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Article

Ab initio post-HF CCSD(T) Calculations for Triplet and Singlet Methylene in Four consecutive Dunning Basis Sets with Extrapolations to Infinite Limits for Various Molecular Properties

by
Alexander Neugebauer
1 and
Günter Häfelinger
2,*
1
Saarland University, Pharmaceutical and Medicinal Chemistry, Im Stadtwald, Building 32, D-66123 Saarbruecken, Germany
2
University of Tuebingen, Institute of Organic Chemistry, Auf der Morgenstelle 18, D-72076, Tuebingen, Germany
*
Author to whom correspondence should be addressed.
Int. J. Mol. Sci. 2005, 6(1), 157-176; https://doi.org/10.3390/i6010157
Submission received: 8 October 2004 / Revised: 10 December 2004 / Accepted: 10 December 2004 / Published: 31 January 2005
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Abstract

:
Stationary points for four geometrically different states of methylene: bent and linear triplet methylene, bent and linear singlet methylene were investigated using the highly reliable post-HF CCSD(T) method. Extrapolations to the CCSD(T) basis set (CBS) limit from Dunning triple to quintuple correlation consistent polarized basis sets were performed for total energies, for the equilibrium CH distances re(CH), for singlet-triplet separation energies, for energy barriers to linearity and for correlation energies. Post-HF calculations with Dunning basis sets of the literature are presented for comparisons.

