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Hydrogen chemisorption on Al, Mg and Na surfaces — calculation of adsorption sites and binding energies
Surface Science 0 North-Holland
81 (1979) 539-561 Publishing Company
HYDROGEN CHEMISORPTION ON Al, Mg AND Na SURFACES CALCULATION OF ADSORPTION SITES AND BINDING ENERGIES
*
H. HJELMBERG Institute of Theoretical Physics, Chalmers University of Technology, Received
4 August
S-41296
Gdteborg Sweden
1978
Chemisorption properties for H on Al, Mg and Na surfaces have been computed. Starting with the results from a self-consistent, Kohn-Sham calculation for a H-jellium system, the effects of the substrate pseudopotential lattice have been included by first order perturbation theory. An improved method, namely inclusion of part of the pseudopotentials in a self-consistent way and only the remainder in perturbation theory, has also been used. Binding energies, bond lengths, vibrational frequencies and activation energies have been calculated. It is found that the three substrates behave qualitatively different, as Al is a segregator, Na is an absorber and Mg is in between. Calculated bond lengths agree in general well with the ones for the corresponding molecules. The binding energy is predicted to be below or close to the dissociation energy of Hz for Al and Na, while it is slightly larger for Mg.
1. Introduction The prospect of making detailed comparisons between experiment and theory for simple chemisorption systems is becoming increasingly realistic. The growing wealth of experimental information concerns primarily adsorption on semiconductor and transition metal surfaces [I]. However, judging from the history of bulk systems, a physical understanding is more likely to come first for simple, freeelectron-like metals, for which a detailed theoretical analysis is possible. Furthermore, the free sp-conduction electrons are present also in transition metals, being responsible for most of the screening as well as a significant part of the chemisorption bond [2]. In addition, within the growing field of energy storage [3], magnesium, or rather some magnesium alloys, turn out to be among the most attractive metals for hydrogen storage. Hydrogen has numerous advantages as an energy carrier [3], and storing H as a metal hydride is both safer and more space-saving than as a gas or liquid [3]. In the storing process, dissociative chemisorption is an important step, as well as the absorption of H in the metal. This thus gives another important impetus for this study. * Supported
by the Swedish
Natural
Science
Research 539
Council.
540
H. Hjelmberg /Hydrogen
chetnisorption
on Al, Mg and Na surfaces
Using an earlier published calculational scheme [4,5], chemisorption properties of the system consisting of a hydrogen atom adsorbed on a semi-infinite jehrum substrate have already been calculated [5-81. The calculational scheme involves solving the Kohn-Sham equations [9] self-consistently with no adjustable parameters. Exchange and correlation are treated in the local density approximation 19,101, discussed in, e.g., ref. ill] together with an estimate of its accuracy. To be able to predict binding energies as well as preferred adsorption sites, the substrate lattice has to be included. This is done in the simplest possible way by using the successful solid state method of replacing the ions with a pseudopotential lattice, and then using perturbation theory. Here, however, only the lowest order perturbation theory is used on the model a results, i.e. the results from the self-consistent jellium calculation. In the following, this procedure will be called mode1 b. The validity of this model of course depends on the strength of the pseudopotentials. One improvement of model b is to use the method of Perdew and Monnier [ 12,131 to include part of the pseudopotential contribution already in the self-consistent (model a-like) calculation. This procedure, called model c, takes account of part of the variation of the substrate electron density perpendicular to the surface in a self-consistent way, while the variation parallel to the surface is included by lowest-order pe~urbation theory (as in model b). An independent investigation along similar lines (using models a and b) is being performed by Lang and Williams [ 14-181. While the present investigation stresses the dependence of the chemisorption properties on the substrate parameters, the study of Lang and Williams seems more to emphasize the dependence on the adatom. In section 2 the calculational method is described. Results are presented in section 3 for H on Al( loo), (110) and (111) surfaces using model b and c. H on Mg(OOOl), where model b is approximately equal to model c, is next discussed, followed by results for H on Na(lOO) and (110). The last part of this section comments on the nume~c~ accuracy and on expected deviations between the present calculations and experimental data. Effects of the pseudopotentials on the adsorbate-induced density of states and dipole moment are also discussed.
2. Theoretical
method and calculational
2.1. The Kahn-Sham
details
scheme
The chemisorption problem is comp~cated from a theoretical point of view, as it has local as well as extended features, low symmetry, and as correlation effects are important. The Kohn-Sham scheme [9] is simple, includes exchange and correlation effects, and has proved to give results with a useful accuracy, also in approximate versions, for atoms [1 13, molecules [ 191, solid [20] and metallic surfaces
H. Hjelmberg /Hydrogen
chemisorption
on Al, Mg and Na surfaces
541
[12,13,21,22]. A short-coming of the scheme is that it applies only to the ground state of the quantum system studied [lo] and to certain excited states [ 111. The scheme gives the density as
PW =
c 1$iW I2, i
where the sum is over occupied orbitals. The $&)‘s are solutions to the Schrodinger like equations (atomic units, with the energy in Rydbergs, are used) c-v2
+ h2ffW>
Gik)
= hi(r)
2
(2)
where d3r’ + u&r) ,
(3)
The external potential is ~,,~(r), and uX,(r) is the exchange-correlation potential, which is here treated in the local density approximation [10,23]. This approximation, which has been used in the applications mentioned above, is discussed in, e.g. ref. [ 111, together with an estimate of its accuracy. The energy parameters ei have no formal relation to oneelectron excitation energies. However, for the systems studied here, which have electron states with an extended nature, the density of states constructed from the ej’s is expected to be similar to the true one-electron excitation spectrum [24]. 2.2. Model a In model a the Kohr-Sham equations are solved self-consistently without adjustable parameters for a system consisting of an adatom on a jellium surface. The jellium, i.e. a system where the positive charge of the ion lattice is homogeneously smeared out to form a positive background, is described by one parameter, the radius r, determined by 4nrz/3 = l/p,, where p0 is the bulk electron density. The surface is formed by terminating the positive charge abruptly. The self-consistent solution to the clean surface, which is the input to the calculation, was taken from ref. [25]. The model is defined by only three parameters, r, giving the bulk electron density, d giving the distance from the adatom to the jellium edge (cf. fig. l), and Z giving the charge of the adatom nucleus. As the clean surface has no structure parallel to the surface, the adatom-jellium system possesses cylindrical symmetry, which reduces the calculational effort. Due to the metallic screening, the adsorbate-induced changes in the electron density are localized to a region near the adsorbate. As described in ref. [4] and in more detail in ref. [5] the calculational scheme uses this fact. The perturbed region is taken to be a sphere. A basis set, {p,}:,, assumed to be complete inside this
542
H. Hjelm berg /Hydrogen
chemisorption
on AI, Mg and Na surfaces
a) yz PLANE metal _
i-th
lattice
plane
-d-[Ii-l/2)
2nd lattfce
plane
-d-3/21
1st lattice
plane
-d-1/2
jellium edge
vacuum Y
-
-d
b) xy PLANE
0
1st lattice
X I-th
plane
lattkce plane
Fig. 1. Definition of the coordinate system by showing (a) the yz plane and (b) the xy plane. The origin is at the adatom, which is a distance d from the jellium edge. The positive charge of the jellium uniformly fills the z < -d half-space. In the yz plane (a), the positions of the lattice planes are shown. The (smallest) interplanar distance is 1. In the xy plane (b), the adatom position is shown relative to the ions in the different lattice planes. The positions of the ions in a lattice plane is given by the primitive lattice vectors 0, and 02, while the vector bi determines the position of the i-th lattice plane relative to the first one. All vectors in (b) are twodimensional.
sphere,
is introduced.
The adsorbateinduced
electron
density,
Ap(r),
as well as the
wave functions, pi (the solutions to eq. (2)), are expanded in this basis set. The equation for the Green function corresponding to eq. (2) is converted to a matrix equation of order NXN. The solution of this equation gives such properties as A&) inside the perturbed region [4,5]. 2.3. Model
b
To remedy an artefact of model a, namely the exaggerated repulsion between the positive background and the adatom nucleus, model b is brought into use. In this model the substrate ions are reintroduced by considering the effects to first order of a lattice of weak pseudopotentials. Such a procedure has been successfully used for calculating the surface energy of free-electron-like metals [21]. The pseudopotential effects are included to first order by adding the term
H. Hjelmberg f Hydrogen 6E~s =s(6
vb(r)
Apa
-
Z6 e(r)
chemisorption 6(r))
on Al,
Mg and Na surfaces
d3r
543
(4)
to the energy found in model a, i.e. Eh=E”+GEbp,.
(5)
(The coordinate system is defined in fig. 1.) The superscripts indicate results from model a and b, respectively. Svb(r) is the change in potential when the positive background is replaced by the pseudopotential lattice: 6 P(r)
= 6 e(r)
+ 6
V+?(r),
(6)
where 6 c is the pure Coulomb part and 6 I$ is the part representing the repulsive effects of the ion cores. E is the change in energy of the adatom upon chemisorption. There are several equilibrium properties of interest. In model b, only the binding energy, Ek= -E’(d&) (d& is the equilibrium distance), is influenced directly by the pseudopotentials. They influence only indirectly the equilibrium values of other properties, as these should be calculated according to model a, but with the equilibrium distance, d&, of model b. For instance, the dipole moment pb is equal to I.ca(&,). In the present paper, the pseudopotential origin is here at the ion nucleus) ups(r) = -2(Zi,&)
of Ashcroft
[26] is used, i.e. (the
O(r - r,) ,
(7)
where Zion is the ionic charge, 6’(x) the step function and rc the core radius. The values of Z.,O,,, r, [26], r, and the Wigner-Seitz radius rws for AI, Mg and Na are found in table 1. Taking away the positive background has the effect of screening the pseudopotentials. To illustrate this, fig. 2 shows the Ashcroft pseudopotentials of Al, Mg and Na, screened by uniform spherical charges with radii equal to rws. It should be noted, however, that the screening is not self-consistent. Fig. 2 indicates, that the Al pseudopotential is the strongest, followed by the ones for Mg and Na. This is also
Table 1 Substrate data; Zion is the ion charge, rs gives the electron density p as l/p = 4nrz/3, Q,S is the Wigner-Seitz radius, rc is the Ashcroft pseudopotential core radius 1261 and Ecoh is the cohesive energy of the metal
Al Mg Na
Zion
rs (au)
rws
3 2 1
2.01 2.65 3.99
2.99 3.34 3.99
(au)
rc (au)
Ecoh
1.12 1.39 1.67
3.34 1.53 1.13
(eV/atom)
544
H. Hjelmberg f Hydrogen
chemisorption
I
on AI, Mg and Na surfaces
I
I
I
LO
Al 30
--... 1 1 4 I
2c
1c
-.-
5 v a F
0
5 a0 -10
-20
-30
-LO ~
0
I I
I
I
e
.,
L
J
1 4
r(a u)
l:ig. 2. The Ashcroft
pseudopotentials (261 for Al, Mg and Na, screened by uniform spherical charges of the corresponding densities and Wigner-Seitz radii (rws).
reflected in the results of 6EE,,whose minimum values are -2.3,-1.5 and -0.5eV for Al, Mg and Na, respectively. The rest of this section will give the main steps of the present calculation of 6 vb(r). Another procedure is found in ref. [ 181. For convenience, the superscripts a and b will be dropped in the following. The coordinate system is found in fig. 1. (1) The electron density of model a has cylindrical symmetry, i.e., the first term in eq. (4) may be written as (8)
jXdXdzjAp(X,z)jd@‘(r). (2) The Ashcroft
pseudopotential
(eq. 7)) is split into Coulomb
and core parts
H. Hjelmberg /Hydrogen
chemisorption
on Al, Mg and Na surfaces
545
(with the origin at the ion nucleus), ups(r) = upS(r) + uy(r)
+ 2(Zi,Jr) e(r, - r) .
