Chapter 8 Approximation Methods, Hueckel Theory

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1 Witer 3 Chem 356: Itroductory Quatum Mechaics Chapter 8 Approximatio Methods, Huecel Theory... 8 Approximatio Methods... 8 The Liear Variatioal Priciple... Chapter 8 Approximatio Methods, Huecel Theory Approximatio Methods A) The variatioal priciple For ay ormalized wave fuctio ψ, the expectatio value of Hˆ, Hˆ E, the exact groudstate eergy. Proof: ψ = c φ with Ĥφ = E φ If we would measure the eergy we would fid E with probability Ĥ = ψ *(τ )Ĥψ (τ )dτ P E PE E P = E ( E E ) This argumet is a bit shay whe Ĥ has degeerate eigevalues. You will do a correct proof i the assigmets. P = c If ψ would ot be ormalized we ca calculate N = ψ *( τ) ψ( τ) dτ ad the Ĥ E N ψ ψ or i the fial form: The variatioal priciple!e = Domai Domai ψ *(τ )Ĥψ (τ )dτ ψ *(τ )ψ (τ )dτ E where E is the exact groud state eergy The variatioal eergy E! is exact whe ψτ () = φ () τ, the exact groud state wavefuctio. Chapter 8 Approximatio Methods, Huecel Theory 8

2 Witer 3 Chem 356: Itroductory Quatum Mechaics Examples: use a trial wavefuctio that depeds o oe or more parameters α, β The miimize the ( ) trial eergy E α,β Some simple (trivial) examples: Ĥ h.o. =!ω d dx + x Tae trial wavefuctio of the type Or Q: what is α? What is E? ψ () x = e αx Ne α x / / A: The exact wavefuctio has the form Ne α x /, E =!ω by miimizig the eergy we should get α =, E! = "ω Q: Tae the Hamiltoia for the Hydroge atom, ad a l = state Ĥ =! d µr dr r d dr e 4πε r Tae the trial wavefuctio e α r - What are the itegrals to evaluate? - What is the optimal value for α? - What is the value for!e? A:!E (α ) = 4π r e αr Ĥe αr dr 4π r e αr e αr dr Miimizig α :!E (α ) = α αopt =!E = E = e agai: exact a a 4πε a = 4πε! µe You ca do those problems yourself ad see if you get the correct aswer Notrivial example: Tae the Hamiltoia for H- atom, s- orbital, ad use the trial wavefuctio e αr Chapter 8 Approximatio Methods, Huecel Theory 9

3 Witer 3 Chem 356: Itroductory Quatum Mechaics E! (α ) = 4π r e αr Ĥe αr dr 4π r e αr e αr dr = 3! α e α m e ε (π ) 3/ =... E α = 3! e m e (π ) 3/ ε α = / α = m e e 3(π ) 3/ ε! m α opt = e e 4 8π 3 ε! 4!E (α opt ) = 4 m e e 4 3π 6π ε "!E (α opt ).44 e E 4πε a e = 4πε a!e > E ow, sice the trial wavefuctio caot be exact for ay α Note: Gaussia trial orbitals (basis sets) are widely used i electroic structure programs. This is because itegrals are easily evaluated over Gaussias. This is the origi of the ame for the Gaussia Program: It uses Gaussia basis fuctios! Chapter 8 Approximatio Methods, Huecel Theory

4 Witer 3 Chem 356: Itroductory Quatum Mechaics The Liear Variatioal Priciple Cosider a trial wave fuctio ψτ ( ) = c f( τ) Let us assume for simplicity - Real coefficiets, fuctios, f( τ ) - Orthoormal expasio fuctios: The we ca try to optimize the coefficiets E The =!E ( " c) = N = f *( τ ) f ( τ ) dτ = δ Domai m m ψ *(τ )Hψ (τ )dτ N ψ *(τ )ψ (τ )dτ D c f (τ )Hc m f m (τ )dτ = c c m f (τ )Ĥf (τ )dτ m,m ch m, m, c m m D c f ( τ) c f ( τ) dτ = = m, m, m m cc f ( τ) f ( τ) dτ m m = ccδ = c m, m N D D N N = = c D D Or N c N N D = D N D = E m = H c + c H m m D = c = c Chapter 8 Approximatio Methods, Huecel Theory

