Molecular energy levels and accurate high-temperature partition functions

96 Pages·2015·12.15 MB·English

Molecular energy levels and accurate
high-temperature partition functions
Thesis
Chemistry Masters
N ´ D ´
ORA ENES
Supervisors:
Prof. Dr. Attila G. Csa´sza´r
Dr. Tibor Furtenbacher
Laboratory of Molecular Structure and Dynamics
Eo¨tvo¨s Lora´nd University, Budapest
Faculty of Science
Institute of Chemistry
Place of defence: Department of Physical Chemistry
2015.
Contents
1 Acknowledgments 4
2 Introduction 5
2.1 Physicalpropertiesofthewatermolecule . . . . . . . . . . . . . . . . . . . 8
3 Ontheimportanceofwaterspectroscopy 9
4 Ontheconstructionofspectroscopicdatabases 10
4.1 Spectroscopicdatabases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4.2 Spectroscopicnetworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.2.1 Basicsofgraphtheory . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.2.2 Spectroscopicnetworks . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2.3 Propertiesofspectroscopicnetworks . . . . . . . . . . . . . . . . . . 15
4.3 TheMARVELprocedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.3.1 Theinputfile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.3.2 Thepre-MARVELprocedure . . . . . . . . . . . . . . . . . . . . . 18
4.3.3 TheMARVELinversion . . . . . . . . . . . . . . . . . . . . . . . . 18
4.3.4 TheMARVELresults . . . . . . . . . . . . . . . . . . . . . . . . . 19
5 Thesimplestmodelstointerpretrotational-vibrationalspectra 20
5.1 Theharmonicoscillatorapproximation . . . . . . . . . . . . . . . . . . . . . 20
5.2 Therigid-rotorapproximation . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.3 Isotopicshifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6 Abinitioenergylevelsforwater 22
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6.2 Thebasicsofvariationalnuclearmotioncalculations . . . . . . . . . . . . . 23
6.3 Potentialenergysurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6.3.1 PESwithadiabaticcorrection . . . . . . . . . . . . . . . . . . . . . 25
6.3.2 PESuptothedissociationlimit . . . . . . . . . . . . . . . . . . . . 25
6.4 TheD2FOPIcode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6.5 Theroleofabinitioenergylevelsinmythesis . . . . . . . . . . . . . . . . . 28
6.6 RigidRotorDecomposition(RRD) . . . . . . . . . . . . . . . . . . . . . . . 29
2
7 Partitionfunctionsandthermodynamicproperties 29
7.1 Partitionfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
7.2 Partitionfunctionsandderivedproperties . . . . . . . . . . . . . . . . . . . 31
7.2.1 ThermochemistrywithintheRRHOapproximation . . . . . . . . . . 31
7.2.2 Thermochemistybasedonindividualenergylevels . . . . . . . . . . 32
8 MARVELanalysisofD 16O 34
2
8.1 Methodsandinputdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
8.2 MARVELenergylevels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
8.2.1 RRDanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
8.3 Size-frequencydistributionofspectroscopicnetworks . . . . . . . . . . . . . 42
9 PartitionfunctionsofD 16O,D 17O,andD 18O 45
2 2 2
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
9.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
9.2.1 Approximaterovibrationalenergylevels . . . . . . . . . . . . . . . . 46
9.3 H 16Oasamodelsystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2
9.4 Computation of the partition functions and the derived thermochemical prop-
erties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
9.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
9.6 Uncertaintyanalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
10 Summaryandconclusions 63
A Appendix 73
A.1 TheRRDmethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
A.2 DatasourcesforD 16O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2
A.3 ConstantsusedintheeffectiveHamiltoniancalculationsforD 16O[1] . . . . 79
2
A.4 Partitionfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3
1 Acknowledgments
First and foremost I would like to express my sincere gratitude to my supervisors, Professor
Attila Csa´sza´r and Dr. Tibor Furtenbacher, for their patience, motivation, encouragement and
expertise. A very special thanks goes to Dr. Tama´s Szidarovszky, how introduces me to the
mysteriesofvariationalnuclearmotioncomputations. IwouldliketothankDr. EszterCzinki
forgivingthoroughandinsightfulfeedbackonthefirstversionofmythesis.
