Potential Functions and Thermodynamic Properties of UC, UN, and UH

Tang, Shuang-Ling; Wang, Yu; Xia, Qi-Ying; Ju, Xue-Hai

doi:https://doi.org/10.1155/2020/7512737

Journal of Chemistry

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Abstract Introduction Results and Discussion Conclusions Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 7512737 | https://doi.org/10.1155/2020/7512737

Potential Functions and Thermodynamic Properties of UC, UN, and UH

Shuang-Ling Tang,¹Yu Wang,¹Qi-Ying Xia,²and Xue-Hai Ju¹

Academic Editor: Teik-Cheng Lim

Received19 Oct 2019

Accepted09 Jan 2020

Published30 Jan 2020

Abstract

Potential energy surface scanning for UC, UN, and UH was performed by configuration interaction (CI), coupled cluster singles and doubles (CCSD) excitation, quadratic configuration interaction (QCISD (T)), and density functional theory PBE1 (DFT-PBE1) methods in coupling with the ECP80MWB_AVQZ + 2f basis set for uranium and 6 − 311 + G for carbon, hydrogen, and nitrogen. The dissociation energies of UC, UN, and UH are 5.7960, 4.5077, and 2.6999 eV at the QCISD (T) levels, respectively. The calculated energy was fitted to the potential functions of Morse, Lennard-Jones, and Rydberg by using the least square method. The anharmonicity constant of UC is 0.0047160. The anharmonic frequency of UC is 780.27 cm⁻¹ which was obtained based on the PBE1 results. For UN, the anharmonicity constant is 0.0049827. The anharmonic frequency is 812.65 cm⁻¹ which was obtained through the PBE1 results. For UH, the anharmonicity constant is 0.017300. The anharmonic frequency obtained via the QCISD (T) results is 1449.8 cm⁻¹. The heat capacity and entropy in different temperatures were calculated using anharmonic frequencies. These properties are in good accordance with the direct DFT-UPBE1 results (for UC and UN) and QCISD (T) results (for UH). The relationship of entropy with temperature was established.

1. Introduction

Among various theoretical simulation methods for molecules and materials, the first principles and molecular dynamic simulation techniques are very powerful for computing the micro and macro properties [1, 2]. The properties and phenomena in materials typically occur at multiple time and length scales. Therefore, to investigate the dynamic behaviors and the time evolution processes, one should resort to the molecular dynamics simulations instead of the first principles [3]. As an alternative solution, molecular dynamics simulation with a molecular force field is a practical method to calculate the dynamic property of the condensed materials [4]. The molecular dynamics uses the Hamilton canonical equation to describe the object system instead of the Schrodinger equation, which requires much smaller computational resources. Furthermore, the kinetic degrees of freedom are easily traversed to obtain the normal frequencies and thus obtain the thermodynamic properties [5]. The computational scale of molecular dynamics methods are on the nanometer and nanosecond scales [6–8]. Consequently, the development of the simulation techniques that couple together with physics on multiple levels is of very importance. In order to realize this goal, one should develop an atom-atom pair potential and obtain the parameters of the force field. The characteristics of the potential energy between diatomic molecules can be described by the corresponding analytical potential energy functions [9]. In particular, the potential function is necessary for establishing and optimizing the force field parameters, which in turn plays an important role for investigating static and dynamic properties of molecules as well as of solid states [10, 11].

The molecular dynamics deals with the atom-to-atom forces and the individual atomic movement. It is equally applicable for both crystalline and noncrystalline materials. Furthermore, molecular dynamics could obtain the total energy composed of all interatomic potential energies and kinetic energies of all degrees of freedom at any loading and temperature conditions. The initial important step in molecular dynamics simulations is the selection of interatomic potentials. For metals, the Morse potential is the most popular one [9, 12]. However, the interatomic potentials or the force field parameters are not available in the commonly used COMPASS force field for the uranium atom except for its oxides. In this paper, we scanned the potential energy surface of UC, UN, and UH with ab initio configuration interaction (CI), coupled cluster singles and doubles (CCSD) excitation, quadratic configuration interaction (QCISD (T)), and density functional theory PBE1 methods [13]. Then, the Morse potential functions were established. We also fit the energy curves into the Lennard-Jones (L-J) and Rydberg potentials. The thermodynamic properties were calculated by using the Morse potential parameters in a temperature of 298.2 K to 1500 K. For comparison, the heat capacity and entropy were also obtained with the usage of the DFT-PBE1 or QCISD (T) methods.

