Ab initio potential energy surface and anharmonic vibration spectrum of NF+3

Yan-Nan Chen(陈艳南), Jian-Gang Xu(徐建刚), Jiang-Peng Fan(范江鹏),Shuang-Xiong Ma(马双雄), Tian Guo(郭甜), and Yun-Guang Zhang(张云光)

School of Science,Xi’an University of Posts and Telecommunications,Xi’an 710121,China

Keywords: ab initio methods, potential energy surfaces, vibration frequencies, coupled resonance, infrared spectra

1.Introduction

NF3is used in the microelectronics industry as a fuel[1]and as an anthropogenic greenhouse gas.Its vibrational levels and molecular spectra have been widely studied.[2–4]However, the vibrational levels and spectra of NF+3received relatively little attention.Initially,the existence of NF+3was questioned, and its potential was studied in terms of the Franck–Condon principle for ionization and dissociation of nitrogen trifluoride by electron impact.[5]Later, it was found to be formed during the synthesis of NF+4salt intermediate by lowtemperature UV photolysis.The NF+3radical ion serves as an intermediate during the low-temperature UV photodegradation of the NF3–F2–AsF5and NF3–F2–BF3systems,a process that has been successfully observed through experimental electron spin resonance(ESR)spectroscopy.[6]The fragmentation of the valence electronic state of NF+3and its photoelectron spectra were studied using threshold photoelectronphoton coincidence spectroscopy.[7]In the 1980s, there was significant interest in the inversion barrier of NF+3.Berkowitz and Greene[8]performed a vibration analysis of the first band of the photoelectron spectrum,[9]revealing the vibrational structure of NF+3.Ab initiocalculations of the NF+3inversion barriers and the inversion vibration dynamics were performed separately.[10,11]The results show that electron correlation is extremely crucial for the inversion barrier.However,its vibration spectrum remains an area of limited research.Molecular vibrational spectroscopy plays an essential role in physicochemical analysis,especially since molecular spectroscopy is an important tool for studying the structure of molecules or ions.[12–14]

In recent years, it is a common phenomenon concerning the use ofab initiocalculations to predict molecular spectra and vibration frequencies.Ab initiocalculations[15]have now achieved accurate prediction of a chemical structure and prediction of vibrational frequencies.The potential energy surface[16–19]is one of the most interesting areas in molecular spectroscopy today as it contains information about the dynamics of a molecular framework.Moreover, a number of methods, including vibrational selfconsistent fields (VSCF),[20–24]vibrational group state interactions(VCI),[25–28]vibrational Møller–Plesset(VMP),[29,30]vibrational coupled-cluster theory (VCC),[25,31]and secondorder perturbation theory (VPT2),[32,33]have been developed in vibrational structure theory (VST).These methods have been applied to the prediction of molecular spectra and vibration frequencies, as the dynamic of the molecular framework directly affects the spectrum.The VSCF/VCI method is widely used inab initiocalculation procedures,and the vibration self-consistent field(VSCF)method can strike a balance between accuracy and computation time.The VSCF method was proposed in the late 1970s by many researchers.[34–37]The VCI method[26,38]can make a extremely accurate approach when it appears that a single Hartree product cannot be used to describe a situation, e.g., Ref.[39].The general VCI calculation procedure is based on the VSCF method.[40–42]Based on the study of VSCF/VCI vibrational wave functions,the spectral intensity can be obtained theoretically by the variational method.For example, the spectra of H2O,[43,44]CH4,[44]CH2O,[6]and C2H2[45]have been calculated using the VSCF/VCI method.(CH3OH)2H+[46]has been successfully investigated in theoretical calculations using VSCF/VCI.The main objective of this paper is to obtain spectral data for NF+3using more reliable and efficient theoretical algorithms.Therefore,the VSCF/VCI method is selected to study the potential energy surface and vibration spectrum of NF+3.

In this paper,we presentab initiocalculations of the anharmonic frequencies of NF+3.Our analysis primarily relies on theoretical calculations, which aim to examine the potential energy surface,anharmonic frequencies,and IR spectrum of NF+3.In Section 3 we provide a thorough presentation and analysis of the results obtained from the NF+3calculations.Finally,we present our conclusions in Section 4.

