Systematic trends in electronic properties of alkali hydrides

0.29 MB·

by  Mireille Aymar

1
Systematic trends in electronic
properties of alkali hydrides
9 Mireille Aymar, Johannes Deiglmayr, and Olivier Dulieu
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n
a
J
Abstract: Obtaining ultracoldsamples of dipolar molecules isacurrent challenge which
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requires anaccurate knowledge of their electronic propertiestoguide the ongoing

experiments. Alkali hydride molecules havepermanent dipole significantlylarger than
]
h those of mixed alkali speciesand, aspointed out byTaylor-Juarroset al. [Eur.Phys. J.D 31,
p 213 (2004)] and by Juarros etal. [Phys. Rev. A 73, 041403 (2006)], arethusgood candidates
- for moleculeformation. Inthispaper, using astandard quantum chemistryapproach based
m
on pseudopotentials for atomic corerepresentation, largeGaussian basissets, andeffective
o core polarization potential, wesystematicallyinvestigate theelectronicproperties of thealkali
t
a hydrides LiH toCsH,inorder todiscussgeneral trends of their behavior. Wecomputed (for
. the firsttimefor NaH,KH, RbH, andCsH) thevariation of their staticpolarizability with
s
c the internuclear distance. Moreover, inaddition topotential curves, wedetermineaccurate
i values of permanent and transitiondipole momentsfor ground and excitedstatesdepending
s
y on theinternuclear distance. Thebehavior of electronicproperties of all alkali hydridesis
h compared toeach other, inthelight of the numerous other dataavailableintheliterature.
p Finally, theinfluence of thequality of therepresentation of thehydrogen electronic affinityin
[ the approach on theresultsisdiscussed.

1 Key words: LiH,NaH, KH, RbH,CsH, alkali hydrides, permanent dipole and transition
v dipole moments, staticdipole polarizabilities.
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PACSNos.: 31.15.ac,31.15.ap,31.50.Bc,31.50.Df
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1 1. Introduction
.
1
Alkalihydrideshavecontinuouslyattractedtheinterestofresearchersinvariousareas.Indeed,they
0
9 are among the molecules with the simplest electronic structure, allowing very detailed comparisons
0 between differenttheoreticalmodels, and with available experimentalresults. The astrophysicaland
: cosmologicalrelevanceofLiHiswellknown[1–3],asitisbelievedtobeformedintheearlyuniverse
v
i byradiativeassociationofHandLi[1].Thedeterminationoftheabundancesofchemicalelementsin
X
stellaratmospheresandthemodelingofthechemistryoftheseelementsintheearlyuniverserequires
r aknowledgeofthecrosssectionsforvariouscollisionalprocessessuchasinelasticcollisionsbetween
a
hydrogenandalkaliatoms(Li,Na)[4,5],radiativechargetransferbetweenalkaliatomsandH+ [6],
dissociativerecombination[7],orneutralizationcollisionsbetweenalkaliionsandH−[8,9].Molecular
MireilleAymar, JohannesDeiglmayr,1 and Olivier Dulieu.2 Laboratoire Aime´ Cotton, CNRS, Baˆt. 505,
UnivParis-Sud,91405OrsayCedex,France
1Present Address: PhysikalischesInstitut,Universita¨t Freiburg, Hermann-Herder-Strasse3, 79104 Freiburg,
Germany
2Corresponding author (e-mail:[email protected]).
unknown 99:1–23(2009) DOI:10.1139/Zxx-xxx (cid:13)c 2009NRCCanada
2 unknownVol.99,2009
reactiondynamicsof alkaliatoms andmolecularhydrogeninvolvealkalihydridesas productsand a
good description of these systems is required to analyze the reaction mechanism [10,11]. Coˆte´ et
al. [12,13] suggested that the formation of Bose-Einstein condensates of a dilute gas of hydrogen
couldbeachievedbycoolinghydrogenatomsinabuffergasofalkaliatoms,whichcriticallydepends
ontherelatedscatteringlengths.RecentlyJuarrosetal.havepredictedthatformationofultracoldLiH
or NaH moleculesis possibleusingvariousprocessessuchas stimulatedRamanphotoassociationor
stimulatedone-photonphotoassociation[14–16].Duetoitslargepermanentdipolemoment,theLiH
molecule[17],andpossiblytheNaHmolecule,maybeconsideredascandidatesforStarkdeceleration.
