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8 The Variational Principle
8.1 Approximate solution of the Schroedinger equation
If we can’t find an analytic solution to the Schroedinger equation, a trick known as the varia- tional principle allows us to estimate the energy of the ground state of a system. We choose an unnormalized trial function Φ(an) which depends on some variational parameters, an and minimise
E[an] =
〈Φ| Hˆ|Φ〉
with respect to those parameters. This gives an approximation to the wavefunction whose accuracy depends on the number of parameters and the clever choice of Φ(an). For more rigorous treatments, a set of basis functions with expansion coefficients an may be used.
The proof is as follows, if we expand the normalised wavefunction
|φ(an)〉 = Φ(an)/〈Φ(an)|Φ(an)〉^1 /^2
in terms of the true (unknown) eigenbasis |i〉 of the Hamiltonian, then its energy is
E[an] =
∑
ij
〈φ|i〉〈i| Hˆ|j〉〈j|φ〉 =
∑
i
|〈φ|i〉|^2 Ei = E 0 +
∑
i
|〈φ|i〉|^2 (Ei − E 0 ) ≥ E 0
where the true (unknown) ground state of the system is defined by Hˆ|i 0 〉 = E 0 |i 0 〉. The inequality arises because both |〈φ|i〉|^2 and (Ei − E 0 ) must be positive.
Thus the lower we can make the energy E[ai], the closer it will be to the actual ground state energy, and the closer |φ〉 will be to |i 0 〉.
If the trial wavefunction consists of a complete basis set of orthonormal functions |χi〉, each multiplied by ai: |φ〉 =
∑ i ai|χi〉^ then the solution is exact and we just have the usual trick of expanding a wavefunction in a basis set. Alternately, we might just use an incomplete set with a few low-energy basis functions to get a |Φ〉 close to the ground state |i 0 〉. In practice, this is how most quantum mechanics problems are solved.
8.2 Excited States
The variational method can be adapted to give bounds on the energies of excited states, under certain conditions. Suppose we choose a trial function Φ 1 (βn) with variational parameters βn. which is made orthogonal to the ground state φ 0 , by imposing the condition 〈φ 0 |φ 1 〉 = 0.
If we know |φ 0 〉 = |i 0 〉, then similar to the above
E[an] =
〈Φ 1 | Hˆ|Φ 1 〉
∑
ij
〈φ 1 |i〉〈i| Hˆ|j〉〈j|φ 1 〉 =
∑
i
|〈φ 1 |i〉|^2 Ei = 0+E 1 +
∑
i=
|〈φ 1 |i〉|^2 (Ei −E 1 ) ≥ E 1
So the variational method gives an upper bound on the first excited-state energy, and so on. We can satisfy 〈i 0 |φ 1 〉 = 0 if |i 0 〉 is known, or if it has a known symmetry from which we can exploit (e.g. if |i 0 〉 has even parity, chosing |Φ 1 〉 to be odd.)
In general, though, we only have a variational estimate of the ground state φ 0 (αn). In this case the expression above, subject to the constraint 〈φ 1 (βn)|φ 0 (αn)〉 = 0, gives an estimate of E 1. However, the error in this approach will be larger than for E 0 because not only is the wavefunction incorrect, but also the constraint 〈φ 1 |φ 0 〉 = 0 is not quite correct; using an approximate ground state does not guarantee that we get an upper bound for the excited states.
If the excited state has different symmetry from those of the lower-lying levels, and we choose trial functions with the correct symmetries, orthogonality is guaranteed and we get an upper bound to the energy of the lowest-lying level with those symmetries, which is the excited state.
8.3 Analytic example of variational method - Binding of the deuteron
Say we want to solve the problem of a particle in a potential V (r) = −Ae−r/a. This is a model for the binding energy of a deuteron due to the strong nuclear force, with A=32MeV and a=2.2fm. The strong nuclear force does not exactly have the form V (r) = −Ae−r/a, unlike the Coulomb interaction we don’t know what the exact form should be, but V (r) = −Ae−r/a^ is a reasonable model.
The potential is spherically symmetric, most attractive at r = 0 and falls rapidly to zero at large r, so we choose a trial wavefunction which does the same, say φ = ce−αr/^2 a. This has only one dimen- sionless variational parameter, α. The value of c follows from normalisation
∫ c^2 e−αr/a 4 πr^2 dr = 1; which gives c^2 = α^3 / 8 πa^3. (The 4πr^2 comes from the problem being three dimensional).
According to the variational principle, our best estimate for the ground state using this trial function comes from minimising 〈φ| Hˆ|φ〉 with respect to α.
