Download Aleksander Herman

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Toward universal embedded-atom method - empirically adjusted and
consistent set of atomic densities for all elements of the periodic table
Aleksander Herman
Department of Chemistry, Gdańsk University of Technology, Gabriela Narutowicza
11/12, 80-952 Gdańsk, Poland, e-mail: [email protected]
Abstract
The Hartree-Fock-Slater model of atom has been modified by using individual values of
the exchange parameter ex for each atom. Each value of ex was adjusted to reproduce
the empirical value of the first ionization energy of the atom considered. The
expectation values of energies and radial functions for all elements of the periodic table
have been evaluated on the basis of the Hartree-Fock-Slater model and individual
exchange parameters. Qualitatively, the expectation values compare well with Mann's
numerical Hartree-Fock values but contain some influence of correlation and relativistic
phenomena. A consistent set of universal atomic electron density tables for all elements
of the periodic table, suitable for embedded-atom method (EAM) type calculations, is
presented. We considered the electron density distribution as the key variable linking the
total energy and inter-particle separation in the EAM molecular dynamics study. The
universal set of atomic densities have been tested using the XMD molecular dynamics
package.
* Presented as poster at Nano and Giga Challenges in Microelectronics 2004 Conference
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Introduction
It is very important to consider the physical properties of materials and nanosystems on the every atom counts level. In principle, we can perform pseudopotential
DFT calculations for such systems. In many cases, however, the complexity of the
system demands more approximate methods. A simpler approach to every atom counts
level, which has been used around for many years, is the pair potential model. However,
the pair potential scheme omits the crucial many-body effects, and therefore falls many
times as reliable tool. The past two decades saw the development of the embedded-atom
method (EAM) [1], which incorporates an approximation to the many-atom interactions
neglected by the pair-potential scheme. The advent of the EAM allowed simulation of a
very large set of interesting problems. The main physical property incorporated in the
EAM is the moderation of bond strength by other bonds (coordination-dependent bond
strength).
The basic equation of the EAM is
E   Fi (  (r ij)) 
a
i
j i
j
1
  ij ( )
2 i , j (i  j ) r ij
(1)
F i (  (r ij)) is the energy required
a
where E is the total internal energy of system,
j i
j
to embed atom i into an environment with an electron density
  (r ) .
a
j i
spherically averaged atomic electron density
j
ij
The
 (r ) is the contribution from an atom j
a
j
ij
to the electron density at an atom i as function of the distance between the two atoms rij.
 ij (r ij) is the two-body, core to core, electrostatic potential between atoms i and j.
The EAM provides very useful and robust means of calculating approximate
structure and energy of large systems. Although the EAM form is calculationally
convenient, it is often conceptually useful to visualize effective N-atom interactions.
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The two physical pictures (the first in terms of embedding energies, the second in terms
of explicit N-atom interactions) are unified under the EAM. On one hand, the
embedding picture (within the approximations made by the EAM) provides a compact
scheme for computing energies. On the other hand, the N-atom interaction picture
provides a more direct way to visualize interactions. The two pictures are equivalent in
the EAM. It is generally acknowledged that the EAM calculations can be roughly
divided into two categories. The first category generally relates to modeling a particular
experimental property of a large system [1]. The second one category focuses on
providing a conceptual framework for considering the organization of the experimental
knowledge (understanding) [1]. Although both categories are based on the EAM
concept, they are notably different in terms of definition of EAM potentials
representation on an atomic basis. The choice of approach is generally a question of
style and interpretation desired.
In general there are (5n + n2)/2 EAM functions for n atom types present. n –
electron density functions (one for each atom type), n – embedding functions (again one
for each type), and n(n+1)/2 pair functions. If for example you have 2 atoms types in the
considered system, then there will be 7 EAM functions: two electron density functions,
two embedding functions, and three pair functions.
