field Non-Born–Oppenheimer calculations of the HD molecule in a strongmagnetic Chemical Physics Letters

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Chemical Physics Letters

j o ur na l h o me p a g e :w w w . e l s e v i e r . c o m / l o c a t e / c p l e t t

Non-Born–Oppenheimer calculations of the HD molecule in a strong magnetic field

Ludwik Adamowicz

, Erik I. Tellgren

, Trygve Helgaker

aDepartmentofChemistryandBiochemistry,UniversityofArizona,Tucson,AZ85721,USA

bCentreforTheoreticalandComputationalChemistry(CTCC),DepartmentofChemistry,UniversityofOslo,P.O.Box1033Blindern,N-0315Oslo,Norway

a rt i c l e i n f o

Articlehistory:

Received7July2015

Infinalform28September2015 Availableonline9October2015

a b s t ra c t

Aneffectivevariationalnon-Born–OppenheimermethodisappliedtocalculatethegroundstateoftheHD moleculeinastrongmagneticfield.TheHamiltonianusedinthecalculationsisobtainedbysubtracting theoperatorrepresentingthekineticenergyofthecenter-of-massmotionfromthetotallaboratory- frameHamiltonian.Orbitalbasissetsareusedforthedeuteron,theproton,andtheelectrons.Basedon thecalculatedexpectationvalues,itisdeterminedthat,withincreasingfieldstrength,thebondlength decreasesandthealignmentofthemoleculewiththefieldincreases.

©2015ElsevierB.V.Allrightsreserved.

1. Introduction

Theinteractionofatomsandmoleculeswithexternalmagnetic fields,includingverystrongfields,hasbeenintenselystudiedfor thelastthreedecades[1–13].Theconventionalapproachforcal- culatinggroundandexcitedboundstatesofatomsandmolecules placedinauniformmagneticfieldwithouttheBorn–Oppenheimer (BO)approximationinvolvestheseparationofthecenterofmass motion.Thisisusuallydonebyselectingacoordinatesystemin whichtheHamiltonianautomaticallyseparatesintoanoperator representingthekineticenergyofthecenter-of-massmotionand aninternalHamiltonianrepresentingthesystem’sinternalstate.

Therehavebeenseveralworksconcerningtheseparationofthe center-of-mass motionfor molecular systems (mostly diatomic systems)interactingwithauniformmagneticfield[14–22].The propertiesofthetotalpseudo-momentumofthesystemexpressed intermsofrelativecoordinateswereexploitedinthoseworksto partitiontheresultingHamiltonianoperatorintoanelectronicpart andanuclearpart.Althoughthemagnetic-fieldmodificationofthe nuclearHamiltonianandtheelectronic–nuclearcouplinghavenot beeninvestigatedindetail,therehavebeenreportsonthevalidity oftheBOapproximationforahydrogenatom[15]andadiatomic molecule[16,20],andonhowthemagneticfieldaffectstherota- tions and vibrationsof neutral diatomicmolecules [21]. Toour knowledge,therehavebeennopracticalapplicationsofthedirect

∗Correspondingauthor.

E-mailaddress:[email protected](L.Adamowicz).

non-BOapproachincalculationsofgroundandexcitedstatesof molecularsystemsinthepresenceofamagneticfield.

Herewepresentanalternativenon-BOmethodfordescribing thebehaviorofamoleculeinamagneticfield.TheHamiltonian operatorusedintheapproachisobtainedbysubtractingtheoper- atorrepresentingthemotionofthecenterofmassexpressedin the Cartesianlaboratorycoordinate systemfromthetotal non- relativisticHamiltonianexpressedinthiscoordinatesystem.The approachisimplementedandusedinmodelcalculationsoftheHD molecule.

