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Comments on chemisorption at a metal surface
Solid State Communications, Vol. 20, pp. 1073—1075, 1976.
Pergamon Press.
Printed in Great Britain
COMMENTS ON CHEMISORPTION AT A METAL SURFACE L.M. Kahn* and M. Rasolttl Battelle Columbus Laboratories, Columbus, OH 43201, U.S.A. (Received 9 August 1976 by L. Hedin)
The effect of the gradient correction to the exchange and correlation energies on chemisorption properties is examined using the evaluation of ~ arrived at by Rasolt and Geldart. Quantities common to the study of hydrogen and alkali chemisorption on a high density metal are calculated for various forms of the energy functional within the linear response approach. It is found that the gradient correction is a quantitatively significant term, lowering binding energies by 10% and increasing the dipole sizably. THE GROUND STATE ENERGY of a non-uniform interacting electron gas of density n(r) can be written as E[n(r)] = f dr v(r)n(r) + VH[n(r)1 + 1’8 [n(r)] + ~ [n(r)]. Where VH is the electrostatic, T~the kinetic, and ~ the exchange and correlation contributions to the energy. Here v(r) is an external potential.’ Practically, this expression involves two major approximations, ~‘8 and ~ are represented via an explicit functional of n(r). Through this treatment of the energy, the electron density, and thus the ground state properties, of a systern can be determined directly (see below). It is, however, possible to treat the kinetic energy exactly. In such a procedure, the density is given directly through a set of self-consistent Euler—Lagrange equations.2 Although the latter procedure can involve a very complex numerical solution (as the symmetry of v(r) is lowered),3 it is evident that the fundamental problem resides in knowing
(e.g. binding energy, and induced dipole moment) are becoming available, the present numerical treatment still neglects many important contributions (e.g. the ions of the substrate),3 making quantitative comparison with experiment unreliable. Taking the view that these numerical procedures can be perfected, the importance of the proper treatment of the exchange and correlation contribution to these various properties is of increasing interest. To examine the magnitude of these effects, it is adequate to take the simpler approach of Ying et al.8 rather than solve the full self-consistent Euler—Lagrange equations. Briefly, the solution involves writing the potential v(r) as v(r) = v+(r) + v~(r)where v÷(r) corresponds to the potential of a step uniform positive background [p+(r) = n+O(— x), where n~is the bulk density] and v~(r r 0) is the potential of the adion the structure of E~~[n(r)].In this note, we wish to in- 4’5 centered at r0 with a corresponding density ~p~(r ro). The induced density n vestigate the effect of~recent refinement of E~0[n(r)], 1(r), due to v~(r),is then deteri.e., the second term in ~ dr [A~~[n(r)]+ mined 2], on several fproperties of an adion at densityby n treating it as a linear order correction to the B~~[n(r)]Vn(r)1 0(r), induced by v÷(r) as given in reference 9. a metal surface. Since this non-local term was found to With n 1(r) known, the various adion 47 (as properties are then easily evaluated. surface interaction make a major contribution to the surface energy large as 40% of the total surface energy), it raises the To calculate n 1 (r), it is obviously necessary to have question of its importance in the properties of chemian explicit form for 7’~[n] and ~ [n]. For 7’~ [n], wex 2/lOm) sorption. use the common approximation T8 [n] (3h We should point out that while more refined experi- (3ir2)2~’3f dr n5”3(r) + (h2/72m) f dr IVn(r)12/n(r). The mental data on various properties of chemisorption exchange contribution to A~~[n]is given by A~[n(r)] = e2 ~(3/ir)1”3f dr n~3(r),and the correlation by * Sponsored by Battelle Institute Program Grant A~[n(r)1 0.056 e2 f dr n~3(r)/[(0.079 + n Number B-2336-1100. For the form of ~ [n(r)], we use the result of reference 4. Knowing an explicit form for E[n(r)], our remaining Sponsored by Battelle Institute Program Grant task is to evaluate the effect of the additional term Number B-1331-1 190. B~~[n(r)]on the induced density n 1(r) and thus the ~ Also at The Ohio State University, Columbus, OH corresponding chemisorption properties. 43210, U.S.A. —
—
—
1073
1074 0.10
COMMENTS ON CHEMISORPTION AT A METAL SURFACE
Table 1. The ionic binding energies and dipole moments are given for hydrogen and the alkalis for the local expression for the exchange and the gradient correction described in the text
—
0.09
..‘
0.08
-
0.06
-
.
