Ab Initio Intermolecular Potential Energy Surfaces and Cross Second Virial Coefficients for the Dimer N2-NO.

Ab initio intermolecular potential energy surfaces (PES) of N2-NO have been constructed at the level of theory CCSD(T) with the augmented correlation-consistent basis sets aug-cc-pVmZ (with m = 2, 3, 4). The nitrogen in the closed-shell electronic configuration X1Σ+ and nitric oxide in the open-shell electronic configuration A2Σ+ were employed to calculate ab initio intermolecular interaction energies. The two new ab initio 5-site intermolecular pair potentials at the theoretical level CCSD(T)/aug-cc-pVmZ (with m = 4, 24) were developed appropriately and are suitable for N2-NO dimer by using the nonlinear least-squares fitting method combining MIO and Levenberg-Marquardt algorithms. The correlation quality of these two potentials was found to be very good with R 2 values in the range of 0.98372 to 0.99775. The cross second virial coefficients B 12(T) of the N2-NO dimer were calculated in the temperature range of 100 to 470 K using the two ab initio 5-site potentials. The discrepancies between the calculated results and the experimental data can be acceptable.

Introduction

Under standard conditions, two nitrogen atoms formed a nitrogen molecule. Nitrogen gas with the formula N2 is colorless and odorless.[1] Many important industrial compounds are synthesized from nitrogen such as ammonia, nitric acid, organic nitrates (propellant and explosives), and cyanide, which contain nitrogen.[1] The nitrogen bond is extremely strong. This makes it difficult to convert N2 into useful compounds.[2] During combustion, the explosion or decomposition of nitrogen compounds forms nitrogen gas with an amount of NO and energy.[3] In biology, nitric oxide gas is an important molecule that plays a signaling role in many physiological and pathological processes.[4,5] Understanding the thermodynamic properties of the N2-NO mixture is important for practical applications. We need to find a way to determine the thermodynamic properties of the N2-NO system. The experimental data of this system are difficult to measure with the experimental techniques. It is also necessary for industrial application.[6]

Today’s computers have become important research media, and they have evolved both in quantity and complexity.[7,8] With the current computer technology, it is possible to solve many subjects on different systems when data are not available or just require approximate research results. The subject of prediction of physical and chemical properties of liquid and solid systems has used the simulation method, which results in the study using intermolecular interaction potential.[9,10] The properties of a system are influenced primarily by intermolecular forces. Consequently, simulation techniques using the interaction potential built from ab initio calculation have become an essential tool for studying the nature of various liquids and liquid mixtures.[11,12] Potential functions built by the experimental way are not very meaningful. By simulating the thermodynamic and structural properties of liquids can be influenced without being limited by experimental measurement techniques. Simulation techniques are instantly becoming an essential tool for predicting the nature of liquid systems.[9] These could allow determination of the thermodynamic properties and fluid–structure without being constrained by practical techniques for calculating multidimensional integral equations in statistical thermodynamics. The ab initio intermolecular potential functions can be used for Monte Carlo simulations.[12,13]

Furthermore, in the atmospheric environment, the intermolecular interactions between molecules NO and N2 in air are known to increase collision absorption in transition regions. Each monomer always has a certain polarity; especially, the NO molecule can exist in various electronic states. This has also been determined based on the structural spectrum of 8000 to 105,000 cm–1 such as X1Σ+, A2Σ–, and A2Σ+.[14] In the case of the electronic state X1∑+, the NO+ molecule can easily interact with other gas molecules in the atmosphere.[15] The potential surface for NO+ in the ground state X1Σ+ was determined by the ab initio theoretical calculation at the Hartree–Fock, CASSCF, MR-CI, and MP2 levels proposed by Fehér and Martin.[15] Their research results for the quadrupole moment and the dipole polarisabilities were consistent with the experimental results of the spectroscopy spectrum of NO Rydberg states.[14,15] In the air environment, the interaction of solar UV radiation with both N2 and NO molecules has become a research interest, which is important for the energy process that includes photoabsorption, photodissociation, and photoionization. In the recent efforts of Lavín et al.’s group, the nitrogen N2 in closed-shell electronic state X1Σ+ and nitric oxide NO in open-shell state X2Π were employed in evaluating the property of the collision absorption interaction to resolve a part of the data not available.[16] The gas molecules N2 and NO have been identified as abundant in the atmosphere of industrial areas. These gas molecules are too abundant in the universe, and there are also significant amounts of N2-NO dimer, and they can be generated by different reactions that can be found in the atmosphere.[17]

