On the HSAB based estimate of charge transfer between adsorbates and metal surfaces

Chemical Physics 393 (2012) 1–12

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On the HSAB based estimate of charge transfer between adsorbates and metal surfaces Anton Kokalj ⇑ Department of Physical and Organic Chemistry, Jozˇef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia

a r t i c l e

i n f o

Article history: Received 14 July 2011 In final form 17 October 2011 Available online 31 October 2011 Keywords: Adsorption Charge transfer HSAB Work functions of metal surfaces Mulliken electronegativity of metals Quantum chemical calculations Density functional theory (DFT) Corrosion inhibitors Benzotriazole Nitrogen Oxygen Chlorine

a b s t r a c t The applicability of the HSAB based electron charge transfer parameter, DN, is analyzed for molecular and atomic adsorbates on metal surfaces by means of explicit DFT calculations. For molecular adsorbates DN gives reasonable trends of charge transfer if work function is used for electronegativity of metal surface. For this reason, calculated work functions of low Miller index surfaces for 11 different metals are reported. As for reactive atomic adsorbates, e.g., N, O, and Cl, the charge transfer is proportional to the adatom valence times the electronegativity difference between the metal surface and the adatom, where the electronegativity of metal is represented by a linear combination of atomic Mulliken electronegativity and the work function of metal surface. It is further shown that the adatom-metal bond strength is linearly proportional to the metal-to-adatom charge transfer thus making the DN parameter a useful indicator to anticipate the corresponding adsorption energy trends. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction A theoretical formalization of the Hard and Soft Acids and Bases (HSAB) principle [1], which is based on conceptual DFT (DensityFunctional-Theory) [2–7], has provided a well founded theoretical chemical reactivity indicators, such as, the electron charge transfer DN and its associated change of energy DE parameters [2] as well as the electrophilicity index [8]. These have found a widespread application in various areas of molecular chemistry [9,10], however, in the field of surface chemistry this subject is less consolidated [11]; to some extent this is due to ambiguities and complexities associated with the existence of energy bands in condensed systems [12,13]. In the latter field, the DN parameter as well as local softness and related Fukui functions have been perhaps the most used among the HSAB based reactivity indicators; the local descriptors were used to explain the preferred surface reaction sites [14–19], while the DN was exploited either to anticipate the strength of the adsorbate–surface interaction or to explain the chemisorption energy trends [11,20–22]. In particular, such an application of DN has found a particularly widespread ⇑ Tel.: +386 1 477 35 23; fax: +386 1 477 38 22. E-mail address: [email protected] URL: http://www-k3.ijs.si/kokalj/ 0301-0104/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2011.10.021

use in quantum-chemical studies of corrosion inhibitors [23–37]. However, there appears to be some uncertainties how to estimate the necessary fundamental parameters used in DN equation that describe the metal surface. In particular, charge transfer between a molecule and metal-surface is driven by their difference in electronegativity and is opposed by the absolute (chemical) hardness. While the vanishing chemical hardness of metal-surfaces has been appreciated [23–31,34–37], the situation concerning the electronegativity is perplexing. In a recent letter [38] this issue has been shortly introduced. Here this subject is more thoroughly considered. Despite the well established and rather obvious fact that the electronegativity of macroscopic metals is given by the work function, e.g., Refs. [39–43], its usage in the context of chemical reactivity indicators is not widely accepted—e.g., it has been rarely used in the quantum-chemical studies of corrosion inhibitors [34– 37]—and some other estimates are frequently used instead. For this reason various estimates that are currently being used in the literature to describe the electronegativity of metal surfaces are reviewed and it is argued that two among them are conceptually wrong. The purpose of this paper is to evaluate the applicability of DN parameter in the case of an adsorbate interaction with metal surfaces and to advocate the use of work function for the electronegativity of metal surface in DN equation. It is shown that

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A. Kokalj / Chemical Physics 393 (2012) 1–12

the use of work-function indeed leads to a reasonable estimated trends of charge transfer for molecular adsorbates (or in the case of not too strong adsorbate–surface interaction). On the other hand, for reactive atomic adsorbates, such as N, O, and Cl, the charge transfer is found to be proportional to the adatom valence times the electronegativity difference between the metal surface and the adatom, where the electronegativity of metal is represented by a linear combination of local and global properties, i.e., atomic Mulliken electronegativity and the work function of metal surface. This paper is organized as follows: in next section the principle parameters involved in the HSAB principle are stated. Section 3 surveys the current status on the use of HSAB based estimate of charge transfer between adsorbates and metal surfaces with a discussion about the work function. The results and conclusions are presented in Sections 4 and 5: molecular adsorbates are considered in Sections 4.1 and 4.2, while atomic adsorbates in Section 4.3. Appendix A describes the computational method, whereas technical details about explicit charge transfer calculations are contained in Appendix B. 2. Basic HSAB equations The formalization of the HSAB principle is based on considering an acid–base chemical reaction, A + :B ? A:B, as a two step process: (i) transfer of some charge from :B to A, and (ii) formation of the actual bond [2] (for more rigorous theoretical foundations of the HSAB principle see, e.g., Refs. [3–7]). The DN and DE parameters emerge from considering the transfer of charge in the first step of the A:B formation, expanding the corresponding change in energy, DE, in Taylor series up to a second order, and applying the Sanderson’s principle of electronegativity equalization [2,44]. The resulting equations are [2]:

DN ¼

lB  lA 2ðgA þ gB Þ

ð1aÞ

and

ðl  lA Þ2 DE ¼  B ; 4ðgA þ gB Þ

ð1bÞ

where DN  DNA = DNB measures the charge transferred from B to A. The l and g, known as the electron chemical potential and absolute hardness, respectively, are the first and second partial derivatives with respect to electron number N at constant external potential v, i.e.:

l¼



 @E @N v

and g ¼

! 1 @2E : 2 @N2 v

ð2Þ

It is worth mentioning that many recent publications omit the factor 1/2 in the definition of g, i.e., gnew = @ 2E/@N2. Note that this definition affects all the parameters that depend on g; this issue and the associated pitfalls that are unfortunately common in the literature are commented in the recent letter [38]. Note, however, that due to integer discontinuities [45,46], the derivatives of Eq. (2) are ill defined [47,48] and to pass from them to operational definitions, the E(N) curve should be postulated; perhaps the most venerable is the ground-state parabola model [49], which yields l = (I + A)/2 and g = (I  A)/2, where I and A are vertical ionization potential and electron affinity, respectively. This operational definition of chemical potential is equivalent to the negative of the Mulliken electronegativity, v, i.e. [50,51]:

l¼

IþA ¼ v: 2

ð3Þ

3. HSAB estimate of charge transfer between adsorbates and metal surfaces Let us now consider the interaction of a molecule with a metal surface. The subscripts A and B will be replaced by ‘‘metal’’ and ‘‘mol’’ to designate a metal surface and a molecule, respectively. With this notation and the use of Eq. (3), the Eq. (1a) can be written as:

DN ¼

ðlmol  lmetal Þ ðvmetal  vmol Þ ¼ : 2ðgmetal þ gmol Þ 2ðgmetal þ gmol Þ

ð4Þ

The so defined DN measures the transfer of electrons from molecule to metal surface if DN > 0 and vice versa if DN < 0. While there is no ambiguity for the molecule, it appears to be much less clear what to use for the metal counterpart, in particularly so for the vmetal, for which various estimates are being used in the literature. These are shortly surveyed below. 3.1. Survey of various electronegativity estimates Various estimates of vmetal that have been used in the literature can be categorized in three classes: (i) the electronegativity of the corresponding metal atom is used; (ii) vmetal is represented by the work function; and (iii) the value of v (or l) for the bulk metal is searched for. 3.1.1. vmetal from electronegativity of metal atoms A popular choice for vmetal is the Mulliken electronegativity of corresponding metal atom [11,20,28,31,32,52], although occasionally the Pauling electronegativity is used instead [33]. However the latter is not appropriate in this context, because the use of electronegativity in Eq. (4) is due to l = vMulliken identity of the groundstate parabola model, Eq. (3). Pauling electronegativity instead uses arbitrary relative scale, hence l – vPauling, as already pointed out by Martinez et al. [53]. 3.1.2. vmetal from work function The work function was also used to represent the vmetal by several authors [34–37,42,54]. As one among the aims of this work is to advocate the use of work function for vmetal, the relation between the electronegativity of metal and work function is elaborated in Section 3.2. 3.1.3. vmetal from free electron gas Fermi energy As for the electronegativity of bulk metals, the value quoted as ‘‘theoretical result’’ is used almost exclusively [23–27,29,37,55], and in all cited articles this value apparently corresponds to free electron gas result from the book of Slater [56]. However, it is demonstrated below that the use of the free electron gas Fermi energy as the vmetal is conceptually wrong. In the free electron gas model the Fermi energy (eF) is given by eF ¼ 12 k2F (in Hartree atomic units), where the kF is Fermi wave number, kF = (3p2n)1/3, and n is a particle density of the free electrons, which in terms of Wigner–Seitz radius, rs, is given by 4 pr3s ¼ 1=n. Translated to eV energy unit the eF can be written in 3 terms of rs as [57]:

eF ¼

50:1 eV; r 2s

ð5Þ

where the rs value in bohr units must be used. As for the particle density of the free electrons, Slater assumed one electron per metal atom, hence the resulting Wigner–Seitz radius for, e.g., Fe and Cu is about 2.67 bohr, which gives the eF  7 eV. This value is frequently used as vmetal of Fe and Cu in Eq. (4) [23–27,29,37,55]; note that at zero temperature eF = l [40]. However, the Mulliken electronegativity

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A. Kokalj / Chemical Physics 393 (2012) 1–12

is opposite to the chemical potential, cf. Eq. (3), which makes the use of vmetal = eF conceptually wrong. Moreover, the v = l identity also implies that the electronegativity of free electron gas is negative. This is because only the kinetic energy of electrons is considered in this 2 model, i.e., v ¼ l ¼  12 kF < 0. The free electron gas model is also chemically inadequate, because it neglects the electron–electron interaction. 3.2. Relation between chemical potential, work function, and dipole barrier

conditions at the surface, such as, its geometry and the presence of net charges (if any). For an uncharged metal, Lang and Kohn [40] demonstrated that the bulk electron chemical potential measured with respect to the vacuum level is given by the negative of the work function, U, which is schematically represented in Fig. 1. This is rather obvious if the potential in vacuum infinitely away from any object is chosen as zero, V+1 = 0. Then the l = (@E/@N)v can be written as:

l ¼ lim ½EðNÞ  EðN  1Þv ¼ U;

ð6Þ

N!1

There is another important caveat in directly using the free electron gas Fermi energy to represent electronegativity of metals. Namely, it is well known that the eF or the electron chemical potential of infinite bulk is ill defined with respect to the vacuum level [58], and for this reason the eF is usually measured/calculated with respect to some arbitrary reference; a common choice is the mean electrostatic potential in the bulk—the l measured with respect to this reference will be designated by lhvi in the following. Such choice is, for example, adopted in electrochemistry (see below). Also the eF of Eq. (5) can be seen as being relative to mean electrostatic potential in the bulk. However, the lhvi and also the eF of Eq. (5) are not appropriate for use in Eq. (4), because the molecular analogue, lmol, is on the contrary measured with respect to the vacuum level. The lmol and lmetal in Eq. (4) must be measured with respect to a common reference, and the vacuum level is a very convenient choice for the zero of potential. Hence, it is necessary to introduce a surface that divides bulk metal from the vacuum; note that the electron chemical potential of semi-infinite metal is well defined with respect to vacuum level [58]. The chemical potential measured with respect to the vacuum level therefore depends on actual

because for macroscopic metal the number of electrons N is indeed exceedingly large number; E(N) and E(N  1) are the ground state energies of the macroscopic metal without and with an electron being removed, respectively. The second equality therefore stems from the definition of work function, which is the minimum work required to remove an electron from the uncharged bulk metal to its rest state in the vacuum. The work function is therefore, by definition, an appropriate measure of the electronegativity of metal surface and should be used as vmetal in Eq. (4). The relation between the work function, U, chemical-potential, lhvi, measured relative to mean electrostatic potential in the bulk, V1, and the dipole barrier, DV = V+1  V1, is [40]:

U ¼ DV  lhvi ;

ð7Þ

which is schematically presented in Fig. 1. The dipole barrier DV is the change of electrostatic potential across the dipole layer created by the spilling-out of electrons at the surface. The DV and lhv i can be therefore seen as a surface and bulk contributions to work function, respectively. It is known that both of them vary significantly as a function of free electron density, yet their difference is affected to

+∞

+∞

Δ

+∞

−

Δ

−∞

+∞

−

−∞

−∞

>0 0 −∞

Fig. 1. Top: schematic representation of work function (U), vacuum level (V+1), dipole barrier (DV), and electron chemical potential in the bulk (lhvi) relative to mean electrostatic potential therein (V1) for (a) high and (b) low free-electron density metals. Bottom: PBE calculated U, lhvi, and macroscopically averaged electrostatic potential V(z) as a function of the distance from the surface layer for (c) high free-electron density Al (111) and (d) low free-electron density Na (100) surfaces. The dipole barrier DV is negligible for the latter, but significant for the former surface. Black dots on abscissa indicate the position of atomic layers.

