Physical Chemistry Third Edition

19.1 The Variation Method and Its Application to the Helium Atom 793

proportional toZ. The electron–electron repulsion term in the Hamiltonian does not

contain the parameterZ, but the effective radius of the orbital region is inversely pro-

portional toZ, making the average distance between the electrons inversely propor-

tional toZand making the expectation value of the repulsion energy proportional toZ.

The result in Eq. (19.1-8) is more positive than the correct value of− 79 .0eV, as

the variation theorem guaranteed. The error of 4 eV is much smaller than the error of

−30 eV obtained with the zero-order approximation. Our wave function is still the zero-

order wave function obtained by complete neglect of the electron–electron repulsion.

The improvement came from using the complete Hamiltonian operator in calculating

the variation energy rather than the zero-order Hamiltonian.

We now use a variation trial function that represents a family of functions. We

choose a modified 1sspace orbital in which the nuclear chargeZis replaced by a

variable parameter,Z′:

ψ′ 100 ψ′ 100 (Z′)

1

√

π

(

Z′

a

) 3 / 2

e−Z

′r/a

(19.1-9)

whereais the Bohr radius. We label this orbital with a prime (′) to remind us that the

orbital depends onZ′, not onZ. The variation trial function is

φφ(Z′)ψ′(1)ψ′(2)

1

√

2

[α(1)β(2)−β(1)α(2)] (19.1-10)

where we omit the subscripts on the orbital symbols.

Shielding

There is a physical motivation for choosing this variation function. As an electron

moves about in the helium atom, there is some probability that the other electron will

be somewhere between the first electron and the nucleus, “shielding” the first electron

from the full nuclear charge and causing it to move as though the nucleus had a smaller

charge. A value ofZ′smaller than 2 should produce a better approximate energy value

than the value of− 74 .8 eV in Eq. (19.1-8).

When the wave function of Eq. (19.1-10) and the correct Hamiltonian are substituted

into Eq. (19.1-1) the variation energy is

W

∫

ψ′(1)ψ′(2)×

[

K̂(1)− Ze

2

4 πε 0 r 1

+K̂(2)−

Ze^2

4 πε 0 r 2

+

e^2

4 πε 0 r 12

]

ψ′(1)ψ′(2)dq (19.1-11)

whereK̂is the kinetic energy operator for one electron. The kinetic energy operator

of electron 1 operates only on the coordinates of electron 1, so that

∫∫

ψ′(1)∗ψ′(2)∗K̂(1)ψ′(1)ψ′(2)d^3 r 1 d^3 r 2



∫

ψ′(1)∗K̂(1)ψ′(1)d^3 r 1

∫

ψ′(2)∗ψ′(2)d^3 r 2 

∫

ψ′(1)∗̂K(1)ψ′(1)d^3 r 1

Z′^2 〈K〉H(1s)−Z′^2 E 1 (H) (19.1-12)

where〈K〉H(1s)is the expectation value of the kinetic energy of a hydrogen atom in

the 1sstate, which is equal to the negative of the total energy, as shown in Section 17.5.