physical chemistry iii (01403342) chapter 3: atomic structure
DESCRIPTION
Physical Chemistry III (01403342) Chapter 3: Atomic Structure. 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 - PowerPoint PPT PresentationTRANSCRIPT
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Physical Chemistry III (01403342)
Chapter 3: Atomic StructurePiti Treesukol
Kasetsart UniversityKamphaeng Saen Campus
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Electronic Structures of Atoms Hydrogenic atoms
Many-electron atoms The orbital approximations Self-consistent Field orbitals Approximation Methods• Variation Method• Perturbation Method
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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
The Hamiltonian for the electron and a nucleus
r
ZeV
0
2
4
r
Ze
mm
VEEH
NN
ee
nucluesKelectronK
0
22
22
2
,,
422
ˆˆˆ
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The Hamiltonian for the internal motion of electron relative to the nucleus
Consider only the internal, relative coordinates
eNe mmmr
ZeH
1111
42 0
22
2
X
Xe
XN
r
Ze
mH internalcmtotal
0
22
22
2
422
Er
Ze
0
22
2
42
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Hydrogenic Wavefunction The wavefunction for
hydrogenic atom is separable into radial and angular components.
,)(,, YrRr
222
22
22
2
222
22
222
22
2ˆ2
2
ˆ2
2
ˆ12
2
rYY
rdr
dRr
dr
Rdr
R
RYRYYr
R
r
R
r
Y
r
RY
RYRYRYrrrr
E
E
E
V
V
V
RY
r 2
multiply through by
YllY
llY
Y
)1(
2
)1(
22
22
2
constant
2
2
0
2
2
22
2
)1(
4ˆ
ˆ2
2
r
ll
r
Ze
RRdr
dR
rdr
Rd
eff
eff
V
V E
Spherical harmonics* Radial Wave Equation
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The Radial Solutions The effective potential
The allowed energy
The radial wavefunctions are in form ofR(r) = (polynomial in r) x (decaying exponential in r)
2
2
0
2
2
)1(
4ˆ
r
ll
r
Ze
effV
Coulombic energy
Centrifugal energy
2220
2
42
32 n
eZEn
2
20
00
2/,,,
42
)()(
ema
a
Zr
eLn
NrR
e
ln
l
lnln
Associated Laguerre polynomial
Bohr radius = 52.9177 pm
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orbital n l Rn,l
1s 1 0
2s 2 0
2p 2 1
3s 3 0
3p 3 1
3d 3 2
Hydrogenic Radial Wavefunctions
2/
2/3
0
2
e
a
Z
4/
2/3
0 2
12
22
1
e
a
Z
4/
2/3
064
1
e
a
Z
6/2
2/3
0 9
126
39
1
e
a
Z
6/
2/3
0 3
14
627
1
e
a
Z
6/2
2/3
03081
1
e
a
Z
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The Radial Wavefunctions The radial wavefunction of
hydrogenic atoms (Z)
R/(
Z/a
0)3
/2
Zr/a0
0 1 2 3
R/(
Z/a
0)3
/2
Zr/a0
0 7.5 12 22.5
R/(
Z/a
0)3
/2
Zr/a0
0 5 10 15
R/(
Z/a
0)3
/2
Zr/a0
0 7.5 12 22.5
R/(
Z/a
0)3
/2
Zr/a0
0 7.5 12 22.5
R/(
Z/a
0)3
/2
Zr/a0
0 5 10 15
1s
2s
3s
2p
3p
3d
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Example A 1s-electron with n = 1, l = 0, ml
= 0
At r = 0
• The probability density
2/12/
2/3
00,00,10,0,1 4
12),()(
e
a
ZYrR
2/12/3
00,00,10,0,1 4
12),()0(),,0(
a
ZYR
30
32 ),,0(
0,0,1 a
Z
pm 1015.2),,0( -362
0,0,1
When Z=1
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Atomic Orbitals and Their Energies An atomic orbital (AO) is a one-
electron wavefunction for an electron in an atom
Each hydrogenic AO is defined by n, l, and ml
An electron described by is in the state and is said to occupy the orbital with n=1, l=0 and ml=0
Electron in an orbital with quantum number n has an energy given by
• Different states with the same n are degenerate
),,0(0,0,1 0,0,1
22220
2
42 1
32 nn
eZEn
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The Energy Levels The energy level of H atom
2220
2
42
32 n
eZEn
1
2
3
Infinite separation (H+
+e-)
Energ
y
Bound State : E is negative
Unbound State: E is positive
ch
emRR
mR
ehcR
e
e
HH
HH
320
4
220
2
4
8
32
Rydberg Constant for H
Rydberg Constant
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Ionization Energies The ionization Energy, IE, is the
minimum energy required to remove an electron from the ground state of one of its atoms.