During construction of the EAM potentials using the first approach, the problem
of potentials is treated very formally, avoiding any reference to their physical meaning
in some cases. We suppose that some or all quantities like density, embedding and pair
functions, which enter the EAM method are assumed empirical functions. Then their
optimal values can be determined from the condition that calculated molecular
properties agree either with the corresponding experimental data or with the results of
accurate calculations. In order to ensure the generality of such functions, we should
select them for a series of distinct systems and for various physical properties [1]. For
that reason the method is sometimes called the “separated parameterization for separate
problem approach”.
In the second approach the electron density, embedding, and/or pair interaction
functions are obtained from first principles [1]. In this approach, one should pay
attention to the physical meaning of the fitted functions. The firm foundation of such
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philosophy was given by Rose et al. [2] in 1984 under the name “universal bindingenergy curve”. That approach use realistic atomic data (electron density distribution) as
the variable linking the total energy and inter-particle separation. It means that the
method does not give all results accurately, but a universal set of the EAM potentials
can be used without suspicion that it is wrong for a particular purpose [3].
The primary goal of this article is generation of an empirically adjusted and
consistent set of electron density functions for all elements of the Periodic Table, in the
spirit of the second approach. In prior EAM calculations, the atomic electron densities
have been assumed to be well represented by the spherically averaged free-atom
densities calculated from the Hartree-Fock theory by Clementi and Roetti [4] and
McClean and McClean [5]. The electron density functions can also be obtained
numerically by solution of the Hartree-Fock (HF) equations [6, 7]. A limiting factor of
existing sources of electron density functions for the EAM interaction analysis remains,
in that they are not consistent and not accurate enough, even when compared with
known experimental atomic properties. To overcome this, we suggest the use of
empirically adjusted Hartree-Fock-Slater model of the atom [8,9] as the source of
accurate data for estimation of electron density functions. The relationship between the
Hartree-Fock and the Hartree-Fock-Slater (HFS) description of atoms is well described,
in the Herman and Skillman’s original book [10].
Evaluation of individual values of statistical exchange parameters αex for atoms
The Hartree-Fock-Slater model of atom entails the self-consistent solution of
(Eqns 2-5)
[- ½ 12 + Veff ] i (1) = i i (1)
(2)
Veff = - Z/r1 + ∫(2)/r12 d2 - 3ex (3/8)1/3 1/3
(3)
(1) = i ni i (1)2
(4)
where
and
The ground-state energy is then determined from the equation
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E [] = i ni i - J [] + Kex []
(5)
Our approach differs from the common method in that, the optimal exchange
parameter αex was estimated for each atom in the periodic table [11,12]. Originally, this
idea was introduced by Slater and Johnson [13]. In section IV of their paper, they have
suggested a criterion for determining a value of the αex parameter. The suggested
scheme was to determine the αex parameter in an atomic calculation and use the same
values in molecular or solid-state calculations. Next, Schwarz have examined two
criteria for determining the exchange parameter αex [14, 15]. These criteria are (i)
adjustment of the statistical total energy to the Hartree-Fock total energy, leading to αHF,
and (ii) satisfaction of the virial theorem, leading to αvt. Numerical calculations of
Schwarz show that, individual αHF and αvt values are almost the same [14, 15] and differ
by not more than 0.001. But the HFS model of atom used with αex values obtained
according to procedure (i) or (ii) has the same disadvantage as the pure HF model [6, 7],
the correlation and relativistic phenomena are not included in such model. It is
consistent but not empirically adjusted. For empirical adjustment, we need an atomic
property, which is known for all atoms in the periodic table. According to the NIST
Standard Reference Data Program [16] only the first ionization energy meets this
constraint, except for astatine. In this work, for the empirical adjustment, the individual
value of the exchange parameter αex was fixed at the value, which gives exact
reproduction of the experimental first ionization energy of the considered atom
according to Koopman’s theorem. In that way, some correlation (very low individual αex
value of the exchange parameter for Li) and relativistic phenomena (very high
individual αex values for Pt, Au, and Hg) are partially included [12].
