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THEORETICAL AND COMPUTATIONAL STUDIES OF
INTERSTELLAR C2nH and SiC2m+1H
Ryan Fortenberry
T. Daniel Crawford, Advisor
Virginia Tech
Abstract between theory and experiment, with quantum me-
chanical models providing accurate estimates of rota-
This work focuses on computation of simulated spectra tional constants for candidate species, laboratory ex-
the C2n H family (n = {1, 2, . . .}) of carbon chain rad- periments searching for the corresponding fingerprints
ical molecules and their silicon analogues, SiC2m+1 H within high-resolution microwave spectra, and finally
(m = {0, n}), that are potential carriers of the Diffuse comparing these to radioastronomical data. Unfortu-
Interstellar Bands (DIBs). High-level quantum chem- nately, a similar approach is problematic in the case
ical computations for the ground- and excited-states of the DIBs in part because, while modern quantum
of these radicals indicate agreement with experimen- chemical methods can provide exceptionally accurate
tal data that the ground-state of C2 H is 2 Σ+ while predictions of molecular structure, electronic spec-
that of C6 H, SiCH, SiC3 H, and SiC5 H is 2 Π. On the tra are significantly more challenging than rotational
other hand, most of the theoretical models employed in spectra due to the greater sensitivity of the former to
this work indicate that the ground state of C4 H is 2 Π, the accuracy of the model, vibronic coupling, etc.
in disagreement with previous spectroscopic interpre-
tations. Simulations of the electronic spectrum of the The purpose of this work is the development and
short chains considered here exhibit a strong π → π ∗ application of high-accuracy theoretical methods for
transition that is too high in energy for these chains to simulating the electronic spectra of radical species rel-
be carriers of any of the DIBs, though longer chains evant to interstellar chemistry and the DIBs. We focus
remain viable candidates. on two families of linear-chain radicals, viz. C2n H (n
= {1, 2, . . .}) and the silicon analogues, SiC2m+1 H (m
= {0, n}), whose complicated electronic spectra offer
Introduction a stringent test of our new models. The theoretical
and computational details and corresponding results
Interstellar chemistry1 is the study of chemical pro- are provided in the following sections.
cesses taking place in the regions around and between
stars — known as the interstellar medium (ISM). It
focuses on the identification of the molecules present
in this harsh environment and how they react with ra- Theoretical Methods
diation and each other.2 Since recreating the vacuum
of space in a terrestrial laboratory is a Herculean task Modern quantum chemistry provides a veritable al-
at best, the experimental analyses of such processes phabet soup of computational models for studying
is problematic.3, 4 Theory and computation, however, the chemistry of molecules in the ISM. One of the
are uniquely suited to help answer the difficult ques- most promising is coupled cluster theory, sometimes
tions posed by interstellar chemistry.5 referred to as the “gold standard” of quantum chem-
One of the longest standing mysteries of interstel- istry for its unprecedented accuracy as compared to
lar chemistry and astronomical spectroscopy is that of experiment for properties such as molecular struc-
the diffuse interstellar bands (DIBs),1, 2 a set of spec- ture, theromchemistry, vibrational spectra, nuclear-
tral lines found from the UV to near-IR seen along magnetic-resonance chemical shieldings, etc. The cou-
multiple sightlines in the ISM.1, 2, 5 To date, not a pled cluster wavefunction is constructed based on an
single such line or peak has been positively linked exponential expansion of electronic configurations (de-
to a specific molecular or atomic carrier.1, 5 Histor- terminantal functions),
ically, the most successful approaches used to identify
molecules in the ISM have involved close collaboration ΨCC = eT̂ Φ0 (1)
Fortenberry 1where T̂ is a so-called “cluster operator” that gener- wave functions. For example, for a transition from
ates excited configurations from the reference config- the ground-state (labelled “0”) to an excited state (la-
uration, Φ0 (often taken from an independent-particle belled “n”) we have:13
model such as the venerable Hartree-Fock theory).6
The cluster operator must typically be truncated at a f0n ∝ µ|Ψn i|2
|hΨ0 |~
selected level of excitation for practical calculations, = hΨ0 |~
µ|Ψn i · hΨn |~
µ|Ψ0 i
e.g. the coupled singles and doubles (CCSD) model,7, 8 = hΦ0 |(1 + Λ)e−T µ
~ eT Rn |Φ0 i ·
which includes only those configurations that differ-
ence from the reference by excitation of one or two hΦ0 |Ln e−T µ
~ eT |Φ0 i (4)
electrons, at most, or the CCSD(T) model in which where Ψn and Ψm are the wavefunctions of the desired
CCSD is augmented by a perturbational approxima- states m and n where n is a more highly excited state
tion of the effects of triple excitations.9 The two key than m, Λ is a set of Lagrangian multipliers, and µ ~ is
advantages of the exponential form of the wave func-
the dipole moment operator. The energy of the tran-
tion are its rapid convergence towards the exact wave
sition and the oscillator strength can then be used
function with the level of truncation of T̂ and that to generate simulated spectra of interstellar species.
