First-principles investigation of the lattice vibrational properties of inorganic double helical XY (X = Li, Na, K, Rb, Cs; Y = P, As, Sb)

Abstract

Inorganic double helical XY (X = Li, Na, K, Rb, Cs; Y = P, As, Sb) materials are newly discovered compounds. Their structures are so simple that we can study their properties with first-principles calculations. Our calculation results show that their structures and lattice vibrational frequencies are sensitive to the cation (Li, Na, K, Rb, Cs) and anion (P, As, Sb). At the Γ point, there are 15 optical vibrational modes, in which the in-phase and out-of-phase breathing modes, dipole mode, and out-of-phase rotating mode are characteristic modes. Owing to the special double helical structure, the frequency of the in-phase breathing mode is always higher than that of the out-of-phase breathing mode. Furthermore, the frequencies of these two breathing modes evolve differently with cation and anion. The dipole mode and out-of-phase rotating mode are A₂ modes. The frequency of the former decreases continuously from LiY to CsY, while the frequency of the latter firstly decreases, then increases, and finally approaches a stable value. Besides these characteristic modes, other modes are sensitive to ions, too. Our systematic calculation and analysis not only give much valuable insight into these newly discovered inorganic double helical XY, but also apply to other double helical compounds.

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1. Introduction

The double helical structure is definitely one of the most charming and landmark structures. Since being revealed as the structure of DNA,¹ the double helix has become a research focus in physics, chemistry, materials research, engineering, and other application areas.^2–5 For example, the double helical structure gives rise to a special copying mechanism for genetic material and plays a particular role in evolution and metabolism.⁶ Compared to organic molecules, the double helical structure is rarely observed in inorganic compounds. Different to organic material, it is very difficult for inorganic material to form two congruent helices simultaneously, which can be combined together.² Although inorganic double helical nanostructures are occasionally observed in experiments, they are usually complex, involving multiple building units with relatively large length scales.^7–11 Even if they are successfully synthesized, it is very difficult to obtain high purity double helical materials and to handle them at the nanoscale level.² Therefore, modeling is indispensable in the investigation of the double helical structure.

A breakthrough of inorganic double helical compounds was recently achieved by Alexander S. Ivanov et al.³ They found an unusually simple double helical chain formed from interpenetrating spirals of lithium and phosphorus. It is expected to exhibit certain special property due to its unique organization of the two helices and their synergistic effect.¹² Double helical LiP is so simple that there are only eight atoms (four Li and four P) in its primitive unit cell. In ref. 3, the authors suggest that inorganic double helical structure may be found in compounds XY (X = Li, Na, K, Rb, Cs; Y = P, As, Sb) obeying the Zintl rule. Inspired by this suggestion, we calculate the structure and lattice vibrational property of double helical XY.

2. Calculation method

We used the Quantum Espresso package¹³ to calculate the properties of double helical XY (X = Li, Na, K, Rb, Cs; Y = P, As, Sb). As we are interested in the calculated Raman spectrum of double helical XY, we can only choose local density approximation and the norm-conserving pseudopotential¹⁴ in Quantum Espresso package. The wave function is expanded with plane wave, and the energy cutoff was set to 100 Ry. The first Brillouin zone of infinite double helical structure was sampled on 1 × 1 × 8 mesh, and the supercell with 10 Å vacuum along x and y directions was employed. Vibrational property was calculated within the framework of density-functional perturbation theory.¹⁵

3. Result and discussion

3.1 Structure

The inorganic double helical XY are composed of interpenetrating spiral of X and Y atoms. As depicted in Fig. 1, there are three characteristic parameters to describe the infinite periodic structure: helical pitch (h), radius of spiral X and Y (R_X and R_Y). It can be seen in the side view that there are as few as eight atoms (X₄Y₄) in one pitch period. In the top view, X and Y have different helical radius: R_X is a little larger than R_Y.

Fig. 1 The side view (a) and top view (b) of double helical XY (X = Li, Na, K, Rb, Cs; Y = P, As, Sb). The helical pitch (h), radius of spiral X (R_X) and Y (R_Y) are depicted.

