ISAS
SISSA
SCUOLA INTERNAZIONALE SUPERIORE DI STUDI AVANZATI
INTERNATIONAL SCHOOL FOR ADVANCED STUDIES
Improving mole ular-dynami s simulations
of
simple ioni systems
Thesis submitted for the degree of
Do tor Philosophi
Candidate: Supervisor:
1 Introdu tion 1
2 Car-Parrinello Mole ular Dynami s 5
2.1 The Car-ParrinelloMethod . . . 6
2.2 Car-ParrinelloDynami sCompared to Born-Oppenheimer Dynami s . . . 8
2.3 The RigidIon Approximation. . . 11
2.4 Testing the Theory . . . 14
2.4.1 Introdu tion . . . 14
2.4.2 Te hni al Details . . . 15
2.4.3 Results . . . 16
2.5 Water . . . 27
2.5.1 Details of the Simulation . . . 27
2.6 Dis ussion . . . 33
3 Simple Models of Ioni Systems 37 3.1 Introdu tion . . . 37
3.1.1 The ShellModel . . . 38
3.2 Many-body Intera tions inSimple Ioni Systems . . . 39
3.3 Ele trostati E e ts . . . 40
3.3.1 Ele trostati Polarization. . . 40
3.3.2 Polarizationby Short-Range Intera tions . . . 41
3.4 Short-Range RepulsiveIntera tions . . . 42
3.5 DispersionFor es . . . 44
3.6 Dis ussion of Existing Approa hes . . . 44
3.7 Our Approa hto Modelling Ioni systems . . . 46
3.7.1 Polarization . . . 47
3.7.2 A Distortable Ion Model . . . 49
3.7.3 Applying the Model . . . 53
4 Parametrizing E e tive Potentials 57 4.1 Introdu tion . . . 57
4.2 Parametrizingfrom Ab-InitioData . . . 58
4.2.1 The For e-Mat hing Method . . . 58
4.3 Ab-Initio Cal ulations . . . 61
4.3.1 ComputationalDetails . . . 62
5 Sili a 65 5.1 Introdu tion: Why study sili a? . . . 65
5.2 Modelling Sili a . . . 66
5.3 Results . . . 68
5.3.1 Fitting toAb-InitioData . . . 68
5.3.2 Testingthe Potential . . . 69
5.4 Dis ussion . . . 74
6 Magnesium Oxide 75 6.1 Introdu tion . . . 75
6.2 Testing Potentialsfor MgO. . . 78
6.3 The Distortable-Ion Model Revisited . . . 84
6.4 The Melting Line of MgO . . . 87
6.4.1 Corre ting the melting slopewith ab initio al ulations . . . 95
6.5 Dis ussion . . . 99
Introdu tion
Although the fundamentalintera tions responsiblefor the ma ros opi properties of ma-terialshavebeen knownforsometime, ourabilitytouse thisknowledgeinorderto quan-titatively map mi ros opi properties onto ma ros opi observables is still very limited. The goal of theoreti al ondensed matter physi s is twofold : to understand mi ros opi me hanisms-the intera tionsof ele tronsand ions- andhowthey relatetoexperimental observables, and to quantitatively predi t the properties of materials. Theory has had many su esses inexplainingexperimentallyobserved phenomena and some notable fail-ures. Goodexamples arethe explanation,more than forty years ago, ofthe phenomenon of super ondu tivity in a wide lass of materials as being the result of ele tron-phonon oupling[1℄ and the more re ent failure of theorists to fully explain higher temperature super ondu tivity in a di erent lass of materials despite intensive e ort. The quanti-tative predi tion of ma ros opi properties is extremely desirable as it would allow the designofnewmaterialswhi h ouldbetailoredtosuitaspe i purposeanditwouldalso allowone togain knowledgeabout materialsunder onditions ina essibletoexperiment. However, despite very rapid progress in the eld of materialsmodellingin the lastthirty years, it has not yet developed to the point where it an pro eed without re ourse to empiri ismor experimental veri ation.
The fundamentalproblem isthat although we an state the prin iples by whi h ele -trons and ions intera t, the nature of these intera tions and the large number of su h intera tions present in real materials means that apart from extremely simple systems su h as the isolated hydrogen atom it is not possible to solve the relevant equations in order to see how the ma ros opi system behaves as a whole. It is ne essary to use omputers to help solve the many-body problem, but unfortunately despite enormous in reases in omputational power in the last three de ades, it is simply not possible to solve the equations exa tly ex ept inan extremely limited number of ases. In fa t,one an showthat tosolveexa tlythe S hrodinger equationinitsstandard formforeven ten ele trons isbeyond any on eivable omputer.
Amorepra ti algoalthereforeistore-expressthemany-bodyprobleminawaywhi h is easier to solve, by making reasonable approximations wherever possible. There are many di erentapproa hes to this and the best approa htouse depends ona wide range of fa torssu hasthematerialofinterest,the onditionsunderwhi hthe materialistobe
studied, the omputationalresour es available and the man-hoursrequired toimplement the solution. Here the fo us is on two of the most widely used approa hes for atomisti materials modelling, namely, the mole ular dynami s te hnique and density fun tional theory within the lo al or semilo al density approximations. Density fun tional theory (DFT)[2, 3℄ , whi h is dis ussed in more detail in hapter 4, allows one to al ulate the ground-stateprobabilitydensityofasystemofele tronsinagivenpotential. Asaresultit maybeused to al ulatethe energyandfor esofasystemofatomswhi hislargeenough toapproximate ertainpropertiesof the bulkmaterial. Centraltothe appli ation ofthis methodistheuse ofwhatisknownasthelo al densityapproximation,orLDA,whi hhas produ ed very a urate results for a wide range of materials. The mole ular dynami s (MD)te hnique was histori allyatoolfor exploringthe statisti alme hani albehaviour ofidealizedsystems. Given aknown,or al ulableintera tionpotentialbetween asystem of atoms it evolves the atoms in time a ording to the for es via Newton's equations of motion. Sin e most atoms are large enough to be onsidered lassi al parti les, this allowsonetomodelthebehaviourofasystematarbitrarytemperaturesorpressures. The ne essity of repeatedly al ulating thefor es between atomshas meant thatthe majority ofmole ulardynami ssimulationshaveusedverysimpli edphenomenologi alintera tion potentials and that these simulationsare qualitative ratherthan quantitative innature.
In1985, ina landmarkpaper[4℄, Carand Parrinellomarrieddensity fun tionaltheory with mole ular dynami s for the rst time to allow the in reased a ura y of the DFT intera tion potentialand the nite temperaturereal-timesystem evolution of MDinthe same simulation. Now known simply as Car-Parrinello Mole ular Dynami s (CPMD), this method has be ome one of the most widely used te hniques for the al ulation of dynami andele troni propertiesof ondensedmatterwiththeoriginalpaperbeing ited many thousands of times to date. Although this method represents a vast improvement over the simpli edpotentials whi h existed beforehand, it is extremely omputationally expensiverelativetothesepotentialsand even withamodernparallelsuper omputerone is generally on ned to system sizes of the order of one hundred atoms and simulation timesofonlyafewpi ose onds. This isamajorproblemasitseverelylimitsthe rangeof properties whi hone an al ulate and the range of phenomena whi h an besimulated. Thelargerthesystemsizethatissimulatedthe loseronegetstothedimensionsobserved by experiment and to the human s ale on whi h most materials are en ountered. The longer that one an run a real-time simulation,the loser one gets to experimental time s alesand tothe humantimes ale ofminutes, hours,days, andyears onwhi h materials are generally used. In pra ti e simulation an never even approa h the relevant size and times alesand itgenerallyfails by many orders of magnitudeinea h ase. However, the errorintrodu edinthisway anbeminimizedbyhavingaslargeandaslongasimulation aspossible. Forexample,thelargerand longerasimulationisthemorepre ision one an a hieve in the al ulation of thermodynami observables su h as pressure, temperature ordensity. Atpresent, the fa t that CPMD and other DFT basedMD te hniques are so limitedby omputational expense means that the pre ision with whi h many properties are al ulated is poor. On the other hand, simpli edphenomenologi al potentialswhi h allowin reases insize and time s ale ofthree to ve orders ofmagnitude, provide avery poordes ription of interatomi intera tions.
