Valence and Rydberg states of CH₃Cl: a MR-CISD study

Abstract

In this work ten singlet and nine triplet states are studied through multi-reference configuration interactions with singles and doubles (MR-CISD), including Davidson extensivity correction (MR-CISD+Q). For the first time the excited states whose energies are larger than ∼9.5 eV have been calculated using highly correlated methods. The energies, spatial extent (〈r²〉), configurations weights and oscillator strengths (f) have been computed. At the MR-CISD+Q level the excited states energies vary from ∼7.51 to 11.98 eV. The lowest (nσ*) excited singlet state is significantly mixed with the n3p_a1 and n3s Rydberg states, while the next (n3s) has a non-negligible mixture with the nσ* state. The next three singlet states obtained result from the (n_Cl)³(3p_e)¹ configuration and are almost degenerate. The next (n3p_a1) singlet state is significantly mixed with the nσ* state, while the last three have σ_C–Cl → Rydberg (3s or 3p) as the main configurations. According to the f values the most intense transition is to the 4¹A₁ state, a σ3p_a1 Rydberg state mixed with the σσ* and σ3s configurations. Our results indicate that the σσ* configuration is responsible for the high f value of the gs → 4¹A₁ transition. The Rydberg-valence mixing is greatly reduced in the triplet states whose singlet counterparts have significant multiconfigurational character. The 2³A₁ state (σσ*) does not have its singlet counterpart, while the 4¹A₁ state (σ3p_a1 + σσ* + σ3s) does not have its triplet counterpart. The obtained results are in good agreement with experimental results and with previous CASPT2 results.

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Introduction

CH₃Cl is one of the most abundant halocarbons and it is estimated that it contributes with ∼16% to the Cl atoms in the stratosphere.¹ It is well known that Cl atoms catalyse O₃ decomposition.² A very significant percentage of CH₃Cl comes from natural sources,³ making it a particularly important compound.

Some photodissociation channels of chlorofluorcarbons (CFCs) and halocarbons involve excitation from chlorine lone pairs (n) to C–Cl antibonding (σ*) orbitals^4–6 and lie in the UV region.⁷ Although a large percentage of these molecules photodissociate while in the stratosphere, the surviving molecules can reach the ionosphere, where vacuum UV (VUV) is present, and in this case the Rydberg states play a very important role.⁸ Thus, a detailed knowledge of its excited states is of fundamental importance to understand its photochemistry.

The photoabsorption spectrum of CH₃Cl has been studied by several authors (see, for instance, ref. 9–13). Four main characteristics can be found in this spectrum: (i) a very weak and relatively broad band between ∼6.8 and 7.5 eV; (ii) many sharp and intense peaks in the region from ∼8.7 to 11.3 eV; (iii) a relatively broad and intense band from ∼10 to 12 eV overlapped with the peaks in (ii); (iv) a very broad and intense band above ∼12 eV. The broad bands are consistent with transitions from bonding or to antibonding orbitals. However, the latter two broad bands are also expected to underlie Rydberg states.

The neutral CH₃Cl has C_3v geometrical structure, and its ground state electron configuration can be represented as¹⁴ (Core) (5a₁)²(6a₁)²(1e)⁴(7a₁)²(2e)⁴(8a₁)⁰, where 7a₁, 2e, and 8a₁ correspond to bonding σ(C–Cl), nonbonding n(Cl, 3p_x, 3p_y ≡ 3p_e) and antibonding σ*(C–Cl) orbitals, respectively. In the ground state geometry the Rydberg orbitals have higher energies than the σ*(C–Cl) orbital. Thus, it is expected that the some valence states are associated with n → σ* and σ → σ* excitations, while some Rydberg states are expected to be associated with n → Ryd and σ → Ryd excitations, in which Ryd represent a set of Rydberg orbitals.

In many cases the energies of Rydberg states can be estimated through the Rydberg formula:^15–17

ΔE = IE − Z/2(n − δ_l)² (1)

where Z, ΔE and IE are the charge of the ionic core (1 for neutral molecules), vertical excitation and ionization energies, respectively (in Hartrees), and n and δ_l are the quantum number and the quantum defect, respectively. In eqn (1), the l subscript indicates the main dependence of the quantum defect on the quantum number l. As can be seen from this equation the energies of Rydberg states for each series (e.g. s, p, d,…) converge to the corresponding ionization energy as n increases. The IE value to be used in eqn (1) refers to that orbital from which the electron is excited. In the present case the first two IE values (11.32 and 13.40 eV (ref. 18)) are associated with the Cl lone pairs and with the C–Cl σ orbital, respectively.^13,19 The type of Rydberg orbital will dictate the n and δ_l values to be used in eqn (1).

