_{1}

^{*}

Recent experimental evidence suggests again the existence of the metastable methane anion in plasma swarms. In order to test the reliability of the complete basis set (CBS) extrapolation scheme with augmented correlation-consistent basis sets for anionic molecules, we study the evolution of the electron affinity of methane with benchmark ab initio calculations with aug-cc basis sets up to aug-cc-pV6Z + diffuse. Geometry optimizations and vibrational analysis were done at the MP2 level. The electron affinity (EA) was calculated at the MP2 and CCSD(T) levels with and without frozen core and including the extrapolations to the CBS limit. Using the aug-cc-pVnZ basis sets it is found that two non-decreasing CCSD(T) CBS limits exist for the EA (0.29 and 0.53 eV) obtained with the n = 3, 4, 5 and n = 4, 5, 6 series, respectively. A new scheme is proposed which can be generalized for very accurate quantum chemical description of molecular anions: the standard aug-cc-pVnZ basis sets can be supplemented with extra-diffuse orbitals using a simple even-tempered scheme. This yields a reliable CBS extrapolation method to develop a (discrete approximation of a) continuum anionic state near ionization, viz., one that closely matches the energy of the corresponding neutral state. These results show that CH4 has no stable anions of 2A1 symmetry, implying that plasma swarms with anionic methane consist of metastable rather than stable methane anions.

It is known that methane is a very stable molecule. It is characterized by five closed shells and its electronic configuration resembles than of the neon atom. Resembling a noble gas atom, it is natural to expect that the methane anion is unstable with respect to the neutral molecule. However, clear experimental evidence of the existence of the methane anion as a metastable species has been provided by several groups [_{4} took place over the range of electron energies 8 - 13 eV, with ^{−17} V cm^{2}). Electron-impact ionization processes have been thoroughly studied for gaseous methane over a very wide range of the density-normalized electric field strength up to 50,000 Td, also suggesting the presence of the methane anion [_{4}/N_{2} mixtures at high pressures (100 - 600 Torr) reveal again the presence of negative methane ions in the avalanche, formed by a three-body collision process [

From the experimental side work is currently in progress through a hybrid swarm/mass spectrometry technique to directly measure the electron affinity of metastable methane [^{−2} eV). However, from the quantum chemical point of view, the fact that anionic methane is metastable (i.e. that the charged _{4} plus a free electron), has a strict logical implication: for a given level of theory, the energy of the anion should approach that of methane as the quality of the atomic basis sets improves. This stems from the fact that larger and more diffuse basis sets will allow the extra electron to move away from the neutral molecule thus lowering the total energy of the (CH_{4}, e-) pair. In this direction we note that that previous electronic structure studies have placed the EA of methane between 6.1 and 1.2 eV [

The aug-cc-pVnZ basis sets were developed applying stringent conditions on the atoms, but their combination yields molecular basis sets that, in general, provide similar asymptotic behavior of the most important molecular quantities to their CBS limit values, such as energy, equilibrium geometries and harmonic vibrational frequencies [

From the work of Jursic [

With present day computational resources, electronic structure calculations at the Coupled Cluster with singles, doubles and perturbative triple excitations -CCSD(T)- level can be done in a few days or weeks even with the largest aug-cc basis sets in parallel computers with large core memories. In this work we shall use the aug-cc-pVnZ basis sets with n = 2, 3, 4, 5, 6 and 6 + diffuse [

^{−4} angstroms. MP2 vibrational analysis of the optimized structures was done for each basis set up to aug-cc-pV5Z. The CCSD(T) (and the unrestricted counterpart for the anion) single-point calculations were performed using the MP2 optimized geometries obtained with and without the frozen core for each basis set, thus the CCSD(T) calculations were also done with and without the frozen core. To give an idea of the computational resources needed, the (U)MP2(full) optimizations and the (U)CCSD(T)-full single point calculations with the aug-cc-pV6Z + diffuse basis sets required 40(17 + 23) and 133(56 + 77) CPU days, respectively. The Gaussian03 parallel code [

