Alexander N. Smirnov^{a},
Victor G. Solomonik^{a},
Sergei N. Yurchenko*^{b} and
Jonathan Tennyson^{b}
^{a}Department of Physics, Ivanovo State University of Chemistry and Technology, Ivanovo 153000, Russia
^{b}Department of Physics & Astronomy, University College London, London WC1E 6BT, UK. E-mail: s.yurchenko@ucl.ac.uk
First published on 2nd October 2019
We report an ab initio study on the spectroscopy of the open-shell diatomic molecule yttrium oxide, YO. The study considers the six lowest doublet states, X^{2}Σ^{+}, A′^{2}Δ, A^{2}Π, B^{2}Σ^{+}, C^{2}Π, D^{2}Σ^{+}, and a few higher-lying quartet states using high levels of electronic structure theory and accurate nuclear motion calculations. The coupled cluster singles, doubles, and perturbative triples, CCSD(T), and multireference configuration interaction (MRCI) methods are employed in conjunction with a relativistic pseudopotential on the yttrium atom and a series of correlation-consistent basis sets ranging in size from triple-ζ to quintuple-ζ quality. Core–valence correlation effects are taken into account and complete basis set limit extrapolation is performed for CCSD(T). Spin–orbit coupling is included through the use of both MRCI state-interaction with spin–orbit (SI-SO) approach and four-component relativistic equation-of-motion CCSD calculations. Using the ab initio data for bond lengths ranging from 1.0 to 2.5 Å, we compute 6 potential energy, 12 spin–orbit, 8 electronic angular momentum, 6 electric dipole moment and 12 transition dipole moment (4 parallel and 8 perpendicular) curves which provide a complete description of the spectroscopy of the system of six lowest doublet states. The Duo nuclear motion program is used to solve the coupled nuclear motion Schrödinger equation for these six electronic states. The spectra of ^{89}Y^{16}O simulated for different temperatures are compared with several available high resolution experimental studies; good agreement is found once minor adjustments are made to the electronic excitation energies.
A considerable number of experimental studies have been performed probing the A^{2}Π–X^{2}Σ^{+},^{10,11,13,19–34} B^{2}Σ^{+}–X^{2}Σ^{+},^{19,21,29,35–37} A′^{2}Δ–X^{2}Σ^{+},^{16,38,39} and D^{2}Σ^{+}–X^{2}Σ^{+}^{37} bands of YO, as well as its microwave rotational spectrum^{40–42} and its hyperfine structure.^{26,41,43–47} Chemiluminescence spectra of YO have also been investigated.^{29,48,49} Many of these spectra were recorded using YO samples which were not in thermodynamic equilibrium, thus, at best, only providing information on the relative intensities. For YO, relative intensity measurements were carried out for the A^{2}Π–X^{2}Σ^{+} system by Bagare and Murthy.^{24} However, the permanent dipole moments of YO in both the X^{2}Σ^{+} and A^{2}Π states were measured using the Stark technique.^{27,45,47}
In the absence of direct intensity measurements, measured lifetimes can provide important information on Einstein A coefficients and hence transition dipole moments.^{50} The lifetimes of some lower lying vibrational states of YO in its A^{2}Π, B^{2}Σ^{+}, and D^{2}Σ^{+} states were measured by Liu and Parson^{12} and Zhang et al.^{37}
YO is a strongly bound system. The compilation by Gaydon^{51} reports its dissociation energy to be 7.0 ± 2 eV, while Ackermann and Rauh^{52} recommended a D_{0} value of 7.290(87) eV based on mass spectrometric determinations.
A few theoretical investigations of YO are available in the literature. The most comprehensive one was carried out by Langhoff and Bauschlicher^{53} who reported the spectroscopic constants for the lowest five doublet, X^{2}Σ^{+}, A′^{2}Δ, A^{2}Π, B^{2}Σ^{+}, C^{2}Π, and fourteen quartet electronic states of YO. The doublets were studied at the multireference single and double excitation configuration interaction (MRCI) level of theory and, in the case of the X^{2}Σ^{+}, A′^{2}Δ, and A^{2}Π states, also using the modified coupled-pair functional (MCPF) method. All the quartet states were considered at the CASSCF level, and that with the lowest energy, reportedly ^{4}Φ, at the MCPF level as well. Zhang et al.^{37} have recently reported the CASPT2 spectroscopic constants and excitation energies for a set of lowest doublet states of YO including the D^{2}Σ^{+} state in addition to the doublets studied previously by Langhoff and Bauschlicher.^{53} In all of the previous theoretical studies, only modest double-ζ^{53} or triple-ζ^{37} basis sets were employed. RKR curves and some Franck–Condon factors of YO were computed by Sriramachandran and Shanmugavel.^{54}
The main objective of the present study is to characterise both the electronic ground state and the plethora of low-lying excited states of YO with high-level ab initio methods, and to accurately describe from first principles the spectroscopy of YO via producing the potential energy curves (PECs) and other data needed to calculate the rovibronic energies and transition probabilities comprising a so-called line list for this molecule. The generation of such line lists is a major object of the ExoMol project.^{55}
Thus far, ExoMol studies of open-shell transition metal (TM) diatomics have struggled due to difficulties in providing reliable ab initio starting points.^{2,56–58} The intrinsic challenge to theory posed by open-shell systems is associated with several types of problems including spin contamination, symmetry breaking in the reference function, strong nondynamical electron correlation effects, avoided crossings between adiabatic potential energy surfaces, etc. (for the discussion, see, e.g., ref. 59 and 60). In the open-shell TM-containing species, these problems are exacerbated by stronger relativistic effects than those in the molecules made up of relatively light main group elements, and greater number of electronic excited states governing the spectroscopic behaviour of a molecule and hence deserving to be taken into account in a study aimed at accurate description of its spectroscopy. Moreover, the low-lying electronic states of TM species are commonly degenerate or near-degenerate, which complicates their theoretical treatment even more. Multireference methods of quantum chemistry best suited for describing closely spaced electronic states might seem to be the natural choice for studying these systems. However, most routine multireference methods, such as MRCI, are incapable of properly handling dynamical electron correlation and therefore do not provide high accuracy description of TM-containing species commonly featuring strong dynamical correlation effects. Such effects are best treated with single reference coupled cluster (CC) theory known for its capability to predict highly accurate properties even for molecules with mild to moderate MR character. Unfortunately, the higher likelihood of severe multireference character in the ground and/or low-lying electronic excited states of open-shell TM-containing species makes their treatment by single reference methods very problematic, if possible at all. Particularly this is true for the studies aimed at a description of the molecular potential energy surfaces over a wide range of geometries. It is therefore not surprising that the high-level coupled cluster studies on the open-shell TM-containing species, where a few excited states are treated on an equal footing with the ground state, are very uncommon and only deal with near-equilibrium regions of these states (see, e.g., ref. 61–64). Such a study on a manifold of electronic excited states of a TM-containing diatomic molecule over a wider bond length range has not been reported so far.
It is thus clear that none of routine methods of modern quantum chemistry are entirely satisfactory in all respects for accurately describing from first principles the spectroscopy of open-shell TM-containing species. Nevertheless, one can try to solve this challenging task via the so-called composite approach by which the desired set of molecular properties is obtained using multiple methods of different nature and sophistication rather than a single method.
