1 Introduction

Nitric oxide, NO, is one of the most common ligands binding to a transition metal center. Particularly, nitric oxide interacting with coinage metals,[1] such as Cu, has found its application in many different areas such as NO decomposition [2], corrosion processes [2], and biochemical catalysis [3]. Since 1991, various experimental techniques have been applied to characterize the CuNO molecule. The gas-phase existence of the molecule was first identified by mass spectrometry by Sulzle et al. [4], while other experimental characterizations have included matrix FTIR [5,6,7,8,9,10] and UV–Vis spectroscopy [5].

In past studies, there have been some controversy about the ground electronic state of the CuNO molecule. In the early study of Hrušák et al. [11], the 3A′′ state was found to be lower in energy than the 1A′ state with the Hartree–Fock (HF), coupled cluster singles and doubles (CCSD), and configuration interaction (CI) methods. CCSD with perturbative triples, CCSD(T), however, gave the opposite conclusion that the singlet should be the ground state, particularly for larger basis sets. Several years later, Uzunova [12] used complete active space second-order perturbative theory (CASPT2) with unrestricted natural orbitals (UNO) and found that the triplet was the ground state, but CCSD(T) calculations again favored the singlet. Meanwhile, several density functional theory (DFT) calculations were also applied to study the CuNO molecule [5, 8, 10, 12,13,14], but failed to achieve consensus; the pure functionals gave a singlet ground state, while the hybrid functionals preferred a triplet ground state. In 2012, Marquardt and co-workers [13, 15] used multi-reference configuration interaction (MRCI) with several different sizes of reference spaces, as well as CCSD(T), to study this molecule. Unlike the CASPT2 study, the MRCI and CCSD(T) calculations both definitively supported a singlet ground state. Throughout the present work, a singlet ground state has been assumed for all three species.

Since they are proposed to have similar catalytic applications as CuNO, an extension down the periodic table has included previous experimental studies of both AgNO [16, 17] and AuNO [18,19,20,21,22,23]. As with CuNO, spectroscopic investigations have been limited to observation in rare gas matrices. In addition, only limited theoretical studies have been performed on these molecules, and these have been carried out nearly exclusively at the DFT level,[16, 18, 21, 23] although Tielens et al. [18] did carry out some calculations of the binding energy of NO to Au at the CCSD(T) level of theory.

In this study, the near-equilibrium potential energy surfaces (PESs) of all three coinage metal nitrosyls (CuNO, AgNO, AuNO) have been accurately calculated using the Feller-Peterson-Dixon (FPD) coupled-cluster-based composite methodology [24] in order to determine their spectroscopic properties. In addition, the electric dipole moment surfaces have also been calculated to obtain pure rotational and infrared intensities.

