Quantum Mechanics – Molecular Structure - CHERIC · 2013-12-19 · Quantum Mechanics –...
Transcript of Quantum Mechanics – Molecular Structure - CHERIC · 2013-12-19 · Quantum Mechanics –...
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Quantum Mechanics – Molecular Structure
Jeong Won KangDepartment of Chemical Engineering
Korea University
Applied Statistical Mechanics Lecture Note - 4
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Subjects
Structure of Complex Atoms - ContinuedMolecular Structure
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Structure of Complex Atoms- Continued
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The Spectra of Complex Atoms
The spectra of atoms rapidly become very complicated as the number of electrons increases.
Spectra of an atom : The atom undergoes transition with a change of energy; ΔE = hνGives information about the energies of electron However, the actual energy levels are not given solely
by the energies of orbitals Electron-Electron interactions
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Measuring Ionization Energy (H atoms)
The plot of wave numbers vs. 1/n2 gives the ionization energy for hydrogen atoms
2
2
nR
hcI
EIhc
EnR
c
hEEE
H
lower
lowerH
lower
−=
=
−−==
=−=Δ
ν
νν
ν
Slope = I
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Quantum defects and ionization energies
Energy levels of many-electron atoms do not vary as 1/n2
The outermost electron Experience slightly more charge than 1e Other Z-1 atoms cancel the charge slightly lower than 1 Quantum Defect (δ): empirical quantity
2
2
2
)(
nR
hcI
nhcRE
nhcRE H
−=
−−=
−=
ν
δ
Rydberg state
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Pauli exclusion principle and the spins
Palui exclusion principle “ No more than two electrons may occupy any given
orbital and, if two occupy one orbital, then their spin must be paired “
“ 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 “
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Pauli exclusion principle and the spins
Two electrons (fermions) occupy an orbital Ψ then ;
Total Wave function = (orbital wave function)*(spin wave function)
)1,2()2,1( Ψ−=Ψ
)2()1( αα
)2()1( ψψ
)2()1( βα )1()2( βα )2()1( ββ
four possibility
{ }{ })2()1()2()1()2/1()2,1(
)2()1()2()1()2/1()2,1(2/1
2/1
αββασαββασ
−=
+=
−
+
We cannot tell which one is α and β
Normalized Linear Combination of two spin wave functions
)2()1()2()1()2,1()2()1()2,1()2()1(
)2()1()2()1(
ββψψσψψσψψ
ααψψ
−
+
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Pauli exclusion principle and the spins
)2()1()2()1()2,1()2()1()2,1()2()1(
)2()1()2()1(
ββψψσψψσψψ
ααψψ
−
+
)1,2()2,1( Ψ−=Ψ
{ }{ } )1,2()2()1()2()1()2/1()2,1(
)1,2()2()1()2()1()2/1()2,1(2/1
2/1
−−
++
−=−=
=+=
σαββασσαββασ
Requirement :
)1,2()2,1()1,2()2,1(
)1,2()2,1()1,2()2,1(
Ψ=ΨΨ−=Ψ
Ψ=ΨΨ=Ψ
)2,1()2()1( −σψψ Only one acceptable
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Singlet and Triplet State
Excited State of He atom 1s2 1s1 2s1
The two electrons need not to be paired Singlet : paired spin arrangement
Triplet : parallel spin arrangement
Hund’s principle : triplet states generally lie lower than triplet state
)2()1( αα
)2()1( ββ
{ })2()1()2()1()2/1()2,1( 2/1 αββασ +=+
{ })2()1()2()1()2/1()2,1( 2/1 αββασ −=−
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Spectrum of atomic Helium
Spectrum of He atom is more complicated than H atom Only one electron is excited
• Excitation of two electrons require more energy than ionization energy
No transitions take place between singlet and triplet states
• Behave like two species
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Spin-Orbit Coupling
Electron spins has a further implication for energies of atoms when l > 0 ( finite orbital angular momentum )
(spin magnetic momentum) + (magnetic moment doe to orbital angular momentum) spin-orbit coupling
21+= lj
Parallel high angular momentum
Opposed low angular momentum
21−= lj
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Spin-orbit coupling
When l=2
Spin-Orbit Coupling constant (A) Dependence of spin-orbit interaction on the value of j
23
21
25
21
=−=
=+=
lj
lj
))1()1()1((21
,, +−+−+= sslljjhcAE sjl
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Spin-Orbit Coupling
Spin-orbit coupling depends on the nuclear charge The greater the nucleus charge the
stronger spin-orbit coupling Very small in H , very large in Pb
Fine structure Two spectral lines are observed The structure in a spectrum due to spin-
orbit coupling Ex) Na (street light)
