Chapter 4 The structure of diatomic molecules. What is a chemical bond?" It's only a convenient fiction, but let'spretend...'There is nosuch thing!"SOMETIMES IT SEEMS to me that abond between two atoms has become soIt is more useful to regard areal, so tangible, so friendly, that I canchemical bond as an effectalmost see it. Then I awake with a littlethat causes certain atoms toshock, for a chemical bond is not a realjoin together to formthing. It does not exist. No one has everenduring structures thatseen one. No one ever can. It is a figmenthave unique physical andof our own imagination."chemical properties-C.A. Couls0n (1910-1974)Chemical bonding occurs when one or more electronsare simultaneously attracted to two nuclei
Chapter 4 The structure of diatomic molecules • What is a chemical bond? It is more useful to regard a chemical bond as an effect that causes certain atoms to join together to form enduring structures that have unique physical and chemical properties. Chemical bonding occurs when one or more electrons are simultaneously attracted to two nuclei. "SOMETIMES IT SEEMS to me that a bond between two atoms has become so real, so tangible, so friendly, that I can almost see it. Then I awake with a little shock, for a chemical bond is not a real thing. It does not exist. No one has ever seen one. No one ever can. It is a figment of our own imagination.” -C.A. Coulson (1910-1974) “ It's only a convenient fiction, but let's pretend
Physicists'viewChemists' viewChemical Bondingan effectthat results fromelectrostaticattractionthat reduces thebetweenpotential energy of two orelectronsnucleimore atomic nucleicausing them to formis an aggregate ofthat can combinea molecule"chemical bonds"to formthat is suffcientlydescribed bylong-livedtopossesscharacterized bypotential energy curvesDistinguishingthat revealobservablepropertiesVVibrational12bond dissociation energybond lengthfrequencies
Chemical Bonding is an effect that reduces the potential energy of two or more atomic nuclei causing them to form “chemical bonds” described by potential energy curves that reveal bond dissociation energy bond length Vibrational frequencies that results from electrostatic attraction between that can combine to form electrons nuclei is an aggregate of a molecule that is suffciently long-lived to possess Distinguishing observable properties characterized by Chemists’ view Physicists’ view
Quantum mechanical theory for description ofmolecular structures and chemical bondingsValence Bond (VB) Theorya) Proposed by Heitler and London in193Os, further developmentsby Pauling and Slater etalb) Finally programmed in later 1980s, e.g., XMVB3.0Molecular Orbital (MO) Theorya) Proposed by Hund, Mulliken, Lennard-Jones et al. in 1930sb) Further developments by Slater, Huckel and Pople et alc) MO-based softwares are widely used nowadays, e.g., Gaussian·DensityFunctional Theorya) Proposed by Kohn et al.b) DFT-implemented QM softwares are widely used, e.g., Gaussic
Quantum mechanical theory for description of molecular structures and chemical bondings • Valence Bond (VB) Theory a) Proposed by Heitler and London in1930s, further developments by Pauling and Slater et al. b) Finally programmed in later 1980s, e.g., XMVB3.0 • Molecular Orbital (MO) Theory a) Proposed by Hund, Mulliken, Lennard-Jones et al. in 1930s. b) Further developments by Slater, Hückel and Pople et al. c) MO-based softwares are widely used nowadays, e.g., Gaussian • Density Functional Theory a) Proposed by Kohn et al. b) DFT-implemented QM softwares are widely used, e.g., Gaussian
SlaterPaulingKohn卢嘉锡
Slater Pauling Kohn 卢嘉锡
S 1 Electronic structure ofH,+ion1. Schrodinger equation ofH,Born-OppenheimerApproximationTheelectrons are much lighterthanthe0BAnuclei.RNuclear motion is much slower than the(Rexp. = 106 pm)electronmotion Neglecting the motion of nuclei!The hamiltonian operatorSchrodinger equation ofH,11H:Hy=EyR2arb+ R?-2r.Rcos0Vrb=12
§1 Electronic structure of H2 + ion 1. Schrödinger equation of H2 + A B e - r r b a R Born-Oppenheimer Approximation • The electrons are much lighter than the nuclei. • Nuclear motion is much slower than the electron motion. Neglecting the motion of nuclei! rb ra R 2ra Rcos 2 2 r r R H a b e 1 1 1 2 1 ˆ 2 The hamiltonian operator H ˆ E Schrödinger equation of H2 + (Rexpt. = 106 pm)
HMolecularOrbital TheoryThe schrodinger equation for H,+ can be solved exactly usingconfocal elliptical coordinates(xi) =(r,+rp)/Rn (eta) = (ra-rp)/Rx@is arotationaround z> R≤(r,+rp) < 80Ap7.-R≤(r-rb) ≤R0≤Φ≤2元;1 ≤≤8;Rra=(=+n)R/2 r, =(=-n)R/2-1≤n≤1RH(r,R)μ(r,R)= E(R)μ(r,RH(r)y(r)= Ey(r)fixed12position of theelectron!Yet very TEDIOUS!
