SPECTROSCOPIC NETWORKSSPECTROSCOPIC NETWORKS

ANDAND

ACTIVE DATABASESACTIVE DATABASES

11th ASA and 12th HITRAN conferenceUniversity of Reims

Reims, France, August 29-31, 2012

Attila G. Császár and Tibor Furtenbacher

Laboratory of Molecular Structure and Dynamics

Institute of Chemistry

Eötvös University

Budapest, Hungary

AcknowledgmentsAcknowledgments

(Dr.) Csaba Fábri (ELTE)

Prof. Jonathan Tennyson, Dr. Oleg Polyansky (U.K.)

Dr. Olga Naumenko, Dr. Kolya Zobov (Russia)

Prof. Alain Campargue (France)

Dr. Larry Rothman, Dr. Linda Brown, Dr. Iouli Gordon, Prof. Peter Bernath (U.S.)

Funding agencies

Inside Hungary

Scientific Research Fund of Hungary (OTKA), NKTH, EU & EuropeanSocial Fund

International cooperations

MC RTN „QUASAAR” supported by the

European Commission (FP6)

NSF-MTA-OTKA

IUPAC

COST

ERA-Chemistry

OUTLINE

Introduction

First-principles nuclear motion computations

Spectroscopic networks

MARVEL (Measured Active Rotational-Vibrational Energy Levels)

Energy levels and transitions of H2O isotopologues

Summary

Spectroscopic databases

High-resolutionTransmissionMolecularAbsorptionDatabase(Harvard-Smithonian)

AtmosphericRadiationAnalysis(GEISA)

QuantitativeInfraredDatabase(NIST)

CologneDatabase forMolecularSpectroscopy(CDMS)

IUPAC database of water isotopologues:JQSRT 110 (2009) 573 and 111 (2010) 2160.

ASA –Atmos.Spectr.Appl.

“Third age of quantum chemistry”

Fourth age of quantum chemistryFront cover of Phys. Chem. Chem. Phys.

(A. G. Császár, C. Fábri, T. Szidarovszky, T. Furtenbacher, E. Mátyus, G. Czakó: Fourth age of quantum chemistry - molecules in motion, 2012, 14, 1085).

• Qualitative understanding of nuclear motionRRHO (rigid rotor – harmonic oscillator)– SQM force fields, GF method, simple model problems

• Interpretation of experimental resultsPerturbative approaches, anharmonic force fields– Local vs normal modes, VPT2, higher-order PT

• Fourth age: moving (much) beyond fixed nucleiVariational approaches– VBR

Grid-based(nearly variational) approaches– FBR ↔ DVR

TreatingTreating qquantumuantum motionmotionss of of nucleinuclei

The D3OPI and D2FOPI algorithms

Discrete Variable Representation of the Hamiltonian

Energy operator in orthogonal (O) coordinate system

Direct-product (P) basis

Diagonalization with an iterative (I ) technique (e.g., Lánczos)____________________________________________________________________ G. Czakó, T. Furtenbacher, A. G. Császár, and V. Szalay, Mol. Phys. (Nicholas C. Handy Special Issue) 102, 2411 (2004).T. Szidarovszky, A. G. Császár, and G. Czakó, PCCP 12, 8373 (2010).

I. I. TailorTailor--mademade approachapproach: : prederivedprederived, , analyticanalytic, , casecase--dependentdependent T,T,

uniqueunique codecode forfor eacheach moleculemolecule

The The simplestsimplest gridgrid--basedbased procedureprocedure toto solvesolve thethe

((triatomictriatomic)) rovibrationalrovibrational problemproblem variationallyvariationally::

II. General Hamiltonian: DEWE(Discrete Variable Representation –Eckart-Watson Hamiltonian –

Exact inclusion of the potential energy)

1. Eckart-Watson vibrational operator (normal coordinates: Eckart-system, orthogonal, rectilinear internal coordinates):

2. Numerically exact inclusion of arbitrary (e.g., valence) coordinatepotential energy surface

3. Matrix of Hamiltonian in DVR representation, on-the-fly(impossible to store even the nonzero elements of the sparseHamiltonian) iterative eigenvalue and eigenvector computation.

