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Methylene: a turning point in the history of quantum chemistry and an enduring paradigm

  • Ashley M. Allen ORCID logo , Allison C. Sargent ORCID logo and Henry F. Schaefer ORCID logo EMAIL logo
Published/Copyright: October 20, 2025
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Pure and Applied Chemistry
From the journal Pure and Applied Chemistry Volume 97 Issue 11

Abstract

Prior to 1970, no successful ab initio electronic structure predictions were made that challenged experiment for polyatomic molecules. For diatomics, the work of Ernest Davidson stands out, in 1960 explaining that two spectroscopically known but misunderstood electronic states of H2 were in fact part of the very same potential energy curve. Another diatomic example was the startling 1968 overturn by Kolos and Wolniewicz of the “known” experimental dissociation energy of H2. Moving to polyatomics, the research in 1970 concerning the structure of triplet methylene captured the imagination of many chemists, and the 1982 success of theory for the singlet-triplet separation of methylene confirmed for many the great usefulness of ab initio theory. In the second half of this paper, the utility of methods based on single Slater determinant reference wavefunctions for both singlet and triplet methylene is demonstrated. In particular, we examine how the optimized geometries and harmonic vibrational frequencies of triplet and singlet CH2 evolve with systematic improvements in basis set size and valence electron correlation treatment. To accurately pinpoint the geometric structures and singlet-triplet splitting of methylene, we perform comprehensive focal point analyses (FPA) that push the level of theory to new heights, leveraging core-valence basis sets up to sextuple-zeta quality and all-electron coupled-cluster methods through CCSDTQ(P) appended with relativistic (MVD1) and non-Born-Oppenheimer (DBOC) corrections. Our final FPA prediction for the singlet-triplet splitting is 9.01 kcal mol−1, in complete agreement with the best empirical estimate of 9.00 ± 0.01 kcal mol−1. The corresponding optimized FPA geometries are [r e(H–C), θ e(H–C–H)] = (1.1063 Å, 102.35°) for ã1A1 CH2 and (1.0756 Å, 133.94°) for X̃3B1 CH2, in close agreement with the best existing experimental and theoretical structures but with a little finer precision. Our outcomes not only affirm the validity of the contemporary single-reference coupled-cluster theory pushed to high order but also provide definitive resolutions for a paradigmatic molecule that has long been emblematic of the challenges and triumphs that have shaped a century of quantum chemistry.

Keywords: Electronic structures; quantum chemistry; quantum science and technology

Historical perspectives

Prior to 1970, chemists often considered ab initio quantum chemistry to be elegant and beautiful, but not able to make reliable predictions of chemical interest. Until that date, an often-cited failure of ab initio methods was the methylene molecule CH2. The pioneering 1960 research 1 of Frank Boys (Cambridge University) predicted the bond angle of triplet methylene to be 129°. However, in his 1961 endowed lecture at Cornell University, future Nobelist Gerhard Herzberg concluded the triplet ground state to be linear. Herzberg’s somewhat belittling remark 2 was that “our experimental values are distinctly different from these (Boys) but not excessively so, when the approximate nature of the calculations is considered.” The CH2 attack on ab initio quantum mechanical methods continued with Christopher Longuet-Higgins, Chair Professor at Cambridge. Using semi-empirical methods (combining theory with experimental constraints), Longuet-Higgins 3 reproduced Herzberg’s linear geometry for CH2. Longuet-Higgins’ assessment was even more pointed than that of Herzberg: “It may be that future theoretical progress will require elaborate variational calculations such as those of Foster and Boys on CH2, but until the results of such machine experiments can be interpreted physically, there would seem to be a place for more empirical theories such as that which we now describe.”

Even within the theoretical community, during the 1960s there seemed to be acceptance of the “fact” that the Foster and Boys prediction of bent CH2 was in error. Future Nobelist John Pople (later a close friend of HFS) reported with G. A. Segal some better semi-empirical computations on CH2, 4 predicting, like Boys, a bent structure (in this case 141°) for the triplet ground state. Unfortunately, these authors accepted Herzberg’s linear CH2 structure and rationalized what they thought was their incorrect theoretical prediction. They stated, “This is a situation where the CNDO approximation is least satisfactory” because it “neglects the one-center exchange integral.” Pople and Segal concluded, “This triplet stabilization is probably a maximum in the linear form, and its inclusion would very likely modify the calculated bond angle.”

Although other sources would confirm the great theoretical “disaster” for CH2, one more will suffice here. In 1969, Jim Harrison and Lee Allen (both excellent theorists) reported the best ab initio CH2 computations to date. 5 Like Frank Boys, Harrison and Allen predicted triplet CH2 to be bent, this time with a bond angle of 138°. However, Harrison and Allen were also intimidated by the Herzberg experiments: “We certainly wish the present results to be regarded with conservatism, and there is certainly room for a greatly improved theoretical treatment.” Focusing on a possible weakness in their methodology, Harrison and Allen added, “Because of the very flat potential curves of the 3B1 and 1B1 states, the predicted equilibrium angles of 138 and 148°, respectively, could be changed significantly with minor changes in the atomic bases.”

During the summer 1968 Gordon Conference on Theoretical Chemistry, HFS met a uniquely gifted young theorist, Charlie Bender. Charlie had been a PhD student with Ernest Davidson at the University of Washington and would accept a position at the University Computing Company in Palo Alto, next to Stanford University, where HFS had carried out doctoral studies. Before HFS began at Berkeley as an Assistant Professor, a collaboration with Bender was agreed upon. Knowing the terrible reputation of theory with respect to the CH2 structure, methylene was attacked with new theoretical methods. During the first year of HFS on the Berkeley faculty, a controversial paper on the CH2 problem was submitted, concluding the molecule to have a bond angle of 135°. The paper was not at all apologetic, stating, “On the basis of the present and previous ab initio calculations and the stated experimental uncertainties, we conclude that the CH2 ground state is nonlinear with a geometry close to r = 1.096 Å, θ = 135.1°.” 6 Five years later, the distinguished organic chemists George Hammond and Peter Gaspar 7 stated in their review, “In 1970 Bender and Schaefer reported by far the most elaborate calculation carried out to date on methylene, or indeed almost any molecule.” For the moment, Charlie Bender and HFS were near the peak of theoretical developments in computational quantum chemistry. Very fortunately for HFS (assistant professorships in chemistry at Berkeley were more often than not terminated after a few years), the controversial prediction of bent triplet methylene was quickly confirmed experimentally by Ed Wasserman (Bell Labs) and by Bob Bernheim and Phil Skell at Penn State. 8 , 9 A fuller description of events following can be found elsewhere. 10