2.Theoretical and computational details

In this paper,the NF+3is calculated from scratch using the Molpro2018 program package.[47]This work aims to construct theab initiopotential energy surface of NF+3as accurately as possible by using the latest electronic structure calculations.The generated potential energy surface is employed to perform calculations of the vibrational structure.Afterward,VSCF and VCI calculations are performed to obtain the wave functions corresponding to the vibration states.Finally, the vibrational Schr¨odinger equation is solved using the wave function to obtain information about the predicted mean vibration geometry and vibration frequency of the NF+3structure.

2.1.Potential energy surface

The balance position, harmonic frequency, and normal mode are calculated before utilizing the VSCF calculation.The SURF program can be used to calculate the potential energy surface (PES) around a reference structure, which is required to determine anharmonic frequencies.Within the surfaces (SURF) program, the PES can be expanded in terms of normal coordinates, the linear combination of normal coordinates, or localized normal coordinates.As a result, it is necessary to perform a harmonic frequency calculation first.Carteret al.proposed then-mode representation(nMR) method.[48–50]Then, the potential can be expressed by a multi-mode expansion,[51]which follows a hierarchical scheme given by

with

whereqidenotes the coordinates and this expansion needs to be aborted after then-body contribution.Based on an iterative algorithm, the SURF procedure adds automatic points to the lattice point representation of the potential function,reaching a convergence threshold after iteration.The final calculation results in the PES in different dimensions, which paves the way for subsequent calculations of harmonic frequency.

2.2.Calculation of energy levels

To better understand the molecular vibrations, we used theab initiocalculation procedure UCCSD(T) energy gradient analysis to optimize the molecule’s equilibrium geometric configuration.Then,we conducted a simple normal mode analysis to determine the vibrational Schr¨odinger equation in mass-weighted normal mode coordinates:[52]

whereVis the full potential function for the system,andnis the number of the vibrational normal modes.However, this method does not consider the effect of rotation-vibration coupling.Equation(6)is derived under the VSCF approximation,which is based on the ansatz

which leads to the single-mode VSCF equations

where the effective VSCF potential for modeQjis given by

For the single-mode wave functions,energies and effective potentials must be solved self-consistently.The VSCF approximation for the total energy is then given by

As the VSCF method applies a mean-field approach, correlation effects must be explicitly taken into account in the post-VSCF calculations.[41]Therefore,higher excitation levels are required in the post-VSCF calculations due to the appearance of high-order many-mode operators.As shown in Ref.[41],quadruple excitations cannot be neglected in most applications.The VCSF/VCI method can be used for calculating the non-harmonic frequencies in code multi-mode implementations.The VSCF calculation is first performed to obtain the optimal Hartree product in the vibration state, and here it is used as a reference for the VCI calculation.The excitation below four times the level is included in the VCI calculation(VCISDTQ).For all basis groups included in the VCI matrix,the sum of up to 10 quanta is used.This matrix is diagonalized to obtain the anharmonic frequencies.[46]

3.Results and discussion

3.1.Spectral data for F2, N2, NF and fundamental frequency data for NF3

We employ the UCCSD(T)/cc-pVTZ method to calculate the spectral constants of F2, N2, NF, as well as the fundamental frequency of NF3.Table 1 presents the calculated values for F2, N2and NF,comparing them with the experimental values.For the fundamental frequency of NF3,the calculated values for theν1,ν2,ν3, andν4modes are 493.86 cm−1,648.24 cm−1,928.34 cm−1,and 1041.42 cm−1,respectively, while the corresponding experimental values[53]are 492 cm−1,647 cm−1,907 cm−1,and 1032 cm−1.We can see that the results show excellent agreement with the experiment.Therefore, we further utilized this method for subsequent calculations of NF+3.

Table 1.The spectral constants of F2,N2,and NF at this benchmark,calculated and compared with the experimental results.