Ij~ energy (eV) local gradient
(a.u.) local
H Li Na
9.0 8.76 Atomic binding 4.50 4.16 2.85 2.60
0.01 moment 0.023 Dipole 0.39 0.71 1.00 1.52
K
2.08
1.94
1.65
2.25
Cs
1.29
1.24
3.15
3.90
E
.~ 0.04 ~ 007
-
,‘// \ :1
/
~,
gradient
/
/ /
0.030.02
Vol. 20, No. 11
/ -
0.01 -I
I
I I
0
2
Rb
1.70 1.58 2.49 3.20 The solution for the induced density, n1(r), 2v and the corresponding potential v1(r), (defined by V 1(r) = 4ire[n1(r) + ~ r0)] due to an arbitrary charge distribution is most easily calculated by a superposition 8 due to a of response point chargefunctions, positionedR(r, at r’. r’)Then and L(r, r’),
I
3
4
Distance (a u)
Fig. 1. The induced electron density due to a point charge of unit magnitude, H~,is illustrated for several forms of the energy functional. The density is given along the surface the perpendicular for the pointjoining charge the 1.1 point a.u. from charge the and surface. The ordinate’s scaled units are such that one scaled unit equals the bulk density ( local exchange +
—
n 1(r)
gradient correction, local exchange + local correlation + gradient correction, no exchange or correlation, local exchange + local correlation.)
=
Jdr’R(r, r’)~z~d(r’ r0), —
(1)
— — —
0.050
and Vi(T) =
J dr’
L(r, r’)lp~(r’— r0).
—
000C
I
2
3
5I
4
6I
7
8
9I
Distance (Cu,) —
0.050
.—
—0100’-’-
2
—0.150
/
-
C
w
—0,200
//
-
/ —0.250
~
/ 0300 \ -~ —0.350
/
/
/
/
/ /
/
/ “H
//
—
Fig. 2. The total interaction energy as a function of ion position is given for hydrogen and potassium. The solid lines represent the results obtained using the local expression for the exchange while the dashed line employs the gradient correction described in the text.
Vol. 20, No. 11
COMMENTS ON CHEMISORPTION AT A METAL SURFACE
The total energy of the adion and substrate can then be given as a function of the adion position r0 by W(ro)
=
f drp~(r—ro)vo(r) + L(r, r’)p~(r’— ro)
-~
ffdr’ drp~(r—ro). (2)
1075
‘adion is similar to that at the surface of a metal. It is thus not surprising that ~ is of similar importance in chemisorption in metallicofsurface energy calcu47 Theand convergence this expansion should, lations. therefore, be equally satisfactory to that found at solid surfaces.’° In Fig. 2, we display the effect of ~ on the binding
where v 0 is the bare surface potential. The induced dipole moment as a function of r0 is given by /1(ro)
=
ff dr dr’(ro
—
r)R(r, r’)~d(r’
—
To).
