Coupled-cluster theory and the
discovery of astronomical molecules:
A pragmatic perspective
Identification of molecules in space
Radio astronomical detection
Very sensitive technique that can allow for
unambiguous detection of molecular species
Searches are usually based on rotational Hamiltonians
based on laboratory data
Sensitivity goes with square of the molecular dipole moment
First radio detection: OH in 1963; NH3, H2O, H2CO before
end of decade (first polyatomics); HCO+ in 1970 (first positive
ion); C6H- in 2006 (first negative ion)
Strength: Observation of several lines at expected frequencies provides excellent
evidence for the existence of particular species in space
Weakness: Inherent bias towards polar molecules. Questions regarding the existence
of fundamental molecules like benzene, C60, etc. will never be answered
by radio astronomy.
Infrared detection
ISO satellite (European Space Agency)
Operating range: 2.5 - 240 µm (42-4000 cm-1)
Allows for the observation of non-polar polyatomic molecules
and condensed phases (ice, for example)
Can (in principle) detect near-IR electronic spectra of radicals
and metastables with low-lying excited states
Some molecules have extremely intense vibrational transitions
A few molecules detected this way: H2CO, HF (recent), ice,
OH, benzene (?), C3, C5
First detection: CO (1979, Kuiper Airborne Observatory Lockheed C141 flying at ca. 12 km); first new molecule:
C2H4 (1983), then SiH4 (1984), C2H2 and CO2(1989),
CH4 (1991), others.
Strength: Observation of several lines at expected frequencies provides excellent
evidence for the existence of particular species in space
Weakness: Atmosphere largely opaque to long-wavelength radiation, satellites required
for good data. Ultimately a less precise (more uncertain) means of molecular
identification than radio astronomy. Identification often based on one (1) line!
The evidence for benzene in the interstellar medium
Optical detection
ISO satellite (European Space Agency)
Operating range: 2.5 - 240 µm (42-4000 cm-1)
Historically the first technique used: CH discovered in space
in 1937. CN and CH+ discovered during WWII. CO and H2 seen
optically in 1970, N2 in 2004.
Can (in principle) detect any molecule; selection rules are not as
restrictive; e.g. H2 and N2 are IR and MW-inactive species.
Some molecules have extremely intense electronic transitions
Strength: Less difficult to obtain data than for IR. Ground-based measurements with CCD
cameras now yield now yield high quality spectra.
Weakness: Laboratory data for complex molecules not easy to acquire, some molecules
do not have intense electronic transitions; dissociative excited states lead to broad features
The general process of identifying new molecules
Obtain precise line
positions and molecular
constants in the laboratory.
Determine or predict potential
candidate line positions
Quality (robustness) of identification
depends on:
PRECISION of observed line positions
AGREEMENT with laboratory lines
Conduct astronomical search
for lines, usually choosing the
strongest lines, or those that
are in an accessible region of
the spectrum.
NUMBER of lines used in identification
What can a quantum chemist do to help?
A different sort of synthetic chemistry
Obtain precise line
positions and molecular
constants in the laboratory.
Determine or predict potential
candidate line positions
C2H2, C4H2
Benzene,Toluene
SiH4, H2S, S vapor
Conduct astronomical search
for lines, usually choosing the
strongest lines, or those that
are in an accessible region of
the spectrum.
N2
C6H6 + 2000 volts
and (unquestionably) many, many, many more: both known and unknown species….
The challenge to quantum chemists:
Guide the laboratory searches for particular compounds
The laboratory data is then used in turn to guide the astronomical search.
Theoretical predictions
Laboratory identification
Astronomical search
Astronomy and astrochemistry:
Chemical physics:
Models of interstellar chemistry
Abundance, isotopic studies
Molecular properties (geometries)
Studies of dynamics (S4), etc.
What ab initio theory can provide to laboratory searches::
Rotational Spectroscopy: Molecular geometries (re) and associated moments of inertia.
Centrifugal distortion constants, dipole moments.
In principle: Observable rotational constants, but not really
important in general for laboratory astrophysics.
