Physical Chemistry and Structural Chemistry

1

Zoltán Rolik

2018 fall

Zoltán Rolik Physical Chemistry and Structural Chemistry

Physical Chemistry

Physical Chemistry I - Equilibrium (phase equilibrium, chemical equilibrium)

Physical Chemistry II - Change (reaction kinetics, transport, electrochemistry)

Physical Chemistry III - Structure (molecular structure,

spectroscopy, materials science)

Curriculum

Introduction

The basics of quantum mechanics The structure of the hydrogen atom Structure of many-electron atoms Optical spectroscopy

Rotational spectroscopy Vibrational spectroscopy

Electronic structure of molecules

3

Curriculum

Photoelectron spectroscopy Lasers and laser spectroscopy Fundamentals of nuclear structure Nuclear magnetic resonance Mass spectrometry

X-ray diraction

The structure of atoms, molecules, and other particles is described by quantum mechanics.

The foundation of quantum mechanics was laid in the 1920´s.

Preliminaries: some experiments which contradict the principles of classical physics

5

Joseph Fraunhofer's experiment, 1815

The sunlight was dispersed by a grating.

Dark lines were observed in the continuous spectrum.

The spectrum of the sun

7

the sun emits continuous radiation

the particles of the gas surrounding the Earth and the Sun absorb only photons of particular wavelength/frequency particle A absorbs light of

,

, ... frequency particle B absorbs light of

,

, ... frequency, etc.

hence the energy of particle A can be changed by quanta of

E

= h

, h

, ... and the energy of particle B can be

changed by E

= h

, h

, ...

11

that is, the corresponding physical quantities have only discrete values.

This realization is reected by the term quantum mechanics

In the non-relativistic case the submicroscopic systems can be described by the Schrödinger equation

i ~ @

@ t (r; t ) = ~

2 m r

+ V (r; t )

(r; t )

Let's start from the beginning. What does i stand for?

13

complex numbers^a

aP. Atkins, J. Paula, R. Friedman, Chapter 2

Natural numbers

negative numbers (Diophantus [200 - c.284 CE]: The solution of the 4 = 4 x + 20 equation is absurd.)

rational numbers (Pythagorean school: all phenomena in the universe can be reduced to whole numbers and their ratios)

but what is p

2?

complex numbers

Real numbers form a closed set for the a + b ; a b ; a b ; a = b ( a ; b 2 R) operations.

But what is p

1? (Cardano, 1545)

15

complex numbers

real line vs. complex plane 1D vs. 2D

( x ) ( x ; y )

x ; y 2 R (ordered pairs) ( x ; y ) 6= ( y ; x )

complex numbers

addition, (a;b) + (c;d) , (a+c;b+d) subtraction, (a;b) (c;d) , (a c;b d) multiplication, (a;b) (c;d) , (ac bd;ad+bc) real numbers have the form of (a; 0) they lie on the real axis:

(a; 0) + (c; 0) , (a+c; 0) (a; 0) (c; 0) , (a c; 0)

(a; 0) (c; 0) , (ac; 0)

17

complex numbers

imaginary numbers have the form of (0; b ) they lie on the imaginary axis:

z = (0; b ) z z = z

(0; b ) (0; b ) , ( b

; 0)

z

= b

z 2 C ; b 2 R

(0; 1) (0; 1) , ( 1; 0)

complex numbers

z = (0; 1) is special, it is denoted by i , and called the imaginary unit ( i

= 1) with its help z = ( a ; b ) = a + bi

complex conjugate of z = a + bi is denoted by a star superscript z

= a bi

z z

= (a + bi ) (a bi ) = a

+ b

= jzj

19

complex numbers

division by a complex number:

a + bi

c + di = a + bi

c + di c di c di

= (a + bi )(c di ) c

+ d

= (ac + bd )

c

+ d

+ (bc ad )

c

+ d

i

complex numbers

polar form of complex numbers

z = a + bi = r (cos ' + i sin ') r = p

a

+ b

tan ' = b a multiplication and division in polar form:

z

= r

(cos '

+ i sin '

) z

= r

(cos '

+ i sin '

) z

z

= r

r

(cos('

+ '

) + i sin('

+ '

))

z

z

= r

r

(cos('