Two recent separate studies on the molecular system NO and N2 were reported by Lozeille et al.[18] and Costen et al.[19] The à ← X̃ transition of mixtures between NO and N2 was explained by generating the REMPI (1 + 1) spectra. The results obtained from ab initio calculations RCCSD(T)/aug-cc-pVQZ//QCISD/6-311+G (2d) have demonstrated the signs that the NO space rotates more freely N2 in the NO-N2 system. Interactions in the NO-N2 system are still not well understood. This is still something to look at more fully, and this will also be a challenge.[18,19] The X̃ state of the interactive geometry for the NO-N2 dimer has also been optimized at theoretical levels from MP2/6-311G* to QCISD/6-311+G(2d). Lozeille et al. have shown the molecular configurations as NO-N2 and ON-N2 straight form and the T-shaped direction (NO-N2 and ON-N2).[18] The theoretical levels QCISD/6-311+G(2d) and RCCSD(T)/aug-cc-pVTZ and aug-cc-pVQZ have been implemented by Lozeille’s group for these configurations. The interaction energy has been corrected for BSSE using the full counterpoise correction. These are the research efforts for the interaction of the NO-N2 dimer of Lozeille’s group.[18]

The studies of Costen et al.[19−22] showed that the à state is useful to calculate for the NO-N2 configuration. The Rydberg 3s orbital state for NO corresponds to an excited electron. Costen’s group noted that, to make a more reliable energy difference, the theoretical level RCCSD(T)/aug-cc-pVTZ was employed for the selected energy calculation. This has been demonstrated through a molecular imaging ion beam velocity imaging study. In their study, the authors also pointed out that angular alignment moments can be extracted as a function of the internal energy of N2 in the system NO + N2. EBustos et al.[13] have also been successful in studying the collision interactions between N2 and NO as well as the electronic transitions based on ab initio calculations at the MQDO theoretical level. As we know, the study of the collision of NO(A2Σ+) with other gases such as Ne, He, and D2 was also done by Costen et al.,[20,21] Steill et al.,[22] and Pajón-Suárez et al.[23] Both the simple classical hard-shell model and quantum scattering were employed to calculate an ab initio potential energy surface (PES) to conduct the theoretical prediction. The obtained results were compared with experimental results. Costen et al.[20,21] performed the same survey using experimental testing of ab initio potential energy surfaces specifically for stereodynamics of the state of NO(A2Σ+) + Ne inelastic scattering at multiple collision energies. The experimental results were consistent with the results of close-coupled quantum scattering calculation on two ab initio potential energy surfaces proposed by Pajón-Suárez et al.[23] and Costen et al..[20,21] Moreover, Costen et al.[21] performed a research in molecule-atom and molecule-molecule rotational energy transfer NO(A2Σ+) + He and D2. These also are successes in proving the existence of collision interactions between NO(A2Σ+) and He. This has helped to understand the spectra of these systems, especially when combining the experimental research with ab initio calculations. From studies on the collision of the excited states of NO with gas molecules, the results of ab initio calculation at the RCCSD(T) level of the theory of the interaction of NO in the electronic state (A2Σ+) with rare gas molecules were found to be a good qualitative agreement with the previously published experimental spectra. We have seen that this is also a somewhat difficult direction to a theoretical study.[23,24] This can result in high quality of the potential energy surface available for systems He-NO and Ne-NO.[22−24]

In this research, we report the construction of different orientations of the N2-NO dimer. The nitrogen in closed-shell electronic state X1Σ+ and nitric oxide in open-shell electronic state (A2Σ+) are employed for ab initio calculation. The ab initio intermolecular energy is calculated at the theoretical level CCSD(T)/aug-cc-pVmZ (m = 2, 3, 4) using Basis Set Superposition Error (BSSE) correction method. Moreover, the ab initio energy of complete basis sets limit aug-cc-pVmZ (m = 23, 24, 34) are extrapolated by a couple of three basis sets aug-cc-pVmZ (m = 2, 3, 4), respectively. The 5-site ab initio intermolecular potentials of the N2-NO dimer are constructed from ab initio energy values using the nonlinear least-squares technique that associates the Maximum Inherit Optimization (MIO) algorithm and the Levenberg–Marquardt method. The virial second cross coefficients of the N2-NO dimer are calculated using the numerical integration technique. The cross second virial coefficients for the N2-NO dimer obtained from the ab initio 5-site potentials are compared with experimental data and with those from the equations of state.

