Photodissociation of carbon dioxide in singlet valence electronic states. I. Six multiply intersecting ab initio potential energy surfaces

The $\sim 7.5$ eV deep adiabatic ground state PES supports three structural isomers: The familiar linear OCO molecule is the global equilibrium, while the carbene-like bent OCO and the linear COO are the two local ones. Table 1 summarizes the characteristic features of the $\tilde{X}^{1}\Sigma_{g}^{+}$ state at the three equilibria and compares them with the previous ab initio studies and with the available experimental data.

The vicinity of the global minimum is of capital importance for the environmental chemistry.P11 The calculated equilibrium CO bond distance in linear OCO, $R_{e}=2.1991\,a_{0}$, agrees well with the experimental value of 2.1960 $a_{0}$. The accuracy of the vibrational zero-point energy (ZPE) and the vibrational transitions frequencies is assessed in Table 2 which compares energies of the low lying vibrationally excited states in rotating CO₂ (the total angular momentum $N_{\rm CO2}\geq 0$) with experimentC79A and with recent electronic coupled cluster/vibrational configuration interaction calculations.RHYHTST07 ; YHH08 Each eigenstate $(v_{s},v_{b}^{l},v_{a})$ is labeled using the quantum numbers of the symmetric stretch $v_{s}$, the bend $v_{b}^{l}$ (with $l$ indicating the vibrational angular momentum, $N_{\rm CO2}\geq l$), and the antisymmetric stretch $v_{a}$. The calculated fundamental frequencies of the infrared active bend ($\omega_{b}=668.6$ cm^-1) and antisymmetric stretch ($\omega_{a}=2350.6$ cm^-1) are accurate to within 1.5 cm^-1. The zeroth order symmetric stretch frequencyNISTDATABASE1 $\omega_{s}^{0}\approx 1333$ cm^-1 is about twice as large as the bending frequency $\omega_{b}$, and the two modes are involved in the accidental Fermi resonance.F31 As a result, the vibrational spectrum is organized in polyads with the polyad quantum number $P=2v_{s}+v_{b}$; states with $P=2$, 3 and 4 are given in Table 2. In the original version of the PES, called ‘PES1’ in Table 2, the energies of states $(1,0^{0},0)$ and $(0,2^{0},0)$, belonging to the lowest polyad $P=2$, are underestimated by 20 cm^-1 and the difference with the observed energies grows rapidly with $P$. This systematic discrepancy is substantially diminished by slightly rescaling the symmetric stretch, $R_{+}=(R_{1}+R_{2})/\sqrt{2}$, and the bend $\alpha_{\rm OCO}$ via

	$\displaystyle R_{+}$	$\displaystyle\rightarrow$	$\displaystyle\sqrt{2}R_{1e}+(R_{+}-\sqrt{2}R_{1e})\cdot 1.023$
	$\displaystyle\alpha_{\rm OCO}$	$\displaystyle\rightarrow$	$\displaystyle 180^{\circ}+(\alpha_{\rm OCO}-180^{\circ})\cdot 1.0035\,.$

The vibrational energies in the scaled ‘PES2’ agree with their experimental counterparts to within 7 cm^-1 and for the most states below 3000 cm^-1 the accuracy is better than 3 cm^-1. The results outperform even the highly accurate calculations of Refs. RHYHTST07, and YHH08, shown in the second column of Table 2, making ‘PES2’ one of the best available ab initio potentials of the $\tilde{X}^{1}\Sigma_{g}^{+}$ state. Since the coordinate dependent dipole moment $\mu_{\tilde{X}}$ has also been calculated, the ab initio intensities of the infrared rovibrational transitions can be directly evaluated.G13C

The other two isomers in Table 1 have never been detected in the gas phase, and the only reference data stem from the previous ab initio studies. For the bent OCO, discovered by Xantheas and Ruedenberg,XR94 the present calculations confirm the $C_{2v}$ symmetric equilibrium with the CO bond lengths of 2.51 $a_{0}$ and the OCO bond angle of 73.2^∘. This minimum is located 6.03 eV above the global one, again in good agreement with the previous findings.XR94 ; HM00A The fundamental excitations in the OCO well, calculated for $N_{\rm CO2}=0$, are $\omega_{a}=680$ cm^-1, $\omega_{b}=720$ cm^-1, and $\omega_{s}=1550$ cm^-1. For the linear COO, the calculated CO and OO bond lengths are identical to the ones given in Ref. HM00A, ; both are elongated compared to free diatoms ($2.2\,a_{0}$ vs. 2.14 $a_{0}$ for CO and 2.45 $a_{0}$ vs 2.28$\,a_{0}$ for OO). The calculations place the COO minimum at 7.35 eV, about 0.1 eV below the lowest dissociation threshold.

The ground electronic state correlates adiabatically with two dissociation channels,

	$\displaystyle{\rm CO}_{2}+h\omega(E_{\rm ph.}\geq\mbox{ }7.41\,{\rm eV})$	$\displaystyle\rightarrow$	$\displaystyle{\rm O}(^{1}\!D)+{\rm CO}(X^{1}\Sigma^{+})$		(1)
	$\displaystyle{\rm CO}_{2}+h\omega(E_{\rm ph.}\geq\mbox{ }11.52\,{\rm eV})$	$\displaystyle\rightarrow$	$\displaystyle{\rm C}(^{3}\!P)+{\rm O}_{2}(X^{3}\Sigma^{-})\,,$		(2)

and the ZPE corrected dissociation energies $D_{0}$ are shown in Table 1. In channel (1), the calculated $D_{0}$ is 0.13 eV less than the experimental value. The deviation might reflect a large basis set superposition error introduced by the diffuse functions and as such is the downside of the highly accurate vibrational spectrum in Table 2. The error is independent of the arrangement channel, and $D_{0}$ in channel (2) [closed between 120 nm and 160 nm] is equally underestimated. The calculations of Hwang and MebelHM00A , using a noticeably smaller basis set, perfectly agree with the experimental dissociation energy for this channel.