5 Witer 3 Chem 356: Itroductory Quatum Mechaics Hmcm Ec + c H Ec = m Sice H = H, this is twice the same equatio. H c Ec = m m This has the form of a matrix eigevalue equatio! H = m H m m H m c m = c E ad We ca also write [ H Eδ ]( c) = This has the form of a liear equatio. m H! c =! c E A! c =, with A = H E This type of equatio oly has a solutio if det ( A ) =. Hece det ( E ) = H equatio for E secular determiat Let us discuss examples later. For ow I wat to draw the aalogy: Schrödiger equatio Ĥψ = Eψ If we mae a basis expasio ψ = c f f f = δ m m The we get a matrix type Schrodiger equatio H c! = E c! With H H m = φ *(τ )Ĥφ m (τ )dτ Such a eigevalue equatio has M solutios for a M M matrix. They represet approximatios to the groud ad excited states. If the basis is ot orthoormal the defie S m = Or det H SE = eigevalues E. φ *(τ )Hφ m (τ )dτ ad H c! = S ce! (see MQ) Chapter 8 Approximatio Methods, Huecel Theory

6 Witer 3 Chem 356: Itroductory Quatum Mechaics Example Liear Variatios: Cosider particle i the box φ (x) = H =! m d dx π x si a Now add to H a liear potetial V a x Use as a trial wave fuctio πx πx c si + c si a a a a i, j =, H ij = a si iπ x! a m d dx + V V 6V E + 9π = 6V V E + 9π a x si jπ x a E = π! ma υ = ma! π V H =! π ma + υ 6υ 9π 6υ 4 + υ 9π The eigevalues of this Hamiltoia are! π ma ε Chapter 8 Approximatio Methods, Huecel Theory 3

7 Witer 3 Chem 356: Itroductory Quatum Mechaics ε = 5+υ ± 9 + 3υ 9π / 5+υ ± 3 + υ This happes to be pretty good solutio, especially if V is small 4 + υ Other istructive example: cosider particle o the rig!, with the degeerate mr ϕ m = solutios cos ϕ π si ϕ π Now apply a magetic field, which adds E = h mr e m e B e ˆLz = γ ˆL z to the Hamiltoia ˆLz = i! ϕ Uder the ifluece of the perturbatio the levels split. Calculate the eergy splittig. γ = e H = H m + γ L z e ψ = c cosϕ+ c siϕ π π γ z ˆL z π cosϕ = γ!i π siϕ γ ˆL z π siϕ = γ!i π cosϕ H E =! mr E iγ! +iγ!! mr E det( H E) =! mr E! mr E = ±γ! + γ! = Chapter 8 Approximatio Methods, Huecel Theory 4

8 Witer 3 Chem 356: Itroductory Quatum Mechaics E ± =! mr ± γ! Ca I fid eigefuctios?! mr iγ! +iγ!! mr! mr iγ! +iγ!! mr i i =! mr + γ! =! mr γ! i i Examples: What are the eigefuctios the? cos isi ~ e i ϕ ϕ + ϕ i cosϕ isi ϕ ~ e i i (we ormalize, π factor)! mr ϕ iγ! e iκ! ϕ mr + γ! ϕ eiϕ! mr ϕ iγ! e iκ! ϕ mr γ! e iϕ Bottom lie: We ca idicate perturbatio H = H + V Diagoalize H over degeerate states. H- atom: H = H (e) + g(v) ( L! S! ) Diagoalize Ĥ over p orbitals p, p 3 eigefuctios from diagoalizig 6 6 Hamiltoia. Everythig comes out by brute force. Example : add i additioal magetic iteractio H = H (e) + g(v)! L! S + diagoalize H over p ad s orbitals e B m z L z + e B e m z Ŝ z e Chapter 8 Approximatio Methods, Huecel Theory 5