2. Themethod

In the present calculations, we employ the effective non- Born–Oppenheimer (non-BO) method introducedby Kozlowski andAdamowicz[23].Thestartingpointisthetotalnonrelativis- ticHamiltonianofthesystemwithoutamagneticfieldwrittenin termsofthelaboratoryCartesiancoordinates,R_l,l=1,...,N,where Nisthenumberofparticlesformingthesystem(thenumberof thenucleiplusthenumberoftheelectrons).Inatomicunits,the Hamiltonianis:

Hˆ=−

!

l

1 2M_l∇²R_l+

!

k>l

Q_kQ_l

R_kl . (1)

In(1),M_landQ_larethemassandthechargeofparticlel,respec- tively,andR_klisthedistancebetweenparticlekandparticlel.In http://dx.doi.org/10.1016/j.cplett.2015.09.051

0009-2614/©2015ElsevierB.V.Allrightsreserved.

thenextstep,thecoordinatesofthecenterofmassofthesystem aredefined:

Rcm=

"

M , (2)

whereM=

"

lM_listhetotalmassofthesystem,andusedtowrite theoperator, ˆT_cm,representingthekineticenergyofthecenter-of- massmotion:

Tˆcm=− 1

2M∇²Rcm. (3)

TheeffectiveHamiltonianrepresentingtheinternalenergyof thesystem, ˆH_int,isthedifferencebetweenthelaboratory-frame Hamiltonian, ˆHand ˆT_cm:

Hˆ_int=Hˆ−Tˆcm. (4)

Notethat,as ˆT_cm= _2M¹ Pˆ²_cm,where ˆP_cmisthecanonicalmomen- tumoperatorassociatedwiththemotionofthecenterofmass,

Tˆcm=(

"

2M , (5)

where ˆp_listheoperatorrepresentingthecanonicalmomentumof particlel. ˆH_intisusedinthecalculationoftheinternalboundstates ofthesystem.

Asonenotices,intheeffectivenon-BOmethod,aseparation ofthelaboratory-frametotalHamiltonianintoaninternalHamil- tonian and a center-of-mass kinetic-energy Hamiltonian is not explicitlyperformed.Instead,aneffectiveinternalHamiltonianis constructedasisthedifferenceoftheformerandthelatter.Note thattheeffectiveinternalHamiltoniandependsonthelaboratory coordinatesofallparticlesformingthesystem.Thewavefunction thereforealsodependsonthesecoordinates.

The wave function in the effective non-BO method, which dependsonthelaboratorycoordinatesoftheparticles(nucleiand electrons)formingthesystemandontheirspins,hastobeproperly symmetrized(forbosons)andantisymmetrized(forfermions)with respecttothepermutationswithineachgroupofidenticalparticles.

Forexample,thewavefunctionrepresentingtheHDmoleculehas tobeantisymmetricwithrespecttothepermutationoftheelec- tronlabels.Moreover,thewavefunctionrepresentingtheground statehastobeaproductofaspatialfunctionthatissymmetricwith respecttothepermutationoftheelectronsandaspinfunctionthat isanantisymmetricfunctionwithrespecttothispermutation.The spatialwavefunctionrepresentingtheground-stateHDmolecule is:(1+P(1,ˆ 2))!(R_d,Rp,R1,R2),whereindicesd,p,1,and2indi- catethedeuteron,theproton,andthetwoelectrons,respectively.

Toobtainthetotalwavefunction,theabovespatialfunctionismul- tipliedbytheproductofaspinfunctionforthedeuteron,aspin functionfortheproton,andanantisymmetricspinfunctionforthe twoelectrons.

Forhighaccuracyinnon-BOcalculationsofboundstatesofa systemwithCoulombicinteractions,thewavefunctionshouldbe expandedinbasisfunctionsthatexplicitlydependonthedistances betweenparticles,knownasexplicitlycorrelatedbasisfunctions [24,25].However,in thisfirst modelfor non-BOcalculationsof amoleculeinamagneticfield,weuseanexpansioninproducts ofone-particlefunctions(orbitals).Asimilarorbitalapproachfor describingthenuclearmotionwasusedbeforeby,forexample, Nakaietal.[26]andNakatsujietal.[27].Threedifferentsetsofsuch functionsareusedfortheHDmolecule,thesystemconsideredin thisinitialmodelapplication.Thefirstsetcontaintheorbitals, ^d_i, usedtoexpandthecomponentofthetotalwavefunctiondescrib- ingthestateofthedeuteron,thesecondsetcontaintheorbitals,

p

i,describingthestateoftheproton,andthethirdsetcontain theorbitals, ^e_i,describingthestateofthetwoelectrons.Thus,the

totalspatialground-statewavefunctionfortheHDmoleculeisa superpositionofproductsoftheorbitalsbelongingtothethreesets:

(1+P(1,ˆ 2))!(Rd,Rp,R1,R2)

=

!

i

!

j

!

k≥l

c_ijkl ^d_i(R_d) ^p_j(Rp)(1+P(1,ˆ 2))

#

k(R1) ^e_l(R2)

$

wherec_ijklarethelinearexpansioncoefficients.

