Transcript PowerPoint 프레젠테이션

6
QUANTUM MECHANICS
AND ATOMIC STRUCTURE
CHAPTER
6.1 Quantum Picture of the Chemical Bond
6.2 Exact Molecular Orbital for the Simplest Molecule: H2+
6.3 Molecular Orbital Theory and the Linear Combination
of Atomic Orbitals Approximation for H2+
6.4 Homonuclear Diatomic Molecules: First-Period Atoms
6.5 Homonuclear Diatomic Molecules: Second-Period
Atoms
6.6 Heteronuclear Diatomic Molecules
6.7 Summary Comments for the LCAO Method and
Diatomic Molecules
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Potential energy diagram for the decomposition of the methyl methoxy radical
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6.1 QUANTUM PICTURE OF THE CHEMICAL
BOND
 Potential energy of H2 (see section 3.7)
V
=
Ven +
Vee +
Vnn
 Using effective potential energy function, Veff or V(RAB)
- At large RAB, Veff → 0, and the atoms do not interact.
- As RAB decreases, Veff must become negative because of attraction.
- At very small RAB, Veff must become positive and large as Veff → ∞.
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Fig. 6.1
E0 : zero-point energy for the molecule
by the uncertainty principle
Dissociation energy: Do or De
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Changes in electron
density on formation
of H2 from 2H
(Fig. 6.2)
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Born-Oppenheimer Approximation: Slow
Nuclei, Fast Electrons
- Nuclei are much more massive than the electrons, the nuclei in the
molecules will move much more slowly than the electrons.
→ decoupling of the motions of the nuclei and the electrons
(A) Consider the nuclei to be fixed at a specific set of positions.
Then solve Schrödinger’s equation for the electrons moving around
and obtain the energy levels and wave functions.
Next, move the nuclei a bit, and repeat the calculation.
Continue this procedure in steps.
Each electronic energy level (E(el)a) is related to the
nuclear coordinates, RAB.
(a: the proper set of quantum numbers)
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- Visualizing a group of electrons moving rapidly around the sluggish
nuclei, to establish a dynamic distribution of electron density (Fig. 6.3).
rapid movement
of the electrons
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effective potential
energy functions
8
Mechanism of
Covalent
Bond Formation
First phase
- When RAB → ∞,
independent H atoms
e-
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- As the two atoms approach one
another,
Second phase
(The ‘particle in a box’ energies
decrease as the size of the box
increases)
e-
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- As bond formation continues,
Third phase
e-
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Final phase
e-
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 For the ionic bond, potential energy alone is essential. (section 3.8)
repulsion attraction
 For the covalent bond, the charge distribution and the kinetic energy
of the electrons are also important.
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6.2 EXACT MOLECULAR ORBITALS FOR THE
SIMPLEST MOLECULE: H2+
 H2+ ion: a single electron bound to two protons
bond length 1.06 Å; bond dissociation energy 2.79 eV = 269 kJ mol-1
- For a fixed value of RAB,
the position of the electron:
- The potential energy has cylindrical
(ellipsoidal) symmetry around the RAB axis.
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 By the Born-Oppenheimer approximation,
- RAB : holding at the equilibrium bond length of 1.06 Å
Omitting f due to the same potential energy for all values of f