In this paper, we have examined efficiency of such an approach taking the example of YO. The PECs for the six lowest doublet electronic states of this molecule, X^{2}Σ^{+}, A′^{2}Δ, A^{2}Π, B^{2}Σ^{+}, C^{2}Π, D^{2}Σ^{+}, were obtained from the extensive high-level coupled cluster calculations addressing core–valence correlation and basis set convergence issues, whereas the spin–orbit curves (SOCs), electronic angular momentum curves (EAMCs), electric dipole moment curves (DMCs), and transition dipole moment curves (TDMCs) were obtained at the MRCI level of theory. These curves, with some simple adjustment of the minimum energies of the PECs, are used to solve the coupled nuclear-motion Schrödinger equation with the program Duo.^{65} The spectroscopic model and ab initio curves are provided as part of the ESI.† Our open source code Duo can be accessed via http://exomol.com/software/.
Potential energy, spin–orbit coupling, and dipole moment curves, as well as electronic angular momentum and transition dipole matrix elements were obtained at the MRCI level for the six lowest doublet states. Moreover, the potential curves were calculated using the extended multi-state complete active space second-order perturbation theory,^{73} XMS-CASPT2, with the basis sets aug-cc-pwCVTZ-PP^{71} on Y and aug-cc-pwCVTZ^{74} on O (henceforth abbreviated as awCVTZ). In the respective SA-CASSCF calculations, the (7e,13o) active space was employed together with averaging over the lowest six doublet states. In order to remedy issues pertaining to intruder states, a level shift of 0.4 and an IPEA (ionisation potential, electron affinity) shift of 0.5 were employed for XMS-CASPT2.
To calculate the molecular Ω states and respective spin–orbit curves, we used the spin–orbit – MRCI state-interacting approach:^{75} the spin-coupled eigenstates were obtained by diagonalizing H_{es} + H_{SO} in a basis of MRCI eigenstates of electrostatic Hamiltonian H_{es}. The matrix elements of H_{SO} were constructed using the one-electron spin–orbit operator accompanying the yttrium pseudopotential.
Spin–orbit effects were also treated more rigorously in relativistic four-component (4c) all-electron calculations employing a Gaussian nuclear model and an accurate approximation to the full Dirac–Coulomb Hamiltonian.^{76} The respective spin-free results were obtained with the spin-free Hamiltonian of Dyall.^{77} In these calculations, the relativistic TZ-quality basis sets of Dyall^{78,79} were used for the Y and O atoms (hereafter referred to as TZ_{D}). The basis sets were kept uncontracted to provide sufficient flexibility. Electron correlation was taken into account via the equation-of-motion CCSD (EOM-CCSD) method^{80} with the Y outer-core (4s and 4p) electrons correlated together with the valence electrons. The EOM-EA scheme (adding 1 electron to the closed shell) was applied with the reference defined by the YO^{+} cation and the active space comprising 12 spinors (Y 5s and 4d). For the YO electronic states inaccessible via the EOM-EA procedure, we employed the EOM-IP scheme (removing 1 electron from the closed shell) with the YO^{−} (Y 5s^{2}) anion taken as the reference and an active space composed of 8 spinors (Y 5s and O 2p). The virtual orbital space was truncated by deleting all virtual spinors with orbital energies larger than 15 a.u. In the relativistic calculations of dipole moments, a finite-field perturbation scheme was employed by adding the z-dipole moment operator as a small perturbation to the Hamiltonian. Perturbations with electric field strengths of ±0.0005 a.u. were applied.
The atomic spin–orbit corrections, ΔE_{SO}, utilised in the calculations of the YO atomization energy were obtained from the experimental J-averaged zero-field splittings of the ground state atomic terms:^{81} ΔE_{SO} = −77.975 cm^{−1} (O) and −318.216 cm^{−1} (Y).
The most sophisticated PEC computations were performed at the coupled cluster singles, doubles, and perturbative triples, CCSD(T), level of theory^{82} with a restricted open-shell Hartree–Fock reference and with an allowance for a small amount of spin contamination in the solution of the CCSD equations, i.e., RHF-UCCSD(T).^{83,84} Symmetry equivalencing of the ROHF orbitals was performed for the degenerate atomic and molecular electronic states. Both valence (4d, 5s Y; 2s, 2p O) and outer-core (4s, 4p Y; 1s O) electrons were correlated. Scalar relativistic effects were treated with the yttrium pseudopotential described above. Sequences of aug-cc-pwCVnZ-PP^{71} (n = T, Q, 5) basis sets for Y were used in conjunction with the corresponding all-electron basis sets aug-cc-pwCVnZ^{74} for the O atom. These combinations of basis sets are denoted below as awCVTZ, awCVQZ, and awCV5Z, respectively.
For each point in a grid of r(Y–O) bond lengths, the CCSD(T) calculated energies were extrapolated to the complete basis set (CBS) limit. Three extrapolation schemes were employed. First, a two-point extrapolation of total energies was performed using the formula:^{85}
(1) |
E_{n} = E_{CBS} + A(n + 1)exp(−9.03n^{1/2}); | (2) |
E_{n} = E_{CBS} + Aexp(−(n − 1)) + Bexp(−(n − 1)^{2}), | (3) |
The spectroscopic constants r_{e}, ω_{e}, ω_{e}x_{e}, and α_{e} of YO were obtained from a conventional Dunham analysis^{88} using polynomial fits of total energies for bond lengths in the vicinity of the minimum for a given electronic state.
The CCSD(T) calculations of the equilibrium dipole moments, μ_{e}, for a few lowest states were carried out at the corresponding CCSD(T)/CBS1 equilibrium bond lengths. The dipole moments were computed by numerical differentiation of the total energy in the presence of a weak electric field. Finite perturbations with electric field strengths of ±0.0025 a.u. were applied. Since hierarchical sequences of basis sets have been used, the dipole moments were also extrapolated to the CBS limit using the three extrapolation schemes described above.
Most of the ab initio calculations were carried out using the MOLPRO electronic structure package.^{89} The relativistic 4c-EOM-CCSD calculations were performed using the DIRAC program.^{90}
In Duo calculations, the coupled Schrödinger equation was solved on an equidistant grid of 301 bond lengths r_{i} ranging from r = 1.2 to 3 Å using the sinc DVR method. Our ab initio curves are represented by sparser and less extended grids (see below). For the bond length values r_{i} overlapping with the ab initio ranges, the ab initio curves were projected onto the denser Duo grid using the cubic spline interpolation. The PECs outside the ab initio range were reconstructed using the standard Morse potential form
f_{PEC}(r) = V_{e} + D_{e}(1 − e^{−a(r−re)})^{2}. |
f^{short}_{TDMC}(r) = Ar + Br^{2}, |
f^{short}_{other}(r) = A + Br, |
f^{long}_{EAMC}(r) = A + Br, |
f^{long}_{other}(r) = A/r^{2} + B/r^{3}, |
The vibrational basis set was taken as eigensolutions of the six uncoupled 1D problems for each PEC. The corresponding basis set constructed from 6 × 301 eigenfunctions was then contracted to include 60 lowest (in terms of the vibrational energy) X functions and 20 from each other state (160 in total). These vibrational basis functions were then combined with the spherical harmonics for the rotational and electronic spin basis set functions. All calculations were performed for ^{89}Y^{16}O using atomic masses.
This study is the first where a Duo calculation has been performed for a system with avoided crossings between curves of the same symmetry. The current version of Duo does not allow for non-adiabatic couplings, and therefore these were ignored in this study. However, despite the expectation that the non-adiabatic coupling effects should be relatively important in the regions near an avoided curve crossing, as we show below, our model neglecting these effects is justified by close agreement with the available experimental spectra.