Wenowintroducetheinteractionwithastaticmagneticfield intotheeffectiveinternalHamiltonian, ˆH_int.Indoingthis,wedistin- guishbetweenthecanonicalmomentumoperator ˆp_l=−i!∇land thegauge-invariantkineticmomentumoperator

ˆ

"_l=−i!∇l−Q_lA(r_l), (7)

whereQ_listhechargeofparticlel.Themagneticvectorpoten- tialA(r)representsanexternalmagneticfieldB=∇×A(r).Inthe presentwork,werestrictattentiontouniformmagneticfieldsand vectorpotentialsoftheform

A(r_l)= 1

2B×(r_l−g), (8)

wheregisthegaugeorigin.Thetotalkineticenergyoperatorinthe presenceofastaticmagneticfieldisthengivenby

Tˆ=

!

l

ˆ

"_l² 2M_l

=

!

l

#

$

l, (9)

where wehave omittedthespin Zeemaninteractions. Thesec- ondtermintheparenthesiscanbeidentifiedasB·ˆL_g;l,with ˆL_g;l= (r_l−g)×pˆ_ldenotingthecanonicalmomentumrelativetog.This termrepresentstheorbitalZeemaneffectandvanishesforstates withzeroangularmomentum.Fortime-reversalsymmetricstates, representedbyreal-valuedwavefunctions,thistermiszero.The termisalsotrivialforstatesthatarerelatedtoatime-reversalsym- metricstatebyagaugetransformation,whichintroducesthesame single-valuedphasefactorintoallparticlecoordinates.

Usingthevectoridentity|u×v|²=u²v²−(u·v)²,weobtainfor thediamagneticcontributionintheHamiltonian

q²_l|A(r_l)|²=q²_l

4B²|r_l−g|²−q²_l

4[B·(r_l−g)]² (10) Takingtheoriginofthecoordinatesystemtocoincidewiththe gaugeorigin andthez-axistocoincidewiththemagneticfield direction,weobtain

g=0, B=Bez=(0,0,B_z); (11) hence,B_x=B_y=0.Effectively,theparticlesarethensubjecttoan externalpotentialoftheform

V_l(xl,y_l,z_l)= q²_l

2M_l|A(rl)|²= q²_l

8M_lB²_z(x_l²+y²_l), (12) wherer_l=(x_l,y_l,z_l).Thepotentialplaceseachparticleinaquadratic (harmonic)wellthatisproportionaltothesquaredparticlecharge, thesquaredfield strength,andtheinverseparticlemass.More- over,thepotentialactingonparticleldependsonlyonitsxand ycoordinates,beingindependentofitszcoordinate.Thenon-BO Hamiltonianrepresentingamoleculeplacedinthemagneticfield alongthezaxisisasumof ˆH_intandB²_z

"

l q²_l

8m_l(x²_l +y²_l).Theorbital Zeemantermsmaybeomittedbecauseourchoiceofbasisseteffec- tivelytargetsstatesforwhichtheyvanish.

Theterm ˆH_intincludessubtractionoftheoperatorrepresenting thekineticenergyofthecenter-of-massmotion(5).Inthepresence ofthemagneticfield,thecanonicalmomentumoperators ˆp_lin ˆTcm

arereplacedbythekineticmomentumoperators ˆ"_l(7),

Tˆcm= (

"

2M . (13)

Inthespiritoftheeffectivecharacterofthepresentmethod,we heremakeanapproximation,representing ˆT_cminthepresenceof thefieldas:

Tˆcm= #ˆ²

2M ≈[ ˆP+Q_totA(Rcm)]²

2M (14)

where Q_tot=

"

lQ_l is the total charge of thesystem and ˆ#=

"

l"ˆ_l.SinceQ_totiszeroforaneutralsystem,thesecondterminEq.

