The transition from the open minimum to the ring minimum on the ground state and on the lowest excited state of like symmetry in ozone: A configuration interaction study

CASSCF

Complete active space self-consistent field

FORS

Full optimized reaction space

CEEIS

Correlation energy extrapolation by intrinsic scaling

SD

Single plus double excitations

SDT

Single plus double plus triple excitations

SDTQ

Single plus double plus triple plus quadruple excitations

CISD

configuration interaction for SD

CISDT

configuration interaction for SDT

CISDTQ

configuration interaction for SDTQ

MRCISD

CISD with respect to a multi-determinant-reference

MRCISDT

CISDT with respect to a multi-determinant-reference

MRCISDTQ

CISDQ with respect to a multi-determinant-reference

VSDNO

Natural orbitals in the virtual space of a MRCISD calculation

The ozone molecule is important in many respects. Its ultraviolet absorption in the stratosphere is beneficial¹ while its presence in the lower atmosphere is harmful.² Its strong oxidative power has made it a heavily used sterilization agent in industry³ and a key oxidizing agent in chemical research laboratories.⁴ A wide range of experimental and theoretical studies have been made on this fundamental molecule.

Diverse spectroscopic observations have yielded significant amounts of experimental information regarding the excited states. Numerous theoretical studies have been performed to address the challenges presented by the excited state potential energy surfaces (PESs). Of particular interest is the potential existence of metastable excited state minima.⁵ Most high accuracy spectroscopic measurements have only been able to infer the existence of such minima.⁶ But metastable excited minima were identified from anion photoelectron spectroscopic measurements,⁷ and these minima were reproduced by accurate theoretical calculations.^8–11 

A significant amount of work has also been performed to elucidate the photodissociation¹² of ozone because of its important role in the ozone-oxygen cycle.^13,14 The wavelength dependences of the quantum yields and branching ratios have been determined by dynamics calculations that are based on adiabatic potential energy surfaces that were specifically calculated for these problems^15,16 and on coupling potential terms for non-adiabatic interactions.¹⁷ 

Another set of issues is associated with the ground state. Its vibrational spectrum presents a challenge in as much as it cannot be accurately recovered by uncorrelated wave functions. Therefore, the reproduction of this spectrum has been used to test the performance of wave functions that include correlation in various ways^9–11,18 and, on the other hand, to assess multi-reference diagnostics.¹⁹ For a considerable time, the bonding structure of ozone was a subject of debate, viz., whether the accurate electronic wave function of the ground state implies a conceptual interpretation as a diradical²⁰ or a genuine closed shell singlet.⁹ A thorough ab initio investigation²¹ of a series of triatomic molecules containing oxygen and sulfur has clarified the various degrees of partial radical character of ozone and its analogues. The description of the ground state dissociation path and the recovery of the experimental dissociation energy have proven particularly challenging and required extremely accurate wave functions.^22–24 

Conversely, the theoretical approach has revealed certain features of the ground state potential energy surface that have posed a great challenge to experimentalists. Many high level ab initio calculations have established beyond any doubt that there exists a metastable ring structure of ozone with D_3h symmetry. But this isomer has so far eluded experimental detection even though interest in the ozone ring is over 100 years old.²⁵ The first quantitative ab initio predictions of a closed ozone conformer were made in the early 1970s by Hay and Goddard and, then, by other authors.²⁶ Extremely accurate calculations were made by Lee²⁷ and by Qu, Zhu, and Schinke,²⁸ where references to further previous work are given. The first to explore the barrier that separates the ozone ring conformer from the open conformer were Xantheas, Atchity, Elbert, and Ruedenberg.^29,30 Their extensive and thorough ab initio investigations (based on multi-configuration-self-consistent-field calculations in the full valence space—FORS-CASSCF) showed that the ring isomer is surrounded by a substantial ridge in C_s symmetry on which three saddle points, each in C_2v symmetry, provide the transition states between the ring minimum and the three open minima on the potential energy surface. Their investigations demonstrated that the ridge is the result of an avoided crossing between the ¹A′ ground state and the lowest excited ¹A′ state and, notably, that the two states in fact touch each other along a closed conical intersection seam in C_s symmetry.^29,30 This intersection seam crosses the three C_2v symmetry planes in conical intersection points that lie near the three transition states.²⁹ Furthermore, the excited ¹A′ state was shown to have three minima that lie extremely close to the transition states of the ground state.²⁹ Analogous features were found by Ivanic, Atchity, and Ruedenberg³¹ on the potential energy surfaces of the ozone homologues S₃, S₂O, and SO₂.

