Coupled-cluster theory and the discovery of astronomical molecules: A pragmatic perspective

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 Hamiltoniansbased on laboratory data

Sensitivity goes with square of the molecular dipole moment

First radio detection: OH in 1963; NH3, H2O, H2CO beforeend of decade (first polyatomics); HCO+ in 1970 (first positiveion); 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

Allows for the observation of non-polar polyatomic moleculesand condensed phases (ice, for example)

Can (in principle) detect near-IR electronic spectra of radicalsand 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!

ISO satellite (European Space Agency)Operating range: 2.5 - 240 µm (42-4000 cm-1)

The evidence for benzene in the interstellar medium

Optical detection

Historically the first technique used: CH discovered in spacein 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 asrestrictive; 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 CCDcameras now yield now yield high quality spectra.

Weakness: Laboratory data for complex molecules not easy to acquire, some moleculesdo not have intense electronic transitions; dissociative excited states lead to broad features

ISO satellite (European Space Agency)Operating range: 2.5 - 240 µm (42-4000 cm-1)

The general process of identifying new molecules

Obtain precise linepositions and molecularconstants in the laboratory.

Determine or predict potentialcandidate line positions

Conduct astronomical searchfor lines, usually choosing thestrongest lines, or those that are in an accessible region of the spectrum.

Quality (robustness) of identificationdepends on:

PRECISION of observed line positions

AGREEMENT with laboratory lines

NUMBER of lines used in identification

What can a quantum chemist do to help?

A different sort of synthetic chemistry

Obtain precise linepositions and molecularconstants in the laboratory.

Determine or predict potentialcandidate line positions

Conduct astronomical searchfor lines, usually choosing thestrongest lines, or those that are in an accessible region of the spectrum.

C2H2, C4H2

Benzene,Toluene

SiH4, H2S, S vapor

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:

Models of interstellar chemistryAbundance, isotopic studies

Chemical physics:

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))

Geometry optimization, harmonic force field calculation, perhaps larger basis set energy calculations.

Evaluation of anharmonic (cubic or quartic)molecular force field, followed by applicationof vibrational perturbation theory.

Ae, Be, Ce, CD constants

Dipole moment (e)

Ao, Bo, Co, CD constants

Dipole moment (o)

Increased treatmentof correlation

Basis setimprovement

Increases bond lengths

Ie Be

Decreases bond lengths

Ie Be

Basis set

STO-3G 6-31G* cc-pVDZ cc-pVTZ cc-pVQZ Exact

Lev

el o

f ca

lcul

atio

nFCI

CCSD(T)

MP2

CCSD

SCF

Be < Beexact

Be > Beexact

Be Beexact

Be =Beexact

In general, CCSD(T)/cc-pVTZ overestimates bond lengths a bit:

Becalc < Be

expt

However, experiments do not measure Be, but rather Bv

Usually positive

Molecular vibration effects tend to extend the structure (rg > re) B0 < Be

Exact


Becalc < Be

expt


Usually positive


Exact

CCSD(T)/cc-pVTZ


Becalc < Be

expt


Usually positive


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).

First molecular anionfound in space(carrier of U1377 in IRC+10216)

Laboratory spectra of C4H- and C8H-

recently recorded (Gupta et al., preprint)

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

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.

These calculations require evaluation of:

the harmonic, cubic and quartic force fieldsthe 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 Taylorand Almlöf work extremely well for vibrational level positions.

But, alas…

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:

Molecular

identity

Laboratory

lines

Astronomical

linesMW

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:

Molecular

identity

Laboratory

lines

Astronomical

linesMW

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 bandsCatalog 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 higherlevels 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)

ISO spectra of the Planet. Neb. NGC 7027 and the PDR region at the Orion Bar

Peeters et al., 2003

%3.1≥Δλλ

Orion Bar

NGC 7027

The Unidentified Infrared Bands (UIRs)

Wavelength (µ)

Flu

x D

ensi

ty (

10-1

3 W

/m2 /

µ)

C-H st

retch

C-C st

retch

C-H in

-plane

bendC-H

out-of-p

lane

bend

The unidentified infrared (UIR) features

4395

4435

Carrier of λ4429?

