Transcript PhysChem 728342 Introduction to Quantum Theory

Physical Chemistry III (01403342)
Atomic
Structure
Chapter 3:
Piti Treesukol
Kasetsart University
Kamphaeng Saen Campus
Electronic Structures of
Atoms
 Hydrogenic atoms




Many-electron atoms
The orbital approximations
Self-consistent Field orbitals
Approximation Methods
• Variation Method
• Perturbation Method
2
Hydrogenic Atoms


A hydrogenic atom is a one-electron
atom (H) or ion of general atomic
number Z (He+, Li2+, etc.)
The coulombic
potential
energy
Ze
V 
2
4 0 r

H  Eˆ
for
Eˆ
 Vˆ electron and
The Hamiltonian
the
a nucleus        Ze
K ,electron
2
2me
K , nuclues
2
2
e
2mN
2
2
N
4 0 r
3

The Hamiltonian for the internal
motion of electron relative to the
nucleus

Ze
1
1
1
H 
 



Xe
2
2
2
X
2
XN
H total

4 0 r

me
mN
1
me
2 2
2 2
Ze2

cm 
internal 
2m
2
4 0 r
Consider only the internal, relative
coordinates
2 2
Ze2

 
  E
2
4 0 r
4
Hydrogenic Wavefunction

The wavefunction for hydrogenic
atom is separable into radial and
angular components.
r, ,   R(r )Y  , 
2  2 2  1 2 
 2 

 2   RY  VˆRY  ERY
2  r
r r r

 2   2 R 2Y R R 2  ˆ
 Y 2 

 2  Y   VRY  ERY
2  r
r r r

r2
multiply through by
RY
2  2 d 2R
dR  ˆ 2  2 2
2
 r



2
r

V
r


Y

E
r
2R  dr2
dr 
2Y
constant
2 2
 2l (l  1)

Y 
2Y
2
2Y  l (l  1)Y
Spherical harmonics*
 2  d 2 R 2 dR  ˆ
 2 
  Veff R  ER

2   dr
r dr 
Vˆeff
Ze 2
l (l  1) 2


4 0 r
2 r 2
Radial Wave Equation
5
The Radial Solutions

The effective potential
Vˆeff
Ze2
l (l  1) 2


4 0 r
2r 2
Centrifugal
energy
Coulombic
energy

The allowed energy

The radial wavefunctions are in form
of

R
(
r
)

N
(  )er) x (decaying
R(r) = (polynomial
  L in
n
exponential
in
r)
4 
2 Zr

a 
Z 2 e 4
En  
32 2 02 2 n 2
l
  / 2
n ,l
n ,l
n ,l
2
Associated
Laguerre
polynomial
0
a0
0
me e 2
Bohr radius = 52.9177 pm
6
orbital
1s
2s
n
1
2
Hydrogenic Radial
Wavefunctions
l
R
n,l
0
0
2p
2
1
3s
3
0
3p
3d
3
3
Z
2 
 a0 
1
2
1 Z
 
2 2  a0 
3/ 2
3/ 2
e / 2
1   / 4

 2   e
2 

1 Z
 
4 6  a0 
1 Z
 
9 3  a0 
3/ 2
3/ 2
e / 4
1 2   / 6

 6  2    e
9 

1 Z
 
27 6  a0 
3/ 2
1   / 6

4

  e

3 

1 Z
 
81 30  a0 
3/ 2
 2e   / 6
7
The Radial Wavefunctions
1
Zr/a0
2
2s
0
5
Zr/a0
0
3
R/(Z/a0)3/2
0
3s
10
15
7.5
12
Zr/a0
2p
0
5
Zr/a0
22.5
3p
0
7.5
10
15
12
Zr/a0
R/(Z/a0)3/2
1s
R/(Z/a0)3/2
R/(Z/a0)3/2
R/(Z/a0)3/2
The radial wavefunction of
hydrogenic atoms (Z)
R/(Z/a0)3/2

