Molecular Dynamics and Density Functional Theory

Transcription

1 Molecular Dynamcs and Densty Functonal Theory What do we need? An account n pemfc cluster: Host name: pemfc.chem.sfu.ca I wll take care of that. Ths can be usually a common account for all of you but please at the begnnng of your work every group makes ts own drectory. pemfc cluster s a lnux based cluster, therefore you should know some lnux commands. I wll gve you a lst of mportant lnux commands later. The followng packages: 1. putty.exe a open shell program for SSH under the wndows. Wth that you can connect to pemfc cluster and run the programs/commands of the cluster.. psftp.exe a open shell program for ftp to transfer your fles from your computer to the pemfc cluster and vse versa. you can download them from the followng address (they are free): 3. rasmol t s a graphc nterface packege n whch you can see the poston of the atoms n your system you can download t from the followng address: (please download the latest verson under the wndows

2 The am of ths project We want to calculate the frst or possblty both of the followng tems: Hydrogen or CO adsorpton energy. Molecular dynamcs smulaton of a hydrogen atom or a CO molecule adsorbed on a Pd slab. Hydrogen and CO adsorpton energes In order to calculate the CO adsorpton energy on a Pd slab we have to do: 1. Calculate the total energy of a Pd slab. The total energy means all nteracton energes between all the Pd-Pd atoms. We call t E slab Note that each atom has proton + electrons and the electrons can nteract wth each other (What s the sgn of the total energy?) Fg.1 shows a slab of Pd(111) wth 4 layers of Pd. Fg.1 shows a slab of Pd(111) wth 4 layers of Pd. When we say Pd(111) that means a Pd bulk whch s cut along the surface (111). We have nfnte number of Pd atoms along x and y drecton (n the pcture t shows only part of that). We have to calculate the total energy of a CO molecule n a gas phase (n the vacuum and n the absence of everythng). It s called E CO. 3. We have to calculate the total energy of a slab of Pd when CO molecule s adsorbed on that (E slab+co ). That means the nteracton of all Pd atoms wth each other ncludng CO molecule. Fg. shows a pcture of the Pd slab when CO s adsorbed on top of t. Fg. CO adsorpton on Pd(111) slab. The total energy contans all the nteractons ncludng the nteracton wth CO.

3 Hence the CO adsorpton energy E ads (or CO-Pd bndng energy) can be obtaned by: E ads = E slab+co E slab - E CO Now the queston s how we can calculate the total energy of for example a slab of Pd? Ab-nto total energy calculaton of a system of nucleons and electrons When we say ab-nto calculaton ths means calculatons n whch we need no expermental data for that as an nput,.e. atomc radus, bndng energes, onzaton energes... In order to calculate the total energy of a system of nucleons and electrons (see Fg.3 for example) we have to consder two facts: Fg.3 System of nucleons (red) and electrons (gray). The nucleons consder as classcal partcles whle electrons consder as quantum partcles 1. In prncpal electrons and nucleons are very small therefore we have to consder them as quantum partcles. When we say quantum partcles that means that at tme t=t 0 we can not say exactly where they are and what energy they have. We have to consder them as a wave packet or wavefuncton. Snce n our case nucleons are Pd and they are much bgger compare to electrons we approxmately consder the nucleons as classcal partcles. Electrons -> quantum partcles -> Wavefuncton -> Charge densty n(r) -> Schrödnger eq. Nucleons -> classcal partcles -> Pont charge -> Newton law. Born-Oppenhemer approxmaton: s based on smallness of the electron mass as compared to the nuclear mass. Wthn ths approxmaton when we want to solve the Schrödnger Eq. for electrons the nucle are regarded as a fxed charges.

4 Suppose that we would lke to calculate the total energy of a system. Frst we have to calculate the Schrödnger equaton for electrons and then determne the total energy of electrons E el and then calculate the energy of the ons E on The total energy wll be gven by: E tot = E el + E on Schrödnger equaton for system of electrons H Ψ = E Ψ where Ψ s the total wave functon of the system and H = T + V e-e + V ext s the Hamltonan. V e-e s electron-electron coulomb nteracton and t s gven by: n(r' )dr' V e e [n] = r r' V ext z r r r-r n(r ) V ext V ext V ext x y Charge densty of electrons n(r) Fg. 4 The electronc charge densty n(r) of a system of electrons. The charge densty s larger n the darker regon. V ext s a source of an external potental.e. a proton. and V ext s the nteracton wth an external potental. In our case the external potental s just electron-proton nteracton: V ext [n] = N = 1 r z R z r R r-r z x y R s the poston of ons. T s the knetc energy operator of the nteractng system. The bad news s that there no analytcal form for T and Ψ therefore there s no soluton for Schrödnger equaton.