The second shoe of the methylene controversy took much longer to fall. The energy separation (T 0) between the ground triplet (3B1) and lowest singlet (1A1) state of methylene is important for organic chemistry because the reactions of singlet carbenes (CR2 or CRR’) with olafins tend to be stereospecific while those of triplet carbenes are not. By the year 1972, there was a wide range of experimental T 0 values from 2 to 10 kcal mol−1. In that year, Hay, Hunt, and Goddard 11 used their newly developed and very promising generalized valence bond (GVB) method with a double-zeta basis to predict a CH2 singlet-triplet separation (T e) of 11.5 kcal mol−1. Very shortly thereafter, Bender and HFS with Donald Franceschetti and Lee Allen from Princeton addressed the same problem. 12 The two papers 11 , 12 thank each other for private communications. The Berkeley-Princeton collaboration 12 used a large triple-zeta basis set and a more complete description of electron correlation. It was concluded, rather daringly, that the true CH2 singlet-triplet separation was 11 ± 2 kcal mol−1. These proposed error bars pointed a big target on the proverbial backs of the authors.

All seemed satisfactory for the Cal Tech and Berkeley theoretical predictions of the singlet-triplet splitting of methylene until 1976. In that year, the first direct experimental measurement was reported by Carl Lineberger and coworkers 13 at Boulder. Lineberger was already recognized both as a brilliant experimental physical chemist and a friend to all in the community. Via the laser photodetachment of the CH2 − anion, Lineberger concluded that T 0 was 19.5 ± 0.7 kcal mol−1. This meant that all the previous experimental estimates (2–10 kcal mol−1) were wrong, as were the Cal Tech and Berkeley theoretical results. Most observers quickly accepted the validity of the conclusions of the Boulder group, and the theorists were subjected to a certain amount of humiliation, 10 which need not be recapitulated here. One exception not yet in print is the statement by legendary statistical mechanician Berni Alder at his April 14, 1981 seminar to the Chemistry Department at Berkeley: “If they can’t even get CH2 right, what hope is there for quantum chemistry?”

Six painful years after the Carl Lineberger experiment, the great quantum chemistry “disaster” was overturned. In 1982, Yuan Lee and his group (including now-prominent Dan Neumark) reported the second direct experimental measurement of the CH2 singlet-triplet energy separation. 14 Their result was 8.5 ± 0.8 kcal mol−1, in agreement with the Berkeley-Princeton result of 11 ± 2 kcal mol−1 from ten years earlier. And thus the reliability of state-of-the-art ab initio quantum chemistry was secured. There were battles yet to come, 10 but the momentum was permanently shifted.

Since the early 1980s, extensive experimental 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 and theoretical efforts 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 have focused on further elucidating the structure and spectroscopic properties of methylene and related species, as nicely summarized in Refs. 28] and 36]. Most relevant to the current work, a series of papers on triplet methylene by Bunker et al. 31 , 32 and Jensen et al. 30 (1982–1986) fit a large body of microwave, infrared, and photodetachment spectroscopic data using nonrigid bender Hamiltonians, resulting in (r e, θ e) = (1.0766 ± 0.0014 Å, 134.037 ± 0.045°) for X̃3B1 CH2. A subsequent 1988 paper by Jensen and Bunker 33 employed the Morse oscillator-rigid bender internal dynamics (MORBID) Hamiltonian to fit all rotation-vibration data for triplet CH2, leading to the benchmark empirical singlet-triplet splitting of 8.998 kcal mol−1. This compares excellently to the 2003 ab initio T 0 result of 8.972 kcal mol−1 computed by Császár and coworkers 36 using a valence focal point method, augmented with corrections resulting from core correlation, relativity, and DBOC. For the ã1A1 CH2 state in 1989, Petek and coworkers 26 determined equilibrium rotational constants and thus the equilibrium structure (r e = 1.107 ± 0.002 Å, θ e = 102.4 ± 0.4°) from infrared flash-kinetic spectroscopy. More recently in 2024, Egorov and coworkers 28 , 29 constructed ab initio potential energy surfaces (PES) for both X̃3B1 and ã1A1 methylene, culminating in comprehensive rovibrational line lists. Each PES was built from all-electron CCSD(T)/aug-cc-pCVXZ (X = T, Q, 5, 6) results and included DBOC and scalar relativistic corrections as well as high-order correlation adjustments, namely CCSDT and CCSDT(Q) with cc-pVQZ, CCSDTQ with cc-pVTZ, and CCSDTQP with cc-pVDZ. The final values for (r e, θ e) for the triplet and singlet states are (1.0756 Å, 133.92°) and (1.1064 Å, 102.31°), respectively, in excellent agreement with the best empirical results cited above, 26 , 32 but significantly more precise in the triplet case.

Theoretical approach

The single-configuration Hartree-Fock method is inadequate for the lowest singlet state of CH2, especially near linear geometries where the ã1A1 and open-shell singlet Ã1B1 states coalesce into a doubly-degenerate 1Δ manifold. As early as 1971, the coefficients for the two most important configurations of the ã1A1 electronic wavefunction in the region of the bent equilibrium geometry were found 45 in a natural orbital representation to be

0 . 962       1 a 1 2   2 a 1 2   1 b 2 2   3 a 1 2

0 . 194       1 a 1 2   2 a 1 2   1 b 2 2   1 b 1 2

Thus, the Hartree-Fock configuration … 3 a 1 2 represents 92.5 % of the wavefunction, while the second configuration … 1 b 1 2 represents 3.8 % of the wavefunction in a natural orbital representation. Although 3.8 % of the wavefunction might be considered not terribly important, it is enough to make single-configuration Hartree-Fock comparisons of the X̃3B1 and ã1A1 states of CH2 very inaccurate. For example, our computations near the complete basis limit show the Hartree-Fock singlet-triplet separation for CH2 to be 25 kcal mol−1, far above the best experimental value 33 , 46 of 9.0 kcal mol−1. At present, coupled-cluster theory is the most reliable electronic structure methodology for the description of molecular systems well-described by a single configuration. 47 , 48 Single-reference coupled-cluster theory will approach the exact solution to the Schrödinger equation if carried to high enough excitation levels. In the present research, we investigate how far the coupled-cluster methodology needs to be pushed to pinpoint the methylene singlet-triplet separation.