'

) + i sin('

'

))

21

Taylor-series

if f (x) is innitely dierentiable at a real or complex number a then f (x) = f (a) + f

(a)(x a) + 1

2 f

(a)(x a)

+ 1

3 2 f

(a)(x a)

+ : : :

= X

n=0

f

( a )

n ! ( x a )

when a = 0 it is called a Mclaurin series

Taylor-series,a= 0

f ( x ) = exp( x ) = e

e

= e

+ ( e

)

( x 0) + 1

2 ( e

)

( x 0)

+ 1

3 2 ( e

)

( x 0)

+ : : :

= X

n=0

1 n ! x

23

Taylor-series,a= 0

f ( x ) = sin( x )

sin x = 0 + x + 0 1

3! x

+ 0 + 1

5! x

+ 0 1

7! x

+ : : :

= X

n=0

1

(2 n + 1)! x

Taylor-series,a= 0

f ( x ) = cos( x )

cos x = 1 + 0 1

2! x

+ 0 + 1

4! x

+ 0 1

6! x

+ : : :

= X

n=0

1

(2 n )! x

25

Euler's formula

i

= 1 i

= i

i

= 1 i

= i

recall that e

= 1 + z +

z

+

z

+ : : : if z = ix

e

= 1 + ix + 1

2 i

x

+ 1

3! i

x

+ 1

4! i

x

+ 1

5! i

x

+ : : :

Euler's formula

e

= 1 + ix + 1

2 i

x

+ 1

3! i

x

+ 1

4! i

x

+ 1

5! i

x

+ : : :

= 1 + ix 1

2 x

i 1

3! x

+ 1

4! x

+ i 1

5! x

+ : : :

= 1 1

2 x

+ 1

4! x

+ ix i 1

3! x

+ i 1

5! x

+ : : :

= (1 1

2 x

+ 1

4! x

+ : : : ) + i ( x 1

3! x

+ 1

5! x

+ : : : )

= cos x + i sin x

27

complex numbers

exponential form of complex numbers

z = a + bi = r (cos ' + i sin ') polar form e

= cos ' + i sin '

z = r e

exponential form

conserved properties

conservation laws

some measurable physical properties do not change

mass (m)

and energy ( E )

electric charge ( q ) linear momentum (p) angular momentum (l)

aThere is always a symmetry behind the conservation laws: conservation of energy is connected to the time-invariance of physical systems.

bconservation of mass is not exact: nuclear fusions

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the ability to do work

the kinetic energy (E

| K ) is due to motion; E

= f (p) a moving object can do work

the potential energy ( E

| V ) is due to position; E

= g (r) stored energy of an object that can do work

E

= E

+ E

or H = K + V

Hamilton function: E=H=H(p, q), where p; q are the

canonical coordinates.

simple manipulations

recall the scalar product of vectors: v v = jvj

= v

E

= 1 2 mv

p = mv p

= m

v

E

= p

2 m

31

E_pot= 0

E

= E

= p

2 m p 2 mE

= p = m dx

dt dx

dt =

r 2 E

Z

m

x(0)

dx =

r 2 E

m

Z

0

dt

x ( t ) = x (0) +

r 2 E

m t p ( t ) = mv ( t ) = m dx

dt = m

r 2E

m p(t ) = p

2mE

restoring force is proportional to the displacement from the equilibrium position

the spring stores the energy as V ( x ) = 1

2 kx

) F

= dV dx F = kx

m d

x

dt

= kx m

e

= ke

( m

+ k ) e

= 0

= k m = i

r k

m = i !

x ( t ) = c

e

+ c

e

= A sin(! t + ') p(t ) = m dx

dt = !Am cos(!t + ')

dt = e^t ) d²x

dt² = ²e^t

33

uniform circular motion, centripetal force,F_cp, and angular momentum, `

F_cp=ma=mr!²=mv² r

` =rp=rmv

s '=2r

2 ) s=r ' v=ds

dt = lim

t!0

s

t =r lim

t!0

' t

=r !

v '=2v

2 ) v=v ' a=dv

dt = lim

t!0

v

t =v lim

t!0

' t

=v ! =r !²

special case of rotational motion, r is xed

x ( t ) = A sin( 2

T t ) = A sin(! t ) v = r !

a = v ! = r!