= -2(Zi,Jr)
(9)
The removed positive background plus the lattice of upS contribute to the potential 6 VI(r), while the lattice of uy gives 6 V,(r) (eq. (6)). (3) J& 6 V*(r) is calculated analytically (the result for one UT is expressed as an elliptical integral of the first kind). (4) &V,(r) is Fourier expanded parallel to the surface, i.e. 6 Vr(X, z) = 5
8 VI (Q, z> exp(iQ . X> ,
SV,(Q, z) =A-‘Jd’X6V,(X,
(10)
z) exp(-iQ*
X) ,
(11)
where Q = (Q,, Q,,) and A is the area of the surface. To ensure charge conservation, (12)
NxyZion = ZAP, > where NxY is the number tance (cf. fig. 1). (5) The contribution
of ions in an ion plane and I the smallest interplanar
dis-
to 6 V,(Q,z) from the i-th ion plane, 6 V:(Q,z),
8 vf(Q. z) = A-’ Jd2X
exp(-iQ
. X) c
up”@’ -R
R
- Xi,
zi)
,
(13)
where (cf. fig. 1) R=m,
fm2,
n,m=-m
,..., -l,O,l,...,
m,
(14)
x;=x*+!+, zi=z+dtI(i-
(15) l/2).
(16)
As can be seen from eq. (16), zi has its origin at the i-th ion plane. The Q = 0 term, which is the only term where the removed positive background enters, is [21] 6 Vf(Q = 0, z) = -4np,(
lzil - 1/2)2 ,
lzil
(17)
The contribution to 6 v’,(Q # 0, z) (eq. (13)) is non-zero, only for Q = G, where G is a two-dimensional reciprocal lattice vector of the ion planes. The integrations in eq. (13) are performed analytically [27], and with the use of eq. (12). 6 Vf(G, z) = -4np,ZG*
exp(Glzjl)
exp(-iG*
Xi),
G f 0.
(18)
546
H. Hjelmberg /Hydrogen
(6) The summation
chemisorption
on AI, Mg and Na surfaces
over i, i.e. over the ion planes, is now performed,
i.e.
W,(G,z)=~Wf(G,z).
(19)
i=1
For points with zi > 0, i.e. outside the i-th ion plane, the sum is calculated straightforwardly. For zi < 0, the following procedure is used: The vector bi only takes on a few different values, e.g., an fcc(lO0) surface has bzj_t = 0 and bzj = (ur + a2)/2, j= 1, 2, . . . . Thus Xi (eq. (15)) takes on the same number of different values, making it possible to split the sum in eq. (19) into a few subsums, each with a constant value of XL. In such a subsum, the only dependence of i left is in zi (eq. (16)), making the subsums infmite geometrical series (for Zi < 0). As an example, taking all bi = 0, eq. (19) reads (G # 0)
SV,(6, z) =
c
i=l
Wf(G,
4711pe exp( -G lZiO z) - ~ G 1 -exp(-GZ)
I)
exp(-iG
. X,)
,
(20)
where ie is the smallest i for which Zi < 0. (7) The last step in calculating 6 V,(r) is to sum over G (cf. eq. (10)) 6 V,(X, z) = c
6 Vr(G, z) exp(iG . X) .
(21)
G
Note that the p-integration
(cf eq. (8)) may be performed
analytically,
as [27]
2n
s 0
dp exp(iG . X) = 27rJ,(GX)
,
(22)
where Jo is the cylindrical Bessel function of order zero. Thus the cp-integration in eq. (8) is made analytically (cf. point 3). As both the G and -G term enter in the sum (21) with the same coefficient (cf. eqs. (18) and (20)), exp(-iG . Xi) may be replaced by cos(G . X9. The same argument can be used to reduce the number of terms in the sum (2 1). Finally, one has to perform the integration in X and z in eq. (8) to get 6E,,. A disadvantage with this method is that the sum (21) diverges for z in an ion plane (Zi = 0) and has a slow convergence nearby z (cf. eq. (20)). As these points Zi * 0 only make up a small volume in space, however, and as the values of 6 V(r) in this volume may be interpolated from values for /Zi] > 0, the resulting error is small. This divergence is not present in the method described in ref. [ 181. 2.4. Model c The obvious shortcoming of model b is that the substrate may still not be well described, i.e. the pseudopotentials might not be weak enough to make the use of
H. Hjelmberg /Hydrogen
chemisorption
on Al, Mg and Ma surfaces
541
first order perturbation theory sufficiently accurate. An improved description can be obtained by using a method of Perdew and Momrier [ 12,131, where part of the pseudopotential contribution is included in the self-consistent calculation and only the remainder in perturbation theory. The method has a variational parameter c, whose optimum value depends on the surface plane considered. This is done by adding a step potential cB(-z - d) to the external potential of the clean jellium surface (eq. (3)). The Kohr-Sham equations (2) and (3) are solved self-consistently, giving a different electron density of the clean surface, p”(z), for every c. The great advantage is that due to the translational symmetry in two dimensions, the self-consistent calculation is as easy to perform as for the pure jellium surface. The value of c was determined by Perdew and Monnier [12,13] by minimizing the expression for the total surface energy (pseudopotentials included), using p”(z) for the electron density. Thus consistently improved results for the surface energy [ 121 as well as the work function [12,13] of clean metal surfaces have been obtained. The role of the step potential cB(-z - d) is to mimic the “real” potential 6 vb(r) in eq. (4). This is well done, as can be concluded from (i) the successful application to clean surface properties, (ii) the good correlation between the value of c and the average of 6 vb(r), (iii) the good correlation between the first “peak” in p”(z) and the first dip in 6 p(z) (the average parallel to the surface) and (iv) the successful application to the vacancy formation problem in the bulk [28]. The values of c [ 12,13,25] for different surfaces are given in table 2 together with the corresponding locations of the bottom of the band. Table 2, indicates that
Table 2 Clean surface parameters of model c (section 2.4); the optimum values of c [ 13,251, the size of the step in the potential &-z-d) introduced by Perdew and Monnier [ 12,131, are given together with the corresponding values for the energy of the bottom of the band, uztt (--) (zero energy at the vaccum level); the value of & (--) for c = 0 gives the result of model b (section 2.3) Metal Al (fee)
Face
c (eV)
u%f (--)
_
0
-15.59 -15.48 -15.40 -15.92
(100) (110) (111)
Mg (hw)
Na (bee)
1.05 3.14 -1.87
_
0
(0001)
0.4
_
0 1.07 0.28 1.45
(100) (110) (111)
-10.83 -10.8 -6.25 -5.91 -6.16 -5.93
(ev)
548
II. Hjelmberg /Hydrogen
chcmisorption
on Al, Mg and Ma surfaces
in the self-consistency procedure, the long-range effects of the potential step are reduced, as the energy of the bottom of the band is shifted with less than 0.4 eV or less than 30% of c. The application of this method to the chemisorption problem is straight-forward, as the actual calculations are those of model b. The difference lies in the external potential in the self-consistent part of the calculation, as the step potential cB(-z - d) (values of c from ref. [25]) is included. This part gives a c-dependent adsorbate-induced electron density, Ape(r), i.e. one that depends on the surface plane. The rest of the effect of the lattice pseudopotentials, s
[S V”(r) A#@) - S V,‘(r) Z&(r)] d3r ,
6 V”(r) = 6 V”(r) -- cq-z
- d) )
(2% (24)
is then added to first order to get the energy he [29]. Consequently, model c is reduced to model b if the parameter c is equal to zero. Thus an improved description of the variation of the electron density normal to the surface has been obtained in model c. Still lacking, however, is the variation of the electron density parallel to the surface.
3. Results Inclusion of the substrate structure by reintroducing the pseudopotential lattice to the jellium (model a) results, makes it possible to predict stable positions, binding energies, vibrational frequencies and activation energies for diffusion along the surface as well as into the substrate (absorption). In this section detailed results for H of Al(lOO), (llO), (111) (section 3.1), Mg(OOO1) (section 3.2), Na(lOO) and (110) (section 3.3) will be presented. For Al, calculations using both model b and c have been performed. In section 3.1, only the results of model c will be discussed, while a comparison of the models is made in section 3.4. As model b and c turn out to be almost the same for Mg(OOO1) [13] (cf. table 2), only the former has been used. Finally, for Na, only results of model b witl be presented. The input data needed from model a calculations have been all taken from ref. [5], whose input in its turn (clean surface data) was taken form ref.
P-51. Selected and prelirnina~ results for H on Al and Na have already been published [S-8]. An independent investigation carried out by Lang and Williams has given selected results for H, Li, 0, Na, Si and Cl on a jellium surface (I, = 2.00) [14-181, Si (for d = 1.5 au) on a jellium surface (rs = 4.00) [18] and for 0, Na, Si and Cl on jellium (r, = 2.00) with perturbing Al pseudopotentials corresponding to the (111) surface [ 16-181 (equivalent to model b).
H. Hjehnberg /Hydrogen
chemisorption
on Al, M, and Na surfaces
549
-2 -
I
-3
I
-2
1
I
-1
0
I
1
I
2
d(au1 Fig. 3. Calculated energy curves for H chemisorbed on three different jellium substrates, rs = 2.07 au (Al), r, = 2.65 au (Mg) and rs = 3.99 au (Na) (51. The distance between the H atom and the jellium edge is denoted by d.
Section 3.4 is devoted to a discussion of models b and c, including their accuracy, while section 3.5 deals with an estimate of the effects on the adsorbateinduced density of states and dipole moment from the pseudopotentials. In the discussion of the results below, it is useful to have in mind the two main factors that determine the bond length: (i) The repulsion due to the increase in kinetic energy, when the adatom enters the conduction electron density of the substrate. This “kinetic repulsion”, discussed in detail in ref. [S], is very r,-dependent. This is illustrated in fig. 3 [5], showing the results for H on three jellium surfaces (model a). The importance of this factor is also illustrated by compa~g results for two different substrate densities. For instance, using r, = 2.00 instead of 2.07 in model b, for H on Al(11 I), would change the energy with about OS eV for d = -1 au and change the equilibrium distances for bridge and center positions with about 0.3 au [30]. (ii) The attraction, when the adsorbate-induced electron density enters the region just outside an ion core, and the strong repulsion, when it penetrates an ion core of the substrate (cf. fig. 2). This penetration would occur for an adsorbate-substrate atom distance of about r, + +t [6], where r, is the core radius (cf. section 2.3) and rH (02 au) the radius of the sphere around the proton that contains one electron. It has been noticed, that the distance r, t TH agrees with the bond length for the molecules AlH, MgH, NaH, KH, RbH and CsH within 5% [6]. if one of the interactions (i) or (ii) above dominates over the other (for relevant distances), two different situations may be distinguished. If the “kinetic repulsion” (i) is dominating the force, all different sites should have almost the same vibrational frequency (wc,,~), as this corresponds to the jellium substrate. If, on the
550
H. Hjelmberg /Hydrogen
on Al, Mg and Na surfaces
I
I
I o-
chemisorption
I
H/AL (110)
b Ai -l -
4
-2 -
-
1.35.a I -1
” ---I
0
I I
I 2
other hand, the force between the adsorbate and the nearest substrate atom(s) (ii) is dominating, the vibrational frequencies should be strongly dependent on the site, following in general the sequence c&r,r > uabr > asbr (A, B and C indicate the site, cf. figs. 4-6). The sequence is determined by the effect that, compared with the A site, the substrate atom-adatom distance varies much more slowly with d (fig. 1) for the B and especially the C site. This effect is in general stronger than the
H. Hjelmberg /Hydrogen
chemisorption
on AI, Mg and Na surfaces
551
Fig. 4. Calculated energy curves for H chemisorbed in several configurations on the (a) AhlOO), (b) (110) and (c) (111) surfaces. Results for both model b (dashed lines) and model c (full lines) are shown for (a) (100) and (b) (110). The distance between the H atom and the jellium edge is denoted by d. The distance from the jellium edge to the first layer of Al ions is given in the lower left corner for each surface. In the upper right corner is a figure of the unit cell of that surface, defining the different configurations. Or
I
I
I I
_I
I
I
I
H/Mg (0001)
I
-1
-I
”
I
dfoul Fig. 5. Calculated energy curves for H chemisorbed face. As results of model b and c should be almost The distance between the H atom and the jellium layer of Mg ions is marked by a vertical dashed defining the different configurations (C standing the configurations are shown with respect to the indicated.
in several configurations on a Mg(0001) surequal, only the results of model b are shown. edge is denoted by d. The position of the first line. Near to this line, the unit cell is shown, for both CA and CC). In the upper left corner, second layer of Mg ions, which position is also
552
H. Hjelmberg /Hydrogen
chemisorption
on Al, Mg and Na surfaces
I
I
I
Nqh-,q
H/Na(lOO)
","u";
”
1
I
o-c-i!l I
a
J
C
-2 ---_
------2.02 au
L
B
.H
/J
-4
.N’
-1 NR
I -1
I 0
I
1
dtau)
\;
H/Na(llO)
0
Noc)-L?-$3 7.01 ,
;,I.$
\
b
A
\
“,$o
\
4
\
“\
“\
I
I -2
I -I
I 0
I I
die u)
Fig. 6. Calculated energy curves for H chemisorbed in several configurations on the (a) Na(100) and (b) Na(l10) surfaces. Only results of model b are shown. The distance between the H atom and the jellium edge is denoted by d. The distance from the jellium edge to the first layer of Na ions is given in the lower left corner for each surface. In the upper right corner is a figure of the unit cell of that surface defining the different configurations.
contributions to the force from the extra substrate atoms present in the B and C configurations. Both models (b and c) are expected to describe the real situation best for the most close-packed surfaces, i.e. Al(lll), Mg(OOO1) and Na(1 lo), as discussed in section 3.4.