3.2.Equilibrium geometry of NF+3

NF+3is a flat, symmetrical top molecule with spatial geometry.In this work, we employ the spherical harmonic function to perform integration calculations.The spherical harmonic function serves as an eigenfunction of the angular momentum operator.This approach of utilizing spherical harmonic functions for integration lays the foundation for the study of vibrations.NF+3was optimally calculated using UCCSD(T)/cc-pVTZ and its equilibrium geometry parameters were obtained in the process of optimization.The optimized bond lengthsr(N1–F2),r(N1–F3),r(N1–F4)are all 1.2839 ˚A.The key angles ∠(F2–N1–F3), ∠(F2–N1–F4), and ∠(F3–N1–F4) are 113.829°, 113.829°, and 113.83°, respectively.This result is in line with the calculated values in the Comprehensive Computational Chemistry Database (CCCBDB),[55]and so far there are no experimental values for comparison.CCCBDB is a database that provides comprehensive and reliable information on a wide range of molecular properties,energetics, and spectroscopic data.It contains calculated data based on quantum chemical calculations for a large number of molecules.

3.3.Potential energy surfaces

In this work, the one-dimensional (1D) and twodimensional(2D)PESs of NF+3are derived by using the SURF program in combination with the ploting(PLOT) command.Hereqirepresents the trend of the distance to the equilibrium position for each atom in different modes.The change in bond length stretching or compression for each atom in different modes is associated with energy.Since the final mode will remain in the new geometry when it is stationary,this energy is potential energy (position dependent).Figures 1 and S1 in the supporting information are both 1D PESs.The observed 1D PESs ofq2(A'') andq6(A'') there are symmetric.The symmetry is due to the integrable representation in these two modes that does not fully describe the geometry in the current vibrational state.However, for the four vibration modes ofq1scissoring,q3symmetric deformation,q4symmetric,andq5asymmetric,the integrability of the symmetry in normal coordinates is expressed as A',which generates asymmetric 1D PESs,as seen in Fig.S1.Asymmetrical PESs arise because of the tendency of their atoms to change (at a distance from the equilibrium position),with different magnitudes of stretching or compression causing them to generate different potential energies.In summary,1D PESs are symmetric when the integrable representation of the vibrational mode is A'',and otherwise, they are asymmetric when the integrable representation is A'.

As shown in Fig.2,the 2D complete PESs are generated by combining six modes pairwise with each other,two forms of PESs are produced, complete symmetric and axisymmetric.The multi-mode expansion of the PESs is truncated after the 3D contribution, which also contains a 3-mode coupling term.The lowest point in the PESs,when both the horizontal and vertical coordinates tend to zero,is the most stable state of the system, i.e., the equilibrium geometry.Figures 2(a)–2(c)are completely symmetrical due to the complementary orientation of the two modes vibrating against each other in the transverse coordinates of the two modes.Figures 2(d)–2(f)are axisymmetric and arise for reasons related to theq6mode.The coupling different vibrational modes has various effects on the variation of the ion potential energy.When the absolute values of the vertical coordinates corresponding toqiandqjreach their maximum values at the same time,their mutual coupling leads to the greatest change in potential energy.It is observed that modeq6and its different modes coupled to each other produce axisymmetric PESs and that all three sets of couplings have a relatively large effect on the energy of the system.This is due to the ion vibrations that tend to be larger in modeq6and therefore have a greater impact on the system when they are coupled to other modes.

Fig.1.One-dimensional PESs of molecular vibrations unfolding along q2(A'')and q6(A'').

Fig.2.Two-dimensional complete PESs with different symmetry forms.

The coupling of different modes to each other also generates relative PESs,which are also plotted as shown in Fig.S2 in the supporting information.Looking at Fig.S2, the vertical coordinates of the potential energy appear to be positive and negative,the reason for this is that the zero PES has been chosen as a reference.By using the zero PES as a reference,it is possible to observe more clearly the variation of the potential energy under different modes of coupling.Observation of Figs.S2(a)and S2(b)reveals the same axisymmetric form of the PESs as the complete PESs.Figures S2(a) and S2(b)are reversed if Fig.S2(a) is assumed to be the front side and Fig.S2(b) is assumed to be the side after reversing it.This phenomenon is caused by the fact that modeq5vibrations are in a different direction from modeq1vibrations.As shown in Figs.S2(c)and S2(d),the PESs are completely asymmetric and their potential energy tends to change less after coupling in Fig.S2(c).As shown in Figs.S2(c)and S2(d),when the modes have the same integrable representation, e.g., A'or A'', their mutual coupling produces the completely asymmetric PESs.