22.5
3d
0
7.5
12
Zr/a0
22.5
8
Example

A 1s-electron with n = 1, l = 0, ml = 0
 1,0,0

Z
 R1, 0 (r )Y0, 0 ( ,  )  2 
 a0 
3/ 2
1/ 2
 1 
e / 2 

 4 
At r = 0
Z
 1,0,0 (0, ,  )  R1,0 (0)Y0,0 ( ,  )  2 
 a0 
• The probability density
 Z3
 1, 0 , 0 (0, ,  )   3
 a0
2
When Z=1
3/ 2
1/ 2
 1 


4






 2 (0, ,  )  2.15106 pm-3
1, 0 , 0
9
Atomic Orbitals and Their
Energies
 An atomic orbital (AO) is a one

electron wavefunction for an electron
in an atom
 (0,is
,  ) defined by1,0n,
,0
Each hydrogenic AO
l, and ml
An electron described by
is
in the state
and is said to
occupy the orbital with n=1, l=0 and
Z e
1
ml=0
E 

32   n
n
Electron in an orbital with quantum
number n has an energy given by
1, 0, 0
2
n

2
4
2
0
2
2
2
10
The Energy Levels
The energy level of H atom

Energy

Infinite separation (H++e-)
3
Z 2 e 4
En  
32 2 02 2 n 2
2
Bound State : E is negative
Unbound State: E is positive
H e4
hcRH 
32 2 02  2
H
1
me e 4
RH 
R R 2 3
me
8 0 h c
Rydberg Constant for H
Rydberg Constant
11
Ionization Energies


The ionization Energy, IE, is the
minimum energy required to remove
an electron from the ground state of
one of its atoms.
hcR
 e
HydrogenEatom,
the

  ground state
n
32  
has n = 1 E  0
4
H
1
2
H
2
2
0
2

IE  E  E1  hcRH
• Ionization energy of H atom is 2.179 x
-18
12
Shells and Subshells
All the orbitals of a given value of n are said to
form a single shell of the atom
• n=

1
K
2
L
3
M
4
N
…
…
0
s
1
ps
2
df
3
f
4
g
The orbital with the same value of n but
different values of l are said to form a subshell
of a given shell
• l=
n

Energy

p
d
[1]
[3]
[5]
[1]
[3]
g
h
5
h
…
…
4
3
2
1
[1]
13
Curvatures and Energy

The hamiltonian operator
 d
Hˆ  Eˆ K  Vˆ  
2
2
2m dx2
 Vˆ
• The sharply curved function corresponds to
a higher EK (and a lower V) than the less
sharply curved function
high EK
high EK
low EK
low EK
kinetic E
E

Hydrogenic
Ze atom
l (l  1)
ˆ
V 

2
eff
4 0 r
2
2r 2
Effective Potential Energy
potential E
l0
l=0
Radius, R
14
s-orbital
• s orbital is spherically
symmetrical
• The ground
state of
1
 
e
hydrogenic
atom is electron in
a 
2 Zr
1 Z 
1 


1s orbital
   2   e

a
a
2
R(r)

Atomic Orbitals
 r / a0
1s
3 1/ 2
0
1s
3s
3/ 2
 / 4
2s
2 2 0 

3/ 2
1 Z 
1
   6  2    2 e   / 6
 3s
9 
9 3  a0  
0
2s
radius
 (r )  0
 (r )
• A radial node is where
• A probability density of
electron is
• A simple way to show the
2
15
The Mean radius of an
orbital
 The mean radius of a 1s orbital
2
r   rd    rd
*
  Rn ,l (r )Yl ,m ( ,  ) d  r 2 dr sin dd
l
r 