F = mv

r

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correspondences

linear momentum p angular momentum ` = r p = I ! velocity v angular velocity ! = r v

r

mass m moment of inertia I = mr

Kinetic energy p

2m

`

2I

model^a

asee also in wikipedia, Wave equation, Hooke's law

elastic, homogeneous string stretched to a length of L endpoints are xed

is the mass of the string per unit length

u ( x ; t ) represents the displacement of the string at a point x at a time t from its equilibrium position

only vertical movements are allowed

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derivation

T

cos = T

cos( + ) := T T

sin( + ) T

sin = m a = x @

u ( x ; t )

@ t

T

sin( + )

T

cos( + )

T

sin T

cos = 1

T x @

u ( x ; t )

@ t

tan( + ) tan = 1

T x @

u ( x ; t )

@t

@ u

@x

@ u

@x = 1

T x @

u ( x ; t )

@t

39

derivation

@ u

@ x

@ u

@ x = 1

T x @

u ( x ; t )

@ t

@ux+x

@x @ux

x

= 1 T

@

u ( x ; t )

@ t

@

u ( x ; t )

@ x

= 1

T = @

u ( x ; t )

@ t

@

u ( x ; t )

@ x

= 1

c

@

u ( x ; t )

@ t

traveling, standing waves and interference

(x;t) =A sin(kx !t) sin + sin = 2 sin( +

2 ) cos(

2 )

(x;t)interference=A sin(kx !t) +A sin(kx !t+ ') = 2A sin(kx !t+' 2) cos('

2) (x;t)standing=A sin(kx !t) +A sin(kx+ !t) = 2A sin(kx) cos(!t)

(x;t)standing=A sin(kx !t) A sin(kx+ !t) = 2A cos(kx) sin(!t)

light

light is electromagnetic radiation: (x;t) =A sin(kx !t) =A sin(²(x ct)) amplitude,A, maximum displacement from the rest position

wavelength, , the distance between two successive maxima

(Einstein, 1905), heat capacity of low temperature isolator crystals (Debye, 1912): energy is quantized, E = h .

Diagram of the maximum kinetic energy as a

function of the frequency of light on zinc.

de Broglie (1924): all matter has wave properties, p =

= ~k

45

H emission spectrum

the experimental emission spectrum of the H-atom

H emission spectrum^a

awikipedia, Hydrogen spectral series

Balmer(n 3)

~ = 1096801 4

1 n²

cm ¹ Rydberg(n₂>n₁)

~ = 109680 1

n²₁ 1 n²₂

cm ¹

Ritz combination rule: spectral lines include frequencies that are either the sum or the dierence of the frequencies of two other lines [the wavenumber (

~ = 1=) of any spectral line is the dierence between two terms

~ =term(i) term(j) ]

47

atomic emission spectra, characteristic for the atoms

Bohr's theory of the H-atom (1913)^a

awikipedia

existence of stationary orbits (xed nucleus and circular orbit), no electromagnetic radiation

frequency condition: E =h(his the Planck constant, 6:626 10 ³⁴J s) angular momentum is quantized: ` =n~, ~ =h=2

49

Bohr's theory of the H-atom (1913)

Felectrostatic=Fcentripetal

e²

4₀r² = mev² r

` =n~ =rmev v= n~

rme

v²= n²~² m²_er² e²

40r² = m_emⁿ²₂^~2

er2

r r = n²~²4₀

m_ee²

Bohr radius,a₀= 0:529 Å, (n= 1)

vacuum permittivity ₀= 8:854187817620::: 10 ¹²A²s⁴kg ¹m ³

Bohr's theory of the H-atom

E_tot =E_kin+E_pot

=1

2m_ev² e² 40r

=1 2

e² 40r

e²

40r = 1 2

e² 40r

= 1

2 e² 40n2~²40

mee²

= m_ee⁴ 80h²

1 n²

e²

4₀r² =mev² r mev²= re²

4₀r² r =n²~²40

mee²

~²= h² 4²

51

Bohr's theory of the H-atom

E =h =hc =hc~ E =En₂ En₁= mee⁴

80h²( 1 n₁²

1 n₂²)

