Download The Schrödinger equation Combining the classical Hamilton

1. Historical introduction
2. The Schrödinger 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 — 2004-05-26
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The Schrödinger equation
Combining the classical Hamilton function for a free particle (with
mass m), H = p2 /(2m), with the de Broglie relation p = ~k and the
Planck relation E = ε = ~ω leads to
H=E=T =
p2
~2 k 2
=
= ~ω ,
2m
2m
k 2 = kx 2 + ky 2 + kz 2
(20)
With eq. (19), this relation is readily transferred into an operator
identity for a differential equation (wave equation?):
~2
2m

~2 
c
H= −
2m


∂ 2
−i
+
∂x
~2
(kx 2 + ky 2 + kz 2 ) = ~ω
2m

!2
2
∂
∂
∂

=~ i
+ −i
−i
∂y
∂z
∂t

!2
2
∂
(−i ~ ∇)2
∂ 2
∂
∂

+
=
= i~
+
∂x
∂y
∂z
2m
∂t
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-26
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This operator identity can be applied to a suitable mathematical object, e.g. a complex-valued scalar function ψ(r , t) of position r and
time t, to give the time-dependent Schrödinger equation:
c = i ~ ∂ψ
Hψ
c− i~ ∂
H
(22)
⇔
ψ=0
∂t
∂t
The function ψ(r , t) is called ‘state function’, or ‘wave function’.
As in classical mechanics, where the Hamilton function includes kinetic
and potential energies, H = T + V , the Hamilton operator is extended
c = Tb + Vb .
to include parts for both kinetic and potential energy, H
For application to scalar functions ψ(r , t) we thus have:
b = − i~∇
p
~2 2
~2
(−i ~ ∇)2
= −
∇ = −
∆
Tb =
2m
2m
2m
Vb = V (r , t) ×
−→
p
p2
2m
V = V (r , t)
T =
−→
−→
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-26
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(24)
(25)
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Gradient operator ∇ and Laplace operator ∆ in cartesian coordinates:


