N.H. March and A. Rubio: Equilibrium Bond Lengths in Lattices of Metal Atoms
311
phys. stat. sol. (b) 207, 311 (1998)
Subject classification: 61.50.Lt; S2; S4
Equilibrium Bond Lengths in Monoatomic Lattices
of Metal Atoms as Function of Coordination Number
N.H. March and A. Rubio
Departamento de F{sica TeoÂrica, Universidad de Valladolid, E-47011 Valladolid, Spain
(Received January 8, 1998)
A recent work of Schmidt and Springborg on one-dimensional chains of metal atoms has
prompted us to use a quantum-chemical model, posed in terms of dimer potential energy curves,
to analyze equlibrium near-neighbour distances r0 as a function of local coordination number c.
For five monoatomic metal atoms, H, K, Al, Pb and Bi, values of r0 are available for three or
more different lattices. The increase of r0 from the dimer value correlates with c1=2 in the same
general manner for four of the five atoms considered, Al being the exceptional case according to
present theoretical calculations.
1. Introduction: Quantum-Chemical Model
Recently [1] we have been concerned with K metal atoms in lattices with different local
coordination numbers c, including one-dimensional chains with c ˆ 2. Our calculations
on the lattices of K metal atoms were carried out by the density functional theory
(DFT), using a local density approximation for the exchange and correlation potential [2].
It was shown in [1] that the gist of the DFT results would be interpreted in terms of a
quantum-chemical model [3] characterized by the K dimer potential energy curves. The
model was succesfully applied to the description of the cohesive energy curves of alkali
metals in different geometrical structures and coordinations (in good agreement with
the ab-initio calculations and experiment). In this simple quantum-chemical model, the
ground-state energy per atom E…r0 ; c† as a function of near-neighbour distance r0 and
coordination number c is the sum of two factorizable terms having the form [1,3]
E…r0 ; c† ˆ 12 cR…r0 † ÿ f …c† g…r0 † :
…1†
For the alkali metals considered in [1], R…r0 † is given by the triplet 3 u potential energy
curve of the free-space dimer, while g…r0 † is the `exchange' part, which is half of the
difference between the triplet 3 u and the singlet 1 g potential energy curves taking as
zero of energy the two atoms infinitely separated.
Evidently the equilibrium value of r0, say r0e, for a given lattice, is determined by
‰@E…r0 ; c†=@r0 Šr0 ˆr0e ˆ 0, while yields directly
1 0
0
…2†
2 cR …r0 † ÿ f …c† g …r0 † r0e ˆ 0 :
Regrouping the terms in eqn. (2) by using the above definition of g…r0 †,
g…r0 † ˆ
R…r0 † ÿ 1 g …r0 †
;
2
…3†
312
N.H. March and A. Rubio
then yields, in an obvious notation, the following equation for the equilibrium distance
in terms of properties of the corresponding dimer:
‰c ÿ f …c†Š R0 …r0e † ÿ f …c† 1 0g …r0e † ˆ 0 :
…4†
2. Discussion
We wish, on the basis initially of the K lattices studied in [1], to note the values of
‰c ÿ f …c†Š appearing in the first term of eqn. (4) which are collected for convenience in
Table 1. What is remarkable to us is that ‰c ÿ f …c†Š has decreased by more than one
order of magnitude in going from the face-centred cubic lattice to the one-dimensional
chain. Evidently, the free-space dimer equilibrium distance rd is given by the condition
1
0g …rd † ˆ 0 ;
…5†
and the smallness of ‰c ÿ f …c†Š for low c in eqn. (4) suggests that the nearest approach
of r0e should come from the lowest coordination structure in the metallic K lattices, i.e.
the one-dimensional chain with c ˆ 2 (linear or zig-zag chains). This is generally the
case for covalent bonded systems due to the presence of a directional bond; however,
this rule seems to hold even for the case of delocalized bonding as in normal metals.
To test the generality of this suggestive argument we show in Fig. 1 along with our
own K data [1], supplemented with the experimental free space K2 dimer bond length,
the data for Bi and Pb from [4]. Though the Bi data are the most limited the trends of
equilibrium bond length versus c1=2 for these metal atoms show considerable similarity.
The idea behind this plot is the dependence with coordination of the energy per bond
in the so-called ªglueº models of solid-state physics, see e.g. [5]1 † and [6] that deals with
metallic cohesion as arising from embedding ions in a free electron gas. Furthermore,
expanding the equilibrium distance r0e around the dimer bond length, that is
r0e ˆ rd ‡ d, we can estimate from eqn. (4) the correction d as
0
c
R …rd †
dˆ
ÿ1
;
…6†
f …c†
}…rd †
Ta b l e 1
Values of universal function f …c† and ‰c ÿ f …c†Š used to describe the K lattices in [1]. They
are needed in the evaluation of the quantum chemical model of eqn. (1) for different
metals. All the other ingredients are obtained through the corresponding singlet and
triplet dimer potential energy curves. For completeness we give the ratio f …c† c1=2 as
would stem from the glue models (see text)
lattice
c
f …c†
‰c ÿ f …c†Š
f …c†=c1=2
f.c.c.
b.c.c.
s.c.
diamond
linear chain
dimer
12
8
6
4
2
1
4
4
3
2.9
1.3
1
8
4
3
1.1
0.7
0
1.2
1.4
1.2
1.4
0.9
1
1
† For this embedded atom potential the pair-wise potential (related to R…r0 †) and many-body
embedding function (related to g…r0 †) are usually fitted to experimental data.