Single-point energies, optimized geometries, and harmonic vibrational frequencies for singlet and triplet methylene were computed using a systematic progression of electronic wavefunction methods, starting with Hartree-Fock theory 49 , 50 and second-order Møller-Plesset (MP2) 51 perturbation theory, and then followed by coupled-cluster methods 52 , 53 proceeding from single and double excitations (CCSD) to full single, double, triple, and quadruple excitations (CCSDTQ). 54 , 55 , 56 Perturbative triple CCSD(T) and quadruple CCSDT(Q) excitation methods 57 , 58 were also employed owing to their popularity and effectiveness. Finally, an additional perturbative pentuple excitation CCSDTQ(P) 59 , 60 energy correction was applied to rigorously approach the full configuration interaction (FCI) limit.

The correlation-consistent polarized-valence orbital basis sets of Dunning 61 , 62 , 63 , 64 were used throughout this research, namely cc-pVXZ (X = D, T, Q, 5), along with the related core-valence basis sets cc-pCVXZ (X = D, T, Q, 5, 6). Thus, definitive conclusions regarding basis set dependence were reached. All computations with CCSD(T) and lower levels were performed using the CFOUR package, 65 , 66 while those of CCSDT and higher levels through CCSDTQ(P) were obtained with the MRCC program 67 , 68 interfaced with Psi4 69 and its OPTKING geometry optimization program when applicable.

A composite focal point analysis (FPA) scheme 70 , 71 was employed to rigorously approach the ab initio limit of the singlet-triplet energy splitting of CH2 as well as the X̃3B1 and ã1A1 equilibrium geometries. Our methodology mirrors that used highly successfully in our recent publication 72 on fulminic acid (HCNO) and HCN. Single-point energies through CCSD(T) used a ROHF reference, whereas higher-order corrections (CCSDT and beyond) used UHF. To reach the complete basis set (CBS) limit, extrapolations of the Hartree-Fock total energies (E HF) and the all-electron MP2, CCSD, and CCSD(T) correlation energies (E corr) were performed according to the following equations: 73 , 74

(1) E HF X = E HF , CBS + b e − c X

(2) E corr X = E corr , CBS + b X − 3

where X is the cardinal number of the correlation-consistent basis set series. The E HF extrapolations used cc-pCV(Q,5,6)Z total energies, while E corr extrapolations were based on all-electron (AE) cc-pCV(5,6)Z results. Within the FPA scheme, higher-order correlation increments (δ) were computed as follows:

(3) δ CCSDT Q = E corr CCSDT Q / cc-pCVQZ − E corr CCSDT / cc-pCVQZ

(4) δ CCSDTQ = E corr CCSDTQ / cc-pCVQZ − E corr CCSDT Q / cc-pCVQZ

(5) δ CCSDTQ P = E corr CCSDTQ P / cc-pCVTZ − E corr CCSDTQ / cc‐pCVTZ

Two auxiliary corrections (Δ), namely the scalar relativistic one-electron mass-velocity and Darwin terms (MVD1) 75 , 76 , 77 and the diagonal Born-Oppenheimer (DBOC) correction, 43 were computed at the AE-CCSD(T)/cc-pCV5Z and AE-CCSD/cc-pCVQZ levels of theory, respectively. Thus, the final FPA energy is evaluated according to the following expression:

E FPA = E HF , CBS + E corr , CBS CCSDT + δ CCSDT Q + δ CCSDTQ

(6) + δ CCSDTQ P + ∆ MVD 1 + ∆ DBOC

Results and discussion

Survey of the valence correlation series

Tables 1 and 2 detail our valence correlation results for the X̃3B1 and ã1A1 states of methylene. The data include optimized bond distances (r e) and bond angles (θ e), total energies, harmonic vibrational frequencies (ω i ), and the singlet-triplet splitting (T e). The intent of these tables is to elucidate the requirements for a precise theoretical description of methylene. Because the structure of methylene has historically been such a contentious issue, it is particularly worthwhile to assess whether frozen-core correlation methods with finite basis sets can accurately converge on the true equilibrium geometries of the singlet and triplet states.

Table 1:

Triplet CH2 (X̃3B1) optimized bond distances (r e, Å) and bond angles (θ e, °), total energies (E tot, E h), and harmonic vibrational frequencies (ω i , cm−1).

r e(H–C) θ e(H–C–H) E tot ω 1 (a 1) ω 2 (a 1) ω 3 (b 2)
UHF
 cc-pVDZ 1.0808 131.45 −38.92684 3281 1199 3497
 cc-pVTZ 1.0700 131.83 −38.93786 3262 1201 3472
 cc-pVQZ 1.0693 131.84 −38.94035 3265 1203 3473
 cc-pV5Z 1.0692 131.84 −38.94097 3266 1203 3474
MP2
 cc-pVDZ 1.0902 131.74 −39.01978 3202 1159 3437
 cc-pVTZ 1.0737 132.79 −39.05548 3203 1143 3437
 cc-pVQZ 1.0722 132.99 −39.06602 3208 1137 3444
 cc-pV5Z 1.0718 133.05 −39.06963 3209 1135 3446
CCSD
 cc-pVDZ 1.0947 132.24 −39.04005 3152 1129 3376
 cc-pVTZ 1.0772 133.42 −39.07446 3155 1112 3380
 cc-pVQZ 1.0759 133.57 −39.08348 3161 1106 3387
 cc-pV5Z 1.0755 133.61 −39.08608 3162 1105 3388
CCSD(T)
 cc-pVDZ 1.0956 132.22 −39.04182 3139 1126 3365
 cc-pVTZ 1.0784 133.48 −39.07785 3140 1105 3366
 cc-pVQZ 1.0772 133.66 −39.08733 3144 1097 3372
 cc-pV5Z 1.0769 133.70 −39.09008 3145 1092 3374
CCSDT
 cc-pVDZ 1.0959 132.19 −39.04220 3135 1126 3362
 cc-pVTZ 1.0787 133.48 −39.07834 3136 1105 3363
 cc-pVQZ 1.0775 133.66 −39.08780 3141 1097 3369
 cc-pV5Z 1.0771 133.71 −39.09052 3141 1094 –
CCSDT(Q)
 cc-pVDZ 1.0960 132.20 −39.04228 3135 1125 3361
 cc-pVTZ 1.0788 133.46 −39.07841 3135 1104 3362
 cc-pVQZ 1.0775 133.67 −39.08790 3140 1096 3368
 cc-pV5Z 1.0772 133.72 −39.09061 3140 1093 –
CCSDTQ
 cc-pVDZ 1.0960 132.20 −39.04229 3134 1125 3361
 cc-pVTZ 1.0788 133.49 −39.07842 3135 1104 3362
Empirical 32 1.0766(14) 134.037(45) – – – –
Table 2:

Singlet (ã1A1) CH2 optimized bond distances (r e, Å) and bond angles θ e, °), total energies (E tot, E h), harmonic vibrational frequencies (ω i , cm−1), and singlet-triplet splitting (T e, kcal mol−1).

r e(H–C) θ e(H–C–H) E tot ω 1 (a 1) ω 2 (a 1) ω 3 (b 2) T e
RHF
 cc-pVDZ 1.1071 102.74 −38.88110 3108 1497 3176 28.70
 cc-pVTZ 1.0957 103.57 −38.89255 3092 1496 3155 28.44
 cc-pVQZ 1.0946 103.72 −38.89536 3098 1492 3161 28.23
 cc-pV5Z 1.0944 103.79 −38.89610 3100 1490 3164 28.16
MP2
 cc-pVDZ 1.1217 100.85 −38.99144 2984 1429 3067 17.78
 cc-pVTZ 1.1044 101.82 −39.03100 2990 1430 3067 15.36
 cc-pVQZ 1.1025 102.09 −39.04330 2999 1425 3079 14.25
 cc-pV5Z 1.1020 102.22 −39.04769 3002 1422 3082 13.77
CCSD
 cc-pVDZ 1.1279 100.54 −39.01964 2913 1418 2987 12.81
 cc-pVTZ 1.1090 101.63 −39.05657 2930 1422 2999 11.23
 cc-pVQZ 1.1070 101.90 −39.06657 2942 1419 3012 10.62
 cc-pV5Z 1.1065 102.01 −39.06951 2945 1418 3016 10.40
CCSD(T)
 cc-pVDZ 1.1291 100.54 −39.02258 2897 1406 2975 12.07
 cc-pVTZ 1.1105 101.61 −39.06138 2912 1407 2983 10.34
 cc-pVQZ 1.1086 101.89 −39.07191 2922 1403 2996 9.68
 cc-pV5Z 1.1081 102.00 −39.07504 2925 1401 2999 9.44
CCSDT
 cc-pVDZ 1.1288 100.67 −39.02330 2898 1399 2978 11.86
 cc-pVTZ 1.1102 101.80 −39.06219 2913 1398 2985 10.13
 cc-pVQZ 1.1084 102.10 −39.07270 2923 1394 2998 9.48
 cc-pV5Z 1.1080 102.19 −39.07579 2925 1393 – 9.24
CCSDT(Q)
 cc-pVDZ 1.1289 100.67 −39.02345 2898 1398 2977 11.81
 cc-pVTZ 1.1103 101.82 −39.06238 2912 1396 2985 10.06
 cc-pVQZ 1.1085 102.12 −39.07291 2922 1392 2998 9.40
 cc-pV5Z 1.1080 102.22 −39.07601 2924 1390 – 9.16
CCSDTQ
 cc-pVDZ 1.1289 100.69 −39.02348 2898 1398 2977 11.80
 cc-pVTZ 1.1104 101.82 −39.06241 2911 1396 2984 10.05
Empirical a 1.107(2) 102.4(4) – – – – 9.37

  1. aEquilibrium geometry taken from Ref. 26]. The T e value arises by adding the ZPE correction 37 of 0.37 kcal mol−1 to the empirical T 0 result 33 of 9.00 ± 0.01 kcal mol−1.

For θ e of the X̃3B1 state, Hartree-Fock theory gives 131.45° with the DZ basis set but quickly converges thereafter to a CBS limit of 131.84°. The MP2 bond angles are somewhat more sensitive to basis set, with (DZ, TZ, QZ, 5Z) results of (131.74, 132.79, 132.99, 133.05)°. The CCSD and CCSD(T) bond angles also increase with basis set size, although with somewhat quicker convergence than for MP2. For CCSD, θ e incrementally increases from 132.24° (DZ) to 133.61° (5Z), whereas the corresponding CCSD(T) results match those of CCSD to within 0.1°. Full treatment of triples excitations (CCSDT) past CCSD(T) changes the bond angle by 0.03° in the case of DZ and 0.01° or less along the rest of the basis set series. In moving from CCSDT to CCSDT(Q), the (DZ, TZ, QZ, 5Z) bond angles are shifted by only (+0.01, −0.02, +0.01, +0.01)°. Further treatment of correlation with CCSDTQ does not yield any deviations more than 0.03° from the corresponding CCSDT(Q) angles. For all methods employed, θ e exhibits strong basis set dependence and is consistently widened in traversing the basis set series. For example, the (DZ → TZ, TZ → QZ, QZ → 5Z) increments in the bond angle for CCSDT are (+1.29, +0.19, +0.04)°. Thus, extending the coupled-cluster series beyond connected triple excitations has very little impact on θ e, while basis set effects are persistent. The [CCSD(T), CCSDT, CCSDT(Q)] results with the largest basis set employed (cc-pV5Z) are (0.34, 0.33, 0.32)° smaller than the empirical result. Such valence correlation treatments thus fail to precisely nail down the bond angle of triplet methylene.