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A. Kokalj / Chemical Physics 393 (2012) 1–12

Table 1 PBE calculated lattice parameter, a0, and electronic properties of bulk metals and their low Miller index surfaces. Mulliken atomic electronegativity, calculated as vatom Me ¼ ðI þ AÞ=2, is also reported (last column). The most densely packed faces for fcc, bcc, and hcp structures are (111), (110), and (001), respectively. Crystal structure

Na K Mg Ca Al Ti Cr Fe Ni Cu Zn a

bcc bcc hcp fcc fcc hcp bcc bcc fcc fcc hcp

a0

K

Vxc

lhvi

DV (eV)

(bohr)

(eV)

(eV)

(eV)

(111)

(001)

7.96 10.01 6.01 10.44 7.66 5.49 5.38 5.37 6.66 6.94 5.23

3.57 4.81 12.37 11.15 16.68 20.51 27.62 23.89 21.24 6.82 17.84

5.93 5.04 8.25 7.76 9.30 11.53 13.75 13.62 13.75 12.26 11.38

2.36 0.22 4.12 3.39 7.38 8.99 13.87 10.26 7.49 5.45 6.47

0.25 2.00

0.28 2.03 7.81 6.10 11.76 13.60 17.97 14.17 12.40 0.85 10.65

6.38 11.55 17.75 14.14 12.53 0.62

vatom Me

U (eV) a

(110)

(111)

(001)

0.24 2.04

2.61 2.22

6.20 11.55

2.99 4.17

18.81 15.08 12.01 1.06

3.88 3.88 5.04 4.83

2.64 2.25 3.69 2.71 4.38 4.61 4.10 3.91 4.91 4.60 4.18

a

(110)

(eV)

2.60 2.26

2.95 2.48 3.73 3.09 3.28 3.56 3.84 4.28 4.58 4.69 4.34

2.81 4.17 4.94 4.82 4.52 4.39

For fcc and bcc metals the (001) surface is equivalent to (100), but the former label is used to match with the label for hcp (001) surface.

a much lower extent [40,59]; majority of metals have the work function in range between 3 and 5 eV [60]. It is instructive to consider the uniform background jellium model in the framework of local density approximation. Following the seminal paper of Lang and Kohn [40], the lhvi is given by:

1 2

lhvi ¼ k2F þ v xc ðnÞ;

ð8Þ

where vxc(n) is the exchange-correlation potential (negative quantity). In this model the lhvi depends solely on the particle density of the free electrons n, and the larger the n the larger the lhvi; for high free electron density metals the lhvi is positive, while at lower densities the vxc(n) overwhelms the kinetic energy term thus making the lhvi negative [40]. On the other hand, the dipole barrier is given by [40]:

DV ¼ 4 p

Z

It is further seen from Table 1 that for majority of considered metals the lhvi is positive, thus demonstrating the inappropriateness of its direct use in DN equation, Eq. (4). Namely, positive lmetal inevitably leads to negative DN—charge transfer from metal to molecule—irrespective of the molecule. Moreover some of the most electropositive metals (Na and K) display the smallest lhvi and its direct use in Eq. (4) would predict the smallest charge transfer from metal to molecule, which is meaningless. On the other hand, metals displaying large work function—referring to densely packed surfaces—also display large lhvi. This is in fact a fortunate circumstance for the naive use of vmetal = lhvi (should be v = l instead); despite the conceptual inappropriateness the corresponding DN would not be completely meaningless although often considerably overestimated in magnitude.

þ1

z½nðzÞ  nþ ðzÞdz;

ð9Þ

1

where n(z) and n+(z) are the electron and positive-background densities, respectively, and z is the surface normal direction. The DV therefore depends on the spill out of electrons over the jellium edge and it turns out that DV is negligible at low electron densities and increases with increasing n [40]. Also within the realistic state-of-the-art full-potential or pseudo-potential model, the chemical potential lhvi can be written analogously to Eq. (8) by using the macroscopically averaged quantities (see Appendix B.1):

lhvi ¼ K þ V xc ;

ð10Þ

where K designates the term related to electron kinetic-energy of 2 the reference Kohn–Sham system (analogous to jellium 12 kF term), and Vxc is the macroscopically averaged exchange-correlation potential. A schematic presentation of bulk and surface contributions to work function and their dependence on the density of free electrons is shown in Fig. 1, whereas Table 1 reports the K, Vxc, lhvi, DV, and U as calculated by the Perdew–Burke–Ernzerhof (PBE) energy functional [61] and plane-wave pseudo-potential method for 11 different metals including simple (alkalis, earth-alkalis, and Al), transition, and noble metals; see Appendix A for computational details. For fcc and bcc metals, the work function and DV of (111), (110), and (100) low Miller index surfaces are reported, whereas for hcp metals only the compact (001) surface is considered. These data confirm the results of simpler jellium model, i.e., both bulk and surface contributions to work function, lhvi and DV, display large variations extending almost over 20 eV range, but cancel each other to a large extent so that the work functions presented in the table vary only from 2 to 5 eV.

3.2.1. Correspondence between electrochemical potential and work function It is interesting to consider the definition of electrochemical po , in electrochemistry, which is for electron defined as: tential, l

l ¼ lhvi  e/;

ð11Þ

where e is the unit charge, and / is the inner potential given as a sum of outer and surface potentials, which are due to net charges and dipoles at the surface of metal, respectively (beware of the difference between two similar symbols, / and U, which designate two different properties). By definition, the electrochemical potential corresponds to the work done to bring an electron from its rest state in vacuum at infinity into the interior of bulk metal [62]. Hence for an uncharged metal in vacuum, the electrochemical potential is, by definition, just the opposite to the work function, l ¼ U ¼ lhvi  DV. It is worth pointing out an amusing issue, namely, it follows from the discussion above that the electrochemical potential of un should be used for lmetal in Eq. (4) to estimate the charged metal l molecule to metal charge transfer and not the chemical potential lhvi as the naming would suggest. 4. Results 4.1. Evaluation of various choices for vmetal in DN equation for molecular adsorbates By using the work function for vmetal and setting gmetal to zero— note that the absolute hardness of bulk metals is related to the inverse of their density of states at Fermi energy [63], which is an extremely small number—the Eq. (4) can be rewritten as:

A. Kokalj / Chemical Physics 393 (2012) 1–12

5

Table 2 PBE calculated vertical ionization potential, I, vertical electron affinity, A, electronegativity, vmol = (I + A)/2, and absolute hardness, gmol = (I  A)/2, of Li2 dimer, benzotriazole (BTAH), and Cl2 molecule.

