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Foundations and Applications of Quantum Chemistry or Frequently Asked Questions about Chemistry Foundations and Applications of Quantum Chemistry Dirk Andrae Theoretical Chemistry Department of Chemistry, University of Bielefeld Summer Term 2004

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Foundations and Applications of

Quantum Chemistry

or

Frequently Asked Questions

about Chemistry

Foundations and Applications of

Quantum Chemistry

Dirk Andrae

Theoretical Chemistry

Department of Chemistry, University of Bielefeld

Summer Term 2004

Outline

1. Historical introduction

2. The Schrodinger equation for one-particle problems

3. Mathematical tools for quantum chemistry

4. The postulates of quantum mechanics

5. Atoms and the ‘periodic’ table of chemical elements

6. Diatomic molecules

7. Ten-electron systems from the second row

8. More complicated molecules

• What are the electrons doing in atoms and molecules?

• What is an orbital? How can it be visualized?

• What is a chemical bond?

• What are the nuclei doing in ordinary matter?

• . . .

‘What is mind? No matter! — What is matter? Never mind.’

Classical mechanics

‘If I have been able to see further, it was only because I stood on the shoulders of giants.’

Isaac Newton in a letter to Robert Hooke

I. Newton (1643-1727)

Philosophiæ naturalis principia mathematica

(1687, English tr. 1729, German tr. 1872)

F = p =dp

dtp = mv

‘For the first time [. . . ] a single mathemati-

cal law could account for phenomena of the

heavens, the tides and the motion of ob-

jects on the earth.’ (Encyclopædia Britan-

nica, 1971) http://www.lib.cam.ac.uk/Ex-

hibitions/Footprints of the -Lion/gravity glory.html

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-05 2/ 52-1

1. Historical introduction

(classical mechanics, electrodynamics & chemistry)

2. The Schrodinger equation for one-particle problems

3. Mathematical tools for quantum chemistry

4. The postulates of quantum mechanics

5. Atoms and the ‘periodic’ table of chemical elements

6. Diatomic molecules

7. Ten-electron systems from the second row

8. More complicated molecules

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-05 1/ 52-1

Newton’s second law of motion (Principle of action):

‘The change of motion [ i.e. momentum ] is proportional to the

motive force impressed; and is made in the direction of the right line

in which that force is impressed.’

F =

FxFyFz

= p =

dp

dtp =

pxpypz

= mv = m

dr

dt(2)

m vt

m v

F t’

m v

t’’t < t’ < t’’

Eq. (2) includes eq. (1) in the special case of constant mass m and

vanishing force (F = o), but applies, in the form F = p, to rockets

(m = m(t)) and even to relativistic dynamics (m→ meff(v2)).

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-05 4/ 52-1

Newton’s first law of motion (Principle of inertia):

‘Every body continues in its state of rest, or of uniform motion in

a right line, unless it is compelled to change that state by forces

impressed upon it.’

F = o ⇔ a = v =dv

dt= o (1)

m v

m v

m vt

t < t’ < t’’

t’’

t’

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-05 3/ 52-1

Given a set of masses mi with known initial positions ri(t) and velo-

cities vi(t) at a time t.

New positions and velocities at a time t′ = t + ∆t are obtainable

through integration, e.g.

vi(t′) = vi(t) + ai∆t ai =

1

miF i (3)

ri(t′) = ri(t) + vi(t)∆t+

1

2ai (∆t)2 (4)

if the total forces F i are assumed to be constant over ∆t.

−→ Molecular Mechanics & Dynamics ←−

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-05 6/ 52-1

Newton’s third law of motion:

‘To every action there is always opposed an equal re-action; or the

mutual actions of two bodies upon each other are always equal, and

directed to contrary parts.’

t t’

t < t’

Attractive and repulsive forces are combined to the resulting total

forces.

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-05 5/ 52-1

Definition of generalized conjugate momenta and forces:

pi =∂L

∂qi, Fi =

∂L

∂qi(7)

Euler-Lagrange equations (2nd order PDE for each qi(t)):

d

dt

(∂L

∂qi

)− ∂L

∂qi= pi − Fi = 0 (8)

Transformation∗ from the Lagrange function L to the Hamilton func-

tion H:

H(qi, pi, t) =∑

i

qipi − L(qi, qi, t) (9)

Hamilton’s equations (pairs of 1st order PDEs for each qi(t)):

pi =dpidt

= − ∂H∂qi

, qi =dqidt

=∂H

∂pi(10)

∗Legendre transformation (A.-M. Legendre, 1752-1833), also known from thermodynamics:Internal energy U = U(S, V ) → Helmholtz free energy A(T, V ) with −A = TS − U , orenthalpy H = H(p, S) → Gibbs free energy G(p, T ) with −G = TS −H,where T = (∂U/∂S)V = (∂H/∂S)p.