H. Hjelmbetg / Hydrogen
ckemisorption
on Al, Mg and Na surfaces
5.53
Table 3 Data for the most symmetric configurations, which are defined in figs. 4-6: Eg is the binding energy [ 3 11, do the distance to the first ion plane, dM_H the bond length and Wvibr the vibrational frequency; values for two CA configurations are given for Mg, CA’ outside and CAi inside the first ion plant; for CA’, dl and dM__H are the distance to the nearest substrate ion in the fist and second layer, respectively; all molecular data are from ref. 132) (EB here without vibrational contributions, cf. ref. [ 3 11) H on Al (molecule) Al (100)
Al(110)
Al(111)
Mg (molecule) Mg(0001)
Na (molecule) Na (100) Na (110)
Position
EB (ev)
dl (au) -
dM -_H (au)
Wvibr (meV)
A B c A SB LB C A B CA cc
3.01 1.9 2.3 1.4 2.3 2.4 1.7 (1.4) 1.9 1.9 1.7 1.8
3.11 3.0 2.0 2.4 3.2 2.2 2.3 2.9 2.0 2.1 1.9
3.11 3.0 3.4 4.5 3.2 3.5 4.5 2.9 3.4 3.8 3.6
209 210 130 70 180 170 110
~2.58 2.2 2.6 2.5 2.7 2.6
3.21 3.3 2.1 2.1 1.9 3.8
3.21 3.3 3.7 4.1 4.0 3.5
186 200 150 110 80 140
2.1 2.4 2.1 1.9 2.4 2.3 2.3
3.57 _ _ 3.5
3.51 3.5 -
14s
A B CAS CC. CA’
B C A SB LB C
300 150 120 120
120 -
Equilibrium properties, like the binding energy Eu, the equilibrium distance to the nearest ion plane dL (= d,, t l/2), the bond length dAr__t, and the vibrational frequencies uvibr, are listed in table 3 for the most symmetric positions. Fig. 4 shows the change in energy upon chemisorption, E, as a function of d. The calculations using model c predict (table 3) that hydrogen is stable in the bridge (B) configuration on the Al(100) and (110) surfaces and atop (A) or bridge on the Al(111) surface, and that the binding energies [31] are 2.3 eV (loo), 2.4 eV
554
H. Hjelmberg /Hydrogen
chemisorption
on AI, Mg and Na surfaces
(110) and 1.9 eV (Ill). The corresponding distances dL are 2.0 au (loo), 2.2 au (110) and 2.9 (A) and 2.0 au (B) (111). Furthermore, the activation energy for diffusion along the surface is estimated to be 0.1-0.2 eV [6], while the activation energy for absorption is high, 1.2-1.4 eV. As the dissociation energy of the Hz molecule [31], calculated with the same Kohr-Sham formalism, is 2.4 eV per atom [33], the model c calculation predicts that the binding energy could be just enough to make a break of the H2 bond thermodynamica~y favourable [34]. However, such a small energy difference is not considered significant, as it is well within the estimated error of the calculations (cf. section 3.4). Furthermore, both H atoms would have to take advantage of the high binding energy in a preferred site, which is geometrically impossible. Together with the expected decrease in entropy [ 1f , dissociative chemiso~tion of H2 could only, if at all, take place at low temperatures. This prediction is in accord with experiments [35]. Absorption of H in Al is improbable, due to the large activation energy (1.2-1.4 eV). This is mainly due to the “kinetic repulsion” (cf. fig. 3). A hydrogen atom put inside a perfect crystal of Al would rather segregate to the surface. The calculated bond lengths (dAl_H) are in good agreement with the corresponding molecular bond lengths, with the exception of some C positions (table 3). A bond length considerably larger than the molecular one, should correspond to a smaller binding energy (cf. the “re t rH” argument in the introduction to section 3 as well as fig. 2).That is the likely reason, why the C positions on the (100) and (110) surfaces and the long bridge (LB) on the (110) are not favoured, as the smallest bond lengths allowed by geometry (= X, in fig. 1) are considerably longer than the molecular one. For the C position on the (111) surface, on the other hand, the distances are equal. The vibrations frequencies follow the sequence w$& > o$br > o$br (table 3), indicating that the adatom-nearest substrate atom force is stronger than the force due to the “kinetic repulsion” (cf. the discussion in the introduction to section 3). Relaxation of the lattice around the adatom is neglected in the calculations. Using the large cohesive energy of Al (cf. table 1) as an indication suggests that this may be a small effect, at least on the more close-packed planes. 3.2. H on Mg(OOU1) The (0001) surface is the most close-packed one of the different Mg surfaces. It turns out that models b and c are almost the same in this case, as the value of c (the “step height”) is very small (table 2). Equilibrium properties are presented in table 3 and the energy curves E($l in fig. 5. The model b calculations predict that the stable position for H chemisorbed on Mg(0001) is the center configuration CC (over a hole also in the second ion plane) with a binding energy [31] of 2.7 eV and with d,= 1.9 au. The activation energy for diffusion along the surface is small, 0.1-0.2 eV (as for Al), and so is the activa-
r%Hjelmberg 1 Hydrogen ckemisorption on Al, Mg and Na surfaces
555
tion energy for diffusion inwards the metal, about 0.4 eV. As the activation energy for this diffusion, and thus for absorption, is small, results are give in fig. 5 also for the hydrogen atom placed inside the first ionic plane. The stable site inside (position CA’ in table 3) is predicted to be near the tetrahedral position, with a binding energy [3 l] of 2.6 eV. The bond lengths to the four nearest Mg atoms are 3.5 and 3.8 au (table 3). In contrast to the result for H on Al, the binding energy on Mg (2.7 eV) is sigmficantly larger than the energy needed to break the H2 bond (cf. ref. [35]). As on Al, the bond lengths for H chemisorbed on Mg are similar to the molecular one (table 3). The vibrational frequencies follow the sequence w&,~ > O& > W& for H chemisorbed on Mg (table 3), as was the case on Al. Some data have been collected for the Mg-H system, due to its importance in the context of storing energy. Magnesium forms a hydride, MgH2, with an energy of formation of 0.77 eV per molecule of Hz [36] or a binding energy [31] of 2.8 eV per atomic H. The structure changes from hcp to tetragonal (similar to rutile, TiO?) [36]. The reaction rate for the formation of the hydride is unfo~unately very low, several months even for small particles, making it unsuitable for direct technical applications [36]. This low reaction rate is probably due to oxidation of the Mg surface. The calculated binding energy for H in Mg metal, 2.6 eV, lies close to the energy of formation of MgHz. Furthermore, inclusion of relaxation of the Mg lattice, which is not present in the c~culation, would lower this binding energy. Some relaxation may be expected, if the small cohesive energy of Mg (table 1) is a good indicator. The bond lengths between the H and the nearest three Mg atoms in MgHs, are 3.6 and 3.7 au [37], i.e. similar to the ones found in the present calculations for the CA’ position inside the metal (table 3).
Only model b has been used in these calculations, because using model c should not significantly change the picture given here, as the Na pseudopotentials are weak (cf. fig. 2). Results for the two most close-packed surfaces, (110) and (loo), are shown in table 3 and fig. 6, and results for the (111) surface are qu~itatively similar. The calculations predict that H is absorbed in Na, with a binding energy [3 1] of 2.4 eV. This energy is just large enough, that the Hz bond might be broken (cf., however, the discussion in section 3.1). The bond length for the atop position on the (110) surface agrees well with the corresponding molecular bond length (table 3). The lack of structure in the energy curves in fig. 6, suggests that the activation energy for diffusion inside the metal is small compared to, e.g., Mg (fig. 5). This is
556
H. Hjelmberg /Hydrogen
chemisorption
on Al, Mg and Na surfaces
in agreement with experimental data for diffusion of several other impurity atoms in Na and Mg, as the activation energy is typically S-10 times larger in Mg [38]. A hydride, NaH, exists, with an energy of formation of 1.2 eV per hydrogen molecule 1361 or [31] 3.0 eV per hydrogen atom. Such an increase in binding energy, as compared to the value given by the present calculation (2.4 eV), could be partly accounted for by the presence of relaxation of the lattice, or, at higher H concentrations, by the structural change from bee (Na) to fee (NaH) [36]. Such effects are not considered in the present calculations. The small cohesive energy of Na (table 1) suggests, that there may be relaxation around a hydrogen atom also at very low H concentrations. The hydride is something quite different from the system studied here. It is interesting to note, however, that this model calculation gives a similar energy for a situation, which might be the embryo of the hydride formation. 3.4. Discussion of models b and c In this section a discussion of the accuracy and physics of models b and c will be made. In comparing the results of models b and c with experimental data, there are essentially only two approximations that should be kept in mind: Firstly, exchange and correlation are described in the local spin-density approximation (LSDA) [ 111. (The spin is important when calculating, e.g., the free H atom properties [39], cf. ref. [ll]). The LSDA is estimated to introduce an error of 0.1-0.5 eV in the binding energy for hydrogen on simple metals [6,11], and a smaller error in energy differences like activation energies. The second and most serious approximation is the model for the substrate. There is no a priori reason why first-order perturbation theory in the pseudopoten&al should give a sufficient accuracy in the present application. The usefulness of models b and c should at this stage rather be judged from the soundness of their results (sections 3.1-3). The reasonable results for the bond lenghts (table 3), with the exception of maybe the most open positions, support the use of these models. For other properties, e.g. the binding energy, a good estimate of the error is hard to get due to the lack of experimental data. A comparison of the binding energy (En) with the formation energy of the corresponding hydride (for Al, it does not exist; MgH, and NaH) [36], suggests that the total error in EB [3 l] is less than 0.6 eV, which should then be a conservative estimate (cf. e.g. the discussion in section 3.3). A further illustration of the error is given by comparing the results of model b with model c, as part of the pseudopotential correction is treated self-consistently in the latter. The energy curves differ with less than about 0.1 eV for Al(lOO) and (111) and 0.4 eV for Al(110) (cf. figs. 4a and 4b). Model c has been introduced to remedy the following artefact of model b: On to section 3) is the close-packed surfaces the “kinetic repulsion” (the introduction exaggerated and therefore probably the equilibrium distances overestimated and the
H. Hjehberg
/Hydrogen chemisorption on Al, Mg and I% surfaces
557