Results and Discussion

Potential Energy Surface

To verify that the calculation method CCSD(T) and the different basis sets can yield adequacy to describe the dimer N2-NO, we approached to perform calculations without restrictions of the bond length of molecules N2 and NO in optimizing T-configuration N2-NO (α = 90, β = 0, ϕ = 0). The calculation results, listed in Figure , show that the CCSD(T) calculation with the extrapolated basis sets gives the acceptable results with the corresponding values ARE,%. The value ARE,% where the bond lengths rcal for N2 and NO have been calculated at the theoretical levels CCSD(T)/aug-cc-pVmZ (with m = 2, 3, 4, 23, 24, 34), the experimental bond length (Exp.) rexp. are taken from the literature,[6,14] and the percentage of the relative deviation of experimental value (Exp.) from the calculated value 100|rexp. – rcal|/rexp. are given. We compared the interatomic lengths Å for the N2 and NO molecules according to different cardinal numbers m of the complete basis set (CBS) aug-cc-pVmZ (with m = 2, 3, 4, 23, 24, 34). We found that the ARE,% values resulting from the basis set aug-cc-pVDZ (i.e., with a cardinal number, m = 2) are much larger than those from other basis sets. Furthermore, we found that the bond lengths of nitrogen and nitric oxide derived from the basis sets aug-cc-pVmZ (with m = 4, 24) are in good agreement with experimental data (Exp.),[14] and these present the lowest values ARE,%, as presented in Figure . Furthermore, the difference between ARE,% values of two basis sets aug-cc-pVmZ (with m = 4, 24) is insignificant. In general, the bond lengths of nitrogen and nitric oxide obtained from the basis sets aug-cc-pVmZ (m = 23, 24, 34), which are extrapolated according to the appropriate equation eq ,[25−29,36,41] are very close to the experimental values (Exp.). Thus, this proves that the basis sets aug-cc-pVmZ (with m = 4, 24) are most suitable for intermolecular energy calculation of the system N2 and NO. Besides, these molecules also demonstrated to be not significantly deformed upon making an interaction with each other, or vibration is not to be expected with low-speed collisions, although the NO molecule can have many different electronic states as demonstrated by Huber and Herzberg[14] and the published articles.[18−24]

Figure 1

Interatomic lengths of N2 and NO are calculated at the theoretical level CCSD(T)/aug-cc-pVmZ (m = 2, 3, 4, 23, 24, 34): (a) bond length N–N; (b) bond length N–O.

It is our experience, however, that the extrapolation scheme is adequate.[34,36,41] The use of extrapolation equation eq is acceptable to the calculation of ab initio energy. Moreover, the extrapolation results proven are well consistent with experimental values, as demonstrated in the published articles.[11,12,25−29] In all cases, the extrapolation to the complete basis set (CBS) limit has a significant effect on the calculated energy values. From the considered results, we can utilize the basis sets aug-cc-pVmZ (m = 4, 24) for all calculations. However, the calculation time of the basis set aug-cc-pVQZ for each configuration can take a lot of hours. So, computational cost will be more expensive than the use of the complete basis set limit aug-cc-pVmZ (m = 24). As well, it is generally shown that the minimum values of the potential energy surfaces resulting from the theoretical levels CCSD(T)/aug-cc-pVmZ (m = 4, 24) are more appropriate, as shown in Figure .

Figure 2

Intermolecular potential energy surfaces of special orientations L, T, H, and X for the dimer N2-NO are calculated at the theoretical level CCSD(T)/aug-cc-pVmZ (m = 2, 3, 4, 23, 24, 34): (a) comparison of the potential depth at theoretical levels; (b.c) contour plot of potential surface vs rotation angles α, β, and ϕ for 15 different orientations with 436 data points.

The interaction between N2 and NO can have many different directions due to the asymmetric properties of the NO molecule. We looked at ab initio calculations for different interaction directions of the N2-NO dimer according to angular coordinates. The equilibrium locations and depths of the potential minima for four special orientations L, T, H, and X (in Figure a) are determined, as also shown in Table . It pointed out that the potential depths of the orientations derived from the basis set aug-cc-pV24Z are lowest. This also showed that the influence of the theory level CCSD(T) with various basis sets for interaction configurations (in Figure a) of the N2-NO dimer is very important.

Table 1

Comparison of Minimum Energies Emin/μEH and Equilibrium Distance rmin/Å of the Dimer N2-NO for Special Orientations in Figure at the Level of Theory CCSD(T)/Aug-Cc-pVmZ (m = 2, 3, 4, 24)

		aug-cc-pVmZ
		m = 2		m = 3		m = 4		m = 24
special dimer		rmin/Å	106Emin/EH	rmin/Å	106Emin/EH	rmin/Å	106Emin/EH	rmin/Å	106Emin/EH
L	N2-NO	5.8	0.035	5.2	–31.592	5.2	–42.476	5.2	–49.916
T	N2-NO	4.6	–123.125	4.6	–137.548	4.4	–147.090	4.4	–150.853
H	N2-NO	4.0	–78.358	3.8	–206.150	3.8	–227.668	3.8	–251.991
X	N2-NO	4.2	–40.661	3.8	–194.061	3.8	–207.954	3.8	–234.755