On the basis of theoretical calculations, Fleming, Wolczanski, and Hoffman suggested that the closed conformer of ozone, as well as its sulfur analogue, might be trapped in transition metal complexes.³² Qu, Zhu, and Schinke reviewed various proposals that have been made for accessing the elusive ring isomer.²⁸ These authors suggested that possible mechanisms might be photochemical excitations followed by radiationless reaction paths that pass through the conical intersection seam. This conjecture motivated DeVico et al. to revisit the structure of the two potential energy surfaces in the region of the transition state of the ground state and the conical intersection.³³ 

Since general experience has shown the ozone PES to be very sensitive to dynamic correlation,²² DeVico et al. examined this region by means of wave functions that account for more electron correlation than was included in the calculations of Refs. 29 and 30. The calculations of DeVico et al. were based on several versions of multi-configurational second-order complete-active-space perturbation theory (MS-CASPT2). Special care had to be taken to avoid erratic behavior of the energy along the reaction path in the transition state region.

In the present study, the transition from the open minimum to the ring minimum is examined by means of wave functions that include correlation contributions up to quadruple excitations beyond multi-configurational reference states. To this end, MRCISDTQ wave functions for the two lowest singlet states are determined. To recover the correlation contributions, the Correlation Energy Extrapolation by Intrinsic Scaling (CEEIS) is used.³⁴ Since the determination of the wave functions in the region of interest requires the state averaged approach, the generalization of the CEEIS method to the simultaneous treatment of several states, which was first used by Bytautas for the B₂ molecule,³⁵ is further developed and generalized to polyatomic molecules. An important aspect of the present work is to determine whether, for a multi-state application in a polyatomic molecule with substantial correlation interactions, the CEEIS extrapolation methodology retains the same high convergence quality that it exhibits when applied to diatomic molecules.³⁶ 

This section contains a brief summary of the CEEIS method. The basis of the CEEIS procedure is an exact expansion of the full configuration interaction (CI) energy, $EkFCI$⁠, of each state as a sum of energy contributions of the various configurational excitation levels,

where the subscript k specifies the electronic state and the excitation levels are indicated in the parentheses. The term E_k(0) denotes the reference energy. The term ΔE_k(1, 2) = E_k(2) − E_k(0) denotes the combined incremental energy contribution due to the inclusion of single and double excitations levels beyond the reference function. The sum over x covers all excitation levels above the double excitations. The term ΔE_k(x) denotes the incremental energy contribution that is due to the addition of excitation level x,

In the CEEIS method, values for the incremental energy contributions of every excitation above x = 3 are extrapolated from the incremental energy contributions that were determined for the single, double, and triple excitations. These extrapolations are based on calculating, at each excitation level x, approximations ΔE_k(x|m), which are obtained by including only the excitations generated by up to a number m of active virtual orbitals, where m < M = the total number of virtual orbitals (the virtual orbital space being the part of the basis orbital space that is unoccupied in the reference function). It was shown in previous papers³⁵ that the convergences of ΔE_k(x|m) with increasing m to the limit ΔE_k(x|M) at different excitation levels x are linearly related. Specifically, as the number of active virtual orbitals, m, is increased, the change that occurs for ΔE_k(x|m) is linearly proportional to the change that occurs for ΔE_k(x*|m), where x* is a lower excitation level than x. This linear proportionality is expressed by the equation