The purpose of the present paper is to describe (singlet and triplet) valence and Rydberg states of the CH₃Cl molecule associated with excitations from the Cl lone pairs and C–Cl σ orbitals. In this way it is possible to extend previous theoretical calculations to include states up to ∼12 eV. Only s and p Rydberg states are considered.

Computational details

The experimental geometry²⁰ of the CH₃Cl molecule has been used in all calculations. C_s symmetry has been used, and the symmetry plane corresponds to the yz plane. The C–Cl bond almost coincides with the z axis. Therefore, the 3p_z lone pair orbital of the Cl atom is involved in the formation of the σ and σ* molecular orbitals. The (3p_x, 3p_y) degenerate pair is designed as 3p_e or n, yielding the nσ* configuration. The 3s(C), 3p_z(C) and 3p_e(C) Rydberg orbitals have also been included in the present study. Although there is a third row atom (Cl) in the molecule the n = 3 notation has been chosen on the basis of the localization of the low Rydberg orbitals in the CH₃ moiety and agrees with that of Rogers et al.²¹ Besides, the Rydberg orbitals shapes are similar to those of the protonated formaldehyde obtained by Antol et al.²²

For the MCSCF calculations a valence CAS with six electrons (four from the two lone pairs Cl orbitals and two from the C–Cl sigma bond) and four orbitals (the 3p_e(Cl) along with the σ and σ* orbitals) was used. Four additional Rydberg orbitals (3s(C), 3p_a1(C) and 3p_e(C)) have been included in the auxiliary (AUX) space and up to three electrons were allowed from the CAS into the AUX orbitals. Preliminary tests performed by our group show that the Rydberg-valence mixing at the MCSCF level is almost independent on the maximum CAS → AUX excitation level. However, very smooth potential energy curves have been obtained, at the MCSCF level, if one includes up to triple excitations from the CAS to the AUX space. Thus, such latter excitation level has been chosen for the MCSCF calculations performed in the present manuscript. Ten singlet states have been included in the state-averaged MCSCF calculations with equal weights for all states. Nine triplet states were treated separately in the state averaging procedure. As C_s symmetry is used in all calculations and the actual symmetry is C_3v one needs to compute the average energy of the correct pairs of states to get the energies of the doubly degenerate E states. The same holds for the 〈r²〉 values, while in the case of oscillator strength the individual values in each pair should be summed. The 1¹E, 2¹E, 3¹E, 4¹E and 5¹E states correspond to the 2¹A′/1¹A′′, 3¹A′/2¹A′′, 5¹A′/3¹A′′, 6¹A′/5¹A′′ and 9¹A′/6¹A′′ pairs of states, respectively. In the case of the triplet states one has the same number of E states, but the corresponding roots of A′ symmetry vary from 1 to 8, while those of A′′ symmetry vary as they do for the singlet states. The 1¹A₁, 2¹A₁, 1¹A₂, 3¹A₁ and 4¹A₁ states correspond to the 1¹A′, 4¹A′, 4¹A′′, 7¹A′ and 8¹A′ states, respectively, while in the case of the 1³A₁, 1³A₂, 2³A₁ and 3³A₁ states the corresponding states in C_s symmetry are 3³A′, 4³A′′, 6³A′ and 7³A′. In order to verify how the inclusion of high energy states (that is, the states whose energies are greater than 10 eV) affect the properties of the low lying ones the calculations were repeated including seven states in the state-averaging MCSCF procedure for both singlet and triplet states. Such tests are based on the fact that additional Rydberg states (not included in the actual calculations) are predicted above the seventh singlet (4¹E) state here considered, as will be discussed later.

For the multi-reference CI calculations with singles and doubles (MR-CISD) the configuration state functions (CSFs) were generated as in the MCSCF calculations, except that only single CAS → AUX excitations were allowed. The total CSF space was constructed by applying single and double excitations from all internal (active plus doubly occupied) orbitals into all virtual orbitals. The K + L shells orbitals of Cl atom along with the K shell orbital of C atom were kept frozen in all CI calculations. Freezing K + L shells for the Cl atom are based on tests performed for the CF₃Cl molecule.⁵ The interacting space restriction²³ was used in all MR-CISD calculations.