We have done all quantum chemical calculations at the theoretical level CCSD(T)/aug-cc-pVmZ (m = 2, 3, 4) with the Gaussian03 program package.[35,42,43]

The ab initio interaction energy values of the N2 (X1Σ+) in the closed-shell electronic state and nitric oxide NO(A2Σ+) in the open-shell electronic state were calculated at theoretical levels CCSD(T)/aug-cc-pVmZ (m = 2, 3, 4, 23, 24, 34) for 15 orientations at different separate distances of 2.8 to 12 Å, as given in Table S1.

Ab Initio Intermolecular Potentials

In several published studies, it has been shown that the 5-site pair models give a reasonable representation.[11,12,26−30] It is very important for us to carefully build the structure of a new 5-site potential function. We have developed the ab initio 5-site Morse-style potentials for mixtures H2-O2, N2-H2, and F2-H2.[26−30] In this work the Morse-style 5-site interaction potentials have been modified newly to obtain the global fit of ab initio energy values for the N2-NO. We constructed successfully the new two-body potential function. To select the most suitable potential function, we have tracked and investigated the influence of terms as well as damping functions that can smooth ab initio data in the long range. We have found the ab initio potential functions that are predominantly different from the Morse-type functions published in the interaction terms and damping functions f1(r) and f(r). These two potential functions built here have been greatly modified compared to the ab initio potentials in the published works.[26−29] The new 5-site potentials obtained here are fully adaptable to the ab initio global potential surface that is specific to the dimer N2-NO. According to our experience and previously published articles,[12,26−29] the adjustable parameters De, α, β, and C are optimized appropriately to utilize for the short-range and long-range interactions. The correlation between ab initio energy calculated from the 5-site potential functions and ab initio energy obtained at ab initio calculation CCSD(T)/aug-cc-pVmZ (m = 4, 24) is presented to be very good, as depicted in Figures S1 and S2. The potential functions fitted for the ab initio energy of N2-NO have been evaluated at a 95% confidence level. This is described by the red area. The correlated quality is expressed at R2 values of 0.98545 and 0.99775 for potential eq and 0.98372 and 0.98398 for potential eq , as shown in Figures S1 and S2. The ab initio potential energy surfaces at the theory levels CCSD(T)/aug-cc-pVmZ (with m = 4, 24) of the dimer N2-NO described too many local extrema of 15 different orientations. We have found that this is a big challenge for building 5-site potential functions using the nonlinear least-squares fitting algorithm. This is not a simple task to get the optimal-adjustable parameters of the 5-site potential functions.

To overcome this challenge, we have succeeded in developing a nonlinear least-squares fitting method. Here, the fitting task has to be carried out by two steps. For the first step, the global extrema are grossly located utilizing maximum inherit optimization (MIO).[45,46] This new optimization algorithm combines fitness inheritance with a genetic algorithm. The fits for the initial population of all the individuals are assessed. Afterward, the harmony of individuals in the succeeding population fitted is inherited. This harmony is named inheritance optimization.

Calculation of the Cross Second Virial Coefficient

Calculation of the Ab Initio Potential

The properties of a system depend heavily on the interaction between particles. The phenomena of the macroscopic system can be determined based on the properties of the microscopic system. The virial coefficients characterized the interacting properties of particles. Here, the cross second virial coefficients B12(T) of the N2-NO mixture can be determined by using the numerical integral. This is also a way to check the accuracy of the constituted ab initio 5-site potentials.

The classical cross second virial coefficients B12(cl)(T) are calculated by a 4D Gauss–Legendre quadrature method[44] using the ab initio 5-site potentials eqs and 4 at the theoretical level CCSD(T)/aug-cc-pVmZ (with m = 2, 3, 4, 23, 24, 34). The calculation results pointed out that the basis sets influenced the second virial coefficients, as shown in Figure a,b and Table S4. We found that the ones generated from basis sets aug-cc-pVmZ (with m = 2, 3) are much different. While using the basis sets aug-cc-pVmZ (with m = 4, 23, 24, 34), the results obtained are closer to the experimental values (Exp.)[55] and those from the Deiters equation of state (D1-EOS)[51,52] and Chang Zhao equation of state (CZ-EOS).[53] Specifically, the ones obtained from the basis set aug-cc-pVmZ (m = 4, 24) are best suited, although there are no additional quantum corrections yet.