Hydrogen atom, the ground state has n = 1
• Ionization energy of H atom is 2.179 x 10-18 J or 13.60 eV
H
HH
hcREEIE
E
e
n
hcRE
1
220
2
4
21
0
32
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Shells and Subshells All the orbitals of a given value of n are
said to form a single shell of the atom• n = 1 2 3 4 …
K L M N … The orbital with the same value of n but
different values of l are said to form a subshell of a given shell• l = 0 1 2 3 4 5 …
s p d f g h …n
1
2
3
4
s p d f g h
[1]
[1] [3]
[1] [3] [5]
En
erg
y
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Curvatures and Energy The hamiltonian operator
• The sharply curved function corresponds to a higher EK (and a lower V) than the less sharply curved function
Hydrogenic atom2
2
0
2
2
)1(
4ˆ
r
ll
r
Ze
effVl = 0
l 0
Eff
ecti
ve P
ote
nti
al En
erg
y
Radius, R
Vdx
d
mVEH K
ˆ2
ˆˆˆ2
22
high EK
low EK
high EK
low EKkinetic E
potential E
E
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Atomic Orbitals s-orbital• s orbital is spherically
symmetrical• The ground state of
hydrogenic atom is electron in 1s orbital
• A radial node is where• A probability density of
electron is• A simple way to show the
boundary surface (high proportion of the electron probability; 90%)
0/
2/130
1
1 ars e
a
R(r
)
radius
1s
2s
3s4/
2/3
02 2
12
22
1
e
a
Zs
6/2
2/3
03 9
126
39
1
e
a
Zs
0
2
a
Zr
0)( r2
)(r
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The Mean radius of an orbital The mean radius of a 1s orbital
• The angular part is normalized
• The mean radius of an orbital is a function of r
0 0
2
0
22
,
2
,
2,,
2*
sin
sin),()(
dddrrYRrr
dddrrdYrR
rddrr
mllln
mlln l
0
2
0
2
, 1sin ddY ml
Z
adrer
a
Zr
ea
ZR
drrRr
aZr
aZr
ln
2
34
2
0
0
/2330
3
/
2/1
30
3
0,1
0
32
,
0
0
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Radial Distribution Functions is the probability in
finding 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 shell or radius r and r+dr• For spherically symmetric
orbital
• In General
d2
drrdrrP 22 4)(
drrRr
ddYdrrRr
dddrrYrRdrrP
22
0
2
0
222
0
2
0
222
)(
sin),()(
sin),()()(
r
d
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The probability density
The radial distribution P(r) of 1s orbital
The most probable radius (r*)
0/2230
34)( aZrer
a
ZrP
0/22 aZre
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 1 2 3 4
P/(
Z/a
0)3
r/a0
The most probable radius of 1s
P(r)(r)2
Z
ar
ea
Zrr
a
Z
dr
rdP aZr
0*
/2
0
2
30
3
02
24)(
0
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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
)(
)(cos
cos)2(4
1
),()(
0
0
2/
2/5
02/1
0,11,2
rzf
rfr
era
Z
YrR
z
aZr
p
)()(cossin)(2
1
)()(cossin)(2
1
)(sin2
1
sin8
1),()(
112/1
112/1
2/1
2/
2/5
02/11,11,2
0
1
ryfrfrPp
rxfrfrPp
rfer
erea
ZYrR
y
x
p
p
i
iaZrp
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d-orbitals d orbitals with opposite values
of ml may be combined in pairs to give real standing waves
)(332/1
)(2
1
)(
)(
)(
22
22
2
22
rfrZd
rfyxd
rzxfd
ryzfd
rxyfd
z
yx
zx
yz
xy
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Radial function R(r) Azimuth function Y(,)
21
,)(,, YrRr
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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
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1s(1) 2s(2) 2pz(3) 2px(4)
The Orbital Approximation Wavefunction of a many-
electron atom is a function of coordinates of all the electrons where ri is the vector from the nucleus to electron i.
