Download PDF 2

H atom solution
Contents
1 Introduction
2
2 Coordinate system
2
3 Variable separation
4
4 Wavefunction solutions
4.1 Solution for Φ . . . . . . . . . .
4.2 Solution for Θ . . . . . . . . . .
4.3 Combined Φ and Θ solution . .
4.4 Solution for R . . . . . . . . . .
4.5 Complete wavefunction solution
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
6
6
7
8
8
10
5 Electron in the H atom
10
5.1 Energy of the electron . . . . . . . . . . . . . . . . . . . . . . 10
5.2 Radial function . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.3 Angular function . . . . . . . . . . . . . . . . . . . . . . . . . 13
6 Charge distribution
6.1 Radial distribution . . . . . . . . . . .
6.2 Three dimensional charge distribution .
6.2.1 s orbitals . . . . . . . . . . . . .
6.2.2 p orbtials . . . . . . . . . . . .
6.2.3 d orbitals . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
13
14
16
16
16
18
7 Quantum numbers
7.1 Principal quantum number, n
7.2 Orbital quantum number, l . .
7.3 Magnetic quantum number, m
7.4 Spin quantum number . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
18
19
19
20
21
1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
MM3010: Physics of Materials
8 Exclusion principle
1
22
Introduction
The H atom is an example of applying the Schrödinger equation to solve the
energy of the electron in a central potential. The solution can also be used
for other one electron systems. It is the only physical system for which a full
solution for the wavefunction is possible, excluding spin. The H atom solutions can be extended, with approximations, to atoms with multiple electrons
to obtain approximate solutions of the electron energies that match with experimental results.
The H atom consists of a nucleus with one proton (unit positive charge,
Z = 1) with one electron. The mass of the nucleus is approximately 1840
times greater than that the electron so that it is a valid approximation to
consider the nucleus to be stationery. A more accurate model would consider
the nucleus stationery but replace the electron mass by the reduced mass give
by
me mn
(1)
µe =
me + mn
where µe is the reduced mass and me and mn are the electron and nucleus
mass respectively. For the H atom, since mn ≈ 1840 me , equation 1 gives
me or 0.99945me . For all practical purposes we can take the mass
µe = 1840
1841
to be the mass of the free electron (i.e. 9.1 × 10−31 kg).
2
Coordinate system
The H atom is an example of a system under a central potential i.e. the
potential experienced by an electron is inversely proportional to the distance.
Thus, an electron located at a distance r from the nucleus, experiences a
potential given by
−Ze2
V (r) =
(2)
4π0 r
where 0 is the permittivity of free space with a value of 8.854 × 10−12 F m−1
and Z is the number of positive charges in the nucleus. For the H atom,
Z = 1. In a 3D carteisan coordinate system, equation 2 is rewritten as
V (x, y, z) =
−Ze2
p
4π0 x2 + y 2 + z 2
2
(3)
MM3010: Physics of Materials
Figure 1: The spherical coordinate system, showing r, θ and φ.
The potential function, V , is then used in the three dimensional time independent Schrödinger equation
~2
∇2 ψ + V ψ = E ψ
2me
(4)
where ψ is the 3D wavefunction of the electron. It is possible to solve this
system using Cartesian coordinates and obtain a solution of ψ in terms of
(x, y, z). But this solution is complicated since variable separation cannot
be done, V (x, y, z) involves all three coordinates. Given that the H atom
potential is spherically symmetric, since it only depends on distance, r, it
would be easier to solve this system in the spherical coordinate system.
In the spherical system, (x, y, z) is replaced by (r, θ, φ), as shown in figure 1.
In the Cartesian system the limits for (x, y, z) are (−∞ to ∞). For the
spherical coordinate system the limits are
r → 0 to ∞
θ → 0 to π
φ → 0 to 2π
3
(5)
MM3010: Physics of Materials
The two coordinates are related by
x = r sin θ cos φ
y = r sin θ sin φ
z = r cos θ
(6)
From equation 6 it is possible to obtain (r, θ, φ) in terms of (x, y, z). This
gives
p
r =
x2 + y 2 + z 2
y
tan φ =
(7)
x
z
cos θ = p
x2 + y 2 + z 2
The advantage of using spherical coordinate system is that the potential is
only a function of r and not θ or φ. Hence it is possible to use variable
separation to separate the electron wavefunction into function of r, θ and φ
alone. This will help in solving the Schrödinger equation to obtain electron
energies and probabilities, i.e. wavefunctions.
