From this study, future work is summarized as follows:
1. To confirm the three-dimensional FEDFT program that computes the helium-like and beryllium-like atoms successfully.
2. To compute the hydrogen molecule that is a two nuclei and two electrons system by the three-dimensional FEDFT program.
3. To compute the carbon atom that is a one nucleus and six electrons system and it has to consider the 2p orbit.
4. If the serial code performs correctly, we will parallelize it to improve the efficiency and to solve more complicated cases.
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APPENDIX A
One-Dimensional FEDFT
In this appendix, we provide the detail of the one-dimensional finite element form of the new DFT formulation from the governing equation to the system equation.
We will use the Galerkin weighted residual method [Burnett, 1987] throughout the research, unless otherwise specified.
The general governing equation is written as [Hsu, 2003]
( ) ( ) ( ) ( ) ( )
computing space, Z is the number of positive charge of the nucleus, the subscript I means the kind of nucleus and N is the number of electrons.
For the system of one nucleus that is the simple atom like hydrogen or helium atom with one or more electrons, Eq. (A.1) can be simplified as, by taking the
spherical symmetry,
( ) ( ) ( ) ( )
( )J where ΨJ is the density function of orbital J that is limited to two electrons with
spin polarization to satisfy the Pauli exclusion principle and r is the radial coordinate originating from the center of the nucleus.
We construct a trial solution to approximate the density function
( )
~( )
;α r U r ≈Ψ .
The typical 1-D element trial solution can always be written in the general form,
( ) ∑ ( )
The coefficients α1,α2,Kαn are undetermined parameters, frequently called degrees
of freedom (DOF). We would say that ~
( )
;α rU in Eq. (A.5) has n DOF. In the following theoretical development, the first three steps are short, formal operations using only the general form Eq. (A.5). It is not until step 4 that we decide on the value of n and the specific form of each of the shape functions Nj
( )
r .Step 1：Write the Galerkin residual equations for a typical element.
The residual for Eq. (A.2) is
We need one residual equation for each DOF in Eq. (A.5):
( ) ( )
r N r dv i n R ; i =0 =1,2,...,∫∫∫ α
(A.7) where the integration integrate over one element and n is the number of nodes in anelement. Substituting Eq. (A.5) into Eq. (A.7) yields
( )
Step 2：Integrate by parts.
The second derivative term is integrated by parts once:
n
The boundary term contains, as usual, the flux,
[ ( ) ]
nAs is characteristic of eigenproblems, there cannot be any loads. Thus the interior load is zero, and the boundary conditions must be zero. For the eigenproblems, the boundary term contains the flux must vanish from the system equations. The boundary term occurs in two different ways in the system equations: at each node on
the boundary of the domain, and as the difference of two such expressions at each node on inter-element boundaries. In the first case, the boundary conditions require that the term vanish at the domain boundary nodes. In the second case, a nonzero difference in flux at an inter-element boundary represents an applied concentrated load; however, the eigenproblems does not permit applied loads. Since the boundary term must vanish from the system equations, we will ignore it right at the outset by eliminating it from the element equations. Therefore Eq. (A.9) may be written as follows:
Step 3：Substitute the general form of the element trial solution into integrals in
residual equations. Inserting Eq. (A.5) into Eq. (A.11) yields
[ ]
These are the element equations for a typical element.
Eq. (A.12) may be written in conventional matrix form that is rearranged to satisfy the matrix solver as:
[ ] [ ]
Step 4：Develop specific expressions for the shape functions Nj
( )
rWe use the two-node linear element to be the 1-D typical element (Fig. 4.1). For
convenience, we repeat those results here:
( ) ∑ ( )
where the shape functions possess the requisite interpolation property,
( )
k j j k jkThis element is frequently referred to as a C0-linear element, in obvious reference to the continuity and completeness properties it process.
Step 5：Substitute the shape functions into the element equations, and transform
the integrals into a form appropriate for numerical evaluation.
Substitute Eq. (A.16) into Eq. (A.14) :
( ) ( ) ( ) ( )
of the integration, H , may expand the solutions in terms of the Legendre Jpolynomials yields
( ) ( )
This completes the six steps for deriving the element equations. After performing the addition operations, all elemental matrix equations are assembled to be system equations that are then solved by J-D matrix solver.
APPENDIX B
Three-Dimensional FEDFT
In this appendix, we provide the detail of the three-dimensional finite element form of the new DFT formulation from the governing equation to the system equation.
We will use the Galerkin weighted residual method [Burnett, 1987] throughout the research, unless otherwise specified.
