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The bound states of He on (0001) graphite with and without screening of the van der Waals interaction
Volume 156, number 9
PHYSICS LETTERS A
8 July 1991
The bound states of He on (0001) graphite with and without screening of the van der Waals interaction J.S. Brown Department of Physics, University of Vermont, Burlington, VT 05405, USA Received 9 November 1990; accepted for publication 8 May 1991 Communicated by A.A. Maradudin
Calculations are reported of the effect on the bound state spectrum of the He adatoms on (0001) graphite of including a screening function f in the atom—lattice van der Waals interaction potential. With three different screening functions of the form f= 1 — P(x) ~ it is shown that very little difference is found in the calculated bound state energies if screening is included. It is also suggested that this result is probably true of otherphysical properties of the He—graphite system.
Recently new measurements of the bound state energy spectrum of He adatoms on the (0001) face of graphite have been reported by groups at Penn State [1] and at Waterloo [2]. Adsorption resonances in the scattering of He atoms from (0001) graphite and from other surfaces of various metals in recent years have provided a wealth of information from which more realistic adatom—lattice interaction potentials may be constructed. In this communication we report calculations of the energy eigenvalues corresponding to the bound states of He adsorbed on (0001) graphite, investigating a certain property ofthe interaction potential between the adatom and the graphite lattice. For this study we have chosen an interaction potential investigated previously in several earlier papers [3,4], a potential which has proven itself capable of providing a rather good fit to the energies measured in the selective adsorption resonance studies cited above. This potential, representing a lateral average over the (0001) crystal face, is given by
by the interlayer spacing of graphite lattice planes. The attractive term represents the van der Waals (vdW) interaction of the He adatom with the various lattice planes ofthe graphite lattice, using a procedure introduced originally by Crowell and Steele [6]. The factor f(~)gives the screening of the vdW interaction due to wave function overlap effects. The parameter ~ in the screening function is given by ~= &z= k~hx,where k~is a cut-off wave-vector entering the screened vdW interaction, as originally discussed by Nordlander and Harris [7]. In the present note we investigate the effect on the calculation ofbound state energies ofthe He adatom in the laterally-averaged potential of eq. (1), using three different screening approximations, discussed in an earlier paper (see ref. [4]). All three screening functions may be cast into the form given by “2 ~ 1 P1~e—2~
(1)
PNH(~)l2~+2~2. (3) This screening function was derived from an integral involving the susceptibility—susceptibility response function approach to the vdW interaction obtained by Zaremba and Kohn [8] using ajellium model for the metal. The second approximation used is that of Ruiz et al. [2] (see ref. [4] for the 1 P(x)e 2x
V(x)=Ae2x_Bf(~))~ (x+n)4.
The repulsive term is obtained by using effective medium theory and makes use ofband calculations for the graphite surface [5] to fix the exponent in e ‘~, wherex= z/h is the ratio of the adatom distance from the first crystal plane of the graphite lattice, divided —
0375-9601/91/S 03.50 © 1991
—
—
‘“
—
—
“
‘
where P(~)is a polynomial in In the Nordlander— Harris (NH) approach we find ~.
Elsevier Science Publishers B.V. (North-Holland)
—
499
Volume 156, number 9
PHYSICS LETTERS A
form) (RSJ) based on a detailed treatment of wave function overlap effects using the core orbitals on the C atoms in the graphite lattice. The RSJ screening function leads to D I~\_ RSJk’s)—
IX 1~2 ~x3 l—.~,+ ~ +3~
~
28~4
855 +~, ,
a result derived in ref. [4] and also introduced earher, as suggested by a referee, in an unpublished thesis by Chow [9]. The third and last form, obtained by the simple requirement that V(x) remain finite as x—~0, was obtained by the author in ref. [4], and is given by P “p’— 1 _2~r+2çe2+~3 (5\ B~~I
3~
—
“
‘
In passing we note that eqs. (3) and (5) are contamed within the higher-power polynomial ofeq. (4), so that there must exist close relationships of these three screening functions, although no investigation of this point has been pursued. The current literature dealing with the physical properties of He adsorbed on (0001) graphite abounds with the use of the NH and RSJ screening functions. For example of this the reader is referred to a recent paper by Chizmeshya and Zaremba [10], who attempt to lay a foundation in wave function overlap effects for the screening function. At the present time however these attempts must still be considered rather ad hoc, and a rigorous treatment of screening effects has not yet emerged. In this note we study the effect of including the screening functions in the vdW part of the interaction potential V( x), using the three screening functions discussed above. In particular we present calculations of the bound state energy eigenvalues for
