1

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

3

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

4

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

5

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

6

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

7

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

8

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

9

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

12

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

13

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

14

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

16

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

17

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

18

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

19

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

20

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

Radial function R(r) Azimuth function Y(,)

21

,)(,, YrRr

22

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(

28

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)

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

32

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.

33

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.

34

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.

35

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?

36

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

37

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

38

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.

39

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

40

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

41

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

42

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

43

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

44

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.

45

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.

46

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( '

47

48

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.