One-dimensional (1D) cuts through the ground state PES are given for several $\alpha_{\rm OCO}$ angles in panels (a,c,e) of Figs. 1 and 2. Black solid circles are the raw adiabatic energies. The O + CO limit is reached smoothly and no barrier is detected towards the asymptote for any orientation of the CO diatom. The same is true for the C + O₂ channel, as illustrated in Fig. 2(e); the potential well in Fig. 2(e) is the COO isomer. Angular dependence of the $\tilde{X}^{1}\!A^{\prime}$ state is shown in panels (a,c,e) of Figs. 3 and 4 for two sets of fixed CO bonds. In Fig. 3, $R_{1}$ is fixed at the FC value; in Fig. 4, it is fixed close to the equilibrium of the bent OCO. Consequently, although the carbene-type minimum is perceptible in all panels, it is best seen in Fig. 4. As CO₂ bends, the adiabatic $\tilde{X}^{1}\!A^{\prime}$ state (black dots) forms a sharp narrowly avoided crossing with the state $2^{1}\!A^{\prime}$ around $\alpha_{\rm OCO}=100^{\circ}$ (see, for example, Fig. 3). A dynamically meaningful representation for such nearly degenerate pairs is diabatic rather than adiabatic. In fact, the black line in Figs. 1 — 4 depicts the $\tilde{X}^{1}\!A^{\prime}$ state locally diabatized at bent geometries as described in Sect. IV.1. This is the reason why the black dots sometimes switch away from the black line and why lines of different colors cross. In this locally diabatic picture, the bent OCO minimum correlates with the state $2^{1}\!A^{\prime}$ [purple line]. The transition state separating the bent and the linear minima, analyzed by Xantheas and RuedenbergXR94 and Hwang and MebelHM00A is thus the signature of this two-state intersection. The two-dimensional (2D) contour plots of the PES of the locally diabatic $\tilde{X}^{1}\!A^{\prime}$ state, used in the above calculations of vibrational states, are shown in Fig. 5.

Potentials of the excited electronic states are shown in Figs. 1 and 2 along one CO bond for $A^{\prime}$ and $A^{\prime\prime}$ symmetries. As with the $\tilde{X}$ state, the solid circles indicate ab initio adiabatic energies. In the CO + O arrangement channel, which is the focus of the present investigation, all calculated states but one converge to the dissociation threshold (1). The state $2^{1}\!A^{\prime\prime}$, correlating with $1^{1}\Sigma_{u}^{-}$ at the FC point, reaches the higher lying threshold

$${\rm CO}_{2}+h\omega(E_{\rm ph.}\geq\mbox{ }11.46\,{\rm eV})\rightarrow{\rm O}(^{3}\!P)+{\rm CO}(a^{3}\Pi)\,.$$

(3)

In the C + O₂ arrangement channel [Fig. 2(e,f)], three electronic states, the $\tilde{X}$ state and the two components of the $\Pi$ state, correlate with channel (2). Three other states ($\Sigma^{-}$ and $\Delta$) converge to the electronically excited fragments:

$${\rm CO}_{2}+h\omega(E_{\rm ph.}\geq\mbox{ }13.75\,{\rm eV})\rightarrow{\rm C}(^{1}\!D)+{\rm O}_{2}(^{1}\Delta)\,.$$

(4)

Topographic hallmarks of the excited states can be exemplified using the states $2,3^{1}\!A^{\prime}$. In the FC region near linearity [Fig. 1(a)], the states $2^{1}\!A^{\prime}$ and $3^{1}\!A^{\prime}$ form two sharp avoided crossings near $R_{1}=2.2\,a_{0}$ and $R_{1}=2.8\,a_{0}$. These crossings are in fact two CIs between states $1^{1}\Pi_{g}$ and $1^{1}\Delta_{u}$.KRW88 ; SFCCRWB92 ; G12A ; GB12 The CIs are not independent: They are connected into a whole line, called a ‘CI seam’,Y04 unusual properties of whichGB12 are discussed in Sect. III.3. As the molecule bends, the gap between the adiabatic states grows. In the lower state $2^{1}\!A^{\prime}$, the intersection cone first turns into a broad barrier along the dissociation path [$\alpha_{\rm OCO}>170^{\circ}$, Fig. 1(c)]. As $\alpha_{\rm OCO}$ decreases further, the local minimum near 2.3 $a_{0}$ deepens and the dissociation barrier disappears [Fig. 1(e) and Fig. 2(a)]. In linear COO [$\alpha_{\rm OCO}=0^{\circ}$, Fig. 2(e)], the sharp avoided crossing between states $2,3^{1}\!A^{\prime}$ reappears again, this time at $R_{1}=3.5\,a_{0}$. Similar to $\tilde{X}$, the state $2^{1}\!A^{\prime}$ supports a COO intermediate, although the local minimum lies 0.7 eV above the C + O₂ threshold (2). The evolution of the uppermost state $3^{1}\!A^{\prime}$ with decreasing angle is different, because its topography is very much influenced by a pronounced barrier located outside the FC zone near $R_{1}=3.6\,a_{0}$ and separating the flat inner region from a steep decline towards the asymptotic limit. This barrier is distinct over a broad angular range and its sharpness suggests a CI with a higher lying state. The nature of this intersection becomes apparent in Sect. III.5 discussing valence/Rydberg crossings.

The $A^{\prime\prime}$ states are similar in many respects. Near linearity, both symmetries mirror the topography of the orbitally degenerate states $1^{1}\Pi_{g}$ and $1^{1}\Delta_{u}$, the states $1,3^{1}\!A^{\prime\prime}$ are involved in the same CIs in the FC region [Fig. 1(b)], and their behavior in the bent molecule up to $\alpha_{\rm OCO}=120^{\circ}$ closely follows that of $2,3^{1}\!A^{\prime}$. The $1^{1}\!A^{\prime\prime}$ state stabilizes with decreasing $\alpha_{\rm OCO}$, while the $3^{1}\!A^{\prime\prime}$ state features a pronounced barrier around $R_{1}=3.6\,a_{0}$ [Fig. 1(d,f) and Fig. 2(b)]. Similarities persist to $\alpha_{\rm OCO}=0^{\circ}$ [Fig. 2(f)]: The states $1,3^{1}\!A^{\prime\prime}$ form a CI at $R_{1}=3.5\,a_{0}$, and the state $1^{1}\!A^{\prime\prime}$ supports a local COO minimum. Differences with the $A^{\prime}$ symmetry are due to the state $2^{1}\!A^{\prime\prime}$, which in the FC region correlates with $1^{1}\Sigma_{u}^{-}$ and which has no counterpart among the calculated $A^{\prime}$ states. This state, which at linearity is accidentally degenerate with $1^{1}\Delta_{u}$, has a single minimum near $2.4\,a_{0}$ and approaches the dissociation limit (3) without a barrier. This simple shape is preserved through most cuts in Figs. 1 and 2. The state $2^{1}\!A^{\prime\prime}$ also supports a COO minimum [Fig. 2(f)].