2 Computational details

Initial calculations on the CuNO molecule were carried out at the coupled cluster singles and doubles with perturbative triples, CCSD(T) [25], level of theory with correlation consistent basis sets, [26,27,28] aug-cc-pVnZ (n = D–5)for N and O and aug-cc-pVnZ-PP (n = D–5) with a small core relativistic pseudopotential (PP) for Cu [29]. It became apparent, however, that CCSD(T) was somewhat problematic. This manifested itself in two ways: (i) after sampling the near-equilibrium PES (see below) the fitting errors were much larger than expected and (ii) the calculated N–O stretching harmonic frequency (calculated with analytical gradients and thus not subject to the possible instabilities leading to the poor fits) was much higher than experiment. Subsequent calculations indicated that these problems were eliminated when the Brueckner coupled cluster doubles with perturbative triples method, BCCD(T) [25, 30, 31], or CCSD with full iterative triples, CCSDT [32, 33], were used. Table 1 shows the results of tests on all three molecules of this study using the aug-cc-pVDZ (N,O) and aug-cc-pVDZ-PP (Cu, Ag, Au) basis sets. These results show that CuNO is the exception, e.g., the CCSD(T) value for the N–O stretching harmonic frequency is larger than BCCD(T) and CCSDT values by more than 70 cm−1. Relatively, large differences are also observed for the Cu–N bond length, nearly 0.02 Å. For the other two molecules, the differences between CCSD(T) and BCCD(T) is nearly negligible for the vibrational frequencies although the CCSD(T) values of r(M–N) appear to be too long, particularly for AgNO. In the case of CuNO, CCSDT(Q) calculations were also carried out with an increased frozen-core definition, i.e., the O 2 s was not correlated, and with just the cc-pVDZ-PP basis set on Cu and aug-cc-pVDZ on N and O. Compared to analogous CCSDT results, the equilibrium geometry was nearly unchanged (–0.0001 Å in re(N–O), –0.0023 Å in re(Cu–N), + 0.4º in the bond angle), while the NO stretching frequency decreased by just 7 cm−1 (about -2 cm−1 in the bend and unchanged for the Cu–N stretch). Also shown in Table 1 for CuNO are two variants of how the (T) correction is included in BCCD(T). One uses a triples correction based on the assumption that canonical orbitals are used and the other does not make this assumption and thus includes all non-Brillouin terms in the determination of the (T) correction. For the properties shown in Table 1, the differences are small but are appreciable for the Cu–N bond length, nearly 0.005 Å. In the remainder of this work, only the non-Brillouin version of BCCD(T) has been utilized. It should be noted that at its equilibrium geometry with this basis set, the T1 diagnostic [34] in the CCSD calculations of CuNO was rather large at 0.082, but there were no doubles amplitudes larger than 0.05, indicating multireference character was not an issue. Likewise, the T1 diagnostics for AgNO and AuNO were 0.077 and 0.065, respectively.

Table 1 A comparison of coupled cluster methods for the determination of the equilibrium structure (Å and degs.) and harmonic vibrational frequencies (cm−1). Unless indicated otherwise, the aVDZ-PP/aVDZ basis set combination was used throughout
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A full set of spectroscopic properties was derived for all 3 species from near-equilibrium potential energy surfaces (PES) of each molecule. The energy of each sampled point on the PES was obtained via the FPD composite approach [24], where in this work the total energy was expressed as:

$$E = E_{{{\text{CBS}}}} + \Delta {\text{CV}} + \Delta {\text{DK}} + \Delta {\text{SO}}$$
(1)

In Eq. (1), the first term on the right-hand side represents the frozen-core total energy extrapolated to the complete basis set limit (CBS) at the BCCD(T) level of theory [25]. Correlation consistent basis sets, aug-cc-pVnZ (n = D–5) [26, 27] for N and O and aug-cc-pVnZ-PP (n = D–5) [28] with small core relativistic pseudopotentials [29] (PPs) for coinage metal elements, were used. For convenience, these are abbreviated as aVnZ below. The Hartree–Fock energies were extrapolated by the formula of Karton and Martin [35],

$$E_{n} = E_{CBS} + A\left( {n + 1} \right)e^{ - 9\sqrt n }$$
(2)

where En are the HF energies calculated by aVQZ and aV5Z sets (n = 4 and 5). The BCCD(T) correlation energies were extrapolated by a two-parameter formula, Eq. (3), also using aVQZ and aV5Z data [36, 37],

$$E_{n} = E_{CBS} + A/\left(n + \frac{1}{2}\right)^{4}$$
(3)

The final CBS limit for a given geometry on the PES, ECBS, is the sum of the results of Eqs. (2) and (3).

The second term in Eq. (1), CV, recovers the correlation energy contribution from outer core electrons. For N and O, the core electrons include 1 s electrons, while for Cu, Ag, and Au the outer core electrons outside the PPs are 3s3p, 4s4p, and 5s5p, respectively. Both the frozen-core and outer-core correlation energies energies were calculated with the BCCD(T) method using aug-cc-pwCVQZ (N and O) and aug-cc-pwCVQZ-PP (Cu, Ag, Au) basis sets. The value of ΔCV is given by the difference in these two calculations.

The third term, ΔDK, provides the scalar relativistic contribution from the light atoms and a correction for the use of the PP for the metals. The all-electron Douglas–Kroll–Hess (DKH) scalar relativistic Hamiltonian [38,39,40] was used for these contributions with aug-cc-pVTZ-DK basis sets (denoted aVTZ-DK) for all atoms [28, 41] at the BCCD(T) level of theory. Second-order DKH was used for CuNO and AgNO, while the 3rd-order DKH Hamiltonian was utilized for AuNO. The energy difference between these calculations and the analogous PP-based ones using aVTZ sets yielded the ΔDK contributions.