it yields energies and properties that have the cor- These simulated spectra may subsequently be com-
rect linear scaling with the number of electrons (un- pared with experimental results taken from the ISM.
like many other methods such as configuration inter-
EOM-CCSD already represents a substantially
action).6–8 On the other hand, the primary disadvan-
time-consuming computation, but improvement of the
tage to coupled cluster methods is their computational
simulation of excited states beyond this level is desir-
expense (CPU time, memory and disk storage, etc.).
able. One method that has much promise for this
For example, the CCSD(T) method scales as O(N 7 ),
purpose is the CC3 approach.14, 15 CC3 is an ap-
where N is a measure of the size of the molecular
proximation to CCSDT like CCSD(T), but involves
system. This steep polynomial scaling implies that
contributions from higher orders of perturbation the-
doubling the size of the molecule requires a 128-fold
ory.14, 16, 17 Hence, one can reasonably expect that
increase in the cost of the computation. While efforts
CC3 would give more accurate excitation energies
are underway to overcome this so-called “polynomial
and excited-state properties than EOM-CCSD. How-
scaling wall”,10, 11 coupled cluster methods are cur-
ever, its use for open-shell molecules has been limited.
rently limited to relatively small molecules, containing
Smith, King, and Crawford15 reported the first imple-
10-20 non-hydrogen atoms at most.
mentation of CC3 for computing excitation energies
In pursuit of the sources of the DIBs, accurate quan- in radicals several years ago, but thus far the method
tum chemical models of electronic spectra are essen- has not been extended to properties such as oscillator
tial. Furthermore, such models must be sufficiently strengths [cf. Eq. (4)], which are necessary for gener-
robust and flexible to provide accurate spectral sim- ating simulated spectra for comparison with results of
ulations for closed- and open-shell molecules, both potential interstellar species taken from the ISM.
of which have been observed in the ISM.1, 2, 12 The
coupled cluster approach to describing electronic ex-
citation computations is called equation of motion Interstellar Chemistry
coupled cluster, EOM-CC.13 By describing the ex-
cited state wavefunction as a parameterization of the DIBs
ground state using Rx and Lx vectors, an eigenvalue
While looking for interstellar spectra from the ISM,
equation can be used to compute the energy of some
the highly structured DIBs absorption lines were first
excited state x (Ex ) in the following form:
noticed in pieces by Heger in the 1920s18 and later by
19, 20
H̄Rx |Φ0 i = Ex Rx |Φ0 i (2) Merrill in the 1930s. Observations over the years
have found bands from the UV to near-IR.12, 21, 22 Var-
hΦ0 |Lx H̄ = hΦ0 |Lx Ex , (3) ious theories as to what the carriers of the DIBs are,
from atoms to dust grains, have been proposed, but
where H̄ = exp(−T̂ )H exp(T̂ ). The left- and right- it is now accepted that these lines are caused by some
hand eigenvalue problems above are distinct due to molecular carrier.2
the use of the non-Hermitian nature of H̄.6, 13
Many types of molecules have been proposed as
The computation of transition strengths, which are the carriers of the DIBs. Although H2 is the most
necessary for computing the corresponding oscillator abundant species in the ISM, it has been ruled out