Potential cuts along bending angle are shown in Figs. 3 and 4. The doubly degenerate $1^{1}\Pi_{g}$ and $1^{1}\Delta_{u}$ states split into $A^{\prime}$ and $A^{\prime\prime}$ components for $\alpha_{\rm OCO}<180^{\circ}$. Evolution of the electronic energies with decreasing $\alpha_{\rm OCO}$ was analyzed by Spielfiedel et al. using Walsh rules.SFCCRWB92 Based on this analysis, the adiabatic excited states are commonly classified as bent or linear. The energy of states $2^{1}A^{\prime}$ and $1^{1}A^{\prime\prime}$ lowers as the molecule bends, while the energy of states $3^{1}A^{\prime}$ and $3^{1}A^{\prime\prime}$ grows [see Fig. 3(a,b)]. The global minima of the ‘bent’ states lie near 120^∘ ($2^{1}A^{\prime}$) and 130^∘ ($1^{1}A^{\prime\prime}$) and are located below the dissociation limit (1) [Fig. 3(a,b) and Fig. 4(a,b)]. Bent equilibrium of the state $2^{1}A^{\prime}$, which at $C_{2v}$ geometries becomes ${}^{1}\!B_{2}$, was predicted by Dixon in his analysis of the CO flame emission bands.D63 Finally, the state $2^{1}A^{\prime\prime}$, which together with $3^{1}A^{\prime}$ and $3^{1}A^{\prime\prime}$ is ‘linear’, is the least anisotropic of all states in a broad vicinity of $\alpha_{\rm OCO}\sim 180^{\circ}$.

Properties of various stationary points in the PESs of excited electronic states are given in Table 3. The $C_{s}$ point group notation is used to label adiabatic states; $D_{\infty h}$ labels refer to the diabatic states at the FC point. Experimental reference data for the excited electronic states are scarce and fit into the footnote $a$. For the bent states, the experimentalD63 ; CLP92 equilibrium CO distances and the bending angle are reproduced within $0.1\,a_{0}$ and $3^{\circ}$, respectively; the bending frequencies are accurate to within 30 cm^-1 or better, and the calculated band origins lie within the experimental uncertainties. Results of the previous ab initio studiesKRW88 ; SFCCRWB92 are also shown in Table 3. Agreement in the equilibrium geometries is excellent, with the exception of the intersection-ridden uppermost states $3^{1}A^{\prime}$ and $3^{1}A^{\prime\prime}$, for which $C_{2v}$-restricted calculations of Refs. KRW88, and SFCCRWB92, miss the local $C_{s}$ minima in the upper CI cones. Vertical excitation energies $T_{v}$ are close to those of Ref. SFCCRWB92, . Slight differences are not surprising for near degenerate states whose ordering is sensitive to the details of the ab initio set up. Peculiarity of the spectral region 120 nm — 160 nm is that $T_{v}$ values are poor approximations to the positions of the absorption maxima because all transitions are electronically forbidden and the TDMs at the FC point are strictly zero. Finally, the ab initio dissociation energies $D_{0}$ in channels (1), (2), (3), and (4), also given in Table 3, agree with the experimental valuesHERZBERG67 ; AMS79 within $\sim 0.15$ eV irrespective of the particular arrangement or electronic channel.

Intersections of electronic states in linear CO₂ follow simple ‘symmetry rules’ KDC84 ; CMD84 which severely restrict the intersection topography. Two types of intersections with regard to their symmetry properties are prominent in Figs. 1(a,b) and 3(a,b):
(1) Renner-Teller (RT) glancing intersectionsR34 ; HERZBERG67 involve the $A^{\prime}$ and $A^{\prime\prime}$ components of the $1^{1}\Pi_{g}$ or the $1^{1}\Delta_{u}$ state. In the rotating molecule, the $A^{\prime}/A^{\prime\prime}$ interaction $\sim\Lambda\Omega/\sin^{2}\alpha_{\rm OCO}$ is proportional to the projections $\Lambda$ and $\Omega$ of the electronic ($\hat{L}$) and the total angular momentum ($\hat{J}=\hat{N}_{\rm CO2}+\hat{L}$) on the molecular axis and diverges for $\alpha_{\rm OCO}\rightarrow 180^{\circ}$.R34 ; HERZBERG67 ; P88 ; GGH93 The $1^{1}\Pi_{g}\leftarrow\tilde{X}^{1}\Sigma_{g}^{+}$ transition is best classified as linear-linear, and the $1^{1}\Delta_{u}\leftarrow\tilde{X}^{1}\Sigma_{g}^{+}$ transition is linear-bent. The ‘degeneracy manifold’ for the intersecting states is the whole $(R_{1},R_{2})$ plane defined by the condition $\alpha_{\rm OCO}=180^{\circ}$.

The CI seam, constructed separately for the $2,3^{1}\!A^{\prime}$ and $1,3^{1}\!A^{\prime\prime}$ states on a fine $(R_{1},R_{2})$ grid,GB12 is depicted in Fig. 6. It has two remarkable properties. First, the intersection along the seam is fivefold. Two CIs and two RT intersections imply four degenerate states. The hitherto ignored state $1^{1}\Sigma_{u}^{-}$ closely follows $1^{1}\Delta_{u}$: The $\Delta/\Sigma$ energy gap falls consistently below 300 cm^-1 which is at the limit of the ab initio accuracy. Thus, the $1^{1}\Pi_{g}/1^{1}\Delta_{u}$ and the $1^{1}\Delta_{u}/1^{1}\Sigma_{u}^{-}$ pairs cross at the same CO bond distances and the total degeneracy is five. Second, the calculated seam traces out a closed loop. Closed CI seams are rarely encountered,ARN97 ; GB12 although arguments have been devised to prove their ubiquity.LHVBZKB08 ; LHVBZKB09 A technical implication of closed or strongly curved seams is that local diabatization schemes failGB12 and global or semiglobal diabatizationKGM01 becomes necessary. The impact of closed seams on photodissociation dynamics has never been systematically investigated.GB12 In CO₂, only a small portion of the loop is directly accessible to UV light. Nevertheless, paper II demonstrates that the seam topology deeply affects the observed absorption spectrum.