In summary, when the PES appears to be symmetrical it saves time and resources for the calculation.The modeq6symmetric stretching vibration has the greatest influence on the potential energy change of the whole system.The completely asymmetric PESs are generated when the integrable representations of the modes are identical.

3.4.Vibration frequency

Molecular vibration is a complex phenomenon,in which an actual molecule oscillates through various normal modal motions.Generally,this vibration is a combination of several simple modes.Figures 1 and S1 show that the NF+3molecule has six modes of vibration,includingν1in-plane shear vibration,ν2in-plane wobble,ν3symmetric deformation,ν4symmetric stretching,ν5antisymmetric stretching, andν6symmetric stretching.The length of the arrows in the diagrams represents the amplitude of the vibrational trend.

Table 2 lists the harmonic vibration frequenciesωi, anharmonic vibration frequenciesνi, and anharmonicity correctionωi −νiof NF+3.Tables 3 and S1 in the supporting information present overtones and combination frequencies, respectively.By observing the fundamental frequencies,we find that the two pairs of vibrational energy levelsν1(522.18 cm−1)andν2(522.17 cm−1),ν5(1249.97 cm−1),andν6(1250.04 cm−1) have a twofold degenerate phenomenon,because the mutual energy band gap between them is less than 1 cm−1.Two of the fundamental frequencies are twofold degenerate and may thus give rise to the Jahn–Teller (JT)effect.Furthermore, we can find a link among modeν1,modeν2, and modeν3ground states bound to overtones and their combinations, as Table 3 overtones and Table S1 combination bands.Mode 2ν1+ν2(1567.06 cm−1) is almost equivalent to the sum of the overtones of modeν1and the fundamental frequency of modeν2.The same is valid for modesν1+2ν2(1567.16 cm−1), 2ν1+ν3(1701.74 cm−1),2ν2+ν3(1701.70 cm−1),ν1+2ν3(1832.48 cm−1),ν2+2ν3(1832.45 cm−1),ν1+ν2(1044.55 cm−1),ν1+ν3(1179.98 cm−1), andν2+ν3(1179.94 cm−1).We also notice that the combination bands resonate with each other in the above-mentioned combinations,e.g.,2ν1+ν2(1567.06 cm−1)andν1+2ν2(1567.16 cm−1).Owing to the degenerate phenomenon, we find that modeν5and modeν6also have their combination bands interwound with each other,e.g.,ν2+2ν6(3004.51 cm−1) andν2+ν5+ν6(3005.44 cm−1).Therefore, the fundamental frequencies’ degenerate phenomenon also leads to this phenomenon between their combined bands.

Table 2.Computed fundamental frequency νi(in cm−1),harmonic vibrational wavenumbers ωi (in cm−1) and anharmonicity corrections ωi −νi (in cm−1)of NF+3.Here ωi and νi are obtained using the UCCSD(T)/cc-pVTZ method.

Table 3.Computed overtones νi (in cm−1), harmonic vibrational wavenumbers ωi (in cm−1) and anharmonicity corrections ωi −νi (in cm−1) of NF+3.Here ωi and νi are obtained using the UCCSD(T)/cc-pVTZ method.

