From Atoms to Materials: Predictive Theory and Simulations. Quantum mechanics in 5 postulates

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1 8/5/15 Fom Atoms to Mateials: Pedictive Theoy and Simulations Basic quantum mechanics of electonic stuctue Ale Stachan School of Mateials Engineeing & Bick Nanotechnology Cente Pudue Univesity West Lafayette, Indiana USA Quantum mechanics in 5 postulates 1. The state of electons is detemined by thei wave function Ψ(, t) We will focus on equilibium popeties: WF does not depend on time Ψ(). Physical obsevables linea opeatos Mathematical objects that act on functions Position: Momentum: p = i = i # x, y, & % ( $ z ' Alejando Stachan Atoms to Mateials 9 1

2 8/5/15 Quantum mechanics in 5 postulates 3. Aveage esults of measuements is given by O = Ψ()OΨ()d 3 Example: = Ψ( ) Ψ( )d 3 = Ψ( ) d 3 Ψ( ) Pobability density of electon being in volume d 3 aound Alejando Stachan Atoms to Mateials 10 Quantum mechanics in 5 postulates 4. The equilibium wave function can be obtained fom the Schödinge equation: Hψ = ( ) Eψ ( ) 5. Pauli s exclusion pinciple Two electons maximum pe obital Electons in one obital must have diffeent spin Alejando Stachan Atoms to Mateials 11

3 8/5/15 The Hamiltonian opeato H is the opeato fo the total enegy of the system We can wite it using the two opeatos we just leaned Fo hydogen the Hamiltonian opeato contains two tems: KineHc enegy: K = p m = m PotenHal enegy: = m V = e % x y ( ' * & z ) Alejando Stachan Atoms to Mateials 19 The Bon Oppenheime Hamiltonian Any mateial is a collechon of electons and nuclei N e electons at posihons i N n nuclei at posihons R i and chage Z i Bon Oppenheime appoximahon Massive nuclei ae classical and fixed in space H = N e i=1 m i N n N e e N Z i j R e n Z i Z j i j i R i R i< j j N e i< j e i j Alejando Stachan Atoms to Mateials 19 3

4 8/5/15 Electonic stuctue and ionic dynamics H ele ψ i ({ }) = Eψ { i } ( ) Hamiltonian and wave funchon depend paametically on ionic posihons Eigenvalue (enegy) then depends on the ionic posihons H ele ({ i };{ R i })ψ { ( i };{ R i }) = E ({ R i })ψ { i }; R i Classical dynamics of ions is then govened by this enegy: H ions ({ Ri },{ P i }) = E R i R P i = i M i P i = F i = Ri E N n ({ }) R i { } ( ) i=1 P i M i ( { }) Alejando Stachan Atoms to Mateials 19 The Schödinge equation Hψ = n ( ) Eψ ( ) A family of solutions E ψ ( ) Eigenvalues need to be eal, so Hamiltonian is an hemitian opeato Wavefunctions ae nomal to one anothe ψ n Popeties of the Schodinge Eq. and WF s n Equivalent to eigenvalue poblem Ax = λx A family of solutions λn x n Symmetic matices have eal eigenvalues Eigenvectos ae nomal to one anothe N ( )ψ m ( )d 3 = δ nm xnxm = x n,i x m,i = δ ij i=1 Alejando Stachan Atoms to Mateials 19 4

5 8/5/15 Nodal theoem The gound state wavefunchon (lowe enegy) has no nodes (except at domain boundaies) The moe nodes a wavefunchon has, the highe its enegy Ψ x Alejando Stachan Atoms to Mateials 0 Fom Atoms to Mateials: Pedictive Theoy and Simulations Quantum well, quantization and optical pocesses Ale Stachan stachan@pudue.edu School of Mateials Engineeing & Bick Nanotechnology Cente Pudue Univesity West Lafayette, Indiana USA 5

