Chapter 1 Atomic Orbitals, Electron Spin,
Linear Combinations
We shall provide here a brief survey of the relevant background quantum mechanics that is required for the chemical bonding treatment presented in this book. In
general, we shall state only the main results, without any derivation of them.
Much, if not all of this material could be familiar to many readers. For fuller
treatments, the reader should consult some of the numerous standard texts
1 on
quantum mechanics and valence.
1-1 Atomic Orbitals
For any atom, there are n
2 atomic orbitals with principal quantum number n (= 1,
2, 3, ...). These orbitals may be classified as ns, np, nd, nf, ... according to the
value of the total orbital angular momentum quantum number l (≡ 0, 1, 2, ... n - 1)
for an electron. For each value of l, there are 2l + 1 orbitals. Thus, there are one
3s, three 3p ( x
3p ,
y
3p and
z
3p ) and five 3d ( xy
3d ,
xz
3d ,
yz
3d ,
2
2
x y
3d and
2
z
3d ) orbitals for l = 0, 1 and 2; here we have assumed that the np and nd orbitals
are all real orbitals
i
. For certain purposes, the nd xy , nd xz and nd yz orbitals are
designated as 2
t g orbitals, and the corresponding designation for the remaining
pair of nd orbitals is e g . Schematic contours for 1s, 2p and 3d orbitals are
displayed in Figure 1-1.
i
The hydrogenic atomic orbitals have the general form (r, , ) R(r) ( ) ( )
, in which r, θ
and φ are the polar coordinates for the electron. For complex atomic orbitals,
z
( ) exp(
)
im
with i= (-1)
and
0
z
m , ±1, ±2, ... ±l. The np+1, np0 and np-1 orbitals have
1
z
m , 0 and -1. The real np orbitals are related to the complex orbitals as follows:
npx = (np+1 + np–1) / 2, npy = (np+1 – np–1) / 2i, npz = np0 . In the absence of a magnetic field,
orbitals with the same n and l values are degenerate, i.e. they have the same energies. For
given n and l values, the number of degenerate orbitals is 2l + 1.
Ó Springer International Publishing Switzerland 2016
1
R.D. Harcourt, Bonding in Electron-Rich Molecules,
Lecture Notes in Chemistry 90, DOI 10.1007/978-3-319-16676-6_1
Linear Combinations
We shall provide here a brief survey of the relevant background quantum mechanics that is required for the chemical bonding treatment presented in this book. In
general, we shall state only the main results, without any derivation of them.
Much, if not all of this material could be familiar to many readers. For fuller
treatments, the reader should consult some of the numerous standard texts
1 on
quantum mechanics and valence.
1-1 Atomic Orbitals
For any atom, there are n
2 atomic orbitals with principal quantum number n (= 1,
2, 3, ...). These orbitals may be classified as ns, np, nd, nf, ... according to the
value of the total orbital angular momentum quantum number l (≡ 0, 1, 2, ... n - 1)
for an electron. For each value of l, there are 2l + 1 orbitals. Thus, there are one
3s, three 3p ( x
3p ,
y
3p and
z
3p ) and five 3d ( xy
3d ,
xz
3d ,
yz
3d ,
2
2
x y
3d and
2
z
3d ) orbitals for l = 0, 1 and 2; here we have assumed that the np and nd orbitals
are all real orbitals
i
. For certain purposes, the nd xy , nd xz and nd yz orbitals are
designated as 2
t g orbitals, and the corresponding designation for the remaining
pair of nd orbitals is e g . Schematic contours for 1s, 2p and 3d orbitals are
displayed in Figure 1-1.
i
The hydrogenic atomic orbitals have the general form (r, , ) R(r) ( ) ( )
, in which r, θ
and φ are the polar coordinates for the electron. For complex atomic orbitals,
z
( ) exp(
)
im
with i= (-1)
and
0
z
m , ±1, ±2, ... ±l. The np+1, np0 and np-1 orbitals have
1
z
m , 0 and -1. The real np orbitals are related to the complex orbitals as follows:
npx = (np+1 + np–1) / 2, npy = (np+1 – np–1) / 2i, npz = np0 . In the absence of a magnetic field,
orbitals with the same n and l values are degenerate, i.e. they have the same energies. For
given n and l values, the number of degenerate orbitals is 2l + 1.
Ó Springer International Publishing Switzerland 2016
1
R.D. Harcourt, Bonding in Electron-Rich Molecules,
Lecture Notes in Chemistry 90, DOI 10.1007/978-3-319-16676-6_1