On the chemical potential of the hydrogen atom
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On the chemical potential of the hydrogen atom P. Fuentealba1,2 · C. Cárdenas1,2 Received: 10 October 2019 / Accepted: 11 May 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract It will be shown that the chemical potential, 𝜇 , of the hydrogen atom is exactly equal to the negative of its ionization potential I. The result is also valid for any hydrogen-like ion. Thereafter, arguments will be presented that support the idea that for 𝛿E any atom 𝜇 = −I + limr→∞ 𝛿𝜌c . Immediate consequences will be presented. keywords Density functional · Chemical potential · Hydrogen atom
1 Introduction In Density Functional Theory (DFT), the chemical potential enters as a Lagrange parameter in the Euler equation for the determination of the electron density 𝜌(⃗r) :
E[𝜌] = 𝜇. 𝛿𝜌
(1)
E[𝜌] yields the energy as a functional of the density. This is to say, the exact ground-state energy is obtained after evaluating the functional at the exact ground-state density. In the same sense, Eq. (1) says that its functional derivative evaluated at the exact density is a constant equal to the chemical potential. Evaluated at any other density, even in the case that it integrates to the correct number of electrons, the right part of Eq. (1) will be not longer a constant but a function without physical meaning. The main difficulty of the theory is that the exact functional is unknown. However, we will show that, for the special case of the hydrogen atom, the functional derivative evaluated at the exact density is known Published as part of the special collection of articles derived from the Chemical Concepts from Theory and Computation. * P. Fuentealba [email protected] * C. Cárdenas [email protected] 1
Departamento de Física, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile
Centro para el Desarrollo de la Nanociencia y la Nanotecnología (CEDENNA), Avda. Ecuador 3493, 9170124 Santiago, Chile
2
for variations of the density that either, keep the number of electron constant or reduce them. Note that in solving Eq. (1) for the chemical potential is not equivalent to know the functional E[𝜌] , but its functional derivative evaluated at the exact density of a given external potential. A very important point is the discontinuity of the chemical potential at integer number of electrons. [11] Therefore, the evaluation of the chemical potential [1] must be handled with care through the boundary conditions set in the choice of the external potential.
2 Theory For any electronic system, the energy functional can be written as
E[𝜌] = min⟨Ψ�T̂ + V̂ ee �Ψ⟩ + Ψ→𝜌
∫
v(⃗r)𝜌(⃗r)d⃗r,
(2)
where T is the kinetic energy functional and the electron electron repulsion functional is defined as
V̂ ee = Ĵ + Ê xc
(3)
J is the classical Coulomb repulsion and Exc is the exchange correlation functional. Note that this is not definition in the Kohn–Sham scheme of DFT. Replacing equations (2) and (3) in Eq. (1) one obtains
𝜇=
𝛿T 𝛿Exc + − V(⃗r). 𝛿𝜌 𝛿𝜌
(4)
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