BioPHYSICAL chemistry

Properties of the general solution

The solutions to the Schrödinger equation for the hydrogen atom should

in principle allow us to calculate all of the properties of molecules. We

will now use the solutions to address three related questions: what are

the physical properties of the orbitals, how do they change when there

is more than one electron, and how do they change when there is more

244 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY

Then

(db12.37)

(db12.38)

Each of these two terms must be zero. The second term yields:

(db12.39)

so

(db12.40)

The first term yields:

(db12.41)

This is the solution for a specific set of quantum numbers (n=1, l=0, ml=0). In general,

the solutions for the Laguerre equation, Lks(x), can be expressed as in the following form:

(db12.42)

where sis the index with integral values starting with 0, and kis a second index greater

than −1.

Lx

ex

sx

ks ex

xk

s

() xsk

!

= ()

−

d −+

d

E

me

=−

4

2

0

32 πεZ^2

Rre

r me

()exp== −

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

−α

Z^2 πε

2

(^40)
α
πε

=

me

Z^2

2

(^40)
re
mE
e
−r e

• 
  

⎛

⎝

⎜⎜

⎞

⎠

α αα⎟⎟+−+−αr

πε

2

2

2

0

2

2

4

2

Z

mm

Z^2

0

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

=

αα

πε

2 αα

2

2

0

2

2

4

ree

me

r

−−rr−+ +Er

⎛

⎝

⎜

⎜

⎞

⎠

⎟

Z ⎟ ee

−αr= 0