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-05 8/ 52-1

Extension to general coordinate systems:

L. Euler (1707-1783)

J. L. d’Alembert (1717-1783)

J. L. de Lagrange (1736-1813)

S. D. Poisson (1781-1840)

C. G. J. Jacobi (1804-1851)

W. R. Hamilton (1805-1865)

Action S and Lagrange function L:

S(t1, t2) =

∫ t2t1L(qi, qi, t) dt (5)

qi, qi — generalized coordinates and

corresponding velocities (i = 1, . . . , f ,

where f is the number of degrees of

freedom)

Principle of stationary action:

δS = 0 (6)

1

2

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-05 7/ 52-1

A suitable choice for L has to be made. Consider, e.g., a free particle

with rest mass m moving with velocity v:

non-relativistic case

L = T =1

2mv2 = L(v2)

p = m v

H = p · v − L = 2T − L

=1

2mv2 = T

= E

relativistic case (β = v/c)

L = −mc2√

1− β2 = L(v2)

p =m v√1− β2

H = p · v − L =mc2√1− β2

= E

H2 = m2 c4 + c2 p2 = E2

Extension to include potential energy V (for conservative potentials):

L = T − V → H = T + V = E (15)

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-05 10/ 52-1

In terms of the Poisson bracket, defined for any pair of physical quan-

tities A and B as

{A,B} =f∑

i=1

(∂A

∂pi

∂B

∂qi− ∂A∂qi

∂B

∂pi

)= − {B,A} (11)

the variation of a physical quantity Q in time is simply

Q = {H,Q} (12)

as follows from the generalized chain rule of differentiation.

Thus Hamilton’s equations read as

qi = {H, qi} , pi = {H, pi} (13)

In general, if {A,B} = 0, A and B are said to commute.

Consider the generalized coordinates and momenta themselves:

{qj, qk}= 0 , {pj, pk} = 0 , {pj, qk} = δjk (14)

Coordinates commute with momenta only when different degrees of

freedom are involved.

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-05 9/ 52-1

Optics and classical electrodynamics

Chr. Huygens (1629-1695)

Traite de la lumiere (1678)

Light is a wave phenomenon (in analogy to water surface waves)

→ Explanation for reflection, refraction, interference, and polarization

of light.

Huygens’ principle: Wave phenomena can be understood by superpo-

sition of spherical waves.

I. Newton (1643-1727)

Opticks (1704)

Light is composed of particles, which travel along straight lines

→ geometrical optics.

Observation of the spectral decomposition of white light into light of

‘all colors’.

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-05 12/ 52-1

Conservation laws and symmetries:

1. If L is translationally invariant in time (homogeneity of time)

→ H is constant: H = E, conservation of energy E

2. If L is translationally invariant in space (homogeneity of space)

→ conservation of linear momentum p

3. If L is invariant with respect to orientation in space (rotational

invariance in space, isotropy of space)

→ conservation of orbital angular momentum l = r × p

These are special cases of the Noether† theorem which states that

with every continuous symmetry of a dynamical system there is related

a conserved physical quantity (see refs. for a rigorous formulation).

† E. Noether (1892-1935)

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-05 11/ 52-1

Observation of dark lines in the visible region of the solar spectrum

1802 — W. H. Wollaston (1766-1828)

1815 — J. Fraunhofer (1787-1826), catalogued over 600 lines

Source: http://www.harmsy.freeuk.com/images/spectrum.jpeg or http://www.coseti.org/solatype.htm

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-05 14/ 52-1

Single slit diffraction — a wave phenomenon

(application of Huygens’ principle)

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.................................................................................................................................................................................................................................................................................................................................

spherical waves

I(θ) = a I0 [ j0(bθ) ]2

j0(x) =sin (x)

x

plane wave

(k, λ, I0)k

θ

I(θ)

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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screen

slit

(d > λ)................................................................................................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................................................................................................

........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-05 13/ 52-1

Maxwell’s equations of electrodynamics

1861-1864 — J. C. Maxwell (1831-1879)

Equations for electric field E(r, t) and magnetic flux density B(r, t)

(in SI units: ε0µ0c2 = 1)

in the absence of dielectric

or magnetic media

∇ × E +∂B

∂t= o

∇ ·B = 0

1

µ0

∇ × B − ε0∂E

∂t= j

ε0∇ ·E = ρ

ρ = 0

j = o

=⇒

in vacuum (no charges, and

no currents)

∇ × E +∂B

∂t= o

∇ ·B = 0

∇ × B − 1

c2∂E

∂t= o

∇ ·E = 0

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-05 16/ 52-1

Law of electrostatic interaction

1785 — Ch. A. Coulomb (1736-1806)

Potential energy V for two charges q1 and q2 at distance r

V = Cq1q2r

(16)

In SI units: C = 1/κ0 with κ0 = 4πε0.