~ = 1 hc

mee⁴ 80h²( 1

n²₁ 1 n²₂)

~ =RH( 1 n²₁

1 n²₂) RH= 1

hc mee⁴

80h² = 109737cm ¹ RH= 109638cm ¹from experiment

Bohr's theory of the H-atom

Bohr(n₂>n₁) : ~ = _hc¹ ₈^me₀^e_h⁴2(_n¹2 1

1 n²₂)cm ¹

Lyman(n₁= 1) Balmer(n1= 2) Paschen(n₁= 3) Brackett(n₁= 4)

1

1On December 1, 2011, it was announced that Voyager 1 detected the rst Lyman-alpha radiation originating from the Milky Way galaxy. Lyman-alpha radiation had previously been detected from other galaxies, but due to interference from the Sun, the radiation from the Milky Way was not detectable.

(Wikipedia)

53

plausibility of Bohr's quantization condition, ` =n~

p_photon= h

(Einstein) p_particle= h

(de Broglie)

= h

p_particle 2r =n 2r =n h

p_electron

` =rp=n h 2

constructive, destructive interferences standing wave - stationary orbit

plausibility of Bohr's quantization condition, ` =n~

Wave-particle duality: "It seems as though we must use sometimes the one theory and sometimes the other, while at times we may use either.

We are faced with a new kind of diculty. We have two contradictory pictures of reality; separately neither of them fully explains the phenomena of light, but together they do." (Einstein)

c = E = h

55

some arguments for the Schrödinger equation

of course there is no proof of it, it is a postulate

Free particle waves: ( x ; t ) = e

! = E =~ (Planck)

@

@t ( x ; t ) = i

~ E ( x ; t ) i ~ @

@ t ( x ; t ) = E ( x ; t )

k = p =~ (De Broglie)

@

@x

( x ; t ) = ( i

~ )

p

( x ; t )

~

2m

@

@x

( x ; t ) = p

2m ( x ; t )

The energy is a classical free particle:

E = p

2 m

@ ~

@

particle in a force eld, time-independent Schrödinger equation

If the particle is not free (3D):

i ~ @

@t (r; t ) = ~

2m @

@x

+ @

@y

+ @

@z

+ V (r)

(r; t ) A particular solution of the time-dependent Schrödinger equation:

(r; t ) = (r) e

i ~ @

@t (r) e

= E (r) e

Using the relations above we obtain the time-independent Schrödinger equation

~

2 m

@

@ x

+ @

@ y

+ @

@ z

+ V (r)

(r) = E (r)

57

Schrödinger equation for the particle in the 1D box model^a

aAtkins, part II, chapter 8

~² 2m

d² (x)

dx² +V(x) (x) =E (x) E_kin+E_pot=E_tot

@² (x)

@x² = 2m(E V(x))

~² (x) d²y

dx² = k²y

y2 fe^ikx; sin(kx); cos(kx)g

Schrödinger equation for the particle in the 1D box model

~² 2m

d² (x)

dx² +V(x) (x) =E (x)

No particle in the innit potential area! ( x ) = 0 if x < 0 or x > L .

@² (x)

@x² = 2mE

~² (x) k=

r2mE

~²

(x) =Ccoskx+Dsinkx

(0) = 0 (L) = 0

) ()

(C= 0

D= 0 or sinkL= 0 kL=n n= (1; 2; )

(x) =Dsinn Lx

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Schrödinger equation for the particle in the 1D box model

V(x) = 8>

><

>>

:

1; 1 <x 0 0; 0 <x<L 1; Lx< 1

k= r2mE

~² =n

L k²=2mE

~² =n²² L²

2mE

~² = n²² L² En= n²h²

8mL²

Born probability interpretation: R₁

1 2(x)dx= 1

properties of the solutions

Born probability interpretation^R11 2(x)dx=1

i.e., probability of nding the particle betweenx andx+dxis ²(x)dx

ifn" then E "

n= 1, zero-point energy

hasn 1 nodes in the 0 <x<Linterval ground and excited states

with increasing mass the energy gap between the levels,En+1 En, decreases

(r) = (r) (r) satises the continuity equation,^@_@t +divj = 0, where

j(r;t) =_2mi^~

(r ) r

is the probability current

.