 The solution of the Schrödinger equation:
- smooth, single-valued, and finite in all regions of space
to define a probability density function of its square
Solutions exists when the total energy and angular momentum
are quantized.
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Electronic Wave Functions for H2+
1*
u
2*u
3 g
3u*
1 g
2g
1 u
1g*
- isosurface comprising the wave function with 0.1 of its maximum value.
- red: + amplitude; blue: - amplitude
- molecular orbital: each of exact one-electron wave functions
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Molecular orbital nomenclature
- Four labels summarize the energy and the shape of each wave function.
1) integer: an index tracking the relative energy
of the wave functions of each symmetry type.
i.e.) 1g: the first (the lowest energy) of the
g wave functions
1 g
2) Greek letter: how the amplitude of the wave
function is distributed around the internuclear axis.
-  : the amplitude with cylindrical symmetry
around the axis
1 u
-  : the amplitude with a nodal plane that contains
the internuclear axis
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3) g or u: how the wave function changes as we
2*u
invert our point of observation through the center
of the molecule (i.e. the wave function at (x, y, z)
and (-x, -y, -z):
g : symmetric, the same at these points
(‘gerade’)
u : antisymmetric, the opposite at these points
(‘ungerade’)
4) * : how the wave function changes when the
point of observation is reflected through a plane
perpendicular to the internuclear axis:
2g
1*
u
no * symbol : no changing sign upon reflection
(bonding MO)
•: changing sign upon reflection (antibonding
MO)
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1 g
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Nature of the Chemical Bond in H2+
antibonding MO
bonding MO
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Summary of the Quantum Picture of
Chemical Bonding for H2+
1. The Born-Oppenheimer approximation: fixing the nuclei positions
2. Molecular orbital: one-electron wave function,
its square describes the distribution of electron density
3. Bonding MO: increased e density between the nuclei,
decreased effective potential energy
4. Antibonding MO: a node on the internuclear axis,
increased effective potential energy
5.  orbital: cylindrical symmetry; cross-sections perpendicular to the
internuclear axis are discs.
6.  orbital: has a nodal plane containing the internuclear axis
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6.3 MOLECULAR ORBITAL THEORY AND THE
LINEAR COMBINATION OF ATOMIC ORBITALS
APPROXIMATION FOR H2+
 LCAO method: selecting sums and differences (linear
combinations) of atomic orbital wave functions to generate
the best approximation to each type of molecular orbital
wave function
- The general form for H2+
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 MOs of the  bonding
 The distribution of electron probability density
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Plots of
wave functions
and electron
probability
density for H2+
MOs
(Fig. 6.7)
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Energy of H2+ in the LCAO Approximation
g1s, RAB = 1.32 Å, D = 1.76 eV
1g, RAB = 1.06 Å, D0 = 2.79 eV
Calculated by Burrau (1927):
from exact solutions of Schrödinger
equation
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 Correlation diagram: the energy-level diagram
within the LCAO-MO model
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6.4 HOMONUCLEAR DIATOMIC MOLECULES:
FIRST-PERIOD ATOMS
For H2 and He2+,
A
B
B

