# the potential energy of single atom in a sphere crystal

Dear all users:

I constructed a sphere single crystal with radius 5nm and orientations x = [1 0 0], y = [0 1 0], z = [0 0 1]. After energy minimization of this crystal at 0 K (minimize 1.0e-50 1.0e-50 100000 1000000), I plotted the potential energy (ei) of each atom against its radial distance (ri) to the sphere center (please see the attached figure). I found that most of the atoms in the interior of the crystal (see ri < 4.0 nm) are of a constant energy -3.36 eV, the rest of the interior atoms are of energies varying from -3.36 to -3.3598 eV, and energies of atoms near the free surface (see ri > 4.6 nm) converge at several discrete values obviously higher than -3.36 eV. Obviously, -3.36 eV represents the energy of an atom in a perfect configuration (or arrangement) for this system.

However, some atoms with ri from 4.1 nm to 4.6 nm are observed to be of energies lower than -3.36 eV (i.e. -3.361 ~ -3.36 eV). To my understanding, this result is obviously unreasonable. It is because that the energy of atom in a perfect configuration is the lowest energy for atoms in any constructed system at 0 K, and any structural deviation from the perfect configuration will lead to a higher energy. I can not figure out which factor leads to this result.

Could anyone give me some comment on this issue?

L.Yang

pe_distribution.tif (181 KB)

Dear all users:

I constructed a sphere single crystal with radius 5nm and orientations x = [1 0 0], y = [0 1 0], z = [0 0 1]. After energy minimization of this crystal at 0 K (minimize 1.0e-50 1.0e-50 100000 1000000), I plotted the potential energy (ei) of each atom against its radial distance (ri) to the sphere center (please see the attached figure). I found that most of the atoms in the interior of the crystal (see ri < 4.0 nm) are of a constant energy -3.36 eV, the rest of the interior atoms are of energies varying from -3.36 to -3.3598 eV, and energies of atoms near the free surface (see ri > 4.6 nm) converge at several discrete values obviously higher than -3.36 eV. Obviously, -3.36 eV represents the energy of an atom in a perfect configuration (or arrangement) for this system.

However, some atoms with ri from 4.1 nm to 4.6 nm are observed to be of energies lower than -3.36 eV (i.e. -3.361 ~ -3.36 eV). To my understanding, this result is obviously unreasonable. It is because that the energy of atom in a perfect configuration is the lowest energy for atoms in any constructed system at 0 K, and any structural deviation from the perfect configuration will lead to a higher energy. I can not figure out which factor leads to this result.

Could anyone give me some comment on this issue?

actually, i think the problem is with your reasoning. when doing a
minimization, you minimize the energy of the *total* system, not that
of individual atoms.
thus the minimization routine may move your system to a state, where
some atoms have a higher (than average) energy and others a lower, if
the sum of these changes leads to a lower total energy. it won't
happen in the bulk part of a system due to (local) symmetry, but at
the surface you can have all kinds of reconstruction operations
happen. some of these may even be activated processes and thus not
necessarily being found by direct minimization of a system cut from
atoms on perfect crystal lattice positions.

axel.

Dear Axel,

According to your explanation, it is actually the minimization routine,
implemented in Lammps, that leads to parts of atoms near the surface
being of lower energies. So, if i construct a sphere bicrystal and minimize it,
then atoms far from the free surface but near the grain boundary may
also be of lower energies because the disordered structure of the boundary
can have different kinds of reconstruction operations during minimization.
But, these atoms should not exist in theory due to the energy-structure
relation mentioned in my previous mail ?

Sincerely, Yang

Dear Axel,

According to your explanation, it is actually the minimization routine,
implemented in Lammps, that leads to parts of atoms near the surface
being of lower energies.

the minimization algorithms in LAMMPS are not different from those in
other programs.

So, if i construct a sphere bicrystal and minimize it,
then atoms far from the free surface but near the grain boundary may
also be of lower energies because the disordered structure of the boundary
can have different kinds of reconstruction operations during minimization.
But, these atoms should not exist in theory due to the energy-structure
relation mentioned in my previous mail ?

sorry, but your "theory" makes no sense. i already explained why.

axel.

Dear Axel,

I tried to construct the same sphere single crystal but without the
minimization operation, and then output its energy. But i still found
that parts of atoms near the surface are of lower energies (see the attached figure),
and energies of these atoms are fixed. Without the minimization,
how could there still be lower energies ?

> According to your explanation, it is actually the minimization routine,
> implemented in Lammps, that leads to parts of atoms near the surface
> being of lower energies.

the minimization algorithms in LAMMPS are not different from those in
other programs.

So, the above issue is common in all molecular dynamic programs.

> So, if i construct a sphere bicrystal and minimize it,
> then atoms far from the free surface but near the grain boundary may
> also be of lower energies because the disordered structure of the boundary
> can have different kinds of reconstruction operations during minimization.
> But, these atoms should not exist in theory due to the energy-structure
> relation mentioned in my previous mail ?

sorry, but your "theory" makes no sense. i already explained why.

The "theory" here is aimed to represent the cases in real experimental material
(or condition) not in MD simulation, i.e. these atoms should not exist in real material.

Sincerely, Yang

Dear Axel,

I tried to construct the same sphere single crystal but without the
minimization operation, and then output its energy. But i still found
that parts of atoms near the surface are of lower energies (see the attached figure),
and energies of these atoms are fixed. Without the minimization,
how could there still be lower energies ?

> According to your explanation, it is actually the minimization routine,
> implemented in Lammps, that leads to parts of atoms near the surface
> being of lower energies.

the minimization algorithms in LAMMPS are not different from those in
other programs.

So, the above issue is common in all molecular dynamic programs.

this is a general property of multivariate optimization.

> So, if i construct a sphere bicrystal and minimize it,
> then atoms far from the free surface but near the grain boundary may
> also be of lower energies because the disordered structure of the boundary
> can have different kinds of reconstruction operations during minimization.
> But, these atoms should not exist in theory due to the energy-structure
> relation mentioned in my previous mail ?

sorry, but your "theory" makes no sense. i already explained why.

The "theory" here is aimed to represent the cases in real experimental material
(or condition) not in MD simulation, i.e. these atoms should not exist in real material.

this statement makes even less sense than what you have mentioned
before. your claim is wrong.
it doesn't apply to individual atoms, but only to the ensemble of
atoms, like i have been telling you before.
you are disregarding the fact that the potential energy attributed to
any single atom is not independent of other atoms and it can only be
computed for tuples of atoms and then be attributed by convention
(i.e. for every n-tupel considered in the potential energy function,
the energy is divided evenly across the n atom, with n typically being
2, 3, or 4).

axel.

Dear Axel,

Thank you for your further explanation.
Now i realize my mistake in analyzing the feature shown in the energy
distribution of the sphere crystal, and have a clear understanding
of the potential energy.

Thanks again.

Sincerely, Yang