The bond distance (r e) of the X̃3B1 state decreases from DZ to TZ by (0.0108, 0.0165, 0.0175) Å for (HF, MP2, CCSD) and this increment settles to 0.0172 Å for CCSD(T) and beyond. Afterwards, the TZ → QZ shift for all methods is approximately −0.001 Å, with that of the QZ → 5Z shift being −0.0004 Å or less. As found for the bond angle, higher-order correlation methods hardly change the CCSDT results, with shifts no more than 0.0001 Å. The CCSD(T)/cc-pV5Z r e = 1.0769 Å is merely 0.0003 Å longer than the empirical result. CCSDT/cc-pV5Z slightly worsens r e, rendering this value 0.0005 Å too long, and the CCSDT(Q) bond distance with the same basis set is 0.0006 Å too long. Thus, the bond distances in Table 1 are essentially converged to the valence FCI limit by CCSDT(Q) and demonstrate good agreement with the empirical benchmark, but this comparison is specious because CBS extrapolation, core correlation, and auxiliary terms have not yet been considered.

For ã1A1 methylene, the [HF, MP2, CCSD, CCSD(T)] bond angles once again systematically increase with increasing cardinality of basis set: the overall DZ → 5Z changes are (+1.05, +1.37, +1.47, +1.46)° with the largest contributor of these overall shifts being the initial DZ → TZ adjustments of (+0.83, +0.97, +1.09, +1.07)°. Interestingly, the HF/cc-pVDZ θ e happens to be within 0.35° of the empirical result, but more robust basis sets increase and worsen the bond angle predictions to an error of +1.4° in the case of 5Z. In contrast, MP2 follows the desired trend of higher quality basis sets producing incrementally more accurate outcomes. With MP2/cc-pVDZ, θ e is 1.5° too small, whereas MP2 with a QZ and 5Z basis improves this discrepancy to merely 0.3° and 0.2°, respectively. The CCSD and CCSD(T) angles with the same basis set differ by no more than 0.02°; in the case of CCSD(T), θ e goes from 100.54° (DZ) to 102.00° (5Z). The [CCSD(T), CCSDT] θ e values for (TZ, QZ, 5Z) are smaller than the empirical result by [(0.79, 0.51, 0.40), (0.60, 0.30, 0.21)]°, showing significant changes with both full inclusion of triple excitations and basis set augmentation. Higher-order θ e results do not differ from CCSDT by more than 0.03°. As is the case for the X̃3B1 bond angle, the best valence correlation treatment in Table 2 remains approximately 0.2° away from the empirical result, and thus a precise theoretical θ e requires a more rigorous treatment.

For HF through CCSD(T), the bond distance is consistently shortened with increasing basis set size, with reduction in magnitude for each increment. For example, the (DZ → TZ, TZ → QZ, QZ → 5Z) shifts for CCSD(T) are (−0.0186, −0.0019, −0.0005) Å. These same CCSD(T) bond distances are longer than the empirical result by (0.0221, 0.0035, 0.0016, 0.0011) Å for (DZ, TZ, QZ, 5Z). In the same basis set series, the CCSDT r e deviations from the empirical benchmark are +(0.0218, 0.0032, 0.0014, 0.0010) Å, demonstrating only minor shifts from CCSD(T). Higher-order increments beyond CCSDT amount to less than 0.0001 Å. The key conclusion from Tables 1 and 2 is that the valence correlation series predict (θ e, r e) to within (0.3°, 0.0003 Å) and (0.2°, 0.001 Å) of the empirical benchmarks for the X̃3B1 and ã1A1 states of methylene, respectively, but pinpointing the geometries of these species requires more rigorous theoretical methods.

Perhaps the most salient issue in our theoretical treatment of methylene is the determination of its singlet-triplet energy gap, which serves as a stringent test for state-of-the-art electronic structure methods. Table 2 reports T e results for the valence correlation series. As mentioned above, the HF predictions are far-removed from the benchmark (T e = 9.37 kcal mol−1) and are quite insensitive to basis set, ranging from 28.7 to 28.2 kcal mol−1 with the DZ to 5Z basis sets. In contrast, the corresponding MP2 results show marked improvements: T e (DZ, 5Z) = (17.8, 13.8) kcal mol−1. This MP2 range is narrowed at CCSD and CCSD(T) to the quite reasonable values (12.8, 10.4) and (12.1, 9.4) kcal mol−1, respectively. Consistent with observations for the equilibrium geometries, basis set effects are paramount to achieving accurate predictions. The T e (DZ → TZ, TZ → QZ, QZ → 5Z) shifts amount to (1.7, 0.7, 0.2) kcal mol−1 in both the CCSD(T) and CCSDT cases. Ascension to CCSDT(Q) and beyond affects T e by less than 0.1 kcal mol−1. Császár and coworkers 36 note that core electron correlation effects on T e are substantial, approximately 0.3 kcal mol−1, meaning that consideration of core correlation impacts T e more than higher-order correlation increments. Moreover, in a landmark 1986 paper, Handy, Yamaguchi, and Schaefer 43 found that the DBOC contribution to T e is 0.11 kcal mol−1. Thus, none of the valence correlation results in Tables 1 and 2 can be considered definitive without inclusion of auxiliary corrections and basis set extrapolation. In particular, the agreement of CCSD(T)/cc-pV5Z and CCSDT(Q)/cc-pVQZ with the empirical benchmark to better than 0.1 kcal mol−1 is a consequence of error cancellation.

Definitive focal point analyses

Tables 3 shows our FPA optimized geometries of the X̃3B1 and ã1A1 states of methylene using all-electron correlation methods and basis set extrapolation appended with scalar relativistic and DBOC effects. We also push the coupled-cluster series to include perturbative pentuple excitations [CCSDTQ(P)]. When applying this composite FPA methodology to the HCN molecule, 72 all bond distances agree with reliable empirical benchmarks within 0.0002 Å. For X̃3B1 and ã1A1 methylene, the AE-CCSD(T)/CBS values are (r e, θ e) = (1.0752 Å, 133.90°) and (1.1060 Å, 102.20°), respectively. These CBS extrapolations differ from the corresponding explicit AE-CCSD(T)/cc-pCV6Z results by no more than 0.0001 Å and 0.03°. Pushing to AE-CCSDT/CBS and AE-CCSDT(Q)/CBS changes r e of the triplet state by +0.0002 and –0.0001 Å, respectively, and θ e is widened by 0.01° with each sequential step. The AE-CCSDTQ/CBS and AE-CCSDTQ(P)/CBS optimized geometric parameters are the same as those of AE-CCSDT(Q)/CBS to the reported accuracy. The singlet methylene r e shifts are the same magnitude of those of the triplet state but in the opposite direction: the AE-CCSD(T)/CBS r e of 1.1060 Å is shifted (–0.0002, +0.0001) Å in going to [AE-CCSDT/CBS, AE-CCSDT(Q)/CBS] with no appreciable changes thereafter. The θ e shift from AE-CCSD(T) to AE-CCSDT is more pronounced at 0.22°, but the next increment is only 0.03° as the bond angle settles to 102.45°. Starting with the AE-CCSDTQ(P)/CBS geometries, inclusion of scalar relativistic effects merely reduces r e by 0.0001 Å for the X̃3B1 state and narrows θ e by 0.02° for the ã1A1 state, while leaving the other parameters unaltered to the precision given. The X̃3B1 methylene (r e, θ e) results are only slightly perturbed with further inclusion of DBOC, as the adjustments from the AE-CCSDTQ(P)/CBS + MVD1 geometry are (+0.0002 Å, +0.02°). However, the ã1A1 state demonstrates more noticeable changes: DBOC addition to the AE-CCSDTQ(P)/CBS + MVD1 results changes (r e, θ e) by (+0.0004 Å, –0.08°).

