Physical Chemistry Third Edition

796 19 The Electronic States of Atoms. III. Higher-Order Approximations

φ(x)

{

Acos(bx)if0<|x|<π/ 2 b

0if|x|>π/ 2 b

wherebis an adjustable parameter.

19.9An oscillating particle of massmhas the potential energy

function

Vcx^4

wherecis a constant with units J m−^4. Find a formula for

the variational energy assuming a trial function

φAe−bx

2

wherebis a variable parameter.

19.10An oscillating particle of massmhas the potential energy
function
Vcx^4

wherecis a constant with units J m−^4. Use the variation

method assuming a trial function

φAe−bx

4

wherebis a variable parameter.

19.11Apply the variation method to the ground state of the
hydrogen atom using the trial function

φ

{

A(1−r/b)if0<r<b

0ifb<r

Compare your result with the correct energy,

− 2. 1787 × 10 −^18 J.

19.12The ionization potential (energy to remove one electron)
of a helium atom in its ground state is 24.58 eV.
a.What effective nuclear charge does this correspond
to? Compare with theZ′value from the simple
variation calculation.

b.What energy in eV is required to remove both

electrons?

19.13. Prove an extended variational theorem, which is that if

the trial functionφis orthogonal toψ 1 , the correct

ground-state wave function, the variational energy

calculated withφcannot be lower than the correct energy

of the first excited state.

19.14.The extended variation theorem states that if a variation

trial function is orthogonal to the exact ground-state wave

function, it provides an upper limit to the energy of the

first excited state. The following trial function is

orthogonal to the ground-state wave function of a particle

in a box of lengtha:

φC

(

x^3 −

3

2

ax^2 +

1

2

a^2 x

)

a.Obtain a formula for the variation energy of the first

excited state of this system. Since this is single

function, no minimization is possible.

b.Evaluate this energy for an electron in a box of length

10.00 Å. Calculate the percent error.

19.15The extended variation theorem states that if a variational

trial function is orthogonal to the exact ground-state wave

function, it provides an upper limit to the energy of the

first excited state.

a.Show that the following function is orthogonal to the

correct ground-state wave function of a harmonic

oscillator:

φ

{

Asin(bx)if−π/b<x<π/b

0ifx>π/b

b.Calculate an upper bound to the energy of the first

excited state of a harmonic oscillator using this

variation function and treatingbas an adjustable

parameter.

19.2 The Self-Consistent Field Method

Theself-consistent field(SCF) method was introduced in 1928 by Hartree.^5 The goal

of this method is similar to that of the variation method in that it seeks to optimize an

approximate wave function. It differs from the variation method in two ways: First,

it deals only with orbital wave functions; second, the search is not restricted to one

family of functions and is capable of finding the best possible orbital approximation.

It is a method that proceeds by successive approximations, or iteration.

(^5) D. R. Hartree,Proc. Camb. Philos. Soc., 24 , 89, 111, 426 (1928).