Introduction

Methylene (CH2), the parent compound of divalent carbon species, is of great interest in organic chemistry related to its importance in synthesis and to the description and determination of its molecular structure. Due to its small size it is a favoured test case for practically any kind of advanced quantum chemical calculations in respect of its unusual bonding situation in the electronically and geometrically different singlet and triplet states. In the early sixties the molecular structure of singlet and especially triplet methylene (CH2) was controversially discussed both from experimental and calculational studies as described below.
Ab initio calculations with large basis sets and quantum chemistry of a high level of theory have greatly contributed to determine the correct structure of this small molecule. Excellent reviews on the historical aspects related to interplay of experiments and calculations with many references to relevant publications up to 1985 are presented by Shavitt [1], Goddard [2], and Schaefer [3,4].
Some milestones of these investigations will be mentioned shortly: Foster and Boys [5] predicted in 1960 by an early ab initio calculation a bent structure of triplet CH2 with a HCH valence angle of 129°, in contrast to an early experimental linear structure determined by Herzberg [6] which was corrected subsequently [7] to the apparent non-linear geometry with an angle around 136°. A highly accurate equilibrium structure of the triplet ground state of CH2 was determined experimentally by Bunker and Jensen [8] in 1983 and refined in 1988 [9] leading to highly accurate values of re(CH) = 1.0753 ° 0.0003 Å with a valence angle θe(HCH) of 133.93 ° 0.06°. The experimental geometry of the lowest singlet state of CH2 as determined by Petek et al. [10] in 1989 with less precise values of re(CH) = 1.107 ± 0.002 Å and θe(HCH) = 102.4 ± 0.4°.
Quantum chemical models of ab initio MO calculations may be classified by the following levels of theory which are treated in detail in the excellent book of Helgaker, Joergensen and Olsen [11]:
1)
The Hartree-Fock (HF) self-consistent-field (SCF) method uses one Slater determinant of LCAO-MO´s describing a single configuration of electrons which may serve as a reference state for most of the following post-HF models.
2)
Configuration-interaction (CI) theory is based on a linear superposition of Slater determinants describing excitations of electrons from a reference state with variational determination of the expansion coefficients. Truncated configurations may use single (one-electron) excitations (S), double (two-electron) excitations (D) or similarly triple (T) or quadruple (Q) excited configurations. Treatment of all possible excited configurations are termed full CI (FCI).
3)
The multireference CI (MRCI) method uses several determinants as reference configurations and generates all excitations up to a given level from each reference configuration, i. e. if all single and double excitations are included results the MRSDCI model. Alternatively this may be termed second order CI (SOCI) for single and double excitations out of a CASSCF reference function.
4)
The multiconfigurational self-consistent field (MCSCF) method uses CI determinants which are variationally optimised simultaneously with the expansion coefficients. The MO space may be partitioned into three subspaces containing inactive (doubly occupied), virtual (unoccupied) and active orbitals (with variable occupancies of 0, 1 or 2 electrons). A MCSCF expansion distributing the active electrons in all possible ways among the active orbitals which is leading to non-integer occupancies is termed complete active space (CAS) method.
5)
The coupled-cluster (CC) model treats excitations between pair-wise correlated electrons (pair clusters) in a non-linear way via a cluster operator acting on a single-determinantal reference state. The cluster operator is partitioned into classes of all single (S), double (D) or triple (T) excitations, In the CCSD(T) method [12, 13] are contributions from triple excitations estimated by a perturbative treatment. The CC methods account well for dynamical electron correlation.
6)
Perturbation theory is applied as Møller-Plesset (MP) perturbation of second, third or fourth order (MP2, MP3 or MP4) related to HF SCF as the unperturbed reference state
All of these methods depend critically on the size and quality of applied Gaussian basis sets for one-electron atomic functions.
The convergence to the basis set limit (this means that the total energy will not change if one adds some more Gaussian basis functions) is generally very slow. Basis set series which comprise systematic improvements of the ground state energy or other properties allow an extrapolation based on the asymptotic behaviour of the series with increasing basis set expansion. Examples for such series of basis sets are the correlation-consistent polarized valence basis sets of Dunning [14] and [15] termed cc-pVXZ with zeta exponents from X = 2 to 5 which we use here, or the atomic natural orbital (ANO) basis sets of Almlöf, Taylor and Helgaker [16, 17, 18].
Ab initio Hartree-Fock (HF) calculations lead to unreliable results for CH2 [1,4], therefore post-HF methods have to be applied, with some important contributions for energy hypersurfaces listed as follows:
In 1971 whole energy hypersurfaces of seven low-lying triplet and singlet states of CH2 were calculated by the group of Schaefer [19] by configuration interaction (CI) methods and extended in 1983 by MCSCF methods with Dunning´s cc-pVDZ basis set [14] for the lowest triplet and singlet states of CH2 by Alexander et al. [20]. Comeau et al. [21] report in 1989 for these states MRCI and MCSCF calculations with a full-valence CAS reference space using an atomic-natural (ANO) basis set of quadruple zeta quality. They calculated vibration-rotation energies using the Morse oscillator rigid bender internal dynamics (MORBID) Hamiltonian to obtain improved fitted potential energy surface parameters. Recently CCSD(T)/cc-pVTZ calculations were fitted by three different analytical functions to determine the global potential energy surface of the triplet ground state of methylene [22]. Important post-HF calculations since 1985 for triplet and/or singlet CH2 will be mentioned here: Bauschlicher and Taylor [23, 24, 25, 26] performed around 1987 various FCI studies with respect to the size of the triplet-singlet separation energy of CH2 with Dunning basis sets [14]. In 1995 and 1997 Dunning et al. [27, 28] applied MP2, MP3 and MP4, coupled cluster (CCSD and CCSD(T)) and MRCI CAS methods for the triplet and singlet states of CH2 using their valence basis sets from cc-pVDZ up to cc-pVQZ [15] and core-valence basis sets cc-pCVXZ [27]. In 1995 Schaefer et al. [4] performed CCSD(T) and frozen-core FCI benchmark calculations for the ground and first excited triplet and singlet states of CH2 using a relatively small DZP basis set of Dunning type which was extended by this group in 1998 to the larger TZ2P basis set calculations [29]. The most extensive ab initio post-HF calculations reported till now for singlet and triplet CH2 but with a fixed geometry using CI methods (CISD, CISDT, CISDTQ up to FCI) and coupled-cluster methods (CCSD, CCSD(T) and CCSDT) with Dunning valence [15] and core-valence [27] basis sets cc-pVXZ and cc-pCVXZ for X = 2 to 6 have been presented in 2003 by Császár et al. [30], when our calculations had already been finished independently.
In our work we concentrate on CCSD(T) calculations with Dunning´s cc-pVXZ basis set series [15] from double-zeta (X = 2) to quintuple-zeta (X = 5) quality which all include appropriate polarization functions on carbon and hydrogen and we will present comparisons to calculated and experimental literature data.
As aim of our publication we present in the first part frozen-core CCSD(T) optimisations to investigate the geometry and energy of stationary points of the bent triplet 3B1 ground state of CH2 (1), the linear triplet state of CH2 (2), the bent singlet 1A1 ground state of CH2 (3) and the linear singlet state of CH2 (4) by the CCSD(T) procedure using four of the already mentioned consecutive Dunning [15] cc-pVXZ basis sets.
From these calculations as an extrapolation to the basis set limit an empirical exponential function of the general form of eqn. 1:
P(X) = P(ℕ ) + a · e - b · X
can be used, where P(X) is an energy or property dependent on the basis set expansion X and P(ℕ) is the predicted value in the corresponding basis set limit (BS limit). Such extrapolations have been applied first for energies by Feller in 1992 [31] and used by Dunning [27] and by Császár [30]. Test extrapolations with alternatively polynomial or potential functions lead to similar results. Here exponential extrapolations are obtained from cc-pVTZ, cc-pVQZ and cc-pV5Z valence basis sets [15], denoted as TQ5-limit for frozen-core CCSD(T) calculations.
Exponential extrapolations for three consecutive basis sets to the infinite BS limit via eqn. 1 for the four different CH2 1 to 4 species have been carried out as usual for total energies, but further concerning equilibrium re CH distances, singlet-triplet separation energies (Te), energy barriers to linearity and correlation energies. A further additional estimation from the best calculated to a predicted experimental equilibrium re CH distance as described in ref. [32] has been performed.

Calculational Procedure

Molecular geometries and energies of CH2 were determined for stationary points of CH2 1 to 4 by ab initio post-HF MO frozen-core optimisations using the CCSD(T) method as very successful approximation for the principally unknown accurate many-electron wave function [11] and alternatively as lower approximation the density functional (DFT) [33] Becke 3-parameter Lee-Yang-Parr [34] (B3LYP) hybrid method. Four consecutive correlation consistent (cc) polarized valence basis sets of Dunning [15] (which are abbreviated by cc-pVXZ in relation to the number of zeta-exponents, where X indicates D = 2, T = 3, Q = 4 and 5 zeta exponent splittings) were applied for the valence electron frozen-core CCSD(T) calculations. The number of contracted Gaussian basis functions for CH2 are: cc-pVDZ = 24, cc-pVTZ = 58, cc-pVQZ = 115 and cc-pV5Z = 201.
All calculations were performed with Pople's Gaussian 98 program system (Rev. A.7) [35] using the unrestricted procedure for the triplet states of methylene. The status of stationary points was checked by frequency calculations. The DFT B3LYP calculations were performed in the cc-pVTZ basis set and the 'fine' integration grid of Gaussian 98 was used for enhanced accuracy. These results are not presented here but may be obtained from the authors.