8 July 1991
the n= 1—5 levels of this system. The n=0 level is not counted because the repulsive coefficient A has in each case been determined by using the WKB method combined with a Schrodinger equation iterative scheme [11] to adjust A until the experimental n=0 etgenvalue has been obtained. Then a self-consistent WKB procedure calculates the n = 1— 5 eigenvalues and the iterative Schrodinger routine refines them a bit. The choice B=215 meV A3 was made for the present calculations, using the upper limit for the choice of this vdW coefficient as discussed by Cole, Goodstein and Frankl [12]. The Waterloo (see ref. [2]) data set was used as representative of the experimental data for these eigenvalues, although use of the Penn State data (see ref. [1]) would have given equally good results. Table 1 gives the bound state eigenvalues for n = 1— 5 with and without the three screening functions. The eigenvalues computed using the screening functions have been carried to a greater number of significant figures than those without screening, because unless this is done there is essentially no difference seen with the three different screening functions. Almost no perceptible differences are found when screening is included. Since these differences are much smaller than the experimental error of ±0.08 meV for each level, we can easily conclude that screening makes essentially no difference in the spectrum of bound states. The reason why screening hardly matters can be found by looking at the values of the three screening functions near the bottom of the potential well, where much of the significant physics enters. At this point we find that fNH=O.999, while f~=O.997, and
Table 1 Bound state energies of a He adatom on (0001) graphite with and without screening (energies are in meV). All calculations were done using the Waterloo dataset (see ref. [2]). We notethat the present approach involves fittingthen = 0 level exactly in each case. Experimental errors in measurement of the energies is ±0.08 meV. n
0 1 2 3 4 5
500
Without screening
—12.27 —6.836 —3.438 —1.531 —0.5835 —0.1789
With screening NH
RSJ
Brown
—12.27 —6.835 —3.437 —1.530 —0.5832 —0.1788
—12.27 —6.834 —3.436 —1.530 —0.5829 —0.1787
—12.27 —6.833 —3.435 —1.529 —0.5827 —0.1786
Volume 156, number 9 fRSJ = 0.982.
PHYSICS LETI’ERS A
In short, all three screening functions could almost be replaced by f= 1 at this distance without affecting anything very much. Only at significantly shorter distances, well up into the repulsive part of the potential does V(x) with screening depart substantially from V(x) without the inclusion of screening. At these distances the energy ofthe system is very large and positive so the He adatom spends essentially no time sampling this part of the potential. Therefore we believe that any significant physics hardly sees the differences between the potential with or without screening effects. We are therefore forced to conclude that the inclusion of such screening functions in treating the He—graphite system amounts to a “tempest in a teapot and, while fashionable, their use does not in any way seriously change the physical properties found in a typical calculation for this system. The variation in the bound state spectrum found by a slight variation in the attractive vdW coefficient B far outweighs any change obtained by including any of these three screening functions in the calculation. As a caveat we must of course state that the above conclusion may not hold for all physical properties, since we have only made this comparison by calculating the bound state energy spectrum. It is also clear, as suggested in the original work of Nordlander and Harris (see ref. [71) that He on Au involves ,
8 July 1991
sufficiently large differences in the region near the potential minimum to possibly affect the calculation of physical properties for that system. But we believe that this is not the case for the He—graphite system. The author wishes to express his thanks to the UVM Academic Computer Center for making time available on the VAX 8600 for the calculations reported in this note.
References [1] S. Chung, A. Kara and D.R. Frank!, Surf. Sci. 171(1986) [2]
J.C.
Ruiz, G. Scoles and H. Jonsson, Chem. Phys. Leti. 129 (1986) 139 [3] J.S. Brown, Phys. Lett. A 126 (1988) 36. [4] J.S. Brown, Phys. Lett. A 147 (1990) 393. [5]M. Weinert, E. Wimmer and A.J. Freeman, Phys. Rev. B26 (1982) 4571. [6] A.D. Crowd! and R.B. Steele Jr., J. Chem. Phys. 34 (1966) 1347 [7] P. Nordlander and J. Harris, J. Phys. C 17 (1984) 1141. [8] E. Zaremba and W. Kohn, Phys. Rev. B 13 (1976) 2270. [9] O.L. Chow, M. Sc. Thesis, Queen’s University, Kingston, Ontario, unpublished (1988). [10] A. Chizmeshya and E. Zaremba, Surf. Sci. 220 (1989) 443. [11] B.R. Johnson, J. Chem. Phys. 67 (1977) 4086. [12] M.W. Cole, D.R. Frank! and D.L. Goodstein, Rev. Mod. Phys. 53 (1981)199.
