Ab initio studies of the Rg–NO⁺(X¹Σ⁺) van der Waals complexes (Rg = He, Ne, Ar, Kr, and Xe)

van der Waals complexes of rare gas (Rg) with NO radical are prototype systems for the interaction of Rg with open-shell species because of their theoretical and experimental accessibility.^1,2 Recently, there has been a great interest in the study of the spectroscopy of highly excited Rydberg states of NO interacting with Rg atoms, up to the electron dissociation leading to the formation of the NO⁺(X¹Σ⁺) cation.^3–5 As a consequence, the interaction of Rg with NO⁺ has also been the subject of numerous theoretical and experimental studies. Previous experimental and theoretical studies determined equilibrium geometries, vibrational frequencies, and ionization and dissociation energies of Rg–NO⁺ complexes.^6–16 

Among the Rg–NO⁺ van der Waals complexes, the Ar–NO⁺ was certainly the most studied. On the experimental side, in 1984, Sato et al.¹⁷ identified it for the first time. Later, Takahashi⁶ measured the adiabatic ionization energy of the Ar–NO complex (73 869 ± 6 cm⁻¹) using multiphoton ionization threshold photoelectron spectra and allow for the determination of the Ar–NO⁺ ground state structure, vibrational frequencies, and dissociation energy. On the theoretical side, Wright and co-workers^9,11,12 investigated the equilibrium geometry, the binding energy, and the vibrational and rotational structures of Ar–NO⁺ using different ab initio methods. A bent structure where the Ar is on the N-side was found. Meanwhile, Robbe et al.¹⁸ determined also a bent structure for this cation, but with Ar on the O-side showing that the determination of the equilibrium structure of the complex is not straightforward. The vibrational structure of the Ar–NO⁺ complex was determined theoretically by Fourre and Raoult.⁷ Very recently, Halvick et al. constructed a new potential energy surface (PES) for the Ar–NO⁺ complex with explicitly correlated coupled cluster method with single, double, and noniterative triple excitations [CCSD(T)-F12] and correlation-consistent, triple-zeta (cc-pVTZ-F12) basis set¹⁴ and confirmed the prediction of Wright and co-workers, i.e., the Ar–NO⁺ equilibrium structure is of skewed T-shape type with Ar atom being closer to the N-side. This new PES was used in quantum close-coupling scattering and variational calculations to treat the nuclear motions. The theoretical bound state energies of the Ar–NO⁺ complex were in good agreement with the available experimental results.

The lightest in the Rg–NO⁺ series, the He–NO⁺ complex, was also the subject of several studies. Robbe et al.¹⁸ computed the interaction PES of the He–NO⁺ complex and found that an equilibrium structure with the He atom lying on the O-side of the NO⁺ moiety. The He–NO⁺ PES was also derived by Soldán et al.⁸ These authors also provided results for the geometry, vibrational frequencies, and dissociation energy of the ground state of the complex. A bent structure where the He is on the N-side was found.

Finally, systematic investigation of the Rg–NO⁺ complexes (including Kr and Xe rare gas atoms) was performed by Lee et al.¹² that determined complexes equilibrium geometries, binding energies, and harmonic vibrational frequencies of all the complexes. They employed different ab initio methods ranging from the Møller-Plesset second order perturbation theory (MP2) to coupled clusters approaches. Recently, Si-sheng et al.³ performed similar studies on Rg–NO⁺ complexes and their neutral Rg–NO counterpart.

In this work, we aim at applying the explicitly correlated coupled cluster treatment to Rg atoms (ranging from He to Xe) interacting with the NO⁺ cation. Using the explicitly correlated coupled cluster methods that include single, double, and non-iterative triple excitations (CCSD(T)-F12)^15,16 with correlation-consistent, triple-zeta cc-pVTZ-F12 and cc-pVTZ–PP–F12 (for Kr and Xe atoms) basis sets, we compute potential energy surfaces, harmonic frequency calculations, and electric dipole moment cut along the radial coordinate and we compare them with available experimental data and previous theoretical work.