The curved intersection seam is responsible for a peculiar shape of the potentials in the FC region. The adiabatic PESs of the states $2,3^{1}\!A^{\prime}$ in linear CO₂ are shown in Fig. 6. The lower adiabatic state $2^{1}\!A^{\prime}$ has a $C_{\rm 2v}$ minimum at $R_{1}=R_{2}=2.41\,a_{0}$, and two $C_{\rm s}$ symmetric saddles near $R_{1}\approx 2.8\,a_{0}$ or $R_{2}\approx 2.8\,a_{0}$, separating the $C_{\rm 2v}$ minimum from the dissociation asymptotics. Outside the area enclosed by the seam line, the $2^{1}\!A^{\prime}$ state has the $\Pi$, inside the $\Delta$ character. The saddles hide portions of cusp lines, along which the states intersect and which are washed out by the spline interpolation. The topography of the upper adiabatic $3^{1}\!A^{\prime}$ state is a literal mirror image of the lower state; one finds a $C_{\rm 2v}$ saddle at $R_{1}=R_{2}=2.41\,a_{0}$ and two $C_{\rm s}$ symmetric minima near $(R_{1},R_{2})=(2.84\,a_{0},2.27\,a_{0})$ and $(2.27\,a_{0},2.84\,a_{0})$. The state character changes from $\Delta$ outside the seam line to $\Pi$ inside. The two minima are cusp-like and correspond to the upper cones of the CIs. The two other saddle points in the $3^{1}\!A^{\prime}$ state, lying outside the FC zone near $(R_{1},R_{2})=(3.6\,a_{0},2.2\,a_{0})$ and $(2.2\,a_{0},3.6\,a_{0})$, result from an avoided crossing with a higher lying state.

Changes of the electronic character of $A^{\prime\prime}$ states across the intersection can be monitored using matrix elements of the electronic angular momentum $\hat{L}_{z}$.DK97 The matrix elements $\left|\langle iA^{\prime\prime}|\hat{L}_{z}|jA^{\prime}\rangle\right|$ along the line passing twice through the closed seam are depicted in Fig. 7 for $i=1,2,3$ and $j=2,3$. The states $|2^{1}\!A^{\prime}\rangle$ and $|3^{1}\!A^{\prime}\rangle$ act as ‘probes’ whose assignment in terms of $\Pi$ or $\Delta$ is known. Outside the seam area, most matrix elements are close to integer values of 0, 1, and 2. For $R_{1}\leq 2.2\,a_{0}$ for example, $\left|\langle 1A^{\prime\prime}|\hat{L}_{z}|2A^{\prime}\rangle\right|=1$ and $\left|\langle 2A^{\prime\prime}|\hat{L}_{z}|2A^{\prime}\rangle\right|=0$ so that $|1A^{\prime\prime}\rangle$ is a $\Pi$ state, while $|2A^{\prime\prime}\rangle$ is a $\Sigma$ state. $\left|\langle 3A^{\prime\prime}|\hat{L}_{z}|2A^{\prime}\rangle\right|$ vanishes too, but the state $|3A^{\prime\prime}\rangle$ is a $\Delta$ state, as confirmed by the matrix element $\left|\langle 3A^{\prime\prime}|\hat{L}_{z}|3A^{\prime}\rangle\right|=2$. As the CI seam is crossed at $R_{1}=2.25\,a_{0}$ into the area interior to the seam loop, the three-state intersection induces violent changes in the electronic labels of the adiabatic states. The state $|3A^{\prime\prime}\rangle$ becomes a $\Pi$ state ($\left|\langle 3A^{\prime\prime}|\hat{L}_{z}|3A^{\prime}\rangle\right|=1$); the state $|2A^{\prime\prime}\rangle$ acquires $\Delta$ character ($\left|\langle 2A^{\prime\prime}|\hat{L}_{z}|2A^{\prime}\rangle\right|\approx 2$), while the state $|1A^{\prime\prime}\rangle$ acquires $\Sigma$ character ($\left|\langle 1A^{\prime\prime}|\hat{L}_{z}|2A^{\prime}\rangle\right|\approx 0.5$), and the non-integer values of the latter two reflect strong mixing of the near degenerate $\Delta_{u}/\Sigma^{-}_{u}$ pair. The next reshuffling occurs as the seam loop is crossed outwards at $R_{1}\approx 2.8\,a_{0}$. The state $|1A^{\prime\prime}\rangle$ becomes $\Pi$ again, while the matrix elements involving $|2A^{\prime\prime}\rangle$ and $|3A^{\prime\prime}\rangle$ vary until $R_{1}\approx 3.6\,a_{0}$ is reached, where $\left|\langle 3A^{\prime\prime}|\hat{L}_{z}|3A^{\prime}\rangle\right|$ finally vanishes indicating that $|3A^{\prime\prime}\rangle$ emerges from the crossing region as a $\Sigma$ state converging towards the upper threshold (3), while $|2A^{\prime\prime}\rangle$ becomes $\Delta$ state correlating with threshold (1).

The positions of cusps, minima, and saddles in the PESs of the $A^{\prime\prime}$ states at linearity are identical to those in $A^{\prime}$ states. Exception is the PES of the state $2^{1}\!A^{\prime\prime}$, correlating in the FC region with $1^{1}\Sigma_{u}^{-}$. Its appearance, illustrated in Fig. 8, is very much simplified by it being tightly linked with the diabatic $1^{1}\Delta_{u}$ state. The PES has a single $C_{\rm 2v}$ minimum at $R_{1}=R_{2}=2.41\,a_{0}$, is free from cusps seen in other adiabatic PESs, and provides a nature’s illustration of the shape of the diabatic $1^{1}\Delta_{u}$ PES.

CIs at linearity can also be recognized in the potentials plotted in the $(R_{1},\alpha_{\rm OCO})$ plane. An example including four electronic states at $R_{2}=2.2\,a_{0}$ is shown in Fig. 9. Although the characteristic features are no longer $C_{2v}$ symmetric, a maximum in the $2^{1}\!A^{\prime}$ state at $\alpha_{\rm OCO}=180^{\circ}$ and $R_{1}=2.8\,a_{0}$ and two minima in the $3^{1}\!A^{\prime}$ state at $\alpha_{\rm OCO}=180^{\circ}$ and $R_{1}=2.3\,a_{0}$ and $R_{1}=2.8\,a_{0}$ are recognizable in panels (a) and (b). Close to linearity, the $A^{\prime\prime}$ states [panels (c) and (d)] maintain the same contour maps as $A^{\prime}$ states.