−+−= ∑

−

=

63

1

11N

kkpk

p

ik

i

pipi Qmm

llccCr

E. Mátyus, G. Czakó, B. T. Sutcliffe, A. G. Császár, J. Chem. Phys. 127, 084102 (2007)

E. Mátyus, J. Šimunek, A. G. Császár, J. Chem. Phys. 131, 074106 (2009)

C. Fábri, E. Mátyus, …, A. G. Császár, J. Chem. Phys. 135, 094307 (2011)

VPHN

k

++−= ∑∑∑−

=

63

1

22

ˆ2

1

8ˆˆ

2

1ˆα

αααβ

βαβα µπµπ η

III. GENIUSH: General rovib code with Numerical, Internal-Coordinate, User-Specified Hamiltonians(general code – full- of reduced-dimensional rovibrational problem– internal

coordinate Hamiltonian withnumerical computation of bothT and V –discrete variable representation)

Initial specifications• body-fixed frame (e.g., Eckart, principal axis, „xxy” gauge)• set of internal coordinates (e.g., Z-matrix, Radau)• constraints for the coordinates if necessary• Cartesian coordinates of the atoms„Main” program• numerical or analytical 1st, 2nd, and 3rd derivatives of Euler angles

and internal coordinates in terms of Cartesians and vice versa• numerically computeG matrix, U, and V at every grid point• determine needed eigenpairs via variants of Lanczos’ algorithm(s)Result: single code for all full- or reduced-dimensionality problems

for molecules with arbitrary number of “interacting” minimaE. Mátyus, G. Czakó, A. G. Császár, J. Chem. Phys. 130, 134112 (2009).

C. Fábri, E. Mátyus, A. G. Császár, J. Chem. Phys. 134, 074105 (2011).

Normal Mode Decomposition (NMD)

HNCO,,Fundamentals”

2HO2JiiJ QNMD ψ=

4631.0 98 0 0 0 0 0 0 0 0 0 0 0 0 … 99

579.5 0 95 0 1 0 0 0 0 0 0 0 0 0 … 97

660.2 0 0 99 0 0 0 0 0 0 0 0 0 0 … 99

780.7 0 2 0 90 0 0 0 0 0 0 2 0 0 … 96

1146.8 0 0 0 0 80 0 6 5 0 0 0 0 0 … 93

1267.1 0 0 0 0 3 0 26 41 20 0 0 0 0 … 94

1275.1 0 0 0 0 0 91 0 0 0 5 0 0 0 … 97

1329.3 0 0 0 2 4 0 11 45 22 0 4 0 0 … 92

1358.6 0 0 0 2 7 0 54 5 15 0 7 0 0 … 93

1476.6 0 0 0 0 0 7 0 0 0 88 0 0 0 … 97

1521.6 0 0 0 2 0 0 2 0 28 0 51 0 0 … 91

1712.7 0 0 0 0 0 0 0 0 0 0 0 69 0 … 90

1843.6 0 0 0 0 0 0 0 0 0 0 0 0 45 … 96

4681.3

572.4

637.0

818.1

1144.8

1209.4

1274.0

1318.3

1390.5

1455.1

1636.2

1717.2

1781.8

Q0

Q5

Q6

Q4

Q5

+5

Q5

+6

Q6

+6

Q3

Q4

+5

Q4

+6

Q4

+4

Q5

+5

+5

Q5

+5

+6

Sum

… … … … … … … … … … … … …

Sum 98 99 98 95 98 99 99 94 98 95 92 95 98

E. Mátyus, C. Fábri, T. Szidarovszky, G. Czakó, W. D. Allen, and A. G. Császár,J. Chem. Phys.2010, 133, 034113.

Rigid-rotor decomposition (RRD)tables for ketene (H2CCO)

Networks within complexsystems(building blocks + interactions

= nodes + links)• Society (human interactions)• Culture (language)• Economy (world economy)• Internet• Telephones• Power stations and electric networks• Brain• Cells (biological networks, immune system,

transport networks, muscle network)• Atoms in molecules

Molecules as complexsystems?

• Size of most molecules studied by quantumchemistry: 2-1000 nuclei (for us 5-6 nuclei…)

• Perhaps 10 times this many electrons

• Fewhundred thousand different energy levelper electronic state and isotopolugue

• Fewbillion allowed transitions for each case

• Different experimental techniques: transitionsbetween greatly different energy level sets

Network theory• graphs (vertices, edges, distributions)• scale dependence• scale free• “small world”(Karinthy(1929))• embedding (principal and subnets)• weak and strong “forces”• network stability (attack/error tolerance)• overall network characteristics (random, scale

free, star, disintegrated)• stochasticity and preferential attachment

Network (graph) theory• graph G: ordered pair, G = (V,E)• degree: no. of edges that connect to a vertex• loop: edge connecting to the same vertex• simple graph: contains neither loops nor multiple edges• undirected graph: no direction of edges• connected graph: there is a path between any pair of

vertices• root: a distinguished vertex• cycle: returning path going through at least three edges• tree (forest): connected (disconnected) graph without

cycles• hub: a vertex with lots of edges

Network (graph) theory

(Erdős-Rényi) (Barabási)

Questionsto be answered• What kind of graphs (networks) would quantum

mechanics build (e.g., dense or sparse, stochastic ordeterministic, with one or with several components)?