The final term is the spin–orbit correction, ΔSO. In this work, two-component (2-c) DFT using the B3LYP functional [42] was used with the spin–orbit potential included with the ECP [29, 43]. Uncontracted aVDZ basis sets were used in these calculations. 2-c DFT was chosen instead of 2-c (or 4-c) CCSD(T) since a PES incorporating the latter displayed large fitting errors just as in the scalar relativistic CCSD(T) case discussed above.

The potential energy functions for each molecule were defined by fits to 84 FPD energies that covered the geometry ranges –0.3ao ≤ R(MN) – Re(MN) ≤  + 0.5ao, –0.3ao ≤ r(NO) – re(NO) ≤  + 0.5ao, and –30º ≤ θ(MNO) – θe(MNO) ≤  + 30º to functions of the form

$$V\left( {Q_{1} ,Q_{2} ,Q_{3} } \right) = \mathop \sum \limits_{ijk} C_{ijk} \left( {Q_{1} } \right)^{i} \left( {Q_{2} } \right)^{j} \left( {Q_{3} } \right)^{k}$$
(4)

where Q1 = R(M–N) – Re(M–N), Q2 = r(N–O) – re(N–O), and Q3 = A0Δθ + A1Δθ2 + A2Δθ3 where Δθ = θ(MNO) –θe(MNO) [44]. For the bending coordinate, the parameters A1 and A2 were chosen to satisfy the boundary conditions at linearity of Q3 = 1 and dQ3/dθ = 0. The value of A0 was roughly optimized in the fitting procedure. The expansion coefficients are given explicitly in Table S1 of the Supplementary Information (SI). Using Eq. (4), the calculated energies were reproduced with root-mean-square errors of less than 1 cm−1 and maximum errors of no more than 2.3 cm−1. The derivatives from Eq. (4) were transformed by the usual L-tensor algebra [45] to a quartic forcefield in dimensionless normal coordinates, and these were then used in standard second-order perturbation theory treatments [45] of the spectroscopic constants. Both the fitting of the surfaces and the latter spectroscopic constant analyses were carried out with the Surfit program of Senekowitsch [46].

The electric dipole moment surfaces are calculated at the BCCD(T) level of theory with the aVTZ basis sets for each molecule at the same 84 geometries as the potential energy surfaces described above. A finite field approach was used with field strengths of ± 0.001 a.u. The resulting dipole moment surfaces were first rotated into an Eckart frame [47] and the resulting a and b principal axes components were then fit to fourth-order polynomials in simple displacement coordinates. The expansion coefficients \({D}_{ijk}^{a}\) and \({D}_{ijk}^{b}\) are given in Table S2 of the SI. The derivatives of the dipole moments corresponding to each vibrational normal coordinate were then obtained via the Surfit program and were used to calculate the intensities for each mode using [48]

$$I_{i} ({\text{cm}}^{ - 2} {\text{atm}}^{ - 1} \;{\text{at}}\;300\;{\text{K}}) = 65.785\times\omega_{i} \times \frac{{\left( {\mu_{i,a}^{{\prime}} } \right)^{2} + \left( {\mu_{i,b}^{{\prime}} } \right)^{2} }}{2}$$
(6)

where ωi is the harmonic frequency (in cm−1) for each mode and \({\mu }_{i,a}^{{\prime}}\) and \({\mu }_{i,b}^{{\prime}}\) are the first derivatives of dipole moments (in a.u.) along the a and b principal axes, respectively, for the ith normal mode.

Except for the spin–orbit calculations, which utilized the DIRAC program [49], all other calculations were carried out using the MOLPRO suite of ab initio programs [50].