∂


 ∂x 


 ∂ 
∂2
∂2
∂2


2
∇=
∆=∇ =∇·∇=
+ 2+ 2
,
 ∂y 
∂x2
∂y
∂z




 ∂ 
∂z
From eq. (22) and its complex conjugate (assuming V to be real),
∂
Tb + Vb − i ~
follows
∂t
ψ=0
∂
Tb + Vb + i ~
∂t
ψ∗ = 0
(26)
∂
∂
ψ Tb + Vb − i ~
ψ=0
ψ Tb + Vb + i ~
ψ∗ = 0
(27)
∂t
∂t
The difference of these two expressions is (the terms with V cancel)
∗
ψ∗
!
!
~2 2
∂ψ
∂ψ ∗
~2 2
∗
∗
−
∇ ψ − i~ψ
−ψ −
∇ ψ − i~ψ
=0
2m
∂t
2m
∂t
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-26
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which gives, after re-ordering
∂ψ
∂ψ ∗
i~ ∗ 2
i~
(29)
+ψ
−
ψ ∇ ψ+
ψ ∇2 ψ ∗ = 0
∂t
∂t
2m
2m
This is the continuity equation (known, e.g., from electrodynamics)
ψ∗
∂ρ
+∇·j =0
(30)
∂t
with real-valued distributions for density ρ and current j (of some
property) defined as
i~
ψ ∗ ∇ψ − ψ ∇ψ ∗
(31)
2m
These are density and current for the quantity ‘probability’ (M. Born),
so that units can be assigned as follows (in 3-dimensional cases):
[ρ] = m−3, [ψ] = m−3/2, and also [jx,y,z ] = (ms−1) m−3.
Normalization of the probability density distribution:
ρ = ψ∗ ψ ,
Z
V
j= −
ρ dr =
Z
V
ψ ∗ ψ dr = N 2 < ∞
(32)
The integration may be extended over the full space R3, if the integral
exists.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-26
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When the Hamilton operator (Hamiltonian) does not depend on time,
c 6= H(t),
c
i.e. H
eq. (22) has solutions of the form∗
ψ(r , t) = φ(r ) exp (−i E t/~)
(33)
which represent stationary states, provided the spatial part φ(r ) satisfies the time-independent Schrödinger equation:
c =Eφ
Hφ
⇔
c−E φ =0
H
(34)
The separation constant E is the energy of the stationary state.
The probability density for stationary states does not depend on time:
ρ = ψ ∗(r , t) ψ(r , t) = φ∗ (r ) φ(r )
(35)
Solving eq. (34) is equivalent to a variation principle, applied to the
integral (functional of φ)
H[φ] =
Z
c dr
φ∗ Hφ
(36)
∗ This is an application of a general theorem from the theory of partial differential equations (PDE):
If the partial differential operator is a sum of terms, where each term acts on a different coordinate
(or set of coordinates), then there are solutions in product form, and the PDE can be split into
lower-dimensional parts.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-26
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under the condition that φ remains normalized (use Lagrange’s method
of undetermined multipliers).
Consider the functional
L[φ] = H[φ] − λ
Z
∗
φ φ dr − 1 =
Z
∗c
φ Hφ dr − λ
(λ is a yet undetermined Lagrange parameter)
Z
∗
φ φ dr − 1
(37)
Condition for stationarity: δL = 0, with
δL =
=
Z
Z
∗c
(δφ) Hφ dr +
Z
∗c
φ Hδφ dr − λ
Z
∗
(δφ) φ dr +
Z h
i
∗ c
c
(δφ) H − λ φ dr +
H − λ φ∗ δφ dr
Z
∗
φ δφ dr
(38)
The first integral is the complex conjugate of the second integral (if
c∗ = H).
c
H
Since δL = 0 shall hold for arbitrary variations δφ, it follows:
c
H −λ φ=0
⇒
λ=E
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-26
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Together with boundary conditions for φ, the time-independent Schrödinger equation thus constitues an eigenvalue problem, which has to
be solved for pairs (E, φ) of eigenvalues E and eigenfunctions φ.
Again a situation comparable to e.g. classical mechanics is found:
There exists a variation principle behind the working equations.
In classical mechanics, the working equations are the Lagrange (or
Hamilton) equations (one equation, or a pair of equations, for every degree of freedom), whereas in quantum mechanics, the working
equation (for stationary states of the system under study) is the timeindependent Schrödinger equation.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-26
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The free particle
This is the quantum mechanical analogue of the situation considered
with Newton’s first law of motion: The potential energy V is constant
(may be set to zero), and the particle (with mass m) has only kinetic
energy, so that
Vb = 0 ,
c = Tb + Vb = Tb = −
H
~2 2
∇
2m
(40)
It follows immediately from our ‘derivation’ of the time-dependent
Schrödinger equation (22), that plane waves are acceptable solutions
(~ k)2
(41)
≥0
2m
which are also eigenfunctions to the linear momentum (vector) operb with eigenvalue (vector) ~ k:
ator p
ψ(r , t) = C exp [ i(k · r − ω t) ] ,
E = ~ω =
b ψ = − i~∇ψ = ~kψ
p
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-26
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The probability density is constant, ρ = ψ ∗ ψ = C ∗ C = |C|2, everywhere and at every time. This is O.K. for a plane wave.
The most general solution is obtained by continuous superposition
(note the relation to Fourier transformation)
Z
~ k2
ψ(r , t) =
ω(k) =
(43)
2m
since integration in k-space can be interchanged with differentiation
with respect to ordinary space coordinates or time.
C(k) ei(k·r−ω t) dk ,
A suitable choice for C(k) leads to a valid representation of a ‘free
particle’ (a ‘well located’ object), which is also known as a ‘wave
packet’.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-26
10/ 29-2
The particle in a box
This is the quantum mechanical analogue of a classical particle (with
mass m) confined to a 1-, 2- or 3-dimensional region of space: The
potential energy V is constant (may be set to zero) within this region,
but infinitely high outside. For a 3-dimensional rectangular box with
lengths Lx , Ly , and Lz :
Vb =