``Adequate’’ levels of theory (i.e. CCSD(T)/cc-pVTZ basis set))
Ae, Be, Ce, CD constants
Dipole moment (e)
Ao, Bo, Co, CD constants
Dipole moment (o)
Geometry optimization, harmonic force field
calculation, perhaps larger basis set energy
calculations.
Evaluation of anharmonic (cubic or quartic)
molecular force field, followed by application
of vibrational perturbation theory.




Increases bond lengths
Increased treatment
of correlation
Basis set
improvement
Ie
Be
Decreases bond lengths
Ie
Be
Be =Beexact
FCI
Level of calculation
CCSD(T)
Be < Beexact
MP2
Be  Beexact
CCSD
Be > Beexact
SCF
STO-3G
6-31G*
cc-pVDZ
Basis set
cc-pVTZ
cc-pVQZ
Exact
In general, CCSD(T)/cc-pVTZ overestimates bond lengths a bit:
Becalc < Beexpt
However, experiments do not measure Be, but rather Bv
Usually positive
Molecular vibration effects tend to extend the structure (rg > re)
B0 < Be
Exact
In general, CCSD(T)/cc-pVTZ overestimates bond lengths a bit:
Becalc < Beexpt
However, experiments do not measure Be, but rather Bv
Usually positive
Molecular vibration effects tend to extend the structure (rg > re)
B0 < Be
Exact
CCSD(T)/cc-pVTZ
In general, CCSD(T)/cc-pVTZ overestimates bond lengths a bit:
Becalc < Beexpt
However, experiments do not measure Be, but rather Bv
Usually positive
Molecular vibration effects tend to extend the structure (rg > re)
B0 < Be
Exact
CCSD(T)/cc-pVTZ
An empirical observation: CCSD(T)/cc-pVTZ Be constants usually within
1% of experimental B0 constants, and are in
better agreement than the theoretical B0 values
Serve as a general-purpose guide
To laboratory searches for new molecules
A more or less trivial application of quantum chemistry
An example: Isomers of C5H2
Theory:
An example: Isomers of C5H2
Theory:
``eiffelene’’
However … heroic calculations can act to support important new claims…
McCarthy, Gottlieb, Gupta and Thaddeus Ap. J. 652, L141 (2006).
Laboratory spectra of C4H- and C8Hrecently recorded (Gupta et al., preprint)
First molecular anion
found in space
(carrier of U1377 in IRC+10216)
Vibrational Spectroscopy: With high quality atomic natural orbital basis sets, CCSD(T)
and a good treatment of anharmonicity, it is now possible to
predict fundamental and two-quantum level positions and
intensities relatively accurately for rigid molecules (frequencies
within 10 cm-1, intensities to a few percent). Laboratory spectra
can be assigned by this means, a field that is well-ploughed and
certainly not limited to laboratory astrophysics.
An example: Overtone spectrum of nitric acid from 4000-6000 cm-1
These calculations require evaluation of:
the harmonic, cubic and quartic force fields
the first three derivatives of the dipole moment components
vibrational second-order perturbation theory for levels and intensities
A practical point and an advertisement:
Combined with CCSD(T) and VPT2, the ANO basis sets of Taylor
and Almlöf work extremely well for vibrational level positions.
But, alas…
While nice, it remains a fact that infrared spectroscopy is not used as often as
microwave spectroscopy as an analytical technique to detect new molecules in
the astrochemistry community. Exceptions, however, exist: Maier, McMahon,
others. Lack of precision in laboratory data (relative to MW) makes it less attractive.
Electronic Spectroscopy: Precise prediction of band positions is extraordinarily difficult
(impossible) here. Accuracy of 0.1 eV (800 cm-1) is in fact
rare. Using calculated band positions to guide laboratory
searches is a dubious proposition. Moreover, the number of
observable vibronic features much less than that typical of a
microwave spectrum.
And with regard to the relationship between astrophysics and
quantum chemistry, things are largely inverted:
Laboratory
Molecular
lines
identity
Astronomical
lines
MW
Elec
Electronic Spectroscopy: Precise prediction of band positions is extraordinarily difficult
(impossible) here. Accuracy of 0.1 eV (800 cm-1) is in fact
rare. Using calculated band positions to guide laboratory
searches is a dubious proposition. Moreover, the number of
observable vibronic features much less than that typical of a
microwave spectrum.