Also, following the ideas of ref. [18] related to alignment and orientation of polar molecules under
the influence of combined static and laser fields, alkali hydrides could be good candidates for such
experiments.Anaccurateknowledgeoftheelectronicpropertiesofthesemoleculesareprerequisiteto
theextensionoftheseinvestigationstoheavieralkalihydrides.
Followingpreviousworkonalkalidimersinourgroup,weperformedsystematiccomputationsof
the electronic propertiesof alkali hydridesfrom LiH to CsH (Section 2) as well as the monocations
formedfromthesemolecules.Newresultsforthedependenceoninternucleardistanceandonthevi-
brationalindexoftheirdipolepolarizabilitiesarepresentedinSection3.Wealsocompareourresults
withnumerousprevioustheoreticaldataforpotentialcurvesandpermanentandtransitiondipolemo-
ments,identifyinggeneraltrendsforthesequantities(Section4),andallowingforananalysisoftheir
accuracy.Inparticular,theissueoftheinclusionofacorrectionforthehydrogenelectronaffinityisdis-
cussed.Mostofthecomputedresultsofthepresentworkareavailableassupplementarymaterialpro-
videdbytheeditor.Inthefollowingwewilluseatomicunitsfordistances(a0=0.052917720859nm)
andenergies(2R =219474.63137cm−1)exceptwhereotherwisestated.
∞
2. Computational details and potential curves for neutral and ionic systems
As in our previousstudies on alkali dimers, the alkali atoms are describedby ℓ-dependentpseu-
dopotentialsfortheioniccores[19,20]includingeffectivecorepolarizationterms[21,22],andbya
large set of uncontractedGaussian functions for the valence electron. For all atoms but Cs we used
the Gaussian basis sets labeled as ”B” in ref. [23,24], while for Cs we use the basis set labeled as
”B’”inref.[25].Wealsousethesamecore-polarizationpotentials(CPP)thaninourpreviouswork,
with the reported valuesfor the cut-off radii. For the H atom we set up a new large [10s5p2d]basis
set of uncontracted Gaussian functions with the following exponents: 195.5, 27.6, 6.3, 1.8, 0.5983,
0.22, 0.09, 0.04, 0.019,0.006 for s basis functions;1.05, 0.22, 0.0669,0.024270,0.00669for p ba-
sis functions;0.34, 0.061for d basis functions.This choice globallyimprovesthe calculated atomic
levelenergiescomparedtopreviousworks.Weobtainthefollowingdiscrepancies:∆E1s =3.2cm−1,
∆E2s =2.1cm−1,∆E2p =7.4cm−1,∆E3s =29cm−1,∆E3p =51cm−1.Forinstance,Huzinaga
etal.[26]usedalargebasissetofcontractedGaussianorbitalswhichgaveexactenergiesforthe1s,
2s, 2p, 3s and 3p levels. In their recent series of papers, Gade´a and coworkers [27–30] designed a
[7s3p2d]Gaussian basis set contractedto [5s3p2d],close to the oneof Geumet al. [31]whoused a
[7s3p2d]contractedto[6s3p2d]basisset.Theformerauthorsobtaineddeviationsof∆E1s ∼11cm−1
and∆E2s ∼ 700cm−1 [27],andthe latter ∆E1s ∼ 64cm−1 and∆E2s ∼ 35cm−1. Moreoverour
basisalsoprovidesanimprovedhydrogenelectronicaffinitywithadeviationofabout140cm−1from
theexactvalueofref.[32],i.e.threetimessmallerthanthevaluequotedbyBoutalibandGade´a[27]
butslightlylargerthanthevalueofGeumetal.(∼111cm−1).
Themolecularcalculationsare performedjustlike in ourpreviouspapers[23,25,33].Molecular
orbitalsare determinedby a restrictedHartree-Fockcalculationincludingcorepolarization,yielding
electronicenergiesandwavefunctionsofthemolecularion.Electronicenergiesandwavefunctionsof
theneutralsystemarededucedfromafullconfigurationinteraction(CI)oftwo-electronconfigurations,
throughtheCIPSIquantumchemistrycodeofthequantumchemistrygroupinToulouse(France)[34].
Our procedure automatically provides potential curves for the related alkali hydride ions treated as
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Aymar,Deiglmayr,andDulieu 3
effective one-electronsystems, and it is worthwhile to examine them as a first check of its accuracy
compared to other approaches. It is striking to see in Table 1 that even for the ground state of such
simplesystems,publishedvaluesvarysignificantlyforthebindingenergyofLiH+ andNaH+,while
onlyfewdataareavailableuptonowfortheheaviestspeciesKH+,RbH+,andCsH+.Toourknowl-
edge only one experimental value has been published up to now, namely the CsH+ binding energy
extracted from differential cross section measurements [35]. Very few measurements have been re-
portedforothermolecularstates.WealsolistinTable1dataforthefirstexcitedstateA2Σ+whichhas
beenwidelystudiedinthecontextofthetheoreticalcalculationofcrosssectionsforchargeexchange
reactions[1,6,36–39].