5 Densty Functonal Theory (DFT) Ths theory was ntroduced n 1964 by Hohenberg and Kohn *. It s based on two mathematcal theores: * W. Kohn and L. J. Sham. Phys. Rev., 140: A1133, 1995 * P. Hohenberg, W. Kohn. Phys. Rev., 136: B864, 1964 Theory1: External potental s a unque functonal of the densty n(r). That means f we have a system of electrons, the ground state densty n 0 (r) correspondng to an external potental (lke the potental from the protons) V ext can not be reproduced usng any other potental V ext. Theory : The correct ground state densty n 0 (r) mnmzes the total energy functonal. Based on the DFT and these two theores the many body problem of fully nteractng partcles n an external potental V ext (or potental from the protons) s smply replaced by a system of non-nteractng partcles n a effectve potental V eff gvng the same ground state charge densty. Interactng partcles (electrons) H = T + V e-e + V ext DFT Non nteractng partcles H s = T s + V eff where H s s the Hamltonan of a non nteractng system and T s s now the knetc operator of a non-nteractng system wth N partcles whch can be easly gven by: T s = N = 1 h d m dr The effectve potental V eff s: V eff = V e-e + V ext + V xc It has one extra term whch contans all the energy contrbuton whch were not takng nto account n the transton from the nteractng system to the nonnteractng system. V xc s called exchange-correlaton potental whch ntroduce quantum mechancal many-body electronc effects nto the model and a porton of the knetc energy whch s needed to correct T s to obtan the true knetc operator of a real system T.

6 The exchange-correlaton potental can be expressed as: Vxc [n] = dexc[n] dn where E xc [n] s the exchange-corralaton energy. The above DFT s formally exact, but as such t s unless n practcal applcatons because all the dffcultes related to the exchange correlaton energy are stll unsolved. To proceed further t s necessary to fnd an approxmaton for the exchange correlaton energy. The most common approach s the Local Densty Approxmaton LDA whch descrbes qute well a large number of systems and has been successfully appled to some ab-nto calculatons. Wthn ths approxmaton the exchange correlaton energy s gven by: E xc [n] = ε xc (n)n(r)dr where ε xc(n) s the energy of a homogeneous electron gas and t s known. Therefore the E xc [n] and hence V xc [n] are all known. Ths approxmaton has been shown capable of dealng on the same ground wth atoms, molecules, clusters, surfaces and nterfaces whch have been successfully reproduced. Nevertheless, besdes these successes there are also some drawbacks of the approxmaton. For nstance, the cohesve energes of solds are systematcally overestmates, whle lattce constants are systematcally underestmated Errors n the structural propertes are usually small for crystals wth covalent or metallc bonds, but t s well known that the hydrogen bond cannot be descrbed accurately wthn LDA In the feld of metals, the ground state structure of crystallne ron s predcted to be paramagnetc fcc, nstead of a ferromagnetc bcc. Varous approxmatons have been ntroduced n the course of the years to mprove LDA. The generalzed gradent approxmaton (GGA) s one of those approxmatons whch s more or less commonly accepted to be an mprovement over LDA. Another approach to obtan a more accurate results s to construct a functonal whch depends not only on charge densty n(r) but also on the magntude of the gradent of the charge densty Gradent Approxmaton GGA. n(n). Ths approach s called Generalzed The choce of the functonal form for the GGA s not unque and many dfferent functonal have been proposed. Among the many types of functonal, n ths work we used Perdew-Wang * 1991 functonal (PW91). John P. Perdew and et.al, phys., Rev. B, 46:6671, 199