The pra ti algoal of this thesis is to nd away in whi h the properties of bulk ioni systems under onditions of low symmetry and high temperatures and pressures an be predi ted with a ura y and a reasonable degree of pre ision. The motivation for this work is the desire to be able to quantitatively simulate systems of geophysi al interest su h as the minerals that are found deep in the earth's mantle whi h are mostly ioni in nature. The extreme onditions of temperature and pressure makeit very diÆ ult for the properties of su h mineralsto be onstrained by dire t measurement and up to now the results of simulation have been of variable quality. Properties, su h as the elasti properties of some of the major onstituents of the earth's lower mantle[5, 6, 7℄, have been simulated with state-of-the-art rst-prin iplesmethods. Although su h simulations frequently have not been veri ed experimentally, it is likely that they are reasonably a urate. The range of problems that an be ta kled in a purely rst-prin iples way is very limited however. Other problems, a prime example being the melting behaviour of oxides under pressure, have been ta kled with a range of for e- elds in a mole ular dynami s framework and the results have been extremely poor. An example, whi h will be dis ussed in detail in hapter 6 is the melting line of MgO. There have been at least ve separate simulations of the pressure dependen e of the melting temperature published. Of these, all have disagreed with experimental measurements of the slope of the zero-pressuremelting urvebyfa torsofbetween2:5 and8. Therehasalsobeen little agreement between di erent simulations, even between those using the same method of al ulating the slope. This level of disagreement learly signi es problems with existing for e- elds.
Inthis thesiswe ta klethe generalproblemof simulatingioni systems su hasoxides under arbitrary thermodynami onditions. Although we have a lear pra ti al goal in mind, we approa hthe problemin averygeneralway and most ofthe methodology that is developed is appli able to almostany system for whi h a urate simulations of simple ioni systems are required.
Webegin,in hapter2bylookingindetailatCar-Parrinellomole ulardynami sasthe most widelyused dynami alab initiote hniqueand dis usssometheoreti alissueswhi h havenotpreviouslybeenfullyaddressed. Weshowthatthestandardunderstandingofthe methodisin ompleteandwedes ribeinamorerigorouswayitstheoreti alunderpinning. Mostimportantly,our theoreti alinvestigationsshowthat ontrarytopopularbelief,the ele troni orbitalsdonottaketheirground-statevaluesonaverageduringaCar-Parrinello simulation. This means that there are errors intrinsi to the method and we show how these errors may be partially orre ted for many systems, but parti ularlye e tively for simple ioni systems.
The omputational expense involved in ab initio mole ular dynami s means that for most of the problems in whi h we are interested, analternativesolution is required. We wouldliketo nda ompromisebetweena ura yandpre isioninour al ulationssothat thermodynami properties an be omputedwith more on den e than hasbeen donein the past. The approa hthat wetaketo ndingthis ompromise istolook forfor e- elds withfun tionalformswhi h apturephenomenologi allymoreofthedynami alele troni e e ts whi h ontribute tointerioni for es and whi h are therefore apable of providing more a ura y. The ompromise liesinthe fa t that this apa ity for improved a ura y
isgenerallyattheexpenseof omputationaleÆ ien y. Theimproveda ura yisa hieved withagivefun tionalformby using datafromdensity fun tionaltheory simulationsina generaland well- ontrolledparametrization pro edure.
The problemof ndinga good fun tionalformfor ioni systems isdis ussed indetail in hapter3. Manyfor e- eldsovermanyyears havebeenproposedfor ioni systemsbut noneof thesefully meetthe riteriathat weset. Allare eithertoo simpletobe a urate, tooslowtobepre iseortoospe i tobeuseful. Weintrodu eanovelgeneralfor e- eld for ioni systems whi h is physi allywellmotivated but orders of magnitude faster than CPMD.
In hapter4webuildonpreviousworkanddes ribehowthesefor e- eldsareparametrized using rst-prin iplesdata. Theresultingfor e- eldisshownin hapters5and6toprovide thermodynami and dynami properties of an extremely high quality while stillallowing the simulation of mu h larger systems over mu h longer times ales than an possibly be ta kled within a fully rst-prin iples approa h. Our method is learly shown to provide pre ision far surpassing that of ab initio methods and an a ura yfar surpassing that of simpleempiri ally-parametrizedfor e- elds.
In hapter 6 this method is applied to the outstanding problem of the melting tem-peratureof MgO underpressure.
Car-Parrinello Mole ular Dynami s
In 1985, in a seminal paper [4℄, Roberto Car and Mi hele Parrinello showed how it was possible toperform a mole ulardynami ssimulationwhilst al ulating the for es within density fun tional theory. In other words, the ele trons were treated expli itly in an ab initio way in the al ulation for the rst time. The importan e of this annot be overstated. Even apart from the fa t that density fun tional theory an be mu h more a urate than e e tive parameter-based for e- elds, any MD simulation whi h does not treat ele trons expli itly requires one to make a hoi e a priori about the nature of the system. Weshowin hapters5and 6howitispossible tomakevery goode e tive for e- elds for ioni systems. However, if the bondinginthe system hangesdue to hangesin pressure, temperature or phase, the a ura y of the for e- eld su ers. Moreover, we are biasing the system fromthe start by the phenomenologythat wein ludein thepotential form,andapartfromtheingredientsthatwein lude,nothingelse anplayarole. Physi s and hemistry are full of surprises and many e e ts and stru tures o ur due todeli ate balan es between many di erent fa tors. It is simply not possible to predi t with any degree of ertainty a priori what e e ts may be ome important under a given set of onditions.
Expli itlytreatingthe ele trons means that, inprin iple, one doesnot make assump-tions about the bonding of the system and this allows surprises to o ur. Spontaneous hanges in bonding an take pla e without loss of a ura y. This means that one an simulate hanges ofphase with mu hmore on den e. Ab initiomole ulardynami s an alsoallowonetomodel hemi alrea tions. Thisissomethingthate e tivefor e eldsare unabletodobe ause, byde nition, hemi alrea tionsinvolve hangesinthebondingand when they o ur itisthe ele trons whi hplay the dominantrole. Unless thedependen e of ele trons onioni positions is expli itly al ulated, the rea tion annot bemodeled.
The method of Car and Parrinello, in its original and most widely used form, works within the Born-Oppenheimer approximation. This means that at any instant the state of the ele troni system an be well des ribed by the ele troni ground-state al ulated for the ioni positions at that instanti and that it responds instantaneously to hanges in ioni positions. For many systems this is anextremely good approximation given the mass of theele tron relativetothat ofthe ions, whi h forhydrogen, thelightest element, is approximately1=1836. When the ele troni ground-state is lose to degenerate,whi h
frequentlyo urs,it an besaidthatthe Born-Oppenheimerapproximationbreaksdown. However it should be valid before and after the o urren e of su h degenera ies and the timeduring whi hdegenera ies are relevant isgenerally small.
The Car-ParrinelloMDmethodstarted the eld of ab initio mole ulardynami sand remainsthe most widelyused methodfor ouplingdensity fun tionaltheory with mole -ulardynami s. Itisnot theonlymethodhowever. Anumberofotherte hniques[8,9,10, 11℄ have been developed whi h are based on minimizationof the ele troni (Kohn-Sham [3℄) orbitals to their ground state at ea h time step. These te hniques will be referred tofrom now on as Born-Oppenheimer (BO) methods to distinguish them from the Car-Parrinello(CP)methodwhi hdoesnotputtheorbitalstotheirgroundstateatea htime step,as we willsee below.
Despitewidespreadappli ationofthe CP methodtomanyareas ofphysi s, hemistry andbiologyanddespiterapiddevelopmentofmanyaspe tsofthemethodology,noserious testingofthea ura yofthemethodhaseverbeenpublishedtoourknowledge. Anumber ofpeople[4,12,13℄,in ludingthe inventors ofthemethod,haveshownthat itreprodu es ground-statefor esandenergiesinsimplesystems,su hastoy-modelsof rystallinesili on orgermanium, extremely well, but sili on and germanium are parti ularlyeasy systems totreatwith mostele troni stru ture methods, and the ability toa hievehigh a ura y forthese systems is noguarantee that the methodworks well inother systems.
In this hapter,we begin by explainingthe Car-Parrinellomethod indetail and some of the reasons that it is generally believed to work. The understanding of the method has evolved somewhat sin e its introdu tion and so no e ort will be made to present the de nitive version of urrent understanding, rather some ommonly held beliefs will be presented. Next, some theoreti al problems with the method will be explained and a partialsolutionof these problems forinert ioni systems willbe presented. Wethen test the theoreti al ideas that have been developed with the simple examples of sili on and MgO,followedby atestof themethodonone ofthe mostfrequentlystudiedsystems,i.e. water, orin this ase \heavy" i e.