binding energies underestimated, while the opposite should be true on the open surfaces. This can easily be envisioned by noticing that the interplanar distance 2 (fig. 1) characterizes the thickness of the electron density fall-off region (at about a distance Z/2 outside the first ion plane, i.e. at the jellium edge, the density is one half of the bulk value). For a close-packed surface, 1 is large, and therefore the “kinetic repulsion” starts to grow too far outside. This effect is most clearly seen for the open surface of Al(1 IO), which is described by a large positive value of the step c (table 2). Compared with model b, model c lowers the energies ford 2 0.8 au and raises them for smaller values of d, thereby giving larger d,, (fig. 4b). In the interior, i.e. for d very large and negative, models b and c always given the same result. A similar, but smaller change is seen for Al(100) (fig. 4a), whose value of c is also positive but smaller (table 2). For the most close-packed surface, Al( 111) (fig. 4c, results of model b not shown), the change is only slightly larger, but with a different sign compared with Al(100). The value of c is accordingly negative (table 2). The deficiency of models a and b still present in model c is the constancy of the electron density of the clean surface parallel to the surface. Due to this, the presence of the “kinetic repulsion” introduces the fo~owing effects. For positions, where the electron density should be smaller than average, model c predicts too large equilibrium distances and probably too small binding energies. The B and, especially, the C sites belong to these positions. The electron density variation parallel to the surface is smallest for the most close-packed planes. Therefore these are the ones, for which the present calculations are expected to give the best results. For the open surfaces, like Al(110) (fig. 4b), on the other hand, results presented here are considered least reliable. This deficiency is significant only for substrates with high densities (small rS), as the “kinetic repulsion” is not very important for rS > 3-4 (cf. tig. 3). The picture of the chemisorption bond is still that of an adatom on jellium, also in models b and c. The adatom electrons have no possibility of forming directed chemical bonds with the nearest substrate atoms, as the ion lattice is not present when the electron density is calculated. 3.5. Density of states and dipole moment In model b effects of the pseudopotentials on the total energy are considered. For, e.g. the adsorbate-educed density of states A&E) and dipole moment P, no direct effects are calculated. The indirect effects are easily included by using the model a values for AP(E) and p, which correspond to equilibrium distances, as calculated in model b (cf. section 2.3). The accuracy of this procedure may be estimated by comparing these results with those of model c. The model a like calculation, which is the first step in model c, includes an average of the pseudopotenti~s self-consistenly and gives A&e) and lz for a number of representative values of the step c (cf. section 2.4). For H on Al with -1
558
H. Hjelmbetg [Hydrogen
chemisorprion
on Al, Mg and Na surfaces
it is found that (i) the energy of the resonance peak in AP(E) is shifted with less than about 1 eV, (ii) the position of this peak in A&) follows the clean surface effective potential (eq. (3)) as in model a [8,.5], and that (iii) the dipole moment is changed with less than about 0.1 au. Delocalized electron states should feel the average 6 V(r) of the pseudopotential correction 6 V(r) rather than its details. As the size of potential step is well correlated with 6 Y(r) [ 12,131, an estimate of the kind described above should be reasonable for adsorbate-induced resonances in the band, e.g., H on free-electron-like metals [5]. For the more localized, deeply bound states, however, the details of 6 V(r) have to be considered [40]. 4. Summary Chem~sorption properties for H on Al, Mg and Na have been calculated with three increasingly more realistic models. The first one, model a, involves using an earlier published calculational scheme [4,5] for solving the Kohn-Sham equations [9] self~onsistently and with no adjustable parameters. The substrate is described by a semi-infinite jellium. Results of model a have been published elsewhere [5-81. The second model (b) adds effects of the substrate pseudopotential lattice as a first order perturbation to the result of modei a. Finally, the third model (c) is an improvement of model b, as part of the pseudopotential contribution is included in the self-consistent calculation [ 12,131. Still lacking, however, is the variation of the clean surface density parallel to the surface. The main predictions of the present calculations are the following: (1) On the low-indexed surface of Al is H chemisorbed in the bridge position, or on the (Ill) surface, possibly in the atop position. The binding energy [3 l] is in the range 1.9-2.4 eV. The calculated values for the binding energy is too small or he too close to the H2 dissociation energy, to make dissociative chemisorption of Hz on these surfaces a likely event. While the activation energy for diffusion parallel to the surface is small (0.1-0.2 eV), it is Iarge for diffusion into the substrate (IL?1.4 eV), making the absorption of H improbable. A H atom placed in a perfect crystal of Al would rather segregate to the surface. (2) On the Mg(OOO1) surface, H is chemisorbed in the center position (with no substrate atom below also in the second layer) with a binding energy /31] of 2.7 eV, large enough to allow dissociation of Hz. The activation energy for diffusion into the bulk is not very large (0.4 eV), making absorption possible for reasonable temperatures. In the bulk the hydrogen atom should sit approximately in the tetrahedral position with a binding energy [31] of 2.6 eV. (3) H should be absorbed by Na at all temperatures, as there is hardly any activation energy for this process, and that it at the same time gains an energy [31] of 2.4 eV. (4) The vibrational frequencies Wtibr for H on Al and Mg decrease in the sequence ~$r,r > w& > w&, where A, B and C denotes the atop, bridge and center positions, respectively.
H. H_jelmberg /Hydrogen
chemisorption
on Al, Mg and Na surfaces
559
(5) The adsorbate-induced density of states and dipole moment of H on free-electron-like metals should not be effected very much by the direct pseudopotential perturbation, i.e., given a good equilibrium distance, these properties may be taken from a model a calculation. For adsorbates with more localized adsorbate-induced states, however, the local field corrections might given strong effects. Unfortunately, no detailed experimental data exist yet for the chem~orption situations considered here (cf. ref. [35]). Comparing the calculated results with other relevant experimental data, one finds that: (1) Bond lengths agree well with the corresponding molecular ones, especially for the atop positions. (2) The total error in the calculated binding energies 1311 is, in a conservative estimate, less than 0.6 eV.
Acknowledgements I am especially grateful to B.I. Lundqvist for his encouragement and for the great interest he has taken in this work. Im also indebted to 0. Gunnarson for many helpful discussions. It is a pleasure to thank S. Andersson, B. Kasemo, J.K. N$rskov, D.M. Newns, J.P. Muscat and all the members of the Solid State Theory Group at the Institute of Theoretical Physics in Goteborg for helpful discussions. The hospitality of the Institute of Physics, Aarhus, where part of this work has been done, is gratefully acknowledged.
References [l] J.P. Muscat and D.M. Newns, Chemisorption on Metals, Prog. Surface Sci. 9 (1978) 1. [2] G. Blyholder, J. Chem. Phys. 62 (1975) 3193; I.P. Batra and 0. Robaux, Surface Sci. 49 (1975) 653; D.J.M. Fassaert and A. van der Avoird, Surface Sci. 55 (1976) 291, 313; D.E. Ellis, H. Adachi and F.W. Averill, Surface Sci. 58 (1976) 497; CF. Melius, J.W. Moskowitz, A.P. Mortola, M.B. Baille and M.A; Ratner, Surface Sci. 59 (1976) 279; K. Sch~nhammer, Inter. J. Quantum Chem. Sll (1977) 517. [3] See, e.g., Proc. Intern. Symp. on Metal Hydrides for Energy Storage, 1977, Geiio, Norway, Intern. J. Hydrogen Energy, to be published. [4] 0. Gunnarsson and H. Hjelmberg, Physica Scripta 11 (1975) 97. [S] H. Hjelmberg, Phys. Scripta 18 (1978). [6] 0. Gunnarsson, H. Hjelmberg and B.I. Lundqvist, Phys. Rev. Letters 37 (1976) 292. [7] 0. Gunnarsson, H. Hjelmberg and B.I. Lundqvist, Surface Sci. 63 (1977) 348. [8] H. Hjelmberg, 0. Gunnarsson and B.f. Lundqvist, Surface Sci. 68 (1977) 158. [9] W.Kohnand L.J.Sham,Phys.Rev. 140(1965)A1133. [ 101 P. Hohenberg and W. Kohn, Phys. Rev. 136 (1964) B864. [ 1 l] 0. Gunnarsson and B.I. Lundqvist, Phys. Rev. B13 (1976) 4274.
560 [ 121 [ 131 [ 141 [ 1.51 [ 161
[ 17) [18] [ 191
[20]
[21]
[ 221 1231 [24] [25] [26] [27] 1281 [29]
[30] [31]
1321 133) [34]
[3.5]
H. Hjelmberg /Hydrogen
chemisorption
on Al, Mg and Na surfaces
J.P. Perdew and R. Monnier, Phys. Rev. Letters 37 (1976) 1286. R. Monnier and J.P. Perdew, Phys. Rev. Bl’? (1978) 2.595. N.D. Lang and A.R. W~Bams, Phys. Rev. Letters 34 (1975) 531. N.D. Lang and A.R. Williams, Phys. Rev. Letters 37 (1976) 212. K.Y. Yu, J.N. Miller, P. Chye, W.E. Spicer, N.D. Lang and A.R. Williams, Phys. Rev. B14 (1976) 1446. N.D. Lang and A.R. Williams,Phys. Rev. B16 (1977) 2408. N.D. Lang and A.R. Williams, Phys. Rev. B18 (1978) 616. 0. Gunnarsson, P. Johansson, S. Lundqvist and B.I. Lundqvist, Intern. J. Quantum Chem. 9s (1975) 83; 0. Gunnarsson and P. Johansson, Intern. J. Quantum Chem. 10 (1976) 307; 0. Gunnarsson, J. Harris and R.O. Jones, J. Chem. Phys. 67 (1977) 3970. 0. Gunnarsson, B.1. Lundqvist and J.W. Wilkins, Phys. Rev. BlO (1974) 1319; J.F. Janak, V.L. Moruzzi and A.R. Williams, Phys. Rev. B12 (1975) 1257; J.F. Janak and A.R. Williams, Phys. Rev. B14 (1976) 4199; V.L. Moruzzi, A.R. Williamsand J.F. Janak, Phys. Rev. B15 (1977) 2854. N.D. Lang and W. Kohn, Phys. Rev. Bl (1970) 4.555; 83 (1971) 1215. N.D. Lang, Solid State Phys. 28 (1973) 225. The explicit form of uxc used in the present calculations is taken from ref. [ 111. See, e.g., the discussion in ref. [S]. R. Monnier, private communication. N.W. Ashcroft, Phys. Letters 23 (1966) 48. I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, New York, 1965). R.M. Nieminen, J. Nucl. Mater. 69.-70 (1978) 633. In the expression for the energy, the function c6(-z - d) enter in both the self-consistent and the pseudopotential part, but with different signs. Thus there will be no discontinuities in the energy k”(d) at d = 0. The value of rs = 2.00 is used in refs. [6] and [7] and in the model b results of ref. [S]. This vaiue is also used by Lang and Williams [14-IS]. No correction due to the zero point vibrational motion has been made. Its effect is to lower the binding energies. For Ha, it is 0.136 eV per H atom f 321 and for chemisorbed H, the values in table 3 divided by 2 should be used. Taking this effect into account would change the energy differences upon dissociative chemisorption of Ha with less than about 0.1 eV per H atom. American Institute of Physics Handbook, Ed. D.E. Gray, 3rd ed. (McGraw-Hill, New York, 1972). From 0. Gunnarsson and P. Johansson, ref. [19]. Refs. [6] and ]7] predicted a binding energy of 2 eV, making dissociative chemisorption of Ha impossible. The present results are, however, regarded as more accurate, due to improvements in the computations as well as the use of the proper density (rs = 2.07) for Al. No indication of adsorption of atomic or molecular 11 has been observed at room temperature on Al and Mg in ultra violet photoemission experiments by S.A. Flodstrom, L.-G. Petersson and S.B.M. Hagstrom, J. Vacuum Sci. Technol. 13 (1976) 280. However, this does not necessarily prove the nonexistence of H chemisorption on these metals, as the structure induced by chemisorbed H in the UPS spectrum could be very weak. Furthermore, an activation energy for desorption of at least about 1 eV is needed, for a chemisorbed layer to stay on the surface during such an experiment, performed at room temperature and very low hydrogen pressure (a prefactor of IOr set-r is used). This corresponds to a binding energy (EB) for chemisorbed H of at least about 2.8 eV [ 311. In this estimate, 2EB -. &.$a, corrected for vibrational contributions (cf. ref. 13111, has been
H. Hjelmberg /Hydrogen
chemisorption
on Al, Mg and Na surfaces
561
used as the activation energy for desorption (DHZ is the dissociation energy of Hz). [36] See, e.g., A.J. Maeland, in: Proc. Intern. Symp. on Metal Hydrides for Energy Storage, 1977, Geilo, Norway, Intern. J. Hydrogen Energy, to be published. [37] Crystal Data, Determinative Tables, Eds. J.D.H. Donnay, G. Donnay, E.G. Cox, 0. Kennard and M.V. King, 2nd ed. (American Crystallographic Association, 1963). [38] Handbook of Chemistry and Physics, Ed. R.C. Weast, 54th ed. (CRC Press, 1973-74) p. F-57. [ 391 Values for the free H atom are taken from J.K. N$rskov, private communication. He has used the same calculational scheme [4,5] as is used in the present paper. [40] This is also discussed in ref. [ 181.