We fully relax the helical pitch and atomic positions of double helical XY. The calculated helical pitch, radius of spiral X and Y (R_X and R_Y) are given in Fig. 2. Similarly to DNA, helical pitch is an important parameter to describe the period structure of double helix. Owing to its complex structure, the DNA helical pitch reaches several nanometers.¹⁶ However, the helical pitches of these XY compounds range from 4.75(LiP) to 6.92(CsSb) Å,^3,17 much smaller than that of DNA. They evolve regularly with ionic radius. As expected, helical pitch increases with cation radius increasing. For example, the helical pitches of XP are 4.75(LiP), 5.29(NaP), 5.53(KP), 5.66(RbP) and 5.76(CsP) Å. Furthermore, it also increases with anion radius increasing. For example, the helical pitches of LiY are 4.75(LiP), 5.00(LiAs) and 5.47(LiSb) Å. As shown in the upper part of Fig. 2(b), the radius of spiral X (R_X) increases with the increase of cation radius. However, R_Y has the opposite trend. Moreover, the anion has large influence on R_Y, but little influence on R_X. For example, the R_Y difference between LiSb and LiP is 0.23 Å, but the R_X difference between them is as small as 0.04 Å. As long as these three parameters (h, R_X and R_Y) are obtained, other structure parameters of double helical XY can be calculated easily. For example, d_XX = [(h/4)² + 2R_X²]^0.5, d_YY = [(h/4)² + 2R_Y²]^0.5 and d_XY = [(h/2)² + (R_X − R_Y)²]^0.5.

Fig. 2 (a) The helical pitch of double helical XY (X = Li, Na, K, Rb, Cs; Y = P, As, Sb). The fifteen compounds are divided into three groups: XP, XAs and XSb. (b) The radius of spiral X(R_X) and Y(R_Y).

We relax the atomic positions of finite linear fragment X₁₂Y₁₂ with 10 Å vacuum along x, y and z directions. The calculated results show that per formula energies of infinite double helical XY are always lower than those of finite linear counterparts. For example, the per formula energy difference of LiP is 0.362 eV, which agree well with ref. 3. The energy difference of NaP, KP, RbP, CsP, LiAs, NaAs, KAs, RbAs, CsAs, LiSb, NaSb, KSb, RbSb, and CsSb are 0.328, 0.292, 0.268, 0.252, 0.342, 0.313, 0.286, 0.269, 0.255, 1.074, 0.294, 0.283, 0.267, and 0.254 eV, respectively.

Following the Fig. 4 in ref. 3, we calculate the 3D structure of packed double helices LiP. After examining the bond lengths carefully, we find the inter helical Li–Li bond length is even smaller than the intra helical ones. This means the inter helical interaction is even stronger than that of the intra helices. Therefore, the packed double helices can not be simply viewed as packed double helices.

3.2 Vibrational property

Infrared and Raman spectra are frequently used in material characterization, especially in nanostructures.^18–22 So as to obtain the symmetry of infrared and Raman active modes of double helical XY, we use the factor group analysis method to classify their vibrational modes.²³ These fifteen double helical compounds have the same symmetry. Their factor groups of the Γ point are isomorphic with the point group D₄.¹⁷ Thus, there are 12 non-degenerate vibrational modes (2A₁ + 4A₂ + 4B₁ + 2B₂) and six double-degenerate modes (6E) at Γ point, including three acoustic modes (A₂ + E) and one rotational mode (A₂). Therefore, there is no silent mode in the 15 optical modes (2A₁ + 2A₂ + 4B₁ + 2B₂ + 5E). Among them, A₁, B₁ and B₂ modes are Raman active, A₂ modes are infrared active and E modes are both infrared and Raman active.

