRS) spectra were calculated with the same functional and basis set at the optimized ground state geometry.

Absolute off and on resonance Raman intensities can be calculated as the differential Raman scattering cross section. For Stokes scattering with an experimental setup of a 90° scattering angle and perpendicular planar-polarized light, the cross section is written as³¹

The ω_in and ω_p are the frequency of the incident light and of the pth vibrational mode, respectively, and S_p is the Raman scattering factor (or Raman activity in unit Å⁴ amu⁻¹),

which is a pure molecular property and independent of experimental setup. α_p and γ_p are the isotropic and anisotropic polarizabilities. In this paper, S_p was directly calculated by Gaussian 09 suite.

The calculations of absorption transition energies and oscillator strengths in absorption were performed by time-dependent DFT (TDDFT) method at the same basis set as geometry optimization, with LC-BP86 functional, which considers the long-range corrected effect.^32–34 CDDs method^35,36 was employed to visualize the orientation of CT for the electronic resonance transitions of Ag₂–PATP–Au₂ junction on graphene sheet.

Results and discussion

1. Geometry and ground state properties

PATP, as a SERS probe, is usually used in SERS studies. It is a typical bi-functional molecule with a π-conjugated benzene ring linked by an electron-donor and an electron-accepter group on each side. Zhou et al.³⁷ fabricated a metal–molecule–metal nano-system by the self-assembly of gold and silver nanoparticles interconnected with PATP molecules. This metal–molecule–metal nanosystem was also theoretically investigated to explore the Herzberg–Teller contribution in our previous work.³⁸ In this work, Ag₂–PATP–Au₂ junction was taken for calculation. For graphene model, two configurations of the junction adsorbed on graphene sheet with Ag₂ or Au₂ cluster close to graphene surface are denoted as G–Ag₂–PATP–Au₂ and Ag₂–PATP–Au₂–G, respectively. For the B-doped graphene model, they are respectively denoted as BG–Ag₂–PATP–Au₂ and Ag₂–PATP–Au₂–BG. Four interaction configurations were adopted with respect to different adsorption sites. Fig. 1 shows the optimized geometries of the junctions supported by graphene sheet and B-substituted graphene sheet. It is noticeable that the junction stands on the graphene sheet with the axes of Ag₂ or Au₂ cluster (near the graphene sheet) almost perpendicular to the surface of graphene, while they slightly tilted on the B-doped graphene sheet. The perpendicular configurations and tilted configurations for graphene and B-doped graphene, respectively, are consistent with the geometries in Kong et al.'s work.²⁴ The former geometry may arises from the metal-on-top stabilization mechanism, but the latter geometry is attributed to the orbital overlapping between the half-occupied s orbital of the Ag or Au atom and the unoccupied p_z orbital induced by the doped B atom.

Fig. 1 Optimized geometries of (a) isolated Ag₂–PATP–Au₂ junction, (b) graphene, (c) B-doped graphene, (d) G–Ag₂–PATP–Au₂, (e) Ag₂–PATP–Au₂–G, (f) BG–Ag₂–PATP–Au₂, and (g) Ag₂–PATP–Au₂–BG.

Table 1 illustrates the bond distance of Ag–N and Au–S in junction and junction supported by graphene. It is noticeable that these bond distances are almost the same as those in the isolated junction, which means the graphene substrate does not change the structure of junction. To investigate the interaction between the junction and graphene sheet, the separation distances (the shortest bond distance between metal atom and graphene's atoms) were calculated. For the B-doped model, the separation distances are shorter than those in graphene model, indicating a stronger adsorption interaction at the interface.

Table 1 The bond distance in isolated Ag₂–PATP–Au₂ junction, G–Ag₂–PATP–Au₂, Ag₂–PATP–Au₂–G, BG–Ag₂–PATP–Au₂, and Ag₂–PATP–Au₂–BG in ground state, respectively

	d (Ag–N)/Å	d (Au–S)/Å	d (junction–graphene)/Å
Ag2–PATP–Au2	2.33	2.43
G–Ag2–PATP–Au2	2.34	2.43	2.66
Ag2–PATP–Au2–G	2.36	2.45	2.73
BG–Ag2–PATP–Au2	2.34	2.43	2.50
Ag2–PATP–Au2–BG	2.36	2.43	2.46