As CO₂ bends and all above degeneracies are removed, the primary topographic features (cusps, minima, and saddles) remain clearly visible up to $\alpha_{\rm OCO}\sim 160^{\circ}$. This is demonstrated in the $(R_{1},\alpha_{\rm OCO})$ plane in Fig. 9 and in the $(R_{1},R_{2})$ plane in Fig. 10. Further decrease of $\alpha_{\rm OCO}$ spawns new avoided crossings, a detailed map of which is given in the angular cuts in Figs. 3 and 4. All pairs of calculated states become near degenerate at various geometries. For example, the $\tilde{X}$ state forms avoided crossings successively with states $2^{1}\!A^{\prime}$ at $\alpha_{\rm OCO}=100^{\circ}$ and $3^{1}\!A^{\prime}$ at $\alpha_{\rm OCO}=80^{\circ}$ (Fig. 4). The state $3^{1}\!A^{\prime\prime}$ approaches closely $2^{1}\!A^{\prime\prime}$ at $\alpha_{\rm OCO}\sim 120^{\circ}-130^{\circ}$ and $1^{1}\!A^{\prime\prime}$ at $\alpha_{\rm OCO}=90^{\circ}$, and the states $1^{1}\!A^{\prime\prime}$ and $2^{1}\!A^{\prime\prime}$ cross between $70^{\circ}$ and $80^{\circ}$. Solid lines in Figs. 3 and 4 cross because they refer to the states locally diabatized through the ‘bent’ avoided crossings (see Sect. IV.1). Potential curves of the states $3^{1}\!A^{\prime}$ and $3^{1}\!A^{\prime\prime}$ bear evidence of strong interactions with the next higher states. Especially pronounced in Fig. 3(d,f) and Fig. 3(b,d) are the sharp near-intersections in $3^{1}\!A^{\prime\prime}$ around 155^∘.

As a result of multiple avoided crossings, local bent equilibria are found in all calculated states. The diabatic origin of a particular local minimum is invariably different from the adiabatic one. Consider the carbene-like bent OCO with $\alpha_{\rm OCO}\sim 70^{\circ}$ in Fig. 4(a). Purely adiabatically (solid dots), this minimum belongs to the ground electronic state. In the locally diabatic picture, bent OCO belongs the state $2^{1}\!A^{\prime}$ (purple line). An avoided crossing between states $2^{1}\!A^{\prime}$ and $3^{1}\!A^{\prime}$, recognizable in panel (a) around $\alpha_{\rm OCO}\sim 95^{\circ}$, implies that another diabatization might re-assign the bent OCO minimum to the $3^{1}\!A^{\prime}$ state (brown line). Furthermore, the broad barrier in $3^{1}\!A^{\prime}$ near $\alpha_{\rm OCO}\approx 140^{\circ}$ indicates another avoided crossing with the next higher $A^{\prime}$ state which thus would be the true owner of the bent OCO minimum. Similar analysis applies to the bent OCO in the $A^{\prime\prime}$ states in Fig. 4(b). Depending on the chosen representation, it can be ascribed to the fully adiabatic state $1^{1}\!A^{\prime\prime}$ (dots), to the locally diabatic state $3^{1}\!A^{\prime}$ (brown line), or — via the sharp near-intersection around $155^{\circ}$ — to the next higher $A^{\prime\prime}$ state. Other bent conformations in the excited states result from state interactions, too. One such isomer with the valence angle of 100^∘ is created via an avoided crossing in $3^{1}\!A^{\prime}$ [Fig. 3(c,e)]. Crossing of the same $3^{1}\!A^{\prime}$ state with the next higher state between 60^∘ and 70^∘ leads to another high-energy bent minimum [Fig. 4(a,e)].

Many local barriers and minima in the states $3^{1}\!A^{\prime}$ and $3^{1}\!A^{\prime\prime}$ are due to intersections with ‘invisible’ higher lying electronic states. Previous detailed ab initio studies exposed these ‘invisible’ intersection partners as mainly Rydberg states.EEW77 ; CJL87 ; SFCRW91 ; SFCRW93 The aim of this section is to illustrate how the Rydberg/valence interaction affects the potential profiles along the dissociation path and to outline the geometries at which valence states can effectively drain population from the Rydberg manifold. To this end, CASSCF calculations of the first 10 states, $1$—$5^{1}\!A^{\prime}$ and $1$—$5^{1}\!A^{\prime\prime}$, have been performed with the d-aug-cc-pVQZ basis set for the CO bond distances $R_{\rm 2}\in[1.7\,a_{0},5.0\,a_{0}]$ and the OCO bond angles $\alpha_{\rm OCO}\in[100^{\circ},179^{\circ}]$. One CO bond is kept fixed at $R_{1}=2.4\,a_{0}$, so that only CO + O arrangement channel is covered.

The CASSCF potentials along $R_{2}$ are shown in Fig. 11. All five $A^{\prime\prime}$ states and four $A^{\prime}$ states converge to thresholds (1) or (3). The states correlating with the lowest threshold (1) are of $\Sigma^{+}$, $\Delta$, and $\Pi$ symmetry. The states correlating with the highest threshold (3) are $\Sigma^{\pm}$, $\Delta$, and $\Pi$. One $A^{\prime}$ state converges towards the asymptote

$\displaystyle{\rm CO}_{2}+h\omega(E_{\rm ph.}\geq\mbox{ }9.64\,{\rm eV})$

$\displaystyle\rightarrow$

$\displaystyle{\rm O}(^{1}\!S)+{\rm CO}(X^{1}\Sigma^{+})\,.$

(5)

At linearity, this state is the Rydberg state $1^{1}\Sigma^{+}_{u}(3\pi_{u})$.NOTE-CO23-1 Compared to Fig. 1, many new avoided crossings are found in Fig. 11(a,b). The $D_{\infty h}$ notation of a state $|i\rangle$ in these panels is related to the expectation value $\langle i|L_{z}^{2}|i\rangle$ which a diabatic state preserves across the intersections. Potential curves in Fig. 11(a,b) are color coded according to the $\langle L_{z}^{2}\rangle$ values: $\Sigma$ states ($\langle L_{z}^{2}\rangle=0$) are shown in green; $\Pi$ states ($\langle L_{z}^{2}\rangle=1$) are blue, and $\Delta$ states ($\langle L_{z}^{2}\rangle=4$) are red/orange. Clearly, most adiabatic curves change color more than once as the CO bond stretches: They are tailored out of several diabatic states.