2
0
0

0
2
2
r Rn ,l Yl ,ml r 2 dr sin dd
• The angular part is normalized

2
0
0

2
Yl ,m sin dd  1
• The mean radius of an orbital is a
function ofr r  R r dr

0
2
3
n ,l
1/ 2
 Z   Zr / a0
R1, 0  2 3  e
 a0 
3a0
4 Z 3  3  2 Zr / a0
r  3  re
dr 
0
a0
2Z
3
16



 d
2
Radial Distribution
Functions
is the probability in finding
d
electron
in a region
Radial Distribution Function P(r)
is the probability density at
radius r of all direction
P(r)dr is the probability of
finding electron in between the
(r )dr   4r dr
shell orPradius
r and r+dr
2
2
• For spherically symmetric orbital
P(r )dr    R(r ) Y ( ,  ) r dr sin dd
 r R(r ) dr  Y ( ,  ) sin dd
• In General
2

2
0
0
2
2
2

2
0
0
r
2
2
 r 2 R(r ) 2 dr
17
The probability density
  e 2 Zr / a
2

0
0.8
The radial distribution P(r)
of 1s orbital
4Z 3 2 2 Zr / a0
P( r )  3 r e
a0

0.9
The most probable radius
(r*)
dP(r ) 4Z 
2Zr 
dr

3
2
 2r 
e
a 
a0 
a
r*  0
Z
3
0
 2 Zr / a0
0
0.7
The most probable
radius of 1s
0.6
P/(Z/a0)3

0.5
0.4
0.3
P(r)
0.2
(r)2
0.1
0
0
1
2
r/a0
3
4
18
p orbitals

A p electron has nonzero orbital angular
momentum
(l  0)
• p orbital has zero amplitude at r = 0
• The centrifugal effect (l >0) tend to put
electron away from the nucleus
p
 p  R2,1 (r )Y1,0 ( ,  )
0
Z
 

4(2 )1/ 2  a0 
 r cosf (r )
 z  zf (r )
1
1
5/ 2
r cose  Zr / 2 a0
1 Z
 R2,1 (r )Y1, 1 ( ,  )   1/ 2  
8  a0 
5/ 2
re  Zr / 2 a0 sin e i
1
 i
r
sin

e
f (r )
1/ 2
2
1
  1/ 2 ( p1  P1 )  r sin  cosf (r )  xf (r )
2
1
  1/ 2 ( p1  P1 )  r sin  cosf (r )  yf (r )
2

p
x
p
y
19
d-orbitals

d orbitals with opposite values of ml
may be combined in pairs to give
real standing waves
d xy  xyf (r )
d yz  yzf (r )
d zx  zxf (r )


1 2
x  y 2 f (r )
2
d z 2  1 / 2 3 3Z 2  r 2 f ( r )
d x2  y2 



20
 r, ,    R(r )Y  ,  

Radial function R(r)

Azimuth function Y(,)
21
Structures of many-electron
atoms
 The Schrödinger equation for many

electron atom is highly complicated
No analytical expression for the
orbitals and energies can be given.
Several approximations are needed
22
The Orbital Approximation

Wavefunction of a many-electron
 (r , r ,)
atom is a function of coordinates
of
all the electrons
where ri
is the vector from the nucleus to
electron i. ψ(r , r ,)  (r ) (r )
The orbital approximation:
1
1

2
1
2
2
• The orbitals resemble the hydrogenic
orbitals
• Each electron occupies its own orbital
• No
interactions
between
electrons
is
2p (3)
2p (4)

1s(1)
2s(2)
accounted
z
x
23

The orbital approximation would be
exact if there is no interactions
between electrons.
• The hamiltonian
of non-interacting
2H H
H
Hψr , r   H  H  r  r 
electron system
1
1
2
1
2
2
1
2
 H1 r1  r2   H 2 r1  r2 
  r2 H1 r1    r1 H 2 r2 
  r2 E1 r1   r1 E2 r2 
 E1  E2  r1  r2 
 E r1  r2 
• Total energy is the sum of each
24
Many-Electron Atoms

The orbital approximation allows us
to express the electronic structure of
an atom by reporting its
configuration
Electronic configuration: the list of
occupied orbitals
He atom (Z=2)

The Pauli exclusion principle


• 1st and 2nd electrons are in a 1s
hydrogenic orbital
• The orbital is more compact than in H
atom
25
Pauli Principle