Fig. 1 CASSCF/aVTZ spin-free potential energy curves of the low-lying doublet (black) and quartet (red) states in YO. |
The lowest six doublet MRCI PECs (X^{2}Σ^{+}, A′^{2}Δ, A^{2}Π, B^{2}Σ^{+}, C^{2}Π, D^{2}Σ^{+}) are shown in Fig. 2. For the states X^{2}Σ^{+}, A′^{2}Δ, A^{2}Π, and B^{2}Σ^{+}, the PECs were obtained in the full bond length range amenable to the underlying CASSCF procedure, 1.58 ≤ r ≤ 2.36 Å, while the C^{2}Π and D^{2}Σ^{+} curves were calculated through r = 2.04 Å and 1.93 Å, respectively. Extending the MRCI curves for the two upper states beyond those distances would require requesting a greater number of states (while exceeding 4 in a single irreducible representation with the chosen active space would make the MRCI computation unfeasible on the hardware used) or selecting the order of the states in the initial internal CI (e.g., 1, 2, 3, 5 rather than 1, 2, 3, 4), leaving some of them out for each MRCI point. This appeared to affect the smoothness of the resulting PECs. Therefore, we refrained from further pursuing the computations with the same number of MRCI roots in the entire bond length range and simply reduced the number of requested states for longer internuclear distances. For all six doublet states, the XMS-CASPT2 PECs could be obtained in the range 1.59 ≤ r ≤ 2.165 Å, Fig. 3; at larger bond lengths the underlying CASSCF procedure failed to converge. As can be seen from Fig. 2 and 3, the A^{2}Π and C^{2}Π curves feature an avoided crossing at bond lengths around 2 Å, as do the B^{2}Σ^{+} and D^{2}Σ^{+} curves in approximately the same region, see Fig. 3. The avoided crossings of both pairs of curves are also seen in the EOM-IP-CCSD calculated PECs, Fig. 4, albeit at shorter distances (1.8–1.9 Å).
Fig. 4 The avoided crossing regions of the EOM-IP-CCSD/TZ_{D} spin-free potential energy curves for the A^{2}Π, B^{2}Σ^{+}, C^{2}Π, and D^{2}Σ^{+} electronic states of YO. |
In order to provide deeper insight into the electronic structure of YO, we analysed the dominant configurations (configuration state functions) in the MRCI wave functions for the lowest electronic states, Table 1, and the leading atomic orbital contributions in the respective molecular orbitals, Table 2.
State | Configurations^{a} | Weights, % | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
10σ | 11σ | 12σ | 13σ | 5π_{+} | 5π_{−} | 6π_{+} | 6π_{−} | 2δ_{+} | 2δ_{−} | 1.79 Å | 2.04 Å | 2.19 Å | |
a The orbital names π_{+}, π_{−}, δ_{+}, and δ_{−} indicate π(b_{1}), π(b_{2}), δ(a_{1}), and δ(a_{2}) orbitals, respectively. The MO occupancies represented by 2, 0 and + or − denote double, zero, and single occupancies with the total spin raised or lowered by 1/2. | |||||||||||||
X^{2}Σ^{+} | 2 | 2 | + | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 79.2 | 71.4 | 63.3 |
2 | + | + | − | 2 | 2 | 0 | 0 | 0 | 0 | 2.1 | 3.1 | 3.5 | |
2 | + | − | + | 2 | 2 | 0 | 0 | 0 | 0 | 2.3 | 2.8 | ||
2 | 2 | + | 0 | 2 | + | 0 | − | 0 | 0 | 3.6 | |||
2 | 2 | + | 0 | + | 2 | − | 0 | 0 | 0 | 3.6 | |||
2 | 2 | + | 0 | − | 2 | + | 0 | 0 | 0 | 2.2 | |||
2 | 2 | + | 0 | 2 | − | 0 | + | 0 | 0 | 2.2 | |||
A′^{2}Δ (a_{1}) | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | + | 0 | 78.9 | 67.3 | 53.1 |
2 | + | − | 0 | 2 | 2 | 0 | 0 | + | 0 | 8.1 | 21.6 | ||
2 | + | 0 | − | 2 | 2 | 0 | 0 | + | 0 | 4.8 | 4.2 | 4.1 | |
A^{2}Π (b_{1}) | 2 | 2 | 0 | 0 | 2 | 2 | + | 0 | 0 | 0 | 79.9 | ||
2 | + | 0 | − | 2 | 2 | + | 0 | 0 | 0 | 3.2 | |||
2 | 2 | 2 | 0 | + | 2 | 0 | 0 | 0 | 0 | 82.6 | 82.3 | ||
2 | 2 | 0 | 0 | + | 2 | 2 | 0 | 0 | 0 | 2.4 | 2.1 | ||
2 | 2 | 0 | 0 | + | 2 | 0 | 2 | 0 | 0 | 2.1 | |||
B^{2}Σ^{+} | 2 | 2 | 0 | + | 2 | 2 | 0 | 0 | 0 | 0 | 73.5 | ||
2 | + | 2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 4.5 | 80.2 | 78.9 | |
2 | + | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2.1 | |||
2 | + | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 2.3 | 2.0 | ||
2 | + | 0 | 0 | 2 | 2 | 0 | 2 | 0 | 0 | 2.3 | 2.0 | ||
C^{2}Π (b_{1}) | 2 | 2 | 2 | 0 | + | 2 | 0 | 0 | 0 | 0 | 83.2 | ||
2 | 2 | 0 | 0 | + | 2 | 2 | 0 | 0 | 0 | 2.8 | |||
2 | 2 | 0 | 0 | + | 2 | 0 | 2 | 0 | 0 | 2.3 | |||
2 | 2 | 0 | 0 | 2 | 2 | + | 0 | 0 | 0 | 68.7 | |||
2 | + | − | 0 | 2 | 2 | + | 0 | 0 | 0 | 4.3 | |||
2 | + | 0 | − | 2 | 2 | + | 0 | 0 | 0 | 3.3 | |||
D^{2}Σ^{+} | 2 | + | 2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 79.1 | ||
2 | 2 | 0 | + | 2 | 2 | 0 | 0 | 0 | 0 | 4.0 | |||
2 | + | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 2.6 | |||
2 | + | 0 | 0 | 2 | 2 | 0 | 2 | 0 | 0 | 2.6 |
MO | 1.79 Å | 2.04 Å | 2.19 Å |
---|---|---|---|
a See footnote to Table 1 for designations. | |||
10σ | 52% 2s O + 43% 4pσ Y | 71% 2s O + 27% 4pσ Y | 84% 2s O + 15% 4pσ Y |
11σ | 63% 2pσ O + 10% 4dσ Y | 72% 2pσ O | 77% 2pσ O |
12σ | 86% 5s Y | 82% 5s Y + 12% 5pσ Y | 81% 5s Y + 13% 5pσ Y |
13σ | 50% 5pσ Y + 36% 4dσ Y | 39% 5pσ Y + 53% 4dσ Y | 34% 5pσ Y + 60% 4dσ Y |
5π_{+(−)} | 94% 2pπ O | 95% 2pπ O | 96% 2pπ O |
6π_{+(−)} | 67% 5pπ Y + 32% 4dπ Y | 46% 5pπ Y + 52% 4dπ Y | 34% 5pπ Y + 63% 4dπ Y |
2δ_{+(−)} | 100% 4dδ Y | 100% 4dδ Y | 99% 4dδ Y |
The analysis indicates that the X^{2}Σ^{+} ground state consists mainly of …10σ^{2}11σ^{2}5π^{4}12σ^{1} electron configuration. The principal contribution to the singly occupied 12σ MO comes from the yttrium 5s atomic orbital, with an admixture of the 5p AO particularly noticeable at longer internuclear distances. The three lowest active MOs, 10σ, 11σ, and 5π_{+(−)}, primarily consist of the oxygen 2s and 2p orbitals whose contributions increase with the bond stretching. Therefore, the bonding in the X^{2}Σ^{+} YO ground state can be roughly described as ionic, Y^{2+}O^{2−}, however, with significant covalent character mainly owing to an appreciable participation of the yttrium 4p_{σ} AO in the 10σ MO. Upon the Y–O bond stretching, there is a rapid decrease in the 4p contribution (see Table 2) reducing the covalency and resulting in a concomitant increase in the magnitude of the electric dipole moment in the YO ground state (see below).