Basis name | No. of contracted functions | Atomic decomposition |
---|---|---|

AVDZ (aug-cc-pVDZ) | 59 | C: 4s3p2d H: 3s2p |

AVTZ (aug-cc-pVTZ) | 138 | C: 5s4p3d2f H: 4s3p2d |

AVQZ (aug-cc-pVQZ) | 264 | C: 6s5p4d3f2g H: 5s4p3d2f |

AV5Z (aug-cc-pV5Z) | 447 | C: 7s6p5d4f3g2h H: 6s5p4d3f2g |

AV6Z (aug-cc-pV6Z) | 697 | C: 8s7p6d5f4g3h2i H: 7s6p5d4f3g2h |

AV6Z + diff (AV6Z + diffuse) | 890 | C: 9s8p7d6f5g4h3i H: 8s7p6d5f4g3h |

The methane anion has a ^{2}A_{1} ground state, the extra electron occupying a molecular orbital which is basically derived from the 3 s virtual orbital of carbon, with small equal contributions from all four hydrogen 2 s atomic orbitals. Thus the anion can be described as arising from the first Rydberg state on the molecule.

As mentioned in the previous section, in order to calculate the EA with the CCSD and CCSD(T) methods, we have used the MP2 optimized geometries of methane and its anion corresponding to a given basis set with and without the frozen core orbital. Both species are tetrahedral (T_{d} point group) so that the C-H distances completely describe their geometries. The evolution of these distances with basis set quality is presented in

The vibrational spectra of methane and its anion have been obtained at the MP2(full) level for the aug-cc-pVnZ (n = 2, 3, 4, 5) basis sets. We found that the vibrational frequencies of both species are already converged to a few (<5) wavenumbers at the MP2/AV5Z level, thus ^{−1}) lower than those of the neutral molecule, even with the largest aug-cc basis sets. A discussion on the reliability of these results is given in the following sections.

MP2 | CH_{4} | ^{ } |
---|---|---|

AVDZ | 1.0986, 1.0975 | 1.1021, 1.1011 |

AVTZ | 1.0861, 1.0841 | 1.0888, 1.0867 |

AVQZ | 1.0848, 1.0823 | 1.0863, 1.0840 |

AV5Z | 1.0846, 1.0807 | 1.0862, 1.0831^{b} |

AV6Z | 1.0845, 1.0807 | 1.0859, 1.0828^{c} |

AV6Z+diff | 1.0845, 1.0807 | 1.0859, 1.0828^{d} |

CBS | 1.0843, 1.0807 | 1.0858, 1.0827 |

MP2/6-311++G(2d, 2p) MP2/6-311++G(3df, 3pd) | 1.0870^{a} 1.0850^{a} | 1.0910^{a} 1.0880^{a} |

^{a}Previous values from [^{b}After 48 steps: Max displacement = 0.000306 a.u., RMS displacement = 0.00016 a.u. ^{c}After 27 steps: Max displacement = 0.000287 a.u., RMS displacement = 0.00017 a.u. ^{d}After 64 steps: Max displacement = 0.000313 a.u., RMS displacement = 0.00014 a.u.

Mode Symmetry | CH_{4} | |
---|---|---|

T_{2} | 1349, 1349, 1349, 3212, 3212, 3212 | 1336, 1336, 1336, 3183, 3183, 3183 |

E | 1585, 1585 | 1571, 1571 |

A_{1} | 3074 | 3036 |

sets used here. For comparison we have also included the previous estimations obtained at different levels of theory.

A careful analysis of this table reveals interesting facts. Jursic [

We note that the aug-cc-pV6Z + diffuse basis sets yield the same optimized MP2 distances and the same electron affinity values as the ones obtained with the aug-cc-pV6Z basis sets, at all levels of theory.

