Computational Chemistry

So from Eq.4.8

Et¼

1

2

mv^2 "mv^2 ¼"

1

2

mv^2 (4.11)

From Eqs.4.7and4.10:

v¼

Ze^2

2 e 0 nh

(4.12)

So from Eqs.4.11and4.12:

Et¼"

Z^2 e^4 m

8 e 02 n^2 h^2

(4.13)

Equation4.13expresses the total (kinetic plus potential) energy of the electron
of a hydrogenlike atom in terms of four fundamental quantities of our universe:
electron charge, electron mass, the permittivity of empty space, and Planck’s
constant. From Eq.4.13the energy change involved in emission or absorption of
light by a hydrogenlike atom is simply

DE¼Et 2 "Et 1 ¼

mZ^2 e^4

8 e 02 h^2

1

n 12

"

1

n 22



(4.14)

whereDEis the energy of a state characterized by quantum numbern 2 , minus the
energy of a state characterized by quantum numbern 1. Note that from Eq.4.13the
total energy increases (becomes less negative) asnincreases (¼1, 2, 3,...), so
higher-energy states are associated with higher quantum numbersnandDE> 0
corresponds to absorption of energy andDE<0 to emission of energy. The Planck
relation between the amount of radiant energy absorbed or emitted and its
frequency (DE¼hn, Eq.4.3), Eq.4.14enables one to calculate the frequencies
of spectroscopic absorption and emission lines for hydrogenlike atoms. The agree-
ment with experiment is excellent, and the same is true too for the calculated
ionization energies of hydrogenlike atoms (DEforn 2 ¼ 1 in Eq.4.14).

4.2.6 The Wave Mechanical Atom and the Schrodinger Equation€

The Bohr approach works well for hydrogenlike atoms, atoms with one electron:
hydrogen, singly-ionized helium, doubly-ionized lithium, etc. However, it showed

96 4 Introduction to Quantum Mechanics in Computational Chemistry