2.1 The Car-Parrinello Method
The Car-Parrinellomethodmakes use of the following lassi allagrangian :
L CP = X i i h _ i j _ i i+ 1 2 X I M I _ R 2 I E[f i g;fR I g℄ (2.1)
togenerate traje toriesfor theioni andele troni degrees of freedomviathe oupled set ofequations of motion M I R I = E[f i g;fR I g℄ R I =F CP I (2.2) i j i i= ÆE[f i g;fR I g℄ Æh i j (2.3) where M I and R I
are the mass and position respe tively of atom I, j i
Kohn-parameters i , and E[f i g;fR I
g℄ is the Kohn-Sham energy fun tional[3℄ evaluated for the set of ioni positions fR
I
g and the set of orbitals f i
g. The fun tional derivative of the Kohn-Sham energy in equation 2.3 is impli itly restri ted to variations of f
i g that preserve orthonormality.
Theideabehindthe methodisthatby puttingtheele trons totheirgroundstateata xed set of ioni positions and then allowing the ions tomove a ordingto equation 2.2, the ele troni orbitals should adiabati ally follow the motion of the ions, performing small os illations about the ele troni ground state. The ele troni orbitals will have a \ titious" kineti energy asso iatedwith theirmotionand the titiousmass parameter
i . If
i
is small enough then the motion of the orbitals will be very fast relative to the motion of the ions. It is generally thought that this motion onsists of os illations about the ground state and so by hoosing a small enough value for
i
one an ensure that the frequen y spe tra of the ele troni orbitalsand the ions are wellseparated from one another if there exists an energy gap between the o upied and uno upied Kohn-Sham orbitals. This is be ause, within a harmoni approximation, the lowest frequen y of os illation of the orbitalsabout the groundstate may be written as[12℄
! 0 = 2( j i ) i ! 1=2 (2.4) where i and j
are the eigenvalues of the highest o upied and the lowest uno upied orbitals respe tively. In lassi al me hani s, systems whi h are well separated from one another infrequen y an beshown toremainenergeti allyisolatedfromone another (see ref.[12℄ and[14℄ and referen es therein). Therefore, it has been thought that by using a small enough value for
i
, one ould isolate the ele trons energeti ally from the ions. In this way one ould ensurethat thermalizationdoes not o ur between ele trons and ions and sotheele troni orbitalsremainatalowtemperature,whi hmeansthattheyremain lose tothe ele troni ground state.
This explanation of the method was originally proposed by Pastore et al. [12℄ and is the standard way in whi h the method is explained (see, for example, the re ent review byMarxandHutter[13℄). Althoughpartsof thisexplanationaretrue,itignoresthefa t that, aswellasthehigh-frequen y os illations,theorbitalshaveaslow omponenttotheir motion. If the ions are moving, the motion of the orbitals ontains a omponent due to the unavoidable response of the ele troni orbitals to the ioni dynami s : as ions move, the groundstate hanges. Byde nition this latter motiono urs onioni times ales and with ioni frequen ies and so it may not be de oupled fromthe ioni motion. In fa t, it willbeshowninthis hapterthatduetothisaspe tofthe ele troni motiontheele trons do notos illateabout the groundstate but aboutanequilibriumwhi hisdispla ed from it. This means that there are systemati errors in the for es on the ions and inthe total
2.2 Car-Parrinello Dynami s Comparedto Born-Oppenheimer
Dynami s
Wewishto ompare CPdynami stotheexa tBOdynami s. Inotherwords,the dynam-i softhe ionswhentheele troni orbitalsremainattheir groundstate. Forthis purpose, wede ompose the CP orbitals as
j i i=j (0) i i+jÆ i i (2.5) wherej (0) i
iarethe groundstate(BO)orbitalswhi hare uniquelyde ned forgiven ioni oordinates asthose that minimizeE[f
i g;fR
I
g℄ . This allows us to onsider separately the evolution of the instantaneous ele troni ground state and the deviations of the CP orbitalsfromthat ground state.
AtthispointwenotethatinthepastsometestingoftheCPmethod[15,16℄ hasrelied ondemonstratingthatthe total ele troni energy al ulated with the CP methodisvery losetothat al ulatedwhentheele troni orbitalstaketheirgroundstatevalues. Thisis not agood riterion touse tovalidate the method. For mole ulardynami ssimulations, the important quantites are the for es on the ions. For a given deviation of the CP orbitalsfromthe groundstate jÆ
i
i the error inthe for e relativeto the size of the for e is generally mu h bigger than the error in the energy relative to the size of the energy. This an easily be seen by writing both of these quantities as Taylor expansions about theirground state values
E E = X i 1 E ÆE Æj i i f (0) i g jÆ i i+hÆ i j ÆE Æh i j f (0) i g +order(Æ 2 i ) (2.6) F I F I = X i 1 F I ÆF I Æj i i f (0) i g jÆ i i+hÆ i j ÆF I Æh i j f (0) i g +order(Æ 2 i ) (2.7)
The rst term onthe righthand side ofequation 2.6 disappears be ause the derivativeis evaluatedatthegroundstate. Howeverthe rsttermontherighthandsideofequation2.7 does not disappear. So, for a given deviation of the ele troni orbitals from the ground state the error in the for e depends to rst order on the fÆ
i
g but the error in the energyonly depends tose ondorder onfÆ
i
g. This meansthat any error inthe orbitals has a mu h larger e e t on the for es than it does on the energies. How mu h greater the relative error in the for es is than the relative error in the energies depends on the system, but, as will be dis ussed in se tion 2.4.3, it an be a few orders of magnitude. When testing the method it is therefore important to see how well the for es reprodu e the Born-Oppenheimer for es.
We now write j _ (0) i i= X I _ R I j (0) i i R I (2.8) and j (0) i i= X R I j (0) i i R I + X _ R I _ R J 2 j (0) i i R J R I (2.9)
A preliminary interesting observation now follows. If the ele troni orbitals are at their ground state values, i.e. jÆ
i
i = j0i , then the right hand side of equation 2.3 vanishes sin ef
i
g=f (0) i
g. However, thelefthandsidedoesnotbyvirtueofequation2.9. Sothe CP orbitals annottaketheirgroundstatevaluesunlessvanishestoo. Asa onsequen e of this, theioni dynami sisa e tedby abias dependent onand, aswewillsee, tothe strength of the ele tron-ion intera tion .
Wenowwishtoexplorethe onsequen esthat su hadeparturefromthegroundstate has on the instantaneous CP for es F
CP
. We therefore al ulate how CP for es deviate fromthe BOfor es F
BO
atagiven pointinphase spa ealongtheCP traje tory. Wemay write, for the -th artesian omponent of the for eon atom I:
F CP I = E[fR I g;f i g℄ R I = dE[fR I g;f i g℄ dR I X i ÆE[fR I g;f i g℄ Æj i i j i i R I + h i j R I ÆE[fR I g;f i g℄ Æh i j (2.10)
Substitution of equation 2.3 yields
F CP I = dE[fR I g;f i g℄ dR I X i i h i j j i i R I + h i j R I j i i (2.11)
Using the expansion
dE[fR I g;f i g℄ dR I = d dR I E[fR I g;f i g℄ f (0) i g + X i ÆE[fR I g;f i g℄ Æj i i f (0) i g jÆ i i + hÆ i j ÆE[fR I g;f i g℄ Æh i j f (0) i g +order(Æ 2 i ) = F BO I +0+order(Æ 2 i ) (2.12)
we an writethe error inthe CP for eas
F I =F CP I F BO I = X i i h i j j i i R I + h i j R I j i i +order(Æ 2 i ) (2.13)
Having established the onne tion, to rst order in Æ i
, between the CP and the BO
for es, we assume adiabati de oupling and look for ontributions tothis di eren e that do not vanish when averaged over time s ales longer than the typi al times ales of the high frequen y part of the titious dynami s of the ele trons (
e
) but shorter than the times alesoftheioni dynami s(
i
). Onlythese ontributionsareexpe tedto ontribute signi antly to the ioni dynami s[12℄. To this end we write
jÆ i i=jÆ (1) i+jÆ (2) i (2.14)
where we have split Æ i
into a term whi h has a very high frequen y relative to ioni frequen ies (Æ
(1) i
) and a term whi h varies onioni times ales (Æ (2) i
) We rewrite equa-tion 2.3, using equation2.5, as
j i i = jÆ (1) i i+jÆ (2) i i+ X I R I j (0) i i R I + X I;J _ R I _ R J 2 j (0) i i R J R I (2.15)
Sin e we are on erned with what happens on ioni times ales, i.e. averaged over the high-frequen y omponentofthe
i
,wemaynegle tthe rsttermontheright-handside. Using equation 2.13 and equation 2.15 we may write the error in the for e to rst orderin Æ i as F I =2 X i i < ( h (2) i j+ X J R J h (0) i j R J + X J;K _ R J _ R K 2 h (0) i j R K R J j (0) i i R I + j (2) i i R I ) (2.16) Any deviation Æ i
from the ground state depends on the titious mass sin e !