Thus, if ΔE_k(x*) = ΔE_k(x*|M) is known, the value of ΔE_k(x) = ΔE_k(x|M) is obtained as follows. Step 1: Values for ΔE_k(x|m) and ΔE_k(x*|m) are calculated for a range of m values that are considerably smaller than M. Step 2: These values of ΔE_k(x|m) and ΔE_k(x*|m) are used in a least-mean-squares fit to determine the coefficients a_k,x and c_k,x in Eq. (2.3). Step 3: The values of a_k,x and c_k,x and the known value of ΔE_k(x*) = ΔE_k(x*|M) are used to extrapolate to the value of ΔE_k(x) = ΔE_k(x|M). For ΔE_k(1, 2), the exact value is calculated. When possible the exact value is also calculated for ΔE_k(3). If the exact calculation of ΔE_k(3) is not affordable, its value is extrapolated from ΔE_k(1, 2) in a similar manner. For all excitation levels x ≥ 4, extrapolations using x* = x − 2 typically require lower values for m than the values that are needed for the extrapolation using x* = x − 1. Hence, for x > 4, ΔE_k(x) is extrapolated using ΔE_k(x − 2). However, for ΔE_k(4), a more reliable estimate is obtained by using ΔE_k(1, 2) rather than only ΔE_k(2) for ΔE_k(x^*).

Various kinds of reference functions will be considered, which are obtained as follows. A preliminary state-averaged CAS MCSCF (complete-active-space multi-configuration-self-consistent-field) calculation is performed in the full valence space in order to identify the dominant electronic configurations for each state. On this basis, more affordable CAS subspaces are then identified that have smaller numbers of determinants while still generating the dominant configurations of every state of interest. Reduced reference functions are then obtained by new state-averaged CAS MCSCF calculation in such subspaces.

Once the occupied orbitals have been optimized, a preliminary MRCISD (multi-determinant-reference singles + doubles configuration interaction) calculation is performed to determine energies and one-particle density matrices for each of the states of interest. The one-particle density matrices are then used to form a state-averaged one-particle density matrix. The virtual-virtual block of this state averaged one-particle density matrix is diagonalized to obtain state-averaged natural orbitals for the virtual orbital space (VSDNOs). The VSDNOs, ordered according to decreasing occupation numbers, provide the basis for calculating the CI energies that are needed for the CEEIS extrapolation of each state’s energy.

Once these preliminary calculations are complete, CI calculations are performed at multiple excitation levels x and using varying numbers m of VSDNOs in order to evaluate all ΔE_k(x|m) that are needed for the CEEIS of each electronic state. Once all ΔE_k(x|m) have been determined, separate CEEIS extrapolations are performed for each electronic state of interest. An automated procedure has been developed for the CEEIS method running through all states of interest. It is included in the GAMESS³⁷ molecular program suite.

The range of the m values to be used for the extrapolations to the CEEIS energies has to be chosen by the user, even if the automated procedure is used. Typically, it is advisable to choose the lowest value for m not to lie significantly below the total number of virtual orbitals that are generated by a double zeta basis set. Below that number of virtual orbitals, the changes in ΔE_k(x|m) and ΔE_k(x*|m) may not be linearly proportional. It is important to examine the plots of the values of ΔE_k(x|m) against the values of ΔE_k(x*|m) to ascertain that the changes in ΔE_k(x|m) and ΔE_k(x*|m) are indeed linearly proportional for the chosen values of m.

Furthermore, when selecting the energies that are used to determine the ΔE_k(x|m) values for the extrapolation, special precaution is required when two or more electronic states are nearly degenerate. A careful analysis of the dominant configurations of each state’s wave function must be performed to ensure that the energy differences that are used in the extrapolations are associated with the same reference wave function (i.e., have the same dominant electronic configurations). This is necessary because the order of the energies of near degenerate states may change as the number of active virtual orbitals is increased or as the maximum excitation level of the calculation is changed.

The full-valence space of ozone consists of 12 orbitals and 18 electrons. State-averaged CASSCF(18,12) calculations were performed using Dunning’s cc-pVTZ and cc-pVQZ basis sets.³⁸ Optimizations were performed to determine the following five geometries:

OM = the open minimum of the 1¹A₁ state,

RM = the ring minimum of the 1¹A₁ state,

TS = the transition state that separates these minima on the 1¹A₁ surface,

XM = the minimum of the 2¹A₁, and

CI = the conical intersection between the two states.

In the subsequent text and tables, the abbreviations OM, RM, TS, XM, and CI will continue to be used to denote these five geometries.

