Calculated Electronic Profiles for Liquid-Metal Surfaces Calculated Electronic Profiles for Liquid-Metal Surfaces

The electronic density profile for a liquid-metal surface can be calculated by solving the self-consistent Lang-Kohn equations for the electronic wave functions. One requires a surface density profile for the ion cores, which enters the electrostatic and pseudopotential parts of the electronic Hamiltonian. We use oscillatory profiles, suggested by those found by molecular-dynamics simulations on a pseudoatom model. Calculating surface potentials and work functions, we obtain excel-lent agreement with experiment (within 0. 2 eV). It is shown that use of either step-function ion profiles or a simple variational method leads to serious errors (1 — 2 eV) for these quantities.


I. INTRODUCTION
Understanding the density 'of conduction electrons at a liquid-metal surface, and how it responds to change in en- vironment and to charging, may be important to formula- tion of realistic models' for the common electrochemical interface.These electrons contribute to the difference in electric potential across the planar interface according to X = -4~J dz j dz'[Zp+(z') -p (z')j, where Zp+(z) is the charge-density profile of the positive ion cores and p (z) that of the conduction electrons (a simple metal, with separation of -core and valence bands, is assumed).
Atomic units are used throughout.
The profile p+(z) should be obtained by a statistical mechanical averaging over ion configuration, the ions in- teracting by a potential which includes screening by the conduction electrons.
To each ion configuration there corresponds an electron density; averaging these densities over ion configurations produces p (z).Since the in- terionic potential includes the interaction of each ion with the perturbation in electron density caused by the others, a complete calculation for the liquid-metal surface requires ' the generation of new interionic potentials, dependent on the electronic distribution, for each ionic configuration until consistency between ionic and electronic profiles is obtained.Calculation of the electronic profile and such properties as the surface potential for a particular ionic distribution is much less difficult.
The average value of a surface property, which should in principle be derived from a series of calculations for different ionic distributions, may be estimated from a sin- gle calculation for a single ionic distribution.This distri- bution should not differ much from the average profile p+(z).Assuming some ionic profile, one can calculate a surface potential and work functions to compare with ex- periment.If this property is sensitive to the ionic profile, agreement with experiment is a test of the quality of the profile.Thus, we assume that p (z) can be obtained from a single calculation, using the average profile p+(z) for the ions.Previous calculations ' assumed a step func- tion or other forms for p+(z).Furthermore, p (z) was obtained ' by a variational method based on local- density-functional theory, with a simple form for the tri- al function.
Recently, however, ion profiles have been calculated by O'Evelyn and Rice from Monte Carlo simulations based on a pseudoatom theory, and we have used the profile p+(z) for mercury to generate p (z) and thence X~.A pseudoatom theory by itself cannot generate charge den- sities, as it implies local neutrality, but when p+(z) for the pseudoatom theory was used in a separate calculation to obtain p (z) we found very satisfactory results for mer- cury, as evidenced by comparison of calculated work functions to experimental results.In addition to the use of the D'Evelyn-Rice profile, our calculation used the self-consistent (Lang-Kohn) equations' instead of a varia- tional method.Even for the step-function profile, this led to a significant change in X Below, we report results for several of the metals for which variational calculations with a step-function ion profile were previously reported.We calculate X and the work functions, to assess (1) the accuracy of the variation- al theory for the step-function profile and (2) the effect of using a highly oscillatory ion profile such as that used for mercury.It will be seen that the fair agreement with ex- perimental work functions previously obtained becomes poor when self-consistent calculations replace variational methods, but becomes very good when the oscillatory ion 4836 J. GOODISMAN profile is introduced.The work function is thus sensitive to the ion profile and should indicate its quality.
Therefore, the bulk term in the work function is p, = 1.8416rb -0.6107rb 0.44(7.8+4rb /3)(7-.g+rb ) +2~&~pb(1 3 I ~o I &m) there being no electrostatic contribution.Here, pb is the bulk electron density and rb is calculated from pb according to (5).The first term in ( 6) is the kinetic-energy part, representing the sum of the kinetic energies of spin orbi- Here, X is the value of the electrostatic potential far in- side the metal minus the value far outside, while p" sometimes called the chemical potential, is the energy per electron at the top of the conduction band, relative to the bottom of the band.Since there are no fields in the bulk metal or in the vacuum outside, X~i s a surface property and p, a bulk property.
The Hamiltonian for the electrons includes the kinetic and Coulomb energies, calculated exactly (see below), the exchange and correlation energies, which are represented by a local density-dependent potential, and the interaction between electrons and ion cores.The last interaction is represented by an energy-independent local model poten- tial of the Heine-Abarenkov form, where the values of the pseudopotential core radius Rã nd the core constant Ao are those previously used, '" and given in Table I, and Z is the ionic charge.The exchange-correlation energy density is given by V", p, where p is the local electron density and the exchange- correlation potential is' where V i.s the electrostatic potential, determined from p+ and p, and Vp, (r) =Z f dr'p+(z')(Ao+s ')6(R -s) with s = ~rr' ~, is the difference between the ion- electron pseudopotential and a purely Coulomb interac- tion, averaged over the profile p+.Combining the squared eigenfunctions of (7) with eigenvalues below the Fermi level Ez [EF ( -, ' )kb.with p--b k~/3m ], we -obtain an electron density p (z).At self consistency, this density should be identical to the electron density used in con- structing the Hamiltonian of (7).Our method for achiev- ing this is discussed elsewhere.' The ion profiles we used are based on those found by D'Evelyn and Rice for mercury and cesium.' They are highly oscillatory within the metal and drop rapidly to zero outside the surface (metal-insulator transition): The contributions to p, are given in Table I.It should be noted that a different exchange-correlation functional was used in our earlier work, which gave -1.5873 rb ' -0.07007 -0.005 167 lapb instead of the second and third terms in (6).As shown in Table I, the results are only slightly changed, but, since the corresponding V", was used in determining the elec- tron density, Eq. ( 6) should be used with the X~c alculat- ed from Lang-Kohn calculations.
The Thomas-Fermi density functional used for kinetic energy gives the same contribution to p, as the correct summing over eigenfunc- tions.
The variational calculations require that all contribu- tions to the electronic energy be expressed as a local densi- ty functional.Then, assuming a form for the electronic density profile p(z), we vary' parameters in it to mini- mize the surface energy.The one-and two-parameter forms used ' were monotonic.Thus when variation is re- placed by a self-consistent solution of the integro- differential equations for electron orbitals, different (and presumably better) results will be obtained because the density functional used for the kinetic energy (~kp ) is not sufficiently accurate and because the variational func- tion is not sufficiently flexible.The self-consistent calcu- lation proceeds as follows: We obtain eigenvalues and eigenfunctions from the Schrodinger equation fixes the center of positive charge at 0, i.e. , f dzp+(z) =Lpb for large L. The parameter a governs the width of the profile, but previous calculations showed results were in- sensitive to its value, so we have used a= -, ' in all cases.
The wavelength of the oscillations is governed by P. For mercury, we chose P=1.225 to fit the profile given by O'Evelyn and Rice.For the other metals, we took P=2kF, corresponding to the wavelength of the Friedel oscillations (2kF 1.444 -for mercury), expecting results to be relatively insensitive to this parameter as well.
In Table II we give first the experimental work func- tions @ for the four metals considered, and then the re- sults for p"X, and N obtained previously, by variation- al calculations with a step-function profile for the ions.
Following that, we give the values for these quantities that result from a self-consistent calculation, again with a step-function profile (the slight change in p, is due to the changed exchange-correlation energy functional, men- tioned above).It is seen that the fair agreement we had with experimental and calculated work functions is des- troyed.The average deviation between experimental and calculated work functions goes from 0.6 to 1.3 eV, the change being greater for the metals of higher electron den- sity.
When the oscillatory profiles with a=0.5 and P=2kF (except for mercury) are used, there is a large increase in X~o ver that for the step-functions.The average devia- tion between experimental and calculated work functions is reduced to about 0.16 eV.Furthermore, 4(calculated) is too high for In, Ga, and Al.For mercury, the profile of D'Evelyn and Rice provided the value of P, which was somewhat below 2kF (1. 255instead of 1.44), and &b(calcu- lated) is slightly low.For the case of Cs, pb --0.0012341 gives 2kF --0.6637, whereas the Monte Carlo simula- tions' give a profile with P about 0.62; for Na, 2kF --0.9478 and the simulations make P about 1.0, slightly larger.It seems reasonable that the actual profiles for the other metals correspond to lower values of P and would bring N(calculated) closer to experiment, or perhaps below.
In Fig. 1 we show part of the ion and electron profiles for aluminum.Those for the other metals resemble this one.The oscillations in p follow those in p+ in position, but not in magnitude, except for the large tail of p ex-

