Physical Chemistry Third Edition

794 19 The Electronic States of Atoms. III. Higher-Order Approximations

The second term in the Hamiltonian operator in Eq. (19.1-11) gives a contribution

analogous to that in Eq. (19.1-6):

Contribution toW

∫

ψ′(1)∗ψ′(2)∗

Ze^2

4 πε 0 r 1

ψ′(1)ψ′(2)dq

ZZ′〈V〉H(1s) 2 ZZ′E 1 (H) (19.1-13)

where〈V〉H(1s)is the expectation value of the potential energy of a hydrogen (not

hydrogen-like) atom in the 1sstate. We have a factor ofZfrom the factorZin the

Hamiltonian, and a factor ofZ′from use of the 1sorbital that corresponds to a nuclear

charge ofZ′e. The final equality comes from Eq. (17.5-9), which states that the expec-

tation value of the potential energy equals twice the total energy.

The third and fourth terms in the Hamiltonian operator in Eq. (19.1-11) are just like

the first two terms except that the roles of particles 1 and 2 are interchanged. After the

integrations these two terms give contributions equal to those of the first two terms.

The fifth term is the same as in Eq. (19.1-6) except that the orbitals correspond to the

nuclear charge ofZ′einstead ofZe, so that its contribution is:

∫

ψ′(1)∗ψ′(2)∗

(

e^2

4 πε 0 r 12

)

ψ′(1)ψ′(2)dq−

5

4

Z′E 1 (H) (19.1-14)

The final result is

WE 1 (H)

(

− 2 Z′^2 + 4 ZZ′−

5

4

Z′

)

(19.1-15)

We find the minimum value ofWby differentiating with respect toZ′and setting

this derivative equal to zero:

0 E 1 (H)

(

− 4 Z′+ 4 Z−

5

4

)

This gives

Z′Z−

5

16

(19.1-16)

ForZ2,Z′ 27 / 16  1 .6875. This corresponds to motion of an electron as

though the nucleus had 1.6875 protons instead of 2 protons, or that the second electron

has a 31.25% probability of being closer to the nucleus than the first electron. Figure 19.1

shows the hydrogen-like 1sorbital withZ2 and the variation orbital withZ′

1 .6875. The variable on the horizontal axis of this figure is the distance from the

nucleus divided by the Bohr radiusa.

For the helium atom (Z2) the minimum value ofWis

W(− 13 .60 eV)[−2(1.6875)^2 +4(2)(1.6875)−

5

4

(1.6875)]− 77 .5eV

(19.1-17)

This value differs from the experimental value of− 79 .0 eV by 1.5 eV, an error of 2%,

1.2

1.0

0.8

Orbital with Z' 5 1.6875

Orbital function (unnormalized)

Orbital with Z 52

0.6

0.4

0.2

0.0

r/a

012

Figure 19.1 Zero-Order and Varia-
tionally Obtained Orbitals for the
Ground State of the Helium Atom.

an improvement over the error of 4 eV or 5% obtained with the unmodified hydrogen-

like orbitals. More nearly accurate values can be obtained by choosing more flexible

variation functions. Hylleraas used the variation function^3

φCe−Z

′′r 1 /a

e−Z

′′r 2 /a

(1+br 12 ) (19.1-18)

(^3) E. A. Hylleraas,Z. Physik, 65 , 209 (1930).