Corannulene, C20H10

Lovas et al., 2005

exp = 2.07(2)DB = 509.8MHz

However, there has not been a single unambiguousdetection of any PAH in space to date, despite considerablepublicity.

Nonpolarity is a problem, although two smallest polarPAH species have not been found

Apart from benzene, there is no evidence of anythingother 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…

Tulej, Kirkwood, Pachkov and Maier , Ap.J.Lett. 506, 69 (1998)

C7- !?

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, butthe carrier of a line measured in a basement laboratory in Massachusetts

Has not proven to be easy - Quantum chemistry not the best approach (we’lldiscuss this shortly), only suggestion for carrier thus far seems a rather unlikelymolecule.

Still not solved definitively - moreover, carrier of λ4429 in the laboratory probablyis 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

AccuracyApplicability

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

AccuracyApplicability

To be useful as a “screener’’ for DIBS candidates requires accuracy of ca. 100 cm-1

Calculate absorption profiles?

Experimental spectrum

Calculated spectrum forcandidate molecule




Assignment




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, particularlyin low-energy part of the spectrum.

Assumptions:(r,R) = (r;R) (R)

Vibronic Electronic Wavefunction Wavefunction

Vibrational Wavefunction[calculated from Veff(R)]

< ’’(r,R) | | ’(r,R) > = < ’’(r,R0) | | ’(r,R0) > “Mij”’

Dipole operator Reference geometry

Vibronic level positions

Electronic energy difference (adiabatic)


Electronic energy difference - ZPE of ground state


Electronic energy difference - ZPE of ground state + vibrational energy in final state


Electronic energy difference + ZPE of ground state + vibrational energy in final state

Origin



Origin

Intensities

Dipole approximation: Intensity |< ’’(r,R) | | ’(r,R)>|2

Mij2 Fk

Transition Franck-Condonmoment factor (FCF)

|< ’’(R)|’(R)>|2



Origin

Intensities

Dipole approximation: Intensity |< ’’(r,R) | | ’(r,R)>|2

Mij2 Fk

Transition Franck-Condonmoment factor (FCF)

|< ’’(R)|’(R)>|2

Origin

…

Origin

Origin

…

0 1 2 3 4 5 6

1 2 3

With several modes …

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

A model potential for a two coupled-mode, two state system

1 = 0.4 eV (323 cm-1) [symmetric]

2 = 0.1 eV (807 cm-1) [non-symmetric]

Parameters:

Δ - Vertical energy gap between the statesλ - Linear coupling constant between the two statesA - Slope of diabatic potential of state A along q1 at q1=q2=0B - Slope of diabatic potential of state A along q1 at q1=q2=0

Hamiltonian corresponding to model (in diabatic basis)

T1 0 A q1 + 1/2 [1 q12 + 2 q2

2] λ q2

H = ( ) + ( ) 0 T2 λ q2 Δ + Bq1 + 1/2 [1 q1

2 + 2 q22]

T + V

(Model due to Köppel, Domcke and Cederbaum)


Diagonalization of potential energy (V) gives adiabatic potential energy surfaces*

*Note that the associated diagonal basis does not necessarily block-diagonalize the Hamiltonian

q2

λ = 0 eV

q2

λ = 0.05 eV

q2

λ = 0.10 eV

q2

λ = 0.15 eV

q2

λ = 0.20 eV


Example: Δ = 0.5 eV; 1 = 0.2 eV; 2 = -0.2 eV; λ = variable

Diabatic surfaces

A state B state

Equivalent minima

Conical Intersection

Pseudorotation transition stateλ = 0.2 eVLowest adiabatic surface (what we calculate in quantumchemistry)

Lower adiabatic surface Upper adiabatic surface

Note that complicated (but realistic) adiabatic surfaces arise from an extremely simple model potential

Conical intersections

More visualization…

TOP VIEW PERSPECTIVE VIEW

Lowest adiabatic surface with different coupling strengths

kk

λ=0.05 eV

λ=0.25 eV λ=0.30eVλ=0.20 eV

λ=0.15 eVλ=0.10 eV

Lowest adiabatic surface with different coupling strengths

kk

λ=0.05 eV

λ=0.25 eV λ=0.30eVλ=0.20 eV

λ=0.15 eVλ=0.10 eV

“A state”

“B state”

In this case, both states areminima on the potential energysurface. But note that two differentelectronic states lie on the samepotential energy surface!!!

