Physical Chemistry Third Edition

740 17 The Electronic States of Atoms. I. The Hydrogen Atom

We have used the fact that the sum of a set of consecutive integers is the mean of the first

integer and the last integer times the number of members of the set (a fact reportedly first

discovered by Gauss when he was seven years old).

The radial factorRfor the hydrogen atom consists of an exponential factor and a

polynomial denoted byG(ρ). These polynomials are related to theassociated Laguerre

functions. Appendix F describes these functions and theLaguerre polynomialsof which

they are derivatives and gives formulas for generating the polynomials. To expressR

in terms ofr, we use Eqs. (17.3-4), (17.3-8), and (17.3-11) to write

ρ 2 αr

2 r

na

(17.3-17)

whereais the Bohr radius andnis the principal quantum number. There is a different

radial factorRfor each set of values of the quantum numbersnandlso we attach these

two subscripts to the symbolRnl.

Table 17.2 gives theRnlfunctions forn1, 2, and 3. The entries in the table are

for thehydrogen-like atom, which is a hydrogen atom with the nuclear charge equal to

a number of protons denoted byZ. The He+ion corresponds toZ2, the Li^2 +ion

corresponds toZ3, and so on. This modification to theRnlfunctions will be useful

when we discuss multielectron atoms in the next chapter. To obtain the radial factors

and the energy levels of a hydrogen-like atom we need to replace the variableρby

ρ 2 αr

2 Zr

na

(17.3-18)

and the energy eigenvalue by

EEn−

h ̄^2 α^2

2 μ

−

Z^2 e^2

2(4πε 0 )an^2

−(13.60 eV)

Z^2

n^2

(17.3-19)

The constants at the first of each formula in the table provide for normalization, which

we discuss later.

Table 17.2 Radial Factors for Hydrogen-Like Energy Eigenfunctions

R 10 (r)R 1 s(r)

(

Z

a

) 3 / 2

2 e−Zr/a

R 20 (r)R 2 s(r)

1

2

√

2

(

Z

a

) 3 / 2 (

2 −

Zr

a

)

e−Zr/^2 a

R 21 (r)R 2 p(r)

1

2

√

6

(

Z

a

) 3 / 2 (

Zr

a

)

e−Zr/^2 a

R 30 (r)R 3 s(r)

1

9

√

3

(

Z

a

) 3 / 2 [

6 −

4 Zr

a

+

(

2 Zr

3 a

) 2 ]

e−Zr/^3 a

R 31 (r)R 3 p(r)

2

27

√

6

(

Z

a

) 3 / 2 (

4 Zr

a

−

2 Z^2 r^2

3 a^2

)

e−Zr/^3 a

R 32 (r)R 3 d(r)

1

9

√

30

(

Z

a

) 3 / 2 (

2 Zr

3 a

) 2

e−Zr/^3 a

Additional functions can be obtained from formulas in Appendix F.