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Mulliken charge distributions on junction and graphene sheet are analyzed in Fig. 2(a). When the metal atom in junction is close to graphene sheet, charge transfer between junction and graphene sheet occurs due to the interaction between junction and graphene sheet. There are rich delocalized π electrons on graphene sheet and free electrons on metal cluster, thus the electron will be transferred from graphene to metal cluster in junction or vice versa by the driving force arising from their different work functions.³⁹ This charge transfer leads to the charge redistributed in Ag₂–PATP–Au₂ junction, which results in the changes of electronic structure. Besides, the amount of transferred electrons is sensitive to the adsorption sites. For the graphene model, the electrons transferred from graphene to junction in Ag₂–PATP–Au₂–G (0.0390 a.u.) are more than those in G–Ag₂–PATP–Au₂ (0.0243 a.u.). This could be interpreted by the diversity in their work function difference. For the B-doped model, the electrons transferred in BG–Ag₂–PATP–Au₂ (0.0076 a.u.) and Ag₂–PATP–Au₂–BG (0.0274 a.u.) were reduced, comparing with those in graphene model, which could be attributed to the doping B atom. The doping B atom in graphene causes the Fermi level of graphene lowered, reducing the work function difference between graphene and metal. As a result, the amount of transferred electrons decreases.

Fig. 2 Mulliken charge distributions on junction and graphene.

To further study the charge redistribution on junction, the Mulliken charge distributions on the metal atoms are illustrated in Fig. 2(b) and (c). The metal atoms closing to the graphene surface are denoted as Ag1, Au1 while Ag2, Au2 represent the other atom far from the graphene surface. Compared to the isolated junction, the electrons were redistributed on the metal clusters, when the graphene sheet is introduced in this system. In the isolated junction, electrons distribute on Ag1 atom and less positive charge distribute on Ag2 atom. In the case of Ag1 atom close to graphene surface (Fig. 2(b)), on Ag1 atom distribute positive charge while on Ag2 atom distribute electrons, indicating electrons are transferred from bottom atom (near graphene) to top atom (near PATP molecule). In the case of Au1 close to graphene surface (Fig. 2(c)), the amount of electrons on Au1 atom decreases whereas it increases on Au2 atom, implying more electrons are gathered on top atom.

2. Normal Raman spectra

Herein, we pay attention to the normal Raman spectrum for the four complexes. Fig. 3 displays the calculated normal Raman spectra. According to Wilson, Jr's notation,⁴⁰ the vibrational modes of benzene ring are classified as in-plane modes a₁, b₂ and out-plane modes a₂, b₁. All these vibrational modes possess Raman activity. As exhibited in Table 2 and Fig. 3, the calculated Raman wavenumber positions agree well with the experimental data⁴¹ and shift slightly as the adsorption sites alter. However, the Raman intensities are significantly enhanced with different enhancement factors. For G–Ag₂–PATP–Au₂ and BG–Ag₂–PATP–Au₂, the Raman intensities are in the order of 10² to 10³ while they are in the order of 10³ to 10⁴ for Ag₂–PATP–Au₂–G and Ag₂–PATP–Au₂–BG. This agrees well with the amount of transferred electrons between graphene and junction, which modified the ground state properties including the static polarizabilities. As shown in Table 3, the static polarizabilities vary along with the changing of the adsorption sites, which directly influence the ground state chemical enhancement. Therefore, the enhancement in NRS spectra could be attributed to the ground state chemical enhancement.

Fig. 3 Calculated normal Raman spectra of (a) G–Ag₂–PATP–Au₂, (b) Ag₂–PATP–Au₂–G, (c) BG–Ag₂–PATP–Au₂, and (d) Ag₂–PATP–Au₂–BG.

Table 2 Calculated vibrational frequencies of four complexes

Species	G–Ag2–PATP–Au2	Ag2–PATP–Au2–G	BG–Ag2–PATP–Au2	Ag2–PATP–Au2–BG	Expt^a
a Experimental assignment of SERS spectra of PATP from ref. 41.
6a (a1)	1008	1008	1005	1008	—
7a (a1)	1055	1055	1053	1055	1074
19b (b2)	1143	1143	1145	1078	1143
9a (a1)	1187	1185	1191	1187	1196
14b (b2)	1347	1346	1357	1350	1396
3 (b2)	1417	1412	1425	1421	1442
19b (b2)	1476	1475	1475	1477	1476
8a (a1)	1578	1578	1573	1576	1580