The orbital approximation:
• The orbitals resemble the hydrogenic orbitals• Each electron occupies its own orbital• No interactions between electrons is accounted
),,( 21 rr
)()(),,(ψ 2121 rrrr
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The orbital approximation would be exact if there is no interactions between electrons.• The hamiltonian of non-interacting 2-electron system
• Total energy is the sum of each electron’s energy
21 HHH
21
2121
221112
221112
212211
212121
,ψ
rrE
rrEE
rErrEr
rHrrHr
rrHrrH
rrHHrrH
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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)• 1st and 2nd electrons are in a 1s hydrogenic orbital• The orbital is more compact than in H atom
The Pauli exclusion principleNo more than two electrons may occupy any given orbital and, if two do occupy one orbital, then their spins must be paired.
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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.
Total wavefunction = Spatial Wavefunction x Spin
),,(ψ),,(ψ
),,(ψ),,(ψ
132312
321312
rrrrrr
rrrrrr
Electrons are
fermions
)(),( )( iiiψ
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Consider possible spins for 2-electron system• There are several possibilities for two spins
• Electrons are indistinguishable so if electrons have different spins, we cannot tell which electron is in which orbital
• The total-wavefunctions of the systems are
)2()1( ),2()1( ),2()1( ),2()1(
)2()1()2()1(2,1
)2()1()2()1(2,1
21
21
)2,1()2()1(
)2,1()2()1(
)2()1()2()1(
)2()1()2()1(
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According to Pauli principle, the wavefunction is acceptable if it changes sign when the electrons are exchanged
The acceptable wavefunction for 2 electrons in the same spatial () orbital is
)1()2()1()2(1,2
)1()2()1()2(1,2
21
21
)2()1()2()1(2,1
)2()1()2()1(2,1
21
21
symmetric
anti-symmetric
)2()1(
)2()1(
)2()1(
)1()2(
)1()2(
)1()2(
symmetric
symmetric
symmetric if both are the same
)2,1()2()1(ψ(1,2)
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Electron exchange•
•
•
29
?,,,
?,,,
,,,,,,,,,
1432
2431
432123414321
rrrr
rrrr
rrrrrrrrrrrr
?,,
,,
1221
12122121
rrrr
rrrrrrrr baba
?,,
?,,
12122121
12122121
rrrrrrrr
rrrrrrrr
baba
baba
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Shielding The subshell orbitals with the
same n are not degenerate in many-electron system
Shielding Effect Electron at a distance r from nucleus experiences a repulsion from other electron that can reduce the positive charge of the nucleus Z to Zeff (the effective nuclear charge)Net effect equivalent to
a point charge at the center
No net effect of these electrons
ZZeff
= shielding constant
Elemen
t
Z Orbital
Zeff
He 2 1s 1.69
C 6 1s 5.67
2s 3.22
2p 3.14
+Z
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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 many-electron atom in general lie in the order s < p < d < f
Rad
ius
Dis
trib
uti
on
funct
ion
, P
radius
3s3p
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Li atom (Z=3)• 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
The electrons in the outermost shell of an atom in its ground state are called the valence electrons and others are called core electrons.
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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 is1s 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• This order is approximately the order of energies of the individual orbitals.• The electron-electron repulsion could have an effect on this order.
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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 stay well apart, which reducing the repulsion.
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Suppose e1 and e2 are described by a(r1) and b(r2)
• Electrons are identical
• Pauli principle (asymmetry under particle interchange)
if r1 = r2 (e1 and e2 are at the same point)
)()( 21 rr ba
)()()()( 122121 rrrr baba
)()()()(
)()()()(
122121
122121
rrrr
rrrr
baba
baba
needs asymmetric spin needs symmetric spin
e– is specified by its position
0
0
There is zero probability of finding 2 electrons at the same point in space when they have parallel spins.
Why?
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Ne: 1S2 2S2 2P6 = [Ne] closed-shell
Na: 1S2 2S2 2P6 3S1 = [Ne] 3S1 Ar: 1S2 2S2 2P6 3S2 2P6 closed-
shell (no e- in 3d) Sc – Zn (21-30) • Energy of 3d is lower than 3s• Sc: [Ar] 3d1 4s2 (spectroscopy)
Energ
y
3d1 4s2
Energ
y
3d1 4s2
due to strong electrons repulsion
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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.V = [Ar] 3d3 4S2 (23 e-)Sc = [Ar] 3d1 4S2 (21 e-)V2+ = [Ar] 3d3 4S0 (21 e-)
Anions• Continuing the building up procedure and adding electrons to the neutral atom.
due to the different Zeffs
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Ionization Energies & Electron Affinities 1st 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.
