101410 Lecture 7

101410 Lecture 7 - Welcome to Chemistry 120A Lecture 7...

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Unformatted text preview: Welcome to:! Chemistry 120A! Lecture 7! Character Tables,! SALCʼs and Molecular Orbital Prof. Joshua Figueroa ! Diagrams! Character Character Tables! What Makes Up a Character Table? Character tables contain informa8on about how func8ons transform in response to the opera8ons of the group Five Parts of a Character Table 1 ­ At the upper le, is the symbol for the point group 2  ­ The top row shows the opera:ons of the point group, organized into classes 3  ­ The le, column gives the Mulliken symbols for each of the irreducible representa:ons 4  ­ The rows at the center of the table give the characters of the irreducible representa:ons (these can not be reduced further) 5  ­ Listed at right are certain func:ons, showing the irreducible representa:on for which the func:on can serve as a basis Character Tables! Character summary for a p ­block atom in C2v symmetry s z px py pz A1 B1 B2 A1 y x Character Tables! Character summary for a p ­block atom in C2v symmetry s px py pz A1 z B1 B2 A1 y x Symmetry Restric8ons on AO’s and MO’s Only orbital of the same character (symmetry can mix) Orbitals of the same symmetry belong to the same irreducible representa8on Any valid molecular orbital must transform according to one of the irreducible representa8ons of the molecular point group Character Tables! Character summary for a p ­block atom in C2v symmetry s px py pz A1 z B1 B2 A1 y x Symmetry Restric8ons on AO’s and MO’s Only orbital of the same character (symmetry can mix) Orbitals of the same symmetry belong to the same irreducible representa8on Any valid molecular orbital must transform according to one of the irreducible representa8ons of the molecular point group Character Tables! Character summary for a p ­block atom in C2v symmetry s px py pz A1 z B1 B2 A1 y x These AO’s could represent the oxygen atom in water O H C2v H Lets consider this further and construct an MO diagram for water Symmetry Adapted Linear Combinations of Orbitals! SALCʼs! We know how the oxygen atoms transform within C2v point symmetry, but what do we with the hydrogen atoms? O H H Symmetry Adapted Linear Combinations of Orbitals! SALCʼs! O We know how the oxygen atoms transform within C2v point symmetry, but what do we with the hydrogen atoms? H z z x y y O H H H x 1s orbitals of hydrogen We must determine the symmetry proper8es of a ‘group orbitals’ and construct ‘symmetry adapted linear combina8ons’ (SALC’s) of orbitals Symmetry Adapted Linear Combinations SALCʼs! Determina8on of the symmetry character for Group Orbitals: If the symmetry opera8on DOES NOT CHANGE the orbital posi8on = +1 If the symmetry opera8on CHANGES the orbital posi8on = 0 If the symmetry opera8on DOES NOT CHANGE the orbital posi8on, but INVERTS its sign =  ­1 Symmetry Adapted Linear Combinations SALCʼs! Determina8on of the symmetry character for Group Orbitals: If the symmetry opera8on DOES NOT CHANGE the orbital posi8on = +1 If the symmetry opera8on CHANGES the orbital posi8on = 0 If the symmetry opera8on DOES NOT CHANGE the orbital posi8on, but INVERTS its sign =  ­1 z y A B x Symmetry Adapted Linear Combinations SALCʼs! Determina8on of the symmetry character for Group Orbitals: If the symmetry opera8on DOES NOT CHANGE the orbital posi8on = +1 If the symmetry opera8on CHANGES the orbital posi8on = 0 If the symmetry opera8on DOES NOT CHANGE the orbital posi8on, but INVERTS its sign =  ­1 z y A B x C2v E Opera8on? Γ E C2 σv(xz) σv’(yz) Symmetry Adapted Linear Combinations SALCʼs! Determina8on of the symmetry character for Group Orbitals: If the symmetry opera8on DOES NOT CHANGE the orbital posi8on = +1 If the symmetry opera8on CHANGES the orbital posi8on = 0 If the symmetry opera8on DOES NOT CHANGE the orbital posi8on, but INVERTS its sign =  ­1 z y A B x C2v C2 Opera8on? E Γ 2 C2 σv(xz) σv’(yz) Symmetry Adapted Linear Combinations SALCʼs! Determina8on of the symmetry character for Group Orbitals: If the