DNUHSAB ¼

vmetal  vmol U  vmol : ¼ 2ðgmetal þ gmol Þ 2gmol

ð12Þ

On the other hand, if the vmetal is represented by the Mulliken electronegativity of corresponding metal atom, vatom Me , the resulting equation is1:

DNvHSAB ¼

vatom Me  vmol : 2gmol

ð13Þ

The performance of these two equations will be evaluated, while the other choice used in the literature, i.e., vmetal = eFfreeelectron, is not considered here, because it is conceptually wrong as shown above; should be vmetal = l = eFfreeelectron instead, which leads to meaningless results, i.e., negative electronegativity and inevitable charge transfer from metal to molecule in all cases. For the purpose of evaluation, three molecules with considerably different electronegativities were chosen: Li2 dimer as an example of electropositive molecule, benzotriazole (BTAH) as a molecule with intermediate electronegativity, and Cl2 as an electronegative molecule. The corresponding PBE calculated vertical ionization potentials (I), electron affinities (A), electronegativities (vmol), and absolute hardnesses (gmol) are presented in Table 2. v The electron transfer as estimated by DN U HSAB and DN HSAB is compared to the explicitly calculated charge transfer, DNDFT, in Fig. 2. For the latter the molecules were kept relatively high above the surface, about 3.5 Å—a situation compatible with the basic assumption in DN derivation; for further details about the explicit charge calculations see Appendix B.3. It is seen that both DN U HSAB v and DN HSAB estimations are reasonable considering the simplicity of Eqs. (12) and (13). For majority of cases they give the direction of charge transfer compatible with the one calculated explicitly (points falling in the white quadrants). It can be however observed that the DN U HSAB equation gives a sharper distribution of points than DN vHSAB , i.e., the former are less scattered around the blue line (fitted slope) than the latter, indicating that the DN U HSAB somewhat v outperforms the DN HSAB . Referring to the corresponding blue lines the maximal deviation is 0.21 and 0.25 electrons for DN U HSAB and DN vHSAB , respectively, whereas the root mean square deviation is 0.08 and 0.10 electrons, respectively. Note that in both cases the blue line is steeper than the unity slope (thin dashed line), hence the DNDFT is systematically smaller in magnitude than the HSAB estimated charge (it is possible this is related to the way the explicit charge transfer has been evaluated). It is not surprising that the U and vatom representations give comparable DN estimations, beMe cause, as seen in Table 1, the Mulliken electronegativities of metal atoms and work functions do not differ substantially. In some cases the direction of the charge transfer as predicted by the HSAB equation is incompatible with the explicitly DFT cal1 If the vatom to use also the gatom Me is used for vmetal then it would be consistent Me for atom 1 @ vMe gmetal and not the gmetal = 0, because the gatom Me ¼  2 @N . The latter was indeed used

by Crawford [11,20]. Nevertheless, in the manuscript the more commonly used gmetal = 0 estimate is taken.

culated direction (points falling in the red rectangular regions in Fig. 2). However, these are limited to relatively small differences in electronegativity between metal surface and molecule. The reason for mismatch can be twofold: it can be either due to (i) the DFT problem of electronic energy levels alignment, i.e., the Kohn–Sham HOMO–LUMO gap severely underestimates the fundamental gap [64], or due to (ii) the failure of the simple HSAB treatment. To tentatively explain the latter failure, three cases are considered and shown schematically in Fig. 3. In the first two cases the difference in electronegativity between the two reactants is rather small, much smaller than the molecular HOMO–LUMO gap (Fig. 3a,b), whereas in the third case the electronegativity difference is considerable (Fig. 3c). In Fig. 3a the metal’s Fermi energy eF (measured with respect to vacuum level) is smaller than the molecular electron chemical potential lmol—taken as arithmetic mean of eHOMO and eLUMO—hence the HSAB equation would predict the flow of charge from molecule to metal. However due to large HOMO– LUMO gap, the position of HOMO electrons is considerably below the eF of metal, which makes the transfer of electron charge from molecule to metal questionable. The situation is oppositely analogous when the eF > lmol (Fig. 3b); in this case the position of LUMO level is considerably above the eF of metal and the flow of electrons from metal to molecule is questionable. In such cases the actual charge transfer is rather small and its direction is given by details of electronic structure. On the other hand, if the difference in electronegativity between the two reactants is large and/or the HOMO–LUMO gap is small the charge transfer picture of HSAB is physically sound and viable; the case where lmol is considerably larger than eF is shown schematically in Fig. 3c.

4.2. Dependence of DN on surface geometry That the value of bulk lmetal measured with respect to the vacuum level depends on the conditions at the surface is not an annoyance but rather an advantage for Eq. (4). The molecule interacts with metal at the surface and not in the bulk. Moreover, it is well known that the molecule-surface interaction depends on the geometry of the surface, and so does the work function. Hence by using the work function for vmetal in Eq. (4), the dependence of surface geometry on the molecule-surface interaction is straightforwardly accounted for. On the other hand, other estimates of vmetal (cf. Section 3.1) does not account for this issue. Nevertheless, the work functions of various low Miller index faces do not differ substantially and in most cases the difference is only a few tens of eV [60]. Yet this difference is rather large for Cr and Fe for which the work functions between densely packed bcc (110) and more open surfaces differ by about 1 eV, see Table 1 and Fig. 4. Such large difference should result in noticeable changes in charge transfer at various surface facets. Among the three investigated molecules, only the Li2 gives a significant electron charge transfer on transition metal surfaces

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A. Kokalj / Chemical Physics 393 (2012) 1–12

(a) χ metal ≡ Work function

(b) χ metal ≡ Mulliken electronegativity

0.5

0.5 Li||2

0.4 0.3

⊥

0.3

BTAH Cl||2

0.2

0.2

Cl⊥ 2

0.1

||

Li2

⊥

Li2

BTAH⊥ || Cl2 Cl⊥ 2

χ

0.1

ΔNHSAB

Φ ΔNHSAB

0.4

⊥

Li2

0.0

0.0

−0.1

−0.1

−0.2

−0.2

−0.3

−0.3

−0.4

−0.4

−0.5 −0.5 −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 0.5 ΔNDFT

−0.5 −0.5 −0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 0.5 ΔNDFT

Fig. 2. Comparison between the explicitly calculated charge transfer from molecule to metal surface, DNDFT, and the HSAB estimated charge transfer, Eqs. (12) and (13): (a) DN UHSAB vs. DNDFT and (b) DN vHSAB vs. DNDFT. The Li2, BTAH, and Cl2 molecules and densely packed metal surfaces from Table 1 are considered. The symbols k and \ stand for molecules orientated parallel and perpendicular to the surface, respectively. Referring to the corresponding blue lines the maximal deviation is 0.21 and 0.25 electrons for DN UHSAB and DN vHSAB , respectively, whereas the root mean square deviation is 0.08 and 0.10 electrons, respectively.

Fig. 3. Schematic representation of the molecule-metal electron charge transfer: when the molecular electron chemical potential lmol (taken as arithmetic mean of eHOMO and eLUMO) is larger then the metal’s Fermi energy eF (measured with respect to vacuum level), the HSAB would predict the flow of electron charge from molecule to metal (a, c) and vice versa if lmol < eF (b). Correspondingly, a fraction of electrons would be emitted from HOMO orbital to metal states at eF in (a, c), while in (b) from eF to LUMO orbital. Note, however, that when the difference between lmol and eF is small—appreciable smaller than the molecule’s HOMO–LUMO gap—the electron charge transfer is small and complicated affair, while its direction as predicted by the HSAB uncertain.