General statement
• When the labels of any two identical
fermions are exchanged, the total
wavefunction changes sign.
• When the labels of any two identical
bosons are exchanged, the total
wavefunction
retains
the
same
sign.
ψ
(
r
,
r
,
r

)


ψ
(
r
,
r
,
r

)
Electrons are
2
fermions

1
3
1
2
3
ψ(r2 , r1 , r3 )  ψ(r2 , r3 , r1 )
Total wavefunction = Spatial
Wavefunction x Spin
ψ (i)
 (i),  (i)
26

Consider possible spins for 2electron system
(1) (2several
),  (1) (2), possibilities
 (1) (2),  (1) (2) for two
• Thereare
spins
• Electrons are
so if
1,2  indistinguishable
 (1) (2)   (1) (2)
electrons have
we
1,2  different
 (1) (2)   (1)spins,
(2) 
cannot tell which electron is in which
orbital
 (1) (2) (1) (2)

1
2

1
2
 (1) (2)  (1)  (2)
 (1) (2)  (1,2)
 (1) (2)  (1,2)
• The total-wavefunctions of the systems
are
27

According to Pauli principle, the
wavefunction is acceptable if it
changes sign when the electrons are
 (1) (2)   (2) (1)
exchanged
 (1) (2)   (2) (1)
symmetric if both  are the same
symmetric
 (1)  (2)   (2)  (1) symmetric
  1,2  12  (1)  (2)   (1) (2)    2,1  12  (2) (1)   (2) (1)
  1,2 

1
2
 (1) (2)   (1) (2)
   2,1 
1
2
symmetric
 (2) (1)   (2) (1) anti-symmetric
The acceptable wavefunction for 2
 (1) (2) (spatial
1,2)
electrons in ψ(1,2)
the same
( )
orbital is

28

Electron exchange
•
•
•
 r1 , r2 , r3 , r4   r1 , r4 , r3 , r2    r1 , r2 , r3 , r4 
  r1 , r3 , r4 , r2   ?
  r2 , r3 , r4 , r1   ?
 r1 , r2    a r1  b r2   r2 , r1    a r2  b r1 
 r1 , r2    r2 , r1 ?
  r1 , r2    a r1  b r2   a r2  b r1    r2 , r1   ?
  r1 , r2    a r1  b r2   a r2  b r1    r2 , r1   ?
29
Shielding


The subshell orbitals with the same
n are not degenerate in manyelectron system
Shielding Effect
Electron at a distance r from nucleus
experiences a repulsion from other
electron that can reduce the positive
Orbital
Zeff
charge of the nucleusElement
Z to ZZ eff (the
He
2
1s
1.69
effective nuclear charge)
+Z
No net effect of these
electrons
Net effect equivalent to a
point charge at the center
Z eff  Z  

= shielding constant
C
6
1s
2s
2p
5.67
3.22
3.14
30
Penetration


The shielding constant is different for s
and p electrons because they have
different radial distribution.
s-electrons has a greater penetration
through inner shells than a p electron.
The energies of subshells in a manyelectron atom in general lie in the order
s<p<d<f
Radius Distribution function, P

3p
radius
3s
31

Li atom (Z=3)

The electrons in the outermost shell
of an atom in its ground state are
• The first two electron occupy a 1s
orbital
• The third electron cannot enter the 1s
orbital (Pauli exclusion) and must
occupy the next available orbital (n=2)
• According to the shielding effect, 2s
and 2p are not degenerate and 2s
orbital is lower in energy than the three
2p orbitals.
• The ground state configuration of Li is
1s2 2s1
32
Aufbau Principle


Aufbau (building up) principle
proposes an order of occupation of
the hydrogenic orbitals that accounts
for the ground-state configurations of
neutral atoms
The occupation is
1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s
…
• Each subshell consists of different
number of orbitals
• Each orbital may accommodate up to
2 electrons
33

Aufbau principle
• Electrons occupy different orbitals of a
given subshell before doubly
occupying any one of them.
 Electrons have a tendency to stay away
from each others.