• What is the degree distribution of SNs (Poisson or scalefree)?

• Is there a significant difference between measured and computed spectroscopic networks?

• Can a graph theoretical approach help spectroscopy and especially improving spectroscopic databases?

• Can one find similarities between spectroscopic (quantummechanical) and human networks and their behavior (aresocial networks unique in their charact-eristics or does„nature” produce similar networks)?

• Etc.

Spectroscopicnetworks (SN):large, finite, weighted,

undirected, and rootedgraphs• energy levels: vertices(nodes, with given,

theoretical or experimental, uncertainties)

• allowed transitions: edges(links)

• transition intensities: weights

A. G. Császár, T. Furtenbacher, J. Mol. Spectrosc. 2011, 266, 99-103T. Furtenbacher, A. G. Császár, J. Mol. Struct. 2012, 1009, 123

Spectroscopic network (SN) ofthe rotational states of HD16O

Degree destributionof HD16O ab initio database

0 1000 2000 3000 4000 5000 6000

0.00

0.02

0.04

0.06

0.08

0.10y = a xb

a 0.12039 ±0.00313b -0.76548 ±0.01416

Cut-off = 1E-20

P(k

)

k

0 2000 4000 6000 8000 10000

0.00

0.01

0.02

0.03

0.04

0.05

0.06 Cut-off = 1E-22

y=axb

a 0.07119 ±0.00202b -0.67310 ±0.01172P

(k)

k

0 2000 4000 6000 8000 10000-0.005

0.000

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040Cut-off = 1E-24

y = axb

a 0.04709 ±0.00128b -0.62284 ±0.00949

P(k

)

k

0 2000 4000 6000 8000 10000 12000

0.000

0.005

0.010

0.015

0.020

Cut-off = 1E-26

y = axb

a 0.03107 ±0.00079b -0.57462 ±0.00751

P(k

)

k

P(k) = Ak−γ

Spectroscopicnetworks arescalefree

Observed First-principles

Number of nodesand links asa functionof the -log of one-photonabsorption intensity

nodes

0

20000

40000

60000

80000

100000

120000

140000

160000

180000

0 10 20 30 40 50 60

links

0

100

200

300

400

500

600

700

800

0 10 20 30 40 50 60 70 80 90

Mill

ions

HD16O

Attack toleranceof SNs

0 20 40 60 80 100

0.0

0.2

0.4

0.6

0.8

1.0

N /

M

deleted nodes in %

measured transitions ab initio (pure rotational) ab initio database

Utility of the network approach

• weak connections (transitions) help to stabilizeSNs

• comparison of computed and measured hubs todesign newexperiments: minimal work(measurements) with maximum yield

• identification of the weakest „link” (in fact hubs ornon-hubs)

• helps to decide which measurements would be most helpful in order to improve the database

• facilitates the assignment of measured spectra and the labeling of lines

MARVEL: An MARVEL: An inverseinverse, ,

HamiltonianHamiltonian--free free approachapproach

toto highlyhighly accurateaccurate

rovibrationalrovibrational energyenergy levelslevels

MMeasuredeasured AActivective

RRotationalotational--VVibrationalibrational

EEnergynergy LLevelsevels

T. Furtenbacher, A. G. Császár, J. Tennyson, J. Mol. Spectrosc. 245, 115 (2007)T. Furtenbacher, A. G. Császár, J. Quant. Spectr. Rad. Transfer 113, 929 (2012)

Ei

Ej

Assignment i,j υij .....+1 ...... -1 ........

υυυυ = a × E

× ......

....

....

Solve for E (in a least-squares sense, taking into account the experimental uncertainties of the υij) to obtain experimentally derived term values Ei, Ej, ....

Database of observedtransition wavenumbersυij with assignments anduncertainties

The υij are determined byterm values Ei, Ej, ....

=

Input of MARVELCompilationof all availablemeasured

transitions, including their systematic andunique labelingsanduncertainties, within a single „active” database.

Wavenumber/cm-1 unc. *10E+6 assignment unique label

0.30077226 0.7659 000 7 4 3 000 7 4 4 83Johns.10.31558743 4.5621 000 10 5 5 000 10 5 6 83Johns.20.31723578 2.2311 000 3 2 1 000 4 1 4 83Johns.30.34827138 0.1998 000 2 2 0 000 2 2 1 83Johns.40.75701556 3.0969 000 7 1 7 000 6 2 4 83Johns.5

Pre-MARVEL validation• Quantummechanical selection rules:

– Spin isomer (ortho-para) distinction (two or more networks)Eliminate forbidden transitions between separate componentsof the SN

– Quantumnumber (J Ka Kc) validationEliminate fake energy levels

– Typos (transcript problems)

• Measurement ‘errors’:– Same assignments with different wavenumbers– Same wavenumbers with different assignments– Duplications– Etc.