Energy (eV)

6

6

4

V(z)

2

Fermi level

Φ

Φ

V(z)

4

Fermi level

2

0 -2

0

μ〈v〉

μ〈v〉

-2

-4

-4

-6

-6

-8

(a) Fe(110)

-10 -12 -15

-10

-5

0

5

10

15

distance from surface layer (bohr)

-8

(b) Fe(100) 20 20

15

10

-10 5

0

-5

-10

-12 -15

distance from surface layer (bohr)

Fig. 4. PBE calculated work function U and electron chemical potential lhvi relative to mean electrostatic potential in the bulk for (a) densely-packed Fe (110) and (b) more open Fe (100) surface. Note that while the lhvi is the same in the two cases, the work functions differ by almost 1 eV. Black dots on abscissa indicate the position of atomic layers, while the energy on ordinate is relative to Fermi level.

and is therefore suitable for further testing the validity of U representation of vmetal. As shown in Fig. 5 the electron transfer from Li2 to the Cr (110) and Fe (110) surfaces is larger than to the Cr (100)

and Fe (100), as would be expected from the corresponding work functions. The amount of charge transferred from Li2 to Cr (100) compares well with that on Al (111) and Zn (001), and all the three

A. Kokalj / Chemical Physics 393 (2012) 1–12

It is reasonable to anticipate that many cases would fall in-between the two limits, where the charge transfer depends on vmetal as given by a combination of work function and Mulliken electronegativity of metal atom, i.e.:

0.16

Cr(110)

0.15

Fe(110)

ΔNDFT

0.14

vmetal ¼ g U þ ð1  gÞvatom Me ;

0.13 0.12 0.11

Cr(100)

0.10

gmetal ¼ ð1  gÞgatom Me :

Fe(100) Mg(001) 3.8

4.0

4.2

4.4

4.6

ð14Þ

where g is a fraction of global property (mnemonic: g for global). Such a postulation of vmetal implies that the chemical hardness is approximately given by2:

Zn(110) Al(100)

0.09 0.08 3.6

7

4.8

5.0

5.2

5.4

work function (eV) Fig. 5. Correlation between the work function and explicitly calculated electron charge transfer, DNDFT, from Li2 molecule to (110) and (100) surfaces of Cr and Fe. For comparison also the DNDFT on Al (111), Zn (001), and Mg (001) is shown.

surfaces have very similar work functions. Indeed, the correlation between the charge transfer and work function is excellent, thus validating the formal arguments about the use of work function for the vmetal. 4.3. Weak and strong coupling limit It is worth pointing out that U is a global property of surface, and its representation for vmetal in DN equation is expected to be more appropriate when the metal surface and the molecule may be viewed as two distinct subsystems, which is the case of the weak molecule-surface interaction or when the distance between the molecule and the surface is kept large—this is also compatible with the basic assumption in the DN derivation, i.e., that bond formation consists of two steps: first at large distances charge transfer between the two reactants takes place and then the electron states of the two reactants hybridize at closer distances. However in the case of a very strong interaction the separation of the two reactants into two distinct subsystems becomes less obvious. In the limit of a very strong coupling of the two subsystems, it is intuitive to expect that charge transfer will depend on local properties of specific atoms involved in the bonding rather than on global properties. This view is compatible with the assessment that adsorption is a phenomenon of local character [65]. For molecular systems Gazquez [66] suggested to replace global softness (inverse of global hardness) by the condensed local softnesses of atoms involved in the considered bond. Although it is known that for metallic systems the local softness is related to the local density of states at Fermi energy [63,67], here a simpler direction is followed. Consider, for example, a reactive atomic adsorbate that interacts strongly with a metal surface. Because of the very strong adatom–metal interaction, the decomposition of the whole system into individual atoms is as appropriate as dividing the adsorption system into the metal substrate and the adatom. Let us now envisage that all individual atoms recombine into the adatom/metal system. From this point of view it may be reasonably expected that the direction of adatom/metal flow of charge is given by the Mulliken electronegativity difference between the adatom and the metal atom. This reasoning may be seen as an intuitive justification for the use of Mulliken electronegativity of metal atom in DN equation. This view is supported by the study of Crawford et al. [20] who found a good correlation between the Mulliken net charge of N adatoms adsorbed on densely packed transition metal surfaces and the HSAB DN estimate, where the Mulliken electronegativities of metal atoms were used.

ð15Þ

To validate this hypothesis, the charge transfer between three electronegative atoms—N, O, and Cl—and the closed packed metal surfaces—subset of those considered in Table 1, from K to Zn—has been calculated explicitly by means of PBE adsorption calculations utilizing Bader charge analysis of optimized adatom/metal structures (see Appendix for technical details). The PBE calculated vertical ionization potentials, electron affinities, electronegativities, and absolute hardnesses of N, O, and Cl atoms are presented in Table 3. The N, O, and Cl atoms interact rather strongly with metal surfaces and the calculated bond strengths are in range between 3 to 8 eV; the PBE results of adsorption calculations are reported in Table 4. The first observation is that the charge transfer is not driven solely by the difference in electronegativity between the metal and the adatom, but also by the elementary chemistry of the latter, namely, its valence. The valences of N, O, and Cl are 3, 2, and 1, respectively, and also the magnitude of charge transfer follows the N > O > Cl order, whereas their electronegativities follow the opposite trend. However for a given adatom the charge transfer at various metal surfaces indeed correlates with electronegativity of the metal. This can be appreciated from Fig. 6; Fig. 6a shows the atomic Mulliken electronegativities and work functions of involved metals, while Fig. 6b shows the charge transfer for the N, O, and Cl adatoms on these surfaces. Comparison of Figs. 6a and b reveals that the shapes of the charge transfer curves approximately follow that of metal electronegativity with the exception of N on Ni (111).3 Further analysis reveals that the charge transfer is proportional to:

DN / v ðvmetal  vad Þ;

ð16Þ

where v is the valence of the atomic adsorbate and vad is its electronegativity. The vmetal is expressed by Eq. (14) with the g = 0.4. The 0:4U þ 0:6vatom curve is shown by bold solid line in Fig. 6a Me and it is seen that it displays rather similar dependence as the charge transfer curves in Fig. 6b. This validates the above proposition that the charge transfer is given by a combination of local and global properties in case of a strong adsorbate–surface interaction . It is worth mentioning that for all three adsorbates, the magnitude of charge transfer is the smallest on Ni (111). This cannot be explained by the Mulliken electronegativity, because the vatom Cu 2

By using the l =  v identity and Eq. (14), the chemical hardness of metal may be h i @ vatom 1 @ lmetal 1 @ vmetal 1 @U Me  ð1  gÞgatom metal ¼ 2 @N ¼  2 @N ¼  2 g @N þ ð1  gÞ @N Me . In the     atom @ v   U was neglected, because @ U  Me last passage, the @@N .  @N @N written as g

3 It may be noted from Fig. 6b that the charge transfer from Ni (111) to N adatom looks anomalous, because it appears as spike in the blue curve, i.e., a larger magnitude of transfer would be anticipated by interpolation from left and right neighboring points. Let us remark that N is the least electronegative and the hardest among the three atoms (cf. Table 3), while Ni surface is the most electronegative (cf. Eq. (14) and purple curve of Fig. 6a), hence the difference in electronegativity between the two reactants is the smallest for N/Ni pair. It is therefore likely that this apparent anomaly is related to arguments presented in Fig. 3.