Table I shows the optimized geometries and the energies of each state at these geometries relative to the respective energies of the open minimum of the 1¹A₁ state. The absolute energies for the two basis sets at the OM geometries are given in footnotes d and e of Table I. The data show that the geometries obtained using the cc-pVTZ and cc-pVQZ basis sets differ from each other by less than 0.004 Å and 0.1° and that their relative energies differ by less than 2 mh. These results justify the use of the more affordable cc-pVTZ basis set.

The theoretical geometries of the open minimum (first row of Table I) can be compared to the experimental data³⁹ listed in footnote b of that table. The deviations of the bond lengths are 0.014 Å and 0.010 Å for the cc-pVTZ and cc-pVQZ basis set calculations, respectively. The corresponding bond angle deviations are 0.3° and 0.2°. The slightly deteriorating effect of state-averaging is exhibited by comparing these deviations with those that are found by the state-specific calculations, which are listed in footnote c of Table I. The state-specific CASSCF(18,12)/cc-pVTZ and /cc-pVQZ calculations yield deviations from the experiment of 0.007 Å and 0.003 Å for the bond length, and 0.1° and 0.0° for the bond angle, respectively.

In Secs. IV–VIII, all correlation energy calculations are performed at the optimized state-averaged CASSCF(18,12)/cc-pVQZ geometries.

The conical intersection point was determined with the cc-pVQZ basis. Figure 1 displays contour plots of the potential energy surfaces for the 1¹A₁ and 2¹A₁ states (panels (a) and (b), respectively) and for the energy difference between these states (panel (c)) in the region near the conical intersection. The region that is shown in panel (c) is indicated by the green box in panels (a) and (b) and is expressed in terms of coordinates q and p, the reaction coordinate and the normal mode that is perpendicular to the reaction coordinate, respectively. The contours show that the position of the minimum of the 2¹A₁ state is nearly identical to the position of the 1¹A₁ transition state. For the 1¹A₁ state, the rate of descent towards the open minimum (the lower right) is larger than the rate of descent towards the ring minimum (the upper left). In contrast, the potential energy surface of the 2¹A₁ state has a rate of ascent in the direction of the 1¹A₁ open minimum that is smaller than its rate of ascent towards the 1¹A₁ ring minimum. This behavior is consistent with the changes that are observed in potential energy surfaces of nonadiabatically coupled states near a conical intersection or avoided crossing. It is a consequence of the switch in the dominant diabatic states. The data in Table I show that, at the conical intersection geometry, the energy gap between the two states is less than 0.005 mh for the cc-pVQZ basis and 0.25 mh for the cc-pVTZ basis.

Since the generation of the dynamic correlation by excitations from the full CAS(18,12) reference space exceeded the available computational resources, the suitability of smaller reference spaces had to be explored. Such spaces must account for the changes in the dominant configurations of the 1¹A₁ and 2¹A₁ states as the geometry of O₃ changes from the open minimum to the ring minimum. To this end, a close examination of the configurational structures of the two states in the full valence space is instructive. Figure 2 displays the 12 canonicalized orbitals obtained from the state-averaged CASSCF(18,12)/cc-pVTZ calculations at the open and ring minima. Between the orbitals of each pair, the orbital label (in C_2v symmetry) is given. Under each orbital contour, the occupation in both the 1¹A₁ state and the 2¹A₁ state as well as its canonicalized orbital energy is listed.

The dominant parts of the wave functions for each state (i.e., the normalized, spin-adapted determinants with coefficients ≥0.1 in magnitude) are displayed in Tables II and III. Table II lists the expansions of the dominant parts of the wave functions in terms of normalized, spin-adapted determinants at each of the discussed geometries. Table III lists the orbital occupations of the 12 determinants that occur in Table II. The results documented in these tables show that the determinants $D1$⁠, $D2$⁠, $D3$⁠, and $D4$⁠, which are indicated by bold faced font in Tables II and III, are the most important determinants for the generation of both states at all of the geometries, with determinants $D1$ and $D2$ usually making contributions that are slightly larger than the contributions from $D3$ and $D4$⁠. Additionally, the last column in Table III shows that all of the determinants listed in Table III, including $D3$ and $D4$⁠, can be generated by single and double excitations from $D1$ or $D2$⁠.