Further experimental study of and theory development for electric and

magnetic phenomena (static and dynamic):

J. B. Biot (1774-1862)

A. M. Ampere (1775-1836)

C. F. Gauß (1777-1855)

H. Chr. Ørsted (1777-1851)

M. Faraday (1791-1867)

F. Savart (1791-1841)

H. Lenz (1804-1865)

W. E. Weber (1804-1891)

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-05 15/ 52-1

Plane wave propagating in the z direction

E and B along the z axis

-1-1

-0,5-0,5

000

2

0,50,5

4

1

6

1

8

10

12

k ‖ E ×B

Isovalue lines in the xz plane

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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x

z

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Experimental generation of electromagnetic waves (oscillating dipole

radiation in the radiofrequency range), and identification of light as

electromagnetic radiation

1887 — H. R. Hertz (1857-1894)

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-05 18/ 52-1

In vacuum, every component Xi of E or B has to be a solution of

the d’Alembert equation

∂2Xi∂x2

+∂2Xi∂y2

+∂2Xi∂z2

− 1

c2∂2Xi∂t2

=

(∂2

∂x2+

∂2

∂y2+

∂2

∂z2− 1

c2∂2

∂t2

)Xi = 0

(17)

Solutions of eq. (17) are plane waves, representable as real part of

Xi(r, t) = X0 ei(k·r−ωt) (18)

They propagate (‘travel’) at the speed of light c in the direction of

the wave vector k (k = |k| = 2π/λ), have amplitude X0 and circular

frequency ω = 2πν.

Behaviour of the function Xi(r, t) under differentiation:

∂xXi(r, t) = i kxXi(r, t)

∂tXi(r, t) = −iωXi(r, t) (19)

so that k2 − ω2/c2 = kx2 + ky2 + kz2 − ω2/c2 = 0, i.e. c = λ ν, with

frequency ν, wavelength λ, and wavenumber ν = ν/c = 1/λ.

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-05 17/ 52-1

The chemical elements

(known before 1800, though not necessarily in elemental form)

element symbols introduced in 1811 by J. J. Berzelius (1779-1848)

H

Be

Mg

Ca

Sr

Ba

Y

U

Ti

Zr

Cr

Mo

W

Mn Fe Co Ni

Pt

Cu

Ag

Au

Zn

Hg

B

Al

C

Sn

Pb

N

P

As

Sb

Bi

O

S

Te

F

Cl

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 20/ 52-1

Chemistry

Law of conservation of mass:

1748 / 1760 — M. W. Lomonossov (1711-1765)

1785 — A. L. Lavoisier (1743-1794)

‘There is no measurable change in mass during a chemical reaction:

the mass of the products is equal to the mass of the reacting sub-

stances.’

3 g hydrogen + 24g oxygen −→ 27g water

Law of constant proportions:

1799 — J. L. Proust (1754-1826)

‘Different samples of a substance contain its elementary constituents

(elements) in the same proportions.’

water: m(hydrogen) : m(oxygen) = 1 : 7.937

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 19/ 52-1

Law of simple multiple proportions:

‘When two elements combine to form more than one compound, the

weights of one element that combine with the same weight of the

other are in the ratios of small integers.’

(derived by J. Dalton from theory, i.e. from his atom hypothesis)

compound AkBl m(A) m(B)

CO 1 · 1.000 : 1 · 1.332CO2 1 · 1.000 : 2 · 1.332N2O 1 · 1.000 : 1 · 0.571NO 1 · 1.000 : 2 · 0.571N2O3 1 · 1.000 : 3 · 0.571(NO2, N2O4) 1 · 1.000 : 4 · 0.571N2O5 1 · 1.000 : 5 · 0.571

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 22/ 52-1

J. Dalton (1766-1844)

A new system of chemical philosophy

(Part I 1808, Part II 1810, Part III 1827)

Atom hypothesis (1805):

‘All substances consist of small particles

of matter, called atomsa, of several diffe-

rent kinds, corresponding to the different

elements.’afrom Greek

����������: ‘undivisible’, ‘indivisible’

Source: http://www.nmsi.ac.uk/piclib/images/preview/10322897.jpg

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 21/ 52-1

Avogadro’s law:

1811 — A. Avogadro (1776-1856)

1858 — S. Cannizzaro (1826-1910)

‘Equal numbers of molecules are contained in equal volumes of all

dilute gases under the same conditions.’

→ the correct chemical formula for water is H2O, not HO

Spectral analysis

1859 — R. W. Bunsen (1811-1899) & G. R. Kirchhoff (1824-1887)

→ discovery of rubidium, Rb, and caesium, Cs

Periodic table of chemical elements

1869 — D. I. Mendeleev (1834-1907) / J. L. Meyer (1830-1895)

→ prediction of new chemical elements

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 24/ 52-1

The table of the chemical elements

(1800–1849)

H

Li

Na

K

Be

Mg

Ca

Sr

Ba

Y

La Ce Nd Tb Er

Th U

Ti

Zr

V

Nb

Ta

Cr

Mo

W

Mn Fe

Ru

Os

Co

Rh

Ir

Ni

Pd

Pt

Cu

Ag

Au

Zn

Cd

Hg

B

Al

C

Si

Sn

Pb

N

P

As

Sb

Bi

O

S

Se

Te

F

Cl

Br

I

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 23/ 52-1

KH G F E D C B

Solar spectrum, visible region (http://mesola.obspm.fr/form spectre.html)

D

Emission spectrum of sodium, Na (http://scidiv.bcc.ctc.edu/wv/spect/sodium em spectrum2.html)

Hγ Hβ

(F)Hα

(C)

Emission spectrum of hydrogen, H (http://astro.u-strasbg.fr/~koppen/discharge/)