Mostofthecalculationsofalkalihydrideionpotentialcurveshavebeencarriedoutwitheffective
core potentials. In a series of recent papers, Magnier [40–42] extended the work of Alikacem and
Aubert-Fre´con[43]byincludingeffectivecorepolarizationpotentialstotheirKlapischmodelpotential
approach[44].Variouskindsofpseudopotentialshavebeenusedbyseveralauthors[36,37,39,45–48],
the closest approach to ours being the work by Berriche and Gade´a [49] for LiH+. A few ab initio
all-electroncalculationshavealso beenperformedmanyyearsago,byRosmusandMeyer[50]with
the coupled electron pair approach, by Vojtik [51] using the MRDCI approach, and by Dalgarno et
al.[1]andOlsonetal.[52]withSCFandCIapproaches,forthetreatmentofradiativeassociationin
anastrophysicalenvironment.Itistedioustoachieveanenlighteningcomparisonofourdatawiththese
works,asmostofthemarequiteoldandwereprobablylimitedbycomputationalfacilitiesatthattime.
Moreover,severalcalculationsreportedbindingenergiesD fortheAexcitedstateonly,withwhich
e
ourresultsreasonablyagree,butwithoutreportingsimilardataforthegroundstate.Ourvaluesforthe
LiH+ groundstate agree wellwith thoseof Magnier[40],butare slightlylargerthanthose obtained
bythesameauthoronNaH+ andKH+ [41,42].FortheAstatethepresentD valuesforLiH+ and
e
NaH+ arealmostidenticalto thosepredictedbyMagnier[40,41]butslightlysmaller forKH+ [42].
Also,thedepthofthepotentialwelloftheAstateofCsH+lieswithintheerrorofthedatareportedin
ref.[35].
Next, the accuracy of our full CI results yielded by our treatment of the neutral molecule as an
effective two-electron system, can be analyzed from Tables 2 and 3, which display the equilibrium
distancesR ,the harmonicconstantω andthe potentialwell depthofthe alkalihydrideselectronic
e e
statestogetherwithavailableexperimentalvaluesandwithseveralotherrecenttheoreticalresults.We
concentrateouranalysisonthefourthlowest1Σ+ andtwolowest1Πelectronicstates,astheycanbe
reachedbyelectricdipoletransitionfromthegroundstate. However,wealso performedcalculations
fortripletstates,andalltheseresultsarecollectedinthesupplementarymaterialattachedtothispaper.
We labelthemolecularstatesaccordingtothestandardspectroscopicnotation,i.e.X,A,C,E corre-
spondingto thefourlowest1Σ+ states, andB ≡ (1)1Π. Similarly,tripletstatesarelabeledwiththe
lowercaseindexesa, b,andc,forthe(1)3Σ+, (1)3Π, and(2)3Σ+, respectively.Inrefs.[27–30],the
authorslabeledthe (4)1Σ+ as D state. Our conventionforthe sign of the permanentdipoleis that a
positivevaluecorrespondstothechargedistributionA+ H−,whereAisanalkaliatom.
The extensive review by Stwalley et al. [54] addresses most of the works already published on
alkalihydrides.Inparticular,thisreviewreportsonexperimentalstudiesofthefirst1Σ+ state(theA
state)fromNaHtoCsH.Vidaletal.[55,56]investigatedthespectroscopyofAandB(thelowest1Π)
states,andYangetal.[57]andHsiehetal.[58]theAstateinKHandinCsH,respectively.