strengths and thus spectral peak heights, takes into as a carrier of the DIBs because it cannot sustain
consideration the differing left- and right-hand CC excitations at the appropriate energies.1, 24–28 Heav-
Fortenberry 2well.47 It was, in fact, computed with spin-restricted
CCSD(T) to be 0.774 Debye.44 For the longer struc-
tures, however, the extension of the chain also yields
more resonance structures that can stabilize the 2 Π
state relative to 2 Σ+ . For C6 H and longer chains, this
stabilization is sufficiently strong as to reverse the or-
dering of these competing terms, producing 2 Π ground
states.41, 46, 47
Figure 1: The DIBs23 Maier and cowork-
ers4, 48, 49 expanded
upon their earlier work
ier diatomics also may be excluded from the list of with C6 H43 and exam-
candidates because their lowest-lying transitions are ined C8 H and C10 H,
at best in the vacuum UV, and thus cannot serve where the latter has
as carriers for longer-wavelength DIBs at 4428-4430Å not yet been found in
and 5780-6614Å.5, 29, 30 Other proposed carriers in- the ISM. They found
clude fullerenes,31 polycyclic aromatic hydrocarbons the same trends in
(PAHs),32 and linear chain molecules29 which include that these two also
carbon-only containing chains,33 single hydrogen con- Figure 2: Structures of C4 H have a 2 Π ground
taining carbon chains,34 cyanopolyactelyne chains,30 state and that their
and the resulting cations and anions of all three.30 HOMO-LUMO transition corresponds to an intense
The work presented here focuses on the C2n H family π → π ∗ band. This has been corroborated with other
of carbon chain radicals and their silicon analogues, experimental data30, 50 and theory.3, 29, 41, 44, 45, 47
SiC2m+1 H where n = {1, 2, . . .} and m = {0, n}.
For C4 H the question is whether or not the reso-
nance between the two 2 Π states, as depicted in Fig. 2,
The C2n H Family lowers the ground state energy below the 2 Σ+ . Dis-
muke, Graham, and Weltner40 originally stated that
The C2n H family of interstellar monoradicals has been
the ground state of C4 H is 2 Σ+ . Further laboratory
selected for this study because of the nexus of a need
experiments also seemed to confirm a 2 Σ+ ground
for theoretical data on their excited states and for de-
state.52–55 However, subsequent higher-level theoreti-
velopment of the methods necessary to do such com-
cal treatments were inconclusive.
putations. As would be expected, the first of this fam-
ily to be discovered in the ISM was C2 H in 1974.35 In 1992, Pauzat and co-workers56 used spin-
C4 H followed in 1978;36 C6 H was 1986;37 and C8 H unrestricted SCF methods to conclude that the first
2 + 2
was 1996.38 All were found using radio astronomy Σ is lower in energy than than the first Π state
techniques. based on weak support of rotational constants. Other
Laboratory experiments examining these molecules studies, as seen in Table 1, in 1992, 1994, 1995, and
go back to around the same time as the first ones were 2001 (when the Giotto spacecraft’s Ion-Neutral Mass
discovered in the ISM. Dismuke, Graham, and Welt- Spectrometer, INMS, found it in the coma of Halley’s
57
ner studied both C2 H39 and C4 H40 and found that comet ) using various SCF, CI, and coupled cluster
their ground states were both 2 Σ+ . It was therefore methods with various basis sets have given mixed re-
assumed that all the C2n H’s had 2 Σ+ ground states. sults as to the ground state of C4 H.