Familiar from the discussion in Sect. III.3 are the fivefold crossings involving $1^{1}\Pi_{g}(5\sigma_{g})$ and the accidentally degenerate pair $1^{1}\Delta_{u}(2\pi_{u})/1^{1}\Sigma_{u}^{-}(2\pi_{u})$. These crossings are marked with arrows in Fig. 11(b). As in the MRD-CI calculations, the $\Delta_{u}/\Sigma_{u}^{-}$ pair is easily recognizable using matrix elements of $L_{z}$. At linearity, $\langle L_{z}^{2}\rangle$ for these two states is either 0 or 4, but already a tiny deviation of 2^∘ causes $\langle L_{z}^{2}\rangle$ to collapse towards an average value of $\sim 2$. This state mixing is stressed in Fig. 11(a,b) with the $\Delta_{u}/\Sigma_{u}^{-}$ label and with the red/green color. The second edition of the mixed $\Delta_{u}/\Sigma_{u}^{-}$ pair is found around 11.5 eV. The Rydberg state $2^{1}\Delta_{u}(3\pi_{u})$, shown with an orange line in Fig. 11(a,b), carries as a satellite the state $2^{1}\Sigma_{u}^{-}(3\pi_{u})$ [green line in Fig. 11(b)]. Again, $\langle L_{z}^{2}\rangle$ for these states assumes a non-integer value between 1.5 and 2.5 in even slightly bent molecule.

The valence pair $1\Delta_{u}/1\Sigma_{u}^{-}$ converges towards the uppermost threshold (3) and successively traverses the higher lying states. The prominent dissociation barrier in the $3^{1}\!A^{\prime}$ and $3^{1}\!A^{\prime\prime}$ states near $R_{2}=3.6\,a_{0}$ is a clear signature of the sixfold valence/Rydberg crossing involving $1\Delta_{u}/1\Sigma_{u}^{-}$, the $1^{1}\Sigma_{u}^{+}$ state [green line in Fig. 11(a)] and a repulsive $R^{1}\Delta_{u}$ state [orange line in Fig. 11(a,b)]. $1^{1}\Sigma_{u}^{+}$ is the optically bright Rydberg state responsible for the strong absorption band around 11.1 eV (111.7 nm).CJL87 The state $R^{1}\Delta_{u}$ corresponds to a pair of strongly repulsive ${}^{1}\!A^{\prime}/^{1}\!A^{\prime\prime}$ states descending towards threshold (1) from very high energies. Two sections of this repulsive potential curve are seen in Fig. 11(a,b) in the intervals $[3.2\,a_{0},3.7\,a_{0}]$ and $[3.7\,a_{0},5.0\,a_{0}]$. Thus, the asymptotic repulsive portions of the $3^{1}\!A^{\prime}$ and $3^{1}\!A^{\prime\prime}$ states diabatically belong to $R^{1}\Delta_{u}$. The sixfold crossing $1\Sigma_{u}^{+}/R^{1}\Delta_{u}/1\Delta_{u}/1\Sigma_{u}^{-}$ near $R_{2}=3.6\,a_{0}$ is not the only one involving $R^{1}\Delta_{u}$ state. At higher energies and shorter CO bonds, $R^{1}\Delta_{u}$ intersects the Rydberg pair $2\Delta_{u}/2\Sigma_{u}^{-}$ (cf. Fig. 11(a,b) near $R_{2}\approx 3.0\,a_{0}$). The diabatic $R^{1}\Delta_{u}$ state distinctly stands out because it strictly preserves the projection $\langle L_{z}^{2}\rangle\approx 4$ even for $\alpha_{\rm OCO}=165^{\circ}$, while $\langle L_{z}^{2}\rangle$ values for the other states become non-integer.

The above discussion is valid for both $A^{\prime}$ and $A^{\prime\prime}$ states. The $A^{\prime\prime}$ symmetry block in Fig. 11(b) contains one more state, namely the Rydberg state $2^{1}\Pi_{g}(6\sigma_{g})$ missing among the $A^{\prime}$ states where it would have been $6A^{\prime}$. This state, materializing out of nowhere at $R_{2}=3.0\,a_{0}$, is involved in the $R^{1}\Delta_{u}/2\Delta_{u}/2\Sigma_{u}^{-}$ crossing — making it a sevenfold intersection.

While the adiabatic gaps in the valence/valence intersection $1\Pi_{g}/1\Delta_{u}/1\Sigma_{u}^{-}$ grow as $\alpha_{\rm OCO}$ deviates from 180^∘, all Rydberg/Rydberg and Rydberg/valence intersections not only remain recognizable in bent CO₂, but clearly sharpen up. This is illustrated in panels (c) — (f) of Fig. 11 drawn for $\alpha_{\rm OCO}=175^{\circ}$ and 160^∘. As a side result, the repulsive $R^{1}\!A^{\prime}$ and $R^{1}\!A^{\prime\prime}$ states, deriving from $R^{1}\Delta_{u}$, remain distinct at all angles, despite new ‘bent’ intersections in the $3,4,5^{1}A^{\prime}$ and $3,4,5^{1}A^{\prime\prime}$ states [see, for example, Fig. 11(e,f)].NOTE-CO23-2 Via these intersections, carbon dioxide excited with vacuum UV light can — at bent geometries — reach any of the shown dissociation channels. Although a detailed analysis of these multiple pathways is beyond the scope of the present work, a brief description of intersections involving the optically bright ${}^{1}\Sigma_{u}^{+}$ state is added at the end of this section.

Potential curves along $\alpha_{\rm OCO}$ are shown in Fig. 12. The crossing patterns, expecially numerous among $A^{\prime\prime}$ states, explain intricate topography of states $3^{1}\!A^{\prime}$ and $3^{1}\!A^{\prime\prime}$ in Fig. 3. For example, the sharp barrier in the state $3^{1}\!A^{\prime\prime}$ at angles between 150^∘ and 170^∘ [Fig. 3(d,f) and Fig. 3(b,d)] originates from a complicated three state crossing around 165^∘ involving states $3,4,5^{1}\!A^{\prime\prime}$ [Fig. 12(b)]. Diabatically, the decreasing branch of the $3^{1}\!A^{\prime\prime}$ state at $\alpha_{\rm OCO}<160^{\circ}$ belongs to $4^{1}\!A^{\prime\prime}$ — the state which at linearity merges into the Rydberg pair $2\Delta_{u}/2\Sigma_{u}^{-}$. This diabatic state can be followed to even smaller angles through another intersection, this time with $2^{1}\!A^{\prime\prime}$ near $130^{\circ}$. Ultimately, as indicated with an arrow in Fig. 12(b), it is this diabatic state which the carbene-like OCO minimum belongs to. The avoided crossings in the $A^{\prime}$ symmetry block are similar and simpler. The three state crossing of $3,4,5^{1}\!A^{\prime\prime}$ is a mere ghost because the adiabatic gap is almost 2 eV wide and the barrier in the $3^{1}\!A^{\prime}$ state is broad and low. The state $4^{1}\!A^{\prime}$, originating from $2^{1}\!\Delta_{u}$, stabilizes upon bending and is easily traced to smaller angles through the intersection with $3^{1}\!A^{\prime}$ near $\alpha_{\rm OCO}=150^{\circ}$. Again, the carbene-like OCO correlates diabatically with the state originating from $2^{1}\!\Delta_{u}$ at linearity.