We brieflyrecall below some of the theoreticalcalculations,mainly those yielding tablesof per-
manent or transition dipole moments, for further reference in the next sections. Ab initio multicon-
figurationself-consistent-fieldcalculationswith configurationinteraction(MCSCF-CI) of permanent
and transition dipole momentsof LiH have been reportedby Docken and Hinze [59] using a Slater-
type basis set, in contrast with Gaussian basis sets employed in the present calculations. Partridge
and Langhoff [60,61] repeated such computationswith an extended Slater-type basis set. Roos and
Sadlej[62]usedthecompleteactivespaceSCF(CASSCF)approachtocalculatethepermanentdipole
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momentandthepolarizabilitiesofthe groundstate asfunctionsoftheinteratomicdistance.Sachset
al. determined the electronic properties of several states of NaH [63,64]. Langhoff et al. extended
theirpreviousinvestigationstoNaH,KH,andRbH[65]usingnearHartree-FockqualitySlater basis
sets and incorporating electron correlation through a coupled-pair formalism. Laskowski et al. [66]
studied the two lowest 1Σ+ states in CsH, using a similar approach to ours: effective ℓ-dependent
pseudopotentials derived from relativistic Hartree-Fock calculations and including core polarization
termsforatomiccorerepresentation,andGaussianbasissetsintheCIcalculation.Carnelletal.[67]
investigatedthe first seventeen states of CsH with ab initio multi-referenceconfigurationinteraction
(MRDCI)calculations.Combiningexistingexperimentaldataandab-initiocalculations,Zemkeetal.
determinedthe dipolemomentforthe A1Σ+ → X1Σ+, B1Π → X1Σ+, and B1Π → A1Σ+ tran-
sitionsandrelatedradiativetransitionprobabilitiesinLiH[68,69],andthepermanentdipolemoment
fortheX andAstatesinNaH[70].Finally,Camachoetal.[71]builttheA1Σ+ →X1Σ+ transition
dipolemomentofKHfromavailableexperimentaldata.
MorerecentlyGade´aandcoworkershaveperformedaseriesofstudiesoftheadiabaticanddiabatic
molecularstatesofalkalihydrides[27–30].Asinthepresentwork,theauthorsusedtheℓ-dependent
pseudopotentialsofDurandandBarthelat[19,20],andcore-valencecorrelationfromref.[21,22].Fur-
thermore,anR-dependentcorrectionoftheion-pairdiabaticcurvesisintroducedtoaccountforbasis
set limitations and to ensure an improved value for the ground state energy. Permanent dipole mo-
mentsforseveral1,3Σ+statesandtransitiondipolemomentsbetween1Σ+ stateshavebeendisplayed
for KH [28], RbH [29], and CsH [30]. Khelifi et al. have determined the radiative and nonradiative
lifetimesoftheA1Σ+ andC1Σ+ vibrationallevelsoftheKHmolecule[72,73].
InTable2,weseethattheoreticalvaluesforR aresystematicallyslightlysmallerthantheexper-
e
imentalones,especiallyforCsH. Thiscouldbe dueto theabsenceof short-rangerepulsiontermsin
theECPcalculations.OurresultingvaluesforD areslightlylargerthansimilarcalculationsofGade´a
e
andcoworkers(column(a)intheTable)[27–30],whichisamanifestationoftheinfluenceofthelarge
basiswe usedin ourwork.HoweveritiswellknownthatECP-typecalculationisnota perfectvari-
ationalapproach,so that it may happenthat the computedwell-depthexceedsthe experimentalone,
especiallywhenlargeioniccoresareinvolved(seetheCsHcase).Thispatternisalsovisibleinresults
fromDolg[74].Theyrelyonfullyrelativisticpseudopotentials,andmolecularcalculationsarebased
on Dirac-Hartree-Fockand CI approaches.Core-polarizationterms have been introduced,as well as
correctionsforcore-corerepulsion.Equilibriumdistancesarealsofoundshorterthantheexperimental
ones, while D are larger than the experimental ones for all species. When an empirical correction
e
forthehydrogenelectronicaffinityisincludedbyGade´aandcoworkersforKH, RbH,andCsH (see
section4),thevaluesforD areincreasedandbecomeclosertotheexperimentalones.However,such
e
a correctioncannotbe consideredas a variationalprocedure,so thattheir D valueforCsH exceeds
e
the experimental one by about 462 cm−1. We will comment more about the effect of their ion-pair
correctionondipolemomentsinsection4.
Wesummarizedthemainspectroscopicconstantsofthefourlowest1Σ+ excitedstates,andofthe
lowest1ΠstateinTable3,comparedtoexperimentalobservations,andtoselectedtheoreticalpredic-
tions.Asalreadypointedout,Gade´aandcoworkers[27–30]haveinvestigatedallsystemsbutNaHand
theirpapersincludeextensivecomparisonswith otherpredictionswhicharenotduplicatedhere.For
NaHourvaluesarecomparedtothoseofLeeetal[76],whousedthelarge-corepseudopotentialofthe
Stuttgartgroup[77]andaccountedforcorepolarizationfollowingref.[21].SCFandCIcalculations
have been performed with the MOLPRO package. The X and A states of KH and RbH have been
investigatedbyGarciaetal[80]usingasmall-corepseudopotential[77],andthedifference-dedicated
configuration interaction (DDCI) method to account for core-valence correlation effects. Our values
aregenerallyingoodagreementwiththoseofGade´aandcoworkersalthoughtheGaussianbasissets
usedinthoseworkfordescribingthealkaliandhydrogenatomsdifferfromours.Itisdifficulttoknow
ifthesomewhatlargedifferenceswiththevaluesforNaHof[76]arerelatedtothedifferencesinthe
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Aymar,Deiglmayr,andDulieu 5
pseudopotentialortothoseinthebasissets.