81 (1979) 539-561 Publishing Company
HYDROGEN CHEMISORPTION ON Al, Mg AND Na SURFACES CALCULATION OF ADSORPTION SITES AND BINDING ENERGIES
*
H. HJELMBERG Institute of Theoretical Physics, Chalmers University of Technology, Received
4 August
S-41296
Gdteborg Sweden
1978
Chemisorption properties for H on Al, Mg and Na surfaces have been computed. Starting with the results from a self-consistent, Kohn-Sham calculation for a H-jellium system, the effects of the substrate pseudopotential lattice have been included by first order perturbation theory. An improved method, namely inclusion of part of the pseudopotentials in a self-consistent way and only the remainder in perturbation theory, has also been used. Binding energies, bond lengths, vibrational frequencies and activation energies have been calculated. It is found that the three substrates behave qualitatively different, as Al is a segregator, Na is an absorber and Mg is in between. Calculated bond lengths agree in general well with the ones for the corresponding molecules. The binding energy is predicted to be below or close to the dissociation energy of Hz for Al and Na, while it is slightly larger for Mg.
1. Introduction The prospect of making detailed comparisons between experiment and theory for simple chemisorption systems is becoming increasingly realistic. The growing wealth of experimental information concerns primarily adsorption on semiconductor and transition metal surfaces [I]. However, judging from the history of bulk systems, a physical understanding is more likely to come first for simple, freeelectron-like metals, for which a detailed theoretical analysis is possible. Furthermore, the free sp-conduction electrons are present also in transition metals, being responsible for most of the screening as well as a significant part of the chemisorption bond [2]. In addition, within the growing field of energy storage [3], magnesium, or rather some magnesium alloys, turn out to be among the most attractive metals for hydrogen storage. Hydrogen has numerous advantages as an energy carrier [3], and storing H as a metal hydride is both safer and more space-saving than as a gas or liquid [3]. In the storing process, dissociative chemisorption is an important step, as well as the absorption of H in the metal. This thus gives another important impetus for this study. * Supported
by the Swedish
Natural
Science
Research 539
Council.
540
H. Hjelmberg /Hydrogen
chetnisorption
on Al, Mg and Na surfaces
Using an earlier published calculational scheme [4,5], chemisorption properties of the system consisting of a hydrogen atom adsorbed on a semi-infinite jehrum substrate have already been calculated [5-81. The calculational scheme involves solving the Kohn-Sham equations [9] self-consistently with no adjustable parameters. Exchange and correlation are treated in the local density approximation 19,101, discussed in, e.g., ref. ill] together with an estimate of its accuracy. To be able to predict binding energies as well as preferred adsorption sites, the substrate lattice has to be included. This is done in the simplest possible way by using the successful solid state method of replacing the ions with a pseudopotential lattice, and then using perturbation theory. Here, however, only the lowest order perturbation theory is used on the model a results, i.e. the results from the self-consistent jellium calculation. In the following, this procedure will be called mode1 b. The validity of this model of course depends on the strength of the pseudopotentials. One improvement of model b is to use the method of Perdew and Monnier [ 12,131 to include part of the pseudopotential contribution already in the self-consistent (model a-like) calculation. This procedure, called model c, takes account of part of the variation of the substrate electron density perpendicular to the surface in a self-consistent way, while the variation parallel to the surface is included by lowest-order pe~urbation theory (as in model b). An independent investigation along similar lines (using models a and b) is being performed by Lang and Williams [ 14-181. While the present investigation stresses the dependence of the chemisorption properties on the substrate parameters, the study of Lang and Williams seems more to emphasize the dependence on the adatom. In section 2 the calculational method is described. Results are presented in section 3 for H on Al( loo), (110) and (111) surfaces using model b and c. H on Mg(OOOl), where model b is approximately equal to model c, is next discussed, followed by results for H on Na(lOO) and (110). The last part of this section comments on the nume~c~ accuracy and on expected deviations between the present calculations and experimental data. Effects of the pseudopotentials on the adsorbate-induced density of states and dipole moment are also discussed.
2. Theoretical
method and calculational
2.1. The Kahn-Sham
details
scheme
The chemisorption problem is comp~cated from a theoretical point of view, as it has local as well as extended features, low symmetry, and as correlation effects are important. The Kohn-Sham scheme [9] is simple, includes exchange and correlation effects, and has proved to give results with a useful accuracy, also in approximate versions, for atoms [1 13, molecules [ 191, solid [20] and metallic surfaces
H. Hjelmberg /Hydrogen
chemisorption
on Al, Mg and Na surfaces
541
[12,13,21,22]. A short-coming of the scheme is that it applies only to the ground state of the quantum system studied [lo] and to certain excited states [ 111. The scheme gives the density as
PW =
c 1$iW I2, i
where the sum is over occupied orbitals. The $&)‘s are solutions to the Schrodinger like equations (atomic units, with the energy in Rydbergs, are used) c-v2
+ h2ffW>
Gik)
= hi(r)
2
(2)
where d3r’ + u&r) ,
(3)
The external potential is ~,,~(r), and uX,(r) is the exchange-correlation potential, which is here treated in the local density approximation [10,23]. This approximation, which has been used in the applications mentioned above, is discussed in, e.g. ref. [ 111, together with an estimate of its accuracy. The energy parameters ei have no formal relation to oneelectron excitation energies. However, for the systems studied here, which have electron states with an extended nature, the density of states constructed from the ej’s is expected to be similar to the true one-electron excitation spectrum [24]. 2.2. Model a In model a the Kohr-Sham equations are solved self-consistently without adjustable parameters for a system consisting of an adatom on a jellium surface. The jellium, i.e. a system where the positive charge of the ion lattice is homogeneously smeared out to form a positive background, is described by one parameter, the radius r, determined by 4nrz/3 = l/p,, where p0 is the bulk electron density. The surface is formed by terminating the positive charge abruptly. The self-consistent solution to the clean surface, which is the input to the calculation, was taken from ref. [25]. The model is defined by only three parameters, r, giving the bulk electron density, d giving the distance from the adatom to the jellium edge (cf. fig. l), and Z giving the charge of the adatom nucleus. As the clean surface has no structure parallel to the surface, the adatom-jellium system possesses cylindrical symmetry, which reduces the calculational effort. Due to the metallic screening, the adsorbate-induced changes in the electron density are localized to a region near the adsorbate. As described in ref. [4] and in more detail in ref. [5] the calculational scheme uses this fact. The perturbed region is taken to be a sphere. A basis set, {p,}:,, assumed to be complete inside this
542
H. Hjelm berg /Hydrogen
chemisorption
on AI, Mg and Na surfaces
a) yz PLANE metal _
i-th
lattice
plane
-d-[Ii-l/2)
2nd lattfce
plane
-d-3/21
1st lattice
plane
-d-1/2
jellium edge
vacuum Y
-
-d
b) xy PLANE
0
1st lattice
X I-th
plane
lattkce plane
Fig. 1. Definition of the coordinate system by showing (a) the yz plane and (b) the xy plane. The origin is at the adatom, which is a distance d from the jellium edge. The positive charge of the jellium uniformly fills the z < -d half-space. In the yz plane (a), the positions of the lattice planes are shown. The (smallest) interplanar distance is 1. In the xy plane (b), the adatom position is shown relative to the ions in the different lattice planes. The positions of the ions in a lattice plane is given by the primitive lattice vectors 0, and 02, while the vector bi determines the position of the i-th lattice plane relative to the first one. All vectors in (b) are twodimensional.
sphere,
is introduced.
The adsorbateinduced
electron
density,
Ap(r),
as well as the
wave functions, pi (the solutions to eq. (2)), are expanded in this basis set. The equation for the Green function corresponding to eq. (2) is converted to a matrix equation of order NXN. The solution of this equation gives such properties as A&) inside the perturbed region [4,5]. 2.3. Model
b
To remedy an artefact of model a, namely the exaggerated repulsion between the positive background and the adatom nucleus, model b is brought into use. In this model the substrate ions are reintroduced by considering the effects to first order of a lattice of weak pseudopotentials. Such a procedure has been successfully used for calculating the surface energy of free-electron-like metals [21]. The pseudopotential effects are included to first order by adding the term
H. Hjelmberg f Hydrogen 6E~s =s(6
vb(r)
Apa
-
Z6 e(r)
chemisorption 6(r))
on Al,
Mg and Na surfaces
d3r
543
(4)
to the energy found in model a, i.e. Eh=E”+GEbp,.
(5)
(The coordinate system is defined in fig. 1.) The superscripts indicate results from model a and b, respectively. Svb(r) is the change in potential when the positive background is replaced by the pseudopotential lattice: 6 P(r)
= 6 e(r)
+ 6
V+?(r),
(6)
where 6 c is the pure Coulomb part and 6 I$ is the part representing the repulsive effects of the ion cores. E is the change in energy of the adatom upon chemisorption. There are several equilibrium properties of interest. In model b, only the binding energy, Ek= -E’(d&) (d& is the equilibrium distance), is influenced directly by the pseudopotentials. They influence only indirectly the equilibrium values of other properties, as these should be calculated according to model a, but with the equilibrium distance, d&, of model b. For instance, the dipole moment pb is equal to I.ca(&,). In the present paper, the pseudopotential origin is here at the ion nucleus) ups(r) = -2(Zi,&)
of Ashcroft
[26] is used, i.e. (the
O(r - r,) ,
(7)
where Zion is the ionic charge, 6’(x) the step function and rc the core radius. The values of Z.,O,,, r, [26], r, and the Wigner-Seitz radius rws for AI, Mg and Na are found in table 1. Taking away the positive background has the effect of screening the pseudopotentials. To illustrate this, fig. 2 shows the Ashcroft pseudopotentials of Al, Mg and Na, screened by uniform spherical charges with radii equal to rws. It should be noted, however, that the screening is not self-consistent. Fig. 2 indicates, that the Al pseudopotential is the strongest, followed by the ones for Mg and Na. This is also
Table 1 Substrate data; Zion is the ion charge, rs gives the electron density p as l/p = 4nrz/3, Q,S is the Wigner-Seitz radius, rc is the Ashcroft pseudopotential core radius 1261 and Ecoh is the cohesive energy of the metal
Al Mg Na
Zion
rs (au)
rws
3 2 1
2.01 2.65 3.99
2.99 3.34 3.99
(au)
rc (au)
Ecoh
1.12 1.39 1.67
3.34 1.53 1.13
(eV/atom)
544
H. Hjelmberg f Hydrogen
chemisorption
I
on AI, Mg and Na surfaces
I
I
I
LO
Al 30
--... 1 1 4 I
2c
1c
-.-
5 v a F
0
5 a0 -10
-20
-30
-LO ~
0
I I
I
I
e
.,
L
J
1 4
r(a u)
l:ig. 2. The Ashcroft
pseudopotentials (261 for Al, Mg and Na, screened by uniform spherical charges of the corresponding densities and Wigner-Seitz radii (rws).
reflected in the results of 6EE,,whose minimum values are -2.3,-1.5 and -0.5eV for Al, Mg and Na, respectively. The rest of this section will give the main steps of the present calculation of 6 vb(r). Another procedure is found in ref. [ 181. For convenience, the superscripts a and b will be dropped in the following. The coordinate system is found in fig. 1. (1) The electron density of model a has cylindrical symmetry, i.e., the first term in eq. (4) may be written as (8)
jXdXdzjAp(X,z)jd@‘(r). (2) The Ashcroft
pseudopotential
(eq. 7)) is split into Coulomb
and core parts
H. Hjelmberg /Hydrogen
chemisorption
on Al, Mg and Na surfaces
545
(with the origin at the ion nucleus), ups(r) = upS(r) + uy(r)
+ 2(Zi,Jr) e(r, - r) .
