Let and be two orthonormal one-electron orbitals ( ) with one-electron energies ,

PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · Identical particles and many-particle quantum mechanics · Identical particles and many-particle quantum mechanics

Deriving the Spectrum, Ch Helium: exchange and the variational method

Problem

Let

u_a

and

u_b

be two orthonormal one-electron orbitals (

a\ne b

) with one-electron energies

E_a

E_b

, and let

\phi_\pm(1,2) = \frac{1}{\sqrt2}\left[u_a(1)u_b(2) \pm u_b(1)u_a(2)\right]

be the normalised symmetric (

+

) and antisymmetric (

-

) two-electron spatial states. Define the direct integral

J_{ab} = \displaystyle\int |u_a(1)|^2|u_b(2)|^2\,\frac{1}{r_{12}}\,d^3r_1\,d^3r_2

and the exchange integral

K_{ab} = \displaystyle\int u_a^*(1)u_b^*(2)\,\frac{1}{r_{12}}\,u_b(1)u_a(2)\,d^3r_1\,d^3r_2

. (a) Expand

\langle\phi_\pm|1/r_{12}|\phi_\pm\rangle

in terms of the four two-electron integrals that appear, and show that two of them equal

J_{ab}

and two equal

K_{ab}

. (b) Hence derive

\langle\phi_\pm|1/r_{12}|\phi_\pm\rangle = J_{ab}\pm K_{ab}

, and so

E_\pm = E_a+E_b+J_{ab}\pm K_{ab}

for the full two-electron energy (kinetic plus nuclear attraction plus repulsion).

Related theory

Quantum Mechanics

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