Size-extensivity corrections have been considered by means of the generalized Davidson method (MR-CISD+Q).^24,25 The COLUMBUS program system^26–29 was used for all calculations. The atomic orbitals (AO) integrals and AO gradient integrals were computed with program modules taken from DALTON.³⁰ The basis set used consists of aug-cc-pVTZ for H and Cl^31,32 and d-aug-cc-pVTZ basis set for C.^33,34 Such choice of a doubly-augmented (d-aug) basis set centered on C atom was based on preliminary calculations concerning potential energy curves of Rydberg states along the C–Cl bond: smooth and continuous curves are obtained if the d-aug basis set is centered on C or on both C and Cl atoms. In both cases the Rydberg orbitals become more localized on CH₃ fragment as the C–Cl bond distance increases, which explains the discontinuous curves obtained when the d-aug basis set is centered only on the Cl atom. Besides, the Rydberg orbitals of the dissociated molecule are very similar to those of the isolated methyl fragment. These results indicate that the Rydberg states are indeed located in the methyl fragment, and the results presented in ref. 35–38 concerning this fragment suggest that the C atom is the Rydberg center.

Additional calculations using the mixed Dunning–Hay DZP + Rydberg” basis set^39,40 for C and aug-cc-pVTZ basis set for the other atoms have also been carried out for the singlet states. The former basis set has been taken from MOLPRO software.⁴¹

Results and discussion

In Table 1 the obtained energies, 〈r²〉 (= 〈x²〉 + 〈y²〉 + 〈z²〉) values and oscillator strengths (f, at the MR-CISD level) for the studied singlet states are shown. As can be seen from this table the 〈r²〉 values at the MCSCF level already show the significant larger diffuseness of the Rydberg states as compared to the ground state. However, the nature of 1¹E, 2¹E, 4¹E and 4¹A₁ states change as dynamic electron correlation is included. Thus the assignments change accordingly. For instance, at the MCSCF level the 1¹E state would be classified as a n3s state, while at the MR-CISD level it should be classified as a (nσ*) state mixed with (n3p_a1) and (n3s) (see Table 1). On the other hand, the 2¹E state would be classified as a (n3p_z) + (nσ*) state at the MCSCF level, while at the MR-CISD level it should be classified as a n3s state (see Table 1). Such states' ordering change is likely to be induced by the Rydberg-valence mixing, as the dynamic correlation effect is larger for the valence components than for the Rydberg ones, due to the smaller number of correlated electrons in the latter. A similar change of nature for these states has been observed by Nachtigallova et al. at the CASSCF and CASPT2 levels.¹⁰ Only for the 4¹E state one has the same major configurations at both MCSCF and MR-CISD levels, but the relative weights change (see Table 1). On the other hand, for the 1¹E state the nσ* and n3p_z configurations are important only at the MR-CISD level, while for the 2¹E and 4¹A₁ states the same holds for the n3s and σσ* configurations, respectively (see Table 1). A similar behavior has also been found for the HCFC-133a (ref. 42) molecule.

Table 1 MCSCF, MR-CISD and MR-CISD+Q results for the singlet states of the CH₃Cl molecule, calculated with the mixed aug-cc-pVTZ(H,Cl)/d-aug-cc-pVTZ(C) basis set

State	Weights (MCSCF)^b	〈r^2〉^c	Weights (MR-CISD)^b	〈r^2〉^c	Etot^d	Vertical excitation energies^a
						MCSCF	MR-CISD	MR-CISD+Q	Expt^e	Previous theoretical results
						−499.129955	−499.518710	−499.565633		CASPT2^f	MCCEPA^g
a In eV.b Configurations whose weights are lower than 0.1 were not included.c 〈r^2〉 expectation values in au.d Ground-state energy in Hartrees.e Experimental results.f Ref. 10.g Ref. 11.h Oscillator strengths (f) calculated at MR-CISD level.i Experimental range modified from that of ref. 45. See text for details.j Ref. 9.k Ref. 12.
1^1A1	0.97gs	39.24	0.86gs	42.04	f (× 10^3)^h	0.00	0.00	0.00	—	—	—
1^1E	0.92(n3s)	84.19	0.56(nσ*) + 0.17(n3pa1) + 0.15(n3s)	66.05	1.85 (5.74–7.77)^i	6.72	7.49	7.51	7.25^j	7.61	7.80
2^1E	0.58(n3pa1) + 0.35(nσ*)	94.29	0.73(n3s) + 0.13(nσ*)	77.04	60.41 (45.93–62.13)^i [56]^g	7.13	7.74	7.89	7.75^j	7.69	8.88
2^1A1	0.98(n3pe)	116.33	0.88(n3pe)	113.83	4.29 [25]^g	7.58	8.66	8.88	8.82^j	8.79	8.11
3^1E	0.98(n3pe)	115.98	0.89(n3pe)	113.88	22.09 [110]^g	7.60	8.67	8.90	8.89^j	8.92	9.00
1^1A2	0.98(n3pe)	116.34	0.88(n3pe)	114.26	0.00 [0]^g	7.63	8.72	8.95		8.98	9.47
4^1E	0.61(nσ*) + 0.38(n3pa1)	76.31	0.69(n3pa1) + 0.18(nσ*)	112.21	40.88 (86.11–116.50)^i [18]^g	8.94	9.21	9.31	9.20^j	9.13	—
3^1A1	0.85(σ3s) + 0.12(σ3pa1)	87.89	0.71(σ3s) + 0.12(σ3pa1)	84.15	30.75	10.15	10.86	10.96	—	—	—
4^1A1	0.74(σ3pa1) + 0.14(σ3s)	110.77	0.48(σ3pa1) + 0.22(σσ*) + 0.17(σ3s)	92.59	340.28	10.75	11.39	11.40	11.64^k	—	—
5^1E	0.99(σ3pe)	115.85	0.88(σ3pe)	113.20	34.57	10.97	11.83	11.98	11.75^k	—	—