= -2(Zi,Jr)
(9)
The removed positive background plus the lattice of upS contribute to the potential 6 VI(r), while the lattice of uy gives 6 V,(r) (eq. (6)). (3) J& 6 V*(r) is calculated analytically (the result for one UT is expressed as an elliptical integral of the first kind). (4) &V,(r) is Fourier expanded parallel to the surface, i.e. 6 Vr(X, z) = 5
8 VI (Q, z> exp(iQ . X> ,
SV,(Q, z) =A-‘Jd’X6V,(X,
(10)
z) exp(-iQ*
X) ,
(11)
where Q = (Q,, Q,,) and A is the area of the surface. To ensure charge conservation, (12)
NxyZion = ZAP, > where NxY is the number tance (cf. fig. 1). (5) The contribution
of ions in an ion plane and I the smallest interplanar
dis-
to 6 V,(Q,z) from the i-th ion plane, 6 V:(Q,z),
8 vf(Q. z) = A-’ Jd2X
exp(-iQ
. X) c
up”@’ -R
R
- Xi,
zi)
,
(13)
where (cf. fig. 1) R=m,
fm2,
n,m=-m
,..., -l,O,l,...,
m,
(14)
x;=x*+!+, zi=z+dtI(i-
(15) l/2).
(16)
As can be seen from eq. (16), zi has its origin at the i-th ion plane. The Q = 0 term, which is the only term where the removed positive background enters, is [21] 6 Vf(Q = 0, z) = -4np,(
lzil - 1/2)2 ,
lzil
(17)
The contribution to 6 v’,(Q # 0, z) (eq. (13)) is non-zero, only for Q = G, where G is a two-dimensional reciprocal lattice vector of the ion planes. The integrations in eq. (13) are performed analytically [27], and with the use of eq. (12). 6 Vf(G, z) = -4np,ZG*
exp(Glzjl)
exp(-iG*
Xi),
G f 0.
(18)
546
H. Hjelmberg /Hydrogen
(6) The summation
chemisorption
on AI, Mg and Na surfaces
over i, i.e. over the ion planes, is now performed,
i.e.
W,(G,z)=~Wf(G,z).
(19)
i=1
For points with zi > 0, i.e. outside the i-th ion plane, the sum is calculated straightforwardly. For zi < 0, the following procedure is used: The vector bi only takes on a few different values, e.g., an fcc(lO0) surface has bzj_t = 0 and bzj = (ur + a2)/2, j= 1, 2, . . . . Thus Xi (eq. (15)) takes on the same number of different values, making it possible to split the sum in eq. (19) into a few subsums, each with a constant value of XL. In such a subsum, the only dependence of i left is in zi (eq. (16)), making the subsums infmite geometrical series (for Zi < 0). As an example, taking all bi = 0, eq. (19) reads (G # 0)
SV,(6, z) =
c
i=l
Wf(G,
4711pe exp( -G lZiO z) - ~ G 1 -exp(-GZ)
I)
exp(-iG
. X,)
,
(20)
where ie is the smallest i for which Zi < 0. (7) The last step in calculating 6 V,(r) is to sum over G (cf. eq. (10)) 6 V,(X, z) = c
6 Vr(G, z) exp(iG . X) .
(21)
G
Note that the p-integration
(cf eq. (8)) may be performed
analytically,
as [27]
2n
s 0
dp exp(iG . X) = 27rJ,(GX)
,
(22)
where Jo is the cylindrical Bessel function of order zero. Thus the cp-integration in eq. (8) is made analytically (cf. point 3). As both the G and -G term enter in the sum (21) with the same coefficient (cf. eqs. (18) and (20)), exp(-iG . Xi) may be replaced by cos(G . X9. The same argument can be used to reduce the number of terms in the sum (2 1). Finally, one has to perform the integration in X and z in eq. (8) to get 6E,,. A disadvantage with this method is that the sum (21) diverges for z in an ion plane (Zi = 0) and has a slow convergence nearby z (cf. eq. (20)). As these points Zi * 0 only make up a small volume in space, however, and as the values of 6 V(r) in this volume may be interpolated from values for /Zi] > 0, the resulting error is small. This divergence is not present in the method described in ref. [ 181. 2.4. Model c The obvious shortcoming of model b is that the substrate may still not be well described, i.e. the pseudopotentials might not be weak enough to make the use of
H. Hjelmberg /Hydrogen
chemisorption
on Al, Mg and Ma surfaces
541
first order perturbation theory sufficiently accurate. An improved description can be obtained by using a method of Perdew and Momrier [ 12,131, where part of the pseudopotential contribution is included in the self-consistent calculation and only the remainder in perturbation theory. The method has a variational parameter c, whose optimum value depends on the surface plane considered. This is done by adding a step potential cB(-z - d) to the external potential of the clean jellium surface (eq. (3)). The Kohr-Sham equations (2) and (3) are solved self-consistently, giving a different electron density of the clean surface, p”(z), for every c. The great advantage is that due to the translational symmetry in two dimensions, the self-consistent calculation is as easy to perform as for the pure jellium surface. The value of c was determined by Perdew and Monnier [12,13] by minimizing the expression for the total surface energy (pseudopotentials included), using p”(z) for the electron density. Thus consistently improved results for the surface energy [ 121 as well as the work function [12,13] of clean metal surfaces have been obtained. The role of the step potential cB(-z - d) is to mimic the “real” potential 6 vb(r) in eq. (4). This is well done, as can be concluded from (i) the successful application to clean surface properties, (ii) the good correlation between the value of c and the average of 6 vb(r), (iii) the good correlation between the first “peak” in p”(z) and the first dip in 6 p(z) (the average parallel to the surface) and (iv) the successful application to the vacancy formation problem in the bulk [28]. The values of c [ 12,13,25] for different surfaces are given in table 2 together with the corresponding locations of the bottom of the band. Table 2, indicates that
Table 2 Clean surface parameters of model c (section 2.4); the optimum values of c [ 13,251, the size of the step in the potential &-z-d) introduced by Perdew and Monnier [ 12,131, are given together with the corresponding values for the energy of the bottom of the band, uztt (--) (zero energy at the vaccum level); the value of & (--) for c = 0 gives the result of model b (section 2.3) Metal Al (fee)
Face
c (eV)
u%f (--)
_
0
-15.59 -15.48 -15.40 -15.92
(100) (110) (111)
Mg (hw)
Na (bee)
1.05 3.14 -1.87
_
0
(0001)
0.4
_
0 1.07 0.28 1.45
(100) (110) (111)
-10.83 -10.8 -6.25 -5.91 -6.16 -5.93
(ev)
548
II. Hjelmberg /Hydrogen
chcmisorption
on Al, Mg and Ma surfaces
in the self-consistency procedure, the long-range effects of the potential step are reduced, as the energy of the bottom of the band is shifted with less than 0.4 eV or less than 30% of c. The application of this method to the chemisorption problem is straight-forward, as the actual calculations are those of model b. The difference lies in the external potential in the self-consistent part of the calculation, as the step potential cB(-z - d) (values of c from ref. [25]) is included. This part gives a c-dependent adsorbate-induced electron density, Ape(r), i.e. one that depends on the surface plane. The rest of the effect of the lattice pseudopotentials, s
[S V”(r) A#@) - S V,‘(r) Z&(r)] d3r ,
6 V”(r) = 6 V”(r) -- cq-z
- d) )
(2% (24)
is then added to first order to get the energy he [29]. Consequently, model c is reduced to model b if the parameter c is equal to zero. Thus an improved description of the variation of the electron density normal to the surface has been obtained in model c. Still lacking, however, is the variation of the electron density parallel to the surface.
3. Results Inclusion of the substrate structure by reintroducing the pseudopotential lattice to the jellium (model a) results, makes it possible to predict stable positions, binding energies, vibrational frequencies and activation energies for diffusion along the surface as well as into the substrate (absorption). In this section detailed results for H of Al(lOO), (llO), (111) (section 3.1), Mg(OOO1) (section 3.2), Na(lOO) and (110) (section 3.3) will be presented. For Al, calculations using both model b and c have been performed. In section 3.1, only the results of model c will be discussed, while a comparison of the models is made in section 3.4. As model b and c turn out to be almost the same for Mg(OOO1) [13] (cf. table 2), only the former has been used. Finally, for Na, only results of model b witl be presented. The input data needed from model a calculations have been all taken from ref. [5], whose input in its turn (clean surface data) was taken form ref.
P-51. Selected and prelirnina~ results for H on Al and Na have already been published [S-8]. An independent investigation carried out by Lang and Williams has given selected results for H, Li, 0, Na, Si and Cl on a jellium surface (I, = 2.00) [14-181, Si (for d = 1.5 au) on a jellium surface (rs = 4.00) [18] and for 0, Na, Si and Cl on jellium (r, = 2.00) with perturbing Al pseudopotentials corresponding to the (111) surface [ 16-181 (equivalent to model b).
H. Hjehnberg /Hydrogen
chemisorption
on Al, M, and Na surfaces
549
-2 -
I
-3
I
-2
1
I
-1
0
I
1
I
2
d(au1 Fig. 3. Calculated energy curves for H chemisorbed on three different jellium substrates, rs = 2.07 au (Al), r, = 2.65 au (Mg) and rs = 3.99 au (Na) (51. The distance between the H atom and the jellium edge is denoted by d.
Section 3.4 is devoted to a discussion of models b and c, including their accuracy, while section 3.5 deals with an estimate of the effects on the adsorbateinduced density of states and dipole moment from the pseudopotentials. In the discussion of the results below, it is useful to have in mind the two main factors that determine the bond length: (i) The repulsion due to the increase in kinetic energy, when the adatom enters the conduction electron density of the substrate. This “kinetic repulsion”, discussed in detail in ref. [S], is very r,-dependent. This is illustrated in fig. 3 [5], showing the results for H on three jellium surfaces (model a). The importance of this factor is also illustrated by compa~g results for two different substrate densities. For instance, using r, = 2.00 instead of 2.07 in model b, for H on Al(11 I), would change the energy with about OS eV for d = -1 au and change the equilibrium distances for bridge and center positions with about 0.3 au [30]. (ii) The attraction, when the adsorbate-induced electron density enters the region just outside an ion core, and the strong repulsion, when it penetrates an ion core of the substrate (cf. fig. 2). This penetration would occur for an adsorbate-substrate atom distance of about r, + +t [6], where r, is the core radius (cf. section 2.3) and rH (02 au) the radius of the sphere around the proton that contains one electron. It has been noticed, that the distance r, t TH agrees with the bond length for the molecules AlH, MgH, NaH, KH, RbH and CsH within 5% [6]. if one of the interactions (i) or (ii) above dominates over the other (for relevant distances), two different situations may be distinguished. If the “kinetic repulsion” (i) is dominating the force, all different sites should have almost the same vibrational frequency (wc,,~), as this corresponds to the jellium substrate. If, on the
550
H. Hjelmberg /Hydrogen
on Al, Mg and Na surfaces
I
I
I o-
chemisorption
I
H/AL (110)
b Ai -l -
4
-2 -
-
1.35.a I -1
” ---I
0
I I
I 2
other hand, the force between the adsorbate and the nearest substrate atom(s) (ii) is dominating, the vibrational frequencies should be strongly dependent on the site, following in general the sequence c&r,r > uabr > asbr (A, B and C indicate the site, cf. figs. 4-6). The sequence is determined by the effect that, compared with the A site, the substrate atom-adatom distance varies much more slowly with d (fig. 1) for the B and especially the C site. This effect is in general stronger than the
H. Hjelmberg /Hydrogen
chemisorption
on AI, Mg and Na surfaces
551
Fig. 4. Calculated energy curves for H chemisorbed in several configurations on the (a) AhlOO), (b) (110) and (c) (111) surfaces. Results for both model b (dashed lines) and model c (full lines) are shown for (a) (100) and (b) (110). The distance between the H atom and the jellium edge is denoted by d. The distance from the jellium edge to the first layer of Al ions is given in the lower left corner for each surface. In the upper right corner is a figure of the unit cell of that surface, defining the different configurations. Or
I
I
I I
_I
I
I
I
H/Mg (0001)
I
-1
-I
”
I
dfoul Fig. 5. Calculated energy curves for H chemisorbed face. As results of model b and c should be almost The distance between the H atom and the jellium layer of Mg ions is marked by a vertical dashed defining the different configurations (C standing the configurations are shown with respect to the indicated.
in several configurations on a Mg(0001) surequal, only the results of model b are shown. edge is denoted by d. The position of the first line. Near to this line, the unit cell is shown, for both CA and CC). In the upper left corner, second layer of Mg ions, which position is also
552
H. Hjelmberg /Hydrogen
chemisorption
on Al, Mg and Na surfaces
I
I
I
Nqh-,q
H/Na(lOO)
","u";
”
1
I
o-c-i!l I
a
J
C
-2 ---_
------2.02 au
L
B
.H
/J
-4
.N’
-1 NR
I -1
I 0
I
1
dtau)
\;
H/Na(llO)
0
Noc)-L?-$3 7.01 ,
;,I.$
\
b
A
\
“,$o
\
4
\
“\
“\
I
I -2
I -I
I 0
I I
die u)
Fig. 6. Calculated energy curves for H chemisorbed in several configurations on the (a) Na(100) and (b) Na(l10) surfaces. Only results of model b are shown. The distance between the H atom and the jellium edge is denoted by d. The distance from the jellium edge to the first layer of Na ions is given in the lower left corner for each surface. In the upper right corner is a figure of the unit cell of that surface defining the different configurations.
contributions to the force from the extra substrate atoms present in the B and C configurations. Both models (b and c) are expected to describe the real situation best for the most close-packed surfaces, i.e. Al(lll), Mg(OOO1) and Na(1 lo), as discussed in section 3.4.