(14)vanishes.Theapproximationin(14)isbasedonthefactthat, classicallyspeaking,thecenter-of-massmotionofthesysteminthe presenceofthemagneticfieldisacombinationofsystem’stransla- tionalmotionandthemotioninducedbytheinteractionofthefield withthetotalchargeofthesystem;theapproximationsomewhat modifiesthecouplingbetweenthemotionsoftheindividualpar- ticlesandthecenter-of-massmotionthatappearsinthemagnetic field[9].Specifically,atermproportionaltothecrossproductofthe magneticfieldandtotaldipolemomentismissingintheparenthe- sisontheright-handside.Intheabsenceofafield,thecoupling vanishesandthetotal nonrelativisticlaboratory-frameHamilto- niancanberigorouslyseparatedintoanoperatorrepresentingthe center-of-massmotionandaninternalHamiltonianindependent onRcm.

Intheabsenceofamagneticfield, ˆH_intisisotropic,i.e.,spher- ically symmetric). Its eigenfunctions are therefore ‘atom-like’, transformingaccordingtotheirreduciblerepresentationsofthe SO(3) rotation group– in particular,theground state ofHD is representedby aspherically symmetric wavefunction.Whena uniform magnetic field B=(0, 0, B_z) is applied, the rotational motionofthemoleculebecomes‘polarized’.Withincreasingfield strength, HD becomes increasingly more aligned along the z- axis, as its ground-state non-BO wave function becomes more cylindrical.

To describethecylindrical symmetryin theorbitalapproxi- mationadoptedhere(6),theorbitalsrepresentingthedeuteron,

d

i(R_d),theproton, ^p_j(Rp),andtheelectrons, ^e_k(R_i),musthave adefinitesymmetrywithrespecttorotationaboutthezaxis.The mostgeneralsuchstateswouldacquirephaseshiftsundersmall rotationsaboutthezaxisandhavemultivaluedphasefunctions (comparesphericalharmonics).Theorbitalsusedinthepresent calculations are spherically symmetric Gaussian functionswith centersshiftedalong thezaxis:g_k(r)=exp[−˛_k(r−s_k)²],where sistheshiftvector(intheHDcalculationss_kx=s_ky=0ands_kz =/ 0).

Asaconsequence,onlystatesofzerocanonicalangularmomentum Lˆg;lcanbedescribed.

In thepresent calculations, we usethe standard variational method. The ground-state wave function and its energy are

obtained by energy minimization with respect to the linear expansioncoefficients, c_ijkl,theorbitalexponents,˛_k, andthez coordinates oftheGaussianshiftvectors, s_k. Sincethesubtrac- tionof ˆTcm in(4)effectivelyeliminatesthetranslationalmotion fromthedescription,optimizationofthenonlinearparametersof theorbitalsrepresentingthedeuteron,theproton,andtheelec- trons,doesnotdelocalizethesystemby‘diffusing’itinspace.Such a diffusing wouldhave occurredif thekinetic energyhad been partoftheenergythatisminimizedbecausetheminimumenergy (zero)wouldthenrepresentasituationwiththecenterofmass completelydelocalizedinspace.

Inpresentingtheresults,weusethenotation(e/p/d)todenote thenumbersoftheelectronorbitals(e), theprotonorbitals (p), andthedeuteronorbitals(d),respectively.Theinitialvaluesofthe orbitalexponentsarechosenusingtheSTO-3Gbasissetforthe electroninthehydrogenatom,theexponentialparametersforthe nucleiderivedfrommolecularnon-BOcalculationsperformedwith explicitlycorrelatedGaussianfunctionsbytheAdamowiczgroup [28],andtheaverageground-stateHDinternucleardistancefrom thesamecalculations.Intheinitialbasisset,theprotonorbitals aredistributed overarelativelyshortintervalofthezaxis.The deuteronorbitalsaredistributedinasimilarwaysuchthattheir locationonthezaxisisonaverageequaltotheaverageinternu- cleardistanceinthegroundvibrationalstateoftheHDmolecule.

Weemphasizethatthepresentcalculationsarepreliminary,being performed to test thepossibility of describing a molecule in a static magnetic field without assuming the BO approximation.

MoreaccuratecalculationswithexplicitlycorrelatedGaussiansare forthcoming.