However, when C6 H was discovered37 in the ISM Even so, it is known that this family of molecules
and quantum chemical techniques were employed,41 it is characterized by a single strong π → π ∗ transition
was determined that the ground state of C6 H was in that dominates its UV/Vis spectrum,4, 49, 50 including
fact 2 Π based on the measured microwave transitions. C4 H52 whose transition is not as strong as the longer
Subsequent studies42, 43 arrived at this same conclu- chains. It is also now believed that the strong DIB
sion in the laboratory, where C6 H was also found to readings around 5780-6614Å may be caused by one
have a π → π ∗ HOMO-LUMO transition. or more π → π ∗ transitions, and many of the other
2 +
It logically follows that C2 H has a Σ ground DIBs around the same range are also believed to be
state3, 29, 44–46 if one first considers the precursor, of this same type of transition due to the observed
acetylene, H-C≡C-H. Removal of one hydrogen would laboratory strength of π → π ∗ transitions of potential
leave a singly occupied σ-type orbital. Also, the lack carriers.4, 43 Therefore, it remains possible that the
of a dipole moment in H-C≡C-H would lead one to longer chains of the C2n H family of molecules could
think that the dipole for C2 H would be small, as still be carriers of the DIBs.
Fortenberry 3Table 1: Reported Energy Differences in kcal/mol between 2 Σ+ and 2 Π States of C4 H
Author Method Basis Set Energy Difference Ground State
Kiefer & co.47 CISD TZ2P 1.30 2
Π
Kiefer & co.47 MR-CISD TZ2P 1.20 2
Σ+
Kolbuszewski45 MRCI (+Q) TZ2P 0.23 2 +
Σ
Natterer & co.46 MRCI ANO[432] 2.54 2 +
Σ
Natterer & co.46 MRCI(+Q) ANO[432] 0.29 2
Π
Natterer & co.46 ACPF ANO[432] 0.53 2
Π
Natterer & co.46 CCSD(T) ANO[4321] 3.25 2
Π
Sobolewski & co.3 CASSCF 4-31G 14.43 2 +
Σ
Sobolewski & co.3 CASPT2 DZVP 0.44 2
Π
Woon44 RCCSD(T) cc-pVDZ 0.75 2
Π
Woon44 RCCSD(T) cc-pVTZ 0.15 2
Π
Woon44 RCCSD(T) cc-pVQZ 0.08 2 +
Σ
Woon44 RCCSD(T) est. CBS 0.21 2 +
Σ
Graf & co.51 MCSCF est. CBSa 9.71 2 +
Σ
Graf & co.51 CASPT2 est. CBSa 0.69 2 +
Σ
51
Graf & co. MRCI(+Q) est. CBSa 0.80 2 +
Σ
a
see source for cc-pVDZ, cc-pVTZ, and cc-pVQZ data.
SiC2m+1 H ods, coupled cluster theory has many advantages when
compared to other methods including size-extensivity,
Silicon containing molecules were first found in the faster convergence to the exact wave function, etc.6 It
ISM in 1971 when a team including Penzias and Wil- also can be approximated at its various levels in order
son (of Cosmic Microwave Background Radiation dis- to find a compromise between accuracy and compu-
covery fame) detected silicon monoxide, SiO.58 To tational expense. Hence, the starting point for this
date, seven other interstellar molecules have been de- study builds upon the previous research of determin-
tected which contain a single silicon atom.12 The ing the ground state of C4 H and seeing how these same
largest is the linear chain radical SiC4 .59 methods match with other accepted conclusions about
Thaddeus and coworkers12 have hypothesized that the entire C2n H family.
silicon-carbon chain radicals may exist in the ISM,
based on structural similarities to the C2n H family. Structural optimizations of each radical were car-
In fact, this group has synthesized and taken Fourier ried out at the CCSD(T) level of theory (simi-
transform microwave (FTM) readings of SiC5 H in the lar to the work by Woon44 ), in conjunction with
laboratory60 and proposed cavity ring-down (CRD) a variety of open-shell reference wave functions
spectral studies of SiC3 H and SiC5 H in preparation based on Hartree-Fock determinants, including spin-
for their eventual astronomical identification. Some unrestricted (UHF),61 spin-restricted (ROHF),62 and
or all of the SiC2m+1 H family of molecules are also quasi-restricted (QRHF)63 formulations. In addition,
candidates for carriers of the DIBs, but more experi- we have carried out specialized equation-of-motion
mental and theoretical study needs to be done on this coupled cluster ground-state computations for ionized
class of molecules. states (EOMIP-CC),64 in which the molecular anions
C2n H− and SiC2m+1 H− provide the initial coupled
cluster wave functions. Subsequent excited-state com-
putations were carried out using the EOM-CCSD and
Current Research CC3 levels of theory with UHF and ROHF reference
With all of the different sources reporting different determinants. All coupled 65 cluster computations were
ground states for C4 H, Neumark and coworkers re-50 carried out with the PSI3 and ACESII66 quantum
ported that the 2 Σ+ and 2 Π states are nearly degen- chemical program packages.