H. Hjelmbetg / Hydrogen
ckemisorption
on Al, Mg and Na surfaces
5.53
Table 3 Data for the most symmetric configurations, which are defined in figs. 4-6: Eg is the binding energy [ 3 11, do the distance to the first ion plane, dM_H the bond length and Wvibr the vibrational frequency; values for two CA configurations are given for Mg, CA’ outside and CAi inside the first ion plant; for CA’, dl and dM__H are the distance to the nearest substrate ion in the fist and second layer, respectively; all molecular data are from ref. 132) (EB here without vibrational contributions, cf. ref. [ 3 11) H on Al (molecule) Al (100)
Al(110)
Al(111)
Mg (molecule) Mg(0001)
Na (molecule) Na (100) Na (110)
Position
EB (ev)
dl (au) -
dM -_H (au)
Wvibr (meV)
A B c A SB LB C A B CA cc
3.01 1.9 2.3 1.4 2.3 2.4 1.7 (1.4) 1.9 1.9 1.7 1.8
3.11 3.0 2.0 2.4 3.2 2.2 2.3 2.9 2.0 2.1 1.9
3.11 3.0 3.4 4.5 3.2 3.5 4.5 2.9 3.4 3.8 3.6
209 210 130 70 180 170 110
~2.58 2.2 2.6 2.5 2.7 2.6
3.21 3.3 2.1 2.1 1.9 3.8
3.21 3.3 3.7 4.1 4.0 3.5
186 200 150 110 80 140
2.1 2.4 2.1 1.9 2.4 2.3 2.3
3.57 _ _ 3.5
3.51 3.5 -
14s
A B CAS CC. CA’
B C A SB LB C
300 150 120 120
120 -
Equilibrium properties, like the binding energy Eu, the equilibrium distance to the nearest ion plane dL (= d,, t l/2), the bond length dAr__t, and the vibrational frequencies uvibr, are listed in table 3 for the most symmetric positions. Fig. 4 shows the change in energy upon chemisorption, E, as a function of d. The calculations using model c predict (table 3) that hydrogen is stable in the bridge (B) configuration on the Al(100) and (110) surfaces and atop (A) or bridge on the Al(111) surface, and that the binding energies [31] are 2.3 eV (loo), 2.4 eV
554
H. Hjelmberg /Hydrogen
chemisorption
on AI, Mg and Na surfaces
(110) and 1.9 eV (Ill). The corresponding distances dL are 2.0 au (loo), 2.2 au (110) and 2.9 (A) and 2.0 au (B) (111). Furthermore, the activation energy for diffusion along the surface is estimated to be 0.1-0.2 eV [6], while the activation energy for absorption is high, 1.2-1.4 eV. As the dissociation energy of the Hz molecule [31], calculated with the same Kohr-Sham formalism, is 2.4 eV per atom [33], the model c calculation predicts that the binding energy could be just enough to make a break of the H2 bond thermodynamica~y favourable [34]. However, such a small energy difference is not considered significant, as it is well within the estimated error of the calculations (cf. section 3.4). Furthermore, both H atoms would have to take advantage of the high binding energy in a preferred site, which is geometrically impossible. Together with the expected decrease in entropy [ 1f , dissociative chemiso~tion of H2 could only, if at all, take place at low temperatures. This prediction is in accord with experiments [35]. Absorption of H in Al is improbable, due to the large activation energy (1.2-1.4 eV). This is mainly due to the “kinetic repulsion” (cf. fig. 3). A hydrogen atom put inside a perfect crystal of Al would rather segregate to the surface. The calculated bond lengths (dAl_H) are in good agreement with the corresponding molecular bond lengths, with the exception of some C positions (table 3). A bond length considerably larger than the molecular one, should correspond to a smaller binding energy (cf. the “re t rH” argument in the introduction to section 3 as well as fig. 2).That is the likely reason, why the C positions on the (100) and (110) surfaces and the long bridge (LB) on the (110) are not favoured, as the smallest bond lengths allowed by geometry (= X, in fig. 1) are considerably longer than the molecular one. For the C position on the (111) surface, on the other hand, the distances are equal. The vibrations frequencies follow the sequence w$& > o$br > o$br (table 3), indicating that the adatom-nearest substrate atom force is stronger than the force due to the “kinetic repulsion” (cf. the discussion in the introduction to section 3). Relaxation of the lattice around the adatom is neglected in the calculations. Using the large cohesive energy of Al (cf. table 1) as an indication suggests that this may be a small effect, at least on the more close-packed planes. 3.2. H on Mg(OOU1) The (0001) surface is the most close-packed one of the different Mg surfaces. It turns out that models b and c are almost the same in this case, as the value of c (the “step height”) is very small (table 2). Equilibrium properties are presented in table 3 and the energy curves E($l in fig. 5. The model b calculations predict that the stable position for H chemisorbed on Mg(0001) is the center configuration CC (over a hole also in the second ion plane) with a binding energy [31] of 2.7 eV and with d,= 1.9 au. The activation energy for diffusion along the surface is small, 0.1-0.2 eV (as for Al), and so is the activa-
r%Hjelmberg 1 Hydrogen ckemisorption on Al, Mg and Na surfaces
555
tion energy for diffusion inwards the metal, about 0.4 eV. As the activation energy for this diffusion, and thus for absorption, is small, results are give in fig. 5 also for the hydrogen atom placed inside the first ionic plane. The stable site inside (position CA’ in table 3) is predicted to be near the tetrahedral position, with a binding energy [3 l] of 2.6 eV. The bond lengths to the four nearest Mg atoms are 3.5 and 3.8 au (table 3). In contrast to the result for H on Al, the binding energy on Mg (2.7 eV) is sigmficantly larger than the energy needed to break the H2 bond (cf. ref. [35]). As on Al, the bond lengths for H chemisorbed on Mg are similar to the molecular one (table 3). The vibrational frequencies follow the sequence w&,~ > O& > W& for H chemisorbed on Mg (table 3), as was the case on Al. Some data have been collected for the Mg-H system, due to its importance in the context of storing energy. Magnesium forms a hydride, MgH2, with an energy of formation of 0.77 eV per molecule of Hz [36] or a binding energy [31] of 2.8 eV per atomic H. The structure changes from hcp to tetragonal (similar to rutile, TiO?) [36]. The reaction rate for the formation of the hydride is unfo~unately very low, several months even for small particles, making it unsuitable for direct technical applications [36]. This low reaction rate is probably due to oxidation of the Mg surface. The calculated binding energy for H in Mg metal, 2.6 eV, lies close to the energy of formation of MgHz. Furthermore, inclusion of relaxation of the Mg lattice, which is not present in the c~culation, would lower this binding energy. Some relaxation may be expected, if the small cohesive energy of Mg (table 1) is a good indicator. The bond lengths between the H and the nearest three Mg atoms in MgHs, are 3.6 and 3.7 au [37], i.e. similar to the ones found in the present calculations for the CA’ position inside the metal (table 3).
Only model b has been used in these calculations, because using model c should not significantly change the picture given here, as the Na pseudopotentials are weak (cf. fig. 2). Results for the two most close-packed surfaces, (110) and (loo), are shown in table 3 and fig. 6, and results for the (111) surface are qu~itatively similar. The calculations predict that H is absorbed in Na, with a binding energy [3 1] of 2.4 eV. This energy is just large enough, that the Hz bond might be broken (cf., however, the discussion in section 3.1). The bond length for the atop position on the (110) surface agrees well with the corresponding molecular bond length (table 3). The lack of structure in the energy curves in fig. 6, suggests that the activation energy for diffusion inside the metal is small compared to, e.g., Mg (fig. 5). This is
556
H. Hjelmberg /Hydrogen
chemisorption
on Al, Mg and Na surfaces
in agreement with experimental data for diffusion of several other impurity atoms in Na and Mg, as the activation energy is typically S-10 times larger in Mg [38]. A hydride, NaH, exists, with an energy of formation of 1.2 eV per hydrogen molecule 1361 or [31] 3.0 eV per hydrogen atom. Such an increase in binding energy, as compared to the value given by the present calculation (2.4 eV), could be partly accounted for by the presence of relaxation of the lattice, or, at higher H concentrations, by the structural change from bee (Na) to fee (NaH) [36]. Such effects are not considered in the present calculations. The small cohesive energy of Na (table 1) suggests, that there may be relaxation around a hydrogen atom also at very low H concentrations. The hydride is something quite different from the system studied here. It is interesting to note, however, that this model calculation gives a similar energy for a situation, which might be the embryo of the hydride formation. 3.4. Discussion of models b and c In this section a discussion of the accuracy and physics of models b and c will be made. In comparing the results of models b and c with experimental data, there are essentially only two approximations that should be kept in mind: Firstly, exchange and correlation are described in the local spin-density approximation (LSDA) [ 111. (The spin is important when calculating, e.g., the free H atom properties [39], cf. ref. [ll]). The LSDA is estimated to introduce an error of 0.1-0.5 eV in the binding energy for hydrogen on simple metals [6,11], and a smaller error in energy differences like activation energies. The second and most serious approximation is the model for the substrate. There is no a priori reason why first-order perturbation theory in the pseudopoten&al should give a sufficient accuracy in the present application. The usefulness of models b and c should at this stage rather be judged from the soundness of their results (sections 3.1-3). The reasonable results for the bond lenghts (table 3), with the exception of maybe the most open positions, support the use of these models. For other properties, e.g. the binding energy, a good estimate of the error is hard to get due to the lack of experimental data. A comparison of the binding energy (En) with the formation energy of the corresponding hydride (for Al, it does not exist; MgH, and NaH) [36], suggests that the total error in EB [3 l] is less than 0.6 eV, which should then be a conservative estimate (cf. e.g. the discussion in section 3.3). A further illustration of the error is given by comparing the results of model b with model c, as part of the pseudopotential correction is treated self-consistently in the latter. The energy curves differ with less than about 0.1 eV for Al(lOO) and (111) and 0.4 eV for Al(110) (cf. figs. 4a and 4b). Model c has been introduced to remedy the following artefact of model b: On to section 3) is the close-packed surfaces the “kinetic repulsion” (the introduction exaggerated and therefore probably the equilibrium distances overestimated and the
H. Hjehberg
/Hydrogen chemisorption on Al, Mg and I% surfaces
557