III. CONCLUSIONS
The variational method and trial functions employed previously are not good enough to reproduce the results of the self-consistent calculation for the electrons in the presence of a step-function ion profile.The fair agree- ment between the work functions obtained from the varia- tional method and from experiment is due to a cancella- tion of errors.The use of the self-consistent method shows that the variational method, either because of the density functional or because of the trial function, is inadequate.
The values of the work functions with a step-function profile are actually several volts below ex- perimental values.Changes in pseudopotential or other parameters in the model cannot resolve the discrepancy, as results are insensitive to their values.
tending into the vacuum.The first and highest maximum in p+ occurring on the tail of p, there is a substantial os- cillation on the rapidly decreasing p here.Note that self consistency between ionic and electronic profiles does not imply coincidence between them; such coincidence, predicted by certain electrostatic models, implies a vanish- ing surface potential.Indeed, it has been argued' that monotonic profiles for ions and electrons violate self- consistency, since a monotonic electronic profile could produce one-body forces on the ions which would lead to oscillations in their distribution.p, (a.u.)   N (eV) Step-function by variation X (eV)   p, (a.u.) Step-functi.on by differential (eV) e (eV) Oscillatory profile by differential equations p, (a.u.) g (eV) @ (eV) tais which are occupied by electrons.
FIG.1.Surface density profiles for aluminum metal.Solid curve is the electronic profile, dashed curve the ionic profile.

TABLE II .
Results.