The analysis of broadening and splittings in the strong absorption band of the Rydberg state $1^{1}\Sigma_{u}^{+}$ focused on the interactions of $1^{1}\Sigma_{u}^{+}$ with valence states.CJL87 ; SFCRW91 ; SFCRW93 Present calculations reveal numerous avoided crossings with both valence and Rydberg states inside and outside the FC zone. At linearity, the $1^{1}\Sigma_{u}^{+}$ state cuts twice through the Rydberg state $2^{1}\!\Delta_{u}$ [Fig. 11(a)], and both crossings persist to smaller angles [Fig. 11(c,e)]. Another crossing occurs near $R_{2}=3.7\,a_{0}$ (for all calculated angles) and mixes $1^{1}\Sigma_{u}^{+}$ with $R^{1}\Delta_{u}$. This interaction strongly perturbs the ab initio $\langle L_{z}^{2}\rangle$ values but vanishes beyond $R_{1}=4.0\,a_{0}$. The potential cut along $\alpha_{\rm OCO}$ in Fig. 12(a) exposes another avoided crossing at $\sim 120^{\circ}$ which involves the states $4,5^{1}\!A^{\prime}$ and leads to a local minimum in $5^{1}\!A^{\prime}$ (the state, correlating with $1^{1}\Sigma_{u}^{+}$ at these R_CO). Finally, the barrier in $5^{1}\!A^{\prime}$ near $150^{\circ}$ implies interaction with the next higher state, which according to the analysis of Ref. SFCRW93, correlates with $2^{1}\Sigma_{u}^{+}$ at linearity. While all these crossings can redirect population from the optically bright $1^{1}\Sigma_{u}^{+}$ state along various linear and bent routes, Fig. 11 clearly demonstrates that the diabatic $1^{1}\Sigma_{u}^{+}$ state, calculated at the CASSCF level, is repulsive at all geometries and thus can dissociate directly.

The vicinity of ‘bent’ CIs is diabatized locally, using the energy-based scheme as described, for example, by Köppel.K04 All considered CIs are shown in Figs. 3 and 4 and include

1. (a)

   the $\tilde{X}^{1}\!A^{\prime}/2^{1}\!A^{\prime}$ pair at $\alpha_{\rm OCO}=90^{\circ}-110^{\circ}$;

2. (b)

   the $2^{1}\!A^{\prime\prime}/3^{1}\!A^{\prime\prime}$ pair at $\alpha_{\rm OCO}=120^{\circ}-150^{\circ}$;

3. (c)

   the $1^{1}\!A^{\prime\prime}/3^{1}\!A^{\prime\prime}$ pair, with $3^{1}\!A^{\prime\prime}$ state diabatized in step (b), at $\alpha_{\rm OCO}=70^{\circ}-100^{\circ}$.

The $\tilde{X}^{1}\!A^{\prime}/2^{1}\!A^{\prime}$ pair is considered as an example. At $C_{2v}$ geometries, these states belong to $A_{1}$ and $B_{2}$ irreps, so that their CI is ‘symmetry allowed’.Y04 Its branching space includes $\alpha_{\rm OCO}$ as the tuning mode (irrep $A_{1}$) and the antisymmetric stretch $R_{-}$ as the symmetry breaking coupling mode (irrep $B_{2}$). The $2\times 2$ diabatic potential matrix ${\bf V}^{d}$ is constructed on the ab initio grid of bond distances $(R_{1},R_{2})$ from the diagonal adiabatic potential matrix ${\bf V}^{a}$ using the orthogonal adiabatic-to-diabatic transformation (ADT)K04

$${\bf V}^{d}={\bf S}^{T}{\bf V}^{a}{\bf S}\,,$$

(6)

with the transformation matrix ${\bf S}$ parameterized by the mixing angle $\Theta$,

$$\Theta(R_{1},R_{2},\alpha_{\rm OCO})=-\frac{1}{2}\arctan{\left(\frac{\alpha_{\rm OCO}-\alpha^{\star}_{\rm OCO}(R_{1},R_{2})}{W_{0}(R_{1},R_{2})}\right)}\,,$$

(7)

which varies between 0 and $\pi/2$; $\alpha^{\star}_{\rm OCO}(R_{1},R_{2})$ denotes the crossing point of the diabatic potentials ($\Theta=\pi/4$); the width $W_{0}$ of the function $\Theta(\alpha_{\rm OCO})$ is evaluated from the adiabatic energy gaps $\Delta V^{a}/2$ and the average slope of the diabatic potentials $\overline{F}$,

$$W_{0}=\left.\frac{\Delta V^{a}}{2\overline{F}}\right|_{\alpha^{\star}_{\rm OCO}}\,,$$

(8)

at each grid point $(R_{1},R_{2})$ for $\alpha^{\star}_{\rm OCO}=100^{\circ}$. With the off-diagonal coupling in Eq. (6) neglected, the ground state $\tilde{X}^{1}\!A^{\prime}$ becomes completely decoupled from other $A^{\prime}$ states. The remaining diagonal matrix element in ${\bf V}^{d}$ is the potential waiting to be further diabatized at linearity. The diabatization is kept local by restricting the interval, over which $\Theta(\alpha_{\rm OCO})$ varies from 0 to $\pi/2$, to the vicinity of $\alpha^{\star}_{\rm OCO}$:

$$\tilde{\Theta}(R_{1},R_{2},\alpha_{\rm OCO})=g_{1}\left[g_{2}\Theta(R_{1},R_{2},\alpha_{\rm OCO})+\frac{\pi}{2}(1-g_{2})\right]\,,$$

(9)

The two switching functions $g_{1,2}(\alpha_{\rm OCO})$,HH00B

$$g_{1,2}(\alpha_{\rm OCO})=\left[1+\exp(-(\alpha_{\rm OCO}-\alpha_{1,2})/\lambda_{1,2})\right]^{-1}\,,$$

(10)

with $\alpha_{1}=95^{\circ}$, $\alpha_{2}=105^{\circ}$, and $\lambda_{1}=-\lambda_{2}=5^{\circ}$, are adjusted to restrict the diabatization to angles 90^∘—$110^{\circ}$. If $(\alpha_{1}-\alpha_{2})\rightarrow 0$ in $g_{1,2}$, the so-called ‘diabatization by eye’ is recovered, which corresponds to a relabeling of ab initio energies at the crossing angle $\alpha_{\rm OCO}^{\star}$.LLYM94 ; GQZSCH07 ; G12A