The first excited state, A′^{2}Δ, mainly consists of …10σ^{2}11σ^{2}5π^{4}2δ^{1} configuration with the 2δ MO clearly assigned to the Y 4d_{δ} orbital. At shorter bond lengths this state can be reasonably described by the single electron excitation from the ground state, 5s^{1} → 4d_{δ}^{1}. Upon bond elongation, the weight of the main configuration gradually decreases, approaching ∼50% at bond lengths of about 2.2 Å, whereas at r > ∼2 Å, a few other large-weight configurations emerge, e.g., the three-open-shell …10σ^{2}11σ^{α}12σ^{β}5π^{4}2δ^{α} configuration (with a weight of 22% at 2.19 Å) that corresponds to the Y^{+}O^{−} bonding (Y 5s^{1}4d^{1}, O 2p^{5}).
At bond lengths up to ∼1.85 Å, the principal configurations of the A^{2}Π and B^{2}Σ^{+} excited states, …10σ^{2}11σ^{2}5π^{4}6π^{1} and …10σ^{2}11σ^{2}5π^{4}13σ^{1}, respectively, include the 6π and 13σ singly occupied MOs mainly composed of the yttrium 5p_{π} and 5p_{σ} atomic orbitals, respectively, yet with a significant admixture of the 4d AOs. Therefore, these states can be viewed as arising from 5s → 5p_{π} and 5s → 5p_{σ} atomic electron promotions. In the same range of bond length values, the C^{2}Π and D^{2}Σ^{+} upper lying states consist mainly of …10σ^{2}11σ^{2}12σ^{2}5π^{3} and …10σ^{2}11σ^{1}12σ^{2}5π^{4} electron configurations, respectively. The YO bonding in the C^{2}Π and D^{2}Σ^{+} states is hence well described by the Y^{+}O^{−} scheme consistent with the Y 5s^{2}, O 2p^{5} electron configuration.
As the Y–O distance approaches the avoided crossing point from below, the principal configurations of the A^{2}Π and B^{2}Σ^{+} states change to those specified above for the C^{2}Π and D^{2}Σ^{+} states, respectively, while, vice versa, the principal configurations of the C^{2}Π and D^{2}Σ^{+} states turn into those being the main ones for the A^{2}Π and B^{2}Σ^{+} states at shorter bond lengths. This alteration of main configurations describes an oxygen-to-metal charge back-transfer, Y^{2+}O^{2−} → Y^{+}O^{−}, in the A^{2}Π and B^{2}Σ^{+} states of YO upon the Y–O bond stretching through the avoided crossing region of bond lengths.
Specifically, the avoided crossing point, r_{ac}, chosen to be the point of closest approach of two curves, for the A^{2}Π and C^{2}Π states amounts to 2.046 Å (XMS-CASPT2) and 1.994 Å (MRCI), with the energy gap, ΔE_{ac}, of 366 cm^{−1} and 243 cm^{−1}, respectively. For the XMS-CASPT2 B^{2}Σ^{+} and D^{2}Σ^{+} curves, r_{ac} = 2.064 Å and ΔE_{ac} = 1484 cm^{−1}. Notably, the principal configuration interchange between the B^{2}Σ^{+} and D^{2}Σ^{+} states occurs at a slightly shorter internuclear distance: ∼1.95 Å (XMS-CASPT2), ∼1.92 Å (MRCI). At the EOM-IP-CCSD level, the avoided crossing characteristics are: r_{ac} = 1.911 Å, ΔE_{ac} = 231 cm^{−1} for A^{2}Π and C^{2}Π curves, and r_{ac} = 1.832 Å, ΔE_{ac} = 272 cm^{−1} for B^{2}Σ^{+} and D^{2}Σ^{+} curves.
The CCSD(T) calculations were carried out for the six lowest doublet states. The reference configurations for each state were as follows:
X^{2}Σ^{+} …10σ^{2}11σ^{2}5π^{4}12σ^{1} |
A′^{2}Δ …10σ^{2}11σ^{2}5π^{4}2δ^{1} |
A^{2}Π …10σ^{2}11σ^{2}5π^{4}6π^{1} |
B^{2}Σ^{+} …10σ^{2}11σ^{2}5π^{4}13σ^{1} |
C^{2}Π …10σ^{2}11σ^{2}12σ^{2}5π^{3} |
D^{2}Σ^{+} …10σ^{2}11σ^{1}12σ^{2}5π^{4} |
The CCSD(T) energies were obtained in the ranges: 1.0 ≤ r ≤ 2.5 Å for X^{2}Σ^{+}, A′^{2}Δ and A^{2}Π; 1.0 ≤ r ≤ 2.4 Å for B^{2}Σ^{+}; 1.4 ≤ r ≤ 2.4 Å for C^{2}Π; 1.74 ≤ r ≤ 2.35 Å for D^{2}Σ^{+}. At longer distances (as well as shorter ones for C^{2}Π and D^{2}Σ^{+}), the coupled-cluster calculations failed due to severe CCSD convergence problems. For the X^{2}Σ^{+}, A′^{2}Δ, A^{2}Π, and B^{2}Σ^{+} states, the distances shorter than 1.0 Å were not considered because relative energies of excited states already exceed 350000 cm^{−1} at this point. The respective CCSD(T)/CBS1 PECs are shown in Fig. 5. Since single reference methods are not suitable for describing avoided crossings, e.g., those between the PECs of the A^{2}Π and C^{2}Π, and B^{2}Σ^{+} and D^{2}Σ^{+} states, the CCSD(T) calculated PECs can be viewed as corresponding to the diabatic presentation of these states. It is clearly seen that the CCSD(T) A^{2}Π and C^{2}Π curves cross each other at 2.031 Å, as do the B^{2}Σ^{+} and D^{2}Σ^{+} ones at 2.013 Å.