0 =)
ÆE Æh i
j
! 0 via equation 2.3. This means that if we are to onsider only terms whi h depend purely tolinear order onthen the terms involvingÆ
(2) i
inequation 2.16 maybenegle ted.
To summarise: If we onsider the dynami sof the ele troni orbitalsto onsistof an adiabati responseoftheele troni orbitalstotheioni dynami sandanindependentfast os illatingpartthen, underthe assumptionthat thetimes ales ofthe fast omponentare mu h shorter than the shortest time period in the ioni system, i.e. assuming adiabati de oupling, the average error in the Car-Parrinello for es to rst order in and Æ
i is given by (usingequations 2.20 and 2.9)
F I = 2 X i i < X J R J h (0) i j R I j (0) i i R J + X J;K _ R J _ R K h (0) i j R I 2 j (0) i i R K R J (2.17)
This orre tion varies on ioni time s ales and therefore does not ne essarily average out as the usual \fast" omponent does. However, its value depends on the ele troni mass. This implies that asimple way to ensure that its ontribution in a CP simulation is negligible onsists of redu ing systemati ally the ele troni mass. Although a smaller impliesa smallertime step forthe integration of the CP equationsof motion, the time step s ales as t
1=2
, whi h means that redu ing by an order of magnitude brings abouta omputationaloverheadofonlyafa tor ofthree. Amorequantitativedis ussion ispresented inse tions IV and V.
We also noti e that if the term proportional to _ R
_
R in the r.h.s. of equation 2.17 vanishes (e.g. by symmetry, see below), and the tensor in the term proportionalto
R is onstant,then the orre tionofequation2.17redu es toares alingoftheatomi masses, whi h is known to leave thermodynami s inta t. This is dis ussed in more detail in the
2.3 The Rigid Ion Approximation.
In order togain insightintothe s ale of this problemwith the CP for es we onsider the simpleexampleof rigidions. Weassumethatea hele tronislo alisedaroundanionand that there isnodistortionofaparti ular ion's hargedistribution asitmoves inthe eld of the other ions. We an refer ea h wavefun tion
i
to aparti ular ionas follows
i (r)= I (r R I ) (2.18)
Where the ele troni states are labelled by anionindex, I,and the index labellingthe ele troni state ofthe ion. The rigidityof the ioni harge distributionmeans that
I (r R I ) R I = I (r R I ) r and I (r R I ) R J =0 8J 6=I (2.19) Equation 2.17 be omes F I = 2 X < R I Z I (r R I ) r I (r R I ) r dr + _ R I _ R I Z I (r R I ) r 2 I (r R I ) r r dr (2.20)
The se ondterminequation 2.20vanishes duetosymmetry,atleastassuminganatomi harge density with spheri al symmetry. The rst term may be written in terms of E
I k the quantum ele troni kineti energy of anele tron instate of atom I as
2 X < R I Z I (r R I ) r I (r R I ) r dr = 2m e 3h 2 R I X E I k (2.21) where m e
is the (real) mass of anele tron. Sin e the ions are rigid the quantum kineti energy asso iated with ea h one is a onstant and equation 2.20 be omes
F I = M I R I (2.22) with M I = 2m e 3h 2 X E I k (2.23)
In this ase the ioni positions and velo ities are updated during a Car-Parrinello simu-lation a ording to (M I +M I ) R I =F BO I (2.24)
In other words, for systems where the rigid ion approximation is valid, the CP approx-imation amounts simply to a res aling of the ioni masses. Sin e the lassi al partition fun tion depends onlyonthe intera tion potential,the thermodynami softhe system as al ulatedwithaCPdynami sisidenti altothethermodynami softheBOsystem. The
CPisgivenby equation2.24, thenthereal temperatureatwhi hthe systemequilibrates, atleast inthe ase of a mi ro anoni al dynami s for the ions, isgiven by
k B T = 1 3N X I; (M I +M I )h(v I ) 2 i (2.25)
wherehisigni estheaverageovertimeandN isthenumberofatoms. Thisdi ersfrom the standard de nition by the addition of a term proportional to M
I
. The additional term in equation 2.25 an be readily tra ed to the additional inertia aused by the rigid dragging of the ele troni orbitals. In fa t, using equations 2.8 and 2.19, we an show that this term oin ides, within the rigid ionmodel, with the titiousele troni kineti energy, when the ontributionfrom the dynami sof the Æ
i is negligiblei.e. T el =T M (2.26) where T el = X i i h _ i j _ i i (2.27) and T M = 1 2 X I; M I h(v I ) 2 i (2.28)
In other words if the ele troni orbitals move rigidly with the ions the a tual inertia of the ions in a CP simulation an be obtained by adding to the \bare" ioni inertia the inertia arriedby the ele troni orbitals. Thisresult has beenpointed out previously [17,18℄andioni massesare ommonlyrenormalizedwhendynami alquantitiesarebeing investigated.
Figure 2.29 illustrates how, within the simpli ed rigid-ion model, the ioni inertia dependsonthekineti energyoftheele trons. Foragivenioni velo ity,thewavefun tion ata point inspa e has to hange more qui kly when it is highlylo alised (and therefore with ahigh quantum kineti energy) than when it is extended. Toa elerate an ionone also needs to in rease the rate of hange of the wavefun tion lo alised on it. Sin e the wavefun tion arriesaninertia() the e e tiveinertia ofthe ionisgreaterthan the bare ioni mass. Inmoregeneral(non-rigid-ion)situations,the olle tivemovementoftheions is a e ted by the requirement that the \heavy" ele troni wavefun tions are rearranged asthe system evolves.
We now explore the onsequen es that su h a modi ation of the ioni inertia has on typi al observables extra ted from CP simulations. First, as already mentioned, the orre t de nition of temperature in a mi ro anoni al CP simulation is given by equa-tion2.25. Similarly,inasimulationwheretemperatureis ontrolled,e.g. through aNose thermostat[19℄, the quantity to be monitored orresponds to the instantaneous value of the temperaturede ned in equation2.25. Dynami alobservables willalsobe a e tedby the additionalinertia, asalready notedin the ase of phonons extra ted fromCP-MD in arbon systems [20, 21℄. In the ase of homogeneous systems (a single atomi spe ies in
ψ
x
ψ
1
.
ψ
.
2
v
v
Figure 2.1: Theme hanism by whi h the e e tiveioni inertia isrelatedto thequantum kineti energyofthelo alisedele troni wavefun tionswithintherigid-ionapproximation: For two ions of the same spe ies whi h are moving with the same velo ity v, the one arryingthe morelo alisedele troni wavefun tion(top)has ahighere e tivemass. The more lo alised ele troni wavefun tion
1
(x) has, on average, a greater slope d (x)=dx (and hen ekineti energy)than themore extended wavefun tion(bottom)
2
(x). Sin e for agiven ioni velo ityvagreaterslopeimpliesagreaterrateof hangewithrespe t to time( _ 1 > _ 2
),thelo alisedwavefun tion hangesmoreperunittimethantheextended wavefun tion. In order to in rease the ion's velo ity one also needs to in rease the rate of hange of the massive wavefun tion lo alised on it. The total inertia asso iated with this required hange of the rate of hange of the wavefun tion is relatedto the quantum kineti energy via equation 2.23 and this quantity must be addedto the bare ioni mass in order toobtain itse e tive mass.
an be simplyres aled using the mass orre tion of equation 2.24. However, for hetero-geneous systems the orre tion is not always trivial, as di erent mass orre tions apply to di erent atomi spe ies due todi erent atomi kineti energies. Inpra ti e, wefound that a onvenient and more general way to express the mass orre tion of ionI is given by M I =f I 2m e E total k 3Nh 2 (2.29) where f I
isadimensionless onstantwhi htakesintoa ount therelative ontributionof spe ies I to the total quantum kineti energy E
total k
. The value of f I
is generally found variationallyas that whi h minimizesthe error inthe for es.
distin tspe ies individually. For temperature, the orre tion forspe ies S is T s = M s M s T s (2.30)
and the orre tion tothe partialthermalpressure of spe ies S is
P t s = M s M s P t s (2.31) Where M s
is the bare ioni mass of spe ies S and P t s
is the thermal ontribution to its partialpressure. Thereisalsoa orre tionwhi hmust beappliedtothe internalpressure due tothe fa t that the ele troni orbitals are not attheir ground state. This orre tion to the internal stress is not trivial to derive sin e the potential energy of the system depends, via the oupling between ions and orbitals,not only onthe ioni positions,but onallhigherorderderivativesoftheioni positionswithrespe ttotime. However, within therigid-ionapproximation,ifweonly onsiderderivativeswithrespe ttoioni positions, usingthe virialtheorem we may write the internalpressure as
P i = 1 X I F I R I (2.32)
and sothe rigid ion orre tion be omes
P i = 1 X I M M F I R I (2.33)
Inpra ti e, forasystemunderperiodi boundary onditions,itisnotpossibletoevaluate this quantity. However, for asystem with asingle spe ies, this be omes
P i = M M P i (2.34)
In pra ti e, in more general situations, in order to get an idea of the true stress of the system, one should perform a large enough number of ele troni minimizations to the ground state and al ulations of the stress at the ground state to get a good average of the true internal pressure.