501
PHYSICS LETTERS A
8 July 1991
The bound states of He on (0001) graphite with and without screening of the van der Waals interaction J.S. Brown Department of Physics, University of Vermont, Burlington, VT 05405, USA Received 9 November 1990; accepted for publication 8 May 1991 Communicated by A.A. Maradudin
Calculations are reported of the effect on the bound state spectrum of the He adatoms on (0001) graphite of including a screening function f in the atom—lattice van der Waals interaction potential. With three different screening functions of the form f= 1 — P(x) ~ it is shown that very little difference is found in the calculated bound state energies if screening is included. It is also suggested that this result is probably true of otherphysical properties of the He—graphite system.
Recently new measurements of the bound state energy spectrum of He adatoms on the (0001) face of graphite have been reported by groups at Penn State [1] and at Waterloo [2]. Adsorption resonances in the scattering of He atoms from (0001) graphite and from other surfaces of various metals in recent years have provided a wealth of information from which more realistic adatom—lattice interaction potentials may be constructed. In this communication we report calculations of the energy eigenvalues corresponding to the bound states of He adsorbed on (0001) graphite, investigating a certain property ofthe interaction potential between the adatom and the graphite lattice. For this study we have chosen an interaction potential investigated previously in several earlier papers [3,4], a potential which has proven itself capable of providing a rather good fit to the energies measured in the selective adsorption resonance studies cited above. This potential, representing a lateral average over the (0001) crystal face, is given by
by the interlayer spacing of graphite lattice planes. The attractive term represents the van der Waals (vdW) interaction of the He adatom with the various lattice planes ofthe graphite lattice, using a procedure introduced originally by Crowell and Steele [6]. The factor f(~)gives the screening of the vdW interaction due to wave function overlap effects. The parameter ~ in the screening function is given by ~= &z= k~hx,where k~is a cut-off wave-vector entering the screened vdW interaction, as originally discussed by Nordlander and Harris [7]. In the present note we investigate the effect on the calculation ofbound state energies ofthe He adatom in the laterally-averaged potential of eq. (1), using three different screening approximations, discussed in an earlier paper (see ref. [4]). All three screening functions may be cast into the form given by “2 ~ 1 P1~e—2~
(1)
PNH(~)l2~+2~2. (3) This screening function was derived from an integral involving the susceptibility—susceptibility response function approach to the vdW interaction obtained by Zaremba and Kohn [8] using ajellium model for the metal. The second approximation used is that of Ruiz et al. [2] (see ref. [4] for the 1 P(x)e 2x
V(x)=Ae2x_Bf(~))~ (x+n)4.
The repulsive term is obtained by using effective medium theory and makes use ofband calculations for the graphite surface [5] to fix the exponent in e ‘~, wherex= z/h is the ratio of the adatom distance from the first crystal plane of the graphite lattice, divided —
0375-9601/91/S 03.50 © 1991
—
—
‘“
—
—
“
‘
where P(~)is a polynomial in In the Nordlander— Harris (NH) approach we find ~.
Elsevier Science Publishers B.V. (North-Holland)
—
499
Volume 156, number 9
PHYSICS LETTERS A
form) (RSJ) based on a detailed treatment of wave function overlap effects using the core orbitals on the C atoms in the graphite lattice. The RSJ screening function leads to D I~\_ RSJk’s)—
IX 1~2 ~x3 l—.~,+ ~ +3~
~
28~4
855 +~, ,
a result derived in ref. [4] and also introduced earher, as suggested by a referee, in an unpublished thesis by Chow [9]. The third and last form, obtained by the simple requirement that V(x) remain finite as x—~0, was obtained by the author in ref. [4], and is given by P “p’— 1 _2~r+2çe2+~3 (5\ B~~I
3~
—
“
‘
In passing we note that eqs. (3) and (5) are contamed within the higher-power polynomial ofeq. (4), so that there must exist close relationships of these three screening functions, although no investigation of this point has been pursued. The current literature dealing with the physical properties of He adsorbed on (0001) graphite abounds with the use of the NH and RSJ screening functions. For example of this the reader is referred to a recent paper by Chizmeshya and Zaremba [10], who attempt to lay a foundation in wave function overlap effects for the screening function. At the present time however these attempts must still be considered rather ad hoc, and a rigorous treatment of screening effects has not yet emerged. In this note we study the effect of including the screening functions in the vdW part of the interaction potential V( x), using the three screening functions discussed above. In particular we present calculations of the bound state energy eigenvalues for