3. Static dipole polarizabilities
If the z axis is chosen along the internuclear axis in the molecule-fixedreference frame (x,y,z),
therearetwoindependentcomponentsofthemolecularpolarizabilitytensor,i.e.,theparallelcompo-
nentα ≡α andtheperpendicularoneα ≡α =α .Tworelatedquantitiesareusuallydefined:
k zz ⊥ xx yy
theaveragepolarizabilityα = (α +2α )/3andthepolarizabilityanisotropyγ = α −α .Asin
k ⊥ k ⊥
ourpreviousworkonthe static dipolepolarizabilitiesofalkalidimers[82],we determinedthestatic
dipole polarizabilitiesof alkalihydridesin their groundstate with the finite-fieldmethod[83],using
electricfieldsbetween0.0003and0.0005a.u.toremainintheperturbativeregime.Wecalculatedboth
componentsofthestaticdipolepolarizabilityforallsystemsasfunctionsoftheinternucleardistance
R(Figure1).Forallsystems,theR-variationsofα andα aresimilar,withanincreaseofα with
k ⊥ k
increasingalkalimass,andasmoothincreaseofα withR.Toourknowledge,nootherR-variationof
⊥
thealkalihydridedipolepolarizabilityhasbeenpreviouslypublished,exceptforLiHinref.[62]which
displaysadependencyinreasonableagreementwiththepresentone(Figure1(c)).LookingatlargeR,
wenotehoweverthatthelithiumpolarizabilityseemstobeslightlyunderestimatedinref.[62].Also,
Kolos and Wolniewicz [84] foundsimilar variationsof the H polarizabilities than the present ones,
2
butwithsmallervalues,asexpected.
Byintegratingthepolarizabilitiesoverthevibrationalwavefunctions,wededucethev-dependency
ofαandofγ,whichregularlyincreasefromLiHtoCsH,justlikefortheseriesofalkalipairs(Figure
2(a)). The anisotropies reach quite large values (up to 967 au. for CsH at v = 24, i.e. two times
largerthanthecesiumatomicpolarizability),magnifyingtheinfluenceofthehydrogenatomwhenthe
moleculeisvibrationallyexcited,incontrastwiththev =0case.Noothervalueshavebeenpublished
inthepast.WealsodisplayinFigure1(b)thepermanentdipolemomentofthealkalihydrideground
state which show similar v-dependence for all the systems, with a maximum followed by a rapid
decreasetozero,asexpectedfromcomparablecalculationsforalkalipairs[23].
Inourpreviousworkaboutthegroundstatepolarizabilitiesofallhomonuclearandheteronuclear
alkalidiatomics[82],bothα andα componentswerefoundtovarylinearlywithaneffectivevolume
k ⊥
V = 4π(R )3/3,α varyingtwotimesfasterthanα withV .Thissuggestedthepictureofan
eff e k ⊥ eff
effectiveelliptic chargedistributionforthe alkalipair atthe equilibriumdistance. Inalkalihydrides,
thisscalinglawremainsvalidonlyforα ,whileα doesnotmonotonicallyvaryalongtheseriesof
k ⊥
alkalihydrides.Thismaybeduetotheweaknessofthehydrogenpolarizability,whichisatleastone
orderofmagnitudesmaller(4.5a.u.[85])thanthealkaliones([82]).Thereforetheeffectiveelliptic
chargedistributionofthemoleculeisprogressivelydominatedbythepolarizabilityofthealkaliatom,
alongtheseriesfromLitoCs.
Mostprevioustheoreticalworksreportedpolarizabilityvaluesattheexperimentalequilibriumdis-
tance Rexp ofthe groundstate, whichareassembledin Table 4,togetherwith the presentresultsfor
e
α ,α ,αandγcalculatedeitheratRexp[54]orforv =0.TheLiHmoleculeisthemostextensively
k ⊥ e
studiedsystem, andwe giveonlythe mostsignificantvalues.Additionalreferencesmaybefoundin
thequotedpapers.CoupledHartree-FockperturbationtheoryhasbeenusedbyLazzerettietal.[86],
and multiconfigurationaltime-dependentHartree-Fock (MCTDHF) method by Sasagane et al. [87].