8

A. Kokalj / Chemical Physics 393 (2012) 1–12

−2.8

(a)

5.0

(c)

Energy (eV)

4.5 −2.4

4.0 3.5 0.4Φ + 0.6χMe Φ χMe

2.5 2.0 −0.5 −1.0 ΔNBader

−1.6

~

(b)

−2.0 ΔN (electrons)

3.0

−1.2

N@Ni(111)

−1.5 −2.0

N O Cl

−2.5 −3.0

N O Cl

−0.8

−0.4 −0.4

K(110) Ca(111) Ti(001) Cr(110) Fe(110) Ni(111) Cu(111) Zn(001)

−0.8

−1.2

−1.6

−2.0

−2.4

−2.8

ΔNBader (electrons)

Metal surface

Fig. 6. (a, b) Similarity between the electronegativity of metals and the charge transfer as calculated from Bader analysis for atomically adsorbed N, O, and Cl on densely e and explicitly calculated charge transfer from Bader analysis, DNBader. The packed metal surfaces. (c) Comparison between the estimated charge transfer by Eq. (18), D N, negative DN stands for the charge transfer from the metal to the adsorbate.

Table 3 PBE calculated ionization potential, I, electron affinity, A, electronegativity, vmol = (I + A)/2, and absolute hardness, gmol = (I  A)/2, of N, O, and Cl atoms.

N O Cl

I (eV)

A (eV)

vmol (eV)

gmol (eV)

14.74 14.04 12.98

0.15 1.67 3.69

7.45 7.85 8.34

7.30 6.19 4.64

Cr(100)

Fe(100)

Cr(110)

Fe(110)

−1.5

ΔNBader

−1.1 −0.9 −0.7 −0.5 3.9

4.0

4.1 4.2 4.3 4.4 0.4Φ + 0.6χMe (eV)

DN /

g U þ ð1  gÞvatom Me  vad ; 2ðð1  gÞgatom Me þ gad Þ

4.5

Fig. 7. Correlation between the electronegativity of metal surface—as calculated by the linear combination of work-function and Mulliken electronegativity of corresponding metal atom, 0:4U þ 0:6vatom Me —and explicitly calculated electron charge transfer from Bader analysis, DNBader, for N, O, and Cl adatoms on densely packed (110) and more open (100) surfaces of Cr and Fe.

surpasses the vatom implying that the magnitude of charge transNi fer should be smaller on Cu(111). On the other hand, the 0:4U þ 0:6vatom representation of vmetal indeed gives the Ni Me (111) as the most electronegative. A further validation of the 0:4U þ 0:6vatom representation is provided by Fig. 7, which shows Me the charge transfer calculated with Bader analysis as a function of 0:4U þ 0:6vatom for the three adatoms on the (110) and Me (100) surfaces of Fe and Cr, for which the work-functions between the two faces differ considerably. The correlation between the Bader charges and the 0:4U þ 0:6vatom is excellent and the Me magnitude of charge transfer is larger on more open (100) surfaces due to their lower work-functions.

ð17Þ

where the equality was replaced by the proportionality relation, because it was already shown that in the current case the DN depends also on other factors, such as, the adatom valence, cf. Eq. (16). The charge transfer is hence reasonably described as:

e ¼ cv DN

N O Cl

−1.3

By plugging Eqs. (14) and (15) into Eq. (4) the DN may be written as:

g U þ ð1  gÞvatom Me  vad ; 2ðð1  gÞgatom Me þ gad Þ

ð18Þ

where the proportionality constant c  2.7. The tilde accent is used e is a modification of the original HSAB derived to indicate the D N e and the one calDN. The correlation between the so estimated D N culated explicitly from Bader charge analysis is presented in Fig. 6c; it is evident that the charge transfer is indeed rather well described by Eq. (18) considering its simplicity. In analogy with Pearson’s study of mononuclear transition metal carbonyls [68], where the DN has been patched by the number of coordinated ligands, the proportionality constant c may be interpreted as an effective coordination number of the adatom on the metal surface. Namely, the adatoms usually adsorb to hollow sites, which involves bonding to three metal atoms on fcc (111) and hcp (001) surfaces, whereas on bcc (110) this implies two shorter and two longer adatom-metal bonds. Indeed, the charge transfer depends on the adsorption site and is smaller on top sites compared to hollow sites. It is rather satisfying the Eq. (18) is that successful in describing the amount of charge transferred, because the HSAB’s DN equation is derived for small changes in N, whereas the current changes are appreciable. Moreover, Pearson stated that the DN is not to be used to calculate the actual charges, but rather that it measures initial effects in the bond formation [69]. Hence, the DN is expected to be applicable to discern trends when comparing similar systems, whereas the current span of chemistry is rather large and can be subdivided at least into three groups, i.e., the interaction of adatom (i) with alkali and earth-alkali metals, where the bonding is predominantly ionic, (ii) with transition metals where the covalent contribution to bonding is significant, and (iii) with Cu and Zn, which have completely full d states resulting in a non-stabilizing hybridization of metal d states with adatom states.

9

A. Kokalj / Chemical Physics 393 (2012) 1–12

Table 4 PBE calculated properties of N, O, and Cl adsorbed on densely-packed metal surfaces. qBader—net Bader charge of adatom (positive numbers indicate electron access, i.e., qBader = DN); Eb—adatom-to-surface binding energy, Eb = Eadatom/surf  Eadatom  Esurf; Dzad—height of the adatom above the nearest neighbor metal atoms on the surface. Negative Dzad indicates subsurface adsorption with the adatom located below the surface layer. The PBE absolute hardness of corresponding metal atoms, calculated as gatom Me ¼ ðI  AÞ=2, is also reported.

gatom (eV) Me

Metal surface

K (110) Ca (111) Ti (001) Cr (110) Fe (110) Ni (111) Cu (111) Zn (001)

N

1.96 3.00 3.01 3.44 3.59 3.47 3.46 5.07

O Eb (eV)

Dzad (Å)

qBader

Eb (eV)

Dzad (Å)

qBader

Eb (eV)

Dzad (Å)

2.72 2.15 1.56 1.27 1.16 0.61 1.01 1.40

2.73 7.02 7.81 7.01 6.34 4.96 3.43 4.31

2.85 1.33 0.69 0.95 0.49 0.97 0.99 0.17

2.09 1.53 1.27 1.11 1.02 0.91 1.02 1.19

5.39 8.40 8.46 7.17 6.19 5.03 4.44 5.19

2.76 0.31 0.96 1.07 0.99 1.11 1.09 0.55

1.16 0.91 0.77 0.63 0.58 0.50 0.56 0.61

5.00 5.14 4.82 4.21 3.94 3.42 3.31 2.86

0.57 1.74 1.84 1.79 1.78 1.77 1.84 1.87

−9

N O Cl

Binding energy (eV)

