19.5 The Density Functional Method 805
configurations.^15 We observe the following pattern: (1) Each triplet state has a lower
energy than the corresponding singlet state. (2) The orbital energies of the 2ssubshell
are lower than the orbital energies of the 2psubshell. The same pattern is generally
followed by higher shells and by atoms with more than two electrons. The orbital ener-
gies of adsubshell lie higher than those of the orbital energies of thepsubshell in
the same shell, the orbital energies of anfsubshell lie higher than those of the orbital
energies of thedsubshell in the same shell, and so on.PROBLEMS
Section 19.4: Excited States of the Helium Atom.
Degenerate Perturbation Theory
19.24Write all of the term symbols for the (1s)(2p) excited
configuration of He.
19.25Write all of the term symbols for the (1s)(3d) excited
configuration of He.
19.26Write all of the term symbols for the (2p)(3p) excited
configuration of He.19.5 The Density Functional Method
This method attempts to calculate the molecular energy and other properties from the
electron probability density, which was introduced in Section 18.3 for a two-electron
system. For a system withNelectrons in a state corresponding to the wave function
Ψ(r 1 ,r 2 ,...,rN), the electron probability density is obtained by integrating over the
spin coordinates of all particles and over the space coordinates of all of the electrons
except one:ρ(r 1 )N∫
Ψ(r 1 ,r 2 ,...,rN)dr 2 dr 3 ...drNds(1)ds(2)...ds(N) (19.5-1)Since the electron probability density depends on only three space coordinates
(represented byr 1 ), it contains much less information than the wave function, which
depends on the 3Nspace coordinates ofNelectrons as well as their spins. However,
Hohenberg and Kohn proved a remarkable theorem: For the ground electronic state of
a molecule in the Born–Oppenheimer approximation, there is only one wave function
and only one energy that correspond to a given electron probability density.^16 This is
surprising, since electron correlation involves effects of the mutual repulsion of pairs
of electrons. It would seem that a function of the six coordinates of two electrons would
be required to describe the energy of the molecule. However, their theorem implies that
the effects of electron correlation are uniquely expressed in a function of only three
coordinates.Walter Kohn,1923–,an
Austrian-Americanphysicist,sharedthe
1998 Nobel Prize inchemistry with
JohnA. Pople for his work ondensity
functional theory.
Since the electronic wave function is uniquely determined by the electron probability
density and since the energy in the Born–Oppenheimer approximation is determined by
the electronic wave function, the ground-state energy is afunctionalof the ground-state
electron probability density. A functional is a rule that associates a numerical value of a
dependent variable with an independent function in much the same way that an ordinary
function associates a numerical value of a dependent variable with a numerical value(^15) Levine,op. cit., p. 270 (note 2).
(^16) P. Hohenberg and W. Kohn,Phys. Rev., 136 , B864 (1964).