The explicitly correlated CCSD(T)-F12 has been proven to be a robust method for the calculations of interaction potentials of van der Waals complexes.¹⁹ Indeed, in the CCSD(T)-F12x (x = a, b) methods,^15,16 functions with the interelectronic distance are explicitly incorporated into the wave function that enables much smaller basis sets to be used for reaching the quality of results comparable to the complete basis set (CBS) limit. As an illustration, in geometry optimizations, the result obtained at the CCSD(T)-F12x/cc-pVTZ-F12 level of theory is similar in accuracy to that obtained with the conventional CCSD(T)/aug-cc-pV5Z calculations which in turn is very close to the CBS limit in terms of equilibrium geometries.^20–24 Approaching the CBS limit at the CCSD(T) level becomes possible as a consequence of these new F12 methods and the corresponding density fitting technique with cc-pVnZ-F12 basis sets.^25–28 

The paper is organized as follows. In the section titled “Computational details,” we present details of our computational methodology followed by discussion of the results and finally we conclude with a summary.

In the present work, the Jacobi coordinate system presented in Fig. 1 was used. The origin of coordinates is placed in the center of mass of the NO⁺ molecule and vector R connects the center of mass of the NO⁺ cation and the Rg atom and θ is the angle between the two distance vectors (θ = 0°: Rg adjacent to nitrogen atom, θ = 180°: Rg adjacent to oxygen atom).

The NO⁺ ground state electronic level (¹Σ⁺) lies approximately 9 eV above the neutral NO(X²Π) level and is well described by a single electronic configuration. The next electronically excited state of the NO⁺ cation, the a³Σ⁺ state, is located ∼6.5 eV above the ground state of the cation and is then well separated energetically.²⁹ Therefore we can expect that the single-reference approach with Hartree-Fock reference wave function followed by the CCSD(T)-F12 method is accurate.

For He–NO⁺, Ne–NO⁺, and Ar–NO⁺, the cc–pVTZ–F12 basis sets of Peterson and co-workers^13,30 were used, whereas, in the Kr–NO⁺ and Xe–NO⁺ calculations, the cc–pVTZ–PP–F12³¹ basis sets were used for both Kr and Xe atoms. The −PP basis sets included a 10-electron relativistic pseudopotential for Kr and 28-electron relativistic pseudopotential for Xe. Basis set was further augmented with a set of bond functions [3s3p2d1f1g]³² at mid-distance between the Rg atom and the NO⁺ center of mass to improve description of the dispersion component of the interaction. The bond functions had the following exponents: sp, 0.9, 0.3, 0.1; d, 0.6, 0.2; and fg, 0.3. The triple excitations do not have direct F12 correction built-in; therefore, we used the scaled version of the triples correction. The scaling is simple multiplication by the ratio of electronic correlation energy calculated at the MP2-F12 level to the one calculated at regular MP2 level (see MOLPRO³³ manual for details). In all CCSD-F12 calculations, the fixed amplitude ansatz is used as implemented in MOLPRO by default, along with the CABS (complementary auxiliary basis set) singles correction for the reference Hartree-Fock energy. The MOLPRO suite of ab initio programs³³ was used to perform the calculations.

Preliminary, we characterize the Rg–NO⁺ complexes by optimizing the equilibrium geometries of the complexes at the CCSD(T)-F12/cc-pVTZ-F12 levels (without mid-bond functions). All internuclear distances including the NO⁺ distance were optimized. The harmonic vibrational frequencies were calculated for the intermolecular bending and stretching modes of the complex and NO⁺ intramolecular stretch. We have also computed harmonic frequency of intramolecular stretch in the isolated NO⁺ cation at the same level of theory. The harmonic frequency is 2374 cm⁻¹ and it compares well to the experimental value of 2376 cm⁻¹.³⁴ This can be compared to the NO⁺ stretching frequency in Rg–NO⁺ complexes. From our 3-D geometry optimization, we find geometric parameters of Rg–NO⁺ equilibrium positions, which we list in Tables I–V along with the De well depths, which we obtain from basis set superposition error (BSSE) corrected calculations for these points.

Calculations of the interaction energies were carried then out for 47 intermolecular distances in the range between 3.5 a₀ and 100 a₀, while the angular grid for the θ variable consisted of 19 values (every 10°). In all the calculations, the internuclear NO⁺ distance was frozen at the equilibrium distance (2.0125 a₀).¹⁹ The interaction energy was corrected for the basis set superposition error (BSSE) for all geometries with the counterpoise procedure of Boys and Bernardi³⁵ such as

where the NO⁺ and Rg subsystems are computed with the full Rg–NO⁺ basis set.