−8

Ti(001) Ti(001) Cr(110)

Cr(110)

−7

Fe(110)

−6

Fe(110)

Ni(111)

−5

Ni(111) Ti(001) Cr(110) Fe(110) Ni(111)

−4 −3 0.4

0.6

0.8

Cl

qBader

1

1.2

1.4

1.6

Bader adatom charge (electron)

certainly worth pursuing in the future with hope to devise a simple equation that would be able to foretell—on the basis of a few simple elementary quantities—the approximate binding energies of adsorbates on metal surfaces. Despite this deficiency, the DN (to some extent also the DE) was nevertheless successfully applied to explain the bonding trends of rather similar systems, e.g., of mononuclear transition metal carbonyls [68] and of adsorbates on the transition metal surfaces [11,20,22]. In these studies, it was found that the larger the charge transfer the stronger the formed bond. Indeed, this is also the case currently if a specific adatom is considered only on transition metal surfaces (i.e. from Ti to Ni, see Table 4). This can be appreciated from Fig. 8, which shows the correlation between the net Bader charge of chemisorbed adatoms and their binding energy on densely packed transition metal surfaces.

Fig. 8. Correlation between the net Bader charge of the chemisorbed N, O, and Cl adatoms adsorbed on densely packed transition metal surfaces and the adatomsurface binding energy.

5. Conclusions 4.4. Shortly on DE parameter Although the HSAB based DN parameter or its modified variant, Eq. (18), gives reasonable estimates for the charge transfer, the corresponding DE of Eq. (1b) tells little about the actual strength of the formed bond. Within the current context, the energy change associated with charge transfer is typically several times smaller in magnitude than the whole reaction energy (or bond strength). Indeed, Pearson remarked that the formalization of the HSAB principle and the development of hardness concept have not helped greatly in evaluation of bond energies [70]. As for the molecular adsorbates, the vmetal  vmol value in range of 1 to 2 eV represents a large electronegativity difference (see Tables 1 and 2) and gmol of 4 eV is a reasonable value for the chemical hardness of soft molecule. The resulting DE of Eq. (1b) is then in range from 0.06 to 0.25 eV. However, the molecule-surface bond energy of, say, BTAH molecule on transition metal surfaces is over 1 eV and about 0.5 eV on noble metals [71,72]. On the other hand, the charge transfer is much larger for electronegative atomic adsorbates (see Table 4 and Fig. 6) due to a much larger electronegativity difference, and the magnitude of the resulting D e E ¼ cv DE can reach values of several electron volts, E and the actual Eb is rather however, the correlation between the D e poor (not shown for brevity); the span of chemistry considered currently is simply too diverse for DE to be applicable. From this perspective, it is not surprising that several HSAB type extensions have been proposed during the years that would yield better estimates of bond energies [73–76,66]. This direction is

The applicability of the HSAB based electron charge transfer parameter, DN (also known as the fraction of transferred electrons), was analyzed for adsorbates on metal surfaces. It was shown that for molecular adsorbates (i.e., in the case of not too strong adsorbate–surface interaction) the DN gives reasonable estimated trends of charge transfer if the work function is used for the electronegativity of metal surface. Despite the fact that the electronegativity of macroscopic metals is, by definition, given by the work function, its usage in the context of chemical reactivity indicators is not widely accepted in the literature and some other estimates are frequently used instead, some of them being—as shown in this paper—conceptually wrong. As for the reactive atomic adsorbates, such as N, O, and Cl, it is shown that the charge transfer is not solely given by the electronegativity difference between the metal surface and the adatom, but also by the elementary chemistry of the latter, namely, its valence. It is argued that in the case of a very strong interaction the electronegativity of metal can be described as a linear combination of atomic Mulliken electronegativity and the work function. The derived modification to DN equation is able to provide reasonable estimates of charge transfer for all the three considered adatoms on alkali, earth-alkali, transition, and noble metal surfaces. It is further shown that if a given adsorbate is considered on similar metals (say 3d transition metals) the strength of the adatom–metal bond is proportional to charge transfer from metal to adatom. The DN estimate is therefore a useful chemical reactivity indicator for anticipating or explaining the chemisorption bonding trends provided that chemically similar systems are considered.

10

A. Kokalj / Chemical Physics 393 (2012) 1–12

(b) Li 2@Mg(001) 0.12

0.10

0.08 Δ−n(z) (e/bohr)

Δ−n(z) (e/bohr)

(a) Cl 2@K(110) 0.15

0.05 0.00 −0.05 a

−0.10

0.04 0.00 −0.04 a’ a’’

−0.08

−0.15

−0.12 0

5

10 15 20

0

distance from surface layer (borh)

5

10

15

20

distance from surface layer (borh)

Fig. 9. Graphical representation of the method used to calculate the net molecular charge, qmol: (a) in simple cases qmol is calculated by integrating the planar charge density  ðzÞ in the blue region, Eq. (23). In cases like the one shown in (b), where the charge accumulates in the region in between the surface and the molecule (green difference Dn region), half of this interstitial charge is attributed to the molecule and other half to the metal, Eq. (24). The integration limits a, a0 , and a00 are also indicated. Dots on abscissa (and the resulting vertical lines) indicate the position of metal layers and the molecule.

Acknowledgement This work has been supported by the Slovenian Research Agency (Grants No. J1-2240 and P2-0148).

were kept fixed to ideal bulk geometry. Other computational parameters were compatible with those for molecular adsorption. Appendix B. Technical details B.1. Planar and macroscopic averages