erate. There need to be more accurate methods em- For comparison to the results from high-level cou-
ployed to help resolve this problem in order to more pled cluster models, excitation energy computations
accurately predict the ground state. This is the start- at the configuration interaction singles (CIS)67 and
ing point of the research presented here. time-dependent density functional theory (TD-DFT)
As mentioned earlier, of the high accuracy meth- – the latter based on the Becke three-parameter ex-
Fortenberry 4change functional68 with the Lee-Yang-Parr correla- HOMO-LUMO excitation. In SiC5 H, the strongest
tion functional (B3LYP)69 – levels were carried out transition still exhibits π → π ∗ character, but is dom-
using the Gaussian03 package.70 Dunning’s series inated by a HOMO-2/LUMO excitation. These can
of correlation-consistent basis sets were used at the be seen in Table 3. As an example, C4 H has other
double- and triple-zeta levels71 for structural opti- transitions computed below 278.0 nm for UHF-EOM-
mizations, and the corresponding sets augmented with CCSD/aug-cc-pVDZ, but these involved significantly
diffuse functions72 were used for excited-state compu- smaller oscillator strengths with a maximum of 0.0910
tations. at 146.1 nm. In addition, only minimal basis-set ef-
fects were observed in the transition energies, with the
Ground States largest shift of 7.9 nm between aug-cc-pVDZ and aug-
cc-pVTZ found at the TD-B3LYP level of theory for
We find that the ground states of C2 H and C6 H are the second excited state of SiC3 H.
2 +
Σ and 2 Π, respectively, in agreement with previ-
ous theoretical and experimental studies.4, 30, 48–50 At
the UHF-CCSD(T)/cc-pVTZ level of theory, the 2 Σ+ Table 3: Strongest UHF-EOM-CCSD/aug-cc-pVDZ
ground state of C2 H lies 9.9 kcal/mol (adiabatically) Transitions
below the lowest 2 Π excited state, while for C6 H the System Wavelength in nm Intensity
2
Π state is lower by 15.2 kcal/mol (ignoring zero-point C2 H 120.5 0.3121
vibrational corrections). For C4 H, however, the two C4 H 164.2 1.1394
states are nearly degenerate, as illustrated in Table 2. C6 H 182.5 1.4838
Using UHF and QRHF reference wave functions, the SiC3 H 227.9 1.0720
2
Π state lies lower by ca. 1 kcal/mol, while the ROHF SiC5 H 246.0 1.5055
reference wave function places the 2 Σ+ state lower by
less than 0.5 kcal/mol. The cc-pVTZ computations
are seen in Table 2.