In the next se tion we demonstrate, with the use of several simple examples, the validityofthetheorydevelopedinthisse tionandthepreviousone. Weshowhowmasses, andthereforedynami alquantities,maybe orre tedusingtheformulaepresented above. Inse tion2.5we lookatthemore diÆ ult aseofwater asanexampleofasystem whose dynami sand thermodynami sone annot orre tso easily.
2.4 Testing the Theory
2.4.1 Introdu tion
Table 2.1: Te hni al Detailsof the Simulations # System Temperature 0 E p E ut t P i i h _ i j _ i i L
Kelvin a.u. Ryd. Ryd. a.u. a.u.10
4 a.u. 1 Si 330 270 1.0 12.0 5.0 4.36 20.42 2 Si 330 270 1.0 12.0 5.0 4.36 20.42 3 Si(liquid) 2000 270 1.0 12.0 10.0 4.35 19.8 4 MgO 2800 400 2.7 90.0 8.0 66.3 14.5 5 Si 330 200 1.0 12.0 5.0 3.23 20.42 6 Si 330 800 1.0 12.0 10.0 12.92 20.42 7 MgO(M O res aled) 2800 100 2.7 90.0 4.0 16.55 14.5 8 MgO(M O res aled) 2800 400 2.7 90.0 8.0 66.4 14.5 9 MgO 2800 200 2.7 90.0 5.65 33.1 14.5 10 MgO 2800 100 2.7 90.0 4.0 16.55 14.5
and on sili on. Among the insulators(we restri t our analysis to insulators as adiabati de oupling isless obviousinmetalli systems and thiswould ompli ate onsiderablyour analysis), MgO and Sili on are extremal ases: MgO is a highly ioni system with large quantum kineti energy asso iatedwiththe stronglylo alised hargedistribution;Sili on on the other hand is a ovalent system where ele tron states are mu h more delo alised. Within our pseudopotential des ription of MgO [22℄, the 1s,2s and 2p states are frozen intothe ore of Mg whereas only the 1s states are frozen into the ore of O. Sin e there is very nearly omplete transfer of the two 3s ele trons from Mg to O (inspe tion of harge density ontour plots reveal no eviden e of any valen e harge anywhere ex ept surrounding O sites) the ele tron quantum kineti energy may to a rst approximation be attributed to ele troni states lo alised on oxygen ions. This makes MgO an ideal system to study within the rigid ion model sin e only the oxygen mass will be res aled. As mentioned in the previous se tion, additional problems arise if one deals with more thanoneele tron- arryingspe iesasthequantumkineti energymustbedividedbetween these spe ies. The large quantum kineti energyof MgO means that the error inthe CP for esshouldbelargerelativetomanymaterials. ThesimulationsofMgOwereperformed at a high pressure (900 kbar) as this enhan ed its ioni ity.
Sili on,on the other hand,is a ovalent/metalli system withrelatively lowquantum kineti energy. As su h it should be one of the systems most favourable to the Car-Parrinelloapproximationbut least favourableto des ription interms of rigid ions.
2.4.2 Te hni al Details
In this se tionwepresentthe resultof tendi erentsimulations. Thete hni aldetailsare summarised in table 2.1.
All simulations were performed with a ubi simulation ell of side L (see table 2.1) under periodi boundary onditions and with 64 atomsin the unit ell. Weused a plane wavebasissetwithanenergy uto forthewavefun tionsofE
ut
ings hemeof Tassone et al.[23℄ andthe parameters 0
and E
p
intable 2.1 are de ned as inRef.[5℄. Withtheuseofapre onditionings heme,wherebytheele troni massiss aled with the kineti energy of the plane wave, the time step an be in reased by a fa tor of 2-3withrespe ttothenon-pre onditioned ase[23 ℄. Theuse ofapre onditionings heme worsens onsiderably theagreementof theCP for eswith the BOones. Inparti ular, we have he ked that using the parameters
0
and E
p
that optimizethe timestep ausesan in rease by about a fa tor of three in the orre tion term of equation 2.17. However, in ordertobringthiserror toitsnon-pre onditioned value,a valueof
0
three timessmaller wouldberequired,with a onsequent redu tion ofthe time step ofonly
p
3. Considering thatthe pre onditioning s heme allows a2-3in rease of the timestep, a redu tionof
p 3 stillmakesthe pre onditioning s heme marginallysuperior.
Liquidsili onismetalli and so,assuggestedby Blo hlandParrinello [15℄,twoNose thermostats were used to ountera t the e e ts of energy transfer between the ions and thejÆ
i
idue tooverlapoftheirfrequen y spe tra.Thevaluesoftheparametersusedwere Q e =21:3 a.u./atom,E kin;0 = 1:6510 4 a.u./atom and Q R
= 244400 a.u.. These were hosen for ompatibilitywith those of Ref.[10℄ by taking intoa ount the slight in rease intemperatureand s aling a ordingly.
In all of these simulations, with the ex eption of simulation 2, the system was rst allowedtoevolveforatleast1psandthistraje torywasdis arded. Forsimulation2,this initialequilibrationtimewas0:5 ps. Allresultsreportedaretakenfromthe ontinuations ofthese equilibrationtraje tories.
In all simulations, the total quantum kineti energy of the system, and hen e the average mass orre tion (see equation 2.29), varied during the simulation by less than 0:3%. It was therefore taken as a onstant infurther analysis.
Thetotalenergyofallthedegrees offreedom(in ludingthethermostatsinsimulation 3)was onserved inall simulationsat least towithin one part in10
5 .
2.4.3 Results
In order to he k the predi tions of the theory developed in Se tions II and III, we have takensegmentsofCP traje tories and al ulatedthetrueBOfor es alongthesesegments by putting the ele troni orbitals to their ground state with a steepest des ent method. We look at the instantaneous error in the
th
artesian omponent of the CP for e on atomI relative tothe root-mean-squared (r.m.s) BOfor e omponent,i.e :
ÆF I (t) = F I (t) r P P J; (F BO J ) 2 3NN (2.35)
and the instantaneous relative error minus the relative error predi ted by the rigid-ion model : [ÆF I (t)℄ orr = F I (t)+M I R I r P P J; (F BO J ) 2 3NN (2.36)
where N
is the number of ioni on gurations at whi h the error in the CP for es was al ulated and
P
is the sum over all su h on gurations. The value of M I
in equa-tion 2.36isdeterminedusing therigid-ion-modelexpression equation2.29,and E
total k
was given its average value during the simulation. The s aling parameters whi h were found to give best resultsfor sili on and oxygen were f
Si =1:0 and f O =1:92 respe tively. We also look at hÆF I i and hÆF I i orr the r.m.s values of [ÆF I ℄ and [ÆF I ℄ orr over all the ions, artesian omponents and on gurations tested.
(c)
(b)
(a)
0
0.01
0.02
0.03
0.04
0.05
0.06
Time [picoseconds]
−0.02
−0.01
0
0.01
Force [atomic units]
×
20
0
0.2
0.4
0.6
0.8
1
Γ
(
t
)
uncorrected
corrected
−3
−2
−1
0
1
2
3
Relative Error [%]
Figure2.2: Simulation1. (a)DistributionamongallatomsIandall artesian omponents of the per entage errors in the CP for es relative to the BO for es at the same ioni positions, 100ÆF
I
(t) (full line) and these errors when the for es have been partially orre ted a ording to a rigid-ion model, 100 [ÆF
I
(t)℄ orr
(dashed line). (b) (t) as de ned by equation 2.37 for the full error in the for es and those as partially orre ted a ording tothe rigid-ionmodel ( )F
BO I ,F CP I and (F CP I F BO I )(multipliedbyafa tor of 20 for visibility) for a typi al for e omponent. Dots indi atethe points at whi h the BO for ewas al ulated (every 5time steps).
the quantity ( ), de ned as ( )= Z 0 1 3N X I; R t0+ t0 F I (t)dt R t 0 + t0 jF I (t)jdt d (2.37)
If we begin our omparison between CP and BO for es at some instant t 0
along the traje tory then inspe tion of ( ) gives a feeling for how large the fast omponent is. If theerrorsinallthefor esofthesystem os illaterapidlywithanaverageofzerothen ( ) de reases very qui kly from the value of one at = 0 to zero at
e
. For systemati errors ( )should de rease gradually fromone tozero on atimes aleof the orderof the period of
i
. In realisti ases ( ) drops from one and levels o to a smaller value for
e
, and then de reases gradually to zero for ex eeding i
. The value of ( ) on the plateau between
e
and i
provides a measure of how mu h the errors al ulated in equations2.35 and 2.36 are attributableto asystemati (i.e. \slow") departure fromthe BOsurfa e.