8 July 1991
the n= 1—5 levels of this system. The n=0 level is not counted because the repulsive coefficient A has in each case been determined by using the WKB method combined with a Schrodinger equation iterative scheme [11] to adjust A until the experimental n=0 etgenvalue has been obtained. Then a self-consistent WKB procedure calculates the n = 1— 5 eigenvalues and the iterative Schrodinger routine refines them a bit. The choice B=215 meV A3 was made for the present calculations, using the upper limit for the choice of this vdW coefficient as discussed by Cole, Goodstein and Frankl [12]. The Waterloo (see ref. [2]) data set was used as representative of the experimental data for these eigenvalues, although use of the Penn State data (see ref. [1]) would have given equally good results. Table 1 gives the bound state eigenvalues for n = 1— 5 with and without the three screening functions. The eigenvalues computed using the screening functions have been carried to a greater number of significant figures than those without screening, because unless this is done there is essentially no difference seen with the three different screening functions. Almost no perceptible differences are found when screening is included. Since these differences are much smaller than the experimental error of ±0.08 meV for each level, we can easily conclude that screening makes essentially no difference in the spectrum of bound states. The reason why screening hardly matters can be found by looking at the values of the three screening functions near the bottom of the potential well, where much of the significant physics enters. At this point we find that fNH=O.999, while f~=O.997, and
Table 1 Bound state energies of a He adatom on (0001) graphite with and without screening (energies are in meV). All calculations were done using the Waterloo dataset (see ref. [2]). We notethat the present approach involves fittingthen = 0 level exactly in each case. Experimental errors in measurement of the energies is ±0.08 meV. n
0 1 2 3 4 5
500
Without screening
—12.27 —6.836 —3.438 —1.531 —0.5835 —0.1789
With screening NH
RSJ
Brown
—12.27 —6.835 —3.437 —1.530 —0.5832 —0.1788
—12.27 —6.834 —3.436 —1.530 —0.5829 —0.1787
—12.27 —6.833 —3.435 —1.529 —0.5827 —0.1786
Volume 156, number 9 fRSJ = 0.982.
PHYSICS LETI’ERS A
In short, all three screening functions could almost be replaced by f= 1 at this distance without affecting anything very much. Only at significantly shorter distances, well up into the repulsive part of the potential does V(x) with screening depart substantially from V(x) without the inclusion of screening. At these distances the energy ofthe system is very large and positive so the He adatom spends essentially no time sampling this part of the potential. Therefore we believe that any significant physics hardly sees the differences between the potential with or without screening effects. We are therefore forced to conclude that the inclusion of such screening functions in treating the He—graphite system amounts to a “tempest in a teapot and, while fashionable, their use does not in any way seriously change the physical properties found in a typical calculation for this system. The variation in the bound state spectrum found by a slight variation in the attractive vdW coefficient B far outweighs any change obtained by including any of these three screening functions in the calculation. As a caveat we must of course state that the above conclusion may not hold for all physical properties, since we have only made this comparison by calculating the bound state energy spectrum. It is also clear, as suggested in the original work of Nordlander and Harris (see ref. [71) that He on Au involves ,
8 July 1991
sufficiently large differences in the region near the potential minimum to possibly affect the calculation of physical properties for that system. But we believe that this is not the case for the He—graphite system. The author wishes to express his thanks to the UVM Academic Computer Center for making time available on the VAX 8600 for the calculations reported in this note.
References [1] S. Chung, A. Kara and D.R. Frank!, Surf. Sci. 171(1986) [2]
J.C.
Ruiz, G. Scoles and H. Jonsson, Chem. Phys. Leti. 129 (1986) 139 [3] J.S. Brown, Phys. Lett. A 126 (1988) 36. [4] J.S. Brown, Phys. Lett. A 147 (1990) 393. [5]M. Weinert, E. Wimmer and A.J. Freeman, Phys. Rev. B26 (1982) 4571. [6] A.D. Crowd! and R.B. Steele Jr., J. Chem. Phys. 34 (1966) 1347 [7] P. Nordlander and J. Harris, J. Phys. C 17 (1984) 1141. [8] E. Zaremba and W. Kohn, Phys. Rev. B 13 (1976) 2270. [9] O.L. Chow, M. Sc. Thesis, Queen’s University, Kingston, Ontario, unpublished (1988). [10] A. Chizmeshya and E. Zaremba, Surf. Sci. 220 (1989) 443. [11] B.R. Johnson, J. Chem. Phys. 67 (1977) 4086. [12] M.W. Cole, D.R. Frank! and D.L. Goodstein, Rev. Mod. Phys. 53 (1981)199.
501