 A

 g1s  Cg  He1
s   He1s   C g 1s  1s 
A
B
A
B



 u*1s  Cu  He1



C
1
s

1
s
s
He1s 
g 
Fig. 6.11. Correlation diagram for H2. Fig. 6.12. Correlation diagram for He2 .
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 H2: Stabilization of bonding MO
by 2  (–E) for H2
compared to the
noninteracting system.
 He2+: Stabilization of bonding
MO by 2  (–E)
compensated by destabilization
of antibonding MO by +E .
 Net stabilization energy =
–E
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 Bond order
Bond order = (1/2) x (number of electrons in bonding MOs
– number of electrons in antibonding MOs)
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6.5 HOMONUCLEAR DIATOMIC MOLECULES:
SECOND-PERIOD ATOMS
 LCAO-MO approximation
 Combination of 2s AOs to form g2s and u2s* MOs
 g 2 s  Cg [2 s A  2 s B ]
 u*2 s  Cu [2 s A  2 s B ]
*
 Combination of 2pz AOs to form  g 2 pand

u 2 pz MOs
z
 g 2 p  Cg [2 pzA  2 pzB ]
z
 u*2 p  Cu [2 pzA  2 pzB ]
z
The core 1s2 electrons are not included – they are too low in
energy
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Rules for LCAO-MO Model (Mulliken)
1. Only AOs of the same symmetry along the bond axis can
be combined to give MOs.
+
+
+
OR
OR
+
OR
OR
+
2. AOs of the same or similar energies will interact more
strongly than those of widely different energies.
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new node
 g 2 pz
Fig. 6.14 Formation of (a) bonding
and (b) antibonding MOs
from 2pz orbitals on atoms A and B.
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 Doubly degenerate
 2 p&  u 2 pMOs
x
y
 u 2 p  Cu [2 pxA  2 pxB ]
x
 g* 2 p  Cg [2 pxA  2 pxB ]
x
 u 2 p  Cu [2 p yA  2 p yB ]
new node
y
 g* 2 p  Cg [2 p yA  2 p yB ]
y
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 Determination of energy ordering
1. Average energy of bonding-antibonding pair of MOs
similar to that of original AO’s
2. Energy difference between a bonding-antibonding pair
becomes large as the overlap of AO’s increases
3. Rule of thumb 1: bonding MOs < antibonding MOs
4. Rule of thumb 2: energy increases with number of nodes
5. Ordering for -group: 2s < 2s* < 2p < 2p* (see slide 35)
6. Ordering for -group: 2p < 2p* (see slide 35)
7. For ordering  and , experimental results must be invoked:
2p < 2p, 2p* < 2p* for all the 2nd-row molecules (except O2 and F2)
2p < 2p, 2p* < 2p* for O2 and F2 (see slide 36, etc)
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Energy ordering of  and  MO types
Energy
increases
with the
no. of
nodes in
the same
symmetry.
σ-MOs
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π-MOs
Box wave functions
35
Fig. 6.16 Energy levels for the homonuclear diatomics Li2 through F2.
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HOMO =
Highest
(energy)
Occupied MO
LUMO
LUMO
HOMO
HOMO
LUMO =
Lowest
(energy)
Unoccupied
MO
Fig. 6.17 Correlation diagrams for second-period diatomic
molecules, N2 & F2.
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Correlation diagram and corresponding
MOs for N2
*u2pz
*u2px , *u2py
g2pz
u2px , u2py
*g2s
g2s
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Correlation diagram and corresponding
MOs for F2
*u2pz
*u2px , *u2py
u2px , u2py
g2pz
*g2s
g2s
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 Cross-over in the correlation diagrams of Li2~N2 and O2~Ne2
 Reversed ordering of energy for Li2 ~ N2
 g 2 p  u 2 p (or u 2 p )
z
x
y
~ Due to large electron-electron spatial repulsions between
*


electrons in g 2 p and u 2 s MO’s.
z
 Normal ordering of energy for O2, F2, Ne2
 g 2 p  u 2 p (or u 2 p )
z
x
y
~ As Z increases, the repulsion decreases since electrons
*

and

in g 2 s
u 2s
MO’s are drawn more strongly toward
the nucleus.
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Fig. 6.18 (a) Paramagnetic liquid oxygen, O2, and (b) diamagnetic
liquid nitrogen, N2, pours straight between the poles of a magnet.
paramagnetic
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Fig. 6.19 Trends in several properties with the number of valence
electrons
in the second-row
diatomic molecules.
General
Chemistry
I
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6.6 HETERONUCLEAR DIATOMIC
MOLECULES
For the 2s orbitals,
2s = CA2sA + CB2sB
2s* = CA’2sA – CB’2sB
In the homonuclear case, CA = CB; CA’ = CB’
If B is more electronegative than A,
CB > CA for bonding  MO,
CA’ > CB’ for the higher energy * MO,
closely resembling a 2sA AO
A
B
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B
O
Fig. 6.20 Correlation
diagram for
heteronuclear
diatomic
molecule, BO. (O is
more electronegative)
LCAO/MO electron
configuration:
(2s)2(*2s)2(2px)2(2py)2(2pz)1
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*
+1sH +2pFz
(d)
nb
+1sH
+2pFx,y
(c)