Table 3:

Triplet (X̃3B1) and singlet (ã1A1) methylene optimized bond distances (r e, Å) and bond angles (θ, °) from rigorous FPA procedures.

Level of theory X̃3B1 CH2 ã1A1 CH2
r e(H–C) θ e(H–C–H) r e(H–C) θ e(H–C–H)
AE-CCSD(T)/cc-pCV6Z 1.0752 133.88 1.1061 102.17
AE-CCSD(T)/CBS 1.0752 133.90 1.1060 102.20
AE-CCSDT/CBS 1.0754 133.91 1.1058 102.42
AE-CCSDT(Q)/CBS 1.0755 133.92 1.1059 102.45
AE-CCSDTQ/CBS 1.0755 133.92 1.1059 102.45
AE-CCSDTQ(P)/CBS 1.0755 133.92 1.1059 102.45
AE-CCSDTQ(P)/CBS + MVD1 1.0754 133.92 1.1059 102.43
AE-CCSDTQ(P)/CBS + MVD1 + DBOC 1.0756 133.94 1.1063 102.35
Egorov et al. (ab initio) 28 , 29 1.0756 133.92 1.1064 102.31
Empirical 26 , 32 1.0766(14) 134.037(45) 1.107(2) 102.3(4)

The final AE-CCSDTQ(P)/CBS + MVD1 + DBOC (r e, θ e) for the X̃3B1 and ã1A1 states of methylene are (1.0756 Å, 133.94°) and (1.1063 Å, 102.35°), respectively. Our r e result for X̃3B1 methylene matches the very recent high-level computations of Egorov et al. 28 , 29 to the precision in Table 3, whereas our θ e is 0.02° larger. A bit more discrepancy is witnessed for the ã1A1 state, as our results differ from those of Ref. 29] by 0.0001 Å and 0.04°. There are a few differences in the methodology of Egorov and coworkers 28 , 29 compared to ours including: no CBS extrapolation for X̃3B1 CH2 and a different extrapolation procedure for ã1A1; use of the Douglas-Kroll-Hess (DKH4) approach for relativistic corrections; higher-order coupled-cluster corrections from frozen-core rather than all-electron computations; and DBOC evaluation from FC-CCSD/cc-pVTZ rather than AE-CCSD/cc-pCVQZ. While none of these differences constitute a significant concern, our new results in Table 3 should be considered slightly more rigorous and preferable.

The empirical benchmark for X̃3B1 methylene is taken from a 1986 paper by Bunker and coworkers 32 whereby 152 rotation and rotation-bending energy levels were determined from a large volume of microwave, infrared, and photodetachment spectroscopic data (0–4500 cm−1) for CH2 and two isotopologues. This data was then fit using two different nonrigid bender Hamiltonians, both of which were obtained using second-order perturbation theory to average over the two stretching vibrations. The equilibrium geometry and shape of the potential surface was then adjusted until the fit of the energy level separations to the experimental data was optimized. Our r e result for X̃3B1 methylene is 0.0010 Å shorter than that of Bunker and coworkers 32 but within the reported uncertainty of ± 0.0014 Å. The θ e presented here is 0.097° smaller than the empirical benchmark, in good agreement albeit outside the reported error bound of ± 0.045°. For the ã1A1 state, the empirical benchmark is taken from a 1989 paper by Petek and coworkers, 26 who measured the symmetric- and antisymmetric-stretch infrared spectra of this state from 2600 to 3050 cm−1 by flash-kinetic spectroscopy. After removing data that were predicted to be triplet-perturbed, the remaining lines for the fundamental bands were fit to a Watson Hamiltonian including Coriolis coupling between the aforementioned vibrational states. This endeavor resulted in high-resolution rotational constants of fundamental vibrational levels, from which equilibrium rotational constants were surmised and hence an equilibrium structure for ã1A1 methylene was obtained. Our reported r e of 1.1064 Å is aligned with the empirical benchmark of 1.107 ± 0.002 Å, and the θ e of 102.35° in this work is in near-perfect agreement with the empirical 102.3 ± 0.4°. We deem our best theoretical (r e, θ e) results to not only be in full accord with experiment but actually more accurate and precise. The successes witnessed here for the historically controversial methylene structure constitute a genuine triumph for modern quantum chemical methods.

In Table 4, we present all-electron FPA results for T e with two-dimensional extrapolation to both the orbital basis set (CBS) and electron correlation (FCI) limits. Trends in Table 4 are similar to those witnessed in Table 2. At 24.7 kcal mol−1, the single-configuration HF/CBS estimate for T e significantly overshoots the converged NET/CBS result by a prodigious 15.4 kcal mol−1. Inclusion of electron correlation at the MP2 level leads to substantial recovery from the deficiencies inherent to HF: T e is reduced by a sizeable 10.9 kcal mol−1 at the MP2/CBS level, although it is still 4.5 kcal mol−1 above the NET/CBS target. Within the MP2/cc-pCVXZ (X = D, T, Q, 5, 6) series, T e is reduced by over 3 kcal mol−1, with values of (17.3, 15.4, 14.5, 14.1, 14.0) kcal mol−1 corresponding to each successive cardinal number. Moreover, an additional reduction of 0.23 kcal mol−1 is achieved by CBS extrapolation, demonstrating the deficiencies of even the best explicit computations.