 0
 ∞
0 ≤ x ≤ Lx , 0 ≤ y ≤ Ly , 0 ≤ z ≤ Lz
elsewhere
(44)
Inside the box, the time-dependent Schrödinger equation has the form
!
~2 2
∂
ψ=0
−
∇ −i~
2m
∂t
which has stationary solutions of the form
c−i~ ∂ ψ =
H
∂t
ψ(r , t) = φ(r ) e−i E t/~ ,
(45)
φ(r ) = X(x) Y (y) Z(z) ,
E = E x + Ey + Ez
(46)
and φ(r ) has to vanish at the boundaries of the box.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-26
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The problem can be reduced to one-dimensional problems of the following form:
!
~2 d 2
− Ex X(x) = 0
−
2m dx2
d2 X
2 X(x) = 0
2 = 2m E
(47)
+
k
k
x
x
x
dx2
~2
with boundary conditions X(0) = 0 and X(Lx ) = 0. The general
solution of the ordinary differential equation (47) is
X(x) = c1 ei kx x + c2 e−i kx x = A cos (kx x) + B sin (kx x)
(48)
with A = c1 + c2 , B = i(c1 − c2). The boundary conditions lead to
A = 0 (from X(0) = 0) and kx Lx = n π with n ∈ Z (from X(Lx ) = 0),
so that the acceptable solutions are
Xn(x) = B sin (kx x) ,
nπ
kx =
,
Lx
(~ kx )2
h2
n 2
Ex =
(49)
=
2m
8 m Lx
with n > 0† .
† n = 0 gives X (x) ≡ 0, while n < 0 gives the same energy and probability density, since X
−n = −Xn .
0
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-26
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The coefficient B is determined by normalization of the probability
density:
1=
Z L
x
0
X ∗(x) X(x) dx = B 2
Z L
x
0
sin2 (kx x) dx
→
B=
s
2
(50)
Lx
The boundary conditions lead to quantization. This is a general
result, as well as the fact, that the number of zeros (nodes) of the
solutions Xn(x) increases with the energy (compare with a string).
The energies (eigenvalues) grow quadratically as a function of n,
Ex(n) ∝ n2, and the lowest energy is higher than the potential energy
minimum, Ex(n = 1) > 0. The ‘particle in a 1-dimensional box’ has at
least this energy, even at temperature T = 0 K (zero-point energy).
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-26
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State functions Xn(x) and probability density distributions
[Xn(x)]∗ Xn(x) (for n = 1, 2, 3) for a particle in a 1-dimensional box
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FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-26
14/ 29-2
Extension to the general case of a 3-dimensional rectangular box is
straightforward:
s
!
8
x
y
z
sin nπ
φnkl (r ) =
sin kπ
sin lπ
Lx Ly Lz
Lx
Ly
Lz
!
h2 n 2
k2
l2
Enkl = Ex + Ey + Ez =
+
+
8m L2
L2
L2
x
y
z
The ground state (i.e. the state with lowest possible energy):
s
8
x
y
sin π
sin π
φ111 (r ) =
Lx Ly Lz
Lx
Ly
!
2
1
1
1
h
+
+
E111 =
8m L2
L2
L2
x
y
z
!
z
sin π
Lz
(51)
(52)
(53)
(54)
Whenever there is a rational relation between any two (or all three)
2
2
squared lengths L2
x , Ly , and Lz , degeneracies occur, i.e. several
state functions belong to the same energy eigenvalue. E. g. in the
case of a cubic box (Lx = Ly = Lz = L): Ennn is non-degenerate,
Ennl (n 6= l) is triply degenerate, and Enkl (n, k, l all different) is
sixfold degenerate.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-26
15/ 29-2
Energy levels Enkl and degrees of degeneracy g for a particle in a
3-dimensional cubic box
h2 2
2
2
Enkl =
n +k +l
8mL2
n
k
l
n2 + k 2 + l 2
g
1
1
1
2
1
2
2
1
1
3
2
1
1
2
2
3
3
1
1
2
1
2
1
2
1
3
1
2
2
3
1
3
1
2
1
2
1
1
2
2
1
3
1
1
2
3
2
3
1
2
1
3
1
6
3
9
3
11
3
12
1
............ E223 (3)
E
h2/(8mL2 )
15 ............
E123
............ E
............ E222
113
10 ............
E122
5
14
(6)
(1)
(3)
(3)
............ E112 (3)
............ E111 (1)
6
0
...............................................................................................................................................
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-26
16/ 29-2
Special functions
The gamma function:
Γ(a) =
Z ∞
0
ta−1e−tdt
Γ(a + 1) = a Γ(a)
πa
Γ(1 + a) Γ(1 − a) =
sin (π a)
Γ(n + 1) = n!
√
Γ(1/2) = π
(a > 0)
(n ∈ N)
The Pochhammer‡ symbol:
(a)k =