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 26/ 52-1

hydrogen(2 vols.)

oxygen(1 vol.)

water vapour(2 vols.)

hydrogen(1 vol.) (1 vol.)

chlorine(2 vols.)

hydrogen chloride

oxygen(1 vol.)(2 vols.)

carbon monoxide(2 vols.)

carbon dioxide

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 25/ 52-1

Rb and Cs discovered by spectral analysis

(Bunsen & Kirchhoff, 1859-1863)

H

Li

Na

K

Rb

Cs

Be

Mg

Ca

Sr

Ba

Y

La Ce Nd Tb Er

Th U

Ti

Zr

V

Nb

Ta

Cr

Mo

W

Mn Fe

Ru

Os

Co

Rh

Ir

Ni

Pd

Pt

Cu

Ag

Au

Zn

Cd

Hg

B

Al

C

Si

Sn

Pb

N

P

As

Sb

Bi

O

S

Se

Te

F

Cl

Br

I

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 28/ 52-1

Source: http://www.physics.brown.edu/Studies/Demo/modern/demo/em.gif

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 27/ 52-1

The ‘periodic’ table of the chemical elements

(1850-1899)

1

2

3

4

5

6

7

IA

1

H

3

Li

11

Na

19

K

37

Rb

55

Cs

IIA

4

Be12

Mg

20

Ca

38

Sr

56

Ba

88

Ra

IIIB

21

Sc

39

Y

∗ 57

La58

Ce59

Pr60

Nd62

Sm63

Eu64

Gd65

Tb66

Dy67

Ho68

Er69

Tm70

Yb89

Ac90

Th92

U

IVB

22

Ti

40

Zr

VB

23

V

41

Nb

73

Ta

VIB

24

Cr

42

Mo

74

W

VIIB

25

Mn26

Fe

44

Ru

76

Os

VIII

27

Co

45

Rh

77

Ir

28

Ni

46

Pd

78

Pt

IB

29

Cu47

Ag

79

Au

IIB

30

Zn

48

Cd80

Hg

IIIA

5

B

13

Al

31

Ga

49

In

81

Tl

IVA

6

C

14

Si

32

Ge

50

Sn

82

Pb

VA

7

N

15

P

33

As

51

Sb

83

Bi

VIA

8

O

16

S

34

Se

52

Te

84

Po

VIIA

9

F

17

Cl

35

Br

53

I

VIIIA

2

He

10

Ne

18

Ar

36

Kr

54

Xe

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 30/ 52-1

Prediction of existence and properties of Sc, Ga, and Ge

(Mendeleev, 1869/1870)

1

H

3

Li

11

Na

19

K

37

Rb

55

Cs

4

Be12

Mg

20

Ca

38

Sr

56

Ba

21

Sc

39

Y

57

La58

Ce60

Nd65

Tb68

Er90

Th92

U

22

Ti

40

Zr

23

V

41

Nb

73

Ta

24

Cr

42

Mo

74

W

25

Mn26

Fe

44

Ru

76

Os

27

Co

45

Rh

77

Ir

28

Ni

46

Pd

78

Pt

29

Cu47

Ag

79

Au

30

Zn

48

Cd80

Hg

5

B

13

Al

31

Ga

6

C

14

Si

32

Ge

50

Sn

82

Pb

7

N

15

P

33

As

51

Sb

83

Bi

8

O

16

S

34

Se

52

Te

9

F

17

Cl

35

Br

53

I

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 29/ 52-1

Oscillators in the wall of the cavity emit or absorb radiation energyonly in quantized form (quantum hypothesis)

∆E = nε , ε = hν = ~ω , ~ =h

2π≈ 1 · 10−34 J s

Previously known approximate laws are included as limiting cases:

hν � kBT : uν(T ) =8πν2

c3kBT (Rayleigh-Jeans)

hν � kBT : uν(T ) =8π hν3

c3exp

(− hν

kBT

)(Wien)

The total energy density of the radiation field inside the cavity isproportional to T4

u(T ) =

∫ ∞0

uν(T ) dν = C T4 , C =8π5

15

kB4

(hc)3, [u] = J m−3

(this is not the Stefan-Boltzmann law)1905 A. Einstein (1879-1955): Explanation of the photoeffect (pho-

tons); theory of special relativity, conservation of mass-energy

E = mc2/√

1− β2 =

√m2c4 + c2p2 , β = v/c

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 32/ 52-1

Development since about 1900

1884/1885 J. Balmer (1825-1898): Empirical formula for visible hy-

drogen spectral line wavelengths (RH Rydberg constant for hy-

drogen)

λ = Cn2

n2 − 4⇔ ν =

1

λ= RH

(1

4− 1

n2

)(C = 4/RH , n > 2)

1895 W. C. Rontgen (1845-1923): Discovery of X rays

1896 A. H. Becquerel (1852-1908): Discovery of natural radioactivity

1897 J. J. Thomson (1856-1940): Cathode rays are beams of free

electrons

1900 M. Planck (1858-1947): Spectral distribution of the radiation

energy density inside a ‘black body’ (cavity at temperature T with

a small hole, h Planck constant)

uν(T ) =8πν2

c3hν

ex − 1, x =

kBT, [uν] = J s m−3

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 31/ 52-1

being located in one focus) and has energies

En = − 1

2

Z2

n2Eh , Eh =

e2

κ0 a0, a0 =

κ0~2

mee2

Electromagnetic radiation is emitted or absorbed according to

energy differences between stationary states

ν =En −Em

hc= R∞Z

2(

1

m2− 1

n2

), R∞ =

mee4

8ε02h3c

(R∞ Rydberg constant for nucleus with infinite mass)

The Balmer formula is included for Z = 1 and n > m = 2.