3
Variable separation
Consider the time independent Schrödinger equations, as shown in equation
4. The potential for the H atom is given by 2 with Z = 1. E is the total
energy of the electron in the H atom, which we want to obtain. ∇2 , called
the Laplacian or the Laplace operator, is a differential operator that depends
on the coordinate system. For the Cartesian system this just becomes
∇2 =
∂2
∂2
∂2
+
+
∂x2
∂y 2
∂z 2
(8)
which can be substituted in the Schrödiger equation. But given that the H
atom solution is in the spherical coordinate system the Laplacian needs to
be converted to this coordinate system. This can be done by using the chain
rule for partial derivatives,
∂
∂ ∂r
∂ ∂θ
∂ ∂φ
=
+
+
∂x
∂r ∂x
∂θ ∂x
∂φ ∂x
(9)
with similar expressions for the y and z terms. Using equation 9 it is possible
to write equation 8 in spherical coordinates and the final expression is given
4
MM3010: Physics of Materials
by
∇2 =
1
∂
∂
1
1 ∂ 2∂
∂2
(r
)
+
(sin
θ
)
+
r2 ∂r ∂r
r2 sin θ ∂θ
∂θ
r2 sin2 θ ∂φ2
(10)
A detailed derivation is given in Planet math website.
An advantage of using the spherical coordinate system is that since the potential only depends on r, see equation 2, it is possible to separate the electron
wavefunction into functions of the individual variables. Thus it is possible to
write
ψ(r, θ, φ) = R(r)Θ(θ)Φ(φ)
(11)
where R(r), Θ(θ), and Φ(φ) are functions of r, θ, φ alone. This will help in
simplifying the Schródinger equation to the individual terms which can be
solved to obtain the electron wavefunction.
Consider the effect of variable separation, equation 11, on the H atom equation, 4. Substituting the various terms, equation 4 becomes
1
d
dΘ
1
d2 Φ
1 d 2 dR
Ze2
~2
(r
)
+
(sin
)
+
=E
−
−
2me r2 R dr
dr
Θ sin θ dθ
dθ
4π0 r
Φ sin2 θ dφ2
(12)
2
2me r2
Rearranging equation 12, by multiplying with ~2 sin θ it is possible to
write
− sin2 θ d 2 dR
sin θ d
dΘ
2Ze2 me r sin2 θ
2Em2e sin2 θ
1 d2 Φ
(r
)−
(sin θ ) −
−
=
R dr
dr
Θ dθ
dθ
4π0 ~2
~2
Φ dφ2
(13)
The left hand side is a function of (r, θ) only while the right hand side is a
function of φ only. By the principle of variable separation the two sides must
be equal to a constant, independent of (r, θ, φ).
It is possible to do a further variable separation to separate the r and the
θ part. This can be done by equating the two sides of equation 13 to a
constant, −m2 . Thus, equation 13 reads
sin θ d
dΘ
2Ze2 me r sin2 θ
2Em2e sin2 θ
− sin2 θ d 2 dR
(r
) −
(sin θ ) −
−
= −m2
2
2
R dr
dr
Θ dθ
dθ
4π0 ~
~
1 d2 Φ
= −m2
Φ dφ2
(14)
There is a particular reason for choosing the constant as −m2 . Later, this
constant m will be called the magnetic quantum number. The top half
5
MM3010: Physics of Materials
of equation 14 containing the r and θ terms can be further rearranged to
separate the two variables.
−
2Ze2 me r
d
dΘ
m2
1 d 2 dR
2Em2e
1
(r
)−
(sin
θ
)
−
(15)
−
=
R dr
dr
4π0 ~2
~2
Θ sin θ dθ
dθ
sin2 θ
Once again we have an equation where the left side is only a function of r and
the right side only a function of θ. Using the concept of variable separation
once again both sides must be equal to a constant, and this constant can be
set to −l(l + 1). Later we shall call this l the orbital quantum number.
Again this particular choice of constant is to make the solution easier to
appreciate. Thus, using equations 14 and 15, it is possible to write the
general Schrödinger equation, containing ψ(r, θ, φ) into three equations, each
containing only one variable.