In the following, we shall only give the relevant computational details for this
particular case. The Hartree-Fock-Slater calculations for atomic systems have been
performed operating the free-electron density approximation program (FDA) [17]. For
comparison of the numerical results, the QCMP102 [18] package has been used. For
evaluation of the individual exchange parameters of the elements, accurate ionization
energy data have been used [16]. The original free electron based universal ex
parameter, introduced by Slater equals 2/3 and yields acceptable results but sometimes
other values are used [10]. Therefore, we have modified the ex parameter individually
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for each atom, to achieve equality of the calculated and experimentally observed first
ionization energy. The results of calculations for the whole Periodic Table are shown in
ref. [12]. The mean value of ex is 0.756 with standard deviation 0.081, the lowest
observed value for Lithium is 0.540 and highest for Gold is 0.956.
Since our goal is the construction of an empirically adjusted and consistent set of
electron density functions for all elements of the periodic table in the spirit of the second
approach above, it may be of particular importance to examine the computed properties
of atoms in comparison with accurate Hartree-Fock results [6, 7]. The questions are how
our total binding energy values compare with results of accurate Hartree-Fock
calculations, and what happens with radial expectation values when we introduced
individual set of ex parameters [12]. The total binding energy values calculated by our
modified method are slightly larger (atoms are more stable) than the Hartree-Fock
values, the modified HFS values are ~ 0.3 % higher than the HF values. The <r>
expectation values calculated by our method are generally equal to or lower than the HF
values by about 6.2 % (atoms are more compact). Both effects can be explained due to
partial inclusion of electron correlation and relativistic phenomena achieved by
individual adjustment of ex. As is seen in [12], computed values of the virial ratio do
not exactly equal -2. The difference shows a kind of price, which we must pay for
empirical parameterization of the HFS model. The results of a basis set free
comparisons with HF results show that our modified HFS model of atomic structure
includes some correlation and relativistic phenomena. The modified HFS model can be
treated as a source of atomic data, for construction of an empirically adjusted and
consistent set of electron density functions for all elements of the Periodic Table.
Evaluation of empirically adjusted and consistent set of electron density
distribution functions for EAM calculations
Routine applications of coordination dependent interactions normally use
empirical potentials, usually within the EAM formalism. These do not require extensive
or highly accurate atomic potential functions. However, a consistent set within and
between members of a series is most desirable. Each numerical electron density function
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obtained by the HF or HFS theory displays oscillatory and exponential falloff
behavior[19]. This behavior, as computed by the HFS theory with individual ex
parameters, is shown in Figures 1 ÷ 4 for all elements of the periodic table. In all cases
the electronic configurations of the elements were those given by the NIST Standard
Reference Data Program [16]. With the real information about electron density
distributions at our disposal, we can view the density profile of heavy, neutral atom as
being composed of seven regions predicted by Lieb [19].
1) The inner core.
2) The core (almost all electrons are in this region).
3) The core mantle.
4) A transition region to the outer shell (this region may or may not exist).
5) The outer shell.
6) The surface (chemistry takes place here).
7) The region of exponential falloff (ρ(r)~K*exp[-2(2mε/ħ2)1/2(r-R)], where ε is the
ionization potential, K is the density at surface, and R is the surface radius) [19].
An upper bound for ρ(r) in the exponential falloff region has been proved by many
people, of whom the first was O’Connor [20]. See also Handy at al. [21], Morrell at al.
[22], Deift at al. [23], Hoffmann-Ostenhof at al. [24] and Silverstone [25] for
interesting discussion of electron density long-range behavior. The result mentioned
above, is crucial for the present work. It shows us why the individual adjustment of αex
gives simultaneous improvement in ionization potential and electron density falloff
outside the surface region of an atom. Moreover, it explains why electronic densities in
this region are quite well approximated by a single exponential term (6) in the range of
distances of interest in the EAM calculations [1].