Results and Discussion

Total Energies

In Table 1 total energies from various frozen-core post-HF calculations from the literature with Dunning valence cc-p-VXZ basis sets [15] for triplet CH2 in forms 1 and 2 are collected in comparison to our CCSD(T) results together with exponential extrapolations via eqn. 1 to estimate infinite basis set (CBS) limits for the final energies obtainable with the CCSD(T) method. Corresponding values for singlet CH2 in forms 3 and 4 are presented in Table 2.
Table
Table 1. Post-HF frozen-core calculations of total energies [Hartree] with Dunning basis sets for triplet methylenes 1 and 2. Own CCSD(T) calculations with exponential TQ5 CBS limits via eqn. 1 and literature data for different methods.
Table 1. Post-HF frozen-core calculations of total energies [Hartree] with Dunning basis sets for triplet methylenes 1 and 2. Own CCSD(T) calculations with exponential TQ5 CBS limits via eqn. 1 and literature data for different methods.
MoleculeMethodBasis setEa this workEb [30]Ec [28]ERef.
3B1 CH2 1CCSD(T)cc-pVDZ-39.041816-39.041248-39.041564-39.046371e[4]
bent cc-pVTZ-39.077854-39.077802-39.077501-39.066184f[29]
cc-pVTZ -39.094104[22]
triplet GS cc-pVQZ-39.087331-39.087285-39.086958
cc-pV5Z-39.090075-39.090029-39.089024d
cc-pV6Z--39.090903-
CBS limit-39.0912-39.092197j-39.09174
CCSDTcc-pVDZ -39.041670
cc-pVTZ -39.078279
cc-pVQZ -39.087741
FCIcc-pVDZ -39.041695 -39.046816e[4]
FCIANO -39.046260[23]
FCIcc-pVTZ -39.078346 -39.066738f[29]
CISDcc-pVDZ -39.036887-39.045381[52]
cc-pVTZ -39.070465-39.080444[52]
cc-pVQZ -39.079271
cc-pV5Z/QZ -39.081170d
MP4cc-pVDZ -39.040477
cc-pVTZ -39.076541
cc-pVQZ -39.085976
cc-pV5Z/QZ -39.088060d
CAS+1+2cc-pVDZ -39.040273
cc-pVTZ -39.074544
cc-pVQZ -39.083477
cc-pV5Z/QZ -39.085397d
CMRCIgcc-pCVDZ -39.041579[27]
cc-pCVTZ -39.075452[27]
cc-pCVQZ -39.083886[27]
cc-pCV5Z -39.086206[27]
CBS limit -39.08796[27]
MCI CAScc-pCVTZ -39.07989[49]
MRCI IShANO QZ -39.083084[21]
MRCI SOiANO QZ -39.083222[21]
MR SOCIANO -39.084972[25]
3Σ- CH2 2CCSD(T)cc-pVDZ-39.030906
Linear cc-pVTZ-39.068449 MCI CAS:-39.07895[49]
cc-pVQZ-39.078196
cc-pV5Z-39.081045
C BS limit-39.0822
Footnotes to Table 1:a) CCSD(T) calculations for optimised geometries (see Table 3), this work.b) CCSD(T) calculations for constant geometries: re (CH) = 1.07598 Å, θe (HCH) = 133.848°, ref. [30].c) CCSD(T) calculations for optimised geometries (see Table 3), ref. [28].d) The Dunning cc-pV5Z/QZ basis set is of type 5Z for C and QZ for H [28].e) DZP basis set used in ref. [4].f) TZ2P basis set used in ref. [29].g) Contracted MRCI CAS CI [27].h) MRCI with full CAS interacting space (IS). [21].i) MRCI with full second order (SO) CI [21].j) Q67 CBS limit.
Due to the quantum mechanical variation principle [11, 36, 37] are these ab initio energies an upper limit for the usually unknown true experimental energies. Thus this value is an indication of the quality of the calculations with respect to applied procedures and basis sets.
We concentrate first on the CCSD(T) total energies in Table 1 and Table 2. Such calculations by Dunning et al. [28] and our results use the same basis sets DZ to QZ, but the 5Z basis sets are different. Both calculations are based on optimised geometries (shown in Table 3 to Table 5) and therefore should be identical. The notable differences in energies by about 0.35 mHartree (0.22 kcal/mol) may be due to the applied different computer programs (MOLPRO in ref. [28] and G 98 by us) and and lead to different geometries (see Table 3 and Table 5). The CCSD(T) calculations of ref. [30] are based on fixed molecular geometries derived from aug-cc-pCVQZ optimisations of 1 and 3 which are very close to experimental geometries. For TZ to 5Z calculations is the difference from [30] values to our energies smaller than above by one order of magnitude. (The presented energies are taken from the supplementary material of ref. [30].)