We begin by lookingatthe for es inSili on inboth the solid at 330 Kand the liquid at2000 K (simulations1,2 and 3). Simulation 1 was pre eded by a short run where the temperature was set toabout 1000 K. Ele trons were then relaxed in their ground state andthe ioni velo ities set tozero. This allows the ele trons tosmoothlya elerate with the ions. A mi ro anoni al simulation followed where the ioni temperature rea hed, after a short equilibration, the value of 330 K. This pro edure was followed in all the simulationsreportedhere, ex eptwheredis ussed. Insolid Siat330 K(Fig.2.2)we nd that the standard deviation of the error in the Car-Parrinello for es is 0:94%. However, most of this error an be attributed to a rigid dragging of the Si atomi orbitals. The standard deviation of the error is in fa t redu ed to 0:24% after the rigid-ion orre tion of equation 2.24 is subtra ted. The 30% drop of (t) ( orre ted) shown in Fig. 2.2b indi ates that 30% of the residual0:24% error an beattributed to \fast"os illations, sothat the overallaverage error introdu edby the CP approximation,on e orre ted for the rigid draggingand under the assumption that the fast omponent is not relevant, is less than 0:2%.
As has been pointed out previously by Remler and Madden [24℄, it is important to begin the dynami s with ele trons and ions moving in a onsistent way as we have done here in all simulations ex ept the one we now dis uss (simulation 2) and in the ase of liquidSi(simulation3). Wefoundthattheerrorinthefor esin reasessubstantiallyifthe simulationisnotstartedfromzeroioni velo ities,apro edure thatwouldotherwisehave the advantage of shortening onsiderably the time needed to rea h thermal equilibrium. Simulation 2 started from the end of simulation 1, but ele trons have been put in the groundstatebeforerestarting(ioni velo itiesandpositionswereinsteadkeptun hanged). For es were tested after0:5ps from the ele tron quen hing.
Thestandarddeviationoftheerror infor es isnow5:7%andthe errorinthe for esas orre teda ordingtotherigidionapproximationat5:68%,isnotsigni antlyimproved. However learly from inspe tion of (t) in Fig. 2.3b and the sample for e omponent in Fig. 2.3 most of this error an be attributed to the high frequen y os illations of the
(c)
(b)
(a)
0
0.01
0.02
0.03
0.04
0.05
Time [picoseconds]
−0.015
0
0.015
Force [atomic units]
0
0.2
0.4
0.6
0.8
1
Γ
(
t
)
uncorrected
corrected
−20
−15
−10
−5
0
5
10
15
20
Relative Error [%]
Figure2.3: Simulation2,CrystallineSiat330Kwhen theele trons re eivea'ki k' atthe beginningof thesimulation. See aptionof Fig. 1for explanation. (F
CP I F BO I )hasnot been s aled for visibility.
the error redu es to about 1.2% for the un orre ted for es and to less than 0.5% for the orre ted for es. The amplitudes of these os illations are nevertheless signi ant and may a e t the thermodynami s in a way that is not easy to predi t. These os illations learlyoriginatefromtheinitialjerkexperien edbythetheele tronsintheirgroundstate and survive for a long time due to the adiabati de oupling. In the liquid ( gure 2.4)
(c)
(b)
(a)
0
0.01
0.02
0.03
0.04
0.05
0.06
Time [picoseconds]
−0.01
0
0.01
0.02
Force [atomic units]
0
0.2
0.4
0.6
0.8
1
Γ
(
t
)
uncorrected
corrected
−15
−10
−5
0
5
10
15
Relative Error [%]
Figure 2.4: Simulation 3, Liquid Si at 2000 K. See aption of Fig. 1 for explanation. (F CP I F BO I
) has not been s aledfor visibility.
the situation is onsiderably worse than in the rystal. The standard deviation of the error in the for es is 3:4% whi h improves only to 3:1% with the rigid-ion orre tion. There do not seem to be high frequen y, high amplitude os illations here despite the simulation being started with nite ioni velo ities. However, there are os illationsof a lowerfrequen y (althoughstillquitehigh relativeto ioni times ales)whi hare probably due to the presen e of the Nose thermostat. It may be that the Nose thermostat has the e e t of damping out the kinds of os illations seen in Fig. 2.3 but the presen e of these other os illations is hardly an improvement. This highlights the need for areful hoi e of parameters for the Nose thermostat, parti ularlythe value of E
kin;0
(c)
(b)
(a)
0
0.003
0.006
0.009
Time [picoseconds]
-0.06
-0.03
0
0.03
Force [atomic units]
0
0.2
0.4
0.6
0.8
1
Γ
(t
)
uncorrected
corrected
-60
-40
-20
0
20
40
60
Relative Error [%%]
Figure 2.5: Simulation 4. For es on the oxygen ions in rystalline MgO at 2800 K. (a) and (b)areasinFig. 1( )Fromtoptobottom: theerrorintheCPfor es (F
CP I F BO I ), the errorinthe CPfor es aspredi tedby the rigid-ionmodel M
O R I
(dottedline),the di eren ebetween thetrue errorintheCP for es andthe predi tederror (F
CP I F BO I + M O R I ), the CP for e F CP I
and the BO for e at the same ioni positions F BO
I for a typi alfor e omponent. Dots indi ate the pointsat whi h the BO for e was al ulated (every time step).
We now look at the for es in rystalline MgO with 0
= 400a.u ( gure 2.5). The relatively high quantum kineti energy asso iated with states atta hed to the O ions means that, a ording to equation 2.21, the errors in the for es are onsiderably larger for the O ions than we have seen for Si. The errors in the CP for es have in fa t a standard deviationas large as32% . However, when this is orre ted asin equation 2.24 by attributing all the quantum kineti energy to states rigidly following the O ions the standarddeviationof the errorredu es to4:8%. Furthermore,the orre tedvalueof (t) indi ates that about 80% of the error on the O for es an els out aftera ount is taken for the high frequen y os illations, suggesting that a more appropriate estimate of the erroris0:4%. The amplitudeof the fastos illationsisa ause for on ern howeverand sin ethesimulationwasbegunatzeroioni velo ityitisnot learhowitmayberedu ed further.
1.1
1.15
1.2
1.25
Time after beginning of simulation [picoseconds]
0
1000
2000
3000
Temperature [Kelvin]
Figure 2.6: Simulation 4. Dashed lines from top to bottom are : Mg temperature, O temperature, T el ,and 10(T el T M O
). The full lineis the Oxygen temperature when itis al ulated with amass whi h isin reased by A
O .
Temperature
We fo us here only on MgO, as the e e ts of the ele troni dragging are enhan ed. A - ording to the results of Se tion III, we should expe t a di eren e between the naive de nition of temperature and the de nition orre ted by the ele troni dragging, equa-tion2.25. Inthe aseofMgO, asnotedinthepreviousse tion,this orre tiona e tsonly theoxygenatoms,asonlyaminoramountofele troni hargeis arriedby theMg
2+ ion. In Fig.2.6 we show the behavior of the instantaneous values of the naive and orre ted temperatures. The orre ted temperature ex eeds the naive de nition by about 500 K.
two atomi spe ies. It is lear that the naive de nition would implythat the twospe ies are not atthermalequilibrium. Onthe otherhand, use ofthe orre ted de nitionforthe oxygentemperaturebringsthetemperatureofthe two spe iesinmu hbetteragreement, supporting the on lusion, based on the rigid ion model, that thermodynami s an be restored byasimpleres alingof theoxygenmass. Themass res aling,as al ulated with equation 2.20 amounts to M
O
7:5 atomi mass units (M O
= 16atomi mass units). We also report in Fig.2.6 the instantaneous value of the titious ele troni kineti en-ergy, the l.h.s. of equation 2.26, and the di eren e between this quantity and the r.h.s. of equation2.26, whi h represents the ontributiondue to the rigid draggingof the ele -troni orbitals. The di eren eisverysmall, implyingthat residual ontributions due, for example, to the fast ele troni os illations are negligible in MgO ompared to the slow dragging of the orbitals.