energy of a proton as a function of position so, the so called potential curve. We note that the equilibrium (3)
The substrate density n~.is chosen to correspond to r8 = 1.5 and we investigate the sensitivity of equations (1)—(3) to the treatment of the exchange and correlation for hydrogen and the full series of alkalis. With ~ ro) = e.5(r ~ and r0 = 1. lao (ao = Bohr radius), corresponding to the equilibrium position of a hydrogen ion, we obtain the induced density displayed in Fig. 1. From the form of the four different induced densities, we conclude that the gradient term ~ has a significant effect. The effect is in fact as large as that of the local exchange A~and much larger than the local correlation (which is negligible), The rapidity of the density variation around an —
—
position [the minimum point of W(ro)] is unchanged but the depth has been altered significantly. In Table 1, we give the equilibrium ionic hydrogen binding energy and dipole moment [equation (3)] with and without ~ Clearly ~ has a sizable effect on both properties. The same study is made for the alkali adions, where p~(r ro) is approximated by a Gaussian taken from reference (11). In Fig. 2, we display the importance of ~ on W(ro) for K, and in Table 1, we give the equilibrium atomic binding energy and dipole moments for the alkalis with and without ~ Again the effect of~ is significant. In conclusion, we have demonstrated the necessity of a careful treatment of the exchange and correlation energies in any refined theoretical study of chemisorption properties. —
REFERENCES 1.
HOHENBERG P. & KOHN W., Phys. Rev. 136, 864 (1964).
2. 3.
KOHN W. & SHAM L.J., Phys. Rev. 140, 1133 (1965). LANG N.D. & WILLIAMS A.R., Phys. Rev. Lett. 34, 531 (1975).
4.
RASOLT M. & GELDART D.J.W., Phys. Rev. Lett. 35, 1234 (1975).
5. 6.
GELDART D.J.W. & RASOLT M.,Phys. Rev. 1477 (1976). RASOLT M. & GELDART D.J.W., Solid State Commun. 18, 549 (1976).
7.
ROSE J.H., SHORE H.B., GELDART D.J.W. & RASOLT M., Solid State Commun. 19,619(1976).
8.
YING S.C., SMITH J. & KOHN W., Phys. Rev. Bi 1, 1483 (1975).
9.
LANG N.D. & KOHN W., Phys. Rev. Bi, 4555 (1970).
10. 11.
RASOLT M., SHY-YIH WANG J. & KAHN L.M. (to be published). KAHN L.M. & YING S.C.,Solid State Commun. 16, 799 (1975).
Pergamon Press.
Printed in Great Britain
COMMENTS ON CHEMISORPTION AT A METAL SURFACE L.M. Kahn* and M. Rasolttl Battelle Columbus Laboratories, Columbus, OH 43201, U.S.A. (Received 9 August 1976 by L. Hedin)
The effect of the gradient correction to the exchange and correlation energies on chemisorption properties is examined using the evaluation of ~ arrived at by Rasolt and Geldart. Quantities common to the study of hydrogen and alkali chemisorption on a high density metal are calculated for various forms of the energy functional within the linear response approach. It is found that the gradient correction is a quantitatively significant term, lowering binding energies by 10% and increasing the dipole sizably. THE GROUND STATE ENERGY of a non-uniform interacting electron gas of density n(r) can be written as E[n(r)] = f dr v(r)n(r) + VH[n(r)1 + 1’8 [n(r)] + ~ [n(r)]. Where VH is the electrostatic, T~the kinetic, and ~ the exchange and correlation contributions to the energy. Here v(r) is an external potential.’ Practically, this expression involves two major approximations, ~‘8 and ~ are represented via an explicit functional of n(r). Through this treatment of the energy, the electron density, and thus the ground state properties, of a systern can be determined directly (see below). It is, however, possible to treat the kinetic energy exactly. In such a procedure, the density is given directly through a set of self-consistent Euler—Lagrange equations.2 Although the latter procedure can involve a very complex numerical solution (as the symmetry of v(r) is lowered),3 it is evident that the fundamental problem resides in knowing
