18-1 Variatioal Method (See CTDL 1148-1155, [Variatioal Method] 252-263, 295-307[Desity Matrices]) Last time: Quasi-Degeeracy Diagoalize a part of ifiite H * sub-matrix : H (0) + H (1) * correctios for effects of out-of-block elemets: H (2) (the Va Vleck trasformatio) *diagoalize H eff =H (0) + H (1) + H (2) coupled H-O s 2 : 1 (ω 1 2ω 2 ) Fermi resoace example: polyads 1. Perturbatio Theory vs. Variatioal Method 2. Variatioal Theorem 3. Stupid oliear variatio 4. Liear Variatio ew kid of secular Equatio 5. Liear combied with oliear variatio 6. Strategies for criteria of goodess various kids of variatioal calculatios 1. Perturbatio Theory vs. Variatioal Method Perturbatio Theory i effect uses basis set goals: parametrically parsimoious fit model, H eff fit parameters (molecular costats) parameters that defie V(x) order - sortig (1) H k (0) E (0) <1 Ek errors less tha this mixig agle times the previous order o zero correctio term ( is i-block, k is out-of block) because diagoalizatio is order (withi block). Variatioal Method best possible estimate for lowest few E, ψ (ad properties derivable from these) usig fiite basis set ad exact form of H.
18-2 Vast majority of computer time i Chemistry is spet i variatioal calculatios Goal is umbers. Isight is secodary. Ab Iitio vs. semi-empirical or fittig [itetioally bad basis set: Hückel, tight bidig qualitative behavior obtaied by a fit to a few microscopic like cotrol parameters] 2. Variatioal Theorem ot ecessarily ormalized If φ is approximatio to eigefuctio of  belogig to lowest eigevalue a 0, the α φ A φ φφ a 0 ay observable PROOF: eigebasis (which we do ot kow but kow it must exist) A = a φ = φ A φ = φφ = φ φ A φ = φ 2, φ φ = φ 2 α φ A φ φφ = completeess a δ eigebasis a φ 2 φ 2 subtract a 0 from both sides α a 0 = a a 0 ( ) φ 2 φ 2 the variatioal Theorem expad φ i eigebasis of A, exploitig completeess 0 a all terms i both sums are 0 agai, all terms i both sums are 0
18-3 because, by defiitio of a 0, a a 0 for all ad all terms i sum are 0. α a 0. QED but useless because we ca' t kow a or φ It is possible to perform a variatioal calculatio for ay A, ot limited to H. 3. Stupid Noliear Variatio Use the wrog fuctioal form or the wrog variatioal criterio to get poor results illustrates that the variatioal fuctio must have sufficiet flexibility ad the variatioal criterio must be as it is specified i the variatioal theorem, as opposed to a clever shortcut. The H atom Schr. Eq. (l = 0) H = 1 2 1 r 2 r r2 r T 1 r V ad we kow 12 / r ψ1s () r = r 1s = π e E1s = 1 / 2 au 1 au = 219475 cm 1 [ ] but try r 3 12 / [ ] ( ) φ = ξ 2π ξr e ξr ormalized for all ξ ξ is a scale factor that cotrols overall size of φ(r) [actually this is the form of ψ 2p (r)] which is ecessarily orthogoal to ψ 1s! STUPID! 12 / ( φ( 0) = 0 but ψ1 s( 0) = π ) ε = φ H φ φφ = 4 ξ 2 3ξ 3 8 skipped a lot of algebra miimize ε: dε dξ = 0 ξ = 3/ 2 ε = 3/ 8 mi mi au FAILURE! 1 c.. f the true values: E1s = 1/ 2 au, E2s = au 8
Try somethg clever (but lazy): What is the value of ξ that maximizes φ 1s? 18-4 ( ) = for the best variatioal ξ = 3 / 2, C1s φ ξ = 3 / 2 1s 0. 9775 if we maximize C1s wrt. ξ : ξ = 5 / 3 C1s = 0. 9826 better? but ε = 0.370 results, a poorer boud tha ξ = 3/2 ε = 0.375 * eed flexibility i φ * ca t improve o d ε by employig a alterative variatioal strategy dξ This was stupid ayway because we would ever use the variatioal method whe we already kow the aswer! 4. Liear Variatio Secular Equatio φ = N =1 c χ KEY TOPIC for this lecture χ H χ = H χ χ = S overlap itegrals (o-orthogoal basis sets are ofte coveiet) φh φ ε = = φφ, mm, cc H c c S m m mm rearrage this equatio ε c m c m = c c m, m m S m, ε = 0 for each j c j H to fid miimum value of ε, take c liear variatio! for each j ad require that because we are seekig to miimize ε with respect to each c j. Fid the global miimum of the ε(c 1,c 2, c N ) hypersurface. j the oly terms that survive c j are those that iclude c j.