FurthermoreIwouldliketothankmyparentsandmysiblings,Lilla,Ba´lintandAndra´sfor
the support they provided me. Last but not least, I would like to thank my boyfriend, Kende.
Hewasalwaystherecheeringmeupandstoodbyme.
4
2 Introduction
The detailed knowledge of rotational-vibrational spectra of molecules is a prerequisite for
solving several physical, chemical, and engineering problems. One important application is
atmosphericmodelling. Thewatermoleculeisresponsibleforthelargestamountofabsorption
regarding both the incoming and outgoing radiation. During the spectroscopic monitoring of
the atmosphere the knowledge of water lines is important because they are present in almost
everyspectralrange.
The large amount of high-resolution spectroscopic data is stored in various information
systems[2,3,4,5,6]. Experimentaldataaremeasuredbydifferentgroupsemployingvarious
techniquesindiverseregionsofthespectrum. Nowadaysitiscomputationallyfeasibletocom-
pute ab initio line lists for small molecules, which include energy levels, quantum numbers,
symmetries, transitions, and intensities. It has to be emphasized that complete rotational-
vibrationalspectracanbedeterminedthroughquantumchemicalcomputations. However,the
accuracy of the computed energy levels can not approach that of the measured ones. At the
same time, only a small fraction of the possible transitions can be determined experimen-
tally. Due to technological reasons only transitions above a certain intensity cut-off value
(10−24cm/molecule) can be determined routinely in absorption. Therefore, the best possible
approach to construct spectroscopic databases is the joint utilization of experimental and the-
oretical data. Both measured and ab initio line lists have their own advantages. Ab initio data
have lower accuracy, their accuracy is 10 to 10000 times worse than that of their experimen-
tal counterparts. For most of the lines, experimental and ab initio transition intensities have
approximately the same accuracy. One great advantage of ab initio line lists is that (almost)
complete line lists can be computed. Modellers often require the knowledge of complete list
of energy levels, e.g.: for computing partition functions and derived properties at high tem-
peratures. This approach raises the question how these experimental and theoretical line lists
can and should be used together in different fields of chemistry. In my Masters thesis I would
like to introduce two fields of chemistry where experimental and theoretical datasets can sup-
plementeachothertogetthebestavailableresults.
The network approach is a hot topic in various fields [7]. Just to mention a few examples,
scientists study communication networks, neural networks, and social networks, as well. In
case of chemistry the graph theoretical approach can be used to understand high-resolution
spectroscopicdatabasesbetterandimprovetheircontent.
My supervisors proposed a graph-theoretical, network-based approach called MARVEL
(Measured Active Rotational-Vibrational Energy Levels) [8, 9, 10, 11, 12] to analyze high-
5
resolution molecular spectra. The MARVEL procedure is based on the concept of spectro-
scopic networks (SNs) [13] introduced by Csa´sza´r and Furtenbacher, and converts measured
transitions into experimental-quality rotational-vibrational energy levels via a linear least-
squaresalgorithm. Itisbeneficialtoconvertthemeasuredtransitionsintoenergylevels,since
considerable data reduction can be achieved. A lot less energy levels are sufficient to repro-
ducealltheoriginalmeasuredtransitions.
The MARVEL energy levels are called active because new transitions can be added to the
databasewhenevertheybecome available. Insuchcases newenergylevelsmayappearorthe
older ones become better defined, more accurate. The improvement is especially pronounced
whenthenewtransitionsestablishcycles[13]. Aprofoundlyimportantfeatureofthedatabase
is that the derived energy levels have experimental uncertainties. With MARVEL energy lev-
els and quantum chemical selection rules, one can determine new, unmeasured transitions
with experimental accuracy. Thus, MARVEL energy levels can facilitate the assignment of
new measured spectra. The MARVEL procedure, through the uncertainties of the energy lev-
els, enables the examination of the self-consistency of the database and to recalibrate certain
measurements.
