Alied Statistical Mechanics Lecture Note - 4 Quantum Mechanics Molecular Structure Jeong Won Kang Deartment of Chemical Engineering Korea University
Subjects Structure of Comlex Atoms - Continued Molecular Structure
Structure of Comlex Atoms - Continued
The Sectra of Comlex Atoms The sectra of atoms raidly become very comlicated as the number of electrons increases. Sectra of an atom : The atom undergoes transition with a change of energy; ΔE hν Gives information about the energies of electron However, the actual energy levels are not given solely by the energies of orbitals Electron-Electron interactions
Measuring Ionization Energy (H atoms) The lot of wave numbers vs. 1/n gives the ionization energy for hydrogen atoms ΔE E E ν ν c I E lower I ν hc R n R n lower H H hν E hc lower Sloe I
Quantum defects and ionization energies Energy levels of many-electron atoms do not vary as 1/n The outermost electron Exerience slightly more charge than 1e Other Z-1 atoms cancel the charge slightly lower than 1 Quantum Defect (δ): emirical quantity hcrh E n hcr E ( n δ ) I R ν hc n Rydberg state
Pauli exclusion rincile and the sins Palui exclusion rincile No more than two electrons may occuy any given orbital and, if two occuy one orbital, then their sin must be aired When label of any two identical fermions are exchanged, the total wavefunction changes sign. When the label of any two identical bosons are exchanged, the total wavefunction retain its sign
Pauli exclusion rincile and the sins Two electrons (fermions) occuy an orbital Ψ then ; Ψ( 1,) Ψ(,1) Total Wave function (orbital wave function)*(sin wave function) ψ ( 1) ψ () four ossibility ψ (1) ψ () α(1) α() ψ (1) ψ () σ (1,) + ψ (1) ψ () σ (1,) ψ (1) ψ () β (1) β () α( 1) α() α ( 1) β () α ( ) β (1) β ( 1) β () σ (1,) (1/ + σ (1,) (1/ 1/ 1/ ) ) We cannot tell which one is α and β { α(1) β () + β (1) α() } { α(1) β () β (1) α() } Normalized Linear Combination of two sin wave functions
Pauli exclusion rincile and the sins Requirement : Ψ( 1,) Ψ(,1) ψ (1) ψ () α(1) α() ψ (1) ψ () σ (1,) ψ (1) ψ () σ (1,) ψ (1) ψ () β (1) β () + Ψ(1,) Ψ(1,) Ψ(1,) Ψ(1,) Ψ(,1) Ψ(,1) Ψ(,1) Ψ(,1) σ (1,) (1/ + σ (1,) (1/ 1/ 1/ ) ) { α(1) β () + β (1) α() } σ + (,1) { α(1) β () β (1) α() } σ (,1) ψ 1) ψ () σ (1,) ( Only one accetable
Singlet and Trilet State Excited State of He atom 1s 1s 1 s 1 The two electrons need not to be aired Singlet : aired sin arrangement σ (1,) (1/ 1/ ) { α(1) β () β (1) α() } Trilet : arallel sin arrangement α( 1) α() { α(1) β () β (1) α() } σ (1,) (1/ 1/ ) + + β ( 1) β () Hund s rincile : trilet states generally lie lower than trilet state
Sectrum of atomic Helium Sectrum of He atom is more comlicated than H atom Only one electron is excited Excitation of two electrons require more energy than ionization energy No transitions take lace between singlet and trilet states Behave like two secies
Sin-Orbit Couling Electron sins has a further imlication for energies of atoms when l > 0 ( finite orbital angular momentum ) (sin magnetic momentum) + (magnetic moment doe to orbital angular momentum) sin-orbit couling Parallel high angular momentum j l + 1 Oosed low angular momentum j 1 l
Sin-orbit couling When l j j l l 1 5 + 1 3 Sin-Orbit Couling constant (A) Deendence of sin-orbit interaction on the value of j 1 E l, j, s hca( j( j + 1) l( l + 1) s( s + 1))
Sin-Orbit Couling Sin-orbit couling deends on the nuclear charge The greater the nucleus charge the stronger sin-orbit couling Very small in H, very large in Pb Fine structure Two sectral lines are observed The structure in a sectrum due to sinorbit couling Ex) Na (street light)
Term symbols and selection rules Ski
Molecular Structure