Post-MARVEL validation

• Search for suspect energy levels; for example, theenergy of a rotational level should be smaller thanthat of the corresponding VBO

• Compare the MARVEL energy levels withab initio or other (effective Hamiltonian) results(preferably linelists) in order to make sure thatonly unique energy levels are present

• Classify the MARVEL energy levels:A+ the best and most reliable energy levelsA−, B+, B− levels of ‘intermediate’ accuracyC the worst energy levels (they can evne be wrong)

Characteristics of theMARVEL approach

– fast

– accurate

– active

– moleculeindependent

– has predictivepower

IUPAC water databaseI.H2

17O and H218OH2

17O H218O

No. of transitions collected 8 463 25 367

Maximum J 17 20

Highest VBO (cm-1) 16 876 16 855

No. of energy levels 2 687 4 849

No. of sources 33 48

J. Tennyson, P. F. Bernath, L. R. Brown, A. Campargue, M. R. Carleer, A. G. Császár, R. R. Gamache, J. T. Hodges, A. Jenouvrier, O. V. Naumenko, O. L. Polyansky, L. S. Rothman, R. A. Toth, A. C. Vandaele, N. F. Zobov, L. Daumont, A. Z. Fazliev, T. Furtenbacher, I. F. Gordon, S. N. Mikhailenko, and S. V. Shirin, IUPAC Critical Evaluation of the Rotational-Vibrational Spectra of Water Vapor. Part I. Energy Levels and Transition Wavenumbers for H2

17O and H218O, J. Quant. Spectr. Rad. Transfer 2009, 110, 573-596.

IUPAC water databaseII.HD16O, HD17O, and HD18O

HD16O HD17O HD18O

No. of transitions collected 54 740 485 8 728

Maximum J 30 11 18

Highest VBO (cm-1) 22 625 1 399 9 930

No. of energy levels 8 819 162 9 627

No. of sources 74 3 14

J. Tennyson, P. F. Bernath, L. R. Brown, A. Campargue, A. G. Császár, L. Daumont, R. R. Gamache, J. T. Hodges, O. V. Naumenko, O. L. Polyansky, L. S. Rothman, R. A. Toth, A. C. Vandaele, N. F. Zobov, S. Fally, A. Z. Fazliev, T. Furtenbacher, I. F. Gordon, S.-M. Hu, S. N. Mikhailenko, and B. Voronin, IUPAC Critical Evaluation of the Rotational-Vibrational Spectra of Water Vapor. Part II. Energy Levels and Transition Wavenumbers for HD16O, HD17O, and HD18O, J. Quant. Spectr. Rad. Transfer 2010, 111, 2160-2184.

IUPAC water databaseIII.H2

16OH2

16O

No. of transitions collected 195 432

Maximum J 42

Highest VBO (cm-1) 41 121

No. of energy levels 18 459

No. of sources 93

Revised version resubmitted to JQSRT

Recalibration of 83GuelachvH2

17O

0,0164

0,0165

0,0166

0,0167

0,0168

0,0169

0,017

0,0171

0,0172

0,9999995 0,9999996 0,9999997 0,9999998 0,9999999 1

H218O

0,0966

0,0968

0,097

0,0972

0,0974

0,0976

0,0978

0,098

0,0982

0,9999995 0,9999996 0,9999997 0,9999998 0,9999999 1

New calibration standards: H2

16O, H218O, D2

16O (15-170 cm-1)• only para-water transitions• measured with internal precision better than 33 kHz• underlying energy levels are involved in at least three

measured transitions

T. Furtenbacher, A. G. Császár, J. Quant. Spectr. Rad. Transfer 109, 1234 (2008)

743,563.526(36)745,320.142(36)1,207,638.714(13)

714,087.313(36)517,181.960(30)987,926.743(16)

692,243.579(36)322,465.170(30)916,171.405(13)

643,247.288(37)203,407.502(30)752,033.104(13)

D216OH2

18OH216O

MARVEL prediction

SummarySummary and and conclusionsconclusions• First-principles nuclear motion computations canprovide highly useful information for spectroscopy.•Network-based MARVEL analyses provide excellent opportunities to understand high-resolution experimental spectra of molecules and turn scatteredinformation into focused knowledge.• Spectroscopic networks (SN) are weighted, undirected graphs, whereby energy levels are the nodes, allowed transitions are the links, and weights are provided by intensities. • Graph theory helps to advance our understanding ofhigh-resolution molecular spectra and design newexperiments.

The 23rd Colloquium onHigh Resolution Molecular Spectroscopy

Margaret Island, Budapest, HungaryAugust 25-30, 2013

Organizing and scientific committee:Attila G. Császár Martin QuackMichel Herman Paolo De Natale