Concise Physical Chemistry

c15 JWBS043-Rogers September 13, 2010 11:28 Printer Name: Yet to Come

244 EARLY QUANTUM THEORY: A SUMMARY

The GAUSSIAN©C program, using a restricted Hartree–Fock procedure, RHF

yields

SCF Done: E(RHF) =-2.80778395662 A.U. after 1 cycles

The value after four cycles by a program named G3 (to be discussed in more detail

later) gives

SCF Done: E(RHF) =-2.85516042615 A.U. after 4 cycles

A Hartree–Fock triple zeta calculation (a linear combination involving three ad-

justableZparameters) gives

SCF Done: E(RHF) =-2.85989537425 A.U. after 3 cycles

The energy – 2.860Ehagrees with the experimental value to within 1.5%.

Example 15.2

The Hamiltonian function for the ground state hydrogen atom (with angular momen-

tum equal to zero) is

Hˆ =−

2

2 me

∇^2 +

e^2

r

=−

^2

2 me

r^2

∂^2

∂r^2

−

e^2

4 π^2 ε 0 r

=−

^2

2 mer

d

dr

(

r^2

∂^2

∂r^2

)

−

e^2

4 π^2 ε 0 r

where there is noθ,φterm because there is no angular momentum in the ground

state and∇^2 =r^2 (∂^2 /∂r^2 )=r^2 (d^2 /dr^2 ) for this one-dimensional operator in which

r^2 appears because the radius vector may be oriented in any direction toward a

surface element of a sphere. The surface area of a sphere goes up as the square of the

radiusA= 4 πr^2 .The4π^2 ε 0 in the denominator of the potential energy arises in the

same way, where 4π^2 ε 0 is the permittivity of free space (essentially a proportionality

constant between coulombs and joules).

Allowing this operator to operate on the trial functionφ=e−αrgives, from the

Schr ̈odinger equation (Hˆφ=Eφ),

[

−

^2

2 mer

d

dr

(

r^2

d^2

dr^2

)

−

e^2

4 π^2 ε 0 r

]

φ(r)=Eφ(r)