The active orbitals that distinguish $D1$ and $D2$ are the orbitals 4b₁ and 2b₂. They are indicated by symbols in red font in Figure 2. For the determinants $D3$ and $D4$⁠, these orbitals remain doubly occupied. Instead, the determinants $D3$ and $D4$ differ in the occupancies of orbitals 6a₁ and 1a₂, which are indicated by symbols in blue font in Figure 2. In Table III, these four orbitals 4b₁, 2b₂, 6a₁ and 1a₂ are all indicated by bold faced font. The determinants $D1$ and $D2$ span the full CAS(2,2) subspace of ¹A₁ symmetry that is generated by the active orbitals 4b₁ and 2b₂. Similarly, the determinants $D1$⁠, $D2$⁠, $D3$⁠, and $D4$ span the full CAS(6,4) subspace of ¹A₁ symmetry that is generated by the active orbitals 6a₁, 1a₂, 4b₁, and 2b₂.

In all the dominant determinants listed in Tables II and III, the valence orbitals 3a₁, 4a₁, and 2b₁ remain doubly occupied, and they are therefore not included in Table III. Figure 2 shows that, in fact, they have occupations nearly equal to 2 in the full CAS(18,12) wave functions for both states. From the images that are displayed for these orbitals in Figure 2, it is apparent that they are dominated by the 2s atomic orbitals on the three oxygen atoms. Based on that fact, the set of the orbitals 3a₁, 4a₁, and 2b₁ will be referred to as the “2s group” in the subsequent discussion.

On the other hand, Table III shows that orbitals 5b₁ and 7a₁ are unoccupied in the determinants $D1$⁠, $D2$⁠, $D3$⁠, and $D4$⁠, and it is apparent from Figure 2 that these two orbitals have very low occupations in the full CAS(18,12) wave functions for both states. For this reason, these two orbitals will be referred as the “weak group” in the subsequent discussion.

In view of the preceding analysis, the following restrictions of the full CAS(18,12) valence space seem reasonable strategies for achieving reductions in the number of correlating excited configurations:

• Move the “weak MO group” (7a₁, 5b₁) from the active space into the virtual space.

• Move the “2s-MO group” (3a₁, 4a₁, 2b₁), which are indicated by green font in Figure 2, from the active space to the inactive occupied space.

• Limit the reference space to the full space generated by the “strong correlation group” of the orbitals 4b₁, 2b₂, 6a₁, 1a₂.

• Limit the reference space to the full space generated by the “minimal reaction group” of the orbitals 4b₁, 2b₂.

In light of these criteria, several combinations of reference and excitation spaces were considered in forming wave functions. They differ with respect to the valence orbitals that are active in the reference space and those that are correlation-active, i.e., subject to excitations into virtual orbitals. Tables IV and V give the specifications of these wave functions in the ORMAS (occupation restricted multiple active space⁴⁰) format, which was used in the calculation. Table IV contains four wave functions, in which the “2s-group” is correlation-active. Table V contains four wave functions, in which the “2s-group” is correlation-inactive, which is a popular assumption. Each table consists of four subtables, each of which describes a wave function. It is identified by the symbol heading in the first column of the subtable. The headings of columns 2, 3, 4, and 5 of each wave function subtable specify, respectively, the following:

• those orbitals that are reference-inactive as well as correlation-inactive;

• those orbitals that are reference-inactive but correlation-active;

• those orbitals that are reference-active as well as correlation-active; and

• the virtual orbitals that are used for this wave function,

as indicated by the overall column headings below the table title. The four numerical rows of each subtable contain the actual information regarding the electron occupations. These rows correspond to the reference space, the reference space + SD excitations, and similarly to adding SDT and SDTQ excitations, as indicated by the abbreviations in the first column of the rows. The numbers in these rows give the minimum number and the maximum number of electrons that can occupy the (reference or virtual) orbitals that are specified in the header of each column.

Table IV lists the specifications for four wave functions, in which the “2s-group” is correlation-active, although not necessarily reference-active. They are as follows:

• CAS(18,12): This is the full valence space wave function with the corresponding virtual space of 75 orbitals.

• CAS(18,10): This space is obtained by moving the “weak MO group” (orbitals 7a₁, 5b₁) from the preceding full valence space into the virtual space, which now has 77 orbitals. The reference space is thus reduced. But all remaining valence orbitals are still active in the reference space and with respect to SD, SDT, and SDTQ excitations into the virtual space.