X^{2}Σ^{+} | A′^{2}Δ | A^{2}Π | B^{2}Σ^{+} | C^{2}Π | D^{2}Σ^{+} | ||
---|---|---|---|---|---|---|---|
a EOM-IP for C^{2}Π and D^{2}Σ^{+}, EOM-EA for the remaining states.b Calculated for each electronic state at the respective CCSD(T)/CBS1 equilibrium bond length.c Calculated at a bond length of 1.7932 Å. | |||||||
T_{e} | XMS-CASPT2/awCVTZ | 0 | 16183 | 16210 | 20521 | 20198 | 24139 |
MRCI/aVTZ | 0 | 16370 | 16287 | 17029 | |||
EOM-CCSD/TZ_{D}^{a} | 0 | 16096 | 16817 | 21654 | 21896 | 23995 | |
MCPF^{53} | 0 | 15288 | 15728 | ||||
MRCI^{53} | 0 | 15853 | 15655 | 20039 | 20743 | ||
CASPT2^{37} | 0 | 15650 | 17340 | 20570 | 21860 | 23800 | |
r_{e} | XMS-CASPT2/awCVTZ | 1.822 | 1.851 | 1.824 | 1.858 | 2.107 | 1.991 |
MRCI/aVTZ | 1.830 | 1.864 | 1.833 | 2.149 | |||
EOM-CCSD/TZ_{D}^{a} | 1.787 | 1.811 | 1.789 | 1.818 | 2.049 | 1.934 | |
MCPF^{53} | 1.811 | 1.842 | 1.813 | ||||
MRCI^{53} | 1.814 | 1.838 | 1.817 | 1.842 | 2.073 | ||
CASPT2^{37} | 1.79 | 1.82 | 1.77 | 1.84 | 1.97 | 1.91 | |
ω_{e} | XMS-CASPT2/awCVTZ | 796 | 726 | 763 | 696 | 592 | 696 |
MRCI/aVTZ | 777 | 693 | 750 | 542 | |||
EOM-CCSD/TZ_{D}^{a} | 881 | 822 | 847 | 804 | 606 | 648 | |
MCPF^{53} | 855 | 785 | 832 | ||||
MRCI^{53} | 866 | 801 | 834 | 789 | 638 | ||
μ_{e} | MRCI/aVTZ^{b} | 4.410 | 7.871 | 4.343 | 2.971 | 1.329 | |
EOM-EA-CCSD/TZ_{D}^{c} | 4.905 | 7.867 | 4.147 | 2.164 | |||
MCPF^{53} | 3.976 | 7.493 | 3.244 |
The results of our EOM-CCSD calculations indicate that the anionic reference is less suitable for describing YO than the cationic one. In Table 3, more accurate cationic-reference EOM-EA-CCSD spectroscopic constants are listed for all states except for the C^{2}Π and D^{2}Σ^{+} ones which are not accessible via the electron attachment procedure and therefore were described at the EOM-CCSD level only via EOM-IP.
The results given in Table 3 are obviously inferior to those obtained from high-level CCSD(T) calculations including core–valence correlation and extrapolation to the CBS limit. The CCSD(T) results are collected in Table 4 together with the experimental data available to date. The spread in the CCSD(T)/CBS results from different CBS extrapolation schemes serve as a rough estimate of the uncertainty in extrapolation. The good agreement between the CBS estimates and experimentally determined spectroscopic properties of the X^{2}Σ^{+}, A′^{2}Δ, A^{2}Π, and B^{2}Σ^{+} electronic states demonstrates the high accuracy in the CCSD(T)/CBS PECs of these states for bond lengths in the vicinity of the PECs minima, and is indicative of the mild MR character of the respective electronic wave functions.
X^{2}Σ^{+} | A′^{2}Δ | A^{2}Π | B^{2}Σ^{+} | C^{2}Π | D^{2}Σ^{+} | a^{4}Π | ||
---|---|---|---|---|---|---|---|---|
D_{0}, T_{e} | awCVTZ | 7.060 | 28924 | |||||
awCVQZ | 7.207 | 14809 | 16555 | 20893 | 21423 | 23261 | 29296 | |
awCV5Z | 7.260 | 14712 | 16538 | 20898 | 21700 | 23528 | 29465 | |
CBS1 | 7.298 | 14633 | 16526 | 20901 | 21925 | 23745 | 29603 | |
CBS2 | 7.298 | 14629 | 16525 | 20897 | 21917 | 23741 | 29592 | |
CBS3 | 7.289 | 29564 | ||||||
Expt | 7.290(87)^{52} | 14701^{38} | 16530^{35} | 20793^{35} | 23972^{37} | |||
r_{e} | awCVTZ | 1.7978 | 2.0902 | |||||
awCVQZ | 1.7927 | 1.8201 | 1.7971 | 1.8268 | 2.0408 | 1.9345 | 2.0841 | |
awCV5Z | 1.7905 | 1.8177 | 1.7950 | 1.8244 | 2.0384 | 1.9323 | 2.0817 | |
CBS1 | 1.7887 | 1.8157 | 1.7932 | 1.8225 | 2.0365 | 1.9306 | 2.0797 | |
CBS2 | 1.7890 | 1.8160 | 1.7935 | 1.8228 | 2.0368 | 1.9308 | 2.0799 | |
CBS3 | 1.7892 | 2.0802 | ||||||
Expt | 1.7882^{11} | 1.817^{39} | 1.7931^{13} | 1.8252^{35} | ||||
ω_{e} | awCVTZ | 855.2 | 546.0 | |||||
awCVQZ | 861.4 | 794.0 | 822.5 | 780.6 | 601.8 | 661.2 | 550.6 | |
awCV5Z | 864.2 | 797.1 | 825.1 | 783.2 | 603.3 | 662.3 | 552.1 | |
CBS1 | 866.5 | 799.7 | 827.1 | 785.3 | 604.6 | 663.1 | 553.3 | |
CBS2 | 866.2 | 799.4 | 826.8 | 785.1 | 604.6 | 663.1 | 553.3 | |
CBS3 | 865.8 | 552.9 | ||||||
Expt | 861.5^{11} | 794^{49} | 820^{35} | 765^{21} | ||||
759^{35} | ||||||||
ω_{e}x_{e} | awCVTZ | 2.79 | 2.52 | |||||
awCVQZ | 2.78 | 3.06 | 3.17 | 2.94 | 2.58 | 2.60 | 2.53 | |
awCV5Z | 2.79 | 3.05 | 3.18 | 2.98 | 2.57 | 2.60 | 2.57 | |
CBS1 | 2.79 | 3.04 | 3.19 | 3.01 | 2.57 | 2.61 | 2.60 | |
CBS2 | 2.79 | 3.03 | 3.18 | 3.01 | 2.57 | 2.61 | 2.60 | |
CBS3 | 2.79 | 2.59 | ||||||
Expt | 2.84^{11} | 3.23^{38} | 3.15^{11} | 3.97^{35} | ||||
3.35^{13} | ||||||||
α_{e} × 10^{3} | awCVTZ | 1.70 | 1.83 | |||||
awCVQZ | 1.68 | 1.85 | 1.90 | 1.86 | 1.79 | 1.85 | 1.83 | |
awCV5Z | 1.68 | 1.85 | 1.91 | 1.87 | 1.80 | 1.86 | 1.83 | |
CBS1 | 1.68 | 1.85 | 1.91 | 1.87 | 1.80 | 1.87 | 1.83 | |
CBS2 | 1.68 | 1.84 | 1.91 | 1.87 | 1.80 | 1.87 | 1.83 | |
CBS3 | 1.68 | 1.83 | ||||||
Expt | 1.73^{11} | 1.7^{38} | 2.01^{13} | 2.49^{35} | ||||
μ_{e} | awCVTZ | 4.615 | 7.595 | 3.711 | 1.749 | 2.059 | 1.256 | 3.605 |
awCVQZ | 4.614 | 7.620 | 3.724 | 1.764 | 2.082 | 1.275 | 3.615 | |
awCV5Z | 4.611 | 7.626 | 3.728 | 1.770 | 2.090 | 1.281 | 3.618 | |
CBS1 | 4.609 | 7.630 | 3.730 | 1.775 | 2.097 | 1.287 | 3.621 | |
CBS2 | 4.609 | 7.630 | 3.731 | 1.777 | 2.097 | 1.287 | 3.621 | |
CBS3 | 4.609 | 7.629 | 3.729 | 1.774 | 2.095 | 1.285 | 3.620 | |
Expt | 4.45(7)^{27} | 3.68(2)^{27} | ||||||
4.524(7)^{45} |
An insight into the reliability of the CCSD(T) PECs over the entire bond length range explored, and for all electronic states considered, including those not yet characterised experimentally, can be provided by using the MR diagnostic criteria commonly employed to identify the suitability of single reference wavefunction-based methods: T_{1},^{95} the Frobenius norm of the coupled cluster amplitude vector related to single excitations, and D_{1},^{96} the matrix norm of the coupled cluster amplitude vector arising from coupled cluster calculations. The utility of different MR diagnostics has been examined in earlier studies^{97,98} on various 3d and 4d TM species. The following criteria have been proposed^{98} as a gauge for the latter to predict the possible need to employ multireference wavefunction-based methods while describing energetic and spectroscopic molecular properties: T_{1} ≥ 0.045, D_{1} ≥ 0.120, %TAE[(T)] ≥ 10%. The symbol %TAE[(T)] denotes here the percent total atomization energy corresponding to a relationship between energies determined with CCSD and CCSD(T) calculations.^{99,100} Obviously, the %TAE[(T)] diagnostic is applicable for judging the SR/MR character of the electronic ground state only. For the YO molecule, the CCSD/awCV5Z calculated %TAE[(T)] of 5.6% is well below the proposed MR threshold. This fact provides further evidence for single reference character of the X^{2}Σ^{+} wavefunction in the near-equilibrium region of Y–O bond lengths.
