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## Lecture 25 Molecular orbital theory I

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**Lecture 25Molecular 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-of-atomic-orbitals (LCAO) MO’s. • It mathematically explains the bonding in H2+ in terms of the bonding andantibonding orbitals.**MO versus VB**• Unlike VB theory, MO theory first combine atomic orbitals and form molecular orbitalsin 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):**MO theory for H2**• Construct an antisymmetric wave function by filling electrons into MO’s**Singlet and triplet H2**(X)2 singlet far more stable (X)1(Y)1 triplet (X)1(Y)1 singlet least stable**Singlet and triplet He (review)**• In the increasing order of energy, the five states of He are (1s)2 singlet by far most stable (1s)1(2s)1 triplet (1s)1(2s)1 singlet least stable**MO versus VB in H2**VB MO**MO versus VB in H2**VB covalent covalent MO ionic H−H+ covalent = ionic H+H− covalent**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 rA rB A R B Parameter**LCAO MO**• MO’s are completely determined by symmetry: A B Normalization coefficient LCAO-MO**Normalization**• Normalize the MO’s: 2S**Bonding and anti-bonding MO’s**φ+ = N+(A+B) φ– = N–(A–B) bonding orbital – σ anti-bonding orbital – σ***Energy**• Neither φ+nor φ–is an eigenfunctionof the Hamiltonian. • Let us approximate the energy by its respective expectation value.**S, j, and k**rB rA A B R rA rB A B R R**Energy**R R**Energy**φ– = N–(A–B) anti-bonding R R φ+ = N+(A+B) bonding**Energy**φ– = N–(A–B) φ–is more anti-bonding than φ+is bonding anti-bonding R E1s φ+ = N+(A+B) bonding**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 βspins. • This explains the bonding in H2+, the simplest paradigm of chemical bond: bound and repulsive PES’s, respectively, of bonding and anti-bonding orbitals.