3. Results

InTable1,thetotalnon-BOenergiesofHDcalculatedinazero magneticfieldandinthemagneticfieldwiththestrengthsof0.15 and0.30atomicunitsalongthez-axisareshown,usingbasissets rangingfromthesmallset(6/1/1/)witha totalof 8orbitals to thelargerset(12/4/4)with20orbitals.Asthenumberoforbitals increases,thezero-fieldenergydecreasesbutisstillsignificantly higherthanthenon-BOground-stateenergyobtainedusingalarge basis ofexplicitlycorrelatedGaussians,withthecenter-of-mass motionexplicitlyseparatedfromtheHamiltonianbyacoordinate transformation[28].Alargenumberofdeuteron,proton,andelec- tronorbitalswouldbeneededtoobtainanon-BOenergycloseto thatobtainedusingexplicitlycorrelatedGaussiansorotherexplic- itlycorrelatedfunctions.Wenotethatorbitalsalignedalongthe field directioncorrectlydescribethenon-sphericalsymmetryof themoleculeinthepresenceofafield.Intheabsenceofafield, however,thetotalmolecularnon-BOwavefunctionisspherically symmetricandtheuseofGaussiansdistributedalongz-axisisfor- mallynotcorrect,althoughtheerrorintroducedinthismanneris muchsmallerthanothererrorsintroducedinthepresentcalcula- tions.

Eventhoughthenon-BOorbital-basedwavefunctionsobtained herearehighlyapproximate,theexpectationvaluescharacterizing Table1

Totalenergies(E)calculatedfortheHDmoleculewithandwithoutthemagneticfieldinhartrees.ThemagneticfieldBisorientedalongthebondofthemolecule(thez axis)andhasthelengthsof0.15and0.30a.u.Thenotation(e/p/d)indicateshowmanyorbitalsareusedfortheelectrons(e),fortheproton(p),andforthedeuteron(d).The energiesareina.u.

No.oforbitals(e/p/d) Ewithoutfield Ewithfield0.15a.u. Ewithfield0.30a.u.

6/1/1 −1.1037636163 −1.0947683818 −1.0690340166

8/2/2 −1.1136166947 −1.1046585541 −1.0791143507

10/3/3 −1.1174273356 −1.1083813961 −1.0830439178

12/4/4 −1.1189995965 −1.1100736458 −1.0845971917

Exact[28] −1.1654719220

Table2

SomeexpectationvaluescalculatedfortheHDmoleculewithandwithoutthemagneticfieldinhartreesusingthenon-BOwavefunctions.ThefieldBisorientedalongthe bondofthemolecule(thezaxis)andhasthelengthsofBz=0.15a.u.andBz=0.30a.u.Theexpectationvaluesshowninclude:theaveragedistancebetweenthedeuteronand theproton,(|⟨rd−rp⟩|),andthesquaresoftheaveragedisplacementsofthedeuteronandtheprotonfromthezaxis(i.e.fromtheaxisalongwhichtheBvectorisnotzero),

⟨x²_d⟩and⟨xp²⟩,respectively.Thenotation(e/p/d)indicateshowmanyorbitalsareusedfortheelectrons(e),fortheproton(p),andforthedeuteron(d).Theresultsareina.u.