The calculated potentials were uniformly shifted by the corresponding asymptotic value (R = 100 a₀) to force the potential to decay to zero. The origin of size-consistency breakdown is in the MP2-F12 scaling of the perturbative triple excitations, as we mentioned before, since the CCSD-F12 itself is size consistent (within its limits of single-reference applicability).^28,36 The size-consistency errors were −17.65, −177.36, −246.82, −51.87, and −62.78 cm⁻¹ for He to Xe–NO⁺, respectively. At R = 100 a₀, there was no observable angular dependence of size-consistency correction.

The calculated interaction energies of the Rg–NO⁺ complexes were fitted by means of the procedure described by Werner et al.³⁷ for the CN–He system. The relative error in the region of the potential well is below 0.1%-0.2%, which for Xe–NO⁺, for example, amounts to 2-3 cm⁻¹ of absolute error. In the long-range and repulsive region, the relative error is of the order of 1%. The largest deviations between the fit and the ab initio points occur primarily in the repulsive region of the PES. In Tables I–V, we also list equilibrium position and well depths of Rg–NO⁺ complexes obtained from 2D fitting procedure.

Finally, we have performed close-coupling calculations of the dissociation energies of the Rg–NO⁺ complexes using the BOUND program.³⁸ The coupled equations were solved using log-derivative method of Manolopoulos.³⁹ The propagator step size was set to 0.005 a₀ in order to converge the bound states. The basis describing the rotation of NO⁺ molecule included 25, 30, 40, 50, and 60 states for the He–NO⁺, Ne–NO⁺, Ar–NO+, Kr–NO⁺, and Xe–NO⁺, respectively. The maximum propagation distance was set to 80 a₀.

We did calculations for both the CCSD(T)-F12a and CCSD(T)-F12b variants and we did not find significant differences between the two sets of results. This is why, hereafter, we only show and discuss the results obtained with the CCSD(T)-F12a approach as it is recommended when using the triple-zeta type of basis sets.

The optimized equilibrium distance was found at r = 2.013 a₀, R_e = 5.28 a₀, and θ_e = 80.2°. These values were compared to previous ab initio calculations^8,12,40 in Table I. The harmonic vibrational frequencies were calculated to be 41, 110, and 2372 cm⁻¹ for the intermolecular bending, stretching, and NO⁺ stretching modes, respectively. The NO⁺ stretching frequency is in very good agreement with the experimental value of 2376 cm^−1 34 and is only 2 cm⁻¹ lower than frequency of the isolated NO⁺ stretch. This shows that the He atom does not perturb the intramolecular stretching mode of the NO⁺.

In Fig. 2, we show contour plots of our CCSD(T)-F12a/cc-pVTZ-F12 PES as a function of R and θ for r = r_e = 2.0125 a₀. There is a single minimum at R_e = 5.25 a₀, θ_e = 79.6° with a well depth of 196 cm⁻¹ with respect to the He–NO⁺ asymptote. The global minimum of the He–NO⁺ complex is of the skewed T-shape type, with the helium atom preferentially positioned towards the nitrogen side of the NO⁺ moiety. This confirms the previous finding of Soldán et al.⁸ We have performed calculations of dissociation energy from harmonic zero-point energy (ZPE) corrections and from anharmonic calculations on the fitted He–NO⁺ PES with the BOUND program. The dissociation energy of the complex corrected for harmonic ZPE is 142 cm⁻¹ and the anharmonic one 132 cm⁻¹. Both values are given in Table VI along with the values for the rest of the complexes and compared to the existing literature data. Our harmonic ZPE corrected dissociation energy compares well with the one calculated by Si-sheng et al.³ for the He–NO⁺.

The equilibrium geometry of the Ne–NO⁺ complex is found to be for r = 2.014 a₀, R = 5.28 a₀, and θ_e = 80.02°, which is close to that of Lee et al.¹² The harmonic vibrational frequencies are similar to those of the He–NO⁺ complex. Specifically, the frequency of the intramolecular stretch, 2368 cm⁻¹, of the NO⁺ turns out to be slightly lower (by 6 cm⁻¹) in comparison to isolated cation, indicating slight influence of the interaction with Ne affecting this vibration. In Table II, we present a comparison of the equilibrium geometries, vibrational harmonic frequencies, and well depth with those obtained in previous studies.