Three independent extrapolations to the CBS limit lead to CCSD(T) values between –39.0912 and –39.092197 Hartree for 1, the largest numbers presented in Table 1, except the energy of –39.09404 quoted in ref. [22] which must be erroneous. The extension to the more elaborate CCSDT method with inclusion of all triplet excitations taken from ref. [30] shows only a small improvement by about 0.45 mHartree for the cc-pVXZ data for X = 2 to 4 . Comparison to the immense extensive full FCI values (i.e. 560 034 determinants with cc-pVDZ in [30]) which are available only for DZ and TZ basis sets also are better than corresponding CCSD(T) data by only 0.447 and 0.544 mHartree for DZ and TZ, respectively. Selecting a higher Dunning basis set in the CCSD(T) method leads to a larger improvement in energy. This shows the superior behaviour of the CCSD(T) method with respect to energies calculated in all the other post-HF procedures shown in Table 1 and Table 2 which are based mainly on calculations of Dunning´s group, ref. [27] and [28]. (All energies in ref. [27] are misprinted too low by 2 Hartree.)
Table
Table 2. Post-HF frozen-core calculations of total energies [Hartree] with Dunning basis sets for singlet methylenes 3 and 4. Own CCSD(T) calculations with exponential TQ5 CBS limits via eqn. 1 and literature data for different methods.
Table 2. Post-HF frozen-core calculations of total energies [Hartree] with Dunning basis sets for singlet methylenes 3 and 4. Own CCSD(T) calculations with exponential TQ5 CBS limits via eqn. 1 and literature data for different methods.
MoleculeMethodBasis setEa this workEb [30]Ec [27]ERef.
1A1 CH2 3CCSD(T)cc-pVDZ-39.022584-39.022029-39.024154-39.025791e[4]
bent cc-pVTZ-39.061384-39.061367-39.062456-39.048005f[29]
singlet cc-pVQZ-39.071913-39.071909-39.072413
GS cc-pV5Z-39.075039-39.075037-39.075244
cc-pV6Z --39.076081j-
CBS limit-39.0764-39.076408k-39.07733
CCSDTcc-pVDZ -39.022748
cc-pVTZ -39.062180
cc-pVQZ -39.072695
FCIcc-pVDZ --39.022937 -39.026635e[4]
FCIANO -39.027183[24]
FCIcc-pVTZ -39.062405 -39.048984f[29]
CMRCIgcc-pCVDZ -39.023177[27]
cc-pCVTZ -39.059919[27]
cc-pCVQZ -39.069396[27]
cc-pCV5Z -39.072067[27]
CBS limit -37.07405[27]
MRCI IShANO QZ -39.068284[21]
MRCI SOiANO QZ -39.068284[21]
SOCIANO -39.070250[25]
1Δg CH2 4CCSD(T)cc-pVDZ-38.967818
linear cc-pVTZ-39.010679
singlet cc-pVQZ-39.022215
cc-pV5Z-39.025820
CBS limit-39.0275
Footnotes to Table 2:a) CCSD(T) calculations for optimised geometries (see Table 4), this work.b) CCSD(T) calculations for constant geometries: re (CH) = 1.106901 Å, θe (HCH) = 102.137°, ref. [30].c) CCSD(T) calculations for optimised geometries (see Table 4). Core-valence basis sets cc-pCVXZ (X = 2 to 5) derived and used in ref. [27]. Energies are misprinted by 2 Hartree.e) DZP basis set used in ref. [4].f) TZ2P basis set used in ref. [29].g) Contracted MRCI using full valence and CAS CI [27].h) MRCI with full CAS interacting space (IS). [21].i) MRCI with full second order (SO) CI [21].j) CCSD(T)/aug-cc-pCV6Z calculation for constant geometries shown in Table 4, ref. [30].k) Largest used basis set of type: aug-cc-pCV6Z with 533 contracted Gaussians [30].
Total energies for the singlet forms 3 and 4 which are presented in Table 2 show a similar behaviour as those in Table 1. Our CCSD(T) energies of 3 are lower than those of ref. [30], decreasing from 0.56 mHartree for DZ to nearly full agreement of 0.002 mHartree for 5Z. Dunning´s energy values cannot be compared because of use of the larger core-valence correlated cc-pVCXZ basis sets [27]. The estimated CBS limit is -39.0764 Hartree for 3 from our calculation and -39.0773 Hartree from [27]. The largest CCSD(T) energy calculated in [30] is -39.076408 Hartree close to our CBS limit. Differences from explicit CCSDT versus lower CCSD(T) calculations [30] are around 0.8 mHartree corresponding to 0.5 kcal/mol. Deviations between FCI and CCSD(T) energies are only 0.091 mHartree and 1.04 mHartree for DZ and TZ basis sets [30], respectively. The CMRCI CBS limit [27] is with -37.07405 Hartree slightly smaller than the above mentioned estimated CCSD(T) limits.
For 2 and 4, the linear forms of CH2, are only few reliable post-HF calculations available. Therefore our energies stay in Table 1 and Table 2 without discussion but they will be used later for calculations of barriers to linearity.