Figure 3

Comparison of classical cross second virial coefficients Bcl(0)(T) of the N2-NO system resulting from the theoretical level CCSD(T)/aug-cc-pVmZ (with m = 2, 3, 4, 23, 24, 34): (a) 5-site potential eq and (b) 5-site potential eq .

Besides, we realize that there is still a certain difference between the ones derived from theoretical calculation and experimental values (Exp.).[55] So, we can deliberate to add the first-order quantum corrections for classical cross second virial coefficients. The first-order quantum correction terms B(1), Ba, (1), and Ba, μ(1) are supplemented according to the equations proposed by Wang,[50] as presented in Table S4 and Figure . We ascertained that, at low temperatures, all cross second virial coefficients B2(T) corrected are closer to the experimental values and made a certain contribution. However, at high temperatures, the cross second coefficients seem to be higher than the experimental values.[55] This may be because the approximate calculation process is not enough. In particular, this can be caused by the processes of integral calculations, which is also an approximate mathematical process. Below, we can use a few other computational methods from equations of state to calculate cross second virial coefficients and compare them with the results obtained by ab initio 5-site potentials.

Figure 4

Comparison of cross second virial coefficients B12(T) for the N2-NO dimer derived from the method CCSD(T)/aug-cc-pVmZ (m = 4, 24) using potentials: (a) eqs ; (b) 4.

Calculation of Deiters Equations of State (D1-EOS)

The equation of state (D1-EOS) proposed by Deiters[51] is known to be an equation built from the perturbed hard chain theory. This equation can be used to calculate critical temperatures, pressures, and densities of pure components.[52] The equation was also proposed by Deiters to extend the calculation of the thermodynamic properties of binary mixtures. The Deiters equation of state has been substantiated to be standard and highly reliable. So, we can use the Deiters equation of state (D1-EOS) to calculate the cross second virial coefficients for the N2-NO dimer.[51,52] In recent studies, we used the Deiters equation of state to calculate the virial coefficients of the various mixtures H2-O2, F2-H2, and N2-H2, and the obtained results proven were very close to the experimental values (Exp.).[26−28] The cross second virial coefficients for the binary N2-NO mixture resulting from this equation of state are also given in Table S4 and described in Figures and 4. We also recognize that the cross second virial coefficients from the ab initio potentials eqs and 4 at the theoretical levels CCSD(T)/aug-cc-pVmZ (with m = 4, 24) are not much different from those of the Deiters equation of state (D1-EOS).[51,52]

Calculation of the Virial Equation of State

To validate the quality of the developed ab initio 5-site potentials, we can operate virial equations of state to calculate the cross second virial coefficients B12(T) for the N2-NO mixture by using the critical parameters of each component.[6,53−55]

The cross second virial coefficients B12(T) are obtained from eq proposed by Zhao et al.,[53] as given in Table S4. The calculated results from the virial equation of state (CZ-EOS)[53,54] in the Taylor series are in good agreement with those from ab initio 5-site potentials eq and eq and experimental values (Exp.).[55]

For the N2-NO mixture, the cross second virial coefficients from the various calculation methods are tabulated in order of temperature increase, and estimation of uncertainty is given in Figures S3 and S4. The uncertainty of each value in a method is compared with the value of the uncertainty of all experimental values. The uncertainties are determined from deviations between calculated virial coefficients B12(T)cal and experimental values B12(T)exp.[55] In general, we notice that most calculation results are within the uncertainty range of the experimental data, as illustrated in Figures S3 and S4. The differences between the cross second virial coefficients and experimental data[55] are acceptable. The N2 and NO molecules are not quantum molecules, so the contribution of quantum correction effects to classical cross second virial coefficients is not much. This shows that the difference between classical virial coefficients and total virial coefficients is not significant, as exhibited in Figures and 4. This is also noticeable in Table S4. However, there are some differences at high temperatures as we have commented above.

In this research, the high-level quantum chemical method CCSD(T) with correlation-consistent basis sets aug-cc-pVmZ (m = 2, 3, 4) was employed successfully to calculate the ab initio intermolecular energies using the basis sets superposition error calculation[41] with the counterpoise correction.[34,36−38] We can confirm selecting the interatomic lengths of the N2 and NO molecules as a necessary and correct object to help the selection of correlation-consistent basis sets aug-cc-pVmZ (m = 4, 24) appropriately. The bond lengths N–N and N–O are extrapolated precisely based on the eq equation scheme, as shown in Figure . To evaluate the accuracy of the calculated results, we use the percentage of the relative error ARE,%, and the mean of the relative error MARE,%. These can be determined by derivation between calculated and experimental values. From Figure , we can remark that these error values for the bond lengths are represented as values ARE,% 0.027 for N–N and 0.070 for N–O and MARE,% 0.0485 for using the basis set aug-cc-pVQZ and values ARE,% 0.056 for N–N and 0.179 for N–O and MARE,% 0.1175 for using aug-cc-pV24Z. Therefore, the calculation results obtained from the complete basis set selected aug-cc-pVmZ (m = 4, 24) is most suitable for calculating the ab initio interactive energy of the N2-NO dimer.