This model failed to describe the simplest two-electron systems

(He, H2), and thus to understand the chemical bond

1920 The name ‘proton’ is given to the nucleus of 1H (E. Rutherford)

1921 O. Stern (1888-1969) & W. Gerlach (1889-1979): A beam of Ag

atoms is split up into two parts by an inhomogeneous magnetic

field. Detection of atomic spin and associated magnetic dipole

moment (to spin - to rotate quickly around the figure axis)

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 34/ 52-1

1910/1913 F. Soddy (1877-1956) & F. W. Aston (1877-1945): Dis-

covery and mass spectrometric detection of isotopes (e.g. 1H, 2H

for hydrogen; 12C, 13C for carbon; 35Cl, 37Cl for chlorine)

1911 E. Rutherford (1871-1937): The atom has a very small nucleus,

which carries almost the total atomic mass and has positive charge

qN = Ze (Z nuclear charge number). This nucleus is surrounded

by electrons (each has charge qe = −e).Size of the atomic nucleus: 10−15 m

1912 M. von Laue (1879-1960): Single-crystal diffraction with X rays

Size of the atoms: 10−10 m = 1 A

1913 H. Moseley (1887-1915): Direct determination of the nuclear

charge number Z from the frequency ν of emitted X rays:

ν ∝ Z2

1913/1916 N. Bohr (1885-1962), A. Sommerfeld (1868-1951):

The one-electron atom as a miniaturized planetary system: In sta-

tionary states, the electron moves on elliptical orbits (the nucleus

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 33/ 52-1

1925 W. Pauli (1900-1958): ‘No two electrons may simultaneously

occupy the same quantum state’ (Pauli exclusion principle)

1925/1926 W. Heisenberg (1901-1976) & M. Born (1882-1970): Ma-

trix mechanics (first version of quantum mechanics)

1926 E. Schrodinger (1887-1961): Wave mechanics (second version

of quantum mechanics)

Proof of equivalence of matrix mechanics and wave mechanics;

Schrodinger equation for stationary states of the one-electron

atom with atomic nucleus of charge number Z fixed at the origin:

− ~2

2me∇2ψ+ V ψ =

(− ~2

2me∇2 + V

)ψ = E ψ , V = − Z e

2

κ0 r

Electron spin is not included, has to be added phenomenologically

1926 E. Fermi (1901-1954) & P. A. M. Dirac (1902-1984): Fermi-

Dirac statistics for electrons and other particles with ‘half-integer

spin’ (spin quantum number s = n/2, n = 2k+1), i.e. for fermions

1927 W. Heisenberg: Uncertainty principle, in modern form:

∆x∆px ≥ ~/2

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 36/ 52-1

1923 A. H. Compton (1892-1962): Elastic scattering of X rays (‘waves’

behave like ‘particles’)

λout = λin + ∆λ , ∆λ = λC (1− cos (θ)) , λC =h

mec

(θ scattering angle, λC Compton wavelength)

1924 S. N. Bose (1894-1974) & A. Einstein: Bose-Einstein statistics

for photons and other particles with ‘integer spin’ (spin quantum

number s = n/2, n = 2k), i.e. for bosons

1925 L. de Broglie [fr.��� ��

] (1892-1987): For the photon (m = 0)

E =

√m2c4 + c2p2 = cp = hν = ~ω ⇒ p =

c=h

λ=

2π~

λ= ~k

Linear momentum p (particle property) associated with wave-

length λ (wave property), suggestion to transfer this relation to

particles

1925 S. A. Goudsmit (1902-1978) & G. E. Uhlenbeck (1900-1988):

Postulate of electron spin (intrinsic non-classical angular momen-

tum)

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 35/ 52-1

Electron spin is naturally included, prediction of anti-particles

(positron)

1932 J. Chadwick (1891-1974): Discovery of the neutron

1932 C. D. Anderson (1905-1991): Discovery of the positron

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 38/ 52-1

The picture of a trajectory, where position and velocity (linear

momentum) are ‘exactly’ known at every instant of time, is wrong!

1927 W. H. Heitler (1904-1981) & F. London (1900-1954): Successful

application of the Schrodinger equation to the H2 molecule, the

‘covalent chemical bond’ is understood for the first time

1927 C. J. Davisson (1881-1958) & L. H. Germer (1896-1971): Single-

crystal diffraction with electron beams (‘particles’ behave like

‘waves’)

1928 P. A. M. Dirac: Dirac equation for stationary states of the one-

electron atom with atomic nucleus of charge number Z fixed at

the origin:

V +mec2 0 c pz c (px − i py)

0 V +mec2 c (px + i py) − c pz

c pz c (px − i py) V −mec2 0

c (px + i py) − c pz 0 V −mec2

ψ = Eψ

p =

pxpypz

= − i~∇ , V = − Z e

2

κ0 r, ψ =

ψ1ψ2ψ3ψ4

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 37/ 52-1

Electrons, protons, and neutrons are fermions with spin quantum

number s = 1/2.