Duringmyresearchleadingtothisthesis,Icollectedalltheavailablemeasuredtransitions,
and constructed a self-consistent database for the doubly deuterated isotopologue of water,
D 16O. To validate the MARVEL energy levels I performed first-principles nuclear motion
2
computations. I carried out all the variational-type computation with the in-house program
called D2FOPI [14], which had to be modified during the course of my work. The D2FOPI
programnumericallysolvesthetime-independentnuclearSchro¨dinger-equation.
Theresultsobtainedshouldinterestmodellers,sinceD 16Oiscurrentlynotincludedinthe
2
main databases storing high-resolution spectroscopic data, such as HITRAN [15] or GEISA
[3].
Boththeengineeringandthescientificcommunitiesshowconsiderableinteresttowardsthe
determination of the fundamental equation of water (as well as its deuterated isotopologues)
from the triple point to 800 K and 1000 MPa pressure. Heavy water is among the compo-
nents that IAPWS (the International Association for the Properties of Water and Steam [16])
takes into consideration when determining the thermodynamic properties of water. The high-
temperature partition function of D 16O is not well studied; thus, it has higher uncertainties
2
thanotherparametersintheIAPWSdatabase. Theothertwodoublydeuteratedisotopologues,
D 17OandD 18O,areevenlessthoroughlystudied. TheMScthesisaimstodeterminethebest
2 2
high-temperaturepartitionfunctionandthederivedthermochemicalpropertiesfortheD 16O,
2
6
D 17OandD 18Omolecules.
2 2
In quantum mechanics one can derive almost all physico-chemical properties from the
wavefunctions. Inthefieldofchemicalthermodynamicspartitionfunctionsplayacentralrole:
allmacroscopicquantitiescanbederivedfromthepartitionfunction. Dataonthermodynamic
propertiesofwateratelevatedtemperaturesisfundamentalinmanyfieldsofchemistry.
To compute accurate partition functions, one has to face different challenges. The general
rigidrotor(RR)andharmonicoscillator(HO)approximationsarenotaccurateenoughtocom-
pute reliable rotational and vibrational partition functions especially at high temperatures. In
thesecondpartofmyMscthesis,Itestdifferentstrategiestocomputeaccuratepartitionfunc-
tions for the D 16O, D 17O and D 18O molecules up to 6000 K. Seemingly the best approach
2 2 2
to compute accurate partition functions at high temperatures is the explicit summation of the
measuredenergylevels(especiallyiftheycomefromMARVELenergylists)augmentedwith
theoretical estimates. If we want to calculate the nearly exact values of the partition function
up to high temperatures, we should know all the energy levels up to and even beyond [17]
dissociation. Despite the recent advances is spectroscopy, the available measured energy lev-
els are sufficient to determine accurate partition functions only at low temperatures, up to a
few hundred K. Thus, I had to investigate several schemes to augment the MARVEL energy
levels. First, I computed the partition function solely based on the MARVEL data. Second,
I included ab initio data derived from first-principles variational computations, as well. In
case of the doubly deuterated isotopologues of water accurate first-principles computations
are feasible only up to 16000 cm−1 above the zero point vibrational energy (ZPVE), which is
far below the dissociation energy 41088 cm−1 [18]. Third, I covered the region from 16000
cm−1 up to dissociation with approximate rovibrational energy levels based on the sum of
ab initio VBOs and approximate rotational energy levels using the third-order Polyansky ex-
pression [19] and the rigid rotor approximation. The critical evaluation of the performance of
the different approaches should guide future attempts to determine high-resolution partition
functionsofothermoleculesandisotopologues.