2me
~2 l(l + 1)
Ze2
1 d 2 dR
(r
) +
−
E +
R = 0
r2 dr
dr
~2
4π0 r
2me r2
dΘ
m2
1 d
(sin θ ) + l(l + 1) −
Θ = 0
sin θ dθ
dθ
sin2 θ
d2 Φ
+ m2 Φ = 0
2
dφ
(16)
These three equations can be solved separately to obtain the values for
(R, Θ, Φ) and combining them together will give the electron wavefunction
i.e. ψ. The solution of the Schrödinger equation for the H atom gives the
result that the energies of the electrons are quantized i.e. they can only take
specific values.
4
4.1
Wavefunction solutions
Solution for Φ
Consider the equation for Φ, shown in equation 16.
d2 Φ
+ m2 Φ = 0
dφ2
(17)
This is a fairly straightforward second order differential equation whose solution can be written in the form
Φ = A exp(±imφ)
6
(18)
MM3010: Physics of Materials
A is a normalization constant which can be obtained by setting
Z 2π
Φ∗ Φdφ = 1
(19)
0
which gives the normalized solution as
1
exp(imφ)
Φ = √
2π
(20)
By convention only the positive value of m is considered. This would be
useful when plotting the wavefunctions and the probabilities.
4.2
Solution for Θ
Consider now the equation for Θ, as shown earlier in equation 16.
dΘ
m2
1 d
(sin θ ) + l(l + 1) −
Θ = 0
sin θ dθ
dθ
sin2 θ
This can be rearranged to write as
1
d
dΘ
sin θ (sin θ ) + sin2 θ l(l + 1) = m2
Θ
dθ
dθ
(21)
(22)
Equation 22 can be simplified by making the substitution µ = cos θ. Hence
dµ = − sin θdθ. And further
d dµ
d
d
=
= − sin θ
dθ
dµ dθ
dµ
Making the substitution, equation 22 becomes
d
m2
2 dΘ
(1 − µ )
+ l(l + 1) −
Θ = 0
dµ
dµ
1 − µ2
(23)
(24)
Solutions to equation 24 are called Associated Legendre equations. These are
named after the French mathematician Adrien Marie Legendre (1752-1833)
and are essentially recursive polynomial solutions to the differential equation.
It is found that the solution depends on both m and l with the constraints
that l is a positive integer and m is also an integer that can take values from
−l to +l i.e. a total of (2l + 1) values.
7
MM3010: Physics of Materials
4.3
Combined Φ and Θ solution
The individual solution to Θ and Φ can be combined into a new function
called the spherical harmonics, Ylm (θ, φ). The normalized form of the spherical harmonics function is given by
m
m
Yl (θ, φ) = (−1)
2l + 1 (l − m)!
4π (l + m)!
1/2
exp(imφ) Plm (cos θ)
(25)
where Plm (cos θ) is the associated Legendre polynomials. There are different
values for the polynomial depending on l and m. These correspond to the
wavefunctions of the different atomic orbitals. The spherical harmonics gives
the angular spread of the electron wavefunction i.e. the θ and φ dependence.
4.4
Solution for R
Consider the radial part, R(r), of the electron wavefunction given in equation
16.
2me
~2 l(l + 1)
Ze2
1 d 2 dR
(r
) +
−
E +
R = 0
(26)
r2 dr
dr
~2
4π0 r
2me r2
Again this can be rearranged to be written as
l(l + 1)~2
~2 d 2 dR
(r
) + V (r) +
−
R = ER
2me r2 dr
dr
2me r2
(27)
where V (r) is given by equation 2. This expression can be simplified by
making the variable substitution u = rR. Then
dR
d2 u
d2 R
d 2 dR
(r
) = 2r
+ r2 2 = r 2
dr
dr
dr
dr
dR
Making this substitution in equation 27 gives
~2 l(l + 1)
~2 d2 u
+ V (r) +
u = Eu
2me dr2
2me r2
(28)
(29)
For the H atom (or H like atom) making the substitution for the potential
function, equation 2, equation 29 becomes
~2 d2 u
Ze2
~2 l(l + 1)
+ −
+
u = Eu
(30)
2me dr2
4π0 r
2me r2
8
MM3010: Physics of Materials
Equation 30 is similar to a one dimensional Schrödinger equation of wavefunction u where the potential function is given by
~2 l(l + 1)
Ze2
+
f or r > 0
V (r) = −
4π0 r
2me r2
(31)
V (r) = ∞ f or r < 0
By convention an electron in an atom is defined to have total energy E < 0
since this electron must be bound to the atom and cannot escape. The solution to the H atom is to get the quantized values of these energies. Rewriting
equation 30 gives
l(l + 1)
d2 u
2me Ze2
2me E
+
+ −
u =
u
(32)
2
2
2
dr
4π~ 0 r
r
~2
This equation can be solved by suitable substitutions. After normalization
the solution depends on two quantum numbers n and l and is given by
1/2
ρ l 2l+1
2Z 3/2 (n − l − 1)!