3 Results and discussion

3.1 Potential energy surfaces and spectroscopic properties

Tables 2, 3 and 4 summarize the basis set convergence, individual FPD contributions, and final composite results for the equilibrium geometries and harmonic frequencies for CuNO, AgNO, and AuNO, respectively. In each case, these properties were derived from separate potential energy functions accurately fit via Eq. (4) to 84 energies representing the stated basis set or FPD contribution, e.g., the \(\Delta {\text{CV}}\) values were derived as the difference between spectroscopic constants calculated from the BCCD(T)/CBS[Q5] PES and the BCCD(T)/CBS[Q5] + \(\Delta {\text{CV}}\) PES, where \(\Delta {\text{CV}}\) in this latter context is the contribution due to outer-core correlation at each of the 84 points as described in the previous section. As seen in these tables, the BCCD(T) geometries and harmonic frequencies are well converged even with the aVQZ basis sets. Further extensions and extrapolation to the CBS limits changes the bond lengths by only a few thousandths of an Angstrom and the frequencies by no more than 3 cm−1.

Table 2 Equilibrium bond lengths (Å), bond angles (deg.) and harmonic frequenciesa (cm−1) of the CuNO molecule calculated from the fitted potentials with respect to the basis set and contributions of different FPD terms. All calculations except \(\Delta {\text{SO}}\) correspond to the BCCD(T) method. See the text
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Table 3 Equilibrium bond lengths (Å), bond angles (deg.), and harmonic frequencies (cm−1) of the AgNO molecule calculated from the fitted potentials with respect to the basis set and contributions of different FPD terms. All calculations except \(\Delta {\text{SO}}\) correspond to the BCCD(T) method. See the text
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Table 4 Equilibrium bond lengths (Å), bond angles (deg.), and harmonic frequencies (cm−1) of the AuNO molecule calculated from the fitted potentials with respect to the basis set and contributions of different FPD terms. All calculations except \(\Delta {\text{SO}}\) correspond to the BCCD(T) method. See the text
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The effects of correlating the outer-core electrons are nearly negligible for CuNO in regard to the equilibrium structure although the frequencies increase by 3–10 cm−1. Correlation of the outer-core has the largest effect on AgNO, decreasing the Ag–N bond length by 0.045 Å and increasing the bending and Ag–N harmonic frequencies by 15.5 and 14.0 cm−1, respectively. The CV effect on these properties in AuNO are intermediate between the Cu and Ag cases, but as in AgNO the largest changes were a decrease in the Au–N bond length and increases in\(\omega_{2}\) and \(\omega_{3}\). The two remaining modifications to the PESs, \({\Delta\text{ DK}}\) and \(\Delta {\text{SO}}\), resulting in only small changes to these properties, particularly in regard to \(\Delta {\text{SO}}\) The largest effect from using the PP is calculated for AgNO where the Ag–N bond length is increased with DK calculations by 0.007 Å. The analogous changes for CuNO and AuNO were just + 0.003 Å and -0003 Å, respectively. In all cases the harmon[48]ic frequencies changed by less than 2 cm−1. Not surprisingly the effects of SO coupling was nearly negligible for these closed-shell molecules. The largest SO effect was for AuNO where Re(Au–N) decreased by 0.005 Å with a concomitant increase in the Au–N harmonic frequency by 2.4 cm−1.

Using the final FPD composite PESs, a large number of spectroscopic constants were calculated via second-order vibrational perturbation theory for all three molecules. The rotational and ro-vibrational constants are shown in Table 5 while the pure vibrational constants are given in Table 6. The bond lengths of Table 5 are expected to be accurate to within about 0.005 Å while the harmonic frequencies of Table 6 have expected accuracies on the order of 15 cm−1. No significant resonances were identified for these molecules, so it is expected that vibrational perturbation theory should be very reliable in these cases. In regard to previous calculations, DFT methods have tended to yielded bond lengths for both the M–N and N–O bonds that are too long, by as much as 0.09 and 0.02 Å, respectively, but with a lot of variation depending on the functional and basis set employed [8, 16, 18, 23]. In regard to the vibrational data of Table 6, the largest variations between the current harmonic frequencies and previous DFT results is in the NO stretching vibration. Consistent with predicting too short NO bond lengths, B3LYP tended to yield values for the NO stretching frequency that are too high by up to 60 cm−1 (CuNO) [8]. The BP86 or BPW91 functionals yielded results much closer to the current BCCD(T)-based values, although the NO stretch for AgNO resulting from BPW91 was smaller by about 60 cm−1.[16] Last, in the matrix isolation spectroscopy work of Krim et al. [5] values were evaluated for the X11 and X22 anharmonicity constants, -16.5 and -4.8 cm−1, respectively. These are in excellent agreement with the results obtained from the current composite PES, -15.9 and -4.7 cm−1.