For classical cross second virial coefficients B12(T) of the N2-NO dimer, we commented that the results were calculated in the temperature range of 100 to 470 K using ab initio potentials eqs and 4 at the theory level CCSD(T) with the complete basis sets limit aug-cc-pVmZ (m = 4, 23, 24, 34) without quantum correction effects and were very close to the experimental values, as shown in Figure . However, in such results, the ones obtained from the theoretical level CCSD(T)/aug-cc-pVmZ (m = 4, 24) are closer to the experimental values. In the case of the addition of quantum correction effects, it has had certain effects on the classical cross second virial coefficients, although these quantum effects are not very much. For the quantum effect terms, the B(1) term contributes more significantly than the angular terms Ba, (1) and Ba, μ(1). The total cross second virial coefficients B12(T) were closer to the experimental values (Exp.),[55] and those were obtained from the Deiters equation of state (D1-EOS)[51,52] and Chang Zhao equation of state (CZ-EOS),[53,54] as can be seen in Figures and 4 and Table S4. However, there are still some small differences, but they are still within the uncertain limits of experimental measurements, as illustrated in Figures S3 and S4.

From the above research results, we can confirm that the accurate calculation of total cross second virial coefficients B12(T) of the dimer N2-NO from the ab initio 5-site potentials without recourse to experimental data can be solved. Furthermore, the cross second virial coefficients of N2-NO derived from ab initio potential eq seem to be more accurate than ab initio potential eq . This can be seen through fitting results, as in Figures S1 and S2. The fitting results of the ab initio potential eq are better than ab initio potential eq because the structure of the potential eq is simpler, and thus, the adjusted parameters of this potential are not complicated, as given in Tables S2 and S3. Therefore, the calculation results derived from the ab initio potential eq are closer to the experimental data[55] and those calculated from the Deiters equation of state (D1 EOS)[51,52] and Chang Zhao equation of state (CZ-EOS).[53] It is found that the differences between them are very small. This denoted that the method used here is suitable for calculating the cross second virial coefficients of the N2-NO dimer.

The most important thing in practice is that we can calculate the cross second virial coefficients B12(T) accurately at different temperatures without the availability of experimental data sources. The CCSD(T) method with basis sets aug-cc-pVmZ (m = 2, 4) allowed an extrapolation to the complete basis sets limit aug-cc-pVmZ (m = 24), where the computational cost can be reduced, and we were able to generate the cross second virial coefficients almost within the uncertainties of the experimental measurements (in Figures S3 and S4). In this case, it is worth noting that the effect of quantum correction has contributed little to the second classical virial coefficients of the N2-NO mixture because the N2 and NO molecules are not the quantum molecules.

Conclusions

We conclude that the two ab initio 5-site potential intermolecular potentials for the N2-NO dimer have been successfully developed from ab initio energy potential surfaces at a high-theory level of CCSD(T) with complete basis sets limit aug-cc-pVmZ (with m = 4, 24). The ab initio potential functions with their adjustable parameters, respectively, are reliable and can be utilized to calculate the cross second virial coefficients for the N2-NO dimer. Therefore, the ab initio 5-site pair potential for N2-NO interactions also becomes important to calculate the thermodynamic properties using the Gibbs Ensemble Monte Carlo (GEMC) simulation that may be useful if the test data is scarce.

Computational Methods

Building Molecular Orientation

The N2-NO dimer can be characterized by a four-dimensional coordinate system. This coordinate system is employed for the ab initio calculation, as shown in Figure . The origin of the coordinates coincides with the center of mass M of the N2 molecule, while the y-axis intersects the center of mass M of the NO molecule. The separation distances between the centers of mass M relate to the coordinates r. The rotation angles α and β wielded depict the orientation of individual molecules in space. These angles are created by the molecular axes of N2 and NO with the y-axis, respectively. The angle ϕ constituted by two planes (xy) and (yz) contain the molecular axes of N2 and NO. The 4D coordinates α, β, and ϕ and r are utilized in the ab initio calculation to generate the potential energy surface. The binding conditions for N2 and NO molecules are assumed as rigid. The bond lengths of N2 and NO molecules employed for the ab initio calculation are taken from experimental data rexp of 115,077 and 109,768 Å for the NO and N2 molecules, respectively.[14]

Figure 5

5-site interaction model and special orientations for the N2-NO dimer.