And with regard to the relationship between astrophysics and
quantum chemistry, things are largely inverted:
Laboratory
Molecular
lines
identity
Astronomical
lines
MW
Elec
The diffuse interstellar bands (DIBS)
First published report of (two) DIBs in 1921
Found along lines of sight associated with reddened stars
Catalog of Herbig (1975): 39 bands
Catalog of Jenniskens and Desert (1994): >200 bands
Source(s) of DIBS remain(s) unknown in late 2006!
Some ideas from the past:
Molecular carriers first suspected (Struve, Russell, Saha,Swings)
ca. 1937-1944
… but population of polyatomics in typical interstellar clouds
thought to be too small. Led to movement towards…
Solid-state carriers
lots of ideas: impurities in dust grains, solid oxygen, etc.
… but lack of characteristic solid-state absorption features
(notably inhomogeneous broadening) led to movement towards
Molecules again!
An excellent review: G.H. Herbig Ann. Rev. Astrophys. 33, 19 (1995)
Bis-pyridyl magnesium tetrabenzoporphyrin
(in part)
Postulated to account for ALL diffuse interstellar bands when just 39 were known (1972)
Crashed and burned in the 1970’s: Occam’s razor, spectra not reproducible.
Some molecular ideas:
Molecular hydrogen, albeit not simply
Gaps between intermediate state and higher
levels fit many DIBs positions
}}
}
BUT
Model required photon fluxes of ca. 1010 s-1
Flux in typical ISM cloud: ca. 1 year-1
… insufficient by 17 orders of magnitude!
Polycyclic Aromatic Hydrocarbons (PAHs)
Polycyclic Aromatic Hydrocarbons (PAHs)
Flux Density (10-13 W/m2/µ)
The Unidentified Infrared Bands (UIRs)

NGC 7027

 1 .3 %
Orion Bar
Wavelength (µ)
ISO spectra of the Planet. Neb. NGC 7027 and the PDR region at the Orion Bar
Peeters et al., 2003
The unidentified infrared (UIR) features
Carrier of 4429?
4435
4395
Corannulene, C20H10
exp
B
= 2.07(2)D
= 509.8MHz
Lovas et al., 2005
However, there has not been a single unambiguous
detection of any PAH in space to date, despite considerable
publicity.
Nonpolarity is a problem, although two smallest polar
PAH species have not been found
Apart from benzene, there is no evidence of anything
other than three-membered ring compounds in the ISM
No hard evidence supports PAH carriers of DIBs
Something quite remarkable transpired in 1998…
Tulej, Kirkwood, Pachkov and Maier , Ap.J.Lett. 506, 69 (1998)
Something quite remarkable transpired in 1998…
C7- !?
Tulej, Kirkwood, Pachkov and Maier , Ap.J.Lett. 506, 69 (1998)
Since this time, sobered experimentalists have not yet advanced another candidate
An interesting puzzle, though…
Molecule produced in a benzene discharge, also with toluene
No REMPI signal (suggests high IP - cation?)
Evidence from isotopic substitution studies:
Similar geometry in lower and upper states
Five hydrogen atoms, two pairs of symmetry equivalent hydrogens
What molecule is this?
An interesting twist:
No longer are we trying to identify the carrier of a line in the ISM, but
the carrier of a line measured in a basement laboratory in Massachusetts
Has not proven to be easy - Quantum chemistry not the best approach (we’ll
discuss this shortly), only suggestion for carrier thus far seems a rather unlikely
molecule.
Still not solved definitively - moreover, carrier of 4429 in the laboratory probably
is not carrier of astronomical line
Is my favorite molecule (NO3) a DIBs carrier?
6616
6235
6278
X
B Absorption spectrum of NO3 (a very strong transition)
From DIBS catalog:
Lines at 6234.3, 6278.3 and 6613.8
v.
6235
6278
6616
What can quantum chemistry do to help?