R fixed Molecular Orbital Theory H2 Hr R r R E R r R e , , , ˆ 1 1 1 The schrödinger equation for H2 + can be solved exactly using confocal elliptical coordinates: (xi) = (ra+rb )/R (eta) = (ra -rb )/R is a rotation around z ra rb z R z x Yet very TEDIOUS! r a ( )R / 2 r b ( )R / 2 position of the electron! R (ra+rb ) < -R (ra -rb ) R 0 2; 1 ; -1 1 1 1 1 H r r E r ˆ
MolecularOrbital TheoryMolecularorbital (MO) of H,Yelec = F(5, n)[(2元)-1/2 imd(m=0, ±1, ±2, ±3,...)eRadial partAngularpart2 =|ml--orbital angular momentum quantum numberEach electronic level with 0 is doubly degenerate, with m = amh or m (in a.u.) -- the z-component of orbital angular momentum.The one-electron wavefunction (MO) is no longer the eigenfunctionof the operator L?, but is the eigenfunction of L,[L, H] + 0;[L,H]=0Types of molecular orbitals are defined by the value of a (=|m)1203入4Type of MO(bond)8中letter12元Y
(m=0, ±1, ±2, ±3,.) • =|m|-orbital angular momentum quantum number. ( , )[(2 ) ] 1/2 i m elec F e Radial part Angular part [ ˆ , ˆ ] 0; [ ˆ , ˆ ] 0 2 L H Lz H Molecular orbital (MO) of H2 + • Each electronic level with 0 is doubly degenerate, with m = ||. • The one-electron wavefunction (MO) is no longer the eigenfunction of the operator L2 , but is the eigenfunction of Lz . • mħ or m (in a.u.) - the z-component of orbital angular momentum. • Types of molecular orbitals are defined by the value of (=|m|). 0 1 2 3 4 letter Type of MO (bond) Molecular Orbital Theory
For diatomics.For atoms,Pele = F(5, n)(2元)-1/2 eimYele = Rn,(r)O1,m, (0)Dmd2 =ml (m=0, ±1, ±2, ±3,...Quantum numbers: n, l, m,3321201a=|m044SdfletterdletterS元pgaYQuantumNumberof OrbitalangularmomentumAtom: = 0, I, 2... and the atomic orbitals are called: s, p, d, etc& each sublevel contains degenerate AOs with m, = l, ..., -l.Diatomics: = 0,1,2, ... and the molecular orbitals are: , 元, , etc& each level contains degenerate MOs with m = ±aQuestion: Supposing MO's are composed of AO's, what is the12relationship between a (MO) and I (AO), or m (MO) and m,(AO)?
=|m| (m=0, ±1, ±2, ±3,.) =|m| 0 1 2 3 4 letter Quantum Number of Orbital angular momentum • Atom: l = 0, 1, 2,. and the atomic orbitals are called: s, p, d, etc. & each sublevel contains degenerate AOs with ml = l, ., -l. For diatomics, For atoms, ( ) ( ) ( ) , , ml ml elec n l l R r Quantum numbers: n, l, ml l 0 1 2 3 4 letter s p d f g Question: Supposing MOs are composed of AOs, what is the relationship between (MO) and l (AO), or m (MO) and ml (AO)? im elec F e 1/2 ( , )(2 ) • Diatomics: = 0,1,2, . and the molecular orbitals are: , , , etc. & each level contains degenerate MOs with m =
XSymmetry of MO1,)eimpfHAdelec2元Z5 = (ra +r)/R1n = (r.-r)/RA'Ri)Inversion:,=,Φ=Φ+A'(, -n, Φ+元)A(E, n, Φ)F(,-n) =BF(,n), B= +1 or -l;im =i[AF(5,n)eimg J = AF(E,-n)eim(s+z)|=Beimam =B'甲,mmYm is an eigenfunction of inversion with B'= +1 or-1!B'= 1, parity(even), (denoted g);B' =-1, disparity (odd), (denoted u);12Notation valid onlyforhomonuclear diatomics!
i m elec F( , )e 2 1 i) Inversion: [ ( , ) ] ˆ ˆ i m m i i AF e ra rb x R z ra rb Symmetry of MO m is an eigenfunction of inversion with B = +1 or -1 ! Notation valid only for homonuclear diatomics! F(,-) = BF(,), B= +1 or -1; ( ) ( , ) i m AF e m m im e B' B = (ra + rb )/R = (ra rb )/R A(, , ) A(, -, +) i A A ( , , ) ' ' ra rb rb ra • B = 1, parity (even), (denoted g); • B =-1, disparity (odd), (denoted u);
TaSymmetryofMOwavefunctionX7F(5,n)eimo4elec2元Ad = (ra+r)/RZn = (ra-r)/Rii)Reflectionby thexz-plane.(operator xz)(ra=ra, r =, =-)A'(, n, -Φ)A(, n, Φ), = AF(E,n)eim(-) =[AF(5,n)e-im ] = m-mXzi.e.When mO, the molecular orbital wavefunction Y itselfis not an eigenfunction of o12
i m elec F( , )e 2 1 ii) Reflection by the xz-plane. m i m i m xz m AF e AF e ( , ) [ ( , ) ] ( ) ra rb x R z Symmetry of MO wavefunction i.e. When m 0, the molecular orbital wavefunctionm itself is not an eigenfunction of xz! = (ra + rb )/R = (ra rb )/R A(, , ) A(, , -) xz ( , , ) ' ' ra ra rb rb (operator xz)