εc m ( S mj + S jm )= c H j + H j m if χ ( ) { } are real S ij = S ji, H ij = H ji ( ) N 0= c H j εs j =1 oe such equatio for each j (same set of ukow {c }) 18-5 These are all of the survivig terms (i.e. those that iclude j). Each j term appears twice i both sums, oce as a bra ad oce as a ket. N liear homogeeous equatios i N ukow c s No trivial {c } oly if H εs = 0 (Not same form as H 1E = 0) The result is N special values of ε that satisfy this equatio. CTDL show: all N ε values are upper bouds to the lowest N E s ad all {φ } s are othogoal! (provided that they belog to differet How to solve H εs = 0 values of E ) 1. Diagoalize S USU = S S ij = siδij (orthogoalize {χ} basis) 2. Normalize S ( S ) 1/2 S ( S ) 1/2 =1 S = T ST 3 diagoal matrices where T = US 1/2 s ƒ 12 / ƒ 12 / ( S ) = S = 0 0 / 0 s2 12 0 0 0 O / 1 12 uitary This is ot a orthogoal trasformatio, but it does ot destroy orthogoality because each fuctio is oly beig multiplied by a costat.
18-6 3. Trasform H to orthoormalized basis set H Sƒ U HU Sƒ 12 / 12 / = ( ) T T U diagoalizes S ot H ew secular equatio H ε S =0 but S =1 H ε1 =0 usual H diagoalized by usual procedure! 5. Combie Liear ad Noliear Variatio typically doe i ab iitio electroic structure calculatios Basis set: χ ( ξ r) ψ c χ ξ r S = ( ) liear variatio where ε is a radial scale factor ( ξ, ξ ), H ( ξ, ξ ) oliear variatio 0. pick arbitrary set of { ξ i } 1. calculate all H ij ξ i,ξ j 2. Solve H - εs =0 ( ) & S ij ( ξ i,ξ j ) a. S S diagoalize S (orthogoalize) b. 12 / S ( ) (ormalize) c. d. H H diagoalize H oliear variatio begis fid global miimum of ε lowest with respect to each ξ i
3. chage ξ from ξ ξ = ξ + δ 4. 5. Solve H- εs 0 to obtai a ew set of ε. Pick lowest ε. 6. calculate 1 () 1 0 () 1 1 () 1 0 recalculate all itegrals i Had Sivolvig χ ε = { } i old ew lowest εlowest εlowest = ξ () 1 1 0 () ξ ξ1 1 i 1 18-7 7. repeat #3 6 for each ξ i (always lookig oly at lowest ε i ) This defies a gradiet o a multidimesioal ε(ξ 1, ξ N ) surface. We seek the miimum of this hypersurface. Take a step i directio of steepest descet by a amout determied by ε/ ξ steepest (small slope, small step; large slope, large step). This completes 1st iteratio. All values of {ξ i }are improved. 8. Retur to #3, iterate #3-7 util covergece is obtaied. Noliear variatios are much slower tha liear variatios. Typically use ENORMOUS LINEAR {χ} basis set. Cotract this basis set by optimizig oliear parameters (expoetial scale factors) i a SMALL BASIS SET to match the lowest {φ} s that had iitially bee expressed i large basis set.
18-8 6. Alterative Strategies * rigorous variatioal miimizatio of E lowest : ab iitio * costrai variatioal fuctio to be orthogoal to specific subset of fuctios e.g. orthogoal to groud state to get variatioal covergece. Applies oly to higher members of specific symmetry class or orthogoal to core: froze-core approximatio. Pseudopotetials (use some observed eergy levels to determie Z eff (r) of froze core) * least squares fittig miimize differeces betwee a set of measured eergy levels (or other properties) ad a set of computed variatioal eige-eergies (or other properties computed from variatioal wavefuctios). { } molecular costats { observed E } parameters i H eff experimetal ψ s i fiite variatioal basis set * semi-empirical model replace exact Ĥ by a grossly simplified form ad restrict basis set to a simple form too. The adjust parameters i H to match some observed patter of eergy splittigs. Use parameters to predict uobserved properties or use values of fit parameters to build isight.