Far too infrequently thought aboutin quantum chemistry.

Real world example: 2A1 and 2B2 states of NO2

An astrophysical example: Propadienylidene

1A1

1A1

1A2

1B1

}

Coupled by modes of b2 symmetry

2A2 state (minimum)

2B1 state (minimum)

Conical intersection

Weak vibronic coupling between dark state and bright state

Lowest adiabatic potential sheet

Vibronic effects on “vibrational” energy levels

Diagonalization of complete Hamiltonian (T+V) gives vibronic energy levels

Model potential again: λ = 0.20 eV; Δ = 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 ( 2n), 752 cm-1 ( 3n), 802 cm -1 ( s),1004 cm-1 ( 4n), 1043 cm-1 ( s+n)

Vibronic effects on “vibrational” energy levels

Diagonalization of complete Hamiltonian (T+V) gives vibronic energy levels

Model potential again: λ = 0.20 eV; Δ = 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)


Diagonalization of potential energy (V) gives adiabatic potential energy surfaces*

n 3ns2n

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

true only if λ=0

= A ci I or = B ci i

“Vibrational level of electronic state A” “Vibrational level of electronic state A”

Electronic states are coupled by the off-diagonal matrix element

“Breakdown of the Born-Oppenheimer Approximation”

Diabatic electronic states

Add more slides here with wavefunctions, densities, projection onto diabatic states, etc.

The Calculation of Electronic Spectra including Vibronic Coupling

|< ’’(r,R) | | ’(r,R)>|2

‘Relative intensities given by

Energies given by

Eigenvalues of model Hamiltonian

Ground state Final state

Model system: Adiabatic perspective

Corresponding Hamiltonian verycomplicated: Potential energy matrixis diagonal but not simple (discontinuities);kinetic energy matrix is clearly not diagonal;transition dipole moment very sensitive wrtgeometry

Green arrow - transition to “bright state”



Red arrow - transition to “dark state”

Green arrow - transition to “bright state”



An aside:

Traditional quantum chemistry assumes:

TA 0 VA 0 H = + 0 TB 0 VB

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

))

) ) Adiabatic potentialenergy surfaces

T1 0 A q1 + 1/2 [1 q12 + 2 q2

2] λ q2

H = ( ) + ( ) 0 T2 λ q2 Δ + Bq1 + 1/2 [1 q1

2 + 2 q22]

Diabatic perspective (KDC Hamiltonian) conceptually (andcomputationally a much simpler approach

Treatment:

1. Assume initial state not coupled to final states (not necessary, but a simple place to start)

2. Assume transition moments between diabatic states are constant

3. Diagonalize Hamiltonian (Lanczos recursion is best choice)

’’ = 0 000…

<0| |A> = MA

<0| |B> = MB

’ = cA0 A 000…+ cB

0 B 000… + ’[ cAi A i + cB

i B i]i

T1 0 A q1 + 1/2 [1 q12 + 2 q2

2] λ q2

H = ( ) + ( ) 0 T2 λ q2 Δ + Bq1 + 1/2 [1 q1

2 + 2 q22]

4. Stick spectra given by

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

[cA0 + cB

0 ]2 (E - )

T1 0 A q1 + 1/2 [1 q12 + 2 q2

2] λ q2

H = ( ) + ( ) 0 T2 λ q2 Δ + Bq1 + 1/2 [1 q1

2 + 2 q22]

4. Stick spectra given by

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

[cA0 + cB

0 ]2 (E - )

Makes entire contribution to intensity -only ONE element of eigenvector matters

Appearance of eigenvectors - pictorial view

A

B

S

N

N

S

Franck-Condon Vibronic coupling

“vibronically allowed level”(weaker)

Appearance of eigenvectors - pictorial view

A

B

S

N

N

S

Franck-Condon Vibronic coupling

“vibronically allowed level”(stronger)

Franck-Condon

Vibronic coupling

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Coupled-cluster theory and the discovery of astronomical molecules: A pragmatic perspective