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Equilibrium Bond Lengths in Monoatomic Lattices of Metal Atoms
Fig. 1. Following the quantum chemical model in eqn. (1) we plot the bond lengths (in at. units)
for different structures and coordinations as a function of the square-root of the coordination number c. The model is able to reproduce the trends of all the metals except of the anomalous behaviour of the aluminum dimer (see text)
where the correction is positive for most dimers. Also, in first order we have f …c† c1=2
which leads to the c1=2 behaviour of the bond length from ªglueº models and indicates
an empirical correlation between the change in bond length and the universal function
f …c†. The deviation from the c1=2 is related to the inclusion of long-range interaction in
the model by the fitting of the universal function to the ab-initio data [1].
To further test this model, it therefore seemed of interest to supplement these data
with results for Al and also, more speculatively, for hydrogen. In the latter case, the
available data of the kind we require go back to the works [7,8]. We summarize their
findings which are relevant in the present context in Table 2 and Fig. 2 (where the
Ta b l e 2
Theoretical results of H metal atom lattice after Min et al. [7]. The data for the diamond
structure are taken from [8]. The results are plotted in Fig. 2 to illustrate the reduction
of bond length with coordination as predicted by the simple quantum-chemical model
lattice
equilibrium Wigner-Seitz
radius (at. units)
equilibrium near-neighbour
distance r0e (at. units)
f.c.c.
b.c.c.
s.c.
diamond
dimer
1.683
1.677
1.707
1.777
±±
3.04
2.95
2.75
2.45
1.40
314
N.H. March and A. Rubio
Fig. 2. Plots of the cohesive energy Ec …r0 ; c† for H atoms with long-range order in b.c.c., diamond,
f.c.c., simple cubic and linear chain structures (coordinations ranging from c ˆ 2 to c ˆ 12) from
the quantum-chemical model in eqn. (1) as a function of the near-neighbour bond length (see inset
for structure labeling). The experimental dimer bond length is indicated by the vertical arrow. The
singlet and triplet potential curves entering in eqn. (1) are taken from ‰13Š. We find the diamond
structure as the most stable structure for a given range of external pressures (measured as changes
in bond lengths) in agreement with quantum Monte Carlo calculations ‰14Š. Note that in the quantum-chemical curves we have not included effects related to the zero-point motion. However, when
considering the relative stability of molecular versus atomic phases of hydrogen, these effects are
relevant
potential energy curves from the chemical model are given for different monoatomic
structures with coordination ranging from 2 to 12). For Al, Robertson et al. [6] have
calculated E…r0 ; c† at the r0e for the f.c.c. lattice for different coordinations and structures, and all their results fit well into the general form of eqn. (1). This has encouraged us to add the results for Al to Fig. 1 (that is, the experimental dimer bond length
and theoretical DFT bond-length calculations for the linear chain [9] and three-dimensional bulk structures).
Before summarizing, we want to mention the initial motivation for the studies of
quasi-one-dimensional systems by Schmidt and Springborg [4] and for our own alkali
metal investigation [1]. In [4] it was noted that the channels in zeolite constrained Pb/Bi
to almost linear chains (these one-dimensional structures are very soft and their geometry can be easily influenced by weak external perturbations such as those of the zeolite
walls or surfaces), while in [1] the motivation was provided by neutron scattering studies of fluid metal Cs along the liquid±vapour coexistence curve towards the critical
point by Hensel and coworkers [10,11]. Their finding was that the major reason for the
lowering of density towards the critical point was the lowering of coordination number
c, the bond length r0 increasing by only tenths of an A. At a given bond length, and
Equilibrium Bond Lengths in Monoatomic Lattices of Metal Atoms
315
sufficiently low density, a regime occurs in which modest zig-zag behavior is found to
stabilize the original linear chains, and therefore these are potential candidates for the
low coordination regimes both in the liquid phase and on semiconductor surfaces.
Though our arguments involving eqns. (1) and (2) were designed for metal lattices, the
theory leading to eqn. (1), discussed for example in [3], involved the assumption of
near-neighbour interactions only. All effects beyond these near-neighbour interactions
stem from the universal function f …c† as it is fitted to ab-initio data [1].