Calculate line positions?
Most techniques have accuracies ca. 0.1-0.5 eV
TDDFT
CASPT, EOM-CCSD
MR-CISD, EOM-CCSDT
MR-AQCC
Applicability
Accuracy
To be useful as a “screener’’ for DIBS candidates requires accuracy of ca. 100 cm-1
What can quantum chemistry do to help?
Calculate line positions?
Most techniques have accuracies ca. 0.1-0.5 eV
TDDFT
CASPT, EOM-CCSD
MR-CISD, EOM-CCSDT
MR-AQCC
Applicability
Accuracy
To be useful as a “screener’’ for DIBS candidates requires accuracy of ca. 100 cm-1
Calculate absorption profiles?
Experimental spectrum

Calculated spectrum for
candidate molecule

Calculate absorption profiles?
Experimental spectrum

Calculated spectrum for
candidate molecule
Assignment

Calculate absorption profiles?
Experimental spectrum

Calculated spectrum for
candidate molecule
Assignment

This sort of assignment can often provide clear picture of the nature of vibronic states
What is required in the simulation of electronic spectra?
The simplest case: The Franck-Condon approximation
Applicable to most transitions from one “isolated” electronic state to another, particularly
in low-energy part of the spectrum.
Assumptions:
Vibronic
Wavefunction
(r,R) = (r;R) (R)
Electronic
Wavefunction
Vibrational Wavefunction
[calculated from Veff(R)]
< ’’(r,R) |  | ’(r,R) > = < ’’(r,R0) |  | ’(r,R0) >
Dipole operator
“Mij”’
Reference geometry
Vibronic level positions
Electronic energy difference (adiabatic)
Vibronic level positions
Electronic energy difference - ZPE of ground state
Vibronic level positions
Electronic energy difference - ZPE of ground state + vibrational energy in final state
Vibronic level positions
Electronic energy difference + ZPE of ground state + vibrational energy in final state
Origin
Vibronic level positions
Electronic energy difference + ZPE of ground state + vibrational energy in final state
Origin
Intensities
Dipole approximation: Intensity
 |< ’’(r,R) |  | ’(r,R)>|2
 Mij2 Fk
Transition
moment
Franck-Condon
factor (FCF)
|< ’’(R)|’(R)>|2
Vibronic level positions
Electronic energy difference + ZPE of ground state + vibrational energy in final state
…
Origin
Intensities
Dipole approximation: Intensity
 |< ’’(r,R) |  | ’(r,R)>|2
 Mij2 Fk
Transition
moment
Origin
Franck-Condon
factor (FCF)
|< ’’(R)|’(R)>|2
With several modes …
…
Origin
1
0
Origin
2
1
3
2
3
4
5
6
To do a Franck-Condon simulation, we need:
Quantum chemical calculation of ground state geometry and force field
“EASY” - CAN BE DONE WITH ANY METHOD
Quantum chemical calculation of excited state geometry and force field
MORE CARE REQUIRED WITH RESPECT TO CHOICE OF METHOD
(CIS, RPA, CIS(D), MCSCF, CASPT2, EOM-CC, MRCI)
(balance is important here)
Quantum chemical calculation of transition dipole moment
NOT SO HARD - ACCURACY USUALLY NOT VERY IMPORTANT
Franck-Condon simulations account for:
Progressions in totally symmetric vibrations
(provides a measure of the geometry change due to excitation)
Even-quantum transitions in nonsymmetric vibrations
(shows up only if there is an appreciable force constant change)
… but do not account for:
Final states that are not totally symmetric
(due to “vibronic” coupling)
Spectroscopic manifestations of non-adiabaticity (BO breakdown)
(effects of conical intersections, avoided crossings etc.)
Low-lying states usually heavily affected by vibronic coupling!
