III-16 G. Brief Review of Grand Orthogonality Theorem and impact on Representations (Γ i ) l i = h n = number of irreducible representations.

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1 III- G. Bref evew of Grad Orthogoalty Theorem ad mpact o epresetatos ( ) GOT: h [ () m ] [ () m ] δδ δmm ll GOT puts great restrcto o form of rreducble represetato also o umber: l h umber of rreducble represetatos aser wth haracters: χ ( ) χ ( ) h for for h δ χ trace of Sce all elemets of a class, have same character, χ, -- related by smlarty trasform N cχ ( ) χ ( ) h δ N c umber class Now: haracters of each rreducble represetato form a vector orthogoal to other rreducble represetatos. These are h-dmesoal vectors but have oly depedet elemets thus these vectors ca defe a -dmesoal space. That the s the umber of rreducble represetatos. or umber of classes umber of rreducble represetatos haracter Table geerato Number of rreducble represetatos umber of classes ν h,,, σ, σ ' -- all class by self aalogy: get set of orthogoal (learly depedet) -vectors over -D vector - space υ h, - class by self classes represetatos, same class + + σ, σ, σ same class x -D + -D -D σ υ σ υ σ υ υ h 8, 5 self, class, ( -) class, σ, σ same class σ υ σ, σ same class σ d (must have oe -D rep) υ 5 σ υ σ d

2 III-7 ffect of symmetry operato,, o wave/fct s ± for -D Makes a ege value relatoshp: φ ±φ Sce symmetry leaves molecule as t was, eergy does ot chage ~ A amltoa operator has total symmetry rep of group ( total symmetry, eergy ot affected by ) so commute,.e.: [,] + the have smultaeous ege fucto kow these propertes wthout solvg the Schroedger equato Solvg QM Itegrals φ φ d Ĥ or α.e. from: f(x)dx ff f(x) symmetrc to be o-zero f(x) φ φ (full argumet of the tegral) must be totally symmetrcal or group theory the product must cota the A totally symmetrc rep ergy operators, sce s A, the for φ φ d τ Ĥ symmetry of φ must be same as φ,.e. Dpole operators: <φ µ φ > the µ A (+ others) If A (typcal for a groud state), the µ or excted state same rep as µ. Now the problem determe represetato of fuctos, ad ther products educble epresetatos smlarty trasform to sum of rreducble represetatos these have the same character, so sum of rred. χ red. χ χ r () a χ () (reducble) (sum. rreducble) Use orthogoalty to setup χ r () χ () a χ () χ () or a h a [ χ () χ ()] χ r () χ () δ h from the GOT by classes a h N k χ r ( k ) χ ( k ) k Meas of reducg a reducble represetato to sum of rreducble represetato educg reducble represetato sum of rreducble represetato χ r () a χ () a h χ r () χ () or a h N k χ r ( k ) χ ( k ) k

3 III-8 xamples osder terchage matrx of s N N σ υ χ σ σ υ c s reducble, what s t composed of? a k N k χ ( k ) χ ( k ) a [ + + ] a [ ] a [ + + ] thus + A + dmesos: D + D D Note from () beg detty matrx the dmesoalty always works esult: A + +, ad, Note always works ths way for obects related by x : O x y z form bases as do rotato matrces xz yz,,, χ xyz, -,, reducble represetato, -D, h rred reps σ υ σ υ a [ + (-) + + ] a [ + (-)(-) + + (-) ] a [ + (-)(-) + (-) + () ] }χ r () a χ () a [ + (-) + (-) + (-) ] so xyz + + A + B + B o A