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Table 3 The calculated static polarizabilities (in a.u.) of isolated Ag₂–PATP–Au₂ junction, G–Ag₂–PATP–Au₂, Ag₂–PATP–Au₂–G, BG–Ag₂–PATP–Au₂, and Ag₂–PATP–Au₂–BG in ground state, respectively

	αxx	αyy	αzz
Ag2–PATP–Au2	1134.5	304.8	195.1
G–Ag2–PATP–Au2	1728.7	1244.9	564.6
Ag2–PATP–Au2–G	2470.3	1488.1	708.9
BG–Ag2–PATP–Au2	2824.3	1232.8	588.3
Ag2–PATP–Au2–BG	2592.9	1436.5	595.9

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3. Excited states properties

To investigate the excited states properties of the four complexes, we calculated the UV-visible absorption spectra using TD-DFT method. As presented in Fig. 4, the profiles of absorption spectra for these four complexes differ from each other, resulting from the different adsorption sites. Previous studies demonstrated that the absorption spectrum of isolate PATP molecule is in the deep UV region. The interaction between the molecule in junction and graphene sheet produces some new excited states in the visible and infrared region. The first excited state appears at 954, 1024, 1531, and 1004 nm for G–Ag₂–PATP–Au₂, Ag₂–PATP–Au₂–G, BG–Ag₂–PATP–Au₂ and Ag₂–PATP–Au₂–BG, respectively. The inset shows the charge difference densities for the corresponding excited states, which were calculated by using CDDs method. The green and red represent holes and electrons, respectively. These insets present the process of electron transition under the incident light. For the first state of G–Ag₂–PATP–Au₂, both the holes and the electrons are localized on the graphene sheet, which means the S₁ state is an intra-substrate charge redistribution state. In the case of Ag₂–PATP–Au₂–G, the holes are localized on graphene sheet while the electrons are localized on Au₂ cluster, indicating that the electrons transfer from graphene sheet to Au₂ cluster. For Ag₂–PATP–Au₂–BG, the holes and electrons are localized on graphene sheet and Au₂ cluster. In brief, the S₁ states of the three complexes mentioned above are intra-substrate charge redistribution states, although the charge redistributes in different form. Nevertheless, the S₁ state of BG–Ag₂–PATP–Au₂ presents a distinctive picture of charge difference densities compared to those of the other three complexes. It is noticeable that all the holes are localized on graphene sheet and Ag₂ cluster while the electrons are all localized on PATP molecule and Au₂ cluster. The S₁ state is a charge transfer excited state, in which electrons transfer from graphene to PATP. When the energy of incident light is near to the energy of these excited states, the intra-substrate charge redistribution and charge transfer from graphene to PATP will be excited. As a result, the Raman intensities are enhanced.

Fig. 4 The absorption spectra of (a) G–Ag₂–PATP–Au₂, (b) Ag₂–PATP–Au₂–G, (c) BG–Ag₂–PATP–Au₂, and (d) Ag₂–PATP–Au₂–BG. The insets are the CDDs for different singlet excitation states of the four complexes (the green and red represent for holes and electrons, respectively).

Herein, we pay attention to the excited states around 632 nm, since the incident light with wavelength at 632 nm is often used in Raman experiment. For the graphene model (Fig. 4(a) and (b)), a charge transfer excited state appears around 632 nm, which is S₅ state (593.14 nm, f = 0.0009) and S₅ state (584.32 nm, f = 0.014) for G–Ag₂–PATP–Au₂ and Ag₂–PATP–Au₂–G, respectively. However, both intensities of the absorption peaks are very weak. On the contrary, for the B-doped graphene model (Fig. 4(c) and (d)), several strong absorption peaks occur near the interesting wavelength. For instance, the S₅ state (716.66 nm, f = 0.5011) and S₇ state (587.99 nm, f = 0.0404) of BG–Ag₂–PATP–Au₂ possess strong oscillator intensity. The CDDs picture shows that all the holes are localized on graphene sheet while all the electrons are localized on PATP molecule and Au₂ cluster, indicating electrons are transferred from graphene to PATP. For Ag₂–PATP–Au₂–BG, the strong absorption peaks occur at S₄ state (616.93 nm, f = 0.1722) and S₅ state (616.09 nm, f = 0.5107), which have the same CDDs picture. As depicted in Fig. 4(d), all the holes are localized on graphene sheet and Au₂ cluster and on the Ag₂ cluster localize all the electrons, meaning the electrons transfer from graphene and Au₂ cluster to Ag₂ cluster by tunneling through PATP molecule. Compared to graphene model, the doping B atom produces more charge transfer excited states with strong oscillator intensity.