No.ofFSGOs(e/p/d) Bz |⟨rd−rp⟩| ⟨x_d²⟩ ⟨x²p⟩

6/1/1 0.0 1.4627 0.0043 0.0091

0.15 1.4526 0.0043 0.0090

0.30 1.4273 0.0042 0.0088

8/2/2 0.0 1.4557 0.0044 0.0095

0.15 1.4464 0.0044 0.0094

0.30 1.4224 0.0043 0.0092

10/3/3 0.0 1.4586 0.0044 0.0097

0.15 1.4442 0.0043 0.0094

0.30 1.4252 0.0043 0.0093

12/4/4 0.0 1.4606 0.0045 0.0098

0.15 1.4504 0.0044 0.0095

0.30 1.4272 0.0043 0.0093

the ground state of HD with and without the magnetic field provideinteresting insightintothebehaviorof themolecule in the field, see Table 2. The first expectation value is the aver- agedeuteron–protondistance.Withincreasingfieldstrength,the averagedistancenoticeablydecreases,asexpectedfromtheobser- vationthattheconfinementpotentialisproportionaltothesquare ofthefield strength.Thesecondtypeof theexpectation values showninTable2correspondstotheaveragevaluesofthesquares ofthexcoordinatesfortheprotonandthedeuteron.Asthexaxis isperpendiculartothefieldvectorB,theexpectationvalueofx² describesthedegreeofthealignmentofthemoleculewiththefield axis(thezaxis).Both⟨x²_d⟩and⟨x²p⟩decreasewithincreasingfield strength,indicatingthatthemoleculebecomesincreasinglymore alignedwiththefieldvector.Onepointtonoteisthatthe⟨x²_d⟩expec- tationvalueissignificantlysmallerthanthevalueof⟨x²p⟩which indicatesthatthedeuteronismorelocalizedinspacethanthepro- ton.Thisisanexpectedbehaviorresultingfromthemassdifference ofthesetwoparticles.

4. Summary

An effective method for describing the behavior a neutral molecule in the presence of a magnetic field without assum- ing the BO approximation is proposed and implemented. The method employs orbitals to represent the states of the nuclei and theelectronsformingthemolecule. Themethodis applied tostudy thenon-BObehavior of theHD molecule inthe mag- neticfield.Theorbitalmethodonlyallowsforobtainingqualitative results concerning the behavior. The results show that, when the field is turned on, the molecule shrinks along both paral- lel and perpendicular directions relative to the bond axis and becomesincreasinglyalignedalongthefieldvector.Theparallel shrinkingmanifestsitselfinthenoticeabledecreaseofthebond length.

Thecontinuationofthisworkwillinvolveimplementationof explicitlycorrelatedGaussianfunctionsfordescribingthebehavior of smallmoleculesin a magneticfield withmuch higheraccu- racy.Assuchanimplementationinvolvesextendedoptimization ofthenon-linearparametersoftheGaussians,theanalyticalgradi- entofthefield-dependentenergydeterminedwithrespecttothese parameterswillbeimplemented.Moreover,theexplicitlycorre- latedGaussiansusedinthefield-dependentimplementationwill havetopossesssufficientflexibilitytodescribethedeformation ofthewavefunctioninthefield.Suchflexibilityisonlypartially achievedwiththeorbitalGaussiansemployedinthepresentcal- culations.

ThecorrelatedGaussiansthatcanbeconsideredforcalculations onmoleculesinmagneticfieldsarethefollowingfunctions:

$_k=exp

#

−(R−s_k)^′A_k(R−s_k)

$

, (15)

whereRisavectorwiththelengthof3NbuiltfromtheR_ivec- tors, s_k is avectorcontainingNGaussians shiftvectors,A_kis a 3N×3Ndimensional,symmetric,positivedefinitematrixofGauss- iansexponentialfactors,andprimedenotestranspositionofthe vector.IfA_k isdiagonalwithfirstthreediagonalelementsequal toeachother,thenextthreeelementsequal toeachother,and soon,and ifonlythezcoordinatesof s_k arenon-zero,then$_k becomesequivalenttoaproductoforbitals,whichisthetypeofthe basisfunctionsusedinthepresentcalculations.Thus,byallowing theAmatrixtohavenon-zerooff-diagonalmatrixelementsand byallowingtheGaussianscenterstomoveawayfromthezaxis, ageneralized explicitly-correlatedformofthebasis functionsis obtained.Suchfunctionscanmuchmoreeffectivelydescribethe stateofthemoleculeunderinfluenceofastaticmagneticfield.An evenmoregeneralformofthecorrelatedGaussianscanbeobtained byallowingfortheexponentialfactorstobecomplexnumbers:

$_k=exp

#

−s_k)^′(A_k+iC_k)(R−s_k)

$

whereC_kisasymmetricmatrixandi=

%

(−1).

Acknowledgements

ThisworkwassupportedbytheNorwegianResearchCouncil throughtheCoECentreforTheoreticalandComputationalChem- istry(CTCC)GrantNo.179568/V30andtheGrantNo.171185/V30 andthroughtheEuropeanResearchCouncilundertheEuropean UnionSeventhFrameworkProgramthroughtheAdvancedGrant ABACUS,ERCGrant AgreementNo.267683.LA thanksCTCCfor supportinghisvisitstoOslo.Thisworkhasbeenalsosupported inpartbytheNationalScienceFoundationwithgrantnumberIIA- 1444127.

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