Quantum mechanics provides the theory to understand why elec-

trons bind to atomic nuclei (without collapsing into the nucleus) to

form stable atoms, and how the atoms thus formed combine to form

molecules and undergo chemical reactions.

Let us consider chemistry as the science which studies the struc-

ture, the properties and the behaviour of electrons distributed around

atomic nuclei (this includes e.g. molecular structure, molecular prop-

erties, and any kind of chemical reactions). One can realize that we

have achieved for chemistry a situation comparable to the situation

of classical mechanics after the discovery of Newton’s laws (compare

the quote on page 2 / 52-1).

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 40/ 52-1

Summary

Matter has a discrete structure, not a continuous structure! It can be

thought of being composed of ‘atoms’ (built from atomic nuclei and

electrons), which somehow form ‘molecules’, crystals, and all other

material things of this world.

The atomic mass is almost completely located in the atomic nucleus.

This nucleus (with mass number A = Z +N) is composed of

- protons (with charge qp = e, their number Z determines the che-

mical element), and

- neutrons (with charge qn = 0, their number N determines the

isotope).

The nucleus is surrounded by

- electrons (with charge qe = −e), which have kinetic energy and

are held to the nucleus by electrostatic attraction.

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 39/ 52-1

Die Quantenchemie maßt sich nicht weniger an, als samtliche chemi-

schen und physikalischen Materialeigenschaften rein theoretisch vor-

ausberechnen zu konnen, nur auf Grund eines einzigen mathemati-

schen Gesetzes, namlich der Schrodinger’schen Differentialgleichung

und der in ihr formulierten Grundeigenschaften der Materie.

Kvantova� himi� stavit svoe� zadaqe� rassqityvat~ qisto teoreti-

qeski himiqeskie i fiziqeskie svo�stva vewestva, ishod� iz odnogo

matematiqeskogo zakona — differencial~nogo uravneni� Xredinge-

ra i teh osnovnyh svo�stv materii, kotorye v nem sformulirovany.

H. Hellmann (1903-1938), Front Nauki i Tehniki (1936) 6:34-48, 7:39-50

(orig. ms. in German ms., 47 p.)

Quantum chemistry claims nothing less than the capability of the

theoretical prediction of all chemical and physical material proper-

ties, simply on the basis of one single mathematical law, i.e. the

Schrodinger differential equation and the basic properties of matter

formulated therein.

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 42/ 52-1

‘The underlying physical laws necessary for the mathematical theory

of a large part of physics and the whole of chemistry are thus com-

pletely known, and the difficulty is only that the exact application of

these laws leads to equations much too complicated to be soluble.

It therefore becomes desirable that approximate practical methods of

applying quantum mechanics should be developed, which can lead

to an explanation of the main features of complex atomic systems

without too much computation.’

P. A. M. Dirac, Proc. R. Soc. London, A 123 (1929) 714-733

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 41/ 52-1

[. . .] Quantum mechanics is essentially mathematical in character, and an under-standing of the subject without thorough knowledge of the mathematical methodsinvolved and the results of their application cannot be obtained. The student notthoroughly trained in the theory of partial differential equations and orthogonalfunctions must learn something of these subjects as he studies quantum mechan-ics. [. . .]Linus Pauling, E. Bright Wilson, Jr.: Introduction to Quantum Mechanics With

Applications to Chemistry , McGraw-Hill, New York, 1935, Preface

In so far as quantum mechanics is correct, chemical questions are problems in ap-plied mathematics. In spite of this, chemistry, because of its complexity, will notcease to be in large measure an experimental science, even as for the last threehundred years the laws governing the motions of celestial bodies have been under-stood without eliminating the need for direct observation. No chemist, however,can afford to be uninformed of a theory which systematizes all of chemistry eventhough mathematical complexity often puts exact numerical results beyond his im-mediate reach. [. . .]H. Eyring, J. Walter, G. E. Kimball: Quantum Chemistry , Wiley, New York, 1944,Preface

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 44/ 52-1

The first textbooks on quantum chemistry

1935 1937 (Russ. & Ger. eds.)

1944

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 43/ 52-1

Naturwissenschaft lasst sich mit Bildern popularisieren, aber nur mit

Mathematik verstehen

H. Tetens: Die Grenze, DIE ZEIT 37/1999 (http://www.zeit.de/archiv/1999/37/public files)

Science can be popularized with pictures, but can be understood only

with mathematics

... this is also a good point to start thinking about what ‘understanding’ in science means. With

Newton’s laws we know how a ‘mass’ behaves, but we still do not know what ‘mass’ is. It seems to

be unimportant to know that. And with the Schrodinger equation we have a tool in our hands to

find out how ‘electrons’ behave in ordinary matter, but we do not know what an ‘electron’ is.