Thermostatting the Ele troni Orbitals
In simulating liquid sili on, whi h is metalli , we have used a standard te hnique for maintaining a low titious kineti energy of the ele trons. This is to use two separate thermostats in the simulation: one for the ele troni orbitalsand one for the ions. This te hnique was rst introdu ed by Blo hl and Parrinello[15℄ and has very re ently been updated by Blo hl [16℄. In referen e [15℄ the re ommended temperature ofthe ele troni thermostat, E
kin;0
, has been determined on the basis of the rigid ion model to be twi e the value of T
M
(as de ned by equation 2.28). The reasoning behind this is that the ele trons should be free to follow the ions and also have room to perform the high fre-quen y os illations. In our simulation of liquid sili on we have used a value of E
kin;0 ompatible withreferen e [25℄howeverwe notethat this is onsiderably smallerthanthe value re ommended inreferen e [15℄. We have also done simulationsusing higher values of E
kin;0
and in all ases the errors in the for es have been greater. It is likely therefore that by de reasing further E
kin;0
we might improve further the for es however this has not been attempted here.
It is importantto note that for some systems, the hoi e of E kin;0
is ru ial and an-not be based on the simple formula of referen e [15℄. This an be seen by inspe tion of gure 2.7 whi h is a magni ation of the lowermost urve in gure 2.6. This is a plot of (T
el T
M O
) during the MgO simulation. One an learly distinguish the high-frequen y os illations of ele troni orbitals from the ioni -times ale os illations due to deviations from the rigid-ion des ription. The amplitudes of the high frequen y os illa-tions is, roughly,1 K.This is3 orders ofmagnitude smaller than T
el
and yetit results inos illationsinthefor ewhi hhaveanamplitudeof 7%. Thissuggests that hoosing the re ommended value for this system would lead tovery large errors inthe for es, and that in order to maintain the errors within a few per ent requires a predi tion of E
kin;0 whi h is orre t to within 0:1%. It is unlikely whether su h pre ision is possible and anyway for MgO the variations in E
kin;0
along the mole ular dynami s traje tory was found tobe 0:3%.
1.1
1.15
1.2
1.25
Time after beginning of simulation [picoseconds]
0
5
10
15
20
Temperature [Kelvin]
Figure2.7: Simulation4 - (T el T M O). Within the rigid-ion approximation, this is the ontribution to the ele troni kineti energy from the high-frequen y os illations. Even thoughthisisverysmall,theos illationsinthefor esareverylargeasshownin gure2.5. Smalldeviations fromthe ground state an lead to large errorsin the for es.
spe i edvalue. Similar problems shouldbeexpe ted for this system however, sin e for a system inwhi h T
el
isin reasing, its value willfrequently be lose tothe spe i edvalue, oratleast of the same order of magnitude.
Phonon Spe tra
We have al ulated the phonon densities of states of rystalline Si and MgO by fourier transformingthe velo ityauto orrelationfun tion. Inall ases,the rstpi ose ondofthe simulationwas dis arded and resultsobtained by averaging over atleast one subsequent pi ose ond. For sili on the velo ity auto orrelation fun tion was al ulated on a time domainof length 1:2ps and forMgO ona time domainof length 0:5ps.
Insili on(Fig.2.8)thedi eren eisreasonablysmall. A ordingtotherigidionmodel the frequen ies shouldbe orre ted using
! orre ted =! CP p 1+M=M (2.38) where ! CP
is the frequen y as extra ted dire tly from the CP simulation. We nd that forsili on this overestimatesbyabouta fa tor of two theamountof the orre tion. This smalldis repan ymaybeduetothelengthofsimulationusedfor al ulatingthefrequen y spe tra or due to a breakdown of the rigid-ion des ription when
0
= 800a:u:. It may alsobe that negle ting the e e t of the fast os illationsis not be ompletelyappropriate
0
400
800
Frequency [
cm
–
1
]
µ
0
=200 a.u.
µ
0
=800 a.u.
Figure 2.8: Phonon density of states of rystalline Sili on for 0 = 200 a.u. (simulation 5) and for 0 =800 a.u (simulation6).
200
400
600
800
1000
1200
Frequency [
cm
–
1
]
µ
0
=
400 a.u
µ
0
=
100 a.u
µ
0
=
400 a.u
M
O
rescaled
µ
0
=
100 a.u
M
O
rescaled
Figure2.9: Phonondensityofstatesof rystallineMgOfor 0
=100a.u. andfor 0
=400 a.u. with res aled (simulations 7 and 8) and unres aled (simulations 10 and 4) oxygen
In MgO, as expe ted the di eren e is mu h larger. We al ulate the phonon spe tra for
0
= 400a.u. and for 0
= 100a.u and nd large di eren es between them (see Fig. 2.9) highlighting again how the dynami s depends on the value of . The fa t that two spe ies are involved ompli ates matters as the mass orre tion is di erent for the two spe ies (it a tually vanishes for Mg). Therefore we should not expe t simply a rigid shiftofthefrequen ies. However, iftherigid-ionapproximationisvalid, onemay on eive to res ale the oxygen mass a priori in equation 2.2 as
~ M O = M O M O , so that the a tualCP dynami sexpressed intermsoftheBOfor es,equation2.24,be omesidenti al to the BO dynami s if the rigid ion approximation holds. We have done this for MgO, again for
0
=400a.u. and 0
=100a.u and we see that the results are mu h improved. There are only small di eren es in the positions of the peaks and the overall shapes of the urves are very similar. We noti e that with
0
=100a.u. (and nomass orre tion) frequen iesarewithin8% the orre tones. This impliesthatinordertoobtainaphonon spe trumofMgOwitha4%a ura yinthepeakpositions(4% isthe typi alun ertainty of a pseudopotential DFT approa h[26℄) the value of
0
should be about 50 a.u., whi h impliesa t2:8 a.u., orabout 1:510
4
timesteps perpi ose ond.
Dependen e of error on
0
100
200
300
400
µ
0
[a.u.]
0
10
20
30
100
×〈
δ
F
I
α
〉
0
0.5
1
100
×〈
δ
F
I
α
〉
corr
Figure 2.10: S aling of the standard deviation of the errors in the for es on the oxygen ionswith 0 (Simulations4,9and 10). hÆF I i orr
has been redu edtoeliminate an elling high frequen y os illationsby inspe tion of (t).
We now try to address the question of how the error in the CP for es depends on the titious ele tron mass. We do not know what the true -dependen e of the error in equation 2.16 is. We have made the assumption in equation 2.17 that it is, to a rst approximation, linear if one assumes that the os illations in jÆ
(1) i
i have a small amplitude. Fig. 2.10 shows hÆF
I i and hÆF I i orr
for the oxygen ion for three di erent values of 0 where hÆF I i orr
Theun orre ted errorisdominatedbythee e tofthedispla ementoftheequilibrium positions of the orbitals from the ground state, this s ales approximately linearly with
0
, thereby vindi ating our negle t of terms in equation 2.17 of higher than linear order in . The small errorwhi hremainsafter the rigid ion orre tionhas been applied ould have ontributions from many di erent sour es in luding deviations from the rigid ion des ription and ontributions fromhigher ordererror terms. Itis alsoof the order ofthe u tuations in E
total k
during the simulation.
2.5 Water
Sofarinthis hapterithasbeenshownthatsystemati errorsarepresentinCar-Parrinello simulations, the magnitude of whi h are proportional to the titious mass parameter. The examples of Siand MgOexaminedwere used mainlytoverify that our derivation of equations 2.17, 2.22 and 2.23 were orre t and to show that for strongly ioni systems su h as MgO, the error in the dynami s an be orre ted and that it does not seriously alter the thermodynami son e properties su hasthe temperatureandthe pressure have been orre ted.
We nowturn our attention towater. Waterinall itsphaseshas been one of the most studied systemswith the Car-Parrinellomethod[27,28, 29, 30, 31,32℄ and itlookslikely to ontinue to be so in the future due its obvious importan ein nature and parti ularly due to the fa tthat itformsthe basis for allof biology. The abilityof the Car-Parrinello method to des ribe water well is therefore of great importan e as mu h of the urrent understanding of itsmi ros opi propertieshas ome fromsu h simulations.
Water isa morediÆ ult ase tosimulatewith CPMDthan eithersili onor MgOand is therefore a better test of the method. Unlike sili on it has a onsiderable quantum kineti energy and unlike MgO there is some subtlety to its bonding in the sense that it annot be onsidered a ompletely ioni system. There is a degree of ovalen y to the intramole ular bonding and intermole ular intera tions are dominated by hydrogen bonds. Here we look at i e, or more spe i ally, at \heavy" i e, D
2
O be ause it is easier to simulate and it is generally what has been simulated in the past. Protons are very lightand this an auseproblems ofenergy transfer between ioni and ele troni degrees of freedom. The lower symmetry and highertemperaturein water and the lower mass of the protonrelativetothedeuteron meansthaterrors shouldbe,ifanything, greaterthan those observed here.