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Term symbols and selection rules
Skip…
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Molecular Structure
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Topics
Valence-Bond TheoryMolecular Orbital Theory
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Born-Oppenheimer approximation
Assumption The nuclei is fixed at arbitrary location H2 molecule
• Nuclei move about 1 pm • Electrons move about 1000 pm
Use different separation solve Schrodinger equation Energy of molecules vary with bond length Equilibrium bond length (Re) Bond Dissociation energy (De)
Structure Prediction, Property Estimation
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Valence-Bond Theory
Consider H2 molecule If electron 1 is on atom A and electron 2
is on atom B
)()( 1111 rrAA SHSH ψψψ =
)2()1( BA=ψ )1()2( BA=ψ
)1()2()2()1( BABA ±=ψ
)1()2()2()1( BABA +=ψ
Simple notation
Linear combination of wave functions
Lower energy
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σ - bond
Cylindrical symmetry around internuclear axis
Rambles s-orbital : called sigma-bond Zero angular momentum around
internuclear axis
Molecular potential energy
Spin : spin paring According to Pauli principle spin must be paired
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π-bond
Essence of valence-bond theory Pairing of the electrons Accumulation of electron
density in the internulear region
N2 atom
1xp
1yp
1112 2222 zyx ppps
1zp σ−bond
π−bondCannot form σ-bond
not symmetricalaround internuclear axis
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Water molecule
1122 2222 zyx ppps
Bond angle : 90 degree Experimental 104.5 degree !
Two σ−bonds
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Promotion
CH4 molecule cannot be explained by valence-bond theory only two electrons can form bonds
Promotion Excitation of electrons to higher energy If bonds are formed (lower energy) , excitation is
worthwhile 4 σ-bonds
112 222 yx pps
1111 2222 zyx ppps
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Hybridization
Description of methane is still incomplete Three s-bond (H1s – C2p)
+ one s-bond (H1s – C2s) Cannot explain symmetrical feature of
methane molecules Hybrid orbital
Interference between C2s + C2p orbital Forming sp3 hybrid orbital
zyx
zyx
zyx
zyx
pppshpppshpppshpppsh
−−+=
−+−=
+−−=
+++=
4
3
2
1
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Hybridization
Formation of σ-bond in methane
)1()2()2()1( 11 AhAh +=ψ
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sp2 hybridization : ethylene
sp2 hybridization
yx
yx
x
ppsh
ppsh
psh
2/12/13
2/12/12
2/11
)23()
23(
)23()
23(
2
−−=
−+=
+=
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sp hybridization : acetylene
sp hybridization
z
z
pshpsh
−=+=
2
1
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Molecular Orbital Theory
Molecular Orbital (MO) Theory Electrons should be treated as spreading throughout the
entire molecule The theory has been fully developed than VB theory
Approach :Simplest molecule ( H2
+ ion) complex molecules
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The hydrogen molecule-ion
Hamiltonian of single electron in H2+
Solution : one-electron wavefunction Molecular Orbital (MO)
The solution is very complicated function This solution cannot be extended to polyatomic molecules
)111(4
2 110
221 Rrr
eVVm
HBAe
−+−=+∇−=πε
Attractions between electron and nuclei Repulsion between the nuclei
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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 superposition of two atomic orbital :
N : Normalization factor Called a σ-orbital
( )
( ) 2/130
/
1
2/130
/
1
0
0
)(
aeB
aeA
BAN
ar
SH
ar
SH
B
B
A
A
πψ
πψ
ψ
−
−±
==
==
±= Linear Combination of Atomic Orbitals (LCO-MO)
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Normalization
( )
( ) 2/130
/
1
2/130
/
1
0
0
)(
aeB
aeA
BAN
ar
SH
ar
SH
B
B
A
A
πψ
πψ
ψ
−
−±
==
==
±=
{ }
56.0
59.0
)211(2
12222*
*
=
≈=
++=++=
=
N
dABS
SNdABdBdANd
d
τ
ττττψψ
τψψ
{ }
10.1
59.0
)211(2
12222*
*
=
≈=
−+=−+=
=
N
dABS
SNdABdBdANd
d
τ
ττττψψ
τψψ
+ψ
−ψ
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LCAO-MO
Amplitude of the bonding orbitalin hydrogen molecule-ion
rA and rB are not independent
{ } 2/122 cos2 θRrRrr AAB −+=
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Bonding Orbital
Overlap density Crucial term Electrons accumulates in the region where atomic
orbital overlap and interfere constructively.