The CCSD/awCV5Z T_{1} plots vs. Y–O bond length are shown in Fig. 6. The similar D_{1} plots are illustrated in Fig. S1 of the ESI.† At shorter bond lengths, the diagnostics amount to 0.02–0.03 (T_{1}) and 0.05–0.12 (D_{1}), remaining below the MR thresholds down to 1.4 Å for most states. Upon bond stretch, T_{1} and D_{1} rapidly increase, typically exceeding the MR threshold at 2.1–2.2 Å. The behaviour of these diagnostics for the C^{2}Π state is a notable exception: the numerical values of both T_{1} and D_{1} remain well below the MR threshold throughout the bond length range studied. The D^{2}Σ^{+} state is also noteworthy: its T_{1} and D_{1} diagnostics are indicative of the CCSD D^{2}Σ^{+} wave function retaining its SR character in much narrower range of bond lengths compared to the other doublet states under study.
The relative importance of SR/MR character of YO can also be guessed with the use of spin contamination appearing in RHF-UCCSD calculations as a result of unrestricted spin at the CCSD level. According to Jiang et al.,^{97} spin contamination with 〈S^{2} − S_{z}^{2} − S_{z}〉 greater than 0.1 in an RHF-UCCSD wave function can be viewed as a strong indication of nondynamical correlation in an open-shell system. Plotting spin contamination vs. bond length, Fig. 7, clearly indicates the severe admixture of higher spin states in the CCSD A′^{2}Δ, A^{2}Π, B^{2}Σ^{+} and D^{2}Σ^{+} wave functions at bond lengths beyond 2.2–2.3 Å. Greater extent of spin contamination at longer internuclear distances can obviously be associated with larger values of T_{1} and D_{1} (Fig. 6 and Fig. S1 of the ESI†) exceeding the established MR thresholds.
Fig. 7 RHF-UCCSD/awCV5Z wave function spin contamination in the low-lying doublet electronic states of YO. |
It is instructive to compare the MR diagnostics discussed above with the weights of the principal configurations, C_{0}^{2}, in the MRCI wavefunctions of YO (see Table 1). At shorter bond lengths, the C_{0}^{2} values amount to ∼0.73 for the B^{2}Σ^{+} state and 0.79–0.83 for the remaining doublet electronic states under study. These values are smaller than the threshold, C_{0}^{2} ≥ 0.90, proposed in ref. 97 and 98 to recognise the wave function strongly dominated by a single configuration. It should, however, be noted that this criterion was established by analyzing the CASSCF wavefunctions, whereas the C_{0}^{2} of the entire MRCI wavefunction differs from that of the CASSCF reference function due to the contributions of external configurations, which make C_{0}^{2} a smaller number.
Upon the YO bond stretch, there is a gradual decrease in the weights of the configurations serving as a reference for the coupled cluster treatment of the X^{2}Σ^{+}, A′^{2}Δ, and A^{2}Π states. This indicates greater multireference character of the respective wave functions at longer bond distances, as do the CC-based MR diagnostics. The reference configuration for the C^{2}Π state has approximately the same weight, C_{0}^{2} ≅ 0.83, in the MRCI wavefunctions of YO throughout the bond length range explored, behaving just like the respective CC-based MR diagnostics. These examples of the CCSD-MRCI correlations imply that the CC-based MR diagnostics can be capable of providing qualitative data about the relative accuracy in the single-reference coupled cluster calculation results not only for near-equilibrium regions of electronic ground states, but also for excited states in a wider range of molecular geometries.
In general, the present analysis indicates essentially single reference character of the YO low-lying doublet states over most part of bond length range explored in our work and hence high accuracy in the respective domains of the CCSD(T) PECs. It may also be indicative of accuracy degradation at larger bond lengths, implying the need for additional adjustments of the CCSD(T) PECs. Nevertheless, the bond length range associated with high-energy sections of PECs is expected to have a limited impact on the simulated spectra.
^{4}Π | ^{4}Φ | ^{4}Σ^{+} | ^{4}Δ | ^{4}Σ^{−} | ||
---|---|---|---|---|---|---|
a The energies of the ^{4}Φ, ^{4}Σ^{+}, ^{4}Δ and ^{4}Σ^{−} states are given here with respect to the ^{4}Π state which is the lowest-lying quartet state of YO.b In ref. 53, the symmetry of the lowest quartet state of YO was reported to be ^{4}Φ rather than ^{4}Π.c In ref. 53, the adiabatic excitation energy of the lowest quartet state was obtained from the MCPF calculations, and the relative energies of various quartet states were determined at the CASSCF level. | ||||||
T_{e}^{a} | CASSCF | 20460 | 68 | 1062 | 1825 | 2542 |
CASPT2 | 24140 | 56 | 1432 | 2158 | 2785 | |
CASPT3 | 23772 | 53 | 1450 | 2175 | 2788 | |
MRCI | 25046 | 55 | 1394 | 2128 | 2748 | |
MRCI + Q | 26274 | 59 | 1357 | 2089 | 2699 | |
CASSCF^{53} | 26975^{b}^{,}^{c} | 52 | 1933 | 2664 | 3359 | |
r_{e} | CASSCF | 2.141 | 2.143 | 2.110 | 2.110 | 2.114 |
CASPT2 | 2.197 | 2.198 | 2.192 | 2.191 | 2.196 | |
CASPT3 | 2.210 | 2.211 | 2.207 | 2.208 | 2.214 | |
MRCI | 2.213 | 2.214 | 2.209 | 2.209 | 2.214 | |
MRCI + Q | 2.218 | 2.219 | 2.218 | 2.218 | 2.222 | |
CASSCF^{53} | 2.121^{b} | 2.122 | 2.108 | 2.109 | 2.114 | |
MCPF^{53} | 2.126^{b} | |||||
ω_{e} | CASSCF | 517 | 516 | 522 | 521 | 519 |
CASPT2 | 515 | 515 | 517 | 494 | 477 | |
CASPT3 | 519 | 518 | 524 | 494 | 474 | |
MRCI | 507 | 507 | 501 | 495 | 492 | |
MRCI + Q | 506 | 506 | 501 | 496 | 493 | |
CASSCF^{53} | 543 | 543 | 524 | 522 | 520 | |
MCPF^{53} | 526^{b} |
State | Configurations^{a} | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
10σ | 11σ | 12σ | 13σ | 5π_{+} | 5π_{−} | 6π_{+} | 6π_{−} | 2δ_{+} | 2δ_{−} | |
a See footnote to Table 1 for designations. | ||||||||||
^{4}Π(b_{1}) | 2 | 2 | + | 0 | + | 2 | 0 | 0 | + | 0 |
2 | 2 | + | 0 | 2 | + | 0 | 0 | 0 | + | |
^{4}Φ(b_{1}) | 2 | 2 | + | 0 | + | 2 | 0 | 0 | + | 0 |
2 | 2 | + | 0 | 2 | + | 0 | 0 | 0 | + | |
^{4}Σ^{+} | 2 | 2 | + | 0 | + | 2 | + | 0 | 0 | 0 |
2 | 2 | + | 0 | 2 | + | 0 | + | 0 | 0 | |
^{4}Δ(a_{1}) | 2 | 2 | + | 0 | + | 2 | + | 0 | 0 | 0 |
2 | 2 | + | 0 | 2 | + | 0 | + | 0 | 0 | |
^{4}Σ^{−} | 2 | 2 | + | 0 | + | 2 | 0 | + | 0 | 0 |
2 | 2 | + | 0 | 2 | + | + | 0 | 0 | 0 |
The results for the YO quartet states obtained in our work agree with those of Langhoff and Bauschlicher,^{53} Table 5, except for the symmetry of the lowest quartet state that was reported^{53} to be ^{4}Φ rather than ^{4}Π.