1.4
Table 2: cc-pVTZ Computed ∆E in kcal/mol between
states of C4 H 1.2
Method ∆E Ground State
2
CCSD-UHF 0.50 Π 1.0
2
CCSD(T)-UHF 0.75 Π
2 +
CCSD(T)-ROHF 0.44 Σ
2
CCSD(T)-QRHF 1.44 Π 0.8
Intensity
2
CCSD-EOMIP 0.94 Π
0.6
For the silicon-containing radical chains, all refer- TD-B3LYP aug-cc-pVDZ
ence wave functions and levels of theory agree qual- TD-B3LYP aug-cc-pVTZ
CIS aug-cc-pVDZ
itatively that the ground state is 2 Π state lies lower 0.4 CIS aug-cc-pVTZ
than the 2 Π state. For SiCH, SiC3 H, and SiC5 H at EOM-CCSD aug-cc-pVDZ
the UHF-CCSD(T)/cc-pVTZ level of theory, the 2 Π
ground state lies (adiabatically) 35.1, 44.2, and 46.4 0.2
kcal/mol, respectively, below the the lowest 2 Σ+ ex-
cited state. The reason for the increased relative sta-
0.0
bility of the 2 Π state is that the hybridization of the
carbon next to the silicon is more favorable in this 120 140 160 180 200 220 240 260 280
Absorption Wavelength (nm)
state than the 2 Σ+ state as a result of the valence s
orbital of the silicon interacting with a p orbital of the
carbon.73
Figure 3: Simulated Spectrum of C4 H
Excited States
C2 H requires the shortest UV wavelengths for elec-
All three levels of theory (B3LYP, CIS, and EOM- tronic excitation, and it has the smallest oscillator
CCSD) agree that the strongest electronic transition stregth as seen in Table 3. On the other hand,
in five of the six molecules corresponds to a π → π ∗ SiC5 H requires the longest UV wavelength and has
Fortenberry 5the strongest oscillator strength. The simulated spec- Conclusions and Future Directions
trum of C4 H is seen in Fig.(3) and is representative of
this entire set of molecules. For each molecule (C2 H, C4 H, C6 H, SiC3 H, and
The quantitative differences in the methods within SiC5 H), a single strong HOMO-LUMO transition was
the respective systems are significant as seen in Table found, but it is too far into the UV for any of these
4. In order for theory to give any help of assignment of transitions to allow for any of the molecules to be car-
the DIBs, the methods used must be convergent as the riers of the DIBs. However, for the longer chains re-
accuracy of the method increases and also arbitrarily main viable candidates to be carriers of the weaker and
close to laboratory values. Only then can they be used more energetic DIBs, given that the HOMO-LUMO
to address spectral readings taken from the ISM. transition energy decreases as the chain length grows.
However, the current limits of quantum chemical tech-
niques need to be extended in order to give more quan-
titative — and thus more conclusive — answers.
Table 4: Comparison of Theoretical Resultsa for the
The research for this study will not only continue
Strongest Transitions (in nm)
to examine novel molecules with interstellar signifi-
System TD-B3LYP CIS EOM cance, but will also extend existing tools. The CC3
C2 H 131.5 117.9 120.5 approach for excited states15 is currently one of the
C4 H 176.2 152.8 164.2 most advanced electron correlation theories, but has
C6 H 203.4 166.9 182.5 not yet been extended to the computation of oscil-
SiC3 H 242.8 180.5 227.9 lator strengths. High accuracy simulation of spectra
SiC5 H 264.0 187.1 246.0 allows for better comparison to experiment, whether it
a is from the laboratory or the ISM. Developing and im-
Computed using the aug-cc-pVDZ basis set.
plementing CC3-level oscillator strengths will involve
equations and code very much like that from EOM-
Experimental studies do report finding strong tran-
CCSD and Eqs.(2,3,4) but with a correspondingly
sitions for some of the carbon chain radicals examined
higher degree of complexity. We will make these new
here,39, 40, 43 but the comparison between theory and
methods and made available within the PSI3 open-
experiment is far from adequate. C2 H, for example,
source quantum chemistry package.
has been studied over the 160-350 nm range,39, 74 far
The higher level of accuracy promised by the
lower in energy than the 120.5 nm strong transition
CC3 model will also allow for computations of other
predicted by UHF-EOM-CCSD/aug-cc-pVDZ. How-
open-shell interstellar molecules like cyanopolyacte-
ever, another computational study of C2 H by Shih,
lyne chains,30 silico-cyanopolyactelyne chains (like
Peyerimhoff, and Buenker75 reported a strong transi-
SiCN), or some planetary species (like SO2 or S2 O)
tion at 129.0 nm, in poor agreement with results from
whose spectra are needed to describe the surface chem-
the UV photolysis experiments mentioned previously.
istry of Io or Titan, for example.78 We remain opti-
A subsequent computational study by Koures and mistic that such computational methods may one day
Harding76 using CI methods was able to resolve some be able to predict the carriers of the DIBs and help
of the discrepancies between theory and experiment. to solve one of the longest standing problems in astro-
For example, a measured peak at 194.6 nm74 was ac- nomical spectroscopy.1, 2, 5, 30
tually refuted to be a transition of C2 H. In addition, a
measured peak at 139.3 nm77 was linked to the third
2
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