• CAS(6,4): Only the orbitals 4b₁, 2b₂, 6a₁, 1a₂ of the “strong correlation group” are active in the reference function. But all valence orbitals are correlation-active. Moreover, the reference-inactive orbitals can also be excited into the reference-active orbitals by SD-excitations. However, the weak MO group is placed into the virtual space, as in the CAS(18,10) case.

• CAS(2,2): Only the orbitals 4b₁, 2b₂ of the “minimal reaction group” are active in the reference function. But all valence orbitals are correlation active. The reference-inactive orbitals will also be excited into the reference-active orbitals by SD-excitations. However, the weak MO group is again placed into the virtual space.

Table V lists the orbital spaces for four corresponding wave functions, in which the “2s-group” is inactive in the reference space and also with respect to correlation excitations, as follows:

• CAS(12,9)*: This wave function is similar to the CAS(18,12) wave function of the preceding paragraph. It differs by excluding the orbitals 3a₁, 4a₁, 2b₁ of the “2s-group” from being active in the reference space and also from excitations into the virtual space.

• CAS(12,7)*: This wave function is similar to the preceding CAS(12,9)* wave function except that the “weak MO group” (7a₁, 5b₁) is moved from the reference space into the virtual space. It can also be deduced from the CAS(18,10) of the preceding paragraph by excluding the orbitals of the “2s-group” from being active in the reference space as well as from excitations into the virtual space.

• CAS(6,4)*: This wave function differs from the CAS(6,4) wave function of the preceding paragraph in that the orbitals of the 2s-group are excluded from the correlating excitations.

• CAS(2,2)*: This wave function differs from the CAS(2,2) wave function of the preceding paragraph in that the orbitals of the 2s-group are excluded from the correlating excitations.

The orders of magnitude of the dimensions of the determinantal configuration spaces for the eight cases specified in Sec. IV B are listed in Table VI for the reference functions, for the reference + SD functions, for the reference + SDT functions, and for the reference + SDTQ functions. For the reference + SDT and the reference + SDTQ subspaces, the table also lists the number of determinants that are generated when only the first 30 or 40 virtual natural orbitals are used in the correlating CI calculations, which are the reduced spaces that are needed in the context of the CEEIS extrapolation. The full values are given in Table T1 of the supplementary material.⁴¹ It should be noted that these dimensions correspond to A₁ symmetry.

The bold font in Table VI indicates dimensions for which the calculations would have been impractical or impossible on the available computational facilities. The italic font indicates calculations that were not made because they would have been of interest only in conjunction with some of the calculations indicated by bold font. These data show that calculations up to the SDT level can be performed for most of these wave functions. On the other hand, for the cases CAS(18,12), CAS(18,10), CAS(6,4), and CAS(12,9)*, the SDTQ energies are beyond the available capabilities, even for the CEEIS extrapolation procedure.

In light of these observations, the following correlation calculations were performed at all five geometries:

• For all eight orbital space cases identified in Sec. IV B, the reference energies and the CISD energies were calculated.

• For the cases identified as CAS(2,2), CAS(2,2)*, CAS(6,4)*, and CAS(12,7)* in Sec. IV B, the CISDT energies were calculated exactly as well as by CEEIS extrapolation.

• For the cases CAS(2,2), CAS(2,2)*, CAS(6,4)*, and CAS(12,7)*, the CISDTQ energies were calculated by CEEIS extrapolation.

In addition, exact SDTQ energies were obtained for the CAS(2,2)* case at two geometries, viz., TS and CI. In all calculations, the valence orbitals were reoptimized for the respective reference spaces. An exception was made for the CAS(12,7)* case, which will be discussed at the end of Section VI B.

The results of these calculations are documented in the supplementary material.⁴¹ In this material, Table T2 contains a list of the number m of orbitals (see Section II) that are used for the reduced virtual spaces in the various CEEIS procedures. The graphs in Figures F1 to F9 of the supplementary material document the linear relationships described in Section II, on which the CEEIS extrapolations are based for all cases. Tables T3–T12 of the supplementary material contain the exact and extrapolated energies obtained for all calculations.⁴¹ 

The reliability of the various CEEIS extrapolations in the cases CAS(2,2), CAS(2,2)*, CAS(6,4)*, and CAS(12,7)*can be assessed from the data documented in Table VII.