Vibronic effects on potential energy surfaces
(Model due to Köppel, Domcke and Cederbaum)
A model potential for a two coupled-mode, two state system
1 = 0.4 eV (323 cm-1)
2 = 0.1 eV (807 cm-1)
[symmetric]
[non-symmetric]
Parameters:
 Vertical energy gap between the states
 - Linear coupling constant between the two states
A - Slope of diabatic potential of state A along q1 at q1=q2=0
B - Slope of diabatic potential of state A along q1 at q1=q2=0
Hamiltonian corresponding to model (in diabatic basis)
T1
A q1 + 1/2 [1 q12 + 2 q22]
0
H= (
)
0
+(
 q2
T2
T
+
 q2
)
 + Bq1 + 1/2 [1 q12 + 2 q22]
V
Vibronic effects on potential energy surfaces
Diagonalization of potential energy (V) gives adiabatic potential
energy surfaces*
 = 0 eV
q2
*Note that the associated diagonal basis does not necessarily block-diagonalize the Hamiltonian
 = 0.05 eV
q2
 = 0.10 eV
q2
 = 0.15 eV
q2
 = 0.20 eV
q2
Vibronic effects on potential energy surfaces
Example:  = 0.5 eV; 1 = 0.2 eV; 2 = -0.2 eV;  = variable
Diabatic surfaces
A state
B state
 = 0.2 eV
Lowest adiabatic
surface (what we
calculate in quantum
chemistry)
Pseudorotation transition state
Equivalent
minima
Conical
Intersection
Lower adiabatic surface
Upper adiabatic surface
Conical intersections
Note that complicated (but realistic) adiabatic surfaces arise
from an extremely simple model potential
More visualization…
TOP VIEW
PERSPECTIVE VIEW
Lowest adiabatic surface with different coupling strengths
kk
=0.05 eV
=0.20 eV
=0.10 eV
=0.25 eV
=0.15 eV
=0.30eV
Lowest adiabatic surface with different coupling strengths
kk
=0.05 eV
=0.20 eV
=0.10 eV
=0.25 eV
=0.15 eV
=0.30eV
In this case, both states are
minima on the potential energy
surface. But note that two different
electronic states lie on the same
potential energy surface!!!
Far too infrequently thought about
in quantum chemistry.
Real world example: 2A1 and 2B2
states of NO2
“B state”
“A state”
An astrophysical example: Propadienylidene
1A
1
1B
1
1A
2
}
1A
1
Coupled by modes of b2 symmetry
Weak vibronic coupling between dark state and bright state
2A
2
state
(minimum)
2B
1
state
(minimum)
Conical intersection
Lowest adiabatic potential sheet
Vibronic effects on “vibrational” energy levels
Diagonalization of complete Hamiltonian (T+V) gives vibronic
energy levels
Model potential again:  = 0.20 eV;  = 0.5 eV; 1 = 0.2 eV; 2 = -0.2 eV
Harmonic A state frequencies from diabatic potential:
806 cm-1 (s); 323 cm-1 (a)
Harmonic A state frequencies from adiabatic potential
806 cm-1 (a); 253 cm-1 (a)
Exact vibronic levels below 1050 cm-1
250 cm-1 (n), 500 cm-1 ( 2n), 752 cm-1 ( 3n), 802 cm -1 ( s),1004 cm-1 ( 4n), 1043 cm-1 ( s+n)
Vibronic effects on “vibrational” energy levels
Diagonalization of complete Hamiltonian (T+V) gives vibronic
energy levels
Model potential again:  = 0.20 eV;  = 0.5 eV; 1 = 0.2 eV; 2 = -0.2 eV
Harmonic A state frequencies from diabatic potential:
806 cm-1 (s); 323 cm-1 (a)
Harmonic A state frequencies from adiabatic potential
806 cm-1 (a); 107i cm-1 (a)
Exact vibronic levels below 1050 cm-1
0, 121, 327, 590, 720, 866, 888, 1046 (symmetric levels)
19, 213, 453, 723, 736, 931, 1045
(nonsymmetric levels)
Vibronic effects on potential energy surfaces
Diagonalization of potential energy (V) gives adiabatic potential
energy surfaces*
n 2n s 3n
Add some slides here with wavefunctions, discussion of nodes, etc.
