Question

The Hohenberg-Kohn theorems show a one-to-one mapping between the density and the molecular Hamiltonian. What information is contained in the Hamiltonian and, conceptually, how can this information be extracted from the density?

Answer 1

The Hohenberg-Kohn Theorems

The Hohenberg-Kohn theorems relate to any system consisting of electrons moving under the influence of an external potential . Stated simply they are as follows:

Theorem 1.
The external potential , and hence the total energy, is a unique functional of the electron density .

The energy functional alluded to in the first Hohenberg-Kohn theorem can be written in terms of the external potential in the following way,

(1.28)

where is an unknown, but otherwise universal functional of the electron density only. Correspondingly, a Hamiltonian for the system can be written such that the electron wavefunction that minimises the expectation value gives the groundstate energy (1.30) (assuming a non-degenerate groundstate),

(1.29)

The Hamiltonian can be written as,

(1.30)

where is the electronic Hamiltonian consisting of a kinetic energy operator and an interaction operator ,

(1.31)

The electron operator is the same for all -electron systems, so is completely defined by the number of electrons , and the external potential .

The proof of the first theorem is remarkably simple and proceeds by reductio ad absurdum. Let there be two different external potentials, and , that give rise to the same density . The associated Hamiltonians, and , will therefore have different groundstate wavefunctions, and , that each yield . Using the variational principle [19], together with (1.31) yields,

			(1.32)
			(1.33)

where and are the groundstate energies of and respectively. It is at this point that the Hohenberg-Kohn theorems, and therefore DFT, apply rigorously to the groundstate only. An equivalent expression for (1.34) holds when the subscripts are interchanged. Therefore adding the interchanged inequality to (1.35) leads to the result:

(1.34)

which is a contradiction, and as a result the groundstate density uniquely determines the external potential , to within an additive constant. Stated simply, the electrons determine the positions of the nuclei in a system, and also all groundstate electronic properties, because as mentioned earlier, and completely define .

Theorem 2.
The groundstate energy can be obtained variationally: the density that minimises the total energy is the exact groundstate density.

The proof of the second theorem is also straightforward: as just shown, determines , and determine and therefore . This ultimately means is a functional of , and so the expectation value of is also a functional of , i.e.

(1.35)

A density that is the ground-state of some external potential is known as -representable. Following from this, a -representable energy functional can be defined in which the external potential is unrelated to another density ,

(1.36)

and the variational principle asserts,

(1.37)

where is the wavefunction associated with the correct groundstate . This leads to,

(1.38)

and so the variational principle of the second Hohenberg-Kohn theorem is obtained,

(1.39)