4. Pre-resonant Raman spectra

To gain a deep insight into the Raman properties of graphene and B-doped graphene model, their pre-resonant Raman spectra were calculated with the incident light at 600 nm, which is near the energies of charge transfer excited states. As described in Fig. 5, the vibrational modes are selectively enhanced with distinct enhancement factors. For the graphene model, the Raman intensities are enhanced within 10 times compared to those in NRS spectra (Fig. 3(a)). Since the charge transfer excited states are near the incident light, the charge transition from graphene to PATP molecule will be excited. However, the oscillator strength of the excited state around the incident light is too weak to produce strong resonance. As a result, the Raman intensities are enhanced weakly. This slightly enhancement can be ascribed to the ground state chemical enhancement. On the contrary, the enhancement in B-substituted model is significant with respect to the NRS spectra. As shown in Fig. 5(c) and (d), the charge transfer excited states near the incident light are of strong oscillator strength, and thus the charge transfer from graphene to metal cluster by tunneling through PATP will be excited, which results in intense resonance. The calculated spectrum in both cases shows the non-totally symmetric modes (b₂) are strongly enhanced as well as the totally symmetric modes (a₁), indicating the contribution of Herzberg–Teller (HT) scattering. Therefore, this enhancement could be attributed to charge transfer resonant enhancement belong to HT contribution. In brief, the enhancement in the graphene model arises from the ground state chemical enhancement, while the Herzberg–Teller contribution plays an important role in the enhancement in B-doped model.

Fig. 5 The calculated pre-resonant Raman spectra of (a) G–Ag₂–PATP–Au₂, (b) Ag₂–PATP–Au₂–G, (c) BG–Ag₂–PATP–Au₂, and (d) Ag₂–PATP–Au₂–BG. The insets are CDDs for the excited states near the incident light, and f denotes oscillator strength.

Conclusion

In summary, this work presented a detailed investigation of the absorption and Raman spectra of Ag₂–PATP–Au₂ junction adsorbed on graphene and B-doped graphene. The interaction between the junction and graphene cause the charge transfer from graphene to junction or vice versa by the driving force arising from their different work functions. As a result, the charge redistributes on the junction leading to the change in static polarizabilities, which directly influence the ground state chemical enhancement. The charge difference densities reveal that the interaction between graphene and junction brings about some new CT excited states, which are pre resonant with the incident light leading to CT resonance enhancement. In pre-RRS spectra, the B-doped model possesses stronger enhancement than graphene model, since there is strong resonance between CT excited states and incident light in B-doped model.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant nos 61137005, 11374045), the Fundamental Research Funds for the Central Universities (Grant no DUT13ZD207), Program for Liaoning Excellent Talents in University (Grant no LJQ2012002), and Program for New Century Excellent Talents in University (Grant no NCET-12-0077).