The single-reference CCSD(T) method is expected to yield quite accurate results for the a^{4}Π state of YO since its MR diagnostics, C_{0}^{2} = 0.90, T_{1} = 0.024, and D_{1} = 0.078, indicate essentially SR character of the a^{4}Π wave function in the vicinity of the a^{4}Π PEC minimum, 2.00–2.25 Å. The very large CCSD(T)/CBS excitation energy of the a^{4}Π state, 29600 cm^{−1}, suggests that the quartet states in YO are too high in energy to significantly affect the spectroscopy of its low-lying doublet states.
r_{e} | Δ_{SO}r_{e} | ω_{e} | Δ_{SO}ω_{e} | T_{e} | Δ_{SO}T_{e} | |
---|---|---|---|---|---|---|
X^{2}Σ^{+}_{1/2} | 1.7866 | 0.0000 | 880.9 | 0.0 | 0 | 0 |
A′^{2}Δ_{3/2} | 1.8109 | +0.0003 | 821.5 | −0.5 | 15937 | −159 |
A′^{2}Δ_{5/2} | 1.8102 | −0.0004 | 822.6 | +0.6 | 16254 | +157 |
A^{2}Π_{1/2} | 1.7892 | +0.0005 | 846.9 | −0.5 | 16591 | −226 |
A^{2}Π_{3/2} | 1.7884 | −0.0003 | 847.8 | +0.4 | 17028 | +212 |
B^{2}Σ^{+}_{1/2} | 1.8181 | −0.0003 | 804.5 | +0.6 | 21671 | +17 |
C^{2}Π_{3/2} | 2.0488 | −0.0004 | 605.7 | +0.1 | 21808 | −88 |
C^{2}Π_{1/2} | 2.0495 | +0.0003 | 605.6 | −0.0 | 21994 | +98 |
Fig. 8 4c-EOM-IP-CCSD/TZ_{D} potential energy curves for the spin-coupled components of the A^{2}Π and C^{2}Π electronic states of YO in the avoided crossing region of bond length values. |
The SOC matrix elements between various doublet states of YO, which also accurately account for the corresponding phases, are shown in Fig. 9 as a function of r(Y–O). The relative phases of the couplings are important when used for solving the nuclear motion problem as part of the coupled Schrödinger equation, see, for example, discussion by Patrascu et al.^{92} The full details of the ab initio coupling curves including the magnetic quantum numbers are provided as part of the ESI.†
To shed more light on the alleged different order of the dipole moment values for the spin–orbit components of the A^{2}Π state in YO compared to ScO and LaO, we performed additional 4c-EOM-EA-CCSD/TZ_{D} computations for the two latter molecules at the experimental equilibrium bond lengths^{104,105} of 1.6826 Å (ScO) and 1.8400 Å (LaO). These resulted in the following values: μ(A^{2}Π_{1/2}) = 4.543 D, μ(A^{2}Π_{3/2}) = 4.532 D (ScO), μ(A^{2}Π_{1/2}) = 3.011 D, μ(A^{2}Π_{3/2}) = 2.907 D (LaO), i.e., the ab initio predicted difference between the two spin–orbit components monotonically increases on passing in the series ScO → YO → LaO: 0.01 → 0.04 → 0.10 D, respectively. The experimental counterparts^{101,103} are: μ(A^{2}Π_{1/2}) = 4.43(2) D, μ(A^{2}Π_{3/2}) = 4.06(3) D (ScO), μ(A^{2}Π_{1/2}) = 2.44(2) D, μ(A^{2}Π_{3/2}) = 1.88(6) D (LaO). Since the 4c-EOM-EA-CCSD dipole moments are expected to be overestimated by at least 0.5 D, one can consider the theoretical results to be in reasonable agreement with experiment. In light of our results, the experimental dipole moments^{27} for the two Ω components of the A^{2}Π state of YO need to be revisited.
The MCPF dipole moments obtained by Langhoff and Bauschlicher,^{53} 3.976 D (X^{2}Σ^{+}), 7.493 D (A′^{2}Δ) and 3.244 D (A^{2}Π), are systematically smaller than our CCSD(T) (Table 4), MRCI, and EOM-EA-CCSD (Table 3) results.
The MRCI DMCs and TDMCs of YO are shown in Fig. 10. The EAMCs are shown in Fig. S2 of the ESI.† All these curves as well as the SOC ones (Fig. 9) exhibit irregular behaviour at bond lengths around r ∼ 2 Å due to strong changes in the A^{2}Π, B^{2}Σ^{+}, C^{2}Π and D^{2}Σ^{+} wave functions over the avoided crossing region.
The Duo rovibronic wavefunctions of YO in conjunction with the ab initio TDMCs were then used to produce Einstein A coefficients for all rovibronic transitions between states considered in this work covering the wavenumber range from 0 to 40000 cm^{−1} and J ≤ 180.5. These Einstein A coefficients and the corresponding energies from the lower and upper states involved in each transition were organised as a line list using the ExoMol format.^{106} This format uses a two file structure with the energies included into the States file (.states) and Einstein coefficients appearing in the Transitions file (.trans). This ab initio line list is available from http://www.exomol.com. The ExoMol format has the advantage of being compact and compatible with our intensity simulation program ExoCross^{107} (see below).