To have a 5-site pair interaction model for the N2-NO dimer, we have to construct a 5-site model for the N2 molecule (two sites on the atoms N, one site in the center of the molecule gravity (M), and two sites (X) placed between the N atom and center (M)) and for NO molecule (two sites on the atoms N and O, one site A placed between atom N and center (M), and one site B placed between atom oxygen (O) and center (M)). The nitrogen (N2) and nitric oxide (NO) molecules are linear molecules. The 5-site intermolecular interaction potential presented is a function of the four-dimensional coordinate system, including separate distance r/Å and three angles, α, β, and ϕ, as explained in Figure .

Interaction energies were calculated for all changing values, r/Å from 2.8 to 12 Å with an increment of 0.2 Å; the angles α, β, and ϕ were varied from 0 to 180° with an increment of 45°. Care was taken to recognize identical configurations to reduce the computational workload.

Construction of the Potential Energy Surface

Calculation of the Basis Set Superposition Error (BSSE)

In this study, we use the theoretical level of CCSD(T) with Dunning basis sets aug-cc-pVmZ (m = 2, 3, 4).[34−40] This method was selected for ab initio calculation based on the published studies[14−24] for the N2 and NO molecules in electronic states X1Σ+ and A2Σ+, respectively, and especially, the NO molecule was explained in the A2Σ+ open-shell electronic state by experimental spectroscopic measurements.[14] To get accurate results, it is necessary to validate the suitability of basis sets aug-cc-pVmZ (m = 2, 3, 4). The coupled-cluster correlation correction method presents one of the most successful approaches to account for the many-electron molecular systems.[34−38] The quantum chemical methods have been developed to include some effects of electron correlation.[34−36] The intermolecular interaction energies can be extrapolated to complete basis set limits (CBS). Details of the calculation procedures can be performed by below eq .[41]

The ab initio potential energy surface of the N2-NO dimer was determined at the coupled-cluster CCSD(T) level of theory with an augmented correlation consistent. Because of the diffuse, wide-ranging nature of dispersion force fields, it is necessary to adopt the appropriate basis sets.[40,41] Here, we utilized the correlation-consistent basis sets aug-cc-pVmZ (with m = 2, 3, 4) proposed by Dunning et al.[34−39] For the nitrogen (N2) molecule, its ground electronic configuration is 1σ21σ22σ22σ21π43σ2X1Σ+.[14,16] For the nitric oxide (NO) A2Σ+ molecule, its ground electronic configuration is the open-shell structure 1σ22σ23σ24σ21π45σ22π1X2Π.[14,16] The results of ab initio calculation were performed using the BSSE correction of Boys and Bernardi techniques.[34−38,41] This is a technique widely used for the ab initio potential surface calculation for the weak binding interactions. To perform the BSSE correction in this study, we have assumed that, for a suitable method and the size of the CCSD(T) method, some properties of the AB dimer system can be given as follows:where E(AB) indicates the total electronic energy of a dimer AB, E(Ab) is the energy of a dimer consisting of an A atom and a B ghost atom (an atom without a nucleus and electrons but with its orbitals), and E(aB) is vice versa.

Extrapolation of the Complete Basis Set Limit (CBS)

The ab initio energy of complete basis set limit (CBS) aug-cc-pVmZ (m = 23, 24, 34) can be extrapolated from the results of ab initio calculation CCSD(T) with basis sets aug-cc-pVmZ (m = 2, 3, 4). This will probably bring more accurate results and reduce the calculation costs after adjusting the BSSE according to the Boys and Bernardi method.[34−38,41] The counterpoise-corrected supermolecular calculations at the CCSD(T) level were performed for nearly 436 different configurations at different coordinate sets. The electronic energies are then extrapolated to the complete basis set limit used[25−29,33,41]where m denotes the cardinal number of the basis sets aug-cc-pVmZ (m = 2, 3, 4), c is the adjustable parameter, ECCSD(T)corr(m) is the ab initio energy resulting from basis sets aug-cc-pVmZ (m = 2, 3, 4), and ECCSD(T)corr(∞) is an approximation to CCSD(T)/aug-cc-pVmZ (m = 23, 24, 34). This ab initio energy obtained corresponds to a couple of three basis sets aug-cc-pVmZ (m = 2, 3, 4).