Hund’s maximum multiplicity rule
• An atom in its ground state adopts a
configuration with the greatest number
of unpaired electrons.
 Electrons with the same spin have
electron correlation effect that make them
34

Suppose e1 and e2 are described by
a(r1) and b
(r2) (r ) (r )
a
1
b
e– is specified by its position
2
 (ridentical
  are
) (r )  (r )
• Electrons
1
2

a
1
b
2
a
2
b
(r1 )
• Pauliprinciple
under
asymmetric spin
  (r ) ((asymmetry
r )  (r ) (r )   needs
particle
 (r ) (r )  (r ) (r )  needs symmetric spin
  interchange)

1
2
a
1
b
2
a
2
b
1

1
2
a
1
b
2
a
2
b
1
  0
There is zero probability of finding 2
electrons at the same point in space
when they have parallel spins.
0
 if r1 = r2 (e1 and
e2 are at the same point)
Why?
35

• Energy of 3d is lower than 3s
• Sc: [Ar] 3d1 4s2 (spectroscopy)
due to strong
Energy


Ne: 1S2 2S2 2P6 = [Ne]
closedshell
Na: 1S2 2S2 2P6 3S1 = [Ne] 3S1
Ar: 1S2 2S2 2P6 3S2 2P6 closedshell (no e- in 3d)
Sc – Zn (21-30)
Energy

3d1 4s2
electrons repulsion
3d1 4s2
36
The Configurations of Ions

Cations
• Electrons are removed from the
ground-state configuration of the
neutral atom in a specific order.
• Electrons in the outer-most shell would
be removed first.
due to the different Z s
 V = [Ar] 3d3 4S2 (23 e-)
 Sc = [Ar] 3d1 4S2 (21 e-)
 V2+
= [Ar] 3d3 4S0 (21 e-)

eff
Anions
• Continuing the building up procedure
and adding electrons to the neutral
atom.
37
Ionization Energies &
Electron
Affinities
st
 1 Ionization Energy: the minimum


energy necessary to remove an
electron from a many-electron atom
in the gas phase.
2nd Ionization Energy: the minimum
energy necessary to remove a
second electron from the singly
charged cation.
The Electron Affinity: The energy
released when an electron attaches
to a gas-phase atom.
38
Electron-Electron
Interactions
 The potential energy of the electrons
in many-electron atom is
Ze2
1
e2
V  
 
2 i  j 4 0 rij
i 4 0 ri
Ze2
e2
V  

i 4 0 ri
i  j 4 0 rij

The Hamiltonian of electrons
2
H 
2me
kinetic
Ze2
1
e2
i   4 i r  4 
0
i
0 i  j rij
2
i
1
e-n attraction
e-e repulsion
• Kinetic energy of a nucleus is omitted.
39
Self-Consistent Field
Orbitals
 The Hartree-Fock Self-Consistent
Field (HF-SCF) hydrogenic
Theory
orbitals
3
• The wave
function
of
many-electron
r
   1 (r1 ) 2 (r2 ) n (rn )
r
system
2
3
2
r1
1
• Focus on electron 1 and regard
electrons
e

V

d
2, 3 ,4 … as being
smeared
out to form

4 r
a
ee 

V 
d


e
d


4
r
r
static distribution
of electric
charge ()
1
2
12
2
2
1 2
0
12
2
12
'2
*
2
2
0
12
2
2
12
 The potential energy of electron 1 due to
electron 2
40
Hartree-Fock Equation

The Hamiltonian for electron 1
H1  

 *j j
2
 2 Ze'
1 
  e' 2 
d j
2me
r1
r1 j
j 1
The Schrödinger equation of
H  E 
electron 1
1
1
1
E
The total energy
ofE n-electron
system E   E    e'2 (i)  ( j) d d
n

i
i 1
n 1
n
i 1
n
i
n

i 1 j i 1
n 1
2
2
i
j
rij
i
j
n
E   Ei    J ij
i 1
i 1 j i 1
coulomb integral
41
Hermitian Operator & Dirac
Notation
P
(
r
)