+1sH –2pFz
(b)
(
+1sH +2sF
(
(a)
a
General
Chemistry I
)
nb
45
Fig. 6.21
Overlap
of
atomic
orbitals
in HF.
Fig. 6.22 Correlation
diagram for HF. The
2s, 2px, and 2py
atomic orbitals of F
do not mix with the
1s atomic orbital of H,
and therefore remain
nonbonding.
LCAO/MO electron
configuration:
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6.7 SUMMARY COMMENTS FOR THE LCAOMO METHOD AND DIATOMIC MOLECULES
The qualitative LCAO-MO method easily identifies the sequence
of energy levels for a molecule, such as trends in bond length
and bond energy, but does not give their specific values.
The qualitative energy level diagram is very useful for
interpreting experiments that involve adsorption and
emission of energy such as spectroscopy, ionization by
electron removal, and electron attachment.
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6
QUANTUM MECHANICS
AND ATOMIC STRUCTURE
CHAPTER
6.8 Valence Bond Theory and the Electron Pair Bond
6.9 Orbital Hybridization for Polyatomic Molecules
6.10 Predicting Molecular Structures and Shapes
6.11 Using the LCAO and Valence Bond Methods Together
6.12 Summary and Comparison of the LCAO and Valence
Bond Methods
6.13 A Deeper Look (Properties of Exact Molecular Orbitals
for H2+)
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6.8 VALENCE BOND THEORY AND THE
ELECTRON PAIR BOND
 Explains the Lewis electron pair model
 VB wave function for the bond is a product of two
one-electron AO wave functions
 Easily describes structure and geometry of bonds
in polyatomic molecules
Nobel Prizes
Chemistry (‘54)
“The Nature of
Chemical
Bonding”
Peace (‘62)
Walther Heitler
Fritz London
John C. Slater
Linus Pauling
(DE, 1904-1981)
(DE, 1900-1954)
(US, 1900-1976)
(US,1901-1994)
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 Single Bonds
- At very large values of RAB,
independent atoms
- As the atoms begin to interact strongly,
(Electrons are now indistinguishable)
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 VB wave function for the single bond in a H2 molecule
 gel  C1 1s A (1)1s B (2)  1s A (2)1s B (1)   Bonding MO
 uel  C1 1s A (1)1s B (2)  1s A (2)1s B (1)   Antibonding MO
Fig. 6.24 (a) The electron density g for gel and u for uel in the simple
VB model for H2. (b) Three-dimensional isosurface of the electron density
for the gel wave function in the H2  bond.
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 For F2  bond,
 For HF  bond,
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 Multiple Bonds
N: (1s)2(2s)2(2px)1(2py)1(2pz)1
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 Polyatomic Molecules
all C-H
bonds
equivalent
(from
experiment)
?
 Electron promotion: electron relocated to a higher-energy orbital
promotion
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hybridization
54
6.9 ORBITAL HYBRIDIZATION FOR
POLYATOMIC MOLECULES
 sp-Hybridization
BeH2
Promotion: Be: (1s)2(2s)2  Be: (1s)2(2s)1(2pz)1
Two equivalent sp hybrid orbitals:
1 (r ) 
1
1
 2s  2 pz  and  2 (r )   2s  2 pz 
2
2
New electronic configuration, Be: (1s)2(1)1(2)1
Wave functions for the two bonding pairs of electrons:
H
H

 bond
(1,
2)

c

(1)1
s
(2)