The contour plot of the 2D fitted PES was shown in Fig. 3. The well depth is D_e = 360 cm⁻¹ for a geometry corresponding to R = 5.38 a₀, and, θ_e = 76.5°, the dissociation energy of the complex is given in Table VI. From the anharmonic bound state calculations, we obtain a value of 300 cm⁻¹ while from the harmonic ZPE correction, the value is 323 cm⁻¹. The harmonic approximation works reasonably well, overestimating the D₀ by only ∼5%. Our harmonic ZPE corrected D₀ is slightly larger than the one calculated by Si-sheng et al.³ 

The optimized equilibrium distance was found at r = 2.015 a₀, R_e = 5.80 a₀, and θ_e = 70°.

The contour plot of the CCSD(T)-F12a PES of the Ar–NO⁺ complex is depicted in Fig. 4. The characteristic points of the PES are listed in Table III along with a comparison to previous studies. This is the only complex for which we can compare to experimental values. The experimental data were obtained from the ZEKE photoelectron experiment,⁶ as mentioned in the Introduction. The values obtained therein were 80.3 and 99.6 cm⁻¹ (harmonic values), which compare very favorably with the values in our calculations in Table III. The interaction with Ar perturbs the NO⁺ intramolecular stretch even to a greater degree than Ne atom as the NO⁺ stretch frequency in the complex is 2359 cm⁻¹, which is by 15 cm⁻¹ smaller than isolated NO⁺ frequency.

The contour plot of the 2D fitted of the Ar–NO⁺ complex is depicted in Fig. 4. The main feature of the PES is a deep and narrow well at the skewed T-shaped geometry, located at θ_e = 66.6° and R_e = 5.88 a₀ and with a well depth of 1024 cm⁻¹. The characteristic points of the PES are listed in Table III along with a comparison to previous studies. As can be seen in Table III, well depth (D_e) in the study conducted by Halvick et al. was found to be ≈45 cm⁻¹ lower compared to our study results. The authors used the same basis set but did not include mid-bond functions. It has emerged that the results of the calculations made using mid-bond functions are closer to the experimental results. Thereby, mid-bond functions improve the accuracy of the ab initio calculations.

In Table VI, we report harmonic and anharmonic ZPE dissociation energies of Ar–NO⁺ that are 930 cm⁻¹ and 927 cm⁻¹, respectively. The harmonic ZPE corrected dissociation energy agrees quite well with the reported literature values by Si-sheng et al.³ Our anharmonic result of 927 cm⁻¹ agrees well with experimental measurement for Ar–NO⁺ of 951 cm⁻¹ by Takahashi.⁶ 

The equilibrium geometry of the Ne–NO⁺ complex is found to be for r = 2.017 a₀, R = 6.02 a₀, and θ_e = 60.2°. The results are shown in the top half of Table IV. The harmonic vibrational frequencies were calculated to be 89, 150, and 2348 cm⁻¹. Here, the intramolecular stretch is 26 cm⁻¹ lower than the isolated NO⁺ value, indicating that the Kr is a strong perturber, which is a consequence of the deeper minimum in comparison with lighter Rg gases.

The 2D fitted PES for this complex is presented in Fig. 5. The minimum energy corresponds to the geometry located at θ_e = 63° and R_e = 6.03 a₀ and with a well depth of 1434 cm⁻¹. It is more skewed towards the nitrogen side of the NO⁺ than previous lighter Rg complexes. Obviously, the 2D PES of this complex presents strong anisotropies with respect to the Kr approach angle. As shown in Table VI, the dissociation energy of Kr–NO⁺ corrected for ZPE from harmonic and anharmonic calculations is 1327 and 1320 cm⁻¹, respectively. It is slightly larger than the value reported by Si-sheng et al.³ but with overall good agreement between these two results.