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As can be also seen from Table 1 the wavefunctions of several states have a significant multiconfigurational character, even at the MR-CISD level. The largest multiconfigurational characters have been obtained for the1¹E and 4¹A₁ states, with a very large Rydberg-valence mixture (see Table 1). Interestingly, the three states resulting from the n³(3p_e)¹ configuration, that is, 2¹A₁, 3¹E and 1¹A₂, have a negligible multiconfigurational character at both MCSCF and MR-CISD levels (see Table 1).

Many members of the Rydberg series are predicted above the here studied 4¹E state. For instance, if one uses a value of 0.01 for the quantum defect^21,43 of carbon d orbitals the Rydberg formula (eqn (1)) predicts a n3d Rydberg state at ∼9.77 eV. The n4s and n4p Rydberg states are predicted at ∼9.80 and 10.13 eV, respectively. Additional members with higher n values are predicted below the first ionization potential (see eqn (1)), although their intensities are expected to decrease steeply as the principal quantum number increases.⁴⁴ It is important to mention that the accuracy of such predictions depends on the absence of Rydberg-valence mixing.¹⁶ As these orbitals (and the associated configurations) can affect the highest lying (3¹A₁, 4¹A₁, and 5¹E or 3³A₁ and 5³E in the case of triplet states) states here studied and as they have not been included in the calculations, due to the prohibitively high computational effort involved, a lower accuracy is expected for these highest lying states, as compared to the lower lying ones. However, to the best of our knowledge there are no other highly correlated theoretical results concerning excited states of CH₃Cl above ∼9.5 eV. Thus, our results cannot be compared with other theoretical results in this energy range. The quantum defect values, estimated from eqn (1) along with the highest level excitation energies values (MR-CISD+Q) provided in Table 1, are ∼1.01 and 0.62 (average value obtained from the 2¹A₁, 3¹E and 1¹A₂ states) for the n3s and n3p_e states, respectively, which can be compared to the corresponding values of 0.98 and 0.58 for the 3s and 3p states of the C atom.²¹ For the 4¹E state the predicted value of ∼0.40 deviates significantly from that expected for a 3p state of C atom, which can be due to the admixture between the n3p_a (Rydberg) and nσ* (valence) configurations and to the non-negligible (though small) weight of the σ orbital in the 3p_a1 orbital. For the remaining states the ionization energy of 13.40 eV (associated with ionization of C–Cl σ orbital) should be used. In the case of the 3¹A₁ state the predicted value of ∼0.64 is slightly higher than the previous one (0.62), which can be explained by a small contribution of a 3p state (see Table 1). On the other hand, for the 4¹A₁ state the significant admixture between valence and Rydberg states leads to a value of ∼0.39, which differ significantly from the value of 0.58 for a 3p state. The excitation energy obtained for the 4¹E (11.98 eV) does not agree at all with the value predicted by the Rydberg's formula for a 3p state, ∼11.08 eV. However, the value of 11.98 eV is relatively close to the value predicted for a 4p state (∼12.24 eV). Thus, such discrepancy can be explained by a possible contribution of a 4p state, not enclosed by the actual calculations.

As can be seen from Table 1 for the states whose nature do not change as dynamic electron correlation is included such effect in the excitation energies is large, leading to increases from 0.64 to 1.09 eV. The effect of dynamic electron correlation at the CASPT2 level is also large, but in this case the excitation energies decrease.¹⁰ Upon inclusion of extensivity correction at the MR-CISD+Q level all excitation energies increase by at most 0.23 eV. This largest increase has been obtained for the 3¹E and 1¹A₂ states (see Table 1). A similar effect has been obtained for the CF₃Cl molecule,⁵ although in that case the maximum increase for the singlet states is 0.16 eV.