Table 4:

Singlet-triplet energy splitting (T e, kcal mol−1) of CH2.a

T e(HF) δ[MP2] δ[Ȼ] δ[Ȼ(T)] δ[ȻT] δ[ȻT(Q)] δ[ȻTQ] δ[ȻTQ(P)] NET
cc-pCVDZ (AE) 25.32 −8.02 −4.51 −0.76 −0.22 −0.05 −0.01 −0.001 11.73
cc-pCVTZ (AE) 24.97 −9.56 −3.94 −0.94 −0.23 −0.08 −0.02 0.001 10.20
cc-pCVQZ (AE) 24.76 −10.21 −3.56 −0.99 −0.24 −0.09 −0.01 [0.001] [9.66]
cc-pCV5Z (AE) 24.70 −10.53 −3.35 −1.01 [–0.24] [–0.09] [–0.01] [0.001] [9.46]
cc-pCV6Z (AE) 24.67 −10.71 −3.25 −1.02 [–0.24] [–0.09] [–0.01] [0.001] [9.37]
CBS [24.66] [–10.94] [–3.10] [–1.03] [–0.24] [–0.09] [–0.01] [0.001] [9.26]
T e = T e(NET/CBS) + ΔMVD1 + ΔDBOC = 9.26 – 0.06 + 0.17 = 9.37 kcal mol − 1
T 0 = T e + ΔZPEb = 9.37 – 0.37 = 9.01 kcal mol −1

  1. aAE-CCSD(T)/cc-pCV5Z reference geometries: [r e(H–C), θ e(H–C–H)] for singlet and triplet methylene are (1.1062 Å, 102.15°) and (1.0753 Å, 133.87°), respectively. Ȼ is shorthand for CCSD. The symbol δ denotes increments in T e with respect to the preceding level of theory in the electron correlation series, beginning with HF and MP2 and followed by CCSD through CCSDTQ(P). Brackets signify results from CBS extrapolations [eqs (1) and (2)] or additivity assumptions. The sum across each row yields the NET column entry, which approximates the FCI result with the corresponding basis set. AE refers to all-electron. Triplet CH2 results were computed with an ROHF reference through CCSD(T), switching to a UHF reference starting with CCSDT. bThe zero-point energies (ZPE) of the ( X ∼ 3B1, a ∼ 1A1) states of CH2 are reported 37 as (10.67 ± 0.03, 10.31 ± 0.04) kcal mol−1.

We now turn to the coupled-cluster electron correlation series to tighten the CH2 singlet-triplet splitting further. At the CBS limit, CCSD corrects the MP2 result by 3.1 kcal mol−1, arriving at a T e result of 10.6 kcal mol−1. The energy gap is significantly reduced by another 1 kcal mol−1 with inclusion of perturbative triples: T e[CCSD(T)/CBS] = 9.6 kcal mol−1. Basis set effects with CCSD are strong but less pronounced than with MP2. For CCSD, the (DZ, TZ, QZ, 5Z, 6Z) core-valence basis set series produces T e values of (12.8, 11.5, 11.0, 10.8, 10.7) kcal mol−1. With ascension to CCSD(T), the incremented CCSD → CCSD(T) corrections δ[Ȼ(T)] are less sensitive to basis set than with either MP2 or CCSD. In particular, δ[Ȼ(T)] = (−0.76, −0.94, –0.99, −1.01, –1.02) kcal mol−1 for (DZ, TZ, QZ, 5Z, 6Z), demonstrating the merit of the essential FPA assumption that high-order correlation increments are less sensitive to basis set than lower-order ones.

As we proceed further up the coupled-cluster hierarchy, the critical question is whether high-order and auxiliary corrections can close the approximately 0.2 kcal mol−1 discrepancy between the CCSD(T)/CBS T e result and the empirical benchmark of 9.37 kcal mol−1. There is further modest improvement in the T e predictions with the full CCSDT method as the gap is narrowed by 0.24 kcal mol−1 at the CBS limit. The CCSDT(Q) and full CCSDTQ methods predict T e to within 0.1 kcal mol−1 of the CCSDT results, cumulatively reducing T e by another 0.1 kcal mol−1. The FPA layout here shows once again that the higher-order correlation contributions become progressively less sensitive to basis set. With a final push toward the FCI limit, AE-CCSDTQ(P)/cc-pCVXZ (X = D, T) values impact T e by a miniscule (−0.001, +0.001) kcal mol−1, respectively. Thus, the dual extrapolation presented in Table 4 yields a strongly converged T e(NET/CBS) = 9.26 kcal mol−1.

For an application this exacting, it is essential to assess the effects of auxiliary corrections on the singlet-triplet splitting. Upon incorporation of first-order scalar relativistic effects (MVD1), T e is further lowered by 0.06 kcal mol−1. However, the diagonal Born-Oppenheimer correction (DBOC) raises this result by 0.17 kcal mol−1. Furthermore, comparison to the empirical T 0 value necessitates a zero-point energy (ZPE) correction. Toward this end, a ΔZPE of 0.37 ± 0.05 kcal mol−1 is ascertained from Furtenbacher and coworkers 37 on the basis of rigorous variational vibrational computations, superseding various earlier results 33 , 34 , 35 , 44 for this quantity. This ΔZPE renders our final singlet-triplet splitting energy as T 0 = 9.01 kcal mol−1, virtually indistinguishable from the best empirical T 0 value 33 of 9.00 ± 0.01 kcal mol– 1. Our FPA computations thus yield a resounding resolution to decades of historical controversy.

Conclusions

As we mark the centenary of quantum mechanics, Paul A. M. Dirac’s famous 1929 declaration still echoes: “The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.” 78 Driven by this vision, theoretical chemistry has made monumental strides in the decades since, with methylene standing as a paradigmatic molecule that has served as a litmus test for both the success and reputation of the field. We have reviewed the remarkable changes (specifically in 1970 and 1982) in the attitudes of non-theoretical chemists toward ab initio computational quantum chemistry. The corollary to these changed attitudes is that today, quantum chemical methods are used in a large fraction of papers published in the best journals in chemistry. In the second half of this report, we witness triumphs of modern quantum chemistry in pinpointing the equilibrium geometry of methylene. The final results for the optimized structures of the ã1A1 and X̃3B1 states of CH2 are in complete accord with the empirical benchmarks, and we deem our values to actually be more accurate and precise. Moreover, our final FPA T 0 result of 9.01 kcal mol−1 achieves near-perfect agreement with the best empirical result of 9.00 ± 0.01 kcal mol–1. These conclusions serve as a compelling example of the extraordinary progress theoretical chemistry has made toward realizing the vision articulated by Dirac nearly a century ago.