symmetry opera8on DOES NOT CHANGE the orbital posi8on = +1 If the symmetry opera8on CHANGES the orbital posi8on = 0 If the symmetry opera8on DOES NOT CHANGE the orbital posi8on, but INVERTS its sign =  ­1 z y A B x C2v σv(xz) Opera8on? E C2 Γ 2 0 σv(xz) σv’(yz) Symmetry Adapted Linear Combinations SALCʼs! Determina8on of the symmetry character for Group Orbitals: If the symmetry opera8on DOES NOT CHANGE the orbital posi8on = +1 If the symmetry opera8on CHANGES the orbital posi8on = 0 If the symmetry opera8on DOES NOT CHANGE the orbital posi8on, but INVERTS its sign =  ­1 z y A B x C2v σv’(yz) Opera8on? E C2 σv(xz) Γ 2 0 0 σv’(yz) Symmetry Adapted Linear Combinations SALCʼs! Determina8on of the symmetry character for Group Orbitals: If the symmetry opera8on DOES NOT CHANGE the orbital posi8on = +1 If the symmetry opera8on CHANGES the orbital posi8on = 0 If the symmetry opera8on DOES NOT CHANGE the orbital posi8on, but INVERTS its sign =  ­1 z y A B x Reducible Representa8on C2v E C2 σv(xz) σv’(yz) Γ 2 0 0 2 Symmetry Adapted Linear Combinations SALCʼs! Determina8on of the symmetry character for Group Orbitals: If the symmetry opera8on DOES NOT CHANGE the orbital posi8on = +1 If the symmetry opera8on CHANGES the orbital posi8on = 0 If the symmetry opera8on DOES NOT CHANGE the orbital posi8on, but INVERTS its sign =  ­1 z y A B x Reducible Representa8on Irreducible Representa8ons C2v E C2 σv(xz) σv’(yz) Γ 2 0 0 2 A1 1 1 1 1 z B2 1  ­1  ­1 1 y Symmetry Adapted Linear Combinations SALCʼs! We now know that the hydrogen 1s orbitals transform as A1 and B2 SALC’s, but what do these look like? C2v E C2 σv(xz) σv’(yz) Γ 2 0 0 2 A1 1 1 1 1 z B2 1  ­1  ­1 1 y Symmetry Adapted Linear Combinations SALCʼs! We now know that the hydrogen 1s orbitals transform as A1 and B2 SALC’s, but what do these look like? C2v E C2 σv(xz) σv’(yz) Γ 2 0 0 2 A1 1 1 1 1 z B2 1  ­1  ­1 1 y z A1 y (Ha + Hb) x Symmetry Adapted Linear Combinations SALCʼs! We now know that the hydrogen 1s orbitals transform as A1 and B2 SALC’s, but what do these look like? C2v E C2 σv(xz) σv’(yz) Γ 2 0 0 2 A1 1 1 1 1 z B2 1  ­1  ­1 1 y z z B2 A1 y (Ha  ­ Hb) y (Ha + Hb) x x Construction of a Molecular Orbital Diagram for H2O! 2b2 2b2 3a1 3a1 B2 A1 1s b1 b1 2px 2py 2pz B1 B2 A1 2a1 2a1 1b2 1b2 1a1 1a1 2s A1 Construction of a Molecular Orbital Diagram for H2O! 2b2 2b2 3a1 Lone Pair 3a1 B2 A1 Lone Pair 1s b1 b1 2px 2py 2pz B1 B2 A1 2a1 1b2 2a1 1b2 2s A1 Lone Pair Lone Pair 1a1 1a1 Another Example: Character Tables! Character summary for a p ­block atom in C3v symmetry N H C3v! H H Another Example: Character Tables! Character summary for a p ­block atom in C3v symmetry s z px py pz N y H x A1 E E A1 E Mulliken Symbol means ʻDoubly Degenerateʼ ! The orbitals transform together under the operation! H H Symmetry Adapted Linear Combinations of Orbitals! SALCʼs! N We know how the oxygen atoms transform within C3v point symmetry, but what do we with the hydrogen atoms? z N H x H H H H H z y y x 1s orbitals of hydrogen We must determine the symmetry proper8es of a ‘group orbitals’ and construct ‘symmetry adapted linear combina8ons’ (SALC’s) of orbitals Symmetry Adapted Linear Combinations SALCʼs! Determina8on of the symmetry character for Group Orbitals: If the symmetry opera8on DOES NOT CHANGE the orbital posi8on = +1 If the symmetry opera8on CHANGES the orbital posi8on = 0 If the symmetry opera8on DOES NOT CHANGE the orbital posi8on, but INVERTS its sign =  ­1 z y C A x B C3v Γ E 2C3 3σv Symmetry Adapted Linear Combinations SALCʼs! Determina8on of the symmetry character for Group Orbitals: If the symmetry opera8on DOES NOT CHANGE the orbital posi8on = +1 If the symmetry opera8on CHANGES the orbital posi8on = 0 If the symmetry opera8on DOES NOT CHANGE the orbital posi8on, but INVERTS its sign =  ­1 z y C A x B C3v E Γ 3 2C3 3σv Symmetry Adapted Linear Combinations SALCʼs! Determina8on of the symmetry character for Group Orbitals: If the symmetry opera8on DOES NOT CHANGE the orbital posi8on = +1 If the symmetry opera8on CHANGES the orbital posi8on = 0 If the