61

Schrödinger equation for the particle in the 1D box model

Born probability interpretation Z₁

1

2(x)dx= 1 ) (x) = r2

Lsin(n L x)

(x) =Dsin(n Lx) D²

Z ₁

1sin²(n

Lx)dx=D² ZL

0 sin²(n

L x)dx= 1 D=

r2 L (x) =

r2 Lsin(n

Lx)

z=n Lx dz=n

Ldx Z_n

0 dz=n L

Z_L 0 dx

sin²z=sin²z+ cos²z+ sin²z cos²z 2

=1 cos 2z 2

Schrödinger equation for the free particle,V(x) = 0,E_kin=E_tot

~² 2m

d² (x)

dx² =E_kin (x)

@² (x)

@x² = 2mE_kin

~² (x) k²= 2mE_kin

~² (x) =A sin(kx)

(x) =A sin( 2x) k= 2

2mEkin= 2m 12mv²=p² p²=k²~²=

2

₂

h

2 ₂

=h p

63

Schrödinger equation for the particle in the 2D box model

~² 2m

@²

@x² + @@y²²

+V =E

V(x;y) =

(0; x 2 (0;L₁) ^y2 (0;L₂)

1; otherwise

Schrödinger equation for the particle in the 2D box model

~² 2m

@²

@x² + @@y²²

=E (x;y) =F(x) G(y)

separation of variables

=F(x) G(y)

@²

@x² =G(y)d²F(x) dx²

@²

@y2 =F(x)d²G(y) dy2

~² 2m

(

G(y)d²F(x)

dx2 +F(x)d²G(y) dy2

)

=EF(x)G(y)

~² 2m

( 1 F(x)

d²F(x) dx2 + 1

G(y) d²G(y)

dy2 )

=E

~² 2m

1 F(x)

d²F(x) dx² =E_x

~² 2m

1 G(y)

d²G(y) dy² =Ey

~² 2m

d²F(x) dx2 =E_xF(x)

~² 2m

d²G(y) dy² =EyG(y)

65

Schrödinger equation for the particle in the 2D box model

~² 2m

d²F(x) dx² =ExF(x)

E_x= n²₁h² 8mL²₁

F(x) = s2

L₁sinn₁ L₁x

~² 2m

d²G(y) dy² =EyG(y)

Ey= n²₂h² 8mL²₂

G(y) = s2

L₂sinn₂ L₂y

(x;y) =F(x) G(y) =

r 4

L₁L₂ sinn₁

L₁ x sinn₂ L₂ y E =Ex+Ey =

(n₁ L₁

₂ +

n₂ L₂

₂) h² 8m

Schrödinger equation for the particle in the 2D box model

(x;y) = s 4

L₁L₂ sinn₁ L₁ x sinn₂

L₂ y

E(n₁;n₂) = n₁

L_{1 2}

+ n₂

L_{2 2}h²

8m

consequence of symmetry,L₁=L₂=L (x;y) =

r4

L² sinn₁

L x sinn₂ L y E(n₁;n₂) = n²₁+n²₂ h²

8mL² E(1; 2) =E(2; 1) but the wavefunctions are dierent degeneracy: same energies dierent wavefunctions

67

Schrödinger equation for the particle in the 2D box model

degeneracy is the consequence of symmetry

Schrödinger equation for the particle in the 3D box model

~² 2m

@²

@x² + @@y²² + @@z²²

=E (x;y;z) =F(x) G(y) H(z)

(x;y;z) =F(x) G(y) H(z) =

r 8

L₁L₂L₃ sinn₁

L₁ x sinn₂

L₂ y sinn₃ L₃ z E =Ex+Ey+Ez =

(n₁ L₁

₂ +

n₂ L₂

₂ +

n₃ L₃

₂) h² 8m

69

Schrödinger equation for the particle in the 3D box model

degenerate case: cubeL₁=L₂=L₃=L (x;y;z) =

r8

L³ sinn₁

L x sinn₂

L y sinn₃ L z E(n₁;n₂;n₃) = n₁²+n²₂+n²₃ h²

8mL² = n²₁+n²₂+n₃² h² 8mV²⁼³

Postulates of Quantum Mechanics

postulate I

aP. Atkins, J. Paula, R. Friedman, Chapter 1

The state of a quantum-mechanical system is completely specied by the so-called wavefunction, (r; t ), that depends on the coordinates of the particles and on time.