Core-Valence Correlation

A further improved treatment for highly accurate energies needs to consider core-valence correlation of all electrons with correspondingly designed core-valence basis sets [27]. This was studied numerically in ref. [26], [27] and [30]. In this case for 1 a CCSD(T) extrapolated CBS limit of -39.14803 Hartree is reached. An all electron core-valence CMRI+Q calculation [27] with the Davidson correction (Q) for CI truncation [38] leads to a CBS limit value of –39.14832 Hartree, the absolutely lowest energy without relativistic effects known for 1.

Equilibrium CH Bond Lengths

The CH bond lengths of molecular states 1 to 4 were optimised via gradient methods at the CCSD(T) level in the frozen-core approximation for each of the four Dunning valence basis sets with numerical re(CH) distances presented in Table 3 and Table 4 in comparison to literature data and experimental values which are available for 1, 2 and 3. The optimised CH distances decrease with each expansion step of the wave functions. Therefore the exponential extrapolation via eqn. 1 could also be applied successfully for prediction of CH distances defining the CBS limit of the CCSD(T) method. The three CBS limit estimations for 1 in Table 3 are rather close around 1.077 Å but still off the experimental value of (1.0753 ± 0.0003) Å.
In ref. [32] we studied the capability of the CCSD(T) CBS limit expansion to predict numerically experimental re distances in bonds of carbon to H, C, N and O and derived eqn. 2:
reexp = r(CCSD(T)/CBS) - 0.0018 Å
with an estimated standard deviation of ± 0.0005 Å as a linear correction for all of the mentioned bonds. For the ground state triplet methylene CH distance of 1 this correction leads to a predicted distance of 1.07526 Å now in perfect agreement to the experimental value listed above. For the linear state 2 is this via eqn. 2 predicted distance 1.0645 Å also in the error limit in agreement to the experimental determination of 1.060 ± 0.005 Å.
For the singlet state CH distance of 3 in Table 4 is our via eqn. 2 predicted CCSD(T) value 1.1059 Å, more distant to the experimental value of (1.1070 ± 0.002) Å than the CBS limit value of 1.1077 Å but both are in the quoted experimental error limit. The calculated distance in [30] is with 1.1069 Å noticeable very close to the experimental distance.
Table
Table 3. Frozen-core by CCSD(T) and other post-HF methods calculated equilibrium distances re(C-H) [Å] of triplet methylenes 1 and 2 in comparison to experimental and literature distances.
Table 3. Frozen-core by CCSD(T) and other post-HF methods calculated equilibrium distances re(C-H) [Å] of triplet methylenes 1 and 2 in comparison to experimental and literature distances.
MoleculeMethodBasis Setre(C-H)are(C-H)bre(C-H)cre(C-H)Ref.
3B1 CH2 1CCSD(T)cc-pVDZ1.09571.09551.09471.0875e[4]
bent triplet cc-pVTZ1.07851.07831.07821.0772f[29]
GS cc-pVQZ1.07731.07711.07711.07793[21]
cc-pV5Z1.07711.0772d1.07671.07114[22]
ANO 1.082[23]
eqn. 1CBS limit1.07706 1.07671.079[25]
eqn. 2Prediction1.07526 1.07598g[30]
experiment 1.0748± 0.0004[8]
experiment 1.0753± 0.0003[9]
CMRCIcc-pCVDZ 1.0951 [27]
cc-pCVTZ 1.0784
cc-pCVQZ 1.0773
cc-pCV5Z 1.0769
CBS limit 1.0769
CISDcc-pVDZ 1.0932
cc-pVTZ 1.0758
cc-pVQZ 1.0744
cc-pV5Z/QZ 1.0744
MP4cc-pVDZ 1.9044
cc-pVTZ 1.0774
cc-pVQZ 1.0762
cc-pV5Z/QZ 1.0762
CAS+1+2cc-pVDZ 1.0959
cc-pVTZ 1.0786
cc-pVQZ 1.0773
cc-pV5Z/QZ 1.0773
3Σ- CH2 2CCSD(T)cc-pVDZ1.0837
Linear cc-pVTZ1.0676 MCI CAS:1.068[49]
Triplet cc-pVQZ1.0665
cc-pV5Z1.0663
eqn. 1CBS limit1.0663
eqn. 2Prediction1.0645
experiment 1.060±0.005[8]
Footnotes to Table 3:a) This work.b) Ref. [28]c) Ref. [27] with core-valence basis sets: cc-pCVXZ (X = 2 to 5).d) cc-pV5/QZ basis set in ref. [28].e) DZP basis set used in ref. [4].f) TZ2P basis set used in ref. [29].g) Contracted MRCI using full valence and CAS CI [27].
For the linear structure 4 of CH2 is no experimental value available. The CH distance of the 5Z calculation and the extrapolation to the CBS limit are the same within four digits (as in the case of 2) and application of eqn. 2 reduces these values by 0.0018 Å.
Table
Table 4. Frozen-core by CCSD(T) and other post-HF methods calculated equilibrium distances re(C-H) [Å] of singlet methylenes 3 and 4 in comparison to experimental and literature distances.
Table 4. Frozen-core by CCSD(T) and other post-HF methods calculated equilibrium distances re(C-H) [Å] of singlet methylenes 3 and 4 in comparison to experimental and literature distances.
MoleculeMethodBasis Setre(C-H)are(C-H)b re(C-H)Ref.
1A1 CH2 3 cc-pVDZ1.12911.1286 1.1199e[4]
Bent cc-pVTZ1.11051.1103 1.1089f[29]
Singlet GS cc-pVQZ1.10861.1086 1.10863[21]
cc-pV5Z1.10801.1080 1.117[23]
1.112[26]
1.110[25]
1.10691[30]
eqn. 1CBS limit1.10771.1079
eqn. 2Prediction1.1059
experiment 1.1112[9]
experiment 1.1070±0.002[10]
CMRCIgcc-pCVDZ 1.1290 [27]
cc-pCVTZ 1.1107
cc-pCVQZ 1.1088
cc-pCV5Z 1.1083
CBS limit 1.1081
1Δg CH2 ‘4 cc-pVDZ1.0789
Linear cc-pVTZ1.0645
Singlet cc-pVQZ1.0640
cc-pV5Z1.0640
eqn. 1CBS limit1.0640
eqn. 2Prediction1.0622 Experiment:1.070[51]
Footnotes to Table 4:a) This work.b) Ref. [27]c) Ref. [28] with cc-pCVXZ (X = 2 to 5) basis set.e) DZP basis set used in ref. [4].f) TZ2P basis set used in ref. [29].g) Contracted MRCI using full valence and CAS CI [27].