Wavefunctions and stationary state energies
Eigenstates of system obtained by diagonalizing Hamiltonian
Given by
Vibrational basis functions
 = A  ci i + B  ci i
Diabatic electronic
states
Wavefunctions and stationary state energies
Eigenstates of system obtained by diagonalizing Hamiltonian
Given by
Vibrational basis functions
 = A  ci i + B  ci i
Diabatic electronic
states
true only if =0
 = A  ci I
“Vibrational level of electronic state A”
or
 = B  ci i
“Vibrational level of electronic state A”
Electronic states are coupled by the off-diagonal matrix element
“Breakdown of the Born-Oppenheimer Approximation”
Add more slides here with wavefunctions, densities, projection onto diabatic states, etc.
The Calculation of Electronic Spectra including Vibronic Coupling
Energies given by
Eigenvalues of model Hamiltonian
‘Relative
intensities given by
|< ’’(r,R) |  | ’(r,R)>|2
Ground state
Final state
Model system: Adiabatic perspective
Corresponding Hamiltonian very
complicated: Potential energy matrix
is diagonal but not simple (discontinuities);
kinetic energy matrix is clearly not diagonal;
transition dipole moment very sensitive wrt
geometry
Model system: Adiabatic perspective
Corresponding Hamiltonian very
complicated: Potential energy matrix
is diagonal but not simple (discontinuities);
kinetic energy matrix is clearly not diagonal;
transition dipole moment very sensitive wrt
geometry
Green arrow - transition to “bright state”
Model system: Adiabatic perspective
Corresponding Hamiltonian very
complicated: Potential energy matrix
is diagonal but not simple (discontinuities);
kinetic energy matrix is clearly not diagonal;
transition dipole moment very sensitive wrt
geometry
Red arrow - transition to “dark state”
Green arrow - transition to “bright state”
An aside:
Traditional quantum chemistry assumes:
0
0
TB
)
VA
+
)
)
H=
TA
0
0
VB
)
Adiabatic potential
energy surfaces
Vibrational energy levels calculated from the Schrödinger equations
(TA + VA) 
= Evib 
(TB + VB) 
= Evib 
and total (vibronic energies) given by:
Eev(A) = (VA)min + Evib
Eev(B) = (VB)min + Evib
Diabatic perspective (KDC Hamiltonian) conceptually (and
computationally a much simpler approach
T1
A q1 + 1/2 [1 q12 + 2 q22]
0
H= (
) + (
0
 q2
T2
 q2
)
 + Bq1 + 1/2 [1 q12 + 2 q22]
Treatment:
1. Assume initial state not coupled to final states (not necessary, but a simple place to start)
’’
= 0 000…
2. Assume transition moments between diabatic states are constant
<0| |A> = MA
<0| |B> = MB
3. Diagonalize Hamiltonian (Lanczos recursion is best choice)
’ =
cA0
A 000…+ cB0 B 000… + ’[ cAi A i + cBi B i]i
T1
A q1 + 1/2 [1 q12 + 2 q22]
0
H= (
) + (
0
 q2
T2
4. Stick spectra given by
[cA0 + cB0 ]2 (E - )
Basis set and symmetry considerations
Direct product basis
A 00, A 01, A 02 …
B 00, B 01, B 02 …
Symmetry of vibronic level
ve = v x e
 q2
)
 + Bq1 + 1/2 [1 q12 + 2 q22]
T1
A q1 + 1/2 [1 q12 + 2 q22]
0
H= (
) + (
0
)
 + Bq1 + 1/2 [1 q12 + 2 q22]
 q2
T2
 q2
Makes entire contribution to intensity only ONE element of eigenvector matters
4. Stick spectra given by
[cA0 + cB0 ]2 (E - )
Basis set and symmetry considerations
Direct product basis
A 00, A 01, A 02 …
B 00, B 01, B 02 …
Symmetry of vibronic level
ve = v x e
Appearance of eigenvectors - pictorial view
S
A
N
S
N
Franck-Condon
Vibronic coupling
(weaker)
B
“vibronically allowed level”
Appearance of eigenvectors - pictorial view
S
A
N
S
N
Franck-Condon
Vibronic coupling
(stronger)
B
“vibronically allowed level”
Franck-Condon
Vibronic coupling