7 Implementaton What we have done so far was to express the effectve potental V eff n terms of V e-e [n], V ext [n] and V xc [n]: V eff [n] = V e-e [n] + V ext [n] + V xc [n] In order to calculate to total energy of the system of electrons we need to solve the followng Schrödnger equaton. H s Ψ = E el Ψ (T s + V e-e [n] + V ext [n] + V xc [n]) Ψ = E el Ψ Snce the partcles n our system does not nteract wth each other the total wave functon can be wrtten as the super poston of the one partcle wave functon and therefore the above equaton can be gven as a set of sngle-partcle equatons known as one-partcle Kohn-Sham equatons. (T s + V e-e [n] + V ext [n] + V xc [n]) φ = ε φ, =1,.,N where N n the total number of partcles n the system, therefore we need to solve N tmes of ths equaton. Note that n order to solve ths equaton we need to know the charge densty of the system n(r). But on the other hand the charge densty of the system s gven by: n(r) = N = 1 ϕ whch depends on φ. The effectve potental depends on the densty, the densty depends on Kohn-Sham orbtals (φ ) and Kohn-Sham orbtals depends on the effectve potental!! Ths set of equatons form a loop whch has to be solved self consstently snce the Hamltonan s a functonal fo wavefunctons and wavefunctons themselves are the soluton of the Hamltonan: One starts.e. wth the approxmate densty, constructs the effectve potental and solve the Kohn-Sham equaton to obtan the orbtals whch are then used to calculate the new densty and new effectve potental and new orbtals etc. Ths procedure s contnued untl the densty, orbtals and hence total energy of electrons no longer change from prevous nteracton. The result of solvng the above self consstent equaton are the total energy of the electrons: E el = ε and Kohn-Sham orbtals φ and charge densty n(r) (see flowchart 1)

8 Tral charge densty n(r) Set up Hamltonan H=T s +V e-e +V ext +V xc calculate the wavefunctons and the total energy E from one-partcle Shrodnger Eq. new ϕ new ϕ,e ϕ new 5 ϕ < 10 Yes new ϕ Output, Eel,n(r) No n(r) = N = 1 ϕ new Flowchart 1

9 Bass set-plane wave In ths secton we are gong to brefly explan how we can solve the Kohn-Sham equaton and obtan the wavefunctons and energes. If we multply both sdes of the Kohn-Sham Eq.? by φ * (r) and take an ntegral from both sdes we gets: h m * d ϕ (r). dr ϕ (r).dr + ϕ * (r) * ( Ve e[n] + Vext [n] + Vxc[n] ) ϕ(r)dr = ε ϕ (r) ϕ(r) dr One way to solve the Kohn-Sham equaton s to expand the wavefuncton φ n a plane wave bass set. Usng plane wave as a bass set has some advantages and some dsadvantage. The advantage of usng the plane wave s that t whch help us to smplfy the above Kohn-Sham equaton. ϕ (r) = k= 1 C,k.exp(k.r) where mathematcally the sum s over a nfnte number of plane wave but of course n practce we see that t already converges for a large number of plane wave. By substtutng ths equaton nto the above Shrodnger equaton and usng the followng mathematcal prncpal: exp ( k r).exp( k' r ) dr = δkk' We end up wth a matrx wth N max x N max elements where N max s the number of the plane waves. N max depends very much to the sze of your system..e. number of atoms n your system and also the elements you use. t can be as bg as 1e+5 number of plane waves. In order to get φ and ε from the matrx we need to dagonal t whch of course needs a powerful computer or even several computers workng as parallel. Remndng: When φ and ε are calculated n the next teraton a new charge densty should be constructed and agan a new matrx should be dagonalzed. Usually teratons n needed to get the total energy of the system self consstently.

10 Total energy of the system The total energy of the system wll be then the sum over the total energy of electrons and protons: E tot = E el + E on where E on s just the culomb nteracton between ons E on =,j, j Z R R j where R and R j are the poston of the ons. After we calculated the total energy we can then calculate the force on each on n the system de F = dr tot Now f you want to perform a geometry optmzaton the ons wll move along the force F n the next onc nteractons untl we approach to a stuaton where there s no sgnfcant force on any ons. In ths case you have found the relaxed poston for ons. In ths pont the total energy of the last onc nteracton s what you need for calculatng the adsorpton energy. Note the you have two types of loop (teraton): nner loop or electronc teracton and outer loop or onc teraton. See flowchart. What s the dfference between Molecular dynamcs and geometry optmzaton? When you perform the geometry optmzaton n every onc teraton the ntal velocty of ons are zero and therefore when you move the ons along the forces you always go along the more mnmum energy and fnally you get to global mnmum energy. It s lke movng a sprng toward to ts equlbrum poston very slowly, so n equlbrum pont (or global mnmum energy) there s no force on ons and snce you have to ntal velocty you stop there. When you perfome a molecular dynamcs n every onc teraton you consder the ntal velocty V 0 0 for the ons and therefore when you move the on along the force you wll approach to the mnmum energy (or equlbrum) but you wll pass ths pont n the next teratons due to havng ntal velocty (t s lke a pendulum where the sold oscllate around ts mnmum