binding energies underestimated, while the opposite should be true on the open surfaces. This can easily be envisioned by noticing that the interplanar distance 2 (fig. 1) characterizes the thickness of the electron density fall-off region (at about a distance Z/2 outside the first ion plane, i.e. at the jellium edge, the density is one half of the bulk value). For a close-packed surface, 1 is large, and therefore the “kinetic repulsion” starts to grow too far outside. This effect is most clearly seen for the open surface of Al(1 IO), which is described by a large positive value of the step c (table 2). Compared with model b, model c lowers the energies ford 2 0.8 au and raises them for smaller values of d, thereby giving larger d,, (fig. 4b). In the interior, i.e. for d very large and negative, models b and c always given the same result. A similar, but smaller change is seen for Al(100) (fig. 4a), whose value of c is also positive but smaller (table 2). For the most close-packed surface, Al( 111) (fig. 4c, results of model b not shown), the change is only slightly larger, but with a different sign compared with Al(100). The value of c is accordingly negative (table 2). The deficiency of models a and b still present in model c is the constancy of the electron density of the clean surface parallel to the surface. Due to this, the presence of the “kinetic repulsion” introduces the fo~owing effects. For positions, where the electron density should be smaller than average, model c predicts too large equilibrium distances and probably too small binding energies. The B and, especially, the C sites belong to these positions. The electron density variation parallel to the surface is smallest for the most close-packed planes. Therefore these are the ones, for which the present calculations are expected to give the best results. For the open surfaces, like Al(110) (fig. 4b), on the other hand, results presented here are considered least reliable. This deficiency is significant only for substrates with high densities (small rS), as the “kinetic repulsion” is not very important for rS > 3-4 (cf. tig. 3). The picture of the chemisorption bond is still that of an adatom on jellium, also in models b and c. The adatom electrons have no possibility of forming directed chemical bonds with the nearest substrate atoms, as the ion lattice is not present when the electron density is calculated. 3.5. Density of states and dipole moment In model b effects of the pseudopotentials on the total energy are considered. For, e.g. the adsorbate-educed density of states A&E) and dipole moment P, no direct effects are calculated. The indirect effects are easily included by using the model a values for AP(E) and p, which correspond to equilibrium distances, as calculated in model b (cf. section 2.3). The accuracy of this procedure may be estimated by comparing these results with those of model c. The model a like calculation, which is the first step in model c, includes an average of the pseudopotenti~s self-consistenly and gives A&e) and lz for a number of representative values of the step c (cf. section 2.4). For H on Al with -1
558
H. Hjelmbetg [Hydrogen
chemisorprion
on Al, Mg and Na surfaces
it is found that (i) the energy of the resonance peak in AP(E) is shifted with less than about 1 eV, (ii) the position of this peak in A&) follows the clean surface effective potential (eq. (3)) as in model a [8,.5], and that (iii) the dipole moment is changed with less than about 0.1 au. Delocalized electron states should feel the average 6 V(r) of the pseudopotential correction 6 V(r) rather than its details. As the size of potential step is well correlated with 6 Y(r) [ 12,131, an estimate of the kind described above should be reasonable for adsorbate-induced resonances in the band, e.g., H on free-electron-like metals [5]. For the more localized, deeply bound states, however, the details of 6 V(r) have to be considered [40]. 4. Summary Chem~sorption properties for H on Al, Mg and Na have been calculated with three increasingly more realistic models. The first one, model a, involves using an earlier published calculational scheme [4,5] for solving the Kohn-Sham equations [9] self~onsistently and with no adjustable parameters. The substrate is described by a semi-infinite jellium. Results of model a have been published elsewhere [5-81. The second model (b) adds effects of the substrate pseudopotential lattice as a first order perturbation to the result of modei a. Finally, the third model (c) is an improvement of model b, as part of the pseudopotential contribution is included in the self-consistent calculation [ 12,131. Still lacking, however, is the variation of the clean surface density parallel to the surface. The main predictions of the present calculations are the following: (1) On the low-indexed surface of Al is H chemisorbed in the bridge position, or on the (Ill) surface, possibly in the atop position. The binding energy [3 l] is in the range 1.9-2.4 eV. The calculated values for the binding energy is too small or he too close to the H2 dissociation energy, to make dissociative chemisorption of Hz on these surfaces a likely event. While the activation energy for diffusion parallel to the surface is small (0.1-0.2 eV), it is Iarge for diffusion into the substrate (IL?1.4 eV), making the absorption of H improbable. A H atom placed in a perfect crystal of Al would rather segregate to the surface. (2) On the Mg(OOO1) surface, H is chemisorbed in the center position (with no substrate atom below also in the second layer) with a binding energy /31] of 2.7 eV, large enough to allow dissociation of Hz. The activation energy for diffusion into the bulk is not very large (0.4 eV), making absorption possible for reasonable temperatures. In the bulk the hydrogen atom should sit approximately in the tetrahedral position with a binding energy [31] of 2.6 eV. (3) H should be absorbed by Na at all temperatures, as there is hardly any activation energy for this process, and that it at the same time gains an energy [31] of 2.4 eV. (4) The vibrational frequencies Wtibr for H on Al and Mg decrease in the sequence ~$r,r > w& > w&, where A, B and C denotes the atop, bridge and center positions, respectively.
H. H_jelmberg /Hydrogen
chemisorption
on Al, Mg and Na surfaces
559
(5) The adsorbate-induced density of states and dipole moment of H on free-electron-like metals should not be effected very much by the direct pseudopotential perturbation, i.e., given a good equilibrium distance, these properties may be taken from a model a calculation. For adsorbates with more localized adsorbate-induced states, however, the local field corrections might given strong effects. Unfortunately, no detailed experimental data exist yet for the chem~orption situations considered here (cf. ref. [35]). Comparing the calculated results with other relevant experimental data, one finds that: (1) Bond lengths agree well with the corresponding molecular ones, especially for the atop positions. (2) The total error in the calculated binding energies 1311 is, in a conservative estimate, less than 0.6 eV.
Acknowledgements I am especially grateful to B.I. Lundqvist for his encouragement and for the great interest he has taken in this work. Im also indebted to 0. Gunnarson for many helpful discussions. It is a pleasure to thank S. Andersson, B. Kasemo, J.K. N$rskov, D.M. Newns, J.P. Muscat and all the members of the Solid State Theory Group at the Institute of Theoretical Physics in Goteborg for helpful discussions. The hospitality of the Institute of Physics, Aarhus, where part of this work has been done, is gratefully acknowledged.
References [l] J.P. Muscat and D.M. Newns, Chemisorption on Metals, Prog. Surface Sci. 9 (1978) 1. [2] G. Blyholder, J. Chem. Phys. 62 (1975) 3193; I.P. Batra and 0. Robaux, Surface Sci. 49 (1975) 653; D.J.M. Fassaert and A. van der Avoird, Surface Sci. 55 (1976) 291, 313; D.E. Ellis, H. Adachi and F.W. Averill, Surface Sci. 58 (1976) 497; CF. Melius, J.W. Moskowitz, A.P. Mortola, M.B. Baille and M.A; Ratner, Surface Sci. 59 (1976) 279; K. Sch~nhammer, Inter. J. Quantum Chem. Sll (1977) 517. [3] See, e.g., Proc. Intern. Symp. on Metal Hydrides for Energy Storage, 1977, Geiio, Norway, Intern. J. Hydrogen Energy, to be published. [4] 0. Gunnarsson and H. Hjelmberg, Physica Scripta 11 (1975) 97. [S] H. Hjelmberg, Phys. Scripta 18 (1978). [6] 0. Gunnarsson, H. Hjelmberg and B.I. Lundqvist, Phys. Rev. Letters 37 (1976) 292. [7] 0. Gunnarsson, H. Hjelmberg and B.I. Lundqvist, Surface Sci. 63 (1977) 348. [8] H. Hjelmberg, 0. Gunnarsson and B.f. Lundqvist, Surface Sci. 68 (1977) 158. [9] W.Kohnand L.J.Sham,Phys.Rev. 140(1965)A1133. [ 101 P. Hohenberg and W. Kohn, Phys. Rev. 136 (1964) B864. [ 1 l] 0. Gunnarsson and B.I. Lundqvist, Phys. Rev. B13 (1976) 4274.
560 [ 121 [ 131 [ 141 [ 1.51 [ 161
[ 17) [18] [ 191
[20]
[21]
[ 221 1231 [24] [25] [26] [27] 1281 [29]
[30] [31]
1321 133) [34]
[3.5]
H. Hjelmberg /Hydrogen
chemisorption
on Al, Mg and Na surfaces
J.P. Perdew and R. Monnier, Phys. Rev. Letters 37 (1976) 1286. R. Monnier and J.P. Perdew, Phys. Rev. Bl’? (1978) 2.595. N.D. Lang and A.R. W~Bams, Phys. Rev. Letters 34 (1975) 531. N.D. Lang and A.R. Williams, Phys. Rev. Letters 37 (1976) 212. K.Y. Yu, J.N. Miller, P. Chye, W.E. Spicer, N.D. Lang and A.R. Williams, Phys. Rev. B14 (1976) 1446. N.D. Lang and A.R. Williams,Phys. Rev. B16 (1977) 2408. N.D. Lang and A.R. Williams, Phys. Rev. B18 (1978) 616. 0. Gunnarsson, P. Johansson, S. Lundqvist and B.I. Lundqvist, Intern. J. Quantum Chem. 9s (1975) 83; 0. Gunnarsson and P. Johansson, Intern. J. Quantum Chem. 10 (1976) 307; 0. Gunnarsson, J. Harris and R.O. Jones, J. Chem. Phys. 67 (1977) 3970. 0. Gunnarsson, B.1. Lundqvist and J.W. Wilkins, Phys. Rev. BlO (1974) 1319; J.F. Janak, V.L. Moruzzi and A.R. Williams, Phys. Rev. B12 (1975) 1257; J.F. Janak and A.R. Williams, Phys. Rev. B14 (1976) 4199; V.L. Moruzzi, A.R. Williamsand J.F. Janak, Phys. Rev. B15 (1977) 2854. N.D. Lang and W. Kohn, Phys. Rev. Bl (1970) 4.555; 83 (1971) 1215. N.D. Lang, Solid State Phys. 28 (1973) 225. The explicit form of uxc used in the present calculations is taken from ref. [ 111. See, e.g., the discussion in ref. [S]. R. Monnier, private communication. N.W. Ashcroft, Phys. Letters 23 (1966) 48. I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, New York, 1965). R.M. Nieminen, J. Nucl. Mater. 69.-70 (1978) 633. In the expression for the energy, the function c6(-z - d) enter in both the self-consistent and the pseudopotential part, but with different signs. Thus there will be no discontinuities in the energy k”(d) at d = 0. The value of rs = 2.00 is used in refs. [6] and [7] and in the model b results of ref. [S]. This vaiue is also used by Lang and Williams [14-IS]. No correction due to the zero point vibrational motion has been made. Its effect is to lower the binding energies. For Ha, it is 0.136 eV per H atom f 321 and for chemisorbed H, the values in table 3 divided by 2 should be used. Taking this effect into account would change the energy differences upon dissociative chemisorption of Ha with less than about 0.1 eV per H atom. American Institute of Physics Handbook, Ed. D.E. Gray, 3rd ed. (McGraw-Hill, New York, 1972). From 0. Gunnarsson and P. Johansson, ref. [19]. Refs. [6] and ]7] predicted a binding energy of 2 eV, making dissociative chemisorption of Ha impossible. The present results are, however, regarded as more accurate, due to improvements in the computations as well as the use of the proper density (rs = 2.07) for Al. No indication of adsorption of atomic or molecular 11 has been observed at room temperature on Al and Mg in ultra violet photoemission experiments by S.A. Flodstrom, L.-G. Petersson and S.B.M. Hagstrom, J. Vacuum Sci. Technol. 13 (1976) 280. However, this does not necessarily prove the nonexistence of H chemisorption on these metals, as the structure induced by chemisorbed H in the UPS spectrum could be very weak. Furthermore, an activation energy for desorption of at least about 1 eV is needed, for a chemisorbed layer to stay on the surface during such an experiment, performed at room temperature and very low hydrogen pressure (a prefactor of IOr set-r is used). This corresponds to a binding energy (EB) for chemisorbed H of at least about 2.8 eV [ 311. In this estimate, 2EB -. &.$a, corrected for vibrational contributions (cf. ref. 13111, has been
H. Hjelmberg /Hydrogen
chemisorption
on Al, Mg and Na surfaces
561
used as the activation energy for desorption (DHZ is the dissociation energy of Hz). [36] See, e.g., A.J. Maeland, in: Proc. Intern. Symp. on Metal Hydrides for Energy Storage, 1977, Geilo, Norway, Intern. J. Hydrogen Energy, to be published. [37] Crystal Data, Determinative Tables, Eds. J.D.H. Donnay, G. Donnay, E.G. Cox, 0. Kennard and M.V. King, 2nd ed. (American Crystallographic Association, 1963). [38] Handbook of Chemistry and Physics, Ed. R.C. Weast, 54th ed. (CRC Press, 1973-74) p. F-57. [ 391 Values for the free H atom are taken from J.K. N$rskov, private communication. He has used the same calculational scheme [4,5] as is used in the present paper. [40] This is also discussed in ref. [ 181.