Our results for the excitation energies at the MR-CISD+Q level are in good agreement with previous CASPT2 results,¹⁰ with a maximum difference of ∼0.2 eV, obtained for the 2¹E state (see Table 1). However, there is a discrepancy between our results and the results of Cossart-Magos et al., obtained at the multiconfiguration coupled electron pair approximation (MCCEPA) level,¹¹ that is, according to their results the 3¹E state is lower in energy than the 2¹A₁ state. Thus, our result for the former state is ∼0.79 eV higher in energy (see Table 1). There is also good agreement between our MR-CISD+Q results and the experimental results from ref. 9 and 12, with a maximum difference of 0.26 eV, obtained for the 1¹E state (see Table 1). However, in the case of the peak at 8.90 eV we suggest a different assignment from that of ref. 9. While the authors assign it as a n4p_a1 state whose Rydberg series converge to the ²E_1/2 state of the cation we alternatively assign it as a n3p_e state (see Table 1), as the relative intensities of the two peaks (at 8.82 and 8.89 eV (ref. 9)) are not compatible with a small spin–orbit coupling expected for this molecule.⁵ Besides, if one compares the intensities of the pair of peaks at 9.2 and 9.32 eV (assigned as n4p_e states by the authors, see Fig. 2b of ref. 9), which converge to the ²E_3/2 and ²E_1/2 states of the cation, respectively, is clear that the intensity of the latter is much lower than that of the former, which is compatible with a small spin–orbit coupling. In the case of the pair of peaks around 7.8 eV (corresponding to n4s transitions) the broadness of the band prevents even a qualitative comparison between the intensities of the two peaks (see Fig. 2a of ref. 9). As there is a large density of states near 11 eV (ref. 12) we were not able to compare the state calculated at 10.96 eV to a experimental peak. The 4¹A₁ state (at 11.40 eV) was compared to a maximum of a very broad band, while the next state (5¹E) was compared to a slight shoulder located at 11.75 eV (see Table 1).

Another important differences between our results and that from ref. 11 are the following: (i) the 2¹E and 3¹E states are highly coupled (what can be explained by their close energies) and mainly composed of the … 2e³4s¹ and … 2e³4p_a1¹ configurations. Apart from the different notation here chosen (n = 3) for the Rydberg orbitals, in our case these two states are not coupled. Instead, the 1¹E and 2¹E states are coupled, but through the n3s and nσ* configurations (see Table 1); (ii) the nσ* configuration is absent from the results of ref. 10; (iii) the trend obtained for the oscillator strength (f) values is significantly different from that here obtained, although in the case of the 2¹E and 1¹A₂ states there is good agreement between our results and that from ref. 11 (see Table 1). In this case the obtained trend for the Rydberg states is 3¹E > 2¹E > 2¹A₁ > 4¹E > 1¹A₂, while in our case is 2¹E > 4¹E > 3¹E > 2¹A₁ > 1¹A₂; (iv) the main configuration obtained for the 4¹E state is … 2e³5s¹, while in our case the main configurations are n3p_a1 and nσ* (see Table 1). However, Rydberg orbitals with higher n values have not been included in our calculations.

As can be seen from Table 1 there is a reasonably good agreement between our f values for the nσ* (1¹E) and n3s (2¹E) states and the f values obtained experimentally in ref. 45. It is important to mention that an incorrect factor of 1.3 × 10⁻⁸ was used in this latter reference, instead of 4.39 × 10⁻⁹, which should be used for the molar absorption coefficient (ε) in L mol⁻¹ cm⁻¹.⁴⁶ Thus, the experimental f values shown in Table 1 have been taken from that of ref. 45 and multiplied by the ratio 4.39 × 10⁻⁹/1.3 × 10⁻⁸. Although in the case of the n3p_a1 state the agreement does not seem to be good the overlap between the vibrational progression of the previous band (with maximum at 8.89 eV) and the band of the n3p_a1 state⁹ indicates that the f value calculated in ref. 45 is likely to be overestimated. However, the Fig. 2b of ref. 9 also indicates that obtaining an accurate f value for the n3p_a1 band is a very difficult task, due to the overlap between vibrational progressions of adjacent bands. The largest f value has been obtained for the 4¹A₁ state, which is compatible with a very broad and intense band, with maximum at 11.64 eV.¹² If one compares the f values obtained for the 3¹A₁ and 4¹A₁ states is clear that the very high intensity of the latter band is due to the σσ*configuration (see Table 1).

The additional calculations with the Dunning–Hay DZP + Rydberg” basis set lead to an incorrect description of the 1¹E and 4¹E states at the MR-CISD level, although the results obtained for the other eight states are in reasonable agreement with those shown in Table 1, at this same level. The main configurations obtained for the 1¹E and 4¹E states are n3p_a1 and nσ*, respectively, which are not correct. It is important to mention that, although the correct configurations (nσ* and n3p_a1, respectively) are not the main configurations, they have non-negligible weights in the wavefunctions obtained for these two states. Further studies are in progress in order to elucidate the reasons for the aforementioned incorrect description.