6 8/5/15 Paticle in a box o quantum well Potential Enegy Fist example: infinite potential well x x=0 x=l To find the possible states (WFs) of the electon: solve Schödinge Eq: m x V ψ = ( x) ( x) Eψ ( x) What is the pobability of the electon being outside [0:L]? Schödinge Eq. inside well: ψ = m x ( x) Eψ ( x) x=0 x=l Ale Stachan Atoms to Mateials 11 Enegy Infinite Potential Well x x=0 x=l Solve the following diffeenhal equahon: ψ Let s ty this function: ψ = m x ( x) Eψ ( x) ( x = 0 ) = 0 ψ( x = L) = 0 ψ ( x) = A sin( kx) Bounday condition Still have to satisfy the bounday conditions Ale Stachan Atoms to Mateials 1 6

7 8/5/15 Enegy Infinite Potential Well x x=0 x=l We have a solution but it need to satisfy the bounday conditions ψ ψ ( x) = Asin( kx) ( x = 0 ) = 0 ψ( x = L) = 0 ψ ( x = 0) = Asin( k0) = 0 ψ ( x = L) = Asin( kl) = 0 ψ n WavefuncHons ( x) = Asin# nπ " L x $ & % Enegies ( ) ( ) E n = k m = nπ ml Ale Stachan Atoms to Mateials 13 Infinite Potential Well: quantization Only some values of k and enegy ae allowed (quantized) Gound state enegy is not zeo Wave functions 16 π ml ψ n nπ L ( x) = Asin x Enegy x/l π 9 ml π 4 ml π 1 ml E n = n π ml Each WF is shifted up accoding to thei enegy The moe wavy a WF is, the highe its enegy (emembe kinetic enegy is popotional to the gadient of the WF squaed) Ale Stachan Atoms to Mateials 14 7

8 8/5/15 Infinite Potential Well: optical popeties Photons ae paticles with enegy popotional to thei fequency E ω = ω = hν c = λν = λ ω π π ml 16 π ml Absoption: A photon can only be absobed if it caies the enegy equied to pomote an electon to an excited state Enegy x/l π 9 ml π 4 ml π 1 ml Emission: An excited electon with enegy E n ) can elax to an empty, lowe enegy state (E n ) and emit a photon with fequency: ω = E n E n' Ale Stachan Atoms to Mateials 15 Optical popeties small molecules cyanine π electons in conjugated molecules ae basically fee to move aound: pinacyanol ( ) π ΔE = N ex N GS ml dicabocyanine G. M. Shalhoub, J. Chem. Edu., 74, 1317 (1997) Ale Stachan Atoms to Mateials 16 8

9 8/5/15 Quantum confinement Ale Stachan Atoms to Mateials 17 Fom Atoms to Mateials: Pedictive Theoy and Simulations The hydogen atom Ale Stachan School of Mateials Engineeing & Bick Nanotechnology Cente Pudue Univesity West Lafayette, Indiana USA 9

10 8/5/15 Now a slightly moe difficult example: H The Schödinge equation: " m x y z Ze ψ = ( ) Eψ ( ) Let s think what we should get befoe doing the math Hydogen atom: Squae well: WF (gound state) WF? x WF (Second enegy) PotenHal Ale Stachan Atoms to Mateials 3 The hydogen-like atom Slightly moe geneal Ze poblem: nuclea chage Ze " ψ ( ) = Eψ ( ) m x y z The potential only depends on the magnitude of that is: = z So we will use Spheical Coodinates z=.cos() φ y y=.sin().sin(φ) A point in 3D space can be epesented by: Thee Catesian coodinates x, y, z OR O two angles ( and φ) and x=.sin().cos(φ) x Ale Stachan Atoms to Mateials 4 10

11 8/5/15 11 ( ) ( ) E Ze z y x m " ψ ψ = The Schödinge Eq. in spheical coodinates, need to wite the Laplacian in spheical coodinates (get it fom a book): ( ) ( ) φ ψ φ ψ φ,,,, 1 sin cos sin 1 E Ze m = ( ) ( ) ( ) ( ) φ φ ψ Θ Φ = R,, We will ty the following type of solution (this is called sepaation of vaiables): Let s focus on finding the gound state (lowest enegy) wavefunction How would the angula pat look like fo the gound state? The hydogen-like atom Ale Stachan Atoms to Mateials 5 ( ) ( ) ER R Ze m = Tial function: ( ) ( ) a A R / exp = Now we plug the tial function into the Schodinge Eq.: = a EA a A Ze a a A a a A m exp exp exp exp And then: E Ze a a m = 1 The hydogen-like atom Ale Stachan Atoms to Mateials 7