Following the suggestion of Peterson et al. [

CH_{4} (^{1}A_{1}) | ^{2}A_{1}) | EA | |
---|---|---|---|

AVDZ MP2(FC) MP2(full) CCSD(T)-FC CCSD(T)-full | −40.367726 −40.370821 −40.395779 −40.398714 | −40.336495 −40.339566 −40.365759 −40.368668 | 0.850 0.850 0.817 0.817 |

AVTZ MP2(FC) MP2(full) CCSD(T)-FC CCSD(T)-full | −40.414459 −40.432995 −40.440894 −40.460316 | −40.388297 −40.406794 −40.415825 −40.435189 | 0.712 0.713 0.682 0.683 |

AVQZ MP2(FC) MP2(full) CCSD(T)-FC CCSD(T)-full | −40.427405 −40.456603 −40.451691 −40.482485 | −40.404025 −40.433197 −40.429318 −40.460039 | 0.636 0.637 0.608 0.610 |

AV5Z MP2(FC) MP2(full) CCSD(T)-FC CCSD(T)-full | −40.431903 −40.467144 −40.454875 −40.492067 | −40.416125 −40.451340 −40.439863 −40.477013 | 0.429 0.430 0.408 0.409 |

AV6Z MP2(FC) MP2(full) CCSD(T)-FC CCSD(T)-full | −40.433700 −40.473910 −40.455948 −40.498371 | −40.414988 −40.455160 −40.438082 −40.480470 | 0.509 0.510 0.486 0.487 |

AV6Z+diff MP2(FC) MP2(full) CCSD(T)-FC CCSD(T)-full^{a } | −40.433700 −40.473913 −40.455950 −40.498377 | −40.414995 −40.455164 −40.438086 −40.480474 | 0.509 0.510 0.486 0.487 |

Previous values^{b} MP2/ 6-31G(d, p) 6-311++G(2d, 2p) 6-311++G(3df, 3pd) MP4/ 6-31G(d, p) 6-311G(2df, p) 6-311++G(2d, 2p) QCISD(T)/6-311G(d, p) QCISD(T)/6-31+G(d) MP2/CBS-B3 G1, G2, G2MP2 | −40.318083 −40.334783 −40.366577 −40.388655 −40.405682 | −40.105892 −40.298017 −40.332108 −40.166351 −40.369066 | 6.1 1.0 0.9 6.0 3.7 1.0 3.7 2.3 1.2 1.2, 1.2, 1.2 |

^{a}These CCSD(T) calculations required 133 CPU days and 96 Gb RAM; ^{b}From [

termine A, B and the asymptotic complete basis set limit value of F, F(CBS). This is usually achieved using the properties obtained with n = 2, 3 and 4. In this case, since we have the larger sequence n = 2, 3, 4, 5, 6, 6 + diffuse, we have chosen to explore the asymptotic EA with two sequences (n = 3, 4, 5) and (n = 4, 5, 6), leading to two estimations labeled EA(CBS-345) and EA(CBS-456), respectively.

Focusing on the results obtained with the most accurate correlation treatment, the CCSD(T)-full method, the table shows that in the first case we obtain 0.29 eV while, quite surprisingly, the second extrapolation yields a

Level of theory | CBS-345 | CBS-456 |
---|---|---|

MP2(FC) | 0.306 | 0.555 |

MP2-full | 0.307 | 0.556 |

CCSD(T)-FC | 0.289 | 0.531 |

CCSD(T)-full | 0.290 | 0.532 |

larger value of 0.53 eV. Although the 0.24 eV difference between these two asymptotic values is energetically small, the percent relative error between them is very large,

We know that in this case the physical problem at hand has to do with the way the extra electron interacts with the neutral neon-like core of methane, which can be translated into what is the exact spatial distribution of the HOMO for the anion? In the following section we address this problem by providing extra degrees of freedom to the unpaired electron of the anion.