Table 2 shows the corresponding MCSCF and MR-CISD results (for the singlet states) obtained from the calculations in which only seven states are considered in the state-averaging procedure at the MCSCF level. As can be seen from this table the excitation energies change by at most 0.15, 0.04 and 0.01 eV at the MCSCF, MR-CISD and MR-CISD+Q levels, respectively, as compared to the corresponding values of Table 1. The changes in the 〈r²〉 values are practically negligible (compare Tables 1 and 2). Although the f values of the two weakest allowed transitions (that is, gs → 1¹E and gs → 2¹A₁) change significantly the general trend is maintained (compare Tables 1 and 2). The main difference between the results of the two tables is the absence of the n3p_a1 configuration in the 1¹E (nσ*) state at the MR-CISD level, although in Table 1 its weight is only 0.17. When a smaller number of states is averaged at the MCSCF level this configuration switches to the 2¹E (n3s) state (see Table 2), at the MR-CISD level. As a consequence the weight of the n3s configuration increases in the 1¹E state and decreases in the 2¹E state (compare Tables 1 and 2). In summary, the highest energy states have a small effect on the properties of the first seven singlet states, at the MR-CISD and MR-CISD+Q levels.

Table 2 MCSCF, MR-CISD, and MR-CISD+Q results for the singlet states of the CH₃Cl molecule, calculated with the mixed aug-cc-pVTZ(H,Cl)/d-aug-cc-pVTZ(C) basis set. Only seven singlet states have been computed

State	Weights (MCSCF)^b	〈r^2〉^c	Weights (MR-CISD)^b	〈r^2〉^c	Etot^d	Vertical excitation energies^a
						MCSCF	MR-CISD	MR-CISD+Q
						−499.1352011	−499.5199922	−499.565633
a In eV.b Configurations whose weights are lower than 0.1 were not included.c 〈r^2〉 expectation values in au.d Ground-state energy in Hartrees.e Oscillator strengths (f) calculated at MR-CISD level.
1^1A1	0.96gs	39.86	0.86gs	42.21	f (× 10^3)^e	0.00	0.00	0.00
1^1E	0.95(n3s)	84.04	0.58(nσ*) + 0.23(n3s)	65.74	2.43	6.85	7.52	7.52
2^1E	0.58(n3pa1) + 0.41(nσ*)	96.33	0.60(n3s) + 0.14(n3pa1) + 0.14(nσ*)	76.16	63.79	7.28	7.76	7.88
2^1A1	0.98(n3pe)	116.07	0.88(n3pe)	113.26	2.78	7.71	8.70	8.88
3^1E	0.98(n3pe)	115.68	0.88(n3pe)	113.32	24.16	7.74	8.71	8.90
1^1A2	0.98(n3pe)	116.08	0.89(n3pe)	113.69	0.00	7.77	8.75	8.95
4^1E	0.56(nσ*) + 0.39(n3pa1)	82.90	0.68(n3pa1) + 0.15(nσ*)	119.75	35.66	8.87	9.21	9.31

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Table 3 shows the results for the nine triplet states studied in the present manuscript. In this case the nature of only two states (1³E and 2³E) change as dynamic electron correlation is included, while in the case of the singlet states the nature of 4¹E also changes (see Table 1). Another important difference between the results for triplet and singlet states is the reduced multiconfigurational character for the states of the former multiplicity at the MR-CISD level, a behavior also obtained for some states of the CF₃Cl molecule.⁵ For instance, while the weights of the n3p_a1 and n3s configurations are negligible in the wavefunction of the 1³E state they cannot be disregarded in the wavefunction of the 1¹E state (compare Tables 1 and 3). Besides, the Rydberg-valence mixing is greatly reduced in the 1³E, 2³E, 4³E and 3³A₁ states, which correspond to triplet states whose singlet counterparts have significant multiconfigurational character (compare Tables 1 and 3). Consequently, there is a clear distinction between the average 〈r²〉 values for the valence (∼50 au), Rydberg 3s (∼78 au) and Rydberg 3p (∼112 au) states. For most of the states the effect of dynamic electron correlation in the excitation energies is large, as in the case of the singlet states, except for the 2³A₁ (σσ*) state (see Table 3). The effect of extensivity correction at MR-CISD+Q level is similar to that obtained for the singlet states, with a maximum increase of 0.24 eV (see Table 3). Through comparison between the excitation energies of the singlet and triplet states is clear that the largest differences have been obtained for the pair of states whose wavefunctions differ most, that is, for the 5³E/5¹E and 2³E/2¹E pairs at the MCSCF level and for the 5³E/5¹E and 1³E/1¹E pairs at the MR-CISD level (compare Tables 1 and 3). It is important to mention that the 2³A₁ (σσ*) state does not have its singlet counterpart, as for this latter multiplicity the σσ* configuration is of secondary importance in the 4¹A₁ state (see Table 1). Similarly to what has been obtained for the singlet states our MR-CISD+Q values are in good agreement with the corresponding CASPT2 results, although in this case the maximum difference is slightly larger, ∼0.33 eV, obtained for the 4³E state. One important difference between our results and the CASPT2 results refers to the energy ordering for the 1³A₂ and 4³E states. According to our results at the MR-CISD+Q (and MR-CISD) level 1³A₂ < 4³E, with an energy difference of 0.33 eV, while at the CASPT2 level 4³E < 1³A₂, with an energy difference of only 0.05 eV. As in ref. 10 this ordering does not change as dynamic electron correlation is included (see Table 3).

