Report

Lecture 25 Molecular orbital theory I (c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed and made available online by work supported jointly by University of Illinois, the National Science Foundation under Grant CHE-1118616 (CAREER), and the Camille & Henry Dreyfus Foundation, Inc. through the Camille Dreyfus Teacher-Scholar program. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the sponsoring agencies. Molecular orbital theory Molecular orbital (MO) theory provides a description of molecular wave functions and chemical bonds complementary to VB. It is more widely used computationally. It is based on linear-combination-ofatomic-orbitals (LCAO) MO’s. It mathematically explains the bonding in H2+ in terms of the bonding and antibonding orbitals. MO versus VB Unlike VB theory, MO theory first combine atomic orbitals and form molecular orbitals in which to fill electrons. MO theory VB theory MO theory for H2 First form molecular orbitals (MO’s) by taking linear combinations of atomic orbitals (LCAO): X A B and Y A B MO theory for H2 Construct an antisymmetric wave function by filling electrons into MO’s X(1)X(2)[a (1)b (2) - b (1)a (2)] Singlet and triplet H2 (X)2 singlet éë X (1) X (2) ùû éëa (1)b (2) - b (1)a (2) ùû far more stable ì a (1)a (2) ï (X)1(Y)1 triplet éë X (1)Y (2) - Y (1) X (2) ùû í b (1)b (2) ï a (1)b (2) + b (1)a (2) î (X)1(Y)1 singlet éë X (1)Y(2) + Y(1) X (2) ùû a (1)b (2) - b (1)a (2) { } least stable Singlet and triplet He (review) In the increasing order of energy, the five states of He are { } Y ( r1 ,r2 ) » j1s ( r1 )j1s ( r2 ) {a (1)b (2) - b (1)a (2)} (1s)2 singlet by far most stable { } (1s) (2s) triplet Y ( r ,r ) » {j ( r )j ( r ) - j ( r )j ( r )} b (1)b (2) Y ( r ,r ) » {j ( r )j ( r ) - j ( r )j ( r )} {a (1)b (2) + b (1)a (2)} Y ( r1 ,r2 ) » j1s ( r1 )j 2s ( r2 ) - j1s ( r2 )j 2s ( r1 ) a (1)a (2) 1 2 1s 1 2s 2 1s 2 2s 1 1 2 1s 1 2s 2 1s 2 2s 1 { } 1 1 Y ( r1 ,r2 ) » j1s ( r1 )j 2s ( r2 ) + j1s ( r2 )j 2s ( r1 ) {a (1)b (2) - b (1)a (2)} (1s)1(2s)1 singlet least stable MO versus VB in H2 VB MO [ A(1)B(2) + B(1)A(2)][a (1)b (2) - b (1)a (2)] éë X (1) X (2) ùû éëa (1)b (2) - b (1)a (2) ùû X(1)X(2) = { A(1) + B(1)} { A(2) + B(2)} = A(1)B(2) + B(1)A(2) + A(1)A(2) + B(1)B(2) MO versus VB in H2 VB [ A(1)B(2) + B(1)A(2)][ spin ] covalent covalent MO [ X(1)X(2)][spin] = [ A(1)B(2) + B(1)A(2) + A(1)A(2) + B(1)B(2)][spin] ionic H −H + covalent = covalent ionic H +H − MO theory for H2+ The simplest, one-electron molecule. LCAO MO is by itself an approximate wave function (because there is only one electron). Energy expectation value as an approximate energy as a function of R. e 2 æ 1 1 1ö e 2 Hˆ = Ñ + - ÷ ç 2me 4pe 0 è rA rB R ø 2 Parameter rA A rB R B LCAO MO MO’s are completely determined by symmetry: y ± = N ± ( A ± B) Normalization coefficient LCAO-MO A B Normalization Normalize the MO’s: N± = = ( ) d t ± ( ò A B dt + ò B Ad t )} * ( A ± B) ( A ± B) d t ò { ò A dt + ò B 2 = ( 2 ± 2S ) - 12 2 - 12 * * 2S - 12 Bonding and anti-bonding MO’s φ+ = N+(A+B) φ– = N–(A–B) bonding orbital – σ anti-bonding orbital – σ* Energy 2 æ 1 1 1ö e 2 Hˆ = Ñ + - ÷ ç 2me 4pe 0 è rA rB R ø 2 Neither φ+ nor φ– is an eigenfunction of the Hamiltonian. Let us approximate the energy by its respective expectation value. Energy 2 2 æ e 1ö 2 ç - 2m Ñ - 4pe r ÷ A = E1s A è e 0 Aø 2 æ e2 1 ö 2 ç - 2m Ñ - 4pe r ÷ B = E1s B è e 0 Bø e2 2 1 e 1 * * j=òA Adt = ò B Bdt 4pe 0 rB 4pe 0 rA 2 1 e 1 * * k=òB Adt = ò A Bdt 4pe 0 rB 4pe 0 rA ˆ ( A ± B )dt A ± B H ( ) ò = e2 * E± 2 ± 2S 2 2 æ 1 1 1 ö üï *ì e ï 2 ò ( A ± B ) íï- 2me Ñ - 4pe 0 çè rA + rB - R ÷ø ýï( A ± B)dt î þ = 2 ± 2S e2 j±k = ... = E1s + 4pe 0 R 1± S S, j, and k A= e- rA /a0 p a03 rB rA A , B= e- rB /a0 p a03 B R rA rB 1.0 S = ò A* Bdt 0.8 A R j = ò A* e2 B 0.6 e2 1 k = ò B* Adt 0.4 4pe 0 rB 1 Adt 4pe 0 rB 0.2 0 2 4 6 8 10 R Energy j±k E± = E1s + 4pe 0 R 1± S e2 e2 1.0 4pe 0 R 3.0 2.5 S = ò A* Bdt 0.8 0.6 j+k 2.0 1.5 0.4 1.0 0.2 0.5 0 2 e2 4 1 * k=òB Adt 4pe 0 rB 6 8 10 e2 R 0 1 * j=òA Adt 4pe 0 rB 2 4 6 j-k 8 10 R Energy j±k E± = E1s + 4pe 0 R 1± S e2 e2 4pe 0 R 3.0 2.5 e2 4pe 0 R j+k 2.0 1.5 - ( j - k) φ– = N–(A–B) anti-bonding 1.0 0.5 0 2 4 6 j-k 8 10 R R e2 ( ) φ+ = N+(A+B) - j+k bonding 4pe 0 R Energy j±k E± = E1s + 4pe 0 R 1± S e2 e2 4pe 0 R - ( j - k) j-k E1s + 4pe 0 R 1- S e2 φ– = N–(A–B) anti-bonding φ– is more anti-bonding than φ+ is bonding R e2 ( ) φ = N+(A+B) - j + k +bonding 4pe 0 R E1s E1s + e2 4pe 0 R - j+k 1+ S Summary MO theory is another orbital approximation but it uses LCAO MO’s rather than AO’s. MO theory explains bonding in terms of bonding and anti-bonding MO’s. Each MO can be filled by two singlet-coupled electrons – α and β spins. This explains the bonding in H2+, the simplest paradigm of chemical bond: bound and repulsive PES’s, respectively, of bonding and anti-bonding orbitals.