HCH Valence Angles

CCSD(T) optimised HCH valence angles of the bent molecular states 1 and 3 are shown in Table 5 in comparison to experimental angles and other reliable calculations.
The calculated angles increase with enlargement of the basis sets. But our CBS extrapolation via eqn. 1 could not be applied for the equilibrium bond angles due to a decrease of our calculated 5Z angle which is not occurring in the calculations of ref. [27] and [28]. This estimated CBS limit for 1 with 133.6° is off the experimental angle of (133.93 ± 0.01)°, but again the optimised angle of ref. [30] is with 133.85° closest to experiment.
For 3 is the estimated CBS limit angle 102.08° [27] and that optimised in [30] is 102.14°, both in the error limit equal to the experimental angle of (102.4 ± 0.04)°. Our 5Z angle with 101.93° is again lower than the QZ angle.
Table
Table 5. Calculated CCSD(T) and other equilibrium bond angles θe (HCH) [°] of bent methylenes 1 and 3 in comparison to experimental and literature values.
Table 5. Calculated CCSD(T) and other equilibrium bond angles θe (HCH) [°] of bent methylenes 1 and 3 in comparison to experimental and literature values.
MoleculeMethodBasis Setθe(HCH)°aθe(HCH)°bθe(HCH)°cθe(HCH)°Ref.
3B1 CH2 1CCSD(T)cc-pVDZ132.198132.10132.13132.15[4]
Bent cc-pVTZ133.565133.33133.33133.29[29]
Triplet GS cc-pVQZ133.648133.51133.51132.900[21]
cc-pV5Z133.539133.57133.56134.127[22]
132.4[23]
133.6[25]
132.7[26]
CBS limit– 133.59133.848[30]
experiment 133.84±0.05[8]
experiment 133.93±0.01[9]
1A1 CH2 3CCSD(T)cc-pVDZ100.542 100.54101.28[4]
Bent GS cc-pVTZ101.617 101.63101.72[29]
cc-pVQZ101.929 101.90102.137[30]
cc-pV5Z101.914 102.01
CBS limit– 102.05
CMRCIgcc-pCVDZ 100.54
cc-pCVTZ 101.63
cc-pCVQZ 101.90
cc-pCV5Z 102.01
CBS limit 102.08
experiment 101.95[9]
experiment 102.4 ±0.4[10]
Footnotes to Table 5:a) This work.b) Ref. [28].c) Ref. [27] with cc-pCVXZ (X = 2 to 5) basis set.e) DZP basis set used in ref. [4].f) TZ2P basis set used in ref. [29].g) Contracted MRCI using full valence and CAS CI [27].