Appendix A. Computational methods Calculations were performed within the framework of densityfunctional-theory (DFT) using a generalized gradient approximation (GGA) of Perdew–Burke–Ernzerhof [61]. Periodic systems (bulk metals and surfaces) were calculated with the pseudo-potential method using ultra-soft pseudo-potentials (US-PP) [77,78] and plane-wave basis set as implemented in the PWscf code from the Quantum ESPRESSO distribution [79]. Vertical ionization potentials and electron affinities were instead calculated using Gaussian09 program [80] and the local Gaussian-type-orbital 6– 311++G (d, p) basis set. Spin polarization was considered where appropriate. Surfaces were modeled by periodic slabs with the lattice spacings fixed at the calculated equilibrium bulk lattice parameters; for hcp metals, the a lattice parameterpwas while the ffiffiffi poptimized, ffiffiffi c/a ratio was kept at the ideal value of 2 2= 3. For the work function calculations, the densely packed fcc (111), bcc (110), and hcp (001) were described by slabs comprised of 19 layers, whereas more open fcc (100), bcc (100), fcc (110), and bcc (111) slabs were composed of 23, 27, 31, and 45 layers, respectively (the so chosen number of layers gives comparable thickness for slabs of various faces). The geometry of the slabs was kept fixed to ideal bulk geometry. The charge transfer between the molecules and metal surfaces was calculated with metal slabs composed of five atomic layers and utilizing big (4  4) supercells, except for Na, K, and Ca surfaces, which were described by (3  3) supercells due to their large lattice parameter. The molecular adsorbates were placed on the upper side of the slab relatively far from the surface and the molecular geometry was kept fixed to the optimized gas-phase geometry; the charge transfer was then calculated from the single point self consistent field calculation (see below). The thickness of the vacuum region—the distance between the top of the adsorbate and the bottom of the adjacent slab—was set to about 16 Å and the dipole correction of Bengtsson [81] was applied to cancel an artificial electric field that develops along the direction normal to the surface due to periodic boundary conditions imposed on the electrostatic potential. On the other hand, the adsorption of N, O, and Cl atoms was calculated utilizing the smaller (3  3) supercells. The structures were fully relaxed, except the two bottom layers of five-layer metal slabs

Several of presented quantities rely on the macroscopic averaging technique [82]. The planar averaged potential is defined as:

v ðzÞ ¼

1 A

Z

v ðx; y; zÞ dx dy;

ð19Þ

A

where z is the surface normal direction and A is the area spanned by the surface cell. The macroscopic averaged potential is defined as:

v ðzÞ ¼

1 d

Z

d=2

v ðz þ z0 Þ dz0 ;

ð20Þ

d=2

where the interlayer spacing is usually taken for d. However especially for slab calculations of alkali-metals, a multiple of interlayer spacing should be taken for d due to Friedel oscillations, i.e., d should be an approximate common denominator of the two. To simplify the notation the macroscopic averaged electrostatic  es ðzÞ is designated by V(z), while exchange-correlation potential v  xc by Vxc. Let us mention that the electrostatic potential potential v V(z) curves in Figs. 1 and 4 are drawn using the macroscopic averaging technique. B.2. Evaluation of work functions From a slab calculation the work function is straightforwardly calculated as the difference between the electrostatic potential in the vacuum region and the Fermi energy. However, the so calculated work function is prone to quantum-size effects and shows oscillation on the order of 0.1 eV as a function of slab thickness. While the Fermi energy depends sensitively on the slab thickness, the size of the dipole barrier is affected to a much smaller extent. Fall et al. [83] therefore suggested to calculate the work function from the combination of slab and bulk calculation as:

U ¼ DV slab  lbulk hvi ;

ð21Þ

where the superscripts indicate that the DV is taken from slab calculation, while the lhvi from the bulk calculation. Both quantities are obtained by utilizing the macroscopic averaging technique. Technically, the lbulk hvi is calculated from the bulk calculation as the difference between the Fermi energy and the mean electrostatic potential potential in the bulk, lbulk hvi ¼ eF  VðbulkÞ. The work functions reported in Table 1 are calculated using this method.

11

A. Kokalj / Chemical Physics 393 (2012) 1–12

−0.30

precise amount of the transferred charge depends on the molecule-surface distance, this dependence is weak (see Fig. 10) and does not affect any of the conclusions. Despite the large molecule-surface distance, the separation of charge between a molecule and metal surface is not always clear, because in some cases the electron charge is transferred from the surface and from the molecule toward the interstitial region in between the two (see Fig. 9b). In such cases, half of the interstitial electron charge, qinterstitial, is attributed to the molecule and other half to the metal, i.e.:

−0.40 3.4

qinterstitial ¼ A

0.20

Cl2@K(110) Cl2@Mg(001)

0.10

Cl2@Al(111)

ΔNDFT

0.00

Cl2@Cu(111) Li2@K(110)

−0.10

Li2@Cu(111)

−0.20

3.6 3.8 4.0 4.2 4.4 distance from surface (Å)

4.6

Fig. 10. Electron charge transfer, DNDFT, from molecule to the metal surface as a function of the molecule’s height above the surface.

B.3. Evaluation of electron charge transfer Decomposing or attributing electron charge to individual fragments (e.g. atoms or group of atoms) is not an easy task, because there is no quantum–mechanical operator for it. There are many schemes and all of them suffer from arbitrariness; perhaps the most versatile and the least arbitrary (if at all) is the Bader charge analysis [84–86], which was used to calculate the charge on atomic adsorbates (N, O, and Cl), because these were calculated with smaller supercells. Corresponding calculations were performed with the PAW (projector-augmented-wave) potentials [87] and 1000 Ry kinetic energy cutoff for the charge density.4 The corresponding calculations are computationally quite demanding and at the limit of our available computational resources. However, molecular adsobates were calculated with bigger supercells, which made Bader analysis currently too expensive. Instead, the charge on the molecule (or metal-slab) was calculated by utilizing the following planar averaged charge density difference scheme. First, the electron charge density difference, Dn(r), is calculated as:

DnðrÞ ¼ nmol=surf ðrÞ  nmol ðrÞ  nsurf ðrÞ;

ð22Þ

where the subscripts mol/surf, surf, and mol stand for the moleculesurface system, bare surface, and isolated molecule, respectively (the geometry is kept compatible in all the three cases). Then the  ðzÞ is calculated, cf. Eq. (19). Finally, the electron planar averaged Dn charge on the molecule, qmol, and metal surface, qmetal, is calculated as:

qmol ¼ A

Z

qmetal ¼ A

b

 ðzÞ dz; Dn

ð23aÞ

 ðzÞ dz; Dn

ð23bÞ

where a is the point in-between the metal surface and the molecule  ðzÞ changes sign (see Fig. 9a for graphical representawhere the Dn tion), b is located somewhere in the vacuum region where the charge density is zero, and C is the size of the simulation cell in the z direction. The electron charge transfer from molecule to metal surface is then given by DN = qmol = qmetal. In must be emphasized that this technique works well when the molecule is kept far enough from the surface, hence the molecules were placed from 3 to 4.5 Å above the surface; in most cases 3.5 Å. Note that charge transfer between the molecule and the surface takes place already at large molecule-surface distances; while the 4 The full scheme of these calculations involves performing a structural relaxation using the US-PP method followed by a self-consistent-field single point PAW calculation of the US-PP optimized structure, and finally computing the Bader adatom charge using the bader program [86].

ð24aÞ

b

ð24bÞ

a00

ð24cÞ

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

a

bC

ðzÞ dz; Dn

a0

where the integration limits a0 and a00 are shown graphically in Fig. 9b. Although this half-half splitting of interstitial charge is arbitrary, an analogous half-half recipe is used also in the highly popular Mulliken population analysis.

a

Z

a00

1  ðzÞ dz þ qinterstitial ; Dn 2 Z a0 1  ðzÞ dz þ qinterstitial ; ¼A Dn 2 bC

qmol ¼ A qmetal

Z

Z

[26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46]

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