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A Matrix Version of the Hartree-Fock
Method Applied to a Helium Atom

The Hamiltonian function for the two electrons of a helium atom is easy to specify. Let p1 and p2 be the momentum of the electrons. The kinetic energy of the atom is then

K = p1²/2m + p2²/2m

where m is the electron mass. Let r1 and r2 be the position vectors of the two electrons with respect to an origin at the center of the nucleus. The magnitudes of r1 and r2 are denoted as r1 and r2. The potential energy of the atom is then

V = −2q/r1 −2q/r2 + q/|r1r2|

where q is the product of the constant for the electrostatic force and the square of the unit charge.

The Schrödinger equation for the system is also easily derived, but obtaining a solution is nearly impossible.

The Hartree Self-Consistent Field Approximation

The Hartree procedure consists of considering a single electron with the effect on it of the other electron being replaced by its effect on the potential energy function.

H = p²/2m − 2q/r + q/R

where R is the average distance between the electron and the average position of the other electron. The average position of the other electron may be the center of the atom, in which case R would be equal to r. However if the other electron is considered as a spherical distribution of charge that part which is closer to the origin than r would have an effect but that which is farther away than r would have no effect. The value of R would then be ½r.

This Hamiltonian function is then converted to its Hamiltonian operator by replacing p with −ih∂/∂r where h is Planck's constant divided by 2π and i is the imaginary unit √−1. The exponent of 2 for p results in the second derivative with respect to r. The time independent Schrödinger equation for the system is then

[−(h²/2m)d²/dr² − 2q/r + ½q/r]ψ(r) = εψ(r)
which reduces to
[−(h²/2m)d²/dr² − (3/2)q/r]ψ(r) = εψ(r)

where ψ is the wave function of the electron and ε is real-valued onstant, the energy of the system. This is converted into matrix form by letting Ψ represent ψ(r) as an infinite dimensional vector. Likewise V is an infinite dimensional diagonal matrix with 1/r on the principal diagonal. This means that points for the arguments of the function must straddle the origin to avoid having a term involving division by zero. This can be done by taking the points nearest the origin to be +δ/2 and −δ/2. Thus the points corresponding to the vector components are …, 2&fract12;δ, 1&fract12;δ, &fract12;δ, −&fract12;δ, −1&fract12;δ, −2&fract12;δ, ….

The second derivative operation can be represented as

(d²ψ/dr²) ≅ [ψ(r+δ)−2ψ(r)+ψ(r-δ)]/δ²

The matrix version of the system is then

[−(h²/(2mδ²))J − (3q/2)V]Ψ = εΨ

where J is a matrix of zeroes except for (… 1, −2; 1, …) centered on the principal diagonal.

As it happens the model with an electron in the same shell shielding a half unit of charge is equivalent to the Bohr model for hydrogen with the central charge being 3/2 rather than 1. The ionization energies can be computed and ccompared with the experimental values. For the details see helium model.

Comparison of Measured Helium Spectrum Lines
with Values Computed from a Modified Version
of the Bohr Model
Measured
wavelength
Computed
wavelength
Error
438.793 nm433.937 nm-1.1%
471.314 nm486.009 nm+3.1%
492.193 nm 486.009 nm-1.3%
501.5675 nm486.009 nm-3.2%
667.815 nm656.112 nm-1.8%

(To be continued.)


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