ψ
(
r
)
ψ
(
r
)
 Probability:
*
1   ψ*ψd  ψ ψ

Eigen Value:A   ψ Aˆ ψd 
*
ψ Aˆ ψ
* ˆ
ˆ ψ * ψd 
ψ
A
ψ
d


A



Overlap integral:
 ψ ψ d 

Schrödinger Equation:
*
i
j
Dirac notation
Hermitian operator
ψ i ψ j   ij
ˆ  d   * E  d  E  * d  E
*H

i
i
i i
i
i

 i i i
 i Hˆ  i  Ei  i  i  Ei
42
Slater Determinants

Consider the ground state of He
not satisfy antisymmetric
2



(
1
)

(
2
)

1
s
(
1
)

(
1
)
1
s
(
2
)

(
2
)
atom (1s )
requirement
•

(anti-symmetric-satisfying
wave fn) the wave
function can be written
1 1s(1) (1) 1s(1) (1)
   (1) (2) 
in the determinant
(2) (2) 1s(2) (2)
2 1sform
1
• Ground
atom
1s(1) (of
  state
1)1s(2He
)  (2) 
1s(2) (2)1s(1)  (1) 
Slater Determinant
2
1s(1)
   (1) (2) (3) 
1s(1)
2s(1)
1
1s(2) 1s(2) 2s(2)
3
1s(3) 1s(3) 2s(3)
• Ground state of Li atom (1s2 2s1)
-spin
-spin
43

Variation Treatments of the Li
Ground State
Applying the Variational method for
the Li atom
1s(1)   1, 0, 0 (1) (1)
• The ground
state of Li atom
1
1s(1)

6
1s(1)
2s(1)
1s(2) 1s(2) 2s(2)
1s(2)   1, 0, 0 (2)  (2)
1s(3) 1s(3) 2s(3)
2s(1)   2, 0, 0 (3) (3)
n with
• The trial
functions
(
wavef
1 b 
  e


 a 
shielding effect)
3/ 2
1
1, 0 , 0
 2, 0, 0 
1/ 2
 b1r / a0
0
 b2 
 
1/ 2 
42   a0 
1
3/ 2

b r
 2  2 e b2 r / 2 a0
a0 

b1 & b2 are the variational parameters representing the effective
nuclear charge for the 1s and 2s electron, respectively.
44
Variational Method

The Variational Theorem: if  is
normalized and satisfied all the
energy
of the then
conditions of the
system
 * Hˆinterested
 d  E1
ground state

 Hˆ  d
• For any trial function

E
   d
*
*

1
Variational theorem allows us to
calculate an upper
H   bound for the
system’s ground
  state
  E energy
1
Trial fn. Real fn.
45
Perturbation Theory*

The Hamiltonian of the complicated
system can be considered as a sum
of simple Hamiltonian with the
perturbation
 d
1
•

H  H with
H'
Hamiltonian
0
2
2
Perturbation
H 
0
 kx 2
2m dx 2 2
(1)
2 ( 2)
Wave functions
and
energies
can
be
 n   n(0)  





n
n
expressed
)  power
( 2)    Eform
( 0)  E (1)
En in
En(0a
En(1)  2 Eseries
n
n
n
 E the
E first-order
E   H' 
d
• EnergyE with
correction
(=1)
n
(0)
n
(1)
n
(0)
n
( 0 )*
n
(0)
n
46
Key Ideas
Electronic
Many-electron
structures
•Hydrogenic atoms
(an electron with a
positive charged ion)
•Many-electron
atoms (interaction
between electrons)
Hydrogenic
atom
•Orbital
wavefunctions
Radial R(r) and
Azimuth Y(,)
functions
atom
•Orbital
approximation
•Electronic
configuration
Pauli exclusion
Hund’s maximum
multiplicity
•Orbital Energies
•Self consistent field
approx.
48