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 46/ 52-1

Sowohl die Materie als auch die elektromagnetische Strahlung kann man sich inkeiner Weise befriedigend anschaulich vorstellen (welches Modell man auch ver-wendet), wenngleich einige Aspekte der beiden physikalischen Entitaten bestimmteAnalogien mit einem korpuskularen Modell aufweisen und andere mit einem Wellen-modell. [...] Wenn man das korpuskulare Modell allzu wortlich nimmt, d. h. wennman Photonen, Elektronen usw. als gewohnliche Korper von sehr kleinen Dimensio-nen betrachtet, macht man den gleichen Fehler, wie wenn man aus den bekanntenAnalogien zwischen elektrischen Stromen und Flussigkeitsstromungen den Schlußzoge, daß Elektrizitat eine gewohnliche Flussigkeit ist.

J. D. Fast: Entropie. Philips, Eindhoven, 1960, § 4.2, p. 175

No completely satisfactory pictorial representation can be made of either matter orelectromagnetic radiation, although some aspects of both physical realities showanalogy with a corpuscular model and others with a wave model. [...] To takethe corpuscular model too literally, i.e. to regard photons, electrons etc. as normalbodies on a greatly reduced scale, is to make the same mistake that would bemade by concluding from the well-known analogy between electrical currents andhydraulic currents that electricity is a normal fluid.

J. D. Fast: Entropy. 2nd ed., Gordon & Breach, New York, 1968, § 4.2, pp.171-172

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 45/ 52-1

The ‘periodic’ table of the chemical elements

(1900-1949)

1

2

3

4

5

6

7

IA

1

H

3

Li

11

Na

19

K

37

Rb

55

Cs

87

Fr

IIA

4

Be12

Mg

20

Ca

38

Sr

56

Ba

88

Ra

IIIB

21

Sc

39

Y

∗∗

∗ 57

La58

Ce59

Pr60

Nd61

Pm62

Sm63

Eu64

Gd65

Tb66

Dy67

Ho68

Er69

Tm70

Yb71

Lu∗∗ 89

Ac90

Th91

Pa92

U93

Np94

Pu95

Am96

Cm

IVB

22

Ti

40

Zr

72

Hf

VB

23

V

41

Nb

73

Ta

VIB

24

Cr

42

Mo

74

W

VIIB

25

Mn

43

Tc

75

Re

26

Fe

44

Ru

76

Os

VIII

27

Co

45

Rh

77

Ir

28

Ni

46

Pd

78

Pt

IB

29

Cu47

Ag

79

Au

IIB

30

Zn

48

Cd80

Hg

IIIA

5

B

13

Al

31

Ga

49

In

81

Tl

IVA

6

C

14

Si

32

Ge

50

Sn

82

Pb

VA

7

N

15

P

33

As

51

Sb

83

Bi

VIA

8

O

16

S

34

Se

52

Te

84

Po

VIIA

9

F

17

Cl

35

Br

53

I

85

At

VIIIA

2

He

10

Ne

18

Ar

36

Kr

54

Xe

86

Rn

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 48/ 52-1

«Plus les sciences physiques ont fait de progres, plus elles ont tendu a rentrer dans ledomaine des mathematiques, qui est une espece de centre vers lequel elles viennentconverger. On pourrait meme juger du degre de perfection auquel une science est

parvenue, par la facilite plus ou moins grande, avec laquelle elle se laisse aborder

par le calcul.»

L. A. J. Quetelet (1796-1874), 1827 — citation d’apres: Edouard Mailly: Essai sur la vie et les

ouvrages de L.-A.-J. Quetelet, F. Hayez, Bruxelles, 1875, p. 55

The more progress physical sciences make, the more they tend to enter the domainof mathematics, which is a kind of centre to which they all converge. We may evenjudge the degree of perfection to which a science has arrived by the facility withwhich it may be submitted to calculation.

A. Quetelet (1796-1874) — ref. for engl. version: GAUSSIAN fortune quotation

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 47/ 52-1

Symbols and names of the chemical elements (Nov. 2003, http://www.iupac.org/)

Ac ActiniumAg SilverAl Aluminium

(Aluminum)Am AmericiumAr ArgonAs ArsenicAt AstatineAu GoldB BoronBa BariumBe BerylliumBh BohriumBi BismuthBk BerkeliumBr BromineC CarbonCa CalciumCd CadmiumCe CeriumCf CaliforniumCl ChlorineCm CuriumCo CobaltCr ChromiumCs Caesium

(Cesium)Cu CopperDb Dubnium

Ds DarmstadtiumDy DysprosiumEr ErbiumEs EinsteiniumEu EuropiumF FluorineFe IronFm FermiumFr FranciumGa GalliumGd GadoliniumGe GermaniumH HydrogenHe HeliumHf HafniumHg MercuryHo HolmiumHs HassiumI IodineIn IndiumIr IridiumK PotassiumKr KryptonLa LanthanumLi LithiumLr LawrenciumLu LutetiumMd MendeleviumMg Magnesium