symmetry opera8on DOES NOT CHANGE the orbital posi8on, but INVERTS its sign =  ­1 z y C A x B C3v E 2C3 Γ 3 0 3σv Symmetry Adapted Linear Combinations SALCʼs! Determina8on of the symmetry character for Group Orbitals: If the symmetry opera8on DOES NOT CHANGE the orbital posi8on = +1 If the symmetry opera8on CHANGES the orbital posi8on = 0 If the symmetry opera8on DOES NOT CHANGE the orbital posi8on, but INVERTS its sign =  ­1 z y C A x B C3v 2C3 Γ UNIQUE OPERATION ONLY E 3 0 3σv Symmetry Adapted Linear Combinations SALCʼs! Determina8on of the symmetry character for Group Orbitals: If the symmetry opera8on DOES NOT CHANGE the orbital posi8on = +1 If the symmetry opera8on CHANGES the orbital posi8on = 0 If the symmetry opera8on DOES NOT CHANGE the orbital posi8on, but INVERTS its sign =  ­1 z y C A x B C3v E 2C3 Γ 3 0 3σv 1 Symmetry Adapted Linear Combinations SALCʼs! Determina8on of the symmetry character for Group Orbitals: If the symmetry opera8on DOES NOT CHANGE the orbital posi8on = +1 If the symmetry opera8on CHANGES the orbital posi8on = 0 If the symmetry opera8on DOES NOT CHANGE the orbital posi8on, but INVERTS its sign =  ­1 z y C A x B C3v 2C3 Γ UNIQUE OPERATION ONLY E 3 0 3σv 1 Symmetry Adapted Linear Combinations SALCʼs! Determina8on of the symmetry character for Group Orbitals: If the symmetry opera8on DOES NOT CHANGE the orbital posi8on = +1 If the symmetry opera8on CHANGES the orbital posi8on = 0 If the symmetry opera8on DOES NOT CHANGE the orbital posi8on, but INVERTS its sign =  ­1 z y C A x B C3v 2C3 Γ UNIQUE OPERATION ONLY E 3 0 3σv 1 How do we! reduce this?! How do we handle E! Representations?! Reduction Formula for Irreducible Representations! # of irreducible Representa>ons = 1 order Of a given type Σ R Character of # opera>ons X Reducible In class representa>on Character of X Irreducible representa>on Reduction Formula for Irreducible Representations! # of irreducible Representa>ons = 1 order Of a given type z Σ R y C A B x C3v E 2C3 Γ 3 0 3σv 1 Character of # opera>ons X Reducible In class representa>on Character of X Irreducible representa>on Reduction Formula for Irreducible Representations! # of irreducible Representa>ons = 1 order Of a given type z Σ R Character of # opera>ons X Reducible In class representa>on Character of X Irreducible representa>on y C A B x C3v E 2C3 Γ 3 0 3σv 1 nA1 = 1/6[(3)(1) + (2)(0)(1) + (3)(1)(1)] = 1 Reduction Formula for Irreducible Representations! # of irreducible Representa>ons = 1 order Of a given type z Σ R Character of # opera>ons X Reducible In class representa>on Character of X Irreducible representa>on y C A B x C3v E 2C3 Γ 3 0 3σv 1 nA1 = 1/6[(3)(1) + (2)(0)(1) + (3)(1)(1)] = 1 nA2 = 1/6[(3)(1) + (2)(0)(1) + (3)(1)(-1)] = 0 nE = 1/6[(3)(2) + (2)(0)(-1) + (3)(1)(0)] = 1 Reduction Formula for Irreducible Representations! # of irreducible Representa>ons = 1 order Of a given type z Σ R Character of # opera>ons X Reducible In class representa>on Character of X Irreducible representa>on y C A B x nA1 = 1/6[(3)(1) + (2)(0)(1) + (3)(1)(1)] = 1 C3v E 2C3 Γ 3 0 1 A1 1 1 1 z E 2  ­1 0 x,y 3σv nA2 = 1/6[(3)(1) + (2)(0)(1) + (3)(1)(-1)] = 0 nE = 1/6[(3)(2) + (2)(0)(-1) + (3)(1)(0)] = 1 Symmetry Adapted Linear Combinations SALCʼs! We now know that the hydrogen 1s orbitals transform as A1 and E SALC’s, but what do these look like? C3v E 2C3 Γ 3 0 1 A1 1 1 1 z E 2  ­1 0 x,y 3σv z y C A B x A1 Ψ = ψ(Ha) + ψ(Hb) + ψ(Hc) Symmetry Adapted Linear Combinations SALCʼs! We now know that the hydrogen 1s orbitals transform as A1 and E SALC’s, but what do these look like? C3v E 2C3 Γ 3 0 1 A1 1 1 1 z E 2  ­1 0 x,y 3σv y C A B x A1 z z z y y A C x B E A C B x E Ψ = ψ(Ha) + ψ(Hb) + ψ(Hc) Ψ = (1/61/2)[ 2ψ(Ha)  ­ ψ(Hb)  ­ ψ(Hc) Ψ = ψ(Hb)  ­ ψ(Hc) Construction of a Molecular Orbital Diagram for NH3! 3a1 3a1 2e 2e E A1 1s 2a1 2a1 2px 2py 2pz E E A1 1e 1e 2s A1 1a1 1a1 Construction of a Molecular Orbital Diagram for NH3! 3a1 3a1 2e Lone Pair 2e E A1 Lone Pair 1s 2a1 2a1 2px 2py 2pz E E A1 1e 1e 2s A1 1a1 1a1 ...
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