Table 3 MCSCF, MR-CISD and MR-CISD+Q results for the triplet states of the CH₃Cl molecule, calculated with the mixed aug-cc-pVTZ(H,Cl)/d-aug-cc-pVTZ(C) basis set

State	Weights (MCSCF)^b	〈r^2〉^c	Weights (MR-CISD)^b	〈r^2〉^c	Vertical excitation energies^a
					MCSCF	MR-CISD	MR-CISD+Q	CASPT2^d
a In eV.b Configurations whose weights are lower than 0.1 were not included.c 〈r^2〉 expectation values in au.d Ref. 10.
1^3E	0.70(n3s) + 0.25(nσ*)	77.35	0.79(nσ*)	53.68	6.55	6.95	6.97	6.70
2^3E	0.42(nσ*) + 0.30(n3pa1) + 0.27(n3s)	73.89	0.85(n3s)	77.52	6.75	7.59	7.79	7.68
1^3A1	0.98(n3pe)	112.96	0.88(n3pe)	110.05	7.45	8.55	8.78	8.70
3^3E	0.98(n3pe)	112.95	0.88(n3pe)	110.82	7.54	8.63	8.87	8.81
1^3A2	0.98(n3pe)	112.97	0.88(n3pe)	111.51	7.62	8.71	8.95	8.90
4^3E	0.64(n3pa1) + 0.33(nσ*)	100.02	0.79(n3pa1)	119.62	8.26	8.99	9.18	8.85
2^3A1	0.90(σσ*)	50.67	0.84(σσ*)	48.13	9.06	9.33	9.37	9.15
3^3A1	0.96(σ3s)	79.94	0.87(σ3s)	78.58	10.10	10.78	10.87	—
5^3E	0.99(σ3pe)	112.59	0.88(σ3pe)	110.20	10.97	11.81	11.95	—

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As aforementioned a smaller number (seven) of triplet states have been calculated as well, in order to check how the inclusion of additional states at the MCSCF level affect the MR-CISD and MR-CISD+Q results of these seven states. However, is also important to point out some non-negligible (though small) differences between the results shown in Table 4 and the corresponding results in Table 3 at the MCSCF level. For the sake of consistency the excitation energies in Table 4 are calculated using the ground state singlet energies from Table 2. As can be seen from these tables the valence character of state 1³E slightly increases while the Rydberg character of the same state slightly decreases (at the MCSCF level), and such changes are accompanied by an increase of the multiconfigurational character of the MCSCF wavefunction (compare Tables 3 and 4). On the other hand, the weight of the nσ* configuration in the 2³E state slightly decreases, while the weight of the n3s configuration increases substantially, such that this state can be clearly classified as a n3s Rydberg state already at the MCSCF level (compare Tables 1 and 3). At the MR-CISD level only the 2³E and 4³E states present non-negligible (though again small) changes in the weights of their configurations, such that these states are now slightly mixed (compare Tables 3 and 4). Small changes have been obtained for the 〈r²〉 values of the seven triplet states. The excitation energies change by at most 0.16, 0.05 and 0.01 eV at the MCSCF, MR-CISD and MR-CISD+Q levels, respectively. Therefore, one can conclude that the inclusion of additional states at the MCSCF level does not alter significantly the main characteristics of the first seven triplet states at the MR-CISD and MR-CISD+Q levels.