Multiconfigurationself-consistentfield(SCF)approacheshavebeenusedatvariouslevelsbyseveral
authors[62,88–90].Inthesepapers,thefinitefieldmethod[83]orthechargeperturbativeapproach[89]
havebeenusedtoextractpolarizabilities.Atime-dependentgaugeinvariantmethod(TDGI)hasbeen
usedbyRe´ratetal.[91],whileVrbiketal.[92]employedadiffusionquantumMonte-Carlo(DQMC)
approach.Sadlejandcoworkersaretheonlyauthorswhoinvestigatedstaticdipolepolarizabilitiesfor
other species than LiH using many-body perturbation theory (MBPT) and coupled-cluster theories
combinedwith finite field methods[93,94]. Most of the above-quotedauthorspresented predictions
obtainedwithdifferentbasissetsoratdifferentlevelsofapproximation,andweonlyreportinTable4
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2000
(a) LiH
NaH
1500
KH
RbH
el CsH
all1000
r
a
P
)
u.
a. 500
(
y
bilit
a
z
ri 0
a
ol
p (b)
c
ati 300
St ar
ul
c
ndi 200
e
p
r
e
P 100
5000
5 10 15 20
)
u. 400 (c)
a.
(
y
bilit 300
a
z
ri
a
ol 200
p
c
ati
St 100
0
5 10 15 20
R (a.u.)
Fig. 1. (a) α and (b) α components of the static dipole polarizability of the alkali hydride ground stateas
k ⊥
functionsoftheinternucleardistance.(c)ForLiH,ourresultsforα (fullline)andα (dashedlines)arecompared
k ⊥
withthoseofRoosandSadlej[62](X-es).
(cid:13)c2009NRCCanada
Aymar,Deiglmayr,andDulieu 7
) (a)
u.
a. 800 LiH
(
es NaH
biliti KH
a RbH
z
ari 400 CsH
ol
p
c
ati
St
0
) (b)
u.
a. 800
(
s
e
biliti
a
z
ari 400
ol
p
c
ati
St
) 0
e
y (c)
b 10
e
D
(
nt
e
m
o
m
ole 5
p
di
nt
e
n
a
m
r
e 0
P
0 5 10 15 20 25 30
Vibrational level
Fig.2. (a)averagepolarizability(dashedlines),(b)polarizabilityanisotropy(fulllines)(inau.),and(b)perma-
nentdipolemoment(inDebye,with1a.u.=2.54158059D)ofthealkalihydridegroundstate,asfunctionsofthe
vibrationallevel.
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8 unknownVol.99,2009
valuesobtainedwiththeirmostsophisticatedmodel.
Valuesforthe polarizabilitiesatR are similar tothose forv = 0forallspeciesbutCsH within
e
2%,simplybecausethesteepnessofthepolarizabilityfunctionsaroundR iscompensatedbythesym-
e
metriccharacterofthev = 0wavefunction.ForCsH,the9%differencecomesfromourequilibrium
distance,whichisslightlysmallerthantheexperimentalone.ForLiHandNaHourpredictionsagree
wellwithothervalueswiththeexceptionofthevaluesofGreadyetal.[88]obtainedwithatoosmall
basisset.Therelativelywidevariationofthetheoreticalvaluesclearlyemphasizesthehighsensitivity
of the polarizabilityto the size of the basis sets and to correlation effects. Our predictionsof α for
k
heavier systems lie between the two quoted values of ref. [93], while our values for α are slightly
⊥
smaller.Notethatthedipolemomentscalculatedwiththeirapproximation(b)inthatpaperarecloser
toourvaluesthantheir(a)values.
Wearenotawareofanypolarizabilitymeasurementinalkalihydrides.Anoldexperimentalvalue
of the polarizabilityanisotropyfor LiH with a largeerrorbar (γ = 1.7±4.au) hasbeen quotedby
StevensandLipscomb[95]asanunpublishedresultfromKlempereretal..
4. Trends of the permanent and transition dipole moments of alkali hydrides
WereportinTable4ourvaluesforthepermanentdipolemomentµ attheexperimentalequilib-
z
riumdistanceofthegroundstate.Wenotethattheyarequiteabitlargerthanthoseofalkalidimers[23]
(ranging between 0.5 and 5 Debye), due to the large asymmetry of the charge distribution. Our val-
ues show good agreement with the limited experimental data available (currently only for LiH and
NaH[96,97]).Table4reflectsthesituationwhichhelduntilrecentlyforthetheoreticalworks:many
ofthemweredevotedtoLiH,andonlyafewtoheavierspecies.Inparticular,weobtainaverygood
agreementwiththemostelaboratemodelofSadlejandUrban[93].