(e.g. binding energy, and induced dipole moment) are becoming available, the present numerical treatment still neglects many important contributions (e.g. the ions of the substrate),3 making quantitative comparison with experiment unreliable. Taking the view that these numerical procedures can be perfected, the importance of the proper treatment of the exchange and correlation contribution to these various properties is of increasing interest. To examine the magnitude of these effects, it is adequate to take the simpler approach of Ying et al.8 rather than solve the full self-consistent Euler—Lagrange equations. Briefly, the solution involves writing the potential v(r) as v(r) = v+(r) + v~(r)where v÷(r) corresponds to the potential of a step uniform positive background [p+(r) = n+O(— x), where n~is the bulk density] and v~(r r 0) is the potential of the adion the structure of E~~[n(r)].In this note, we wish to in- 4’5 centered at r0 with a corresponding density ~p~(r ro). The induced density n vestigate the effect of~recent refinement of E~0[n(r)], 1(r), due to v~(r),is then deteri.e., the second term in ~ dr [A~~[n(r)]+ mined 2], on several fproperties of an adion at densityby n treating it as a linear order correction to the B~~[n(r)]Vn(r)1 0(r), induced by v÷(r) as given in reference 9. a metal surface. Since this non-local term was found to With n 1(r) known, the various adion 47 (as properties are then easily evaluated. surface interaction make a major contribution to the surface energy large as 40% of the total surface energy), it raises the To calculate n 1 (r), it is obviously necessary to have question of its importance in the properties of chemian explicit form for 7’~[n] and ~ [n]. For 7’~ [n], wex 2/lOm) sorption. use the common approximation T8 [n] (3h We should point out that while more refined experi- (3ir2)2~’3f dr n5”3(r) + (h2/72m) f dr IVn(r)12/n(r). The mental data on various properties of chemisorption exchange contribution to A~~[n]is given by A~[n(r)] = e2 ~(3/ir)1”3f dr n~3(r),and the correlation by * Sponsored by Battelle Institute Program Grant A~[n(r)1 0.056 e2 f dr n~3(r)/[(0.079 + n Number B-2336-1100. For the form of ~ [n(r)], we use the result of reference 4. Knowing an explicit form for E[n(r)], our remaining Sponsored by Battelle Institute Program Grant task is to evaluate the effect of the additional term Number B-1331-1 190. B~~[n(r)]on the induced density n 1(r) and thus the ~ Also at The Ohio State University, Columbus, OH corresponding chemisorption properties. 43210, U.S.A. —
—
—
1073
1074 0.10
COMMENTS ON CHEMISORPTION AT A METAL SURFACE
Table 1. The ionic binding energies and dipole moments are given for hydrogen and the alkalis for the local expression for the exchange and the gradient correction described in the text
—
0.09
..‘
0.08
-
0.06
-
.
Ij~ energy (eV) local gradient
(a.u.) local
H Li Na
9.0 8.76 Atomic binding 4.50 4.16 2.85 2.60
0.01 moment 0.023 Dipole 0.39 0.71 1.00 1.52
K
2.08
1.94
1.65
2.25
Cs
1.29
1.24
3.15
3.90
E
.~ 0.04 ~ 007
-
,‘// \ :1
/
~,
gradient
/
/ /
0.030.02
Vol. 20, No. 11
/ -
0.01 -I
I
I I
0
2
Rb
1.70 1.58 2.49 3.20 The solution for the induced density, n1(r), 2v and the corresponding potential v1(r), (defined by V 1(r) = 4ire[n1(r) + ~ r0)] due to an arbitrary charge distribution is most easily calculated by a superposition 8 due to a of response point chargefunctions, positionedR(r, at r’. r’)Then and L(r, r’),
I
3
4
Distance (a u)
Fig. 1. The induced electron density due to a point charge of unit magnitude, H~,is illustrated for several forms of the energy functional. The density is given along the surface the perpendicular for the pointjoining charge the 1.1 point a.u. from charge the and surface. The ordinate’s scaled units are such that one scaled unit equals the bulk density ( local exchange +
—
n 1(r)
gradient correction, local exchange + local correlation + gradient correction, no exchange or correlation, local exchange + local correlation.)
=
Jdr’R(r, r’)~z~d(r’ r0), —
(1)
— — —
0.050
and Vi(T) =
J dr’
L(r, r’)lp~(r’— r0).