References

1. M. Fleischmann, P. J. Hendra and A. J. McQuilla, Chem. Phys. Lett., 1974, 26, 163–166.
2. M. Moskovits, Rev. Mod. Phys., 1985, 57, 783–826.
3. A. Otto, I. Mrozek, H. Grabhorn and W. Akemann, J. Phys.: Condens. Matter, 1992, 4, 1143.
4. M. Sun, Z. Zhang, L. Chen and H. Xu, Adv. Opt. Mater., 2013, 1, 449–455.
5. M. Sun, Z. Zhang, L. Chen, S. Sheng and H. Xu, Adv. Opt. Mater., 2014, 2, 74–80.
6. K. Kneipp, H. Kneipp, I. Itzkan, R. R. Dasari and M. S. Feld, Chem. Rev., 1999, 99, 2957–2975.
7. S. M. Nie and S. R. Emery, Science, 1997, 275, 1102–1106.
8. M. Sun, Z. Zhang, P. Wang, Q. Li, F. Ma and H. Xu, Light: Sci. Appl., 2013, 2, e112.
9. M. Sun, Z. Zhang, L. Chen, Q. Li, S. Sheng, H. Xu and P. Song, Advanced Materials Interfaces, 2014 DOI:10.1002/admi.201300125.
10. A. Otto, J. Raman Spectrosc., 2005, 36, 497–509.
11. L. L. Zhao, L. Jensen and G. C. Schatz, Nano Lett., 2006, 6, 1229–1234.
12. J. R. Lombardi and R. L. Birke, J. Phys. Chem. C, 2008, 112, 5605–5617.
13. S. M. Morton and L. Jensen, J. Am. Chem. Soc., 2009, 131, 4090–4098.
14. N.-J. Kim, J. Phys. Chem. C, 2010, 114, 13979–13984.
15. X. Ling, L. Xie, Y. Fang, H. Xu, H. Zhang, J. Kong, M. S. Dresselhaus, J. Zhang and Z. Liu, Nano Lett., 2010, 10, 553–561.
16. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos and A. A. Firsov, Nature, 2005, 438, 197–200.
17. K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. L. Stormer, U. Zeitler, J. C. Maan, G. S. Boebinger, P. Kim and A. K. Geim, Science, 2007, 315, 1379.
18. L. Xie, X. Ling, Y. Fang, J. Zhang and Z. Liu, J. Am. Chem. Soc., 2009, 131, 9890–9891.
19. Q. Hao, B. Wang, J. A. Bossard, B. Kiraly, Y. Zeng, I. K. Chiang, L. Jensen, D. H. Werner and T. J. Huang, J. Phys. Chem. C, 2012, 116, 7249–7254.
20. S. Sun and P. Wu, Phys. Chem. Chem. Phys., 2011, 13, 21116–21120.
21. C. Qiu, H. Zhou, H. Yang, M. Chen, Y. Guo and L. Sun, J. Phys. Chem. C, 2011, 115, 10019–10025.
22. W. Xu, N. Mao and J. Zhang, Small, 2013, 9, 1206–1224.
23. X. Kong and Q. Chen, J. Mater. Chem., 2012, 22, 15336.
24. X. Kong, Z. Sun and Q. Chen, Phys. Chem. Chem. Phys., 2012, 14, 13564–13568.
25. M. T. Sun and H. X. Xu, ChemPhysChem, 2009, 10, 392–399.
26. M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X. Li, H. P. Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, J. A. Montgomery, Jr, J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, T. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega, J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V. Bakken, C. Adamo, J. Jaramillo, R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W. Ochterski, R. L. Martin, K. Morokuma, V. G. Zakrzewski, G. A. Voth, P. Salvador, J. J. Dannenberg, S. Dapprich, A. D. Daniels, O. Farkas, J. B. Foresman, J. V. Ortiz, J. Cioslowski and D. J. Fox, Gaussian 09, Revision A.02, Gaussian, Inc., Wallingford C.T., 2009.
27. P. Hohenberg and W. Kohn, Phys. Rev., 1964, 136, B864–B871.
28. J. P. Perdew, Phys. Rev. B: Condens. Matter Mater. Phys., 1986, 33, 8822–8824.
29. A. D. Becke, Phys. Rev. A: At., Mol., Opt. Phys., 1988, 38, 3098–3100.
30. P. J. Hay and W. R. Wadt, J. Chem. Phys., 1985, 82, 270–283.
31. J. Neugebauer, M. Reiher, C. Kind and B. A. Hess, J. Comput. Chem., 2002, 23, 895–910.
32. H. Iikura, T. Tsuneda, T. Yanai and K. Hirao, J. Chem. Phys., 2001, 115, 3540–3544.
33. J. D. Chai and M. Head-Gordon, J. Chem. Phys., 2008, 128, 084106.
34. M. A. Rohrdanz, K. M. Martins and J. M. Herbert, J. Chem. Phys., 2009, 130, 054112.
35. L. Xia, M. Chen, X. Zhao, Z. Zhang, J. Xia, H. Xu and M. Sun, J. Raman Spectrosc., 2014, 45, 533–540.
36. B. Dong, L. Liu, H. Xu and M. Sun, J. Raman Spectrosc., 2010, 41, 719–720.
37. Q. Zhou, X. Li, Q. Fan, X. Zhang and J. Zheng, Angew. Chem., Int. Ed., 2006, 45, 3970–3973.
38. S. Liu, X. Zhao, Y. Li, X. Zhao and M. Chen, J. Chem. Phys., 2009, 130, 234509.
39. G. Giovannetti, P. Khomyakov, G. Brocks, V. Karpan, J. van den Brink and P. Kelly, Phys. Rev. Lett., 2008, 101, 026803.
40. E. B. Wilson, Jr, Phys. Rev., 1934, 45, 427.
41. Y. Zhang, L. Sen, L. Wang, X. Qin, J. Tian, W. Lu, G. Chang and X. Sun, RSC Adv., 2012, 2, 538–545.

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