)
exp(−
) ρ Ln+l (ρ)
Rnl (r) = (
na0
2n(n + l)!3
2
(33)
where a0 is a constant and is given by
a0 =
4π~2 0
= 0.053 nm
me e2
(34)
This constant is called Bohr radius and represents the most probable distance
for finding the electron in the H atom. The dimensionless quantity, ρ, is given
by
2Z
r
(35)
ρ =
na0
The quantity L2l+1
n+l (ρ) is called the Laguerre polymonial and arises because of the recursive nature of the solution to the second order differential
equation involving r, equation 30. The quantum numbers n and l involved in
the solution to R(r) are called the principal and orbital quantum numbers
respectively with the added constraint that l can only take values from 0
to (n − 1). This is because (n − l − 1) must be positive since factorial of a
negative number cannot exist, see equation 33.
9
MM3010: Physics of Materials
Table 1: Electron energies in the H atom
n Binding energy (eV ) Spectral series
1
-13.6
Lymann
2
-3.4
Balmer
3
-1.51
Paschen
4
-0.85
Brackett
5
-0.54
4.5
Complete wavefunction solution
The complete solution for the electron wavefunction can be combined by
combining the individual solutions for R, Θ, and Φ or R and Y . These solutions are a function of three integers n, l, m which cause quantization of the
electron wavefunctions. While it is cumbersome to write the general normalized expression for the electron wavefunction it is possible to write specific
wavefunctions for different values of these quantum numbers. Substituting
the wavefunction in the H atom equation 4, it is possible to calculate the
energy of the electron in the H atom. This energy depends only a single
quantum number n and is given by
E = −
Z2
me e4 Z 2
=
−13.6
eV
820 h2 n2
n2
(36)
The negative sign means that the electron is bound to the nucleus and cannot escape without supplying an external energy. This energy is called the
binding energy of the electron and depends on the value of n.
5
5.1
Electron in the H atom
Energy of the electron
Consider the energy of the electron in the H atom, given by equation 36,
with Z = 1. The values of n, the principal quantum number are 1,2,3.... For
the first 5 values of n the electron energies are given in table 1. As the value
of n increases the energies are closer to 0 i.e. the electron is less under the
influence of the nucleus. The binding energy corresponds to the minimum
energy required remove an electron from that n value, i.e. to ionize the atom.
For an electron transition from a higher value of n to lower value of n the
difference in energies is given out as light. These electronic transitions form
the characteristic H spectra. Thus transition from n = 2 to n = 1 has an
energy difference of 10.2 eV and the wavelength is 122 nm. This lies in the
10
MM3010: Physics of Materials
R(r)/(Z/a0)3/2
R(r) for 1s
2
1.5
1
0.5
0
0
1
2
r/a0
3
4
5
Figure 2: Plot of the radial function, R(r), vs r for the 1s H atom orbital.
The axes are normalized. Plot generated using MATLAB.
UV region and is part of the Lymann series (transitions to n = 1). Similar
transitions are shown in table 1.
5.2
Radial function
Consider the radial part of the electron wavefunction, given by R(r), equation
33. For n = 1 the only value of l is 0 and the function is given by
R10 (r) = 2 (
Z 3/2
Zr
)
exp(− )
a0
a0
(37)
This corresponds to the 1s orbital, with Z = 1 for H, plotted in figure 2.
Equation 37 is an exponential function, as seen from figure 2.
For n = 2, there exists two values for l i.e. 0 and 1. These are the 2s (n = 2,
l = 0) and 2p (n = 2, l = 1) orbitals. For H atom, these orbitals are
Z 3/2
Zr
Zr
) (1 −
) exp(−
) ⇒ 2s
2a0
2a0
2a0
Z 3/2 Zr
Zr
R21 (r) = 2 (
)
exp(−
) ⇒ 2p
2a0
2a0
2a0
R20 (r) = 2 (
(38)
The wavefunctions are plotted in 3. The 2s wavefunction passes through
zero at r = 2a0 . This point is called a node. For the 2p there is a node at
r = 0 corresponding to the nucleus, but this usually not considered since an
electron cannot exist in the nucleus.
