Physical Chemistry Third Edition

18.7 Atoms with More Than Three Electrons 785

The time-independent Schrödinger equation corresponding to the zero-order

Hamiltonian can be solved by separation of variables, using the trial function

Ψ(0)ψ 1 (1)ψ 2 (2)ψ 3 (3)ψ 4 (4)···ψZ(Z)

∏Z

i 1

ψi(i) (18.6-11)

where the symbolΠstands for a product of factors, just as theΣsymbol stands for

a sum of terms. The factorsψ 1 (1),ψ 2 (2),ψ 3 (3),..., are all hydrogen-like orbitals

corresponding to the correct value ofZ:

ψi(i)ψnilimi,msi(i) (18.6-12)

We must antisymmetrize the orbital wave function of Eq. (18.6-11). This can be done

by writing a Slater determinant with one row for each spin orbital and one column for

each electron:

ψ(0)

1

√

Z!

∣ ∣ ∣ ∣ ∣ ∣ ∣

∣∣

∣

∣

∣

ψ 1 (1) ψ 1 (2) ψ 1 (3) ψ 1 (4) ··· ψ 1 (Z)

ψ 2 (1) ψ 2 (2) ψ 2 (3) ψ 2 (4) ··· ψ 2 (Z)

ψ 3 (1) ψ 3 (2) ψ 3 (3) ψ 3 (4) ··· ψ 3 (Z)

ψ 4 (1) ψ 4 (2) ψ 4 (3) ψ 4 (4) ··· ψ 4 (Z)

··· ··· ··· ··· ··· ···

ψZ(1) ψZ(2) ψZ(3) ψZ(4) ··· ψZ(Z)

∣ ∣ ∣ ∣ ∣ ∣ ∣

∣∣

∣

∣

∣

(18.6-13)

where the 1/

√

Z! factor normalizes the wave function if all orbitals are normalized and

orthogonal to each other. The Pauli exclusion principle is automatically followed. If

two spin orbitals are the same function the determinant vanishes.

The values ofML,MS,L, andScan be computed in the same way as with the

helium and lithium atoms. The contributions toMLandMSfor any completely filled

subshell cancel. Only partly filled subshells need to be counted. For example, the only

term symbol that occurs for the ground state of an inert gas (He, Ne, Ar, and so on) is

(^1) S 0 and the only term symbol that occurs for an alkali metal in its ground state is (^2) S 1 / 2.
The zero-order energy eigenvalue is a sum of hydrogen-like orbital energies with
the appropriate value ofZ:
E(0)En 1 (HL)+En 2 (HL)+...

∑Z

i 1

Eni(HL) (18.6-14)

In the next chapter, we will discuss approximations beyond the zero-order approxi-

mation. We will find that the orbitals in different subshells of the same shell do not

correspond to the same energy, and will be able to identify ground terms by use of the

Aufbau principle.

PROBLEMS

Section 18.7: Atoms with More Than Three Electrons

18.21Write the possible term symbols for the ground
configuration of the boron atom.

18.22Write the antisymmetrized orbital wave function for one
of the states of the carbon atom in its ground

configuration (1s)^2 (2s)^2 (2p)^2. Use the Slater determi-

nant notation. How many terms are in the wave function

written as a sum of terms without determinant notation?

18.23Write the antisymmetrized orbital wave function for one

of the states of the nitrogen atom in its ground