2. Computational Methods

Computational methods usually produce more accurate results when the higher level method is used. However, it is computationally too expensive or impractical to use methods such as CASSCF to establish the potential energy surface of UC, UN, and UH. Then, we selected the less expensive ab initio CI, CCSD, QCISD (T), and DFT-PBE1 methods to determine the potential energy surface. The calculations described in this paper were performed with the Gaussian 09 package [14]. The DFT method is at the PBE1 level. The basis sets for uranium is ECP80MWB_AVQZ + 2f and 6 − 311 + G for carbon, nitrogen, and hydrogen [15]. The basis set of ECP80MWB_AVQZ + 2f includes a quasirelativistic effective core potential, which represents the relativistic effects largely confined to the core, and an augmented correlation consistent valence quadruple aug-cc-pVQZ (AVQZ level) basis set [16] for the valence, together with 2f in the valence. As a whole, the results from the f-in-core and f-in-valence pseudopotentials are in good agreement [17], and the quality of AVQZ is so good that it is comparable to the complete basis set (CBS) in most cases. This basis set guarantees its well behavior for U. The CI, CCSD, QCISD (T), and DFT methods are reliable for the investigation of transition metal compounds [18, 19]. After the potential energies being obtained, a Morse function was fit as follows:

The eigenvalue of the Morse potential isandwhere ω_e is the harmonic vibrational frequency, D_e is the minimum point in the Morse curve that equals to the dissociation energy in a diatomic system, β is a parameter that is related to the width of the Morse potential curve, μ the reduced mass, and χ_e the anharmonic constant. For comparison, we also fit the Lennard-Jones and the Rydberg interatomic potential functions, respectively, as follows:

By neglecting the contribution of the electronic energy of the excitation states to its thermodynamic properties, the heat capacity and the entropy were calculated. The heat capacity and entropy were obtained on the basis of the vibrational frequencies from DFT-PBE1 or QCISD (T) calculations and the fitted Morse parameters.

3. Results and Discussion

3.1. Potential Energy

The potential energy surfaces of UC, UN, and UH were obtained by the CCSD, CI, QCISD (T), and DFT-PBE1 methods. It is worth noting that the lowest lying states were selected in the potential energy scanning. The triplet was adopted for UC, and the doublet was adopted for UN and UH after comparing the energies of different multiplicities. For instances, the total energies of UC are −75.22439079 a.u. (r₀ = 2.25 Å) and −75.28543016 a.u. (r₀ = 2.00 Å) for the heplet and triplet, respectively, at the PBE1 level by using the ECP80MWB_AVQZ + 2f basis set for U and 6 − 311 + G basis set for carbon. The triplet UC is more stable than its heplet state. The dissociation energies of UC, UN, and UH are 5.7960, 4.5077, and 2.6999 eV at the QCISD (T) levels, respectively.

Table 1 lists the parameters of potential functions of UC, UN, and UH. As can be seen from Table 1, the data from PBE1 were well fitted to the Morse function for UC and UN, in view of larger values of R-square and smaller values of RMSE. However, the best results for UH are from the QCISD (T), which could be well fitted to the Morse or Rydberg functions. Figures 1–3 show the calculated results from PBE1 or QCISD (T) and the fitted Morse curves. As can be seen from these figures, the fitted Morse functions behave almost exactly as the potential energies of PBE1 or QCISD (T). Similarly, the fitted Rydberg functions behave almost as Morse functions. For UC and UN, the Morse functions are slightly better than the Rydberg functions, but the latter is slightly better for UH judged by the values of R-square and RMSE. However, the correlation coefficients for the fitted Lennard-Jones functions are in the range of 0.6199 to 0.9253, and root mean square errors are in range of 1.326 to 3.608 eV, indicating the L-J is not a good model for UC, UN, and UH potentials. The L-J potential is generally good approximation to describe the dispersion and overlap interactions in molecules. However, it is not good to a strong U-X bond, since the twelfth-power repulsive term appearing in the Lennard-Jones potential is chosen for its ease of calculation for simulations (by squaring the sixth-power term) and is not physically based. For brevity, only the drawings from the Morse functions are shown in Figures 1–3. In addition, the potential curve from the CI method is not continuous for UH. Its potential drops all of a sudden at r = 6.3 Å, indicating that the CI method is not applicable for UH. Therefore, we did not apply this method to establish a potential function for the UH system.

For UC, the values of D_e and β of the Morse function fitted from DFT-PBE1 results are 5.152 eV and 1.583 Å⁻¹, respectively. Thus, ω_e = 783.97 cm⁻¹ after substituting the β and D_e values into equation (3). By substituting these values into equation (4), the anharmonicity constant χ_e = 0.004716 and χ_eω_e = 3.70 cm⁻¹. The anharmonic frequency is 780.27 cm⁻¹, which is slightly smaller than the scaled frequency (831.58 cm⁻¹) obtained from DFT-PBE1 when it is scaled by a factor of 0.96.