Table 5 Calculated rotational and ro-vibrational constantsa obtained with the final composite PESs for CuNO, AgNO, and AuNO (most abundant isotopomers)
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Table 6 Calculated vibrational constantsa obtained with the final composite PESs for CuNO, AgNO, and AuNO (most abundant isotopomers)
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Table 7 compares the calculated anharmonic vibrational frequencies of this work as derived from the final composite PESs with the available experimental values obtained in rare gas matrices. In every case the present ab initio values are higher than the values from the matrix IR experiments. As anticipated, the present results do agree better overall with the Ne matrix results, generally within less than 1% for the NO stretch (\(\nu\)1). Larger matrix shifts, at least on a percentage basis, are observed for the lower frequency \(\nu\)2 and \(\nu\)3 modes.

Table 7 Comparison of the composite FPD anharmonic vibrational frequencies (cm−1) with the available experimental values
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3.2 Dipole moment surfaces and normal mode intensities

The expansion coefficients for the Eckart frame dipole moment functions of CuNO, AgNO, and AuNO are shown in Table S2. In these cases the molecules at equilibrium are oriented with the N–O bond nearly aligned along the a-axis with the O atom in the positive ab quadrant (origin at the molecular center of mass). A positive sign of an equilibrium dipole moment component implies the positive end of the dipole vector lies along the positive a or b axis. In all cases, the equilibrium values are relatively small: µa =  − 0.408  D, µb =  − 0.132 D (CuNO); µa = 0.041 D, µb = 0.004 D (AgNO); µa = 0.741 D, µb = − 0.088 D (AuNO). The corresponding total equilibrium values are given in Table 8. With such small magnitudes, these molecules are expected to have relatively weak pure rotational spectra and AgNO will be particularly challenging to observe. The infrared intensities of the fundamental vibrational normal modes for each molecule are also listed in Table 8. The N–O stretching modes have the largest intensities by far, followed by the bending modes and then the M–N stretching modes. The intensity of \(\nu\)1 decreases along the series from CuNO to AuNO. Based on the relatively small magnitudes of the dipole moment second derivatives with respective to dimensionless normal coordinates shown in Table S3, most of the overtones are not expected to have unusually large intensities. An exception may be the bending mode since the second derivative terms \({\mu }_{22}^{{{\prime\prime}}}\) are larger (CuNO and AgNO) or nearly as large (AuNO) as the corresponding first derivatives.

Table 8 Calculated BCCD(T)/aVTZ equilibrium dipole moments µe (in Debye) and infrared intensities Ii (in cm−2 atm−1).The latter are calculated within the double harmonic approximation for the ith vibrational modea
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4 Conclusions

Accurate near-equilibrium potential energy functions have been calculated for the coinage metal nitrosyl molecules CuNO, AgNO, and AuNO using the Feller–Peterson–Dixon (FPD) composite methodology that included CBS extrapolations and contributions from outer-core correlation, all-electron relativistic corrections, and spin–orbit coupling. The present work also found that the BCCD(T) level of theory was much more reliable compared to CCSD(T), particularly for CuNO. This correlated with the relatively large value for the T1 diagnostic in CuNO but otherwise the apparent lack of multideterminantal character. A large set of spectroscopic constants, both rotational and vibrational, was determined for all three molecules which will hopefully facilitate their observation by experiment in the gas phase. Full three-dimensional dipole moment functions were also determined in this work at the BCCD(T) level of theory. The pure rotational spectra are predicted to be relatively weak for all three molecules, particularly AgNO. In agreement with the available IR experiments, the \(\nu\)1 stretch is predicted to be much more intense than the bend or M–N stretch.