 1
k=0
Γ(a + k)
=
 a(a + 1) · . . . · (a + k − 1) k > 0
Γ(a)
(1)n = n!
(−a)k = (−1)k (a − k + 1)k
‡ L. Pochhammer (1841-1920)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-26
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The confluent hypergeometric function (Kummer§ function):
1F1(a; c; x) =
∞
X
(a)k xk
k=0
(c)k k!
=1+
a
a(a + 1) x2
x+
+ ...
c
c(c + 1) 2!
is a solution of the Kummer differential equation
x y 00 + (c − x) y 0 − a y = 0
The Gauß’ hypergeometric function:
2F1(a, b; c; x) =
∞
X
(a)k (b)k xk
k=0
(c)k
k!
a(a + 1)b(b + 1) x2
ab
x+
+ ...
=1+
c
c(c + 1)
2!
is a solution of the Gauß differential equation
x (1 − x) y 00 + [ c − (a + b + 1) x ] y 0 − ab y = 0
§ E. E. Kummer (1810-1893)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-26
18/ 29-2
The harmonic oscillator
The 1-dimensional harmonic oscillator is the quantum mechanical
analogue of a classical particle (with mass m), moving in a parabolic
potential or, equivalently, moving under a restoring force given by
Hooke’s law (with force constant K)
V (x) =
1
K x2
2
Fx = −
⇔
dV
= −Kx
dx
(55)
Classically, the particle can oscillate with arbitrary energy, the oscillating frequency ν being related to the force constant K through
ω = 2πν =
s
K
m
⇒
K = m ω2
(56)
Quantum mechanically, the Schrödinger equation, corresponding to
the potential energy V (x) given above, has to be solved.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-26
19/ 29-2
Stationary states φ(x), with −∞ ≤ x ≤ ∞, are solutions of
!
~2 d 2
1
2
−
K
x
− E φ(x) = 0
+
2m dx2
2
!
d2
+ k2 − λ2 x2 φ(x) = 0
2
dx
(57)
(58)
with
2m
mω
,
λ
=
E
~2
~
√
The variable transformation ξ = λ x gives
k2 =
!
d2
+ ε − ξ 2 φ(ξ) = 0 ,
2
dξ
ε=
k2
2E
=
λ
~ω
(59)
Physically acceptable (i.e. normalizable) solutions of this ordinary
differential equation exist only for
~ω
(60)
2
Quantization, due to boundary conditions (φ(ξ) → 0 for |ξ| → ∞).
εn = 2n + 1 , n = 0, 1, 2, . . .
⇒
En = (2n + 1)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-26
20/ 29-2
These solutions are, in explicit form,
1
φn(x) = Nn Hn(ξ) exp − ξ 2 ,
2
r
ξ=
s
mω
x,
~
ω=
s
K
m
(61)
m ω 1/2
1
~ω
Nn =
,
E
=
(2n
+
1)
, n = 0, 1, 2, . . .
n
2n n! π ~
2
As the mass m increases, the energy spacing En+1 − En = ~ ω decreases. Thus, the classical behaviour is recovered in the limit m → ∞
(and also for h → 0).
The Hermite¶ polynomials Hn (ξ) can be expressed in terms of confluent hypergeometric functions, and obey a recurrence relation:
(2k)!
2
1F1(−k; 1/2; ξ )
k!
(2k
+ 1)!
H2k+1(ξ)= (−1)k
(2ξ) 1F1(−k; 3/2; ξ 2 )
k!
Hn+1(ξ) = 2ξHn (ξ) − 2nHn−1(ξ)
(n ≥ 1)
H2k (ξ) = (−1)k
(62)
(63)
(64)
¶ Ch. Hermite (1822-1901)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-26
21/ 29-2
The first Hermite polynomials
n
Hn (ξ)
0
1
2
3
4
5
1
2ξ
4ξ 2 − 2
8ξ 3 − 12ξ
16ξ 4 − 48ξ 2 + 12
32ξ 5 − 160ξ 3 + 120ξ
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-26
22/ 29-2
The state function for the ground state (n = 0) is a Gauß func√
tion φ0 (x) = exp (−ξ 2/2)/ 4 π, with energy E0 = ~ ω/2 (zero-point
energy).
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-26
23/ 29-2
State functions φn(x) and probability density distributions
[φn (x)]∗φn(x) (for n = 0, 1, 2, 3, 4, and 5) for the
1-dimensional harmonic oscillator
.
V (x) ....
..
...
.
...
φ5(x), E5 = 11~ ω/2
..
n=5
...
...
...
..
...
...
...
.
φ4(x), E4 = 9~ ω/2
n=4
..
...
.
.
...
.
...
..
.
...
.
φ3(x), E3 = 7~ ω/2
..
n=3
...
.
...
...
...
.
..
...
.
...
φ2 (x), E2 = 5~ ω/2
..
n=2
...
...
...
.
..
...
.
...
..
φ1 (x), E1 = 3~ ω/2
n=1
...
...
...
..
.
...
..
...
.
.
...
φ0(x), E0 = ~ ω/2
n=0
.... .....
............
−5
5 x
............................
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......
.
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FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-26
24/ 29-2
The general molecule
After these one-particle examples the step to the case of an arbitrary
molecule can be done quite easily, if we restrict ourselves to the nonrelativistic treatment, and omit external (electric or magnetic) fields.
Hamilton operator for an arbitrary molecule, built from N nuclei (with
charges ZK and masses MK ), and n electrons (definition of the system
under study, in space-fixed coordinates):
c = Tb + V
b
H
(65)
Kinetic energy operator (based on single-particle kinetic energy):
Tb = Tbn + Tbe ,
Tbn =
N
X
K=1
!
~2
−
∆K ,
2 MK
Tbe = −
n
~2 X
∆i
2 me i=1
(66)
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-26
25/ 29-2
Potential energy operator (based on the Coulomb potential energy):
Vb = Vbnn + Vben + Vbee
−1
N
X
e2 NX
ZK ZL
b
Vnn = +
κ0 K=1 L=K+1 RKL
n X
N
e2 X
ZK
b
Ven = −
κ0 i=1 K=1 riK
n
X X
e2 n−1
1
b
Vee = +
κ0 i=1 j=i+1 rij
(67)
(68)
(69)
(70)
One can try to separate motions related to the various degrees of
freedom, as in classical mechanics. This can be done exactly for
the overall translation of the center of total molecular mass, but a
separation of the internal nuclear motions (molecular rotation and
nuclear vibration) from the electronic motion is possible only in an
approximate way.
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-26
26/ 29-2
Switching to molecule-fixed coordinates, and making a sequence of
well-defined steps (adiabatic ansatz, adiabatic approximation, BornOppenheimer approximation) leads to the so called ‘electronic Schrödinger equation for space-fixed (or clamped) nuclei’:
c Ψ =E Ψ ,
H
k k
el k
k = (0), 1, 2, . . .
(71)
The ‘electronic Hamiltonian’ can be split into zero-, one- and twoelectron parts:
c =H
c +H
c +H
c
H
0
1
2
el
(72)
−1
N
X
e2 NX
ZK ZL
c
H0 =
κ0 K=1 L=K+1 RKL