9 Witer 3 Chem 356: Itroductory Quatum Mechaics all the splittig from diagoalizatio The liear variatioal priciple is a very powerful tool to calculate approximate eigefuctios It is widely used to calculate the splittig of eergies i a degeerate maifold, whe addig a perturbatio. Whe the eergies of a Hamiltoia H are ot degeerate, oe ca get a good estimate of the eergy correctio due to a perturbatio V, by calculatig V. Hece if H φ = E φ, the eigevalues of Hˆ = Hˆ + Vˆ are give to first approximatio by () φ Ĥ φ = φ *(τ ) H +V φ (τ )dτ = E () + φ *(τ ) Vφ (τ )dτ These are just the diagoal elemets of the Hamiltoia matrix. = First order Perturbatio Theory: If we go bac to box + liear field φ = φ H =! d m dx + V a x a Ĥ π x si a E () =! π ma Vˆ φ = V a a a e ˆv si π x a = V a a a 4 = V x dx E E () + V all eergies are shifted by V If zero- order states are degeerate, first- order perturbatio theory is useless. Istead use liear variatioal priciple Example L ˆz i cos(mx), si( mx ) basis choose other basis: results im e ϕ, e imϕ always diagoalize over zeroth- order states: degeerate first- order perturbatio theory Chapter 8 Approximatio Methods, Huecel Theory 6

10 Witer 3 Chem 356: Itroductory Quatum Mechaics Aother example of H! c =! ce : Hucel π - electro theory I orgaic chemistry, may molecules are essetially plaar. The plae cotais sp carbo, oxyge, itroge atoms. The out of plae p z - orbitals costitute the π - orbitals. The molecule s π - orbitals are liear combiatios of the atomic p z - orbitals. Oe ca parameterize a oe- electro effective Hamiltoia matrix as follows. Let us restrict ourselves to sp carbos. H α β = β α H H H D α β β α β = β α β β α α β β β β α = β α β α α β β β α β = β α β β β α Rule: α o diagoal β for ay two adjacet atoms coected by a π - bod β < ~ Followig the variatioal priciple we p z () V Ne p z () dτ a) Diagoalize the Hamiltoia orbital eergies E, eigevectors c! b) Fill up orbital levels from the bottom up puttig a α ad a β electro i each level. Occupy as may levels as you have π - electros c) If levels are degeerate, fill them up with α - electros first, the add additioal β - electros Chapter 8 Approximatio Methods, Huecel Theory 7

11 Witer 3 Chem 356: Itroductory Quatum Mechaics d) Total eergy: E = occupied orbitals ε λ e) Desity Matrix Dl = c λ cl λ E = Hl D (see McQuarrie) l occupied l This procedure would wor fie i MathCad How do we do it o paper? Tae det ( H E ) α E β ethylee = β α E ( ) α E β ( α E) =± = E = α ± β β π - electros Usig α ad β is a bit tedious for larger problems α E divide each colum by β ad defie x = β E = α βx x x x = = x =± E = α ± β Or x x = x x x 3 3 x x x x x x + + = 3 + = x = 3+ = x = = Chapter 8 Approximatio Methods, Huecel Theory 8

12 Witer 3 Chem 356: Itroductory Quatum Mechaics Always: x = i i x = is double solutio. ( x ) ( x+ ) = ( x x+ )( x+ ) 3 = x 3x+ o E = α βx 3π - electros (4- fold degeerate) Triplet (3- fold degeerate) Siglet What if we would loo at the siglet state of the aio? This is ot a stable structure, the molecule would distort Chapter 8 Approximatio Methods, Huecel Theory 9

13 Witer 3 Chem 356: Itroductory Quatum Mechaics Jah- Teller Distortio (the picture is ot clear, my apologies. Follow otes i my lecture) I might as questios of 4*4determiat too hard to solve I would give you the solutio x, x, x3, x 4 You would show that ( x x)( x x)( x x3)( x x4) is your secular determiat You ca guess the orbitals (phases) from symmetry argumets: The orbitals are always symmetric or atisymmetric with respect to plae or axis of symmetry If you ow value of x you ca solve x a x b = xc x = : a+ b+ c = ( - ) orthogoal combiatio - x = : ( ) = Chapter 8 Approximatio Methods, Huecel Theory