Toics Valence-Bond Theory Molecular Orbital Theory
Born-Oenheimer aroximation Assumtion Use The nuclei is fixed at arbitrary location H molecule Nuclei move about 1 m Electrons move about 1000 m different searation solve Schrodinger equation Energy of molecules vary with bond length Equilibrium bond length (R e ) Bond Dissociation energy (D e ) Structure Prediction, Proerty Estimation
Valence-Bond Theory Consider H molecule If electron 1 is on atom A and electron is on atom B ψ ψ H 1S ( r1 ) ψ H1S ( r1 ) A ψ A( 1) B() ψ A( ) B(1) A Simle notation ψ A ( 1) B() ± A() B(1) Linear combination of wave functions Lower energy ψ A ( 1) B() + A() B(1)
σ - bond Cylindrical symmetry around internuclear axis Rambles s-orbital : called sigma-bond Zero angular momentum around internuclear axis Molecular otential energy Sin : sin aring According to Pauli rincile sin must be aired
π-bond Essence of valence-bond theory Pairing of the electrons Accumulation of electron density in the internulear region N atom 1 z 1 1 1 s x y z σ bond 1 x 1 y Cannot form σ-bond not symmetrical around internuclear axis π bond
Water molecule 1 1 s x y z Two σ bonds Bond angle : 90 degree Exerimental 104.5 degree!
Promotion CH 4 molecule cannot be exlained by valence-bond theory only two electrons can form bonds Promotion 1 1 s x y Excitation of electrons to higher energy If bonds are formed (lower energy), excitation is worthwhile 1 1 1 4 σ-bonds s1 x y z
Hybridization Descrition of methane is still incomlete Three s-bond (H1s C) + one s-bond (H1s Cs) Cannot exlain symmetrical feature of methane molecules Hybrid orbital Interference between Cs + C orbital Forming s 3 hybrid orbital z y x z y x z y x z y x s h s h s h s h + + + + + + 4 3 1
Hybridization Formation of σ-bond in methane ψ h 1 ( 1 ) A ( ) + h 1 ( ) A (1 )
s hybridization : ethylene s hybridization y x y x x s h s h s h 1/ 1/ 3 1/ 1/ 1/ 1 ) 3 ( ) 3 ( ) 3 ( ) 3 ( + +
s hybridization : acetylene s hybridization h h 1 s + s z z
Molecular Orbital Theory Molecular Orbital (MO) Theory Electrons should be treated as sreading throughout the entire molecule The theory has been fully develoed than VB theory Aroach : Simlest molecule ( H + ion) comlex molecules
The hydrogen molecule-ion Hamiltonian of single electron in H + H e 1 1 1 + V V ( + me 4πε 0 ra 1 rb 1 1 ) R Attractions between electron and nuclei Reulsion between the nuclei Solution : one-electron wavefunction Molecular Orbital (MO) The solution is very comlicated function This solution cannot be extended to olyatomic molecules
Linear Combination of Atomic Orbitals (LCAO-MO) If an electron can be found in an atomic orbital belonging to atom A and also in an atomic orbital belonging to atom b, then overall wavefuntion is suerosition of two atomic orbital : ψ ± N( A ± B) Linear Combination of Atomic Orbitals (LCO-MO) A ψ B ψ H1S H1S A B 3 ( πa ) 3 ( πa ) 1/ N : Normalization factor Called a σ-orbital e e r 0 r A B 0 / a 1/ / a 0 0
Normalization ( ) ( ) 1/ 3 0 / 1 1/ 3 0 / 1 0 0 ) ( a e B a e A B A N a r S H a r S H B B A A π ψ π ψ ψ ± ± { } 0.56 0.59 ) 1 (1 1 * * + + + + N ABd S S N ABd d B d A N d d τ τ τ τ τ ψ ψ τ ψ ψ { } 1.10 0.59 ) 1 (1 1 * * + + N ABd S S N ABd d B d A N d d τ τ τ τ τ ψ ψ τ ψ ψ ψ + ψ
LCAO-MO r A and r B are not indeendent r B { r + R r R cosθ} 1/ A A Amlitude of the bonding orbital in hydrogen molecule-ion
Bonding Orbital ψ + N ( A + B + AB) An extra contribution to the density (Overla density) Probability density if the electron were confined to the atomic orbital B Probability density if the electron were confined to the atomic orbital A Overla density Crucial term Electrons accumulates in the region where atomic orbital overla and interfere constructively.