n
N
2
2
X
X
e
~
ZK 
c =
−
H
∆i −
1
2 me
κ0 K=1 riK
i=1
n
X X
1
e2 n−1
c
H2 =
κ0 i=1 j=i+1 rij
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-26
27/ 29-2
c contributes only an additive constant to
The zero-electron part H
0
c ).
the total energy (and is thus sometimes excluded from H
el
The electronic energies Ek (eigenvalues) and the corresponding electronic state functions Ψk (eigenfunctions) depend parametrically on
the internal coordinates of the nuclear configuration.
In addition to this parametrical dependence on nuclear configuration,
the state functions are functions of the (space and spin) coordinates
of all n electrons (i.e. all those coordinates which have to or may
c ):
appear due to the form of the Hamilton operator H
el
Ψk = Ψk (x1, x2, . . . , xn) ,
(73)
x = (r , σ) combines space and spin coordinates (the latter can take
1
only two values: σ = + 1
2 for ↑ or α spin, σ = − 2 for ↓ or β spin).
The electronic energy Ek , also known as potential energy hypersurface
(PES), is a scalar function in an f -dimensional space, where f depends
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-26
28/ 29-2
on the number of nuclei N :
non-linear molecule: f = 3N − 6 ,
linear molecule: f = 3N − 5 .
For a diatomic molecule (N = 2) f = 1, and the electronic energy
simply reduces to potential energy curves Ek (R), where R denotes
the internuclear distance.
All quantum chemical program codes only determine approximate solutions (Ek , Ψk ) to eq. (71). A quantum mechanical treatment of
nuclear motion, where the PES Ek would appear as (a part of) an
effective potential energy operator, is not attempted.
Most of present-day quantum mechanical methods only differ
(1) in the degree of sophistication applied to approximate the state
functions Ψk and/or
(2) the presence or absence of further approximations made with rec and the evaluation of the
spect to the electronic Hamiltonian H
el
energies Ek
(this statement excludes methods based on density functional theory,
DFT).
FAQC — D. Andrae, Theoretical Chemistry, U Bielefeld — 2004-05-26
29/ 29-2