Singlet-Triplet Energy Gap

Experimental and calculated energies of the singlet-triplet energy gap (T) of CH2 up to 1985 are reviewed in ref. [1] with listing of experimental To values (related to the zero vibrational levels of the states 3 and 1) between 2 to 20 kcal/mol and calculated Te energies (which refer to the minima 1 and 3 of the potential energy hypersurface) between 10 and 30 kcal/mol. Compilations of later publications 1997 are presented in ref [4] and [29]. As direct experimental determination [39] is the energetic difference between the singlet state 3 and the triplet state 1 observable from laser photo-detachment spectroscopy of the radical anion CH2- which leads by removal of one electron simultaneously to transitions between various vibration-rotation levels of the triplet or the singlet states of CH2. This experiment led to a value of (19.5 ± 0.7) kcal/mol for the energy difference To, close to 22.2 kcal/mol from an early ab initio CI calculation [19].
Table
Table 6. Adiabatic singlet–triplet separation energies (Te) [kcal/mol] based on data of Table 1 and Table 2, referring to minima of potential energy curves without vibrational corrections (1 Hartree = 627.5095 kcal/mol).
Table 6. Adiabatic singlet–triplet separation energies (Te) [kcal/mol] based on data of Table 1 and Table 2, referring to minima of potential energy curves without vibrational corrections (1 Hartree = 627.5095 kcal/mol).
cc-pVDZcc-pVTZcc-pVQZcc-pV5Zcc-pV6ZΔ CBSaMethodbRef.
12.06810.3359.6759.435-9.287CCSD(T) VEc
12.06010.3139.6499.4089.301(9.299)dCCSD(T) VEe,f[30]
12.01910.5279.9989.7999.661-CCSD(T) AEe,f[30]
11.79110.0549.4089.184-9.042CCSD(T) VEe[27]
11.86210.3309.7929.590-9.475CCSD(T) AEe[27]
11.5479.7479.0939.6618.8728.729CMRI VEe[27]
11.65210.0519.5029.307-9.193CMRI AEe[27]
11.4969.6208.9318.699-8.553CMRI+Q VEe,f[27]
11.5909.9019.3159.106-8.980CMRI+Q AEe,f[27]
11.77110.003 FCI VEe,f[30]
11.729 FCI AEe,f[30]
12.664 [4]11.141 [29] FCI VE
11.97 [23] FCI VE[23]
9.287 (IS)9.373 (SO) MRCI VEg[21]
9.592 (IS)9.740 (SO) MRCI AEg[21]
9.238 SOCI9.111 (SO+Q) MRCI VEg[25]
10.608 MCPF AEg[26]
9.436 CISD[52]
Experimental determinations of Teexp [kcal/mol]:1) 8.56 ± 0.09 [44] 2) 9.292 ± 0.57 [45] 3) 9.281 ± 0.086 [46]4) 9.215 ± 0.0036 [9] 5) 9.032 ± 0.057 [47]Footnotes to Table 6:a) Difference of CBS limit values.b) VE = valence electron correlation, frozen-core. AE = all electron core-valence correlation.c) CCSD(T) calculations for optimised geometries, this work.d) CBS limit from eqn. 1.e) Core-valence basis set: cc-p-CVXZ defined in ref. [27].f) CCSD(T) calculations for constant geometries shown in Table 1 to Table 4 [30].g) ANO basis sets.
That mentioned experiment is a historical important example of the controversial and fruitful interplay between computational and experimental chemistry which is nicely described in ref. [2]. The experimental value of To was in contrast to other ab initio calculations leading to Te = 11.5 kcal/mol [40] and 13 kcal/mol [41]. This discrepancy was interpreted due to accidentally observed experimental hot bands [42] resulting in a reassignment for To of (8.99 ± 1.15) kcal/mol which could be verified later by different experiments [43] leading now to To = 9.6 kcal/mol. An other accurate experimental To energy of (9.05 ± 0.06) kcal/mol was measured by McKellar et al. [44] by far-infrared laser magnetic resonance. This is in perfect agreement to a recently calculated To value of 9.025 kcal/mol [52].
In Table 6 we show the basis set and method dependence of calculated Te values derived as difference between the total energies of the minima 3 and 1 presented in Table 2 and Table 1 and some additional references [45], [46] and [47].
Our CCSD(T) calculated Te energies follow again an exponential trend and thus the extrapolation to the CBS limit via eqn. 1 is possible. This CBS (TQ5) limit is 9.299 kcal/mol, somewhat larger than the 9.287 kcal/mol obtained as the difference of the CBS limit values of Table 1 and 2.
Experimental Te energies must be derived from To by adjustment to the potential energy curves. This can be done by different approximations. Therefore several experimental Te energies are shown in Table 6. Most reliable experimental energies are (9.215 ± 0.004) kcal/mol [9] and (9.032 ± 0.057) kcal/mol [47] which fit the range of the calculated CCSD(T) ΔCBS limit energies between 9.04 and 9.48 kcal/mol.
All Te energies of Table 6 decrease with increase of Dunning basis sets: from about 12 kcal/mol for DZ calculations to approximately 10 kcal/mol for TZ, 9.6 for QZ, 9.4 for 5Z and 9.3 kcal/mol for 6Z CCSD(T) calculations.
Additional data in Table 6 allow a comparison of valence electron (VE) frozen-core calculations with all electron (AE) core-valence calculations, which use corresponding different Dunning basis sets (cc-p-CVXZ versus cc-p-CVXZ [27]). Mostly are the AE Te energies larger than those of VE calculations.
The few available full CI (FCI) DZ and TZ based Te energies are in the range of 10 to 12.7 kcal/mol, but slightly lower than the corresponding CCSD(T) values. The elaborate ANO calculations [21] and [25] lead to values in the interval of 9.11 to 9.74 kcal/mol.
The numbers presented in Table 6 show clearly that large basis sets (at least QZ) are necessary to reach the range of experimental Te determinations.

Barriers to Linearity

The total energies of Table 1 and Table 2 allow the study of the basis set dependence of the barrier to linearity (ΔEBL) as the difference of energies between the linear and bent conformations of the triplet states (1 versus 2) and singlet states (3 versus 4). These barriers to linearity have been determined rather seldom by calculations [48], [49] and experimentally [50], [51] and [52] with values shown in Table 7 in comparison to our calculations.
Ijms 06 00157 g001 1024
Figure 1. Exponential behaviour of barriers to linearity for the cc-pVXZ (X = 2 - 5) basis set expansion. Above: triplet CH2, below: singlet CH2
Figure 1. Exponential behaviour of barriers to linearity for the cc-pVXZ (X = 2 - 5) basis set expansion. Above: triplet CH2, below: singlet CH2
Ijms 06 00157 g001
Again an exponential extrapolation via eqn. 1 to the CBS limit is possible which is depicted in Figure 1. The CBS limit barrier of the triplet state between 1 and 2 is with 5.623 kcal/mol rather small, in good agreement to the experimental value of 5.48 kcal/mol [8]. The derived CBS limit barrier for the singlet state is with 30.842 kcal/mol substantially higher but in disagreement to an experimental value around 24.6 kcal/mol.
Table
Table 7. Barriers to linearity ΔEBL [kcal/mol] of triplet CH2 (difference of energies between bent 2 and linear 1 from Table 1) and singlet CH2 (ΔE of bent 4 and linear 3 from Table 2).
Table 7. Barriers to linearity ΔEBL [kcal/mol] of triplet CH2 (difference of energies between bent 2 and linear 1 from Table 1) and singlet CH2 (ΔE of bent 4 and linear 3 from Table 2).
CCSD(T)
Basis set		ΔEBL (2 - 1)
Triplet		ΔEBL (4 - 3)
Singlet
cc-pVDZ6.846034.3662
cc-pVTZ5.901731.8179
cc-pVQZ5.732331.1860
cc-pV5Z5.666030.8850
CBS limit5.623430.8422
MCI CAS AE:5.9 [49]CMCI: 25.160 [48]
Experiment5.54 ± 0.23 [8]24.59 ± 1.14 [50]
Experiment5.478 [9]26.88 [51]

