foundations and applications of quantum chemistry and applications of quantum chemistry or...
TRANSCRIPT
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 =
hν
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 =
hν
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