The optimized equilibrium distance was found at r = 2.025 a₀, R_e = 6.25 a₀, and θ_e = 60.2°. The stationary points of Xe–NO⁺ PES are given in Table V. The harmonic vibrational frequencies were calculated to be 100, 216, and 2289 cm⁻¹, for the intermolecular bend, stretch, and NO⁺ stretch, respectively. The intramolecular stretch of NO⁺ in the complex with frequency of 2289 cm⁻¹ is the most different from the isolated value in the series of Rg atoms, and it is 85 cm⁻¹ lower than the diatomic alone, showing that the Xe interaction slows down the NO⁺ vibration.

A contour plot of the fitted PES is shown in Fig. 6. The global minimum of the Xe–NO⁺ PES is located at θ_e = 59.1° and R_e = 6.20 a₀ with a well depth of 2141 cm⁻¹. Resulting dissociation energies from ZPE harmonic correction and anharmonic BOUND calculations shown in Table VI are 2025 and 1994 cm⁻¹, respectively.

Dipole moments are necessary to simulate the emission spectra of the complexes and they also can reveal changes in the electron density distribution. Electric dipole moments as a function of R(Rg–NO⁺) are depicted in Fig. 7. The angular orientation of the NO⁺ during the radial scan corresponded to the equilibrium position of each complex. For these calculations, we used F the internally contracted Configuration Interaction with Single and Double (CISD) exitations method^41,42 with the Hartree-Fock reference wave function and the cc-pVTZ-F12 basis set. As shown in the figure, the electric dipole moment functions for the Ar–NO⁺ and Kr–NO⁺ and Xe–NO⁺ systems are positive for the large distance but suddenly there is a change in sign of the slope for the distance smaller than 2.75 a₀. We can conclude that at large distance, there is no charge transfer between target molecule (NO⁺) and incoming Rg atom. However, at short distances around R = 3-4 a₀, where some of the heavier Rg atoms overlap with the NO⁺ considering their van der Waals radii, there might be in effect charge transfer to some degree between the Rg and the NO⁺ cation. This does not affect the region of the bound states as it takes place at highly repulsive wall of the potential.

We have studied the interaction of the NO⁺ molecule with Rg atoms forming the Rg–NO⁺ van der Waals complex. The two dimensional PESs of the ground electronic states of the Rg–NO⁺ cations were mapped with the explicitly correlated CCSD(T)–F12a method. Increase in interaction energies is expected as long as the polarizability of the rare gas atom increases. The range of values varies between 196 cm⁻¹ for He–NO⁺ and 2141 cm⁻¹ for Xe–NO⁺. The fact that the bonding in He–NO⁺ molecule is still rather weak in spite of the presence of a positive charge on NO is caused by the small polarizability of He with a high probability. The interaction for Xe–NO⁺ strengthens yet it is not enough for chemical bonding. Concerning the harmonic normal modes of the complexes, it is obvious that the size increase of the Rg atom leads to a higher increase of the intermolecular stretch vibration ω₂, compared to the increase of the intermolecular bending frequency ω₁. All of the complexes contain an equilibrium geometry in which Rg atom is on the N side of the complex confirming previous findings. Dissociation energies of the Rg–NO⁺ were also calculated from the harmonic zero point energy corrections of the complex and target molecule and from the anharmonic bound state calculations with the BOUND program employing the full 2D PESs and compared with the literature as shown in Table VI. The agreement between available experimental and previously published theoretical data was found to be very good. In particular, for the Ar–NO⁺ complex, we obtained the best agreement with experimental data ever published. The CISD method was used to calculate electric dipole moments. Afterwards, it was found that the electric dipole moments for the Ar–NO⁺ and Kr–NO⁺ and Xe–NO⁺ systems are positive for the large distance but suddenly change sign for the distance shorter than 2.75 a₀.

It is shown that CCSD(T)-F12/cc-pVTZ-F12 level of theory is an accurate approach for the generation of highly correlated potential energy surfaces. We can use the explicitly correlated quantum chemistry approach confidently in order to interpret state-to-the-art experiments relating to molecular clusters.

N.B. and C.O. acknowledge the support by the Scientific Research Projects Coordination Unit of Firat University, Project No. FF.15.03. J.K. acknowledges financial support from the U.S. National Science Foundation through Grant No. CHE-1213332 to M. H. Alexander.