3.5.Vibration spectrum

Molecular spectroscopy is widely used as a “fingerprint of molecules” for the structure of molecules and the chemical composition of substances.The coupled clustering(UCCSD(T))method in this paper provides accurate predictions of molecular properties and fits the spectral data as shown in Figs.3 and 4.All the IR absorption peaks in the graph appear in the mid-IR region 400–4000 cm−1.In Fig.3,the front part of the plot represents the vibrational IR spectrum of NF+3and the back part of the plot represents the vibrational spectrum at 200–1400 cm−1with four distinct peaks, which are marked in the plot with lower case letters.The vibrational spectrum in 200–1400 cm−1represents the absorption peak generated by the fundamental frequency mode.For two different modes whose frequency values are almost close to each other (coupled resonance), intensity borrowing will occur in the vicinity of this frequency.Intensity borrowing is generated at letters a and d in the diagram.The letter d in Fig.3 indicates the strength borrowed from anti-symmetric stretchν5and symmetric stretchν6.It shows a large peak near 1250 cm−1with an intensity of 341.58 km/mol.Similarly, the letter a indicates the intensity borrowed between the in-plane shear vibrationν1and the in-plane wobbleν2,where the peak is located near 522 cm−1with an intensity of 13.54 km/mol.Symmetric deformation vibrationν3and symmetric stretching vibrationν4produce peaks of 23.74 km/mol and 15.62 km/mol around 974 cm−1and 658 cm−1, respectively, which are represented in the diagram by the letters b and c,respectively.To facilitate the observation of the remaining weaker overtones and combination bands, IR intensity is plotted in the ranges 1400–2000 cm−1and 2000–3500 cm−1,respectively, as shown in Fig.4.In combination with the predicted vibration frequencies, it is found that there are vibrationally coupled overtones and combinations of frequencies,which also have intensity borrowing phenomena between them.The peaks at the letters H,N,G,and K in the diagram are intensity borrowed and occur atν2+ν4andν1+ν4,ν2+ν6andν1+ν5,ν3+ν6andν3+ν5,ν4+ν6andν4+ν5, representing intensities of 2.41 km/mol,0.59 km/mol,0.34 km/mol,and 9.94 km/mol at 14993 cm−1,1768 cm−1,1904 cm−1and 2214 cm−1,respectively.The letters I,M,and L represent the intensities atν3+ν6,ν5+ν2, andν5+ν6, which represent intensities of 0.13 km/mol,0.56 km/mol,and 2.20 km/mol at 1622 cm−1,1763 cm−1,and 2493 cm−1,respectively.Fig.4.The infrared vibrational spectra ofin the ranges 1400–2000 cm−1and 2000–3500 cm−1.

Fig.3.The infrared spectra produced by the NF+3 vibrations and the infrared vibrational spectrum in the range 200–1400 cm−1.

4.Summary and conclusion

This paper presents calculations of the PESs, vibrational frequencies,and spectra of NF+3using the UCCSD(T)method in conjunction with the cc-pVTZ basis set within the MOLPRO software package.Highly accurate PESs are fitted toab initiodata and constructed with spherical harmonic expansion.Symmetrical potential energy curves are generated in the process of generating 1D PESs when the integrability of the normal transverse coordinate is expressed as A''.Conversely,asymmetric potential curves arise when the integrability is expressed as A'.The coupling of different modes creates 2D relative and complete PESs.The modes coupled to each other have different effects on system potential energy, with modeν6symmetric stretching vibrations having the greatest effect.When irreducible representations of the normal transverse coordinate patterns are all identical, completely asymmetric PESs are created.

In the calculation of the PESs,the Hessian is used to calculate the harmonic frequencies of NF+3.In the subsequent calculation of the anharmonic frequencies,the VCI is used to obtain anharmonic corrections for the harmonic frequencies.The fundamental vibrational frequencies of NF+3appear as two pairs of twofold degenerate phenomenon with mutual energy band gaps of less than 1 cm−1,which are out-of-plane shearν1and in-plane oscillationν2, respectively, and anti-symmetricν5and symmetric stretchingν6.Two of them are twofold degenerate and may thus give rise to the Jahn–Teller(JT)effect.This degeneracy is also reflected in the combined and overtone frequencies.Finally,the IR spectrum of the ion is plotted in the mid-infrared region 400–4000 cm−1.Intensity borrowing occurs around this frequency as coupled resonances occur at similar frequencies in different modes, resulting in a shift towards two different modes.Observing the entire spectrum,the greatest intensity in the infrared occurs near the symmetric stretching vibrationν6.The research in this paper provides data to guide experiments with NF+3.

Acknowledgment

Project supported by the National Natural Science Foundation of China(Grant Nos.52002318 and 22103061).