υ | J | Ω | Duo | Obs. | ||
---|---|---|---|---|---|---|
a Band centres. | ||||||
ref. 109 | ref. 13 | |||||
X^{a} | 0 | 0 | 0 | 0 | 0 | |
1 | 0 | 860.879 | 855.2 | 855.7463(52) | ||
2 | 0 | 1716.156 | 1704.4 | 1705.8339(90) | ||
3 | 0 | 2565.836 | 2547.9 | 2550.2684(65) | ||
4 | 0 | 3409.931 | 3385.4 | 3389.0242(90) | ||
5 | 0 | 4248.450 | 4217.1 | 4222.085(11) | ||
6 | 0 | 5081.364 | 5049.454(13) | |||
ref. 13 | ||||||
A | 0 | 0.5 | 0.5 | 16295.492 | 16295.453 | |
0 | 1.5 | 1.5 | 16724.499 | 16724.541 | ||
1 | 1.5 | 1.5 | 17117.400 | 17109.845 | ||
1 | 1.5 | 1.5 | 17545.141 | 17538.459 | ||
2 | 1.5 | 1.5 | 17931.536 | 17916.880 | ||
2 | 1.5 | 1.5 | 18358.848 | 18345.768 | ||
3 | 1.5 | 1.5 | 18740.505 | 18716.674 | ||
3 | 1.5 | 1.5 | 19169.245 | 19146.593 | ||
4 | 1.5 | 1.5 | 19547.495 | 19510.064 | ||
4 | 1.5 | 1.5 | 19964.360 | 19940.488 | ||
5 | 1.5 | 1.5 | 20350.928 | 20296.636 | ||
5 | 1.5 | 1.5 | 20732.561 | 20727.595 | ||
ref. 39 | ||||||
A′ | 0 | 2.5 | 1.5 | 14500.074 | 14502.010 | |
ref. 21 | ref. 35 | |||||
B | 0 | 0.5 | 0.5 | 20741.630 | 20741.688 | 20741.6877 |
1 | 0.5 | 0.5 | 21516.945 | 21492.4773 | ||
2 | 0.5 | 0.5 | 22294.044 | |||
3 | 0.5 | 0.5 | 23102.252 | 22941.71 | ||
4 | 0.5 | 0.5 | 23850.742 | 23615.3 | ||
ref. 37 | ||||||
D | 0 | 0.5 | 0.5 | 23969.916 | 23969.940 | |
1 | 0.5 | 0.5 | 24659.745 | 24723.766 |
To allow for a direct comparison with the observed spectra of ^{89}Y^{16}O, we generated a line list covering rotational excitations up to J = 190 and the energy/wavenumber range up to 40000 cm^{−1}, with a lower state energy cutoff of 16000 cm^{−1}.
Fig. 11 Partition functions of YO: solid line is from this work computed using the energies of the six lowest electronic states; filled circles represent the partition function values by Vardya^{112} generated using spectroscopic constants of 3 lowest electronic states X, A and B (multiplied by a factor of 2 to account for the nuclear statistics); open squares represent values by Barklem and Collet^{110} (times the factor 2). |
An overview of the YO absorption spectra in the form of cross sections at the temperature T = 2000 K is illustrated in Fig. 12. Here, a Gaussian line profile with a half-width-at-half-maximum (HWHM) of 5 cm^{−1} was used. This figure shows contributions from each electronic band originating from the ground electronic state. The strongest bands are A^{2}Π–X^{2}Σ^{+} and B^{2}Σ^{+}–X^{2}Σ^{+}. The visible A–X band is known to be important for the spectroscopy of cool stars. The C state is of the same symmetry as A, however, the corresponding band C–X is much weaker due to the small Franck–Condon effects. The A′^{2}Δ–X^{2}Σ^{+} band is forbidden and barely seen in Fig. 12, however, it is strong enough to be experimentally known.^{39}
Fig. 13 shows a simulated emission spectrum of the strongest orange system YO (A^{2}Π–X^{2}Σ^{+}, (0,0)), which is compared to the experiment of Badie and Granier^{31} (from the plume emission close to the liquid Y_{2}O_{3} surface). It is remarkable that even pure ab initio calculations (after modest adjustment of the corresponding T_{e} value by +9.509 cm^{−1}) provide very close reproduction of experiment. It shows that our line list at the current, ab initio quality should be useful for modelling spectroscopy of exoplanets and cool stars in the visible region.
Fig. 13 Comparison of the computed A^{2}Π–X^{2}Σ^{+} orange band with the observations of Badie and Granier.^{31} Our simulations assume T = 3000 K and Gaussian line profile of HWHM = 1 cm^{−1}. |
Fig. 14 illustrates the A′^{2}Δ–X^{2}Σ^{+} (0,0) forbidden band in emission simulated for T = 77 K compared to the experimental spectrum of Simard et al.^{39} Here, a shift of +81.096 cm^{−1} was applied to the T_{e} value of the A′^{2}Δ state. In spectral simulations, this region appears to be contaminated by the dipole-allowed hot A–X transitions, which are not necessarily very accurate in this region. We therefore applied a filter to select the A′^{2}Δ–X^{2}Σ^{+} transitions only. The difference in shape of the spectra can be attributed either to the non-LTE (Local Thermal Equilibrium) effects present in the experiment or broadening effects, which we have not attempted to model properly. This figure is only to illustrate the generally good agreement of the positions of the rovibronic lines in this band.
Fig. 14 Comparison of the computed emission A′^{2}Δ–X^{2}Σ^{+} (0,0) band with the measurements of Simard et al.^{39} at T = 77 K and Gaussian line profile of HWHM = 0.1 cm^{−1}. |
Fig. 15 shows a series of absorption bands compared to the measurements of Zhang et al.^{37} Zhang et al.^{37} who observed bands in both the B^{2}Σ^{+}–X^{2}Σ^{+} and D^{2}Σ^{+}–X^{2}Σ^{+} systems in a heavily non-thermal environment where the vibrations were hot and the rotations cooled to liquid nitrogen temperatures. In this case of multi-band system it was important to include at least some non-LTE effects by treating it using two temperatures, vibrational and rotational, assuming that the corresponding degrees of freedom are in LTE. The rotational temperature T_{rot} = 77 K was set to value specified by Zhang et al.,^{37} while the vibrational temperature was adjusted to T_{vib} = 2000 K to better reproduce the experimental spectrum. The spectrum is divided into five spectroscopic windows (I–V) which are also detailed in Table 9. In order to match the positions of the vibronic bands in the experiment, some of the windows were shifted. For example, the D^{2}Σ^{+}–X^{2}Σ^{+} (1,0) band was shifted by about 76.5 cm^{−1}. This shift is an indication of the inaccuracy with which our model reproduces the vibrationally excited states of D^{2}Σ^{+}. This is not surprising considering the complexity of the quantum-chemistry part of these systems as well as of the nuclear motion part. The avoided crossing with the B^{2}Σ^{+} state leads to very complex shapes of the D^{2}Σ^{+} PEC and of the SO and electronic angular momentum coupling curves with the A and C states. The corresponding SOCs of the B and D states with the nearby state C are also relatively large, ∼30 cm^{−1} and 80 cm^{−1}, respectively (see Fig. 9), and therefore important. Besides, the D PEC is rather shallow with the equilibrium in the vicinity of the avoided crossing point, which also complicates the solution. An accurate description of the B and D curves would require diabatic representations before attempting any empirical refinement by fitting to the experiment. In all cases our simulations, while not perfect, show striking agreement with the observed spectra.
Fig. 15 Comparison of our computed emission spectra to the measurements of Zhang et al.^{37} Our simulations assumed a cold rotational temperature of T_{rot} = 77 K and a hot vibrational temperature of T_{vib} = 2000 K. The Gaussian line profile of HWHM = 0.1 cm^{−1} was used. |
Experiment | Theory | Band | |
---|---|---|---|
I | 20714.5–20753.5 | 20715–20754 | B(0,0) |
II | 23078.5–23117 | 23073–23112 | D(0,1) |
III | 23837.5–23874.5 | 23769–23806 | D(1,1) |
IV | 23934.5–23973 | 23934.5–23973 | D(0,0) |
V | 24689–24730 | 24625.5–24666 | D(1,0) |
YO is one of the few molecules with the strong potential for laser-cooling techniques,^{18} which have widely ranging applications, from quantum information and chemistry to searches for new fundamental physics. The results of this work will help to model the cooling properties of YO and thus will be important for designing and implementing laser-cooling experiments.
The ab initio curves of YO obtained in this study are provided as part of the supplementary material to this paper along with our spectroscopic model in a form of a Duo input file. The computed line list can be obtained from http://www.exomol.com. This is given in the ExoMol format^{106} which also includes state-dependent lifetimes. The line list can be directly used with the ExoCross program to simulate the spectral properties of YO.
Footnote |
† Electronic supplementary information (ESI) available. See DOI: 10.1039/c9cp03208h |
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