)2( 2222 ABBAN ++=+ψ
Probability density if the electron were confinedto the atomic orbital A
Probability density if the electron were confinedto the atomic orbital B
An extra contribution to the density (Overlap density)
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Bonding Orbital
The accumulation of electron density between the nuclei put the electron in a position where it interacts strongly with both nuclei the energy of the molecule is lower than that of separate atoms
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Bonding Orbital
Bonding orbital called 1σ orbital σ electron
The energy of 1σ orbital decreases as R decreases However at small separation, repulsion becomes large There is a minimum in potential energy curve
Re = 130 pm (exp. 106 pm) De = 1.77 eV (exp. 2.6 eV)
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Antibonding Orbital
Linear combination ψ- corresponds to a higher energy
Reduction in probability density between the nuclei (-2AB term)
Called 2σ orbital (often labeled 2σ *)
)2( 2222 ABBAN −+=−ψ
Amplitude of antibonding orbital
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Antibonding orbital
The electron is excluded from internuclear region destabilizing
The antibonding orbital is more antibonding than the bonding orbital is bonding
sHsH EEEE 11 −>− +−
Molecular orbital energy diagram
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The Structure of Diatomic Molecules
Target : Many-electron diatomic molecules Similar procedure Use H2
+ molecular orbital as the prototype Electrons supplied by the atoms are then
accommodated in the orbitals to achieve lowest overall energy
Pauli’s principle + Hund’s maximum multiplicity rule
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Hydrogen and He molecules
Hydrogen (H2)
• Two electrons enter 1σ orbital • Lower energy state than 2 H atoms
Helium (He)
• The shape is generally the same as H• Two electrons enter 1σ orbital • The next two electrons can enter 2σ* orbital• Antibond is slightly higher energy than bonding• He2 is unstable than the individual atoms
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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
Characteristics The greater the bond order, the shorter the bond The greater the bond order, the greater the bond
strength
)(21 *nnb −=
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Period 2 diatomic molecules- σ orbital
Elementary treatments : only the orbitals of valence shell are used to form molecular orbital Valence orbitals in period 2 : 2s and 2p
σ-orbital : 2s and 2pz orbital (cylindrical symmetry)
From an appropriate choice of c we can form four molecular orbital
zzzz pBpBpApAsBsBsAsA cccc 22222222 ψψψψψ +++=
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Period 2 diatomic molecules- σ orbital
Because 2s and 2p orbitals have distinctly two different energies, they may be treated separately.
Similar treatment can be used 2s orbitals 1σ and 2σ* 2pz orbitals 3σ and 4σ*
sBsBsAsA cc 2222 ψψψ ±=
zzzz pBpBpApA cc 2222 ψψψ +=
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Period 2 diatomic molecules- π orbital
2px, 2pz perpendicular to intermolecular axis Overlap may be constructive or distructive Bonding or antibonding π orbital Two πx orbitals + Two πy orbitals
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Period 2 diatomic molecules- π orbital
The previous diagram is based on the assumption that 2s and 2pz orbitals contribute to completely different sets
In fact, all four atomic orbitals contribute joinlty to the four σ-orbital
The order and magnitude of energies change :
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The variation of orbital energies of period 2 homonuclear diatomics
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The overlap integral
The extent to which two atomic orbitals on different atom overlaps : the overlap integral
ψA is small and ψB is large S is small ψA is large and ψB is small S is small ψA and ψB are simultaneously large S is
large• 1s with same nucleus S=1 • 2s + 2px S = 0
= τψψ dS BA*
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Structure of homonuclear diatomic molecule
Nitrogen1σ2 2σ∗2 1π4 3σ2
b = 0.5*(2+4+2-2) = 3Lewis Structure
Oxygen1σ2 2σ∗2 3σ2 1π4 2π2∗
b = 0.5*(2+2+4-2-2) = 2last two electrons occupy different orbital : πx and πy (parallel spin)
angular momentum (s=1) Paramagnetic
:NN: ≡
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Quiz
What is spin-orbit coupling ? What is the difference between sigma and pi bond ? Explain symmetrical structure of methane molecule.
112 222 yx pps