Mn ManganeseMo MolybdenumMt MeitneriumN NitrogenNa SodiumNb NiobiumNd NeodymiumNe NeonNi NickelNo NobeliumNp NeptuniumO OxygenOs OsmiumP PhosphorusPa ProtactiniumPb LeadPd PalladiumPm PromethiumPo PoloniumPr PraseodymiumPt PlatinumPu PlutoniumRa RadiumRb RubidiumRe RheniumRf RutherfordiumRh RhodiumRn RadonRu Ruthenium

S Sulphur

(Sulfur)

Sb Antimony

Sc Scandium

Se Selenium

Sg Seaborgium

Si Silicon

Sm Samarium

Sn Tin

Sr Strontium

Ta Tantalum

Tb Terbium

Tc Technetium

Te Tellurium

Th Thorium

Ti Titanium

Tl Thallium

Tm Thulium

U Uranium

V Vanadium

W Tungsten

(Wolfram)

Xe Xenon

Y Yttrium

Yb Ytterbium

Zn Zinc

Zr Zirconium

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 50/ 52-1

The ‘periodic’ table of the chemical elements

(1950-1999)

1

2

3

4

5

6

7

1

1

H

3

Li

11

Na

19

K

37

Rb

55

Cs

87

Fr

2

4

Be12

Mg

20

Ca

38

Sr

56

Ba

88

Ra

3

21

Sc

39

Y

∗∗

∗ 57

La58

Ce59

Pr60

Nd61

Pm62

Sm63

Eu64

Gd65

Tb66

Dy67

Ho68

Er69

Tm70

Yb71

Lu∗∗ 89

Ac90

Th91

Pa92

U93

Np94

Pu95

Am96

Cm97

Bk98

Cf99

Es100

Fm101

Md102

No103

Lr

4

22

Ti

40

Zr

72

Hf

104

Rf

5

23

V

41

Nb

73

Ta

105

Db

6

24

Cr

42

Mo

74

W106

Sg

7

25

Mn

43

Tc

75

Re

107

Bh

8

26

Fe

44

Ru

76

Os

108

Hs

9

27

Co

45

Rh

77

Ir

109

Mt

10

28

Ni

46

Pd

78

Pt

110

Ds

11

29

Cu47

Ag

79

Au

111

X

12

30

Zn

48

Cd80

Hg112

(X)

13

5

B

13

Al

31

Ga

49

In

81

Tl

14

6

C

14

Si

32

Ge

50

Sn

82

Pb114

(X)

15

7

N

15

P

33

As

51

Sb

83

Bi

16

8

O

16

S

34

Se

52

Te

84

Po116

(X)

17

9

F

17

Cl

35

Br

53

I

85

At

18

2

He

10

Ne

18

Ar

36

Kr

54

Xe

86

Rn118

(?)

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 49/ 52-1

Fundamental physical constantsa

Quantity Symbol Value in SI unitsb

speed of light in vacuum c, c0 299792458 m s−1

magnetic constant µ0 4π × 10−7 N A−2

(vacuum permeability)electric constant 1/(µ0 c

2) ε0 8.854187817 . . . × 10−12 F m−1

(vacuum permittivity)Planck constant h 6.6260693(11) × 10−34 J s

h/(2π) ~ 1.05457168(18) × 10−34 J selementary charge e 1.60217653(14) × 10−19 Celectron mass me 9.1093826(16) × 10−31 kgproton mass mp 1.67262171(29) × 10−27 kgneutron mass mn 1.67492728(29) × 10−27 kgproton-electron mass ratio mp/me 1836.15267261(85)Sommerfeld fine-structure constant α 7.297352568(24) × 10−3

4πε0~c/e2 α−1 137.03599911(46)Compton wavelength h/(mec) λC 2.426310238(16) × 10−12 mBohr magneton e~/(2me) µB 927.400949(80) × 10−26 J T−1

nuclear magneton e~/(2mp) µN 5.05078343(43) × 10−27 J T−1

Rydberg constant α2mec/(2h) R∞ 10973731.568525(73) m−1

Avogadro constant NA, L 6.0221415(19) × 1023 mol−1

atomic mass constant 112m(12C) mu 1.66053886(28) × 10−27 kg

Faraday constant NAe F 96485.3383(83) C mol−1

molar gas constant R 8.314472(15) J mol−1 K−1

Boltzmann constant R/NA kB 1.3806505(24) × 10−23 J K−1

aCODATA recommended values 2002 (http://physics.nist.gov/constants/).

bThe standard deviation uncertainty in the least significant digits is given in parentheses.

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 52/ 52-1

Prefixes for physical units

n Powers of ten (10±n), prefixes, and symbols

1 10−1 deci- d 10+1 deca- da2 10−2 centi- c 10+2 hecto- h3 10−3 milli- m 10+3 kilo- k6 10−6 micro- µ 10+6 mega- M9 10−9 nano- n 10+9 giga- G

12 10−12 pico- p 10+12 tera- T15 10−15 femto- f 10+15 peta- P18 10−18 atto- a 10+18 exa- E21 10−21 zepto- z 10+21 zetta- Z24 10−24 yocto- y 10+24 yotta- Y

FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-12 51/ 52-1