〈φ|H|φ〉/〈φ|φ〉 =
−¯h^2 2 m
∫ (^) ∞
0
c^2
( e−αr/^2 a∇^2 e−αr/^2 a
) 4 πr^2 dr − A
∫ (^) ∞
0
c^2 exp [−(α + 1)r/a] 4πr^2 dr
¯h^2 α^2 8 ma^2
− A
( α α + 1
) 3
From this we find the minimum for E(α) at α 0
dE dα
h¯^2 α 4 ma^2
− 3 A
( α^2 (α + 1)^4
) = 0 =⇒
(α 0 + 1)^4 α 0
= 12Ama^2 /¯h^2
Solving for α 0 gives α 0 = 1.34, and substituting back into 〈φ|H|φ〉 gives E 0 = − 2. 14 M eV.
This is fairly close to the exact solution for this potential, which can be obtained analytically as a Bessel function of
8 mA(a/¯h)e−r/^2 a^ if you manage to spot that change of variables! The exact solution gives E 0 = − 2. 245 M eV.
8.4 Quantum forces: the Hellmann-Feynman Theorum
For many systems one is often interested in forces as well as energies. If we can write the energy of a in state φ as E = 〈φ| Hˆ|φ〉 and differentiate with respect to some quantity α then
dE dα
dφ dα
| Hˆ|φ〉 + 〈φ|
d Hˆ dα
|φ〉 + 〈φ| Hˆ|
dφ dα
But since Hˆ|φ〉 = E|φ〉 and 〈φ|φ〉 is 1 for normalisation:
dE dα
= 〈φ|
d Hˆ dα
|φ〉 + E
d dα
〈φ|φ〉 = 〈φ|
d Hˆ dα
|φ〉
This result is called the Hellmann-Feynman theorem: the first differential of the expectation value of the Hamiltonian with respect to any quantity does not involve differentials of the wavefunction.
e.g. if α represents the position of a nucleus in a solid, then the force on that nucleus is the
expectation value of the force operator d^ Hˆ dα. It can be applied to any quantity which is a differential of the Hamiltonian provided the basis set does not change.
Caveat: if we use an incomplete basis set which depends explicitly the positions of the atoms, then we have |φ〉 =
∑ n,i |un,i(r)〉. This give spurious so-called “Pulay” forces if^ φ^ is not an exact eigenstate.
8.8 Kohn-Sham functional
For solids, we have 10^26 electron states. Analytic solution becomes impossible. In the past 20 years the density functional theory has come to dominate condensed matter physics, extending to chemistry, materials, minerals and beyond.
A popular form of DFT functional was introduced by Nobel laureate Walter Kohn and Lu Sham:
F (ρ) = T [ρ] +
∫ (^) ρ(r)ρ(r′)
4 π² 0 |r − r′|
d^3 rd^3 r′^ + Exc[ρ] +
∑
i
∫ Z
ieρ(r′) 4 π² 0 |Ri − r′|
d^3 r′
Nobody has found a satisfactory functional for T. What is generally used is:
¯h^2 2 m
∑
i
∫ φi∇^2 i φid^3 r
which is the kinetic energy of non-interacting “quasiparticles” and depends explicitly on the wave- functions. The integrals represent electrostatic interactions between the electrons and between electrons and ions, and Exc is ‘everything else’. The advantage of this form is that it can be recast to give a set of one-particle equations with non-interacting fermions moving in an effective potential:
Vef f =
∑
ion
Ze 4 π² 0 |Rion − r′|
∫ (^) ρ(r′)
4 π² 0 |r − r′|
d^3 r +
δExc[ρ] δρ(r)
Since Vef f depends on ρ(r) these equations must be solved self-consistently.
Thus the density functional theorem shows that the problem of solving the Schroedinger equation for a collection of interacting electrons can be transformed to that of a system of non-interacting ‘quasiparticles’, with the cost that the Hamiltonian depends on the electron density ρ(r):
H[ρ(r)]φi = Eiφi where ρ(r) =
∑
i
|φi(r)|^2
Thus the Schroedinger equation is a nonlinear differential equation of many variables. Thus we must turn to the variational method. The most general approach here is to use a Fourier Series (plane wave basis set). The wavefunction for the ith electron is then written as
φi =
∑
k
cik exp(−ik.r) and the variational equation becomes : E 0 = Min
∑
i
〈φi| Hˆ(ρ)|φi〉
The accuracy of the ground state energy of the electrons is determined by the number of Fourier components used. The wavefunctions are expanded in a computer-friendly basis set and the variational principle is used to transform the problem from a set coupled non-linear differential equations into a minimisation of a single function of many variables. Most structural properties of materials depend only on the electron ground state.
The single particle eigenstates of Kohn-Sham functional are not proper single electron states: indistinguishability means there is no such thing. Nevertheless, they are Bloch states, and they do exhibit well defined symmetry and energy “band-structure” which can help with interpretation of the electronic structure
