Page 186 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

6.4 Atomic orbitals 177

fundamental, originate from an outdated description of spectral lines. The
letters which follow are in alphabetical order with j omitted.

s orbitals
The 1s atomic orbital j1si is

3=2
1 Z
j1siˆj100iˆ R 10 (r)Y 00 (è, j) ˆ e ÿ Zr=a 0 (6:58)
ð 1=2 a 0
where R 10 (r) and Y 00 (0, j) are obtained from Tables 6.1 and 5.1. Likewise, the
orbital j2si is
(Z=a 0 ) 3=2 Zr
j2siˆj200iˆ p  2 ÿ e ÿ Zr=2a 0 (6:59)
4 2ð a 0
and so forth for higher values of the quantum number n. The expressions for
jnsi for n ˆ 1, 2, and 3 are listed in Table 6.2.
All the s orbitals have the spherical harmonic Y 00 (è, j) as a factor. This
spherical harmonic is independent of the angles è and j, having a value
p 
ÿ1
(2 ð) . Thus, the s orbitals depend only on the radial variable r and are
spherically symmetric about the origin. Likewise, the electronic probability
2
density jøj is spherically symmetric for s orbitals.

p orbitals
The wave functions for n ˆ 2, l ˆ 1 obtained from equation (6.56) are as
follows:

(Z=a 0 ) 5=2
j2p 0 iˆj210iˆ p  re ÿ Zr=2a 0 cos è (6:60a)
4 2ð
5=2
1 Z ij
j2p 1 iˆj211iˆ re ÿ Zr=2a 0 sin è e (6:60b)
8ð 1=2 a 0
5=2
1 Z
j2p ÿ1 iˆj21 ÿ1iˆ re ÿ Zr=2a 0 sin è e ÿij (6:60c)
8ð 1=2 a 0
The 2s and 2p 0 orbitals are real, but the 2p 1 and 2p ÿ1 orbitals are complex.
Since the four orbitals have the same eigenvalue E 2 , any linear combination of
them also satis®es the Schrodinger equation (6.12) with eigenvalue E 2 . Thus,
È
we may replace the two complex orbitals by the following linear combinations
to obtain two new real orbitals