12 8/5/15 The hydogen-like atom 1 Ze ma = E ma Only solution is fo both sides to be zeo: Ze ma = 0 and 0 = E ma 1 a = = Z e m Z a 0 Boh adius (size of the atom) 1 E = m a 1 1 Ze = ma a a 1 Z = m a 0 = 1 Z e = a 0 Ale Stachan Atoms to Mateials 8 Hydogen gound state Gound state wave function: ψ ( ) ( ) = Aexp $ # a 0 ψ " Boh s adius % ' & a 0 = e m = Å (a 0 ) 4π ψ ( ) (a 0 ) E = 1 e a 0 = ev Hatee enegy: e a 0 = ev Ale Stachan Atoms to Mateials 30 1

13 8/5/15 Excited states of Hydogen Afte sepaation of vaiables and assuming no angula dependency: ψ (,ϕ, ) = R( )Φ( ϕ)θ( ) Φ( ϕ) = Θ( ϕ) =1 We obtain a diffeential equation fo the adial pat: m Ze R ( ) = ER( ) As befoe, thee is a family of adial solutions: 13.6eV R n ( ) E n = Z n n is called pincipal quantum numbe Ale Stachan Atoms to Mateials 36 Radial solutions n=3 These states ae called s nodes E ~ ev Enegy (ev) n= 1 node E ~ -3 4 ev n=1 No nodes E ~ ev n is called pincipal quantum numbe and denotes the numbe of nodes in the wave function moe nodes - > moe wiggles - > moe enegy Ale Stachan Atoms to Mateials 37 13

14 8/5/15 Fom Atoms to Mateials: Pedictive Theoy and Simulations Excited states of Hydogen & multi-electon atoms Ale Stachan School of Mateials Engineeing & Bick Nanotechnology Cente Pudue Univesity West Lafayette, Indiana USA Hydogen excited states n=3 Enegy (ev) n= n=1 p x p y p z Ale Stachan Atoms to Mateials 8 14

15 8/5/15 Hydogen geneal solution (, φ) = R ( ) Y ( φ ) ψ, n, l, m, nl l, m Thee quantum numbes n pincipal quantum numbe Enegy depends on n: E n 13.6eV = n Z l angula momentum quantum numbe Values limited by pincipal quantum numbe: 0, 1,, n-1 m magnetic quantum numbe (pojection of l along z axis) Values limited by l: m=-l, -l1,,l-1, l Ale Stachan Atoms to Mateials 9 Summay: hydogen solution n=3 m = 0 l = 0 m = 1, 0,1 l =1 m =, 1, 0,1, l = Enegy (ev) n= m = 0 l = 0 m = 1, 0,1 l =1 (, φ) = R ( ) Y ( φ ) ψ, n, l, m, nl l, m E n = 1 e 1 0 n ~ 13.6eV n m = 0 n=1 l = 0 Ale Stachan Atoms to Mateials 30 15

16 8/5/15 Wavefunctions Ma/n Kaplus and Richad N. Pote Atoms and molecules: an intoduc/on fo students of physical chemisty Ale Stachan Atoms to Mateials 31 Atoms with multiple electons Obitals maintain the shape of hydogen ones We ll use hydogen-like obitals fo othe atoms n=3 s p states d states n= n=1 Ale Stachan Atoms to Mateials 3 16

17 8/5/15 Lithium and shielding Electons in inne shells shield the nuclea potential to oute electons Enegy depends not only on n, but also on l The lage the angula momentum numbe l, the highe the enegy 4π R 10 4π R 1 4π R 0 Ale Stachan Atoms to Mateials 33 Enegy levels Shielding cause diffeence in enegies fo obitals with the same n and diffeent l (highe l highe enegy) Enegy p s d p s f d p s Hydogen s n = Ale Stachan Atoms to Mateials 34 17

18 8/5/15 H He Li Be B C N O F Ne Hund s ule and exchange inteaction 1s s p How do we fill obitals? Hund s ule: All obitals with a given l have to be filled with e- with the same spin fist webelements.com Ale Stachan Atoms to Mateials 35 18