It's well known that atomic vibrational amplitude equals dynamic matrix eigenvector divided by (atomic mass)^0.5. So as to reduce the mass influence, we choose CsSb as the representative of double helical XY, because the mass of Cs (132.9) and Sb (121.8) are close to each other. The calculated vibrational modes of CsSb are depicted in Fig. 3. The peculiar double helical structure gives rise to special vibrational modes. It is obvious that Fig. 3(a) and (b) are breathing modes, in which all atoms vibrate along the radial direction. Breathing mode not only appears in double helical structure, but also appears in bct C4,¹⁹ cagelike diamondiod nitrogen,²⁴ nanotubes,^22,25 DNA,^26–30 and etc. In Fig. 3(a), anion and cation vibrate out-of-phase with each other. Therefore, it is an out-of-phase breathing mode. Similarly, Fig. 3(b) is an in-phase breathing mode, as anion and cation vibrate in-phase with each other. In DNA, the breathing mode corresponding to two bases oscillating in opposite direction, stretching and compressing the hydrogen bonds. The hydrogen bond breathing mode is very important in the dynamical property of DNA, such as pair opening, replication, transcription, melting and other processes.^16,31 Similarly, breathing modes of double helical XY may play an important role in their dynamic property.

Fig. 3 The optical vibrational modes of double-helix CsSb. Symmetry and calculated frequency are labeled and arrows represent the movement directions.

There are two kinds of atomic interactions in double helical compound: inter-chain and intra-chain interactions. Accordingly, their vibrational modes can be classified into internal and external modes. The two A₂ modes (Fig. 3(c) and (d)) are external modes. Fig. 3(c) is an out-of-phase rotating mode, in which anion and cation rotate around the screw axis out-of-phase with each other. Fig. 3(d) is a dipole mode, in which anion and cation move along the screw axis in opposite direction. Besides of the breathing mode, dipole mode is another main feature in the lattice dynamics of DNA.²⁸

There are four B₁ modes, which are depicted from Fig. 3(e)–(h). Fig. 3(e) and (g) look like each other. The Cs movement in Fig. 3(e) is similar to that of Fig. 3(g), but the Sb movement in Fig. 3(e) is opposite to that of Fig. 3(g). Furthermore, the movement amplitude of Cs is larger (smaller) than that of Sb in Fig. 3(e) (Fig. 3(g)). In Fig. 3(f), neighboring Cs (Sb) atoms move out-of-phase with each other, along the screw axis. In Fig. 3(h), neighboring Cs atoms move toward each other, while Sb atoms almost keep static.

The two B₂ modes are similar to each other. The Cs movement in Fig. 3(i) is similar to that of Fig. 3(j), but the Sb of Fig. 3(i) move opposite to that of Fig. 3(j). The Cs movements dominate in Fig. 3(i), while the Sb movements dominate in Fig. 3(j). It is quite similar to the relation between Fig. 3(e) and (g). Similarly to Fig. 3(c), atoms of Fig. 3(k) move almost along tangential direction. However, half Cs(Sb) atoms rotate clockwise and the left half Cs(Sb) atoms rotate anti-clockwise, which is different to Fig. 3(c). The Cs movements dominate Fig. 3(l) and (m), while the Sb movements dominate in Fig. 3(n) and (o). The last four vibrational modes in Fig. 3 are complex. They are not purely along tangential, radial or screw axis.