—
000C
I
2
3
5I
4
6I
7
8
9I
Distance (Cu,) —
0.050
.—
—0100’-’-
2
—0.150
/
-
C
w
—0,200
//
-
/ —0.250
~
/ 0300 \ -~ —0.350
/
/
/
/
/ /
/
/ “H
//
—
Fig. 2. The total interaction energy as a function of ion position is given for hydrogen and potassium. The solid lines represent the results obtained using the local expression for the exchange while the dashed line employs the gradient correction described in the text.
Vol. 20, No. 11
COMMENTS ON CHEMISORPTION AT A METAL SURFACE
The total energy of the adion and substrate can then be given as a function of the adion position r0 by W(ro)
=
f drp~(r—ro)vo(r) + L(r, r’)p~(r’— ro)
-~
ffdr’ drp~(r—ro). (2)
1075
‘adion is similar to that at the surface of a metal. It is thus not surprising that ~ is of similar importance in chemisorption in metallicofsurface energy calcu47 Theand convergence this expansion should, lations. therefore, be equally satisfactory to that found at solid surfaces.’° In Fig. 2, we display the effect of ~ on the binding
where v 0 is the bare surface potential. The induced dipole moment as a function of r0 is given by /1(ro)
=
ff dr dr’(ro
—
r)R(r, r’)~d(r’
—
To).
energy of a proton as a function of position so, the so called potential curve. We note that the equilibrium (3)
The substrate density n~.is chosen to correspond to r8 = 1.5 and we investigate the sensitivity of equations (1)—(3) to the treatment of the exchange and correlation for hydrogen and the full series of alkalis. With ~ ro) = e.5(r ~ and r0 = 1. lao (ao = Bohr radius), corresponding to the equilibrium position of a hydrogen ion, we obtain the induced density displayed in Fig. 1. From the form of the four different induced densities, we conclude that the gradient term ~ has a significant effect. The effect is in fact as large as that of the local exchange A~and much larger than the local correlation (which is negligible), The rapidity of the density variation around an —
—
position [the minimum point of W(ro)] is unchanged but the depth has been altered significantly. In Table 1, we give the equilibrium ionic hydrogen binding energy and dipole moment [equation (3)] with and without ~ Clearly ~ has a sizable effect on both properties. The same study is made for the alkali adions, where p~(r ro) is approximated by a Gaussian taken from reference (11). In Fig. 2, we display the importance of ~ on W(ro) for K, and in Table 1, we give the equilibrium atomic binding energy and dipole moments for the alkalis with and without ~ Again the effect of~ is significant. In conclusion, we have demonstrated the necessity of a careful treatment of the exchange and correlation energies in any refined theoretical study of chemisorption properties. —
REFERENCES 1.
HOHENBERG P. & KOHN W., Phys. Rev. 136, 864 (1964).
2. 3.
KOHN W. & SHAM L.J., Phys. Rev. 140, 1133 (1965). LANG N.D. & WILLIAMS A.R., Phys. Rev. Lett. 34, 531 (1975).
4.
RASOLT M. & GELDART D.J.W., Phys. Rev. Lett. 35, 1234 (1975).
5. 6.
GELDART D.J.W. & RASOLT M.,Phys. Rev. 1477 (1976). RASOLT M. & GELDART D.J.W., Solid State Commun. 18, 549 (1976).
7.
ROSE J.H., SHORE H.B., GELDART D.J.W. & RASOLT M., Solid State Commun. 19,619(1976).
8.
YING S.C., SMITH J. & KOHN W., Phys. Rev. Bi 1, 1483 (1975).
9.
LANG N.D. & KOHN W., Phys. Rev. Bi, 4555 (1970).
10. 11.
RASOLT M., SHY-YIH WANG J. & KAHN L.M. (to be published). KAHN L.M. & YING S.C.,Solid State Commun. 16, 799 (1975).