Correlation Energies

The energy of correlation between electrons (ΔEcorr) is defined as ΔEcorr = Epost-HF - EHF, where Epost-HF is the total energy calculated by post-HF methods at maximum approaching the exact non-relativistic total energy of the system of interest and EHF its the reference Hartree-Fock energy calculated under same conditions. In Table 8 we present the basis set dependence of our CCSD(T) values for correlation energies (ΔEcorr) of the methylenes 1 to 4. In each case increases this energy with increase of basis set. This demonstrates the fact that increase of basis sets in the CCSD(T) methods leads to more effective treatment of electron correlation.
Table
Table 8. Correlation energies obtained as the difference between CCSD(T) and Hartree–Fock total energies [Hartree] (ΔEcorrel = EHF – E cc-pVXZ) from our cc-pVXZ (X= 2 – 5) calculations and extrapolations to the CBS limit.
Table 8. Correlation energies obtained as the difference between CCSD(T) and Hartree–Fock total energies [Hartree] (ΔEcorrel = EHF – E cc-pVXZ) from our cc-pVXZ (X= 2 – 5) calculations and extrapolations to the CBS limit.
MoleculeBasis setΔEcorrel [Hartree]
3B1 CH2 bent 1cc-pVDZ-0.114976
tripletcc-pVTZ-0.139995
(ground state)cc-pVQZ-0.146980
cc-pV5Z-0.149101
CBS limit-0.150026
3Σ- CH2 2cc-pVDZ-0.116565
tripletcc-pVTZ-0.142558
(linear)cc-pVQZ-0.149845
cc-pV5Z-0.152062
CBS limit-0.153032
1A1 CH2 bent 3cc-pVDZ-0.141485
singletcc-pVTZ-0.168839
(ground state)cc-pVQZ-0.176555
cc-pV5Z-0.178935
CBS limit-0.179997
1Δg CH2 4cc-pVDZ-0.144840
Singletcc-pVTZ-0.173029
(linear)cc-pVQZ-0.181233
cc-pV5Z-0.183755
CBS limit-0.184875
Again the exponential extrapolation procedure of eqn. 1 could be used to obtain CBS limits of the correlation energy. This behaviour is presented graphically in Fig. 2.
Ijms 06 00157 g002 1024
Figure 2. Exponential behaviour of correlation energies of the triplet 3B1 ground state of CH2 (1) and the singlet 1A1 lowest state of CH2 (3).
Figure 2. Exponential behaviour of correlation energies of the triplet 3B1 ground state of CH2 (1) and the singlet 1A1 lowest state of CH2 (3).
Ijms 06 00157 g002
The correlation energy of the ground state singlet CH2 (3) is substantially larger than that of the ground state triplet CH2 (1) leading to a difference of the correlation energy between 1 and 3 of 18.81 kcal/mol. The same behaviour is observed for the linear conformations of singlet (4) and triplet (2) CH2. The difference of the correlation energy in the basis set limit is by 19.98 kcal/mol larger for 4 compared to 2. The linear singlet CH2 4 shows the largest amount of correlation energy of the four considered species.

Conclusions

The calculations of energies and structural parameters of methylene stationary points 1 to 4 attempt to complete existent coupled cluster CCSD(T) values of the literature by extrapolated Dunning values in the basis set limit. Exponential extrapolations via the empirical eqn. 1 are possible for equilibrium distances, total energies, barriers to linearity, singlet-triplet separation energies and correlation energies but not for our valence angles.
Detailed comparison to the Dunning basis set dependence of other published post-HF calculations shows that the CCSD(T) method is in its Dunning basis set extrapolations a highly effective and reliable method for treatment of electron correlation of post-HF calculations. Only core-valence all electron correlations lead to lower total energies which proves the quality of the here used CCSD(T) calculations.

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Neugebauer, A.; Häfelinger, G. Ab initio post-HF CCSD(T) Calculations for Triplet and Singlet Methylene in Four consecutive Dunning Basis Sets with Extrapolations to Infinite Limits for Various Molecular Properties. Int. J. Mol. Sci. 2005, 6, 157-176. https://doi.org/10.3390/i6010157

AMA Style

Neugebauer A, Häfelinger G. Ab initio post-HF CCSD(T) Calculations for Triplet and Singlet Methylene in Four consecutive Dunning Basis Sets with Extrapolations to Infinite Limits for Various Molecular Properties. International Journal of Molecular Sciences. 2005; 6(1):157-176. https://doi.org/10.3390/i6010157

Chicago/Turabian Style

Neugebauer, Alexander, and Günter Häfelinger. 2005. "Ab initio post-HF CCSD(T) Calculations for Triplet and Singlet Methylene in Four consecutive Dunning Basis Sets with Extrapolations to Infinite Limits for Various Molecular Properties" International Journal of Molecular Sciences 6, no. 1: 157-176. https://doi.org/10.3390/i6010157

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