We calculate the Γ point vibrational frequencies of double-helix XY and discuss their evolution below. There are two A₁ modes: the in-phase and out-of-phase breathing modes. We can see from Fig. 4(a) that the in-phase breathing mode frequency is always higher than the out-of-phase breathing mode frequency. Breathing mode is frequently observed in crystal and nanostructure. In bct C4 and cagelike diamondiod,^19,24 the out-of-phase breathing mode frequency is higher than the in-phase breathing mode frequency, whereas the opposite is true in DNA and double helical XY. For the in-phase breathing mode of DNA, two bases linked by a hydrogen bond oscillate in opposite direction, stretching and compressing their hydrogen bond. However, the out-of-phase breathing mode does not involve in the deformation of hydrogen bond, which reduces its frequency. This mechanism applies to double helical XY as well. Cation and anion atoms with the same z coordinate oscillate in the same direction in the out-of-phase breathing mode. But they oscillate in the opposite direction in the in-phase breathing mode, which increases its frequency. As shown in Fig. 4(a), the in-phase breathing mode frequency does not decrease continuously with cation evolution from Li to Cs (solid symbol). It has a plateau from NaY to CsY. For example, the frequency of in-phase breathing mode is 349(NaP), 355(KP), 352(RbP) and 349(CsP) cm⁻¹. After checking their eigenvectors, we find that anions dominate these vibrational modes, but cations almost do not involve. For example, NaP, KP, RbP and CsP have the same anion (P). As shown in the solid symbols of Fig. 2(b), the spiral radius of P anoin decreases slowly from NaP to CsP. This indicates the P–P interaction changes little with cation. Consequently, the frequencies of this anion dominated mode almost have the same values. This analysis applies to the in-phase breathing mode of XAs and XSb as well. Differently from the in-phase breathing mode, the frequency of the out-of-phase breathing mode (open symbol in Fig. 4(a)) decreases from LiY to CsY continuously. Their eigenvectors show that cations dominate these modes. As atomic mass increases from Li to Cs and the spiral radius of cations increases sharply from LiY to CsY (open symbols of Fig. 2(b)), the out-of-phase breathing mode frequency decreases from LiX to CsX. As shown in Fig. 4(b), the two A₂ modes have different trend. The high frequency one (solid symbol in Fig. 4(b)) is a dipole mode, in which cation and anion vibrate mainly along the screw axis oppositely. As shown in Fig. 2(a), the helical pitch increases sharply from LiY to CsY, which certainly decreases the Coulomb force between anion and cation. Consequently, its frequency decreases continuously from LiY to CsY. The low frequency one is out-of-phase rotating mode. Its frequency firstly decreases, then increases, and finally approach stable value.

Fig. 4 The A₁ and A₂ mode frequencies of double helical XY (X = Li, Na, K, Rb, Cs; Y = P, As, Sb). Their corresponding vibrational modes are depicted.

The vibrational frequencies of B₁ and B₂ modes are shown in Fig. 5. The B₁ modes in Fig. 5(a) have two clear characteristics. One is that their vibrational frequencies decrease obviously from LiY to CsY. The other is that the frequency difference among XP, XAs and XSb are little. For example, the frequency of Fig. 3(e) mode for LiP is 289 cm⁻¹, which is 256 cm⁻¹ higher than that of CsP, but only 29 cm⁻¹ higher than that of LiSb. This condition occurs in Fig. 3(f) mode as well. As shown in the open symbols of Fig. 2(b), the spiral radius of cation is sensitive to cation, but is less influenced by anion. The spiral radius trend of cation is similar to the characteristics of Fig. 5(a). The eigenvector show that Cs movement dominates the mode of Fig. 3(e) and Sb has little contribution. Therefore, its frequency has close relation with cation and has little relation with anion. As shown in the solid symbols of Fig. 2(b), the spiral radius of Cs is less influenced by anion. Therefore, the frequencies of Fig. 5(a) decrease with cation evolution from Li to Cs, but changes little with anion evolution from P to Sb. Differently from Fig. 5(a), the frequencies in Fig. 5(b) change relatively little with cation, but are sensitive to anion. For example, the frequency of Fig. 3(g) mode for LiSb is 91 cm⁻¹, which is only 6 cm⁻¹ higher than that of CsSb, but 124 cm⁻¹ lower than that of LiP. As shown in Fig. 3(g), Sb dominates this mode and Cs has little contribution. Consequently, its frequency has close relation with anion and has little relation with cation. This analysis applies to Fig. 3(h) mode as well.

Fig. 5 The B₁ and B₂ mode frequencies of double helical XY (X = Li, Na, K, Rb, Cs; Y = P, As, Sb). Their corresponding vibrational modes are depicted.