The appearance of a non-monotonic behavior of the electron affinity with increasing n in the aug-cc-pVnZ series at all levels of theory casts serious doubts on the reliability of these basis set for a truly accurate description of the anion. In particular, knowing that the electron affinity must decrease with basis set quality, it is completely against the core philosophy of the aug-cc-pVnZ basis sets that larger estimations of the EA are obtained with the n = 4, 5, 6 series than with the n = 3, 4, 5 series. Therefore we have decided to add one to four diffuse orbitals on the carbon atom to study the convergence of the energy for methane and for the anion at the CCSD(T)-full level. These diffuse orbitals were chosen with spherical symmetry for two reasons: 1) the HOMO of the anion closely approaches a spherical distribution with increasing basis set quality and, 2) there is only one orbital per added exponent against three or five harmonics for p and d symmetries so that these very large CCSD(T) calculations are faster (recall the O(n^{6}) behavior of the method). The four extra-diffuse s exponents (diff2 = 0.012, diff3 = 0.004, diff4 = 0.0013, diff5 = 0.0004) have been chosen using an even-tempered scheme with α = 1/3, taking as starting point the exponent (diff1 = 0.0354) of the last s orbital of the aug-cc-pV6Z + diffuse carbon basis set. In this way a sequence of aug(m)-aug-cc-pV6Z basis sets can be defined with a new cardinal number m = 7, 8, 9, 10 and 11, where m is the sum of n = 6 plus the number of diffuse s orbitals added to the aug-cc-pV6Z basis set of carbon; therefore, m = 7 corresponds to the published aug-cc-pV6Z + diffuse basis set.

As expected, the CCSD(T) energy of methane shows much smaller changes (ca. 10^{−6} a.u.) from the aug(7)- AV6Z to the aug(11)-AV6Z basis set, while the energy decrease of the anion is three orders of magnitude larger. With the largest basis set the energy of the anion is only 13 meV higher than that of the neutral molecule thus yielding an EA nearly 40 times smaller than that estimated with the aug-cc-pV6Z + diffuse basis set.

Although not strictly zero, using the same three-parameter CBS extrapolation scheme we obtain now −0.003 eV as the limit for the electron affinity of methane with the m = 9, 10, 11 sequence. This is consistent with the expected asymptotic behavior of the EA towards zero with increasing basis set quality. Accordingly, note in Ta- ble 6 that the energy of the HOMO of the anion monotonically decreases towards zero, thus confirming the CH_{4} + e^{−} asymptotic description of the anion. With this knowledge, we have performed a new MP2 optimization of

CH_{4} (^{1}A_{1}) | ^{2}A_{1}) | EA | ||
---|---|---|---|---|

AV6Z + diff1 (m = 7) 890 MO | −40.498377 | −40.480474 | 0.487 | 0.02176 |

AV6Z + diff1 + diff2 (m = 8) 891 MO | −40.498378 | −40.487940 | 0.284 | 0.01247 |

AV6Z + diff1 + diff2 + diff3 (m = 9) 892 MO | −40.498378 | −40.493945 | 0.120 | 0.00510 |

AV6Z + diff1 + diff2 + diff3 + diff4 (m = 10) 893MO | −40.498379 | −40.496832 | 0.042 | 0.00173 |

AV6Z + diff1 + diff2 + diff3 + diff4 + diff5 (m = 11) 894 MO | −40.498380 | −40.497909 | 0.013 | 0.00054 |

the anion with the aug(11)-aug-cc-pV6Z basis set and, as expected, the C-H distance decreases to 1.0815 Å, in closer agreement with that obtained for neutral methane with the aug-cc-pV6Z basis set. However, we note that after 16 optimization steps the Max Displacement = 0.000319 a.u. and RMS displacement = 0.00016 a.u parameters are found but the geometry optimization has not been completely achieved due to small oscillations (<0.001 and <0.0008) of the Max-force and RMS-force parameters; this leads to non-compliance of the convergence criteria thus precluding the calculation of the vibrational spectrum of the anion with the aug(11)-aug-cc- pV6Z basis set. These oscillations are due to an unbalanced growth of the l = 0 diffuse basis set and other higher angular momentum orbitals might be needed to stabilize the geometry optimization procedure with such a large basis set.