WealsocomputedtheR-variationofthepermanentandtransitiondipolemomentsforaselection
ofelectronicstates.Itiswell-knownthatsuchfunctionsrepresentsensitivetestsfortheaccuracyofthe
computedelectronicwavefunctions.Asmentionedearlier,manytheoreticalworkshavebeenalready
published for these quantities, and we concentrate on systematic trends in their behavior along the
series ofalkalihydrides.Itis worthnotingthatsuch ananalysiscan be performeddueto the wealth
of data concerning alkali hydrides, in contrast with the few data available for alkali dimers (see for
instanceref.[33]).
In the followingfigures, we display the computedR-variationof the permanentdipole moments
fortheX,A,B,C,a,andbstatesofalkalihydrides,comparedtootheravailablestudies.Asexpected
the most studied state is the X groundstate (Fig. 3(a)).Its permanentdipole momentshows similar
regularvariationswithRforallspecies,reachingahighermaximumamplitudeatlargerdistancewith
increasingmassandpolarizability.Theagreementbetweenallmethodsisgenerallysatisfactory,with
theexceptionofref.[67]wherethemaximumvalueisfoundtobeabout20%larger.Thepermanent
dipolemomentsfortheAstate(Fig.3(b))varyliketheonefortheX state,reachingtheirmaximum
value at larger distances. Severalother determinationsexist in the literature for the LiH and NaH A
state. We founda remarkableagreementfor LiH with refs. [59,60].In contrast,the maximumvalue
foundbySachsetal.[63]isalmosttwotimessmallerthanours.Asforthegroundstate,themaximum
valuefoundbyCarnelletal.forCsHislarger(byabout25%)thanours,whiletheonebyLaskowskiet
al[66]isabout25%smallerthanours.Toourknowledge,theonlyothercalculationsforKH,RbH,and
CsHhavebeenreportedbyGade´aandcoworkersinrefs[28],[29],and[30],respectively.Notablesof
numericaldataareavailableinthesepapers,butasatisfactoryagreementisvisible,aslongastheeye
canjudgefromtheirfigures(seethediscussionforRbHlaterinthissection).Finallyourcalculations
for boththe X andA states of NaH are in excellentagreementwith those of Zemke et al. [70]who
extractedthedipolemomentfunctionsfromananalysiscombiningab-initioandexperimentalresults.
Asexpected,theR-variationoftheA−X transitiondipolemoment(Figure4a)reflectsbothvaria-
tionsoftheX andApermanentdipolemomentsthroughthecorrespondingelectronicwavefunctions:
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Aymar,Deiglmayr,andDulieu 9
20
(a)
) Docken&Hinze (1972)
e
y Partridge&Langoff (1981)
b
e 15 Langoff et al (1986)
D
( Sachs et al (1975)
ent Zemke et al (1984)
m Carnell et al (1989)a
o
m 10 Carnell et al (1989)b
e Laskowski et al (1981)
ol
p
di
nt
ne 5 X-state
a
m
r
e
P
0
40
(b)
A-state
)
e
y
b 30
e
D
(
nt
e
m
o 20
m
e
ol
p
di
nt 10
e
n
a
m
r
e
P 0
5 10 15 20
R (a.u.)
Fig.3. ComputedpermanentdipolemomentsofX1Σ+(upperpanel)andA1Σ+(lowerpanel)statesasfunctions
oftheinternuclear distance.Ourresultsare:LiH(blackfullline),NaH(reddashedline),KH(greendot-dashed
line),RbH(bluedottedline),CsH(orangelong-dashed line).Otherdeterminationsaredisplayedfromref.[59]
(blackfullcircles),ref.[60](blackopencircles),ref.[63](opensquares),ref.[70](plussigns),ref.[65](stars),
ref.[66](fulltriangles),andref.[67]withoutorwithrelativisticeffects(dottedlineandfullcircles,ordottedline
withopencircles,respectively).