11 Tral charge densty n(r) Set up Hamltonan H=T s +V e-e +V ext +V xc calculate the wavefunctons and the total energy E from one-partcle Shrodnger Eq. new ϕ new ϕ,e ϕ new 5 ϕ < 10 Yes Caculate the force on ons F on No n(r) = N = 1 ϕ new Yes F on < 10-3 No End Move the ons along the forces Flowchart

12 Geometry optmzaton In each teraton the ntal velocty s zero (T=0K) F on Next onc teraton F on Next onc teraton

13 Molecular dynamcs In each teraton the ntal velocty s not zero. It s calculated accordng to the temperature of the system Frst confguraton F on Next onc teraton = F on Next onc teraton =3 =6 =5 =4 F on F on F on

14 Problem wth Kohn-Sham equatons: The one-partcle Kohn-Sham equatons can be solved for an solated system wth maxmum few hundred number of partcles. In a crystal or a slab however we are dealng wth a order of 10 3 number of atoms. Smulatng such an amount of partcles s a mpossble task. Therefore a smplfcaton needs to be done. If the system s fully perodc a approprate choce would be to use perodc boundary condtons. Ths can be done by usng the defnton of a Bravas lattce.

15 Bravas lattces: A Bravas lattce s a fundamental concept n the descrpton of any crystallne sold, whch specfes the perodc array n whch the repeated unts of crystal are arranged. A Bravas lattce s an nfnte array of dscrete ponts wth an arrangement and orentaton that appears exactly the same on all Bravas lattce ponts. It consst of all ponts wth poston R of the form: R = n 1 a 1 + n a + n 3 a 3 where a 1, a and a 3 are any three vectors not all n the same plane and n 1, n and n 3 are all nteger values. Therefore f there s an atom or a specfc electronc structure at poston r the same atom or structure s repeated along three bass vectors a 1,,3 (see Fg1 for two dmenson). The calculaton savng comes snce only the atoms and electrons nsde the unt cell of the calculaton - called super-cell and shown n the Fg 1 wth thck sold lnes - needs to be explctly consdered. Fg 1. replcaton of super-cell c along x, y and z drecton. Thre are 6 atoms of type 1 and four atoms of type exst n the super cell.

16 The super cell and the Bravas lattce vectors a 1 and a n Fg. are as followng: a a a 1 = a = (1 0 0) a a = a j = (0 1 0) a a 1 y x Fg. There are two atoms n the super cell. The poston of these two atoms are (000) and (½ ½ 0). Note that we stll consder a two dmenson Bravas lattce therefore the poston of atoms along z-drecton are zero. Prmtve unt cell: s a super cell n whch contans only one atom. The Bravas lattce of a prmtve super cell (or unt cell) of fg. s gven by: y x a a 1 a 1 a = j o o ( a( cos( 45 ) + sn( 45 ) = a + = j o o ( a( cos( 45 ) + sn( 45 ) = a + Fg.3

17 Now wth that overvew we are gong to three-dmenson Bravas lattce. Lets assume we have fcc structure whch s cut along surface(100).. a a 3 a 1 Fg.4 If we consder the followng Bravas lattce vectors: a 1 = a a = a j (Equaton. A) a 3 = a k then the poston of the atoms n the frst layer (where a 3 =0)wll be: Atom 1: (0 0 0) Atom : ( ½ ½ ½ ) Frst layer (a 3 =0) a (½ ½ ½ ) a 1 (0 0 0) And accordng to Fg.4 the poston of the atoms n second layer (where a 3 = ½ ) wll be: Atom1: ( ½ 0 ½ ) Atom: (0 ½ ½ ) Second layer (a 3 = ½ a) (0 ½ ½ ) (½ 0 ½ )

18 And snce we have ndcated a 3 = a j therefore the 3 rd layer (where a 3 = a) s the same as frst layer (where a 3 = 0). Ths s n agreement wth Fg.4. Exercse 1: Consder the same surface (100) and try to ndcate the poston of the atoms and Bravas lattce vectors for a prmtve unt cell. Ths result wll be very useful for your project. Exercse : Do the same as exercse 1 for surface (110). Exersce 3: Try to specfy the Bravas lattce vectors for a prmtve unt cell of a (111) surface. By consderng the Barvas lattce vectors equaton A we wll have a bg problem. The problem s that they do not represent any surface. They just show a fcc bulk (see Fg.5)! However n order to show a surface we should be able to break the symmetry along the z axes! How can we do that? Top-vew Sde vew 1 st layer (blue) + a 3 nd layer (red) a a 3 - a 1 Fg. 5