According to what was said in Section IV C, the actual errors of the CEEIS extrapolations with respect to the exact calculations are available for the SDT calculations in these cases. They are listed in the first and fourth numerical data columns of Table VII for the 1¹A₁ and 2¹A₁ states, respectively. The second and fifth numerical data columns list the predicted uncertainties for the respective SDT extrapolations, which were obtained using the algorithm described in Section III B 4 of Ref. 34(a). (Of the two options given there, the more conservative estimate was used.) It is seen that, for the CAS(2,2) case, all of the actual errors are less than the estimated uncertainties. Most of the actual errors for the CAS(12,7)* case are also lower than the predicted uncertainties. (The difference between CAS(12,7)*a and CAS(12,7)*b in Table VII will be discussed in Section VI A.) For the CAS(2,2)* and CAS(6,4)* cases, on the other hand, the actual errors are somewhat larger than the predicted uncertainties. With very few exceptions, the actual error is less than 1.5 mh. Only for the 2¹A₁ state at the ring minimum, some errors are about 3 mh.

The predicted uncertainties of the extrapolated CISDTQ energies are listed in the third and sixth numerical data columns of Table VII for the 1¹A₁ and 2¹A₁ states, respectively. In view of the results found for the CISDT calculations, the actual errors of the ΔE (4) increments predicted for the CAS(2,2) and CAS(12,7)* cases are likely to be smaller than the listed predicted uncertainties. For the cases CAS(2,2)* and CAS(6,4)*, on the other hand, it is possible that the actual errors of the extrapolated ΔE (4) values are somewhat larger than the listed uncertainties. However, as mentioned in Section II, in all past experiences, the extrapolation of ΔE (4) versus ΔE(1, 2) has consistently converged considerably more rapidly than the extrapolation of ΔE(3) versus ΔE (1,2). In view of the small errors that are observed for the extrapolation of the CISDT energies, it seems thus likely that the CISDTQ energies that are extrapolated for the CAS(2,2)* and CAS(6,4)* active spaces should lie within 1.5 mh of the exact values (except possibly for the values of the 2¹A₁ state at the ring minimum). These conclusions are supported by the results for the CISDTQ energies of the CAS(2,2)* case at the TS and CI geometries where, as mentioned in Section IV C, exact CISDTQ energies were calculated for both states. As noted in footnotes c and d of Table VII, the CEEIS extrapolated CISDTQ energies for the 1¹A₁ and 2¹A₁ states deviate from the exact values by 0.40 mh and −0.52 mh, respectively, at the TS geometry and by 0.41 mh and −1.19 mh, respectively, at the CI geometry.

In summary, the discussed deviations and the graphs in Figures F1 to F9 of the supplementary material⁴¹ demonstrate that the linear relationships, which are the basis of the CEEIS extrapolations, remain valid under the severe present conditions, viz., the simultaneous extrapolations of two states in a molecule of three closely interacting atoms. Even for the 2¹A₁ state at the ring minimum, which lies more than 200 mh higher than the 1¹A₁ state, the discrepancy reaches 3 mh only for two data points in Table VII. The savings of the CEEIS procedure are apparent for the increments Δ(4) = (CISDTQ–CISDT), some with values up to 90 mh, which were typically recovered with millihartree accuracy by using only 40 of the 77 virtual orbitals, requiring less than 10% of the total number of determinants.

Tables VIII and IX list the incremental amounts of correlation energy that are, respectively, recovered in all cases for which the SDT and SDTQ excitations were calculated, viz., CAS(2,2), CAS(2,2)*, CAS(6,4)*, and CAS(12,7)*. Table VIII lists the increments due to the successive additions of single, double, triple, and quadruple excitations to the reference spaces for the 1¹A₁ state. Table IX lists the values for the 2¹A₁ state. (The difference between CAS(12,7)*a and CAS(12,7)*b in these tables is discussed below in this section.) For both states, the following implications regarding the rate of convergence with increasing excitation levels to the full CI energy are apparent.