For n = 3 there are 3 values for l i.e. 0,1, and 2. These correspond to the
11
MM3010: Physics of Materials
R(r) for 2s and 2p
R(r)/(Z/a0)3/2
0.8
0.6
0.4
2s
2p
0.2
0
−0.2
0
2
4
r/a0
6
8
10
Figure 3: Plot of the radial function, R(r), vs r for the 2s and 2p H atom
orbital. The axes are normalized. Plot generated using MATLAB.
R(r) for 3s, 3p, and 3d
R(r)/(Z/a0)3/2
0.4
0.3
3s
0.2
0.1
3p
3d
0
−0.1
0
5
10
r/a0
15
20
Figure 4: Plot of the radial function, R(r), vs r for the 3s, 3p, and 3d H
atom orbital. The axes are normalized. Plot generated using MATLAB.
orbitals 3s, 3p, and 3d respectively. The radial functions are
1
Z
2Zr
2Zr 2
Zr
√ ( )3/2 (6 − 6(
) + (
) ) exp(−
) ⇒ 3s
3a0
3a0
3a0
9 3 a0
1
Z
2Zr 2Zr
Zr
R31 (r) = √ ( )3/2 (4 −
)(
) exp(−
) ⇒ 3p (39)
3a0
3a0
3a0
9 6 a0
1
Z
2Zr 2
Zr
R32 (r) = √ ( )3/2 (
) exp(−
) ⇒ 3d
3a0
3a0
9 30 a0
R30 (r) =
Once again these can be plotted, see figure 4. For the 3s orbital there are
2 nodes, also seen from equation 39 which has a quadratic equation in the
expression. For the 3p there is one node. Thus in general Rnl (r) will have
(n − l − 1) nodes, excluding the node at r = 0.
12
MM3010: Physics of Materials
5.3
Angular function
The function Ylm , equation 25, gives the angular distribution of the electron
wavefunction i.e the distribution in θ and φ. This depends on two quantum
numbers l and m with the constraint that m can take integer values from −l
to l i.e (2l + 1) values. Consider the case when l = 0, the only value of m is
also 0 and the angular distribution function for this, equation 25 is
r
1
(40)
Y00 (θ, φ) =
4π
This corresponds to the s orbital and it is spherically symmetric since there is
no dependence on θ and φ. The radial distribution for the s orbitals depends
on the value of n, seen from equations 37, 38, 39. Overall the product of
both is also spherically symmetric.
Consider the case when l = 1, this is the p orbital and it can take 3 values
for m, i.e. -1,0,1. The corresponding values of Yml are
r
√
6
exp(−iφ) 1 − cos2 θ
Y1−1 =
4π
r
3
0
(41)
cos θ
Y1 =
4π
r
√
3
1
Y1 = −
exp(iφ) 1 − cos2 θ
8π
Similarly for l = 2 there are 5 values for m and hence 5 functions for Ylm . It
is possible to look at the angular distribution of the electron probability (i.e.
charge distribution).
6
Charge distribution
For an electron defined by a wavefunction, ψ, the charge distribution is given
by
Z
ψ ∗ ψdv
Charge distribution = −e
(42)
V
where ψ ∗ is the the complex conjugate of the normalized electron wavefunction, ψ and the integration is over a volume V and dv is the volume element
in the appropriate coordinate system used. Since −e is a constant the term
we are interested is in the integral. For the electron in the H atom since the
wavefunction is obtained by converting the system into spherical coordinates
the volume element term is given
dV = r2 sin θ dr dθ dφ
13
(43)
MM3010: Physics of Materials
The limits for the coordinates are given by equation 5. For a spherically
symmetric wavefunction, like the s orbital (l = 0 and hence m = 0) the
volume element, equation 43 simplifies to
dV = 4πr2 dr
(44)
In the H atom solution, there are two parts to the electron wavefunction,
one is the radial part given by Rnl (r) and the other the angular distribution
given by Ylm (θ, φ). Since the total electron distribution is three dimensional
it is easier to consider these two parts separately first before putting together
the final wavefunction.