Bonding Orbital The accumulation of electron density between the nuclei ut the electron in a osition where it interacts strongly with both nuclei the energy of the molecule is lower than that of searate atoms
Bonding Orbital Bonding orbital called 1σ orbital σ electron The energy of 1σ orbital decreases as R decreases However at small searation, reulsion becomes large There is a minimum in otential energy curve Re 130 m (ex. 106 m) De 1.77 ev (ex..6 ev)
Antibonding Orbital Linear combination ψ - corresonds to a higher energy ψ N ( A + B AB) Reduction in robability density between the nuclei (-AB term) Called σ orbital (often labeled σ *) Amlitude of antibonding orbital
Antibonding orbital The electron is excluded from internuclear region destabilizing The antibonding orbital is more antibonding than the bonding orbital is bonding E EH1s > E+ EH1s Molecular orbital energy diagram
The Structure of Diatomic Molecules Target : Many-electron diatomic molecules Similar rocedure Use H + molecular orbital as the rototye Electrons sulied by the atoms are then accommodated in the orbitals to achieve lowest overall energy Pauli s rincile + Hund s maximum multilicity rule
Hydrogen and He molecules Hydrogen (H ) Two electrons enter 1σ orbital Lower energy state than H atoms Helium (He) The shae is generally the same as H Two electrons enter 1σ orbital The next two electrons can enter σ* orbital Antibond is slightly higher energy than bonding He is unstable than the individual atoms
Bond order A measure of the net bonding in a diatomic molecule n : number of electrons in bonding orbital n* : number of electrons in antibonding orbital b 1 * ( n n Characteristics The greater the bond order, the shorter the bond The greater the bond order, the greater the bond strength )
Period diatomic molecules - σ orbital Elementary treatments : only the orbitals of valence shell are used to form molecular orbital Valence orbitals in eriod : s and σ-orbital : s and z orbital (cylindrical symmetry) ψ + casψ As + cbsψ Bs + ca ψ A cb ψ B From an aroriate choice of c we can form four molecular orbital z z z z
Period diatomic molecules - σ orbital Because s and orbitals have distinctly two different energies, they may be treated searately. ψ ± casψ As cbsψ Bs ψ + ca ψ A cb ψ B Similar treatment can be used s orbitals 1σ and σ* z orbitals 3σ and 4σ* z z z z
Period diatomic molecules - π orbital x, z erendicular to intermolecular axis Overla may be constructive or distructive Bonding or antibonding π orbital Two π x orbitals + Two π y orbitals
Period diatomic molecules - π orbital The revious diagram is based on the assumtion that s and z orbitals contribute to comletely different sets In fact, all four atomic orbitals contribute joinlty to the four σ-orbital The order and magnitude of energies change :
The variation of orbital energies of eriod homonuclear diatomics
The overla integral The extent to which two atomic orbitals on different atom overlas : the overla integral * S ψ ψ dτ A ψ A is small and ψ B is large S is small ψ A is large and ψ B is small S is small ψ A and ψ B are simultaneously large S is large 1s with same nucleus S1 s + x S 0 B
Structure of homonuclear diatomic molecule Nitrogen 1σ σ 1π 4 3σ b 0.5*(+4+-) 3 Lewis Structure N Oxygen 1σ σ 3σ 1π 4 π : b 0.5*(++4--) N : last two electrons occuy different orbital : π x and π y (arallel sin) angular momentum (s1) Paramagnetic
Quiz What is sin-orbit couling? What is the difference between sigma and i bond? Exlain symmetrical structure of methane molecule. 1 1 s x y