Table 4 MCSCF, MR-CISD, and MR-CISD+Q results for the triplet states of the CH₃Cl molecule, calculated with the mixed aug-cc-pVTZ(H,Cl)/d-aug-cc-pVTZ(C) basis set. Only seven triplet states have been computed

State	Weights (MCSCF)^b	〈r^2〉^c	Weights (MR-CISD)^b	〈r^2〉^c	Vertical excitation energies^a
					MCSCF	MR-CISD	MR-CISD+Q
a In eV.b Configurations whose weights are lower than 0.1 were not included.c 〈r^2〉 expectation values in au.
1^3E	0.36(nσ*) + 0.36(n3pa1) + 0.27(n3s)	75.06	0.78(nσ*)	53.31	6.68	6.95	6.96
2^3E	0.64(n3s) + 0.33(nσ*)	73.01	0.77(n3s) + 0.10(n3pa1)	77.44	6.84	7.62	7.78
1^3A1	0.98(n3pe)	111.96	0.88(n3pe)	108.96	7.59	8.58	8.78
3^3E	0.98(n3pe)	111.95	0.88(n3pe)	109.76	7.69	8.67	8.87
1^3A2	0.98(n3pe)	111.96	0.88(n3pe)	110.48	7.78	8.76	8.95
4^3E	0.62(n3pa1) + 0.30(nσ*)	104.93	0.69(n3pa1) + 0.11(n3s)	120.88	8.34	9.02	9.18
2^3A1	0.92(σσ*)	48.90	0.84(σσ*)	47.87	9.19	9.36	9.35

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Fig. 1 below summarizes the best results obtained for the energies of the singlet and triplet states studied for the CH₃Cl molecule.

Fig. 1 Energies (in eV) of the excited singlet and triplet states of the CH₃Cl molecule studied in this manuscript. Only the highest level (MR-CISD+Q) results are shown. Please refer to Tables 1 and 3 for the nature of the excited states.

It is important to mention that the vertical excitation here discussed is, in the Franck–Condon approximation, only the first step in the description of the photodissociation dynamics. Just after the vertical excitation the wavepacket will evolve on the potential energy surface (PES) according to its topography, which controls the photodissociation dynamics. The PES shape is a very important feature to be considered if one wants to understand the type of geometry relaxation adopted by the system during the photodissociation. For instance, after excitation to E states the system (initially in a C_3v geometry) has its geometry distorted to C_s or C₁ symmetry (depending on the magnitude of the Jahn–Teller effect), as the C_3v ground state geometry belongs to a conical intersection seam. Rydberg states usually relax to equilibrium geometries which are close to the geometry of their ionic core, having a C_s distorted geometry. Other types of geometry relaxations, dictated by the orbitals involved in the electronic excitations, should also be taken into account. For instance, the 1¹E (nσ*) state is also subjected to a breakage of the C–Cl bond. Apart from the geometry relaxations the crossings between electronic states of different nature also govern the photodissociation.

Conclusions

In this work ten singlet and nine triplet states of the CH₃Cl molecule are studied through multi-reference configuration interaction with singles and doubles (MR-CISD), including Davidson extensivity correction (MR-CISD+Q). The vertical excitation energies, spatial extent (〈r²〉), configurations weights and oscillator strengths (f) have been computed. For the first time the excited states whose energies are larger than ∼9.5 eV have been calculated using highly correlated methods. At the MR-CISD+Q level the excited states energies vary from ∼7.51 to 11.98 eV. The lowest (nσ*) excited singlet state is significantly mixed with the n3p_a1 and n3s Rydberg states, while the next (n3s) has a small admixture with the nσ* state. The next three singlet states obtained result from the (n_Cl)³(3p_e)¹ configuration and are almost degenerate, with energies of 8.88, 8.90 and 8.95 eV. The next singlet state (n3p_a1) has a small admixture with the nσ* state, while the last three have σ_C–Cl → Rydberg (3s or 3p) as the main configurations. According to the f values the most intense transition is to the 4¹A₁ state, a σ3p_a1 Rydberg state significantly mixed with the σσ* and σ3s configurations. Our results indicate that the σσ* configuration is responsible for the high f value of the gs → 4¹A₁ transition. The Rydberg-valence mixing is greatly reduced in the triplet states whose singlet counterparts have significant multiconfigurational character. The 2³A₁ state (σσ*) does not have its singlet counterpart, while the 4¹A₁ state (σ3p_a1 + σσ* + σ3s) does not have its triplet counterpart. The results obtained for the six and seven first singlet and triplet excited states, respectively, are in good agreement with previous CASPT2 results, while in the case of the singlet states some discrepancies with previous MCCEPA results from ref. 11 have been found. Our results are also in good agreement with experimental results from ref. 9 and 12.

Acknowledgements

The authors are thankful to CNPq and CAPES for financial support. They also thank the reviewers for the very valuable suggestions.

Notes and references

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