 /
e
 exp[   (r l / r le  1)]
(6)
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In the formula (6)

is atomic electron density at distance
electron density at the equilibrium distance
r
le
r
l
,

e
is the atomic
. Parameter values of the equation (6) are
given for the whole periodic system in Table I. For consistency reasons, values of
equilibrium distances
r
le
of E2 molecules are taken from the periodic table of diatomic
molecules of Boldyrev and Simons [26]. Unknown values have been estimated from the
atomic bond radii as in the UFF MM model [27].
Insert Table I here!
Qualitatively, the parameters of equation (6) compare well with ones calculated on
the basis of the Mann’s numerical HF data [6, 7]. The differences are due to the
correlation and relativistic phenomena included by empirical estimation of the exchange
parameters ex for each atom under consideration. According to the HFS model
parameterization scheme, the computed ionization matches exactly experimental
ionization energy of the considered atom and gives the best total electron density
distribution. These new electronic density distributions are particularly useful for the
EAM atomic interactions considerations according to the idea introduced by Daw and
Baskes [1]. To date, no complete consistent and empirically adjusted set of atomic
densities is available. Together with the free software available [28], the proposed set of
atomic densities promises, that the vast majority of nano-systems may be qualitatively
understood by judicious use of the very simple EAM interaction calculations.
In practice most of the EAM potentials, from any source, are stored as text tables.
Such tables of atomic densities, in XMD molecular dynamics program format [28], have
been generated for all elements of the periodic table and can be obtained from the author
on request.
Concluding remarks
In early EAM calculations, the density, embedding, and pair functions were
determined by a complex fitting procedure. Rose at al.[2] have shown that, for a broad
range of materials, the energy as function of nearest-neighbor distance is well
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approximated by universal scaling relation. The direct physical parameters required for
such EAM model are the equilibrium atomic volume, the cohesion energy, as well as the
lattice structure. A two-body potential and an electron density function must also be
specified to define the model completely. Obviously, empirically adjusted and consistent
set of atomic densities, in form of density tables presented above, can be treated as next
step toward an universal EAM method. The cohesive energy, bulk modulus, and
equilibrium condition are all automatically satisfied by the use of the function of Rose at
al. [2].
References
[1] Daw M S, Folies S M and Baskes M I, Mat. Sci.Rep. 9 251 (1993)
[2] Rose J H, Smith J R, Guinea F and Ferrante J, Phys. Rev B., 29 2963 (1984)
[3] Banerjea A and Smith J R, Phys. Rev B., 37 6632 (1988)
[4] Clementi E and Roetti C, At. Data Nucl. Data Tables 14 177 (1974)
[5] McLean A D and McLean R S, Data Nucl. Data Tables 26 197 (1981)
[6] Mann J B 1967 Atomic Structure Calculations I. Hartree-Fock Energy Results for
the Elements Hydrogen to Lawrencium (Los Alamos - New Mexico: Los Alamos
Scientific Laboratory of the University of California; LA-3690; UC-34, PHYSICS;
TID-4500)
[7] Mann J B 1967 Atomic Structure Calculations II. Hartree-Fock Wavefunctions and
Radial Expectation Values: Hydrogen to Lawrencium (Los Alamos - New Mexico:
Los Alamos Scientific Laboratory of the University of California; LA-3691; UC-34,
PHYSICS; TID-4500)
[8] Slater J C, Phys. Rev., 36 57 (1930)
[9] Slater J C 1960 Quantum Theory of Atomic Structure (McGraw-Hill Co. Inc., New
York )
[10] Herman F and Skillman S 1963 Atomic Structure Calculations (Englewood Cliffs,
New Jersey: Prentice-Hall, Inc.)