Experimental evidence for the existence of anionic methane has accumulated over the last 50 years in plasma swarm experiments and the best previous theoretical extrapolated estimations have placed the electron affinity of methane in the 1.2 - 0.5 eV range. However, the various extrapolation schemes used were not based on the augmented and correlation-consistent pVnZ series of basis sets, which are known to yield the correct asymptotic behavior with basis set quality for a wide variety of molecular systems.

In order to test the reliability of the electron affinity with the widely used complete basis set extrapolation scheme in this complex case, geometry optimizations and vibrational calculations for methane and its anion were performed at the MP2 level with the aug-cc-pVnZ (n = 2, 3, 4, 5, 6 and 6-diffuse) basis sets. Using these optimized geometries we have performed single-point CCSD(T) calculations in order to obtain the electron affinity of methane with each basis set. The MP2 and CCSD(T) calculations were done with and without the frozen core approximation and, as expected, yield practically identical energetic results.

As reported in previous studies, the methane EA systematically decreases with basis set quality at the MP2, CCSD and CCSD(T) levels of theory. Using the aug-cc-pVnZ (n = 3, 4, 5) series, the CBS limit for the electron affinity is 0.29 eV at the CCSD(T) level. However, with the aug-cc-pVnZ (n = 4, 5, 6) series the CBS limit yields an even larger (0.53 eV) electron affinity, thus revealing the inadequacy of the CBS extrapolation scheme with the aug-cc-pVnZ series of basis sets for the anionic molecule. However, when even-tempered diffuse and extra-diffuse orbitals are added to the aug-cc-pV6Z + diffuse basis set this leads to further lowering of the CCSD(T) energy of the anion, so that the new CBS energy limit correctly converges to that of CH_{4} + e^{−}. MP2 optimizations for the anion using these aug(m)-aug-cc-pV6Z correctly leads to a shorter C-H distance, in closer agreement with that estimated as the CBS limit for the neutral molecule. Thus the G1, G2, G2MP2 and CBS extrapolations for the energy and geometry of the methane anion previously reported [

A practical scheme for very accurate quantum chemical descriptions of molecular anions has been proposed here: the standard aug-cc-pVnZ basis sets can be supplemented with extra-diffuse orbitals using a simple even- tempered scheme. With this scheme the EA of methane has been shown to follow the asymptotic behavior towards zero.

From the more general perspective concerning the experimental detection of the methane anion in plasma swarms, these calculations confirm that if true electronic capture occurs, the negatively charged methane molecule will rather swiftly relax towards methane plus a free electron. However, given the experimental evidence [^{−} anionic Rydberg states, and these could briefly exist before the anionic methane decomposition takes place. Further exploration of the right experimental conditions (pressure, density-normalized electric field strength and perhaps even the geometry/dimensions of the Townsend chamber) of the ionic avalanche might lead to relaxation times that could give experimentalists the opportunity to obtain refined data of a yet undiscovered meta-stable anionic structure of mehane in those special conditions. This work yields a reliable CBS extrapolation method to develop a (discrete approximation of a) continuum anionic state near ionization, viz., one that closely matches the energy of the corresponding neutral state.

Finally, since the ground state of the neutral methane is ^{1}A_{1}, the metastable anion (long-lived to be detectable in experiments) is unlikely to be ^{2}A_{1}, as this spatial symmetry is identical with that one of the neutral molecule and one free electron, giving rise to a shape resonance [^{−10} - 10^{−15} s, too soon to be detected. This work shows that CH_{4} has no stable anions of ^{2}A_{1} symmetry, implying that plasma swarms with anionic methane consist of metastable rather than stable anions.

The author is indebted to Prof. Jaime de Urquijo for interesting discussions on methane electron attachment and for pointing out the previous and new experimental results showing the presence of the metastable methane anion in Townsend plasma swarms. Support from CONACYT Sabbatical Program is acknowledged. Financial support from the CONACYT-México Basic Science project No. 130931 is also acknowledged.