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10 unknownVol.99,2009
two extremumarevisible,whosepositionscorrespondto thoseobtainedfortheX andApermanent
dipolemomentfunctions.ThesamefigurealsodisplaysresultsconcerningthetransitionsB−X and
B −Ainvolvingthe lowestB1Π excitedstate (Figure4b,c).Anexcellentagreementis foundagain
with refs.[59,60]forallthese transitionsinLiH. Quiteunexpectedly,the agreementwith the values
ofSachsetal[63]forNaHlooksbetterthanfortheApermanentdipolemoment.ForCsH,thecurve
by Laskowski et al [66] is close to ours in the region of the maximum value of the X dipole mo-
ment (below 10 a.u.), and notin the regionof the one of the A dipole moment(around15 a.u.). As
inthepreviousfigure,resultsfromref.[67]disagreewithours.Again,theagreementfortheA−X
transitiondipolemomentsseemsatisfactorywhenlookingatfiguresfromrefs.[28–30]forKH,RbH,
andCsH.Whilenotdisplayedhere,resultsarealsoavailablefortransitiondipolemomentsamongthe
lowest triplet Σ+ and Π states (c−a, b−a, c−b) in LiH [59] and NaH [63], in good agreement
withthepresentresults.ExperimentalvaluesareonlyavailableforA−X andB−X transitionsin
LiH[68,69],andourvaluesareinexcellentagreementwiththem.AdeterminationoftheA−X tran-
sitiondipolemomentvariationinKHhasbeenrecentlyextractedfromtheexperimentbyCamachoet
al.[71](whichistoourknowledgetheonlyexperimentaldataforthismolecule)showingareasonable
agreementwithourvalue.
InFigure5we displaythepermanentelectricdipolemomentforhigherexcitedsingletstates (B
and C), and for the two lowest triplet a and b states. While less data are available for comparison,
we observethe same trendthan in the previousfigures:the systematic investigationof Carnellet al.
ref. [67]really seemsto missmost ofthe characterof the excited electronicwave functionsof CsH,
whileourresultsareinagreementwiththoseofref.[60]fortheBstateofLiH,andofref.[63]forthe
B,a,andbstatesinNaH.Avisualsatisfactoryagreementisalsoobservedwiththefiguresdisplayed
inrefs.[28–30].
Asexpected,allofthesedipolemomentsvanishwithincreasingdistance,withasmallerratewith
increasingalkalimass.TheC stateisthemostexcitedstatedisplayedinthefigure-correlatedtothe
secondexcited1s+3s,1s+4s,1s+5s,1s+4d,and1s+5ddissociationlimitsforLiH,NaH,KH,RbH,
and CsH, respectively-so its magnitudereacheslargervalues than for the other states. For all alkali
hydrides,theC dipolemomentexhibitsirregularvariations,whichisthewell-knownmanifestationof
the contributionof the ion-pair configurationin the electronic wave function(see Figure 6): as long
as this ion-pair character dominates the wave function (between the avoided crossings marked with
circlesinFigure6),thepermanentdipolemomentcantakelargevalues,untilthedistancewherethe
covalent character shows up again. In LiH and NaH, the presence of the avoided crossing C and E
states atshortdistancesisalso visiblein Figure5c(see arrows).Quite unexpectedly,the CsH dipole
momentisnotthelargestoneoftheseries,astheion-paircharactercannotdeveloptowardsdistances
largerthan22a.u.duetothesomewhatsmallerenergyspacingbetweentheexciteddissociationlimits,
comparedtotheotherspecies.
Theinfluenceoftheion-paircharacterisalsovisibleinthetransitiondipolemomentsamongthese
excitedelectronicstates, asvisible onFigure 7 forthe C −A andE −C transitions.The transition
dipole moment functions change very abruptly with the internuclear distance (see arrows), which is
theexpectedmanifestationoflocalchangeoftheelectronicwavefunctionsatthelocationofavoided
crossingsbetweentherelatedpotentialcurves.
The treatment of the ion-pair contribution is actually a sensitive issue for discussing the quality
oftherepresentationoftheelectronicwavefunctionsofthemoleculesinvariousapproaches.Indeed,
suchacharacterisnotexplicitlyintroducedinthestandardGaussianbasissetsforneutralatoms,and
wouldneedinprincipleaninfinitenumberofbasisfunctionstobeaccuratelyrepresented.Inthepresent
work,wesetupalargebasisforthehydrogenatom,inordertoreproducetheatomicenergylevels,and
hopefullytotakeinaccountasmuchaspossibleoftheion-paircharacter.Anindicationforthisisour
computedvalueofthehydrogenelectronicaffinityEA = 5938cm−1,comparedtotheexperimental
valueEAexp = 6083cm−1[32].Letusnotethatinref.[76]mentionedinTable3theauthorsuseda
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