6.1
Radial distribution
The radial distribution of the electron is given by
Radial distribution = r2 R∗ (r)R(r) dr = r2 R2 (r) dr
(45)
This is because R(r) is a real expression and hence its complex conjugate is
the same as the function itself i.e. R∗ (r) = R(r). It is possible to plot this
distribution as a function of r for the different orbitals. Consider the radial
function for the 1s orbital given in equation 46.
R10 (r) = 2 (
Zr
Z 3/2
)
exp(− )
a0
a0
(46)
The radial distribution function for this, using equation 45, is given by
2
r2 R10
(r) = 4r2 (
Z 3
2Zr
) exp(−
)
a0
a0
(47)
This function is plotted as a function of r in figure 5. There is a maximum
at r = a0 . This corresponds to the most probable location of the 1s H atom.
Also, the probability of finding the electron at the nucleus is zero, since there
is a r2 term in equation 47. Hence the solution to the electron wavefunction
naturally prevents the electron from ‘falling’ into the nucleus.
It is possible to similarly plot the 2s and 2p orbitals. The radial distribution
functions for these, based on the R(r) functions in equation 38 is
Z 3
Zr 2
Zr
) (1 −
) exp(− )
2a0
2a0
a0
Z
Zr
Zr
r2 R21 (r) = 4r2 (
)3 (
)2 exp(− )
2a0
2a0
a0
r2 R20 (r) = 4r2 (
14
(48)
MM3010: Physics of Materials
1s radial distribution
2 2
r R (r)*(a0/Z)
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
r/a0
3
4
5
Figure 5: Plot of the radial distribution function, R(r), vs r for the 1s orbital.
The axes are normalized. Plot generated using MATLAB.
2s and 2p radial distribution
2 2
r R (r)*(a0/Z)
0.8
0.7
2p
0.6
0.5
0.4
0.3
2s
0.2
0.1
0
0
2
4
r/a0
6
8
10
Figure 6: Plot of the radial distribution function, R(r), vs r for the 2s and
2porbital. The axes are normalized. Plot generated using MATLAB.
15
MM3010: Physics of Materials
The distribution functions are plotted in figure 6. There is one node for the
2s orbital (at r = 2a0 ), similar to that seen in figure 3 and the most probable
location of the electron is at r = 5a0 . Similar plots could be drawn for higher
order orbitals. It is harder to visualize the angular distribution separately.
But this can be combined with the radial distribution to give the total three
dimensional charge distribution.
6.2
Three dimensional charge distribution
The three dimensional charge distribution of the electron in the various orbitals is given by equation 42. Ignoring the −e terms the charge distribution
is a plot of ψ ∗ ψdV , where ψ is the electron wavefunction and dV is the volume element. To understand this charge distribution we can look at charge
distribution for different values of the quantum number, l i.e. the angular
quantum number.
6.2.1
s orbitals
Consider the electron in the 1s orbital (n = 1,l = 0). This is the ground
state of the electron in the H atom and the normalized wavefunction is given
by
Zr
1 Z
ψ100 = √ ( )3/2 exp(− )
(49)
a0
π a0
Taking the complex conjugate and multiplying the charge distribution is
given by
2Zr
1 Z
ψ ∗ ψ dV = 4πr2 ( ) ( )3 exp(−
)dr
(50)
π a0
a0
This is a spherical symmetric function and hence the 3D plot should be a
sphere. This is seen in figure 7, This is a 3D plot of a constant probability
surface represented by equation 50. Similar plots for other s orbitals will all
be spheres since there is no radial dependence.
6.2.2
p orbtials
For the electron in the p orbital the value of l = 1. Hence the smallest value
of n that is possible is 2 and for each l value there are 3 m values, i.e. -1,0,1.
Hence there are 3 p orbitals and are usually denoted px , py , and pz . The
plot of the 2p orbitals is shown in figure 8. There are 3 2p orbitals and they
are oriented along the axes i.e. x, y, and z. They are respectively called px ,
16
MM3010: Physics of Materials
Figure 7: 3D plot of the 1s charge distribution. The plot was generated using
MATLAB and is a plot of a constant probability surface.
Figure 8: 3D plot of the 2p charge distribution. The plot was generated using
MATLAB and is a plot of a constant probability surface.