For UN, the values of D_e and β of the Morse function fitted from DFT-PBE1 results are 5.08 eV and 1.79 Å⁻¹, respectively. Thus, ω_e = 816.72 cm⁻¹ after substituting the β and D_e values into equation (3). Then, the anharmonicity constant χ_e = 0.004983 from equation (4), and χ_eω_e = 4.07 cm⁻¹. The anharmonic frequency is 812.65 cm⁻¹, which is also slightly smaller than the scaled frequency (887.18 cm⁻¹) obtained from DFT-PBE1 when it is scaled by a factor of 0.96.

For UH, the values of D_e and β of the Morse function fitted from QCISD (T) results are 2.643 eV and 1.235 Å⁻¹, respectively. Thus, ω_e = 1475.3 cm⁻¹ after substituting the β and D_e values into equation (3). Consequently, the anharmonicity constant χ_e = 0.01730 from equation (4) and χ_eω_e = 25.5 cm⁻¹. The χ_eω_e value of UH is much larger than that of UC or UN. The anharmonic frequency of UH is 1449.8 cm⁻¹, which is slightly smaller than the scaled frequency (1468.46 cm⁻¹) obtained from QCISD (T) after being scaled by a factor of 0.96. Luo et al. [20] used the B3LYP/SDD method to derive D_e = 2.886 eV and ω_e = 1540.403 cm⁻¹, which are close to our results. Since the SDD basis set is small and the RMSE from the DFT method is larger than that from QCISD (T) (Table 1), our results are expected to be more reliable.

3.2. Thermodynamic Properties

Tables 2–4 list the entropy and heat capacity of UC, UN, and UH, respectively, in a temperature of 298.2 K to 1500 K. The results from the Morse potentials for UC and UN are almost identical to those of DFT-PBE1 calculations. The results from Morse and QCISD are similar for UH. The C_P values increase very slightly with increasing temperature. The entropies also increase as temperature rises. The relationship between entropy and temperature can be fitted as a linear equation with large correlation coefficient. From the Morse results and DFT-PBE1 results of UC, the established equations are S = 264.6 + 0.0262 T–10150.0 T⁻¹ and S = 263.9 + 0.02624 T–10010.0T⁻¹, respectively. The correlation coefficient R² is 0.999 for both, with a root mean square error of 0.694 J mol⁻¹ K⁻¹ for both Morse and DFT-PBE1 results. From the Morse results and DFT-PBE1 results of UN, the established equations are S = 264.6 + 0.02623 T–10060 T⁻¹ and S = 263.7 + 0.02626 T–9859 T⁻¹, respectively. The correlation coefficient R² is 0.999 for both, with a root mean square error of 0.692 J mol⁻¹ K⁻¹ for Morse results and 0.687 J mol⁻¹ K⁻¹ for DFT-PBE1 results. From the Morse results and QCISD results of UH, the established equations are S = 239.1 + 0.02575 T–8818 T⁻¹ and S = 263.7 + 0.02626 T–9859 T⁻¹, respectively. The correlation coefficient R² is 0.999 for both, with a root mean square error of 0.596 J mol⁻¹ K⁻¹ for Morse results and 0.598 J mol⁻¹ K⁻¹ for QCISD results.

4. Conclusions

Many conventional and standard basis sets fail to converge and/or produce irrational results for the U-containing system. The basis set of ECP80MWB_AVQZ + 2f is a well behaved basis set for U. The anharmonicity constant of UC and UN is small, but that of UH is large. The Morse function is suitable for the UC, UN, and UH potentials, so is the Rydberg function. The entropy and heat capacity of UC and UN in a temperature of 298.2 K to 1500 K from the Morse potential are close to those from the DFT-PBE1 results. The entropy and heat capacity of UH from the Morse potential are comparable to those from QCISD (T). The C_P values increase slightly, but the entropies increase greatly with increasing temperature for UC, UN, and UH. The predicted functions provided useful parameters for establishing the force field of the U-containing system.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare no conflicts of interest.

Authors’ Contributions

Xue-Hai Ju conceptualized the study; Shuang-Ling Tang and Yu Wang were involved in data curation; Xue-Hai Ju was responsible for project administration; Qi-Ying Xia was involved resource management; Shuang-Ling Tang wrote the original draft; and Xue-Hai Ju and Qi-Ying Xia wrote, reviewed, and edited the manuscript.

Acknowledgments

This work was financially supported by the National Science Foundation of China (Grant no. 21101070).

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Copyright © 2020 Shuang-Ling Tang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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