Corresponding author: Henry F. Schaefer, III, Center for Computational Quantum Chemistry, University of Georgia, Athens, GA 30602, USA, e-mail:
Article note: A collection of invited papers to celebrate the UN’s proclamation of 2025 as the International Year of Quantum Science and Technology. Ashley M. Allen and Allison C. Sargent: authors contributed equally and share primary authorship.

Funding source: U.S. Department of Energy, Basic Energy Sciences, Division of Chemistry, Computational and Theoretical Chemistry (CTC) Program

Award Identifier / Grant number: DE-SC0018412

  1. Research ethics: Not applicable.

  2. Informed consent: Not applicable.

  3. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  4. Use of Large Language Models, AI and Machine Learning Tools: None declared.

  5. Conflict of interest: The authors state no conflict of interest.

  6. Research funding: This research was supported by the U.S. Department of Energy, Basic Energy Sciences, Division of Chemistry, Computational and Theoretical Chemistry (CTC) Program under Contract No. DE-SC0018412.

  7. Data availability: Not applicable. (All raw data is included in text).

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Received: 2025-08-15
Accepted: 2025-10-01
Published Online: 2025-10-20
Published in Print: 2025-11-25

© 2025 IUPAC & De Gruyter

Articles in the same Issue

  1. Frontmatter
  2. IUPAC Recommendations
  3. Experimental methods and data evaluation procedures for the determination of radical copolymerization reactivity ratios from composition data (IUPAC Recommendations 2025)
  4. IUPAC Technical Reports
  5. Kinetic parameters for thermal decomposition of commercially available dialkyldiazenes (IUPAC Technical Report)
  6. FAIRSpec-ready spectroscopic data collections – advice for researchers, authors, and data managers (IUPAC Technical Report)
  7. Review Articles
  8. Are the Lennard-Jones potential parameters endowed with transferability? Lessons learnt from noble gases
  9. Quantum mechanics and human dynamics
  10. Quantum chemistry and large systems – a personal perspective
  11. The organic chemist and the quantum through the prism of R. B. Woodward
  12. Relativistic quantum theory for atomic and molecular response properties
  13. A chemical perspective of the 100 years of quantum mechanics
  14. Methylene: a turning point in the history of quantum chemistry and an enduring paradigm
  15. Quantum chemistry – from the first steps to linear-scaling electronic structure methods
  16. Nonadiabatic molecular dynamics on quantum computers: challenges and opportunities
  17. Research Articles
  18. Alzheimer’s disease – because β-amyloid cannot distinguish neurons from bacteria: an in silico simulation study
  19. Molecular electrostatic potential as a guide to intermolecular interactions: challenge of nucleophilic interaction sites
  20. Photophysical properties of functionalized terphenyls and implications to photoredox catalysis
  21. Combining molecular fragmentation and machine learning for accurate prediction of adiabatic ionization potentials
  22. Thermodynamic and kinetic insights into B10H14 and B10H14 2−
  23. Quantum origin of atoms and molecules – role of electron dynamics and energy degeneracy in atomic reactivity and chemical bonding
  24. Clifford Gaussians as Atomic Orbitals for periodic systems: one and two electrons in a Clifford Torus
  25. First-principles modeling of structural and RedOx processes in high-voltage Mn-based cathodes for sodium-ion batteries
  26. Erratum
  27. Erratum to: Furanyl-Chalcones as antimalarial agent: synthesis, in vitro study, DFT, and docking analysis of PfDHFR inhibition
https://doi.org/10.1515/pac-2025-0586
Keywords for this article
Electronic structures; quantum chemistry; quantum science and technology

Articles in the same Issue

  1. Frontmatter
  2. IUPAC Recommendations
  3. Experimental methods and data evaluation procedures for the determination of radical copolymerization reactivity ratios from composition data (IUPAC Recommendations 2025)
  4. IUPAC Technical Reports
  5. Kinetic parameters for thermal decomposition of commercially available dialkyldiazenes (IUPAC Technical Report)
  6. FAIRSpec-ready spectroscopic data collections – advice for researchers, authors, and data managers (IUPAC Technical Report)
  7. Review Articles
  8. Are the Lennard-Jones potential parameters endowed with transferability? Lessons learnt from noble gases
  9. Quantum mechanics and human dynamics
  10. Quantum chemistry and large systems – a personal perspective
  11. The organic chemist and the quantum through the prism of R. B. Woodward
  12. Relativistic quantum theory for atomic and molecular response properties
  13. A chemical perspective of the 100 years of quantum mechanics
  14. Methylene: a turning point in the history of quantum chemistry and an enduring paradigm
  15. Quantum chemistry – from the first steps to linear-scaling electronic structure methods
  16. Nonadiabatic molecular dynamics on quantum computers: challenges and opportunities
  17. Research Articles
  18. Alzheimer’s disease – because β-amyloid cannot distinguish neurons from bacteria: an in silico simulation study
  19. Molecular electrostatic potential as a guide to intermolecular interactions: challenge of nucleophilic interaction sites
  20. Photophysical properties of functionalized terphenyls and implications to photoredox catalysis
  21. Combining molecular fragmentation and machine learning for accurate prediction of adiabatic ionization potentials
  22. Thermodynamic and kinetic insights into B10H14 and B10H14 2−
  23. Quantum origin of atoms and molecules – role of electron dynamics and energy degeneracy in atomic reactivity and chemical bonding
  24. Clifford Gaussians as Atomic Orbitals for periodic systems: one and two electrons in a Clifford Torus
  25. First-principles modeling of structural and RedOx processes in high-voltage Mn-based cathodes for sodium-ion batteries
  26. Erratum
  27. Erratum to: Furanyl-Chalcones as antimalarial agent: synthesis, in vitro study, DFT, and docking analysis of PfDHFR inhibition
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