Cross Second Virial Coefficients

Construction of Ab Initio 5-Site Potentials

In this study, the Morse-style 5-site potentials are chosen to have a global fit of the potential surface of N2-NO dimer, which are similar to the ab initio potential functions developed by van Tat et al. and Van and Deiters.[26−29] However, they are much different from the Morse-style potentials devepoled by van Tat et al.[26−29] We developed two new 5-site potentials eqs and 4. The two 5-site potentials are reconstructed by the long-range terms and Damping function as

The damping functions f1(r) and f(r) are similarly selected in published studies.[11,12,30−33]

The overall 5-site intermolecular interaction function for the N2-NO system is the sum of all 5-site interactions between the two moleculesHere, r is the separate distance between site i in the first molecule and site j at the second molecule, D presents the depth of potential, C coefficients specify the dispersion interaction in the long range, α and β parameters denote the width of the potential, Ω is a set of angular coordinates (α, β, ϕ), ε0 is the dielectric constant, and q and q are electric charges of sites i and j, respectively. These potentials differ in the use of the damping function f1(rij) proposed by Bock et al.[30] and the Tang and Toennies function f(r).[39]q and q are calculated by fitting to the electrostatic potential. The value δ in the damping equation f1(r) can be changed manually. Besides, the power value of the damping equation f1(r) is adjusted in the range of 10 to 20.[30] The best value for smoothing ab initio potential is selected to be 15.

To fit the 5-site potentials 5-site eqs and 4 for ab initio interaction energy, we have to determine the initial parameters by combining the genetic algorithm (GA) with the maximum inherit optimization (MIO) method.[45,46] For the genetic algorithm, the remaining individuals receive appraised fitnesses.[45] Subsequently, the fitness variety in the population can disappear rapidly, and the population will abort convergence so that the GA can be the failed prophecy to search the global optimum.[45] As a result, it is significant to have an optimal inheritance proportion so that a maximum speedup can be produced. The initial parameters resulted from the MIO method.[46] These can be optimized with the Marquardt–Levenberg algorithm.[47,48] The parameters of the 5-site interaction potentials are given in Tables S2 and S3. The fitting quality of the 5-site potentials eqs and 4 for the N2-NO dimer are depicted in Figures S1 and S2, respectively.

Using Ab Initio 5-Site Potentials

The cross second virial coefficient is a function of temperature dependence. In the case of the N2-NO dimer, we can consider supplementing the quantum correction effects using the Pack expansion eq .[49] Thus, the total cross second virial coefficients B12(T) of the N2-NO dimer can be calculated as the sum of the classical term and first-order quantum correction terms eq . Quantum terms can be calculated using the analytical expression of Pack[49]

In Eq., the classical cross second virial coefficient is calculated using the equation below[31−33]where α, β, and ϕ are angular coordinates (in Figure ), r is the separate distance, NA is Avogadro’s number, kB is the Boltzmann constant, and T is the absolute temperature in kelvin.

The first-order quantum correction terms B(1), Ba, (1), and Ba, μ(1) are calculated using Wang’s equations[50] that have been analyzed from the Pack expansion.[49] We have used these terms to calculate the cross second virial coefficients for several mixtures in published works.[26−29]

Using Equations of State

As we all know, the Deiters equation of state (D1-EOS)[51,52] is widely used for determining the properties of gas and liquid systems. In this study, we also used the Deiters equation of state (D1-EOS) to validate the cross second virial coefficients for the N2-NO dimer.

Besides, we use the virial equation of state,[6,53−55] which is presented in the following formwhere Bmix is the second virial coefficient for mixture, Z is the compressibility factor, P is the pressure, Vm is the molar volume, T is the temperature (K), and R is the gas constant.

For a binary mixture, the mixing rule for Bmix becomeswhere B(T) is the cross second virial coefficient.

This equation can be combined with the virial equation of state in the form of the Taylor series proposed by Zhao et al.[53,54] to calculate the cross second virial coefficients for the mixture N2-NO using the critical parameters that we have already known.[6] This equation of state is widely used in practice to calculate properties of different gases and hydrocarbons. It gives high-reliability results when using critical parameters and acentric factors of corresponding components. The cross second virial coefficients B12(T) for the N2-NO dimer can be obtained by eq using the critical parameters and acentric factors[6,54] as

To calculate the cross second virial coefficient, B12(T), a combining rule can be used to determine the cross second virial coefficients using the critical properties of two components 1 and 2. The easier method is usually used in engineering applications. Moreover, the combining rules using the corresponding state correlations are developed for pure components according to T and P.[54] The combining rules employed to determine the cross critical properties are

The parameter k′12 is the binary interaction parameter. If experimental data is not available, it is customary to set k′12 = 0.

These combining rules (eq ) and (eq ) lead to

Then, TC,12, PC,12, and ω12 are used in the virial coefficient correlation to obtain B12(T), which is subsequently incorporated into eq for the mixture.[53]