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[11] Herman A 2001 Substantial Improvement of Hartree-Fock-Slater Part of any
Hybrid Functional at no Computational Cost (VIII-th Scientific Workshop of Polish
Computer Simulation Society, Gdańsk)
[12] Herman A, Modelling Simul. Mater. Sci. Eng., 12 21 (2004)
[13] Slater J C, and Johnson K H, Phys. Rev., 5 844 (1972)
[14] Schwarz K, Phys. Rev., 5 2466 (1972)
[15] Schwarz K, Theoret. Chim. Acta, 34 225 (1974)
[16] U.S. Department of Commerce, Technology Administration, National Institute of
Standards and Technology (http://www.nist.gov)
[17] Heisterberg D J 1995 FDA version 0.10 (Department of Chemistry, The Ohio State
University; e-mail: [email protected] or [email protected])
[18] Herman A 1992 The Hartree-Fock-Slater System of Programs for Qualitative
Atomic Structure Calculations on IBM PC and Compatible Systems QCPE Bulletin
12 24 (Herman and Skillman programs [13] converted to IBM PC)
[19] Lieb E H, Rev. Mod. Phys. 53 603 (1981)
[20] O’Connor A J, Commun. Math. Physc., 32 319 (1973)
[21] Handy N C, Marron M T and Silverstone H J, Phys. Rev., 180 45 (1969)
[22] Morrell M M, Parr R G and Levy M, J.Chem. Phys.,62 549 (1975)
[23] Deift P, Hunziker W, Simon B and Vock E, Commun. Math. Phys., 64 1 (1978)
[24] Hoffmann-Ostenhof M, Hoffmann-Ostenhof T, Ahlrichs R and Morgan J 1980, in
Mathematical Problems in Theoretical Physics, Proceedings of the International
Conference on Mathematical Physics, Lausanne, Switzerland. August 20-25, 1979.
(Springer-Verlag, Berlin, Heidelberg, New York)
[25] Silverstone H J, Phys. Rev. A, 23 1030 (1981)
[26] Boldyrev A I and Simons J 1997 Periodic Table of diatomic molecules (John Wiley
& Sons Ltd)
[27] Rappĕ A K, Casewit C J, Colwell K S, Goddard III W A and Skiff W M J. Am.
Chem. Soc. 114 10024 (1992)
[28] Rifkin J 2002 XMD – Molecular Dynamics Program, Center for Materials
Simulation,
University
of
Connecticut,
http://www.ims.uconn.edu/centers/simul/xmd/
Institute
of
Materials
Science;
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Acknowledgments
Financial support from CLUSTERIX (National CLUSTER of LInuX Systems)
project (No 6 T11 2003 C/06098 – tasks: 14, 15 ) is gratefully acknowledged. In
addition, I am very grateful to the staff of the CI TASK Computing Network in Gdańsk
for help and the possibility of using facilities.
Keywords
EAM potentials, periodic table of elements, RHF densities, HF densities, HFS
densities, exchange parameters, electron density distribution functions of atoms.
Figure captions
Fig. 1 The schematic plot of electron density for elements of IUPAC group 1 and 2 of
the periodic table (13 elements included). The densities have been calculated using
individual αex parameter for each atom. The inner core, the core, its mantle and the outer
shell are not always visible for light elements. The exponential falloff region is the only
visible region for hydrogen (the most left and upper line).
Fig. 2 Schematic plot of electron density for elements of IUPAC group 3 of the periodic
table including Lanthanides and Actinides (32 elements included). The densities have
been calculated using individual αex parameters for each atom. Electron density
distributions are very similar for these elements.
Fig. 3 Schematic plot of electron density for elements of IUPAC groups 4 ÷ 12 of the
periodic table (27 elements included). The densities have been calculated using
individual αex parameter for each atom.
Fig. 4 Schematic plot of electron density for elements of IUPAC groups 13 ÷ 18 of the
periodic table (31 elements included). The densities have been calculated using
individual αex parameter for each atom. The exponential falloff region is the only visible
region for helium (the most left and upper line). Electron density distributions are very
different for these elements.
Supplementary materials
EAM density files in XMD format
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Fig. 1
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Fig. 2
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Fig. 3
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Fig. 4
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Tab. 1