2.5.1 Details of the Simulation
Wehaveusednorm- onservingpseudopotentialstodes ribetheoxygen[22 ℄andhydrogen[33, 34℄ atoms respe tively. We have used a plane-wave uto of 70 Ryd. and a gradient- orre ted ex hange- orrelation fun tional (BLYP)[35℄. Only the -point was used to sample the Brillouin zone. Simulations were performed on a 242412 a.u. simu-lation ell ontaining 32D
2
state for es with respe t to the ioni masses in a preliminary simulation of liquidwater, we have obtained the rigid-ionmass orre tions ( in a.u. ) for oxygen and deuterium of 6:766 and 0:213 respe tively. We have later veri ed that these mass orre tions are optimal for i e also. We have res aled the ioni masses a priori by subtra ting these orre tionsfrom them.
We have begun our simulations by doing a long simulationof i e at low temperature ( 100 K) using an ab initio parametrized polarizable e e tive potential of the same form as that des ribed for sili a in hapter 5. This does not provide a very realisti des riptionof water but it was deemed preferableto randomizingthe positions. In order to minimizethe errors due to high frequen y os illations of the orbitals, as dis ussed in se tion2.4.3,wethenbeganthesimulationwiththeele tronsattheirgroundstateandthe ions atzero velo ity. We used a titiousmass of =900 a.u. No mass-pre onditioning s hemewasused. Afterhalf api ose ond, when the ions were ata temperature of 120 K, a Nose thermostat[19℄ was atta hed to the ions and the temperature was in reased to approximately 250 K during half a pi ose ond of simulation. The system was then equilibratedwithoutathermostatforapproximately3:5pi ose onds. Thetemperaturesof theoxygenand hydrogensubsystems were al ulatedindependentlyusingthe orre tions given by equation 2.30 and a ording to this de nition of temperature the subsystems remainedat di erent temperatures, indi ating that the system was not wellthermalized. This an be seen in gure 2.5.1. It is worth noting that somebody unaware of the issue of mass res aling would, in this ase, see a thermalized system. The simulation was now ontinued, using the ioni and orbital velo ities from the previous MD run, in two separateMDruns: oneusingavalue=100 a.u. and theother remainingwith=900 a.u (see gure 2.5.1). Changing the value of during a simulation perturbs the orbital dynami s only very slightly and has no signi ant lasting e e t on the for es su h as are seen when more severe sho ks take pla e (see se tion 2.4.3). For a system where the rigid-ionapproximationisvalid, thefa tthat wehaveres aled apriorithe massesmeans thatthe simulationsat the di erent values of should beidenti al.
It was found that after more than a further 2:5 ps the simulation with = 900 a.u. still showed no sign of thermalization. The deuterons remained at a low temperature relativetothe oxygenions. Inthesimulationwith=100a.u.,however, thesystemvery qui klyshowed signs ofthermalizationandthe subsystems were atthesame temperature after1:5 2 ps. Sin e wewould liketo ompare the phononspe tra forthe twodi erent values of and we would like to dothis for reasonably well equilibrated systems, atthe sametemperature, itwasde ided to ontinue bothsimulationsfromtheend ofthis initial 2:4 ps run with = 100 a.u. Both for = 100 a.u. and = 900 a.u., a further 5:5 ps of simulation were arried out during whi h time the oxygen and deuterium subsystems remainedat the same temperatures inboth simulations. Wemayspe ulateat this point that the diÆ ulty whi h the = 900 a.u. simulation has in thermalizing is due to the presen e of inertia between the oxygen and deuterium ions whi h impedes the motion of the deuterons. The di eren ein temperature has to bedue tothe deuterons movingtoo slowlyrelative tothe oxygen ions. The bondingbetween them has adegree of ovalen y andthis ovalentbond arriesaninertiawhi honeshouldnotexpe ttobewella ounted for in the rigid-ion approximation. Therefore, the fa t that when the ovalent bond is
0
0.5
1
1.5
2
2.5
3
3.5
Time [picoseconds]
100
150
200
250
Temperature [Kelvin]
Oxygen
Deuterium
150
200
250
300
350
Temperature [Kelvin]
Figure 2.11: Temperatureof the oxygen and deuterium subsystems as afun tion of time a ording to the orre t de nition of temperature in whi h the extra mass due to the ele troni orbitals is a ounted for (top) and a ording to the na^ve de nition (bottom) in whi hthe temperature isnot orre ted.
less \heavy" thedeuterons an movefaster suggests that thismay beane e t due tothe orbital's titiousinertia.
For es
After1psofthe =900MDrun,the for eswere omparedwiththegroundstatefor es. The for eon anindividual oxygen ordeuterium ion is dominated by the intra-mole ular for e, thefor eduetotheothertwoionsintheD
2
Omole ule. Sin ethemole uleremains relativelyrigid,itismoresensibletoexaminethemoresubtleinter-mole ularfor eswhi h are of primary on ern to those studying the stru ture of water. In order to do this we look at the sum of the for es on a mole ule. In gure 2.5.1 we plot some sample for es on D
2
O mole ules. It is learly seen thatthere are verylarge di eren es between the CP for es and the ground-statefor es but that mu hof this di eren e an be orre ted with
3.5
4
4.5
5
5.5
6
Time [picoseconds]
150
200
250
300
350
Temperature [Kelvin]
µ
= 900
150
200
250
300
350
Temperature [Kelvin]
Oxygen
Deuterium
µ
= 100
Figure2.12: Corre ted temperatures of the oxygen and deuterium subsystems as a fun -tion oftime for=100 a.u. (top) and =900 a.u. (bottom).
bea ause for on ern. The error before orre tion amountsto45% of the root-mean-square ground-state for e omponent. After orre tion, this number redu es to 12:5%.
Although for es are very important in mole ular dynami s, they are not generally the quantity whi h are extra ted from simulationsand it is not obvious how, ingeneral, errors in for es map onto errors in thermodynami quantities. In order to get a feel for the s ale of these errors we look at the for es along the same traje tory for the lo al density approximation(LDA) [36, 37℄ and the Perdew-Burke-Ernzerhof(PBE) ex hange- orrelation fun tionals. There has been mu h dis ussion in the literature about what ex hange- orrelation fun tional one should use for water[29℄. It is generally a epted that the LDA performs very poorly for water and most people use generalized gradient approximationswhi hseemtogivebetterresults. Di erentgradient orre tedfun tionals have been shown togivequitedi erent radialdistribution fun tionsfor liquidwater [29℄. Figure 2.5.1 shows a omparison of the for es from BLYP, LDA and PBE fun tionals. Wefound thatthe averagedi eren ebetween PBEand BLYPfor es was7:8%andthe
0
5
10
15
20
Time [fs]
-0.006
-0.004
-0.002
0
0.002
Force[a.u]
0.002
0.004
0.006
0.008
0.01
Force[a.u]
Ground state force
Car-Parrinello force
Corrected Car-Parrinello force
0.002
0.004
0.006
0.008
0.01
Force[a.u]
-0.008
-0.006
-0.004
-0.002
0
Force[a.u]
Figure2.13: Somesamplefor es onD 2
Omole ules. TheCar-Parrinellofor eis ompared to the ground state for e and the Car-Parrinello for e on e orre ted using the rigid-ion orre tion.
general,although thereare leardi eren es inthe magnitudeofthe for es fromLDA and BLYP, for example, the time-derivative is generally quite similar for all the fun tionals tested. This is not the ase forthe Car-Parrinellofor e, where the derivativeof the for e di ers quitestrongly from the groundstate for e.
Finally, we have looked at the for es after very brief simulations in whi h velo ity res aling was employed. Velo ity s aling is ommonly used for CP simulations and has been used in the past to equilibrate liquid water[30 ℄. However, it gives an equivalent sho k to the ele troni degrees of freedom as beginning a simulation with ions at nite velo ities does. In this test,the velo ity res alingfollowed the initial0:5 pssimulationat 120 Kand the velo ities were res aled tobringthe temperatureto220 K.Velo itieswere adjusted only4 times in total and then the system was equilibrated for 1 ps. Wefound, on e again, that there were large-amplitude os illations in the for es with frequen ies typi al of the ele troni orbitals. These os illations also appeared mu h less harmoni than the os illations whi h were seen in the ases of Si and MgO in se tion 2.4.3. The magnitude of these os illations highlights the fa t that it is dangerous to perturb the ele troni system by hanging abruptly the ioni velo ities, parti ularly onsidering the