(r; t) (r; t)dxdydz is the probability that the particle lies in the volume element d = dxdydz located at r at time t.

71

postulate I

properties of (r; t ) continuous

contiguously dierentiable (if the V (r) potential is realistic ...) nite (square integrable for bound states, i.e.,

R

1

j j

d < 1)

single valued

postulate II

To every observable in classical mechanics there exists a

corresponding linear, Hermitian operator in quantum mechanics.

73

correspondences

observables ^ operators

position x x^ multiplication byx

r ^r multiplication by r potential energy V(x) V^(^x) multiplication byV(x)

V(r) V^(^r) multiplication byV(r)

momentum px p^x i~_@^@_x

p ^p i~(ex @

@x + ey @

@y + ez @

@z) kinetic energy Kx K^x ~²

2m @²

@x²

K K^ ^~_2m²(_@^@_x²2 +_@^@_y²2+_@^@_z²2)

total energy E H^ T + ^^ V

postulate III

In any measurement of the observable associated with the operator , the only values that will ever be observed are the eigenvalues ! ^

which satisfy the eigenvalue equation ^

= !

75

linear Hermitian operators

~

2 m

@

@ x

+ V

( x ) = E ( x ) H ( ^ x ) = E ( x ) eigenvalue equation ^ = !

for an operator there can be more than one eigenfunction usually dierent eigenfunctions have dierent eigenvalues (non degenerate)

an operator is called linear if ^ ( + ) = ^ + ^ an operator is called Hermitian if R

i

^

d = nR

j

^

d o

eigenvalues for Hermitian operators are real

postulate IV

If the state of the system is described by a normalized wave function , then the average value of the observable corresponding to the operator ^ can be calculated as h!i = R

^ d

77

postulate V

The wave function of a system evolves in time according to the

time-dependent Schrödinger equation: ^ H (r; t ) = i ~

required properties: measurable physical quantities are real

postulate III: ^ i = !i i ) R

i ^ _id = !i

(we assumed that i is normalized,R₁

1

i id = 1) eigenvalue equation

= ! ^ Z ₁

1

d^ = ! Z ₁

1 d Z ₁

1

d^ = !

its complex conjugate ^ = ! Z ₁

1 ^ d = ! Z ₁

1 d Z ₁

1 ^ d = !

! is real if ! = !, i.e.,R₁

1 d =^ R₁

1 ^ d

79

required properties

hermitian operators R ^ d =R

^ d

let be a linear combination of functions and and ^ is hermitian =ca +cb ) =ca+cb

Z d^ = Z

(ca+cb) ^(ca +cb)d

= Z

(ca +cb) ^(ca+cb)d = Z

^ d

required properties

Z

(c_a+c_b) ^(ca +cb)d =Z

(ca +cb) ^(c_a+c_b)d Z

c_a^cad +Z

c_a^cbd +Z

c_b^cad +Z

c_b^cbd Z =

ca ^c_ad + Z

ca ^c_bd + Z

cb ^c_ad + Z

cb ^c_bd

81

required properties

Z

c_a^cad +Z

c_a^cbd +Z

c_b^cad +Z

c_b^cbd Z =

ca ^c_ad + Z

ca ^c_bd + Z

cb ^c_ad + Z

cb ^c_bd Z

c_a^cbd +Z

c_b^cad =Z

ca ^c_bd +Z

cb ^c_ad c_acb

Z

^ d Z

^d

=cac_b Z

^d Z

^ d

complex number = its complex conjugate Z

^ d =Z

^d and Z

^d =Z

^ d

hermitian operators

general denition Z

^ d = Z

^

d special case ( = )

Z

^ d = Z

^

d

83

properties of hermitian operators

the sum of hermitian operators is also a hermitian operator = ^^ i+ ^j

Z ^i d =Z

^i d and Z

^j d =Z

^j d Z ^ d =Z

n

^_i+ ^_io d

=Z

^i d +Z

^j d

= Z

^i d + Z

^j d (hermitian property)

=Z n ^i

+ ^io

d

= Z

^ d