Physical Chemistry Third Edition

798 19 The Electronic States of Atoms. III. Higher-Order Approximations

This equation is solved numerically and the resulting solution is used under the integral

for the next iteration, and so forth. The equation for thejth iteration is:

−

h ̄^2

2 m

∇ 12 ψ( 1 js)(1)−

Ze^2

4 πε 0 r 1

ψ( 1 js)(1)+

[∫

e^2

4 πε 0 r 12

|ψ( 1 js−1)(2)|^2 d^3 r 2

]

ψ( 1 js)(1)

E( 1 js)ψ( 1 js)(1) (19.2-5)

When additional iterations produce only negligible changes in the orbital function and

the energy, we say that the integral term provides a self-consistent energy of electron 1,

or aself-consistent field. At this point, the iteration is stopped and we assume that our

result resembles the best possible orbital. Hartree carried out this solution prior to the

existence of programmable computers.

DouglasRaynerHartree,1897–1958,
wasProfessorofAppliedMathematics
at the University of Manchesterandwas
laterProfessorof Mathematical Physics
atCambridge University. He becamea
computerpioneer,working with ENIAC,
one of the early electronic computers.
ShortlyafterWorldWarII he saidthat
computers wouldhaveasgreatan
effect oncivilizationas wouldnuclear
energy.

In the SCF method, the expectation value of the energy is not the sum of the

orbital energies, because the electron–electron repulsion has been included in solving

Eq. (19.2-3) for each of the electrons. The sum of the two orbital energies therefore

includes the electron–electron repulsion energy twice. We correct for this double inclu-

sion by subtracting the expectation value of the electron–electron repulsion energy from

the sum of the orbital energies. Ifniterations have been carried out, the energy of the

helium atom in its ground state is

E(atom) 2 E( 1 ns)−

∫

e^2

4 πε 0 r 12

|ψ( 1 ns)(1)|^2 |ψ 1 (ns)(2)|^2 d^3 r 1 d^3 r 2

 2 E( 1 ns)−J 1 s 1 s (19.2-6)

The integralJ 1 s 1 sis called aCoulomb integralbecause it represents an approximate

expectation value of a Coulomb (electrostatic) repulsion energy between two electrons.

The Hartree–Fock–Roothaan Method

Fock modified Hartree’s SCF method to include antisymmetrization.^7 Roothaan further

modified the Hartree–Fock method by representing the orbitals by linear combinations

of basis functions similar to Eq. (16.3-34) instead of by tables of numerical values.^8

In Roothaan’s method the integrodifferential equations are replaced by simultaneous

algebraic equations for the expansion coefficients. There are many integrals in these

equations, but the integrands contain only the basis functions, so the integrals can be

calculated numerically. The calculations are evaluated numerically. This work is very

tedious and it is not practical to do it without a computer.

VladimirAleksandrovich Fock,
1898–1974,wasaRussianphysicist
who taughtat the University ofSt.
Petersburgforover40 years. He made
manycontributionstoquantum physics
andto otherfields of physics.

ClemonsC.J.Roothaan,1918–,
Dutch-Americanphysicist,first
developedhis methodinhisdoctoral
dissertationinthe late 1940sat the
University of Chicago,afterbeing
imprisonedinaGermanconcentration
campduringWorldWarII. He becamea
professorof physicsat the University of
Chicago.

Various kinds and numbers of basis functions have been used. Hydrogen-like basis

functions turn out to consume a lot of computer time.Slater-typeorbitals(STOs) require

less computer time and are a common choice. Each Slater-type orbital is a product of

three factors:rraised to some power, an exponential factor, and the correct spherical

harmonic function. Using a large basis set of STOs, Clementi and Roetti obtained

an approximate energy for the ground state of the helium atom equal to− 77 .9eV,

compared with the experimental value of− 79 .0eV.^9 Basis sets ofGaussian functions

are also used, in which the radial factor in the orbital is a Gaussian function. Some

basis sets include sums of several Gaussian functions that approximate Slater-type

orbitals.

TheSlater-type orbitalsarenamedafter
the same JohnC.Slaterafterwhom the
Slaterdeterminantisnamed.

(^7) V. Fock,Zeits. f. Phys., 61 , 126 (1930).
(^8) C. C. J. Roothaan,Rev. Mod. Phys., 23 , 69 (1951).
(^9) E. Clementi and C. Roetti,At. Data Nucl. Data Tables, 14 , 177 (1974).