3. Concluding Remarks
In summary, the work of Ref. [4] on Pb and Bi, when combined with our own K
studies, has motivated us to prepare Fig. 1 which shows equilibrium bond lengths as a
function of the square root of coordination number. While the choice of c1=2 rather
than c is inessential, we had in mind glue models of interatomic forces [6] in using
this particular independent variable. The addition of H results to Fig. 1 is worthy of
independent comment. It was on H that Poshusta and Klein [12] first suggested the
use of dimer potential energy curves. Therefore, it seemed of obvious interest to construct in Fig. 2 the energy curves E…r0 ; c† for H, from the ab-inito computed dimer
potential energy curves [13]. In accordance with eqn. (4) and Table 1, it is immediately clear that it is for diamond and for the H chain that the free-space dimer bond
length is being approached most closely. As noted in the earlier quantum Monte
Carlo study of Natoli et al. [14], the diamond lattice has the lowest energy of the
structures considered, by a substantial margin, as in our quantum-chemical results in
Fig. 2. Nevertheless, under the conditions of Fig. 2, the molecular solid is still the
ground state (see also [15]), and monoatomic hydrogen solid should appear as a high
pressure phase. It is worthy of note that in the atomic region, solid hydrogen tends
towards low-symmetry, low-coordination structures; however, this tendency is balanced
by the zero point energy which favors high coordination and symmetry. The quantum-chemical model is able to give qualitatively the correct picture for the atomic
phases of hydrogen, but quantitatively it works less well for hydrogen than for the
alkali metals [1].
The final comments lead us back to Fig. 1. There, Al appears anomalous, in that,
with available theoretical data from density functional theory, the linear chain equilibrium near-neighbour distance lies below the dimer experimental bond length. At
present, we consider that this behaviour may be due to using the experiment for the
free-space dimer, and DFT plus a local density approximation for the chain. However, we must note the anomalous behaviour of aluminum dimer as regards both
bond lengths and optical polarizabilities. In the first case, the computed dimer bond
length depends strongly on the symmetry of the electronic state (5.1 and 4.67 at.
units for the 3 u and 3 ÿ
g states, respectively) with an average value larger than the
Al3 (4.65 at. units) and Al4 (4.75 at. units) bond lengths [16]. Furthermore, for all
simple metals aluminum is the only one in which the dimer polarizability per atom is
larger than the atomic one [17]. Thus, it could be expected that the simple quantumchemical model of eqn. (1) would not be adequate in this regime. However, with this
sole exception, the pattern of behaviour of the metal atoms in Fig. 1 is in accord with
the philosophy underlying the quantum-chemical model of the metal lattices subsumed in eqns. (1) to (4).
316
N.H. March and A. Rubio: Equilibrium Bond Lengths in Lattices of Metal Atoms
Acknowledgements A.R. acknowledges financial support from DGES (Grants PB950720 and PB95-0202) and Junta de Castilla y Leon (Grant VA72/96). One of us
(N.H.M.) performed most of his contribution to this study during a visit at the University of Valladolid in 1997. He wishes to thank Prof. J.A. Alonso and his colleagues for
the very stimulating environment they provided and for generous help. N.H.M. also
acknowledges partial financial support from the Leverhulme Trust, U.K. for work involving density functional theory.
References
[1] N.H. March and A. Rubio, Phys. Rev. B 56, 13865 (1997).
[2] D.M. Ceperley and B.J. Alder, Phys. Rev. Lett. 45, 1196 (1980).
J.P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981).
[3] N.H. March, M.P. Tosi, and D.J. Klein, Phys. Rev. B 52, 9115 (1995).
[4] K. Schmidt and M. Springborg, Solid State Commun. 104, 413 (1997).
[5] M.W. Finnis and J.E. Sinclair, Phil. Mag. A50, 45 (1984).
[6] I.J. Robertson, V. Heine, and M.C. Payne, Phys. Rev. Lett. 70, 1944 (1993).
[7] N.I. Min, H.J.F. Jansen, and A.J. Freeman, Phys. Rev. B 30, 5076 (1984); 33, 6383 (1986).
[8] T.W. Barbee III and M.L. Cohen, Phys. Rev. B 44, 11563 (1991).
T.W. Barbee III, A. Garca, M.L. Cohen, and J.L. Martins, Phys. Rev. Lett. 62 1150 (1989).
[9] A. Rubio, Y. Miyamoto, X. Blase, M.L. Cohen, and S.G. Louie, Phys. Rev. B 53, 4023 (1996).
[10] R. Winter, F. Hensel, T. Bodensteiner, and R. GlaÈser, Ber. Bunsenges. Phys. Chem. 91, 1327
(1987); J. Phys. Chem. 92, 7171 (1988).
[11] R. Winter and F. Hensel, Phys. Chem. Liquids 20, 1 (1989).
[12] R.D. Poshusta and D.J. Klein, Phys. Rev. Lett. 48, 1555 (1982).
[13] W. Kolos and L. Wolniewicz, Chem. Phys. Lett. 24, 457 (1974); J. Chem. Phys. 43, 2429
(1965).
[14] V. Natoli, R.M. Martin, and D. Ceperley, Phys. Rev. Lett. 70, 1952 (1993); Phys. Rev. Lett.
74, 1601 (1995).
[15] M. Amato and N.H. March, Laser Part. Beams (UK) 14, 685 (1996).
[16] R.O. Jones, Phys. Rev. Lett. 67, 224 (1991).
[17] W.A. de Heer, Rev. Mod. Phys. 65, 611 (1993).