The main advantage of local diabatization, the results of which are shown in 1D in Figs. 3 and 4, is that smooth potentials are created for strongly bent geometries at relatively low cost. The quality of the resulting PESs in 2D, either in the $(R_{1},R_{2})$ or in the $(R_{1},\alpha_{\rm OCO})$ plane is illustrated for all states in Figs. 5, 8, 9, 10, and 13. For the lowest excited states in each symmetry block, $2^{1}\!A^{\prime}$ and $1^{1}\!A^{\prime\prime}$, the global bent equilibrium at 120^∘ or 130^∘ [Fig. 9(a,c)] is not affected by the local diabatization. In contrast, the carbene-like bent OCO minima change the owner: They appear in the state $2^{1}\!A^{\prime}$ [local minimum near 70^∘ barely visible in Fig. 9(a) but pronounced in Fig. 13(a)] and in the state $3^{1}\!A^{\prime\prime}$ [Fig. 13(d)]. Barriers and local minima arising in the avoided crossings with Rydberg states are not diabatized within this scheme. Examples are the sharp barrier in $3^{1}\!A^{\prime\prime}$ near 150^∘ [Fig. 9(d) and Fig. 13(d)] and the deep local minimum in $3^{1}\!A^{\prime}$ near 100^∘ [Fig. 9(b) and Fig. 13(b)].

In the second step, the CIs in the FC region are diabatized; kinematic singularities due to glancing intersections between $A^{\prime}$ and $A^{\prime\prime}$ states are not considered. Accurate diabatization schemes rely on direct numerical differentiation of the electronic wave functions with respect to nuclear coordinates or on the orbital rotation methodDW93 ; SHW99 as implemented in MOLPRO.MOLPRO-FULL Both types of calculations become prohibitively expensive with the d-aug-cc-pVQZ basis set, and further approximations are needed in order to find the diabatic potential matrix, consisting of a $2\times 2$ block of $A^{\prime}$ and a $3\times 3$ block of $A^{\prime\prime}$ states:

$${\bf V}^{d}=\left(\begin{array}[]{cc|ccc}V_{\Delta^{\prime}}&V_{\Pi^{\prime}\Delta^{\prime}}&&&\\ V_{\Pi^{\prime}\Delta^{\prime}}&V_{\Pi^{\prime}}&&&\\ \hline\cr&&V_{\Sigma^{\prime\prime}}&V_{\Sigma^{\prime\prime}\Pi^{\prime\prime}}&V_{\Sigma^{\prime\prime}\Delta^{\prime\prime}}\\ &&V_{\Sigma^{\prime\prime}\Pi^{\prime\prime}}&V_{\Pi^{\prime\prime}}&V_{\Pi^{\prime\prime}\Delta^{\prime\prime}}\\ &&V_{\Sigma^{\prime\prime}\Delta^{\prime\prime}}&V_{\Pi^{\prime\prime}\Delta^{\prime\prime}}&V_{\Delta^{\prime\prime}}\\ \end{array}\right)\,.$$

(11)

The five diagonal matrix elements are the five diabatic states $\Sigma^{\prime\prime}$, $\Pi^{\prime}$, $\Pi^{\prime\prime}$, $\Delta^{\prime}$, and $\Delta^{\prime\prime}$, with the smooth intersecting potentials in the $(R_{1},R_{2}|\alpha_{\rm OCO}=180^{\circ})$ plane constructed to coincide at $\alpha_{\rm OCO}=180^{\circ}$ with the ab initio PESs for $1^{1}\Sigma^{-}$, $1^{1}\Pi_{g}$, and $1^{1}\Delta_{u}$, respectively. The semi-global diabatization scheme, akin to the vibronic coupling modelKDC84 ; K04 and adjusted to the topography of the closed fivefold CI seam, has already been introduced in Ref. GB12, . Diabatization proceeds in two steps. First, a model diabatic matrix of the form Eq. (11) with elements $V_{ij}^{(0)}$ is constructed. Due to orbital degeneracy, $V^{(0)}_{\Pi^{\prime}}(R_{1},R_{2})=V^{(0)}_{\Pi^{\prime\prime}}(R_{1},R_{2})$ and $V^{(0)}_{\Delta^{\prime}}(R_{1},R_{2})=V^{(0)}_{\Delta^{\prime\prime}}(R_{1},R_{2})$. Moreover, the accidental near-degeneracy of states $1^{1}\Sigma_{u}^{-}$ and $1^{1}\Delta_{u}$ implies $V^{(0)}_{\Sigma^{\prime\prime}}(R_{1},R_{2})\approx V^{(0)}_{\Delta^{\prime\prime}}(R_{1},R_{2})$ in a broad vicinity of the closed CI seam — the property which substantially simplifies diabatization of $A^{\prime\prime}$ states. Deviations from linearity, measured by the coordinate $Q_{u}\sim\sin\alpha_{\rm OCO}$, are included in the model via off-diagonal matrix elements represented as symmetry adapted expansions in $Q_{u}$:

	$\displaystyle V^{(0)}_{\Pi^{\prime}\Delta^{\prime}}$	$\displaystyle=$	$\displaystyle\sum_{k=0}^{N^{\prime}}\alpha_{k}(R_{1},R_{2})Q_{u}^{2k+1}$		(12)
	$\displaystyle V^{(0)}_{\Sigma^{\prime\prime}\Pi^{\prime\prime}}$	$\displaystyle=$	$\displaystyle V^{(0)}_{\Pi^{\prime\prime}\Delta^{\prime\prime}}=\sum_{k=0}^{N^{\prime\prime}}\beta_{k}(R_{1},R_{2})Q_{u}^{2k+1}\,.$		(13)

Couplings of the accidentally degenerate $\Sigma^{\prime\prime}$ and $\Delta^{\prime\prime}$ states to $\Pi^{\prime\prime}$ are set equal, while the matrix element $V^{(0)}_{\Sigma^{\prime\prime}\Delta^{\prime\prime}}$ for the RT-like $\Sigma^{\prime\prime}/\Delta^{\prime\prime}$ interaction is neglected. The model is complete after the expansion coefficients in $V_{ij}^{(0)}$ are calculated from a non-linear least-squares fit to ab initio energies. In the second step, the regularized diabatic states approach is invoked,K04 and the matrix elements $V_{ij}^{(0)}$ are used to define the orthogonal ADT matrix, which is applied to the adiabatic matrix ${\bf V}^{a}$ via Eq. (6) giving the desired diabatic matrix elements of Eq. (11) on the full ab initio grid. Final interpolation is performed using 3D splines.