Several thermochemical quantities of interest can be calculated as derivatives of the parti-
tion function. I calculated the entropy (S), the Gibbs enthalpy function (gef), the Helmholtz
function(hcf)andtheheatcapacityatconstantpressure(C )forthethreewaterisotopologues
p
studied.
7
2.1 Physical properties of the water molecule
Water is the third most abundant molecule in the universe after H and CO; thus, it is the
2
most abundant polyatomic molecule. The electronic ground state of the water molecule is
wellseparatedfromtheelectronicallyexcitedstates. Toreachthefirstexcitedstate,itrequires
morethan50000cm−1 energy. Thus,high-resolutionmolecularspectroscopyconcentrateson
the understanding of the rotational-vibrational transitions of the electronic ground state of the
watermolecule[20,21]. Purerotationaltransitionscanbemeasuredinthemicrowaveregion,
while rotational-vibrational transitions are measured in the visible and the infrared region of
thespectrum.
Every N-atomic molecule is defined by 3N coordinates in space. In case of the water
molecule,itmeansninedegreesoffreedom. Thetranslationalmotionofthemoleculearound
thecenterofmasscanbedescribedbythreedegreesoffreedom.
Thewatermoleculeisatriatomicasymmetrictopmoleculewithsmallmomentsofinertia.
Asymmetric rotors have three different moments of inertia, labelled by I , I , and I . The
A B C
rotational motion is described by three degrees of freedom. The rotational quantum number
corresponding to the overall rotation of a molecule is J, it is related to the eigenvalues of
the rotational Hamiltonian and it is considered to be an exact quantum number. The other
twoasymmetricrotorquantumnumbers,K andK ,areapproximatequantumnumbers,they
a c
correspond to |K| for the prolate and oblate symmetric-top limits of the asymmetric-top rigid
rotor. Table 2 shows the rotational constants I used for the rigid rotor decomposition and the
asymmetricrigidrotorcalculations.
Thevibrationaldegreesoffreedomis3N−6fornon-linearmoleculesand3N−5forlinear
molecules. Everynormalmodecanbecharacterizedbyacertaintypeofsymmetry. Thewater
molecule has C point-group symmetry. A normal mode is an independent, synchronous
2v
motion of atoms when all particles oscillate with the same frequency and phase, but with
different amplitudes. The vibrational motion of symmetrically substituted isotopologues of
water is usually described by three quantum numbers: ν , ν and ν describe the symmetric
1 2 3
stretching,thesymmetricbendingandtheantisymmetricstretchingmotion,respectively. The
energy of the bending motion is approximately half of the energy of the stretching motions
withthesameexcitation.
As suggested in the above discussion, the quantum states of water isotopologues can be
describedbysixnumbers: threedefiningrotations(J,K ,K )andthreecharacterizingvibra-
a c
tions(ν ,ν ,ν ).
1 2 3
Vibrations of water isotopologues have large amplitude due to the two light H (or D)
8
atoms, in case of rotations there is a big centrifugal distortion. Strong rotational-vibrational
couplingischaracteristicofthewatermolecule,whichleadstoacomplexandhighlyirregular
spectrum.
3 On the importance of water spectroscopy
The relative abundance of D 16O compared to H 16O is 20ppb in the Earth’s atmosphere.
2 2
Despitebeingatracecomponent,D 16O’satmosphericabsorptionhasseveralimplications,as
2
describedbelow.
One of the main emission components of nuclear power plants and chemical enterprises
processing and enriching nuclear fuel is D 16O, while super heavy water in the atmosphere
2
indicates the production of plutonium [22]. Thus, detection of heavy and super-heavy water
in the atmosphere is of considerable relevance to the safety of the world. The presence indi-
catesthedegreeofenvironmentalpollutionaswellasillegalnuclearweaponproduction. The
emphasis in these applications is on real-time measurements with the detection limit under
ppb.