Compared to the B₁ modes, the evolution of B₂ modes in Fig. 5(c) are relatively simple. As depicted in Fig. 3(i) and (j), both cation and anion take part in the vibration of B₂ modes. Consequently, their frequencies not only decrease with the evolution of cation from Li to Cs, but also decrease with the evolution of anion from P to Sb.

Fig. 6 gives the vibrational frequencies evolution of E modes. Similarly to the B₁ modes in Fig. 5(a) and (b), some E modes frequencies changes largely with cation (Fig. 6(a)), while others are relatively insensitive to cation (Fig. 6(b)). We check their eigenvectors and find that cation dominates the vibrational modes depicted in Fig. 6(a). As shown in the open symbols of Fig. 2(b), the spiral radius of cation is sensitive (insensitive) to cation (anion). Consequently, their vibrational frequencies have close relation with cation. As shown in the solid symbols of Fig. 2(b), the spiral radius of anion is insensitive (sensitive) to cation (anion). Therefore, the frequencies of three E modes in Fig. 6(b) are relatively less influenced by cation, but change large with anion.

Fig. 6 The E mode frequencies of double helical XY (X = Li, Na, K, Rb, Cs; Y = P, As, Sb). Their corresponding vibrational modes are depicted.

We calculate the non-resonant Raman spectrum of infinite double helical XY at 300 K with 532 nm excitation laser.¹⁷ As can be seen in Fig. 7, all the Raman peaks are located below 500 cm⁻¹. The infrared intensities of most vibrational modes are very small, except for the vibrational mode of Fig. 3(d). It is a dipole mode with cation and anion moving in opposite direction. The calculated infrared intensities of dipole modes are 33 (LiP, 467 cm⁻¹), 11 (NaP, 261 cm⁻¹), 5 (KP, 205 cm⁻¹), 6 (RbP, 175 cm⁻¹), 7 (CsP, 158 cm⁻¹), 31 (LiAs, 433 cm⁻¹), 10 (NaAs, 222 cm⁻¹), 4 (KAs, 161 cm⁻¹), 4 (RbAs, 128 cm⁻¹), and 4 (CsAs, 112 cm⁻¹) (D Å⁻¹)² amu⁻¹.

Fig. 7 Calculated Raman spectrum of infinite double helical XY with random orientation.

It has been shown that the dispersion correction is important for the accurate evaluation of the interaction energies in DNA.³² For example, after the dispersion correction, the stabilization of base pairs increases 1.0–4.5 kcal mol⁻¹.³² The inorganic double helical XY are mainly bonded by ionic bond. Therefore, the dispersion correction is small compared with static Coulomb interaction and will not influence the lattice vibrational property evolution with cation and anion.

4. Conclusion

We study the structure and lattice vibrational property of inorganic double helical XY systematically. Their helical pitches increase with anion and cation diameters increasing. At Γ point, there are 15 optical modes, in which the A₁, B₁ and B₂ (A₂) modes are Raman (infrared) active, and the E modes are both infrared and Raman active. The two A₁ modes are in-phase and out-of-phase breathing modes. The in-phase breathing mode frequency is always higher than the out-of-phase breathing mode frequency. The frequency of the out-of-phase breathing mode decreases continuously with the evolution of cation from Li to Cs, but the frequency of the in-phase breathing mode has a plateau from NaY to CsY. There are two A₂ modes: dipole mode and out-of-phase rotating mode. The frequency of former decreases from LiY to CsY continuously, but the frequency of the latter firstly decreases, then increases, and finally approaches a stable value. The evolutions of B₁, B₂ and E modes are complex. Their vibrational frequencies are sensitive to either cation or anion or both of them. The evolutions of structure and vibrational frequency provide much insight into these newly discovered double helical compounds.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (NSFC, Grant No. 11404096, U1404609, 11504089, U1404111), the Young Scientist Foundation of Henan University of Science and Technology (2014QN036), the key Project for Education Department of Henan Province (Grant No. 16A140008), the Innovation Team of Henan University of Science and Technology (No. 2015XTD001), Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund (the second phase). The crystal structure was drawn using the VESTA software.³³

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