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Electron-Electron Interactions The potential energy of the
electrons in many-electron atom is
The Hamiltonian of electrons
• Kinetic energy of a nucleus is omitted.
ji iji i r
e
r
ZeV
0
2
0
2
42
1
4
ji iji i r
e
r
ZeV
0
2
0
2
44
ji iji ii
ie r
e
r
Ze
mH
2
0
2
0
22
4
1
4
1
2
kinetic e-n attraction e-e repulsion
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Self-Consistent Field Orbitals
212
2
0
112 4
dr
eV
)()()( 2211 nn rrr hydrogenic orbitals
r1
1
r22
r3
3
212
2*22'
212
2
2
0
2112 4
d
red
r
eeV
The Hartree-Fock Self-Consistent Field (HF-SCF) Theory• The wave function of many-electron system
• Focus on electron 1 and regard electrons 2, 3 ,4 … as being smeared out to form a static distribution of electric charge ()The potential energy of electron 1 due to electron 2
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Hartree-Fock Equation The Hamiltonian for electron 1
The Schrödinger equation of electron 1
The total energy of n-electron system
1 1
*2
1
2211 '
'
2 jj
j
jj
e
dr
er
Ze
mH
111 EH
n
iiEE
1
1
1 11
1
1 1
22
1
)()(2'
n
i
n
ijij
n
ii
ji
n
i
n
ij ij
jin
ii
JEE
ddr
jieEE
coulomb integral
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Hermitian Operator & Dirac Notation Probability:
Eigen Value:
Overlap integral: Schrödinger Equation:
ψˆψψˆψ* AdAA
)(ψ)(ψ)( * rrrP
ψψψψ1 * d
ijd jij*i ψψψψ
iiiiii EH Eˆ
iiiiiiiii EddEdH *** Eˆ
Dirac notation
dAdA ψψˆψˆψ ** Hermitian operator
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Slater Determinants Consider the ground state of He
atom (1s2)•
Slater Determinant (anti-symmetric-satisfying wave fn) the wave function can be written in the determinant form• Ground state of He atom
• Ground state of Li atom (1s2 2s1)
not satisfy antisymmetric requirement
)2()2(1)1()1(1)2()1( ss
)2()2(1)2()2(1
)1()1(1)1()1(1
2
1)2()1(
ss
ss
)1()1(1)2()2(1)2()2(1)1()1(12
1 ssss
)3(2)3(1)3(1
)2(2)2(1)2(1
)1(2)1(1)1(1
3
1)3()2()1(
sss
sss
sss
-spin -spin
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Variation Treatments of the Li Ground State Applying the Variational method
for the Li atom• The ground state of Li atom
• The trial functions (wavefn with shielding effect)
)3(2)3(1)3(1
)2(2)2(1)2(1
)1(2)1(1)1(1
6
1
sss
sss
sss
)3()3()1(2
)2()2()2(1
)1()1()1(1
0,0,2
0,0,1
0,0,1
s
s
s
02
01
2/
0
2
2/3
0
22/10,0,2
/
2/3
0
1
2/10,0,1
224
1
1
arb
arb
ea
rb
a
b
ea
b
b1 & b2 are the variational parameters representing the effective nuclear charge for the 1s and 2s electron, respectively.
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Variational Method The Variational Theorem: if
is normalized and satisfied all the conditions of the interested system then
• For any trial function
Variational theorem allows us to calculate an upper bound for the system’s ground state energy
H
1* ˆ EdH
1*
* ˆE
d
dH
energy of the ground state
1E
Trial fn. Real fn.
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Perturbation Theory* The Hamiltonian of the
complicated system can be considered as a sum of simple Hamiltonian with the perturbation• Hamiltonian with Perturbation
Wave functions and energies can be expressed in a power series form
• Energy with the first-order correction (=1)
'HHH 0
)1()0()2(2)1()0(nnnnnn EEEEEE
)2(2)1()0(nnnn
22
220
2
1
2kx
dx
d
mH
dHEEEE nnnnnn)0()*0()0()1()0( '
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Key IdeasElectronic structures•Hydrogenic atoms (an electron with a positive charged ion)•Many-electron atoms (interaction between electrons)
Hydrogenic atom•Orbital wavefunctionsRadial R(r) and Azimuth Y(,) functionsSeparation of variables
•Orbital Energies•Radial distribution
Many-electron atom•Orbital approximation•Electronic configurationPauli exclusionHund’s maximum multiplicity
•Orbital Energies•Self consistent field approx.