17
MM3010: Physics of Materials
py , and pz . The corresponding wavefunctions (used to calculate the charge
distributions) are
Zr
Zr
1
Z
√ ( )3/2 ( ) exp(−
) cos θ
a0
2a0
4 2π a0
1
Z
Zr
Zr
= √ ( )3/2 ( ) exp(−
) sin θ exp(±iφ)
a0
2a0
8 π a0
ψ210 =
ψ21±1
(51)
For purpose of plotting the orbitals the exp(±iφ) function is converted into
a summation of cos and sin functions so that all three wavefunctions are
written as
Zr
Zr
1
Z
√ ( )3/2 ( ) exp(−
) cos θ
a0
2a0
4 2π a0
1
Z
Zr
Zr
ψ211 = √ ( )3/2 ( ) exp(−
) sin θ cos φ
a0
2a0
4 2π a0
Zr
Zr
1
Z
) sin θ sin φ
ψ21−1 = √ ( )3/2 ( ) exp(−
a0
2a0
8 π a0
ψ210 =
(52)
The fact that p orbitals have charge (or electron) densities in certain directions is of great importance in bonding. This is responsible for the directional
bonding in many of the covalently bonded compounds.
6.2.3
d orbitals
For the d orbitals the value of l = 2. Hence there are 5 d orbitals and they are
usually denoted as dz2 , dxy , dyz , dxz , and dx2 −y2 . Again the subscripts indicate
the predominant directions of the electron distribution in space. The charge
distributions are shown in figure 9. A completely filled d or p or f orbital will
be spherically symmetric. This can be seen by taking the charge distribution
for all the individual orbitals and adding them together. f orbitals have an
even more complex charge distribution and the electrons are tightly confined.
There are a total of 7 f orbitals (l = 3 and m = −3 to 3). f orbitals rarely
participate in bonding due to their confined nature. For certain transition
metals the d orbitals can mix with the s orbitals and influence the bonding.
7
Quantum numbers
The solution to the one electron atom (H or H like) generates 3 quantum
numbers. There is also an additional quantum number that is needed in
order to explain the behavior of electrons in a magnetic field (fine structure
18
MM3010: Physics of Materials
Figure 9: 3D plot of the 3d charge distribution. The plot was generated using
MATLAB and is a plot of a constant probability surface.
of the spectroscopic lines). Together these four quantum numbers define the
electron in the H atom.
7.1
Principal quantum number, n
In the one electron system, the principal quantum defines the energy of the
electron, by equation 36. With increasing n, the energy reduces since the
maximum probability of finding the electron moves away from the nucleus
(larger r).
7.2
Orbital quantum number, l
For a given value of n the orbital quantum number along with the magnetic
quantum number (m) define the spatial distribution of the electron in space.
l = 0 and m = 0 corresponds to a spherically symmetric distribution while for
all other values the distribution is more complex. The orbital quantum number is present in the expression for the potential experienced by an electron
in the H atom, see equation 31. For l = 0 the potential is similar to a normal
central potential, while for other values of l the potential is lowered due to
shape of the charge distribution, given by l. This represents the centrifugal
barrier that keeps the electron away from the nucleus. For l = 0, there is a
theoretical possibility of the electron wavefunction being found in the nucleus
(since ψ(r) is not zero at r = 0, though the probability term r2 ψ ∗ ψ is still
zero) but for other orbitals the l term makes sure that the electron cannot
19
MM3010: Physics of Materials
be found near the nucleus. The orbital quantum number does not affect the
total energy, since this depends only on n for a one electron system. But for
a multi electron system, there will be splitting in the energies of the electrons
for different values of l, (same n).
The orbital quantum number defines the angular momentum of the electron
in the H like atom. The angular momentum, L, of the electron is quantized
and is given by
p
(53)
L = ~ l (l + 1)
For l = 0 (s orbital) the angular momentum is zero since the distribution is
spherically symmetric. For other values of l there exists a finite value for L.
7.3
Magnetic quantum number, m
The magnetic quantum number is related to l and defines the behavior of the
electron in an external magnetic field. Experiments on atomic spectroscopy
show that in an external magnetic field the emission lines from an atom split.
This splitting is related to the quantization of the orbital angular momentum
in an external field. If the direction of the field is taken to be z, then the
angular momentum in that direction is given by
Lz = m ~
(54)
For the value of m = 0, corresponding to the pz or dz2 the quantization is zero
since the charge distribution is parallel to the external field. For other charge
distributions the orbital angular momentum is at angle to the external field
and hence the component of L along the z is given by the non-zero value of
m. Values of m can be positive or negative since for an external field Lz can
be positive of negative. For non-zero values of m the net angular momentum
L will be at an angle to the z axis and the value of the angle, θ, is given by
cos θ = p
|m|
l(l + 1)
(55)
Consider a value of l = 2 then the values of m are −2, −1, 0, 1, 2 the corresponding angles are 90◦ (for m = 0), 66◦ (for |m| = 1) and 35◦ (for |m| = 2).
