Ab initio simulation of charged slabs at constant chemical potential

N. D.

Lang

and

W.

Kohn

, , ().

K.-M.

Ho

,

B. N.

Harmon

, and

S. H.

Lui

, , ().

P.

Gies

and

R. R.

Gerhardts

, , ().

J. E.

 

Inglesfield

,  , ();

G. C.

Aers

and

J. E.

Inglesfield

, , ().

S. C.

Lam

and

R. J.

Needs

, , ().

M. W.

Finnis

,

R.

Kaschner

,

C.

Kruse

,

J.

Furthmüller

, and

M.

Scheffler

, , ().

G. V.

Wolf

and

D. V.

Fedorov

, , ().

C. L.

Fu

and

K.-M.

Ho

, , ().

K. P.

Bohnen

and

D. M.

Kolb

, , ().

D. M.

 

Eigler

,

P. S.

 

Weiss

,

E. K.

 

Schweizer

, and

N. D.

 

Lang

,  , ();

N. D.

Lang

, , (), and references therein.

S. C.

Lam

and

R. J.

Needs

, , ().

S.

Heinze

,

X.

Nie

,

S.

Blugel

, and

M.

Weinert

, , ().

E. A.

 

Colbourn

and

J. E.

 

Inglesfield

,  , ();

S.

 

Clarke

,

J. E.

 

Inglesfield

,

M.

 

Nekovee

, and

P. K.

 

de Boer

,  , ();

S.

Clarke

,

M.

Nekovee

,

P. K.

de Boer

, and

J. E.

Inglesfield

, , ().

M. T. M.

Koper

,

R. A.

van Santen

,

S. A.

Wasileski

, and

M. J.

Weaver

, , ().

J. W.

 

Halley

and

D. L.

 

Price

,  , ();

D. L.

 

Price

and

J. W.

 

Halley

,  , ();

J. W.

Halley

, , ().

E.

 

Leiva

and

W.

 

Schmikler

,  , ();

C.

Sanchez

,

E.

Leiva

, and

J.

Kohanoff

, , ().

P.

 

Hohenberg

and

W.

 

Kohn

,  , ();

W.

Kohn

and

L. J.

Sham

, , ().

N. D.

Mermin

, , ().

R. F.

 

Nalewajski

and

R. G.

 

Parr

,  , ();

see also R. G. Parr and W. T. Yang, Density Functional Theory of Atoms and Molecules (Oxford U.P., New York, 1989).

P. W.

Ayers

and

R. G.

Parr

, , ().

Strictly speaking, the “$μ=const$” mode corresponds to the semi-grand canonical ensemble (the grand canonical ensemble with respect to electrons, and the canonical ensemble with respect to nuclei), i.e., the number of nuclei (ions) in the system is fixed. That coincides with the standard DFT definition of the electron chemical potential μ as the partial derivative of the total energy with respect to the number of electrons at fixed external potential; the latter is the Coulomb potential due to nuclei (ions).

C.

Bureau

and

G.

Lécayon

, , ().

The only exception which we are aware of is the work by Bohnen and Kolb (Ref. 9) in which the authors, studying the reconstruction of the Au(100) surface in an electrochemical environment, calculated first the total energy curves in the “$Ne=const$” mode and then compared the energies at the same voltage rather than at the same charge, using the experimentally obtained capacitance of the double layer as a function of the applied voltage.

N. D.

Lang

and

W.

Kohn

, , ().

The work function is the energy required to remove an electron from an infinite slab of definite orientation and put it at “infinity.” An apparent paradox arises if we do not consider the infinite nature of the slab, for it appears that we can remove an electron from one face, and pay an energy φ, and insert it through a different face, and gain an energy $φ′.$ In other words, it would appear that we could extract useful work of magnitude $φ′−φ$ from the system by undergoing a cyclic process. The resolution of this paradox is that, in performing this cyclic process, we implicitly assume a finite slab in which a common $V∞$ must be assumed for both faces. In this sense, the chemical potential (the energy required to remove an electron and put it at this common infinity) becomes a true bulk quantity, independent of the face exposed.

The property that the work function can be effectively separated into the bulk and surface contributions can even be used in practice in ab initio calculations to improve the convergence of the work function with respect to the slab thickness [

C. J.

Fall

,

N.

Binggeli

, and

A.

Baldereschi

, , ()].

A note on the terminology should be added here. Speaking about the Helmholtz free energy F, we imply the free energy with respect to the electronic subsystem, characterized by a certain electron temperature $Te$ (denoted in the paper with T), whereas the ionic positions are perhaps somehow optimized and then fixed, which means that we refer to the zero ionic temperature, and thus the Helmholtz free energy is in fact the free energy for the electronic subsystem, but just the total energy for the ionic subsystem. Similarly, the grand potential Ω is again the grand potential with respect to electrons, but the total energy with respect to ions (Ref. 28). Such a “mixed” description of the system is possible in the Born–Oppenheimer approximation because of a separation of time scales between nuclear and electronic degrees of freedom. We can exploit the separation to compute the Helmholtz free energy for the electrons as a function of nuclear positions for given number of electrons $Ne$ and electronic temperature $Te,$ or to compute the grand potential Ω for the electrons as a function of $Te$ and the chemical potential of electrons μ at again fixed arbitrary positions of nuclei.

We use hereafter the term ions to denote the ionic cores, i.e., the nucleus plus core electrons. This terminology is common in ab initio pseudopotential calculations, where the core electrons are integrated into the ionic core, and thus not considered explicitly. It should not be confused with the real ions, i.e., anions or cations, present in electrolytic solutions, e.g., $SO4=.$

A.

Alavi

,

J.

Kohanoff

,

M.

Parrinello

, and

D.

Frenkel

, , ().

A. Alavi, in Monte Carlo and Molecular Dynamics of Condensed Matter Systems, edited by K. Binder and G. Ciccotti (SIF, Bologna, 1996).

CPMD, version 3.3, written by J. Hutter, A. Alavi, T. Deutsch, and the group of M. Parrinello at the MPI in Stuttgart and IBM research laboratory in Zürich.

This is known as integer preference principle [see, e.g.,

J. P.

 

Perdew

,

R. G.

 

Parr

,

M.

 

Levy

, and

J. L.

 

Balduz

,Jr. ,  , ()].

It has been shown that integer preference, absent both in LDA and GGA, is recovered when the exact exchange functional is used [see

J. B.

Krieger

,

Y.

Li

, and

G. J.

Iafrate

, , ()]. This issue, which is at the origin of several problems in the description of insulators and semiconductors, e.g., the underestimation of the energy gap, is less relevant in the case of metals.

R.

Vuilleumier

,

M.

Sprik

, and

A.

Alavi

, , ().

M.

Leslie

and

M. J.

Gillan

, , ().

G.

Hummer

,

L. R.

Pratt

, and

A. E.

Garcı́a

, , () and references therein.

G.

Makov

and

M. C.

Payne

, , ().

P. A.

Schultz

, , ().

P. A.

Schultz

, , ().

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

For example, the voltage at which the lifting of the missing row reconstruction of the Au(110) surface occurs is known to depend strongly on the electrolyte [

K. P.

 

Bohnen

and

K.-M.

 

Ho

,  , ();

D. M.

Kolb

, , ()].

W.

Schmickler

, , ().