(2)1
s
(1) 
1
  1
1
H
H


 bond
2 (3, 4)  c   2 (3)1s (4)   2 (4)1s (3) 
A pair of  bonds at an angle 180o apart  linear molecule
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Fig. 6.28 Formation, shapes, and bonding of the sp hybrid orbitals in the
BeH2 molecule. (a) The 2s and 2pz orbitals of the Be atom. (b) The two sp
hybrid orbitals formed from the 2s and 2pz orbitals on the Be atom. (c) The
two  bonds that form from the overlap of the sp hybrid orbitals with the
H1s orbitals, making two single bonds in the BeH2 molecule. (d) Electron
density in the two  bonds.
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 sp2-Hybridization
BH3
Promotion: B: (1s)2(2s)2(2px)1  B: (1s)2(2s)1(2px)1(2py)1
Three equivalent sp2 hybrid orbitals:
1 (r)  2s  (1/ 2)1/ 2 2 py
2 (r)  2s  (3/ 2)1/ 2 2 px  (1/ 2)1/ 2 2 py
3 (r)  2s  (3/ 2)1/ 2 2 px  (1/ 2)1/ 2 2 py
New electronic configuration, B: (1s)2(1)1(2)1(3)1
Three bonds at an angle 120o in a plane  trigonal planar
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Fig. 6.29
Formation,
shapes, and
bonding of
the sp2
hybrid
orbitals in
the
BH3
molecule
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 sp3-Hybridization
CH4
Promotion: C: (1s)2(2s)2(2px)1(2py)1  C: (1s)2(2s)1(2px)1(2py)1(2pz)1
Four equivalent sp3 hybrid orbitals:
1
2
1
 2 (r )   2s  2 px  2 p y  2 pz 
2
1
 3 (r )   2s  2 px  2 p y  2 pz 
2
1
 4 (r )   2s  2 px  2 p y  2 pz 
2
1 (r )   2s  2 px  2 p y  2 pz 
New electronic configuration, C: (1s)2(1)1(2)1(3)1 (4)1
Four bonds at an angle 109.5o generating tetrahedral geometry
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Fig. 6.30 Shapes and relative orientations of the four sp3 hybrid orbitals
in CH4 pointing at the corners of a tetrahedron with the C atom at its center.
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Summary of Hybridization Results
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Hybridization and Lone Pairs
 NH3: sp3 hybrid orbitals
trigonal pyramid with three equivalent
bonds
 H2O: sp3 hybrid orbitals
two lone pairs → bent or angular
structure
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 Multiple Bonds in Organic Compounds
 Ethylene, C2H4
Promotion, C: (1s)2(2s)2(2px)1(2py)1  C: (1s)2(2s)1(2px)1(2py)1(2pz)1
Formation of three sp2 hybrid orbitals from the 2s and two 2p orbitals
New electronic configuration, C: (1s)2(1)1(2)1(3)1(2pz)1
Five  bonds: C11-H1s, C12-H1s, C21-H1s, C21-H1s, C13- C23
One  bond: C12pz-C22pz
One double bond (C=C): (C13- C23) + (C12pz-C22pz)
Note:  = sp2
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Fig. 6.33 Formation of  bonds in ethylene.
Fig. 6.34 Formation of a  bond in ethylene.
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 Acetylene, C2H2
Linear, triple bond, H-C-C bond angles of 180o
Promotion, C: (1s)2(2s)2(2px)1(2py)1  C: (1s)2(2s)1(2pz)1(2px)1(2py)1
Formation of two sp hybrid orbitals from the 2s and the 2pz orbitals
New electronic configuration, C: (1s)2(1)1(2)1(2px)1(2py)1
Three  bonds: C11-H1s, C21-H1s, C12- C22
Two  bonds: C12px-C22px, C12py-C22py
One triple bond :  (C12- C22) +  (C12px-C22px, C12py-C22py)
Note:  = sp
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Fig. 6.35 Formation of  bonds in acetylene.
Fig. 6.36 Formation of two  bonds in acetylene.
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6.10 PREDICTING MOLECULAR
STRUCTURES AND SHAPE
 Description of the structure and shape of a molecule
(1) Determine the empirical formula.
(2) Determine the molecular formula.
(3) Determine the structural formula from a Lewis diagram.