4 III-9 I geeral ths partcular decomposto wll prove to be very useful for aalyses of vbratoal ormal modes ad for orbtals,.e. LAO MO s χ xyz ofte character tables recall µ ~ (x, y, z) mportat for determg dpole trastos Geeral behavor multplyg represetatos D x D D rreducble fd by specto D x D (D) ()D usually rreducble obvous by specto D x D (or hgher) dmeso s product (x, x9,...) eed to reduce haracters oly determed by dagoal whch oly depeds o dagoal rreducble Bottom le ca ust use character multply from Z - AB Z ' AB same characters a χ () χ AB () χ A () χ B () (trace s varat to smlarty tras) v example χ χ, (-)(-),,, A + A + χ χ A, (-)(), (-), -, Use to evaluate tegrals: φ φ dτ A + A + A φ α φ A α A A oly f α ~ a show a A whe A B A AB ff A B a A whe A B ths s from products of, -, square all s So for f A f B f dτ eed a) f s all A symmetry b) product fucto of same symmetry eg: A B ƒ A ƒ B or: A B for f A f B f dτ, etc.

5 III- Subgroup v example: :,, ad σ s :, σ (there are,,,) combe groups σ tmes: σ σ σ σ σ Drect Product Group v dmeso s product (x) of ad σ s (Drect Product groups -- PJS otes) Ths same multplcato ca be see to apply to groups (as before) osder h σ h Subgroups : (, ) ad (, ) h A Ag B Au Now take the product (, ), (, ), σ so sum s h, same for characters A B A B g g u u σ aalogous to matrx drect product doe for reducble. Note symmetry haracter table. Fucto Space ad form of symmetry QM We re fshg up techque developmet before gog o to applcatos Above talked about products of represetatos to lear how to evaluate f f dτ. a) f A B AB AB A (totally symmetrcal) ff A B a b) Nature of AB determed from reducg t : AB or χ AB χ to lear combato rreducble represetato by usg characters Now f kow what symmetres are some combed represetato, for example a product of wave fuctos, wat to kow form of result that case of terest. a Proecto operators used to gve lear combato of ( our case, fuctos) that gve a represetato Symmetry Adapted Lear ombato Thk about fuctos QM obects: {f} forms fucto space f for all fuctos g that space: g c f -dmesoal: {ƒ} l, ƒ depedet fucto space a scalar product s tegral: f g ƒg dτ

6 III-

7 III-. Proecto Operators Now that we kow whch rreducble represetatos reducble represetato how fd what the actual fuctos for each represetato are?.e. f symmetry adapted wave fuctos ca make smpler tegrals ( or ) what are they? O eg: O stretch O- f use ( + ) + ( ), each has sym. these lear combatos are represetatos: A B We obtaed ths by specto what about more complex? z N px, py -- ether s a represetato mxes p x, p y must belog to same rep. but how? p x c x p x + c y p y N ll x What are c x + c y? ad how make p c x p x + c y p y, p c x p x + c y p y orthogoal? Symmetry Adapted Lear ombatos I fucto space collect all fuctos tercoected by symmetry operatos If h Σ c f the h depedet of { f }, f spa space xamples from math: f(x) Σ a s x + b cos x fourer space dmeso a b a + b vector space (actually descrbes sp ½ system, α,β) fucto space: {f } f lear depedet bass fucto g Σ c f ad h Σ d f a(g + h) Σ a(c + d ) f f orthoormal: the from: c f g f c f f f c (proect out f part) Soluto to Schrödger qu. set of orthogoal bass fucto - spa a fucto space Subspaces fuctos that tercoect uder a operato or have commo character e.g. Fourer δ Σ b cos x all eve fucto Schrödger q ψ Σ c φ all fucto same eergy ψ ψ m oversely ay lear combato of ege fucto that have same ege value are also ege fucto of Subsets equvalet eergy: φ Σ c φ o chage from [, ] e.g. atoms p + s, p Na

8 III-7 Method symmetry operato yeld lear combato -D φ ±φ (depedet o rreducble represetato) mult φ c φ geerally set of {φ } wll be rreducble represetato (uless accdetally degeerate) geeral φ k t k k st () φs st () m x m matrx s set of matrces k st () (all s) form represetato of group reformulate: φ t φs ()st s multply by () sum over s t ()s t φ t φ s () () s t st S δ δss δtt () φ s t t h δ S l δ ss δtt φs Proecto operator: l Pˆ () ˆ s t h s t Pˆ s φ t t δ δ tt' φ s