To achieve a better understanding of global climate change it is critical to understand
the complex atmospheric processes of Earth. The atmospheric ratio of water isotopologues
(HD16O / H 16O) [23, 24] is intensely studied, since various processes affect this ratio: the
2
vaporization of oceans, the rain, the vegetation, and the Earth’s surface. Recondensation is a
goodexampletohighlightthechangesintheisotopologicratio: whenwarmhumidairreaches
colder parts of the atmosphere, the ratio of heavier isotopologues (e.g., HD16O) decreases.
The atmospheric isotopologue ratio contains information about the history of atmospheric
humidityandhelptogetabetterunderstandingofwatercirculation.
The astrophysical studies of D 16O also have many interesting aspects. The (D/H)
2 water
ratioonVenusisexceptionallyhigh,100timeslargerthanonEarth. Thisleadstotheassump-
tionthattheyoungplanetVenususedtohaveawetenvironment[25,26].
D O is also essential to verify the evolution models of the solar system. According to the
2
currentlyacceptedtheory,deuteriumwasproducedinthefirstfewminutesaftertheBigBang.
D is burnt rapidly in stellar interiors, its abundance has been decreasing since that time. At
low temperatures, below 20 K the effect of the difference among the bond energies of H O,
2
HDO, and D O increases, which leads to a much higher abundance of deuterated molecules
2
[27].
The doubly [28] and the triply [29, 30] deuterated ammonia molecules have already been
detected in dark cold clouds, as well as the doubly deuterated formaldehyde [31] , the D H+
2
9
[32],theD CS[33],the c-C D [34]andtheD Sis[35]. D 16Ohasalreadybeendetected in
2 3 2 2 2
2013[36].
As an added benefit, the detailed knowledge of the spectra of the D O isotopologues can
2
helpthespectroscopyoftheHDOisotopologues[37]. PureHDOspectracannotberecorded,
asitpartiallydisproportionatesintoD OandH O.50%isthehighestHDOratiowhichcanbe
2 2
achieved. SinceHDOsamplescontainasignificantamountofD OandH O,theinvestigation
2 2
oftheirspectraisalsonecessarytostudyHDOspectra.
4 On the construction of spectroscopic databases
4.1 Spectroscopic databases
Even for small molecules the number of bound rovibrational transitions is on the order of
billions, and even for the most thoroughly studied molecules the number of measured transi-
tions is only several hundred thousand [38, 39, 40, 41]. To handle such an enormous amount
of information, it is a prerequisite to effectively sort and store the data. The data should de-
fine a self-consistent database. Thus, it is important to filter out wrong, badly measured or
misassignedexperimentaltransitions.
Several databases have been constructed to store spectroscopic data [3, 4, 5, 6]. These
databases contain mainly measured data, which are sometimes mingled with theoretical data.
A further complication is that the data are usually not uniquely labelled and often it is impos-
sibletotracktheirhistory.
The Quantitative Infrared Database run by NIST (National Institute of Standards and
Technology)[4] contains the absorption spectra of volatile organic compounds. The spectra
wererecordedusingprimarygasstandardsproducedandsurveillancedatNIST.Thedatabase
containstheassumeduncertaintiesofabsorptioncoefficients.
The collection of spectroscopic parameters is a prerequisite for modelling of the atmo-
sphere of Earth and other planets. This is the aim of the GEISA (Gestion et Etude des Infor-
mations Spectroscopiques Atmosphe´riques) [3, 42, 43] database. The accuracy of the models
highly depends on the accuracy of the spectroscopic parameters. The database contains tran-
sitions, infrared, visible and UV cross sections, and the microphysical and optical parameters
ofatmosphericaerosols.
The HITRAN (High-Resolution Transmission Molecular Absorption Database) [2, 44,
45, 15] database collects those parameters that can facilitate the modelling of atmospheric
absorption and emission spectra. It is the canonical database of atmospheric modelling; thus,
10