(4) Determine the molecular shape from experiments.
(5) Identify the hybridization scheme that best explains
the shape predicted by VSEPR.
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EXAMPLE 6.4
Hydrazine: Elemental analysis shows its mass per cent composition to be
87.419% nitrogen and 12.581% hydrogen. The density of hydrazine at 1 atm
And 25 oC is 1.31 gL-1. Determine the molecular formula for hydrazine.
Predict the structure of hydrazine. What is the hybridization of the N atoms?
(1) Elemental analysis: N(87.419%), H (12.581%)
(2) Empirical formula: NH2  (Molar mass)emp
(3) Molar mass calculated from the ideal gas law (with known )
(4) Molar mass / (Molar mass)emp = 2, Molecular formula: N2H4
(5) Lewis diagram, steric number (4)  sp3 hybrid orbitals
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The electrostatic
potential energy map
Fig. 6.38 (a) Space-filling
models. (b) 0.002 e/(a0)3
electron density surfaces,
and (c) electrostatic
potential energy surfaces.
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6.11 USING THE LCAO AND VALENCE BOND
METHODS TOGETHER
 Linear Triatomic Molecules (CO2, N2O, NO2+)
 Central atom  sp hybridization from s and pz orbitals
Two sp orbitals form  bonds with outer atoms
Remaining px and py orbitals form  bonds
  bonds  pz orbital (outer atom) with sp hybrid (central atom)
  bonds  linear combinations of the px (or py) orbitals give
bonding, antibonding, and nonbonding
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Fig. 6.40
-bonding
in linear
triatomic
molecules
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Fig. 6.41 Correlation diagram for electrons in linear triatomic molecules
(for CO2)
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 Nonlinear Triatomic Molecules (NO2, O3, NF2, NO2-)
 Central atom  sp2 hybridization from s, px, and py orbitals
One of sp2 orbitals holds a lone pair
Two of sp2 orbitals form  bonds with outer atoms
Remaining pz orbital forms  bond
 Outer atom
p orbital pointing toward the central atom forms a  bond
pz orbital forms a  bond with pz orbitals of other atoms
Remaining p orbital and s orbital  nonbonding
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General Chemistry I
Fig. 6.42 (a) Ball-and-stick model (bottom) and 
molecular orbitals for bent triatomic molecule, NO2–,
with three sp2 hybrid orbitals on the central N atom
that would lie in the plane of the molecule.
(b) Correlation diagram for the  orbitals. There is
only one of each of MO (, nb, *).
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6.12 SUMMARY AND COMPARISON OF THE
LCAO AND VALENCE BOND METHODS
 LCAO method for H2 forming a  bond:
 g1s  Cg 1As  1Bs  
 1s A  1s B 
 Molecular electronic wave function in the LCAO approximation:
el
 MO
  g1s (1) g1s (2)  1s A (1)  1s B (1)  1s A (2)  1s B (2) 
el
 MO
 1s A (1)1s B (2)  1s A (2)1s B (1)   1s A (1)1s A (2)  1s B (1)1s B (2) 
General Chemistry I
76
 Molecular electronic wave function in the VB model:
el
el
 VB
(r1A , r2B )  c1A (r1A )B (r2B )  c2A (r2A )B (r1B )   VB
 1sA (1)1sB (2) 1sA (2)1sB (1)
 Comparison of MO and VB theories:
el
 MO
 1s A (1)1s B (2)  1s A (2)1s B (1)   1s A (1)1s A (2)  1s B (1)1s B (2) 
el
 VB
purely covalent structure
H–H
 Improved VB wavefunction:
 improved  VB   ionic
General Chemistry I
 ionic
mixture of ionic states,
HA-HB+ and HA+HB-
This is over-emphasized
in original LCAO-MO
method and is absent
in original VB model
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General Chemistry I
78
10 Problem Sets
For Chapter 6,
2, 18, 28, 48, 52, 58, 62, 64, 66, 72
General Chemistry I
79