theorems are true for specific cases where they can be proven, but we believe them
to hold more generally and efforts continue to find more general proofs.
2.2.1 Runge–Gross Theorem
This theorem states, with two caveats, that the time-dependent external potential
v(1) is determined up to an arbitrary function of time by the initial wavefunction
Ψ 0 ¼ Ψ t 0
ð Þat some time t 0 and by the time-dependent charge density ρ(1). Here we
have enriched our notation to include time, i ¼ i; t i
ð Þ ¼ r i ; σ i ; t i
ð
Þ. The statement
that the external potential is only determined up to an arbitrary function of time
simply means that the phase of the associated wave function is only determined up
to a spatially-constant time-dependent constant. This is because two external
potentials differing by an additive function of time e v 1
ð Þ ¼ v 1
ð Þ þ c t 1
ð Þ lead to
associated wave functions e
Ψ t
ð Þ ¼ e
Àiα t
ð Þ
Ψ t
ð Þ where dα t
ð Þ=dt ¼ c t
ð Þ. A consequence of the Runge–Gross theorem is that expectation values of observables A ˆ (t)
are functionals of the initial wavefunction and of the time-dependent charge
density,
A ρ; Ψ 0
½
Š t
ð Þ ¼ Ψ ρ; Ψ 0
½
Š t
ð Þ
^
A t
ð Þ
Ψ ρ; Ψ 0
½
Š t
ð Þ
:
ð5Þ
The proof of the theorem assumes (caveat 1) that the external potential is
expandable in a Taylor series in time in order to show that the time-dependent
current density determines the time-dependent external potential up to an additive
function of time. The proof then goes on to make a second assumption (caveat 2)
that the external potential goes to zero at large r at least as fast as 1/r in order to
prove that the time-dependent charge density determines the time-dependent current density.
2.2.2 van Leeuwen Theorem
Given a system with an electron–electron interaction w(1, 2), external potential
v(1), and initial wavefunction Ψ 0 , and another system with the same timedependent charge density ρ(1), possibly different electron–electron interaction
e
w 1; 2
ð Þ, and initial wavefunction e
Ψ 0 , then the external potential of the second
system v ˜(1) is uniquely determined up to an additive function of time. It should be
noted that we recover the Runge–Gross theorem when w 1; 2
ð Þ ¼ e
w 1; 2
ð Þ and
Ψ 0 ¼ e
Ψ 0 . However, the most interesting result is perhaps when e
w 1; 2
ð Þ ¼ 0 because
this corresponds to a Kohn–Sham-like system of noninteracting electrons, showing
us that the external potential of such a system is unique and ultimately justifying the
time-dependent Kohn–Sham equation
8
M.E. Casida and M. Huix-Rotllant