The quantization of angular momentum is important because it affects the
rules governing the interaction of electrons in an atom with light i.e. atomic
spectroscopy. When an electron interacts with light of sufficient energy it
can get excited from a lower energy level (lower n) to a higher energy level.
While energy is conserved in this process angular momentum must also be
conserved. Since the photon has a finite non-zero momentum given by ~, the
20
MM3010: Physics of Materials
value of l must also change. These are called selection rules and for atomic
transition (due to absorption of light) the rules are given by
∆l = ±1 and ∆m = 0, ±1
(56)
Thus an electron can be excited from 1s to 2p since according to equation
56 ∆l = 1 but not to 2s since ∆l = 0. Similarly an electron from 2s can be
excited to 3p but not to 3s (∆l = 0) or 3d (∆l = 2). The same rules are
also valid for photon emission when the excited electron comes back to the
ground state.
7.4
Spin quantum number
The orbital quantum number l and magnetic quantum number m come naturally from the solution to the Schrödinger equation for the H atom. These
also explain the splitting in atomic spectral lines with and without a magnetic field. A further splitting of the spectral lines, called fine structure,
was observed by Stern and Gerlach which could only be explained by considering that the electrons had an intrinsic angular momentum apart
from the orbital angular momentum. This intrinsic angular momentum is
denoted as Spin, with symbol S. The magnitude of S is given by a new
spin quantum number, s and is given by
p
S = ~ s(s + 1)
Sz = ms ~
(57)
1
s = ±
2
ms is called the spin magnetic quantum number (sometimes magnetic
quantum number is denoted ml ) and represents the z component of the intrinsic angular momentum. Thus, s can have 2 values ± 21 .
It is possible to combine both L and S and define a total angular momentum, represented by J. Again J can be defined by a quantum number j
such that
J = L + S
p
J = ~ j(j + 1)
Jz = mj ~
(58)
Like l and s, j is also quantized but it can take fractional values since it is a
vector addition of two angular momentum terms. The concept of J becomes
important when considering the behavior of a solid in a magnetic field.
21
MM3010: Physics of Materials
8
Exclusion principle
The four quantum numbers n, l, ml and ms define the electron in the H or
H like system. For the H atom the quantum number n defines the total
energy, per equation 36, while n, l, and ml define the spatial distribution of
the wavefunction. For each set of these three values there can only be two
values of the spin quantum number ms (i.e. ± 12 ). Thus each electron in a
H atom has an unique set of four quantum numbers and no two electrons
can have the same set of these quantum numbers. This is called Pauli’s
exclusion principle. The exclusion principle determines the filling up of
electrons in a multi electron system (an atom or a solid) and is the basis for
determining the periodic table. Thus for n = 1 there can be a maximum
of two electrons, with quantum numbers, (1, 0, 0, 21 ) and (1, 0, 0, − 12 ). The
corresponding elements are H and He. For n = 2, there are 8 electrons (since
there can be s and p orbitals). For n = 3, there are 18 electrons (s, p,
and d orbitals). But the 4s level fills up before the 3d levels so that the 3d
transition elements are found in the fourth row of the periodic table. This
is because for a multi electron system the interaction between the individual
electrons also matter and hence equation 36 is only an approximation. For
n = 4 there are 32 electrons, since this also includes the f orbitals. This in
turn can explain the arrangement of atoms in the periodic table.
References
The following source materials were used for preparing this handout
1. Principles of modern physics by Robert B. Leighton, McGraw-Hill
Book company, 1959.
2. Introduction to quantum mechanics by Linus Pauling and E. Bright
Wilson, CBS Publishers, 2008.
3. Principles of electronic materials and devices by S.O. Kasap, McGrawHill India, 2007
The PDF was prepared in LATEX, using Texmaker and MiKTeX compiler.
The plots were generated in MATLAB. The MATLAB code for figures 7, 8,
and 9 were adapted from a code by Peter van Alem from MATLAB Central
(file name Orbital.m). Thanks to Prof. John H. Weaver, University of Illinois
at Urbana-Champaign, for comments and suggestions.
22