Introduction to Density Functional Theory. Jeremie Zaffran 2 nd year-msc. (Nanochemistry)

Transcription

1 Introducton to Densty Functonal Theory Jereme Zaffran nd year-msc. (anochemstry)

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3 A- Hartree appromatons Born- Oppenhemer appromaton H H H e The goal of computatonal chemstry H e??? Let s remnd H e T e et ee Key-problem T e et Z 0 R r j H at and ee j r r j H e H at ee

4 Frst Hartree appromaton: Instead of consderng the operator ee let s consder each electron n the mean feld eperenced from all the other electrons. j ee r where r r r r d 3 r Effectve potental: eff et ee

5 Second Hartree appromaton: Due to the Paul ecluson prncple the system wave functon s supposed to be a sngle Slater determnant: S! where s r From Hartree-Fock model to DFT

6 Consequences o aratonnal prncple o Orthonormalsaton of the spn orbtals bass set E e H e H at j jj j j jj r j Jj Coulomb ntegral j j r j j Kj Echange ntegral Hartree-Fock potental: HF J K

7 Lmtatons of the Hartree-Fock Model. Hartree-Fock model deals wth a non-nteractng reference system and thus the correlaton energy s not taken n account.. The wave functon s a functons reles on 3 varables where s the number of electrons n the system. ery tme-consumng only small systems.b: The correlaton energy could be reached usng post Hartree-Fock methods epandng the wave functon on a bass of several Slater determnants (Confguraton Interacton-CI ) or perturbaton method (MP ) 3. The wave functon has no physcal sense only ts square has one!

8 B- Densty Functonal Theory (DFT) Let s set bass H e T e et ee Functonal: Mathematcal applcaton gong from the functons space to the scalars space. otaton: F[f]= whch means f F Any chemcal system s utterly defned provded one knows ts electrons number and ts eternal potental et and thus ground state energy could be reached such that E 0 E et

9 Hohenberg-Kohn Theorems (964) ) The eternal potental ets a unque functonal of the ground state densty 0 Consequence: Provded one knows the ground state densty one gets n turn the eternal potental and thus the hamltonan resultng n the ground state wave functon and energy (and all the system propertes). et H E (and all other propertes)

10 Hohenberg-Kohn Theorems (964) E ) The ground state energy wll be reached f and only f one use the ground state densty 0 0 n the energy functonal. n other words the well known varatonal prncple! E E E whch means tral tral 0 0 or E 0 mn E

11 DFT key ponts From Hartree-Fock model to DFT The electronc densty becomes the fundamental varable! Interest: s only a functon of 4 varables (yzs) and no more of 3 varables as wth. s an observable. Any DFT algorthm should am to reach only the ground state and no ected state! The energy mnmsaton algorthms have to take care about two man constrants lyng on the densty: must be -representable whch means assocated to an acceptable wave functon : square ntegrable functons The Slater determnant s only an eample of such a set! must be et -representable whch means gvng rse to a fnte eternal potental. ote that to ths date we don t know what makes a densty et -representable on the mathematcal pont of vew. Levy constraned search scheme

12 Epresson of the energy functonal and lmtatons of the Hohenberg-Kohn theorems E T et E ee F HK Feature of the system Unversal functonal et e dr F HK???? ( T s not a functonal of the densty and E ee s not completely known)

13 Kohn-Sham approach F T HK E ee J E c Coloumbc repulson (known)? Idea The major part of the Hohenberg-Kohn functonal s the knetc energy the remander could be just appromated. So let s fnd a way to epress T

14 Owng to the Hartree-Fock theory T s eactly known for a non-nteractng reference system S! : Kohn-Sham (KS) orbtals And thus (teratve resoluton) Kohn-Sham equatons KS f Where the KS operator s j j S j KS r f Effectve or Sham potental ee et j S r

15 Hghlghts of KS approach KS-orbtals have no physcal meanng! The target s only to reach the densty. The KS-orbtals could be epressed as atomc orbtals or as Bloch waves accordng to the calculaton code. An ntal electronc densty nput s necessary Self-Consstent-Feld (SCF)

16 SCF scheme Intal nput densty Sham potental calculaton Sham equatons resoluton Densty output etracton Self-Consstent- Feld? O YES OUT

17 Focus on correlaton energy Correlaton: Mathematcal defnton: electron at r and electron at r are correlated f the followng relaton s OT verfed r r r r ~ Physcal meanng: Classcal and non classcal effects due to the many-body nteractng system. To ths date no useful epresson of the correlaton s known! If an eact functonal of the correlaton energy was known the Schrodnger equaton could have been solved EXACTLY-wthout any appromaton

18 But unfortunately the only mathematcal (and not so useful) formalsm we have s E c T TS Eee J T E C ncl wth T TS T C E ncl : Knetc energy of the real system : Knetc energy of the reference system : Resdual knetc energy : on-classcal energy The Kohn-Sham approach s eact only the echange-correlaton functonal to be appromated! E c has

19 C- The echange-correlaton problem How to appromate ths functonal? Local Densty Appromaton (LDA): Based on the homogeneous electrons gas model. Echange-correlaton densty functonal s eactly known owng to the Thomas-Ferm model. Gradent Generalzed Appromaton (GGA): PBE Applcaton of the gradent operator on the prevous model. Meta-GGA: BB95 Applcaton also of the laplacan operator. Hybrd functonal: HSE06 B3LYP Introducton of an eact Hartree-Fock part n the Echange functonal. E c % E ( %) E HF X GGA X E GGA C

20 Jacob s ladder (Pedrew metaphor) Heaven: Eact soluton? Hybrd MGGA GGA LDA Earth: HF model