The general governing equation is written as [Hsu, 2003]
( ) ( ) ( ) ( ) ( )
of the computing space, Z is the number of positive charge of the nucleus, the subscript I means the kind of nucleus and N is the number of electrons.
For the system of one nucleus with one or more electrons, Eq. (B1) can be
rewritten for density function of orbital J in three-dimensional form as,
( ) ( ) ( ) ( )
( )=0
where ΨJ is the density function of orbital J that is limited to two electrons with
spin polarization to satisfy the Pauli exclusion principle and r is the radial coordinate originating from the center of the nucleus.
We construct a trial solution to approximate the density function
( )
~( )
;α r U r ≈Ψ .
The typical 3-D element trial solution can always be written in the general form,
( ) ∑ ( )
In the current study, linear shape function for 3-D tetrahedral elementis used for 3-D program throughout the research, unless otherwise specified.
Step 1：Write the Galerkin residual equations for a typical element.
The residual for Eq. (B.2) is where the integration integrate over one element and n is the number of nodes in an
element. Substituting Eq. (B.5) into Eq. (B.7) yields
Step 2：Integrate by parts.
The second derivative term is integrated by parts once:
( ) ( ) ( ) ( ) ( ) ( )
As the same in Appendix A, the boundary term contains the flux must vanish from the system equations for the eigenproblems. We will ignore it right at the outset by eliminating it from the element equations. Therefore Eq. (B.9) may be written as
follows:
Step 3：Substitute the general form of the element trial solution into integrals in
residual equations. Inserting Eq. (B.5) into Eq. (B.10) yields
( ) ( ) ( ) ( ) ( ) ( )
These are the element equations for a typical element.
Eq. (B.11) may be written in conventional matrix form that is rearranged to
satisfy the matrix solver as:
[ ] [ ]
Step 4：Develop specific expressions for the shape functions Nj
( )
rWe use the linear tetrahedral element to be the 3-D typical element (Fig. 4.2).
The typical 3-D element trial solution can be written
( ) ∑ ( )
The special coordinates are introduced defined by (Fig. B.1) :
4
4 Solving Eq. (B.15) gives
c
the linear element are simply
2 .
2 1
1 L N L etc
N = = (B.16)
where Vc represents the volume of the tetrahedron.
4
Step 5：Substitute the shape functions into the element equations, and transform
the integrals into a form appropriate for numerical evaluation.
Substitute Eq. (B.16) into Eq. (B.13) :
( )
by Gauss Quadrature (GQ):
( ) ( )
where G is the total numbers of the Gauss point, wk is the weighting factor and the subscript k means that the related values on the Gauss point (Table B.1) [Zienkiewicz and Taylor, 2000]. We will use the quadratic of GQ throughout the research, unless otherwise specified.
( )
The nonlinear term, Π , in the Eq. (B.2) describes as Eq. (B.3). The integration, J
H , integrate by Gauss Quadrature. J
This completes the six steps for deriving the element equations. After performing the addition operations, all elemental matrix equations are assembled to be system equations that are then solved by J-D matrix solver.
Table 4.1 The radial domain data of all models for one-dimensional FEDFT in this research. Cutoff radius is in units of Bohr radius.
Atom cutoff radius nodes elements
H 20 20001 20000
He 10 10001 10000
Li+ 6 6001 6000
Be+2 5 5001 5000
B+3 4 4001 4000
C+4 3 3001 3000
Be 6 6001 6000
B+ 5 5001 5000
C+2 4 4001 4000
Table 4.2 The computational domain data of all models for three-dimensional FEDFT in this research. Cutoff radius is in units of Bohr radius.
Atom cutoff radius nodes elements
H 20 165150 796588
He 5 3668 14629
Table 4.3 Hydrogen atom. The energies of the electron obtained by one-dimensional FEDFT program compared with experiment. Energy is in units of eV.
Atom Z exp n=1 F n=2 F
H 1 13.6 13.60002372 3.400007348
Table 4.4 Helium-like atoms. The energies of the electron obtained by
one-dimensional FEDFT program compared with experimental ones and numerical ones obtained by Hsu. Energy is in units of eV.
Atom Z exp n=1 H n=2 H n=1 F n=2 F
He 2 79 78 19 77.10271484 19.46709836
Li+ 3 198 196 50 196.0677136 49.34028086
Be+2 4 371 370 93 369.4731368 92.84837742
B+3 5 600 598 150 597.2802483 149.9571332
C+4 6 882 880 220 879.4880575 220.6660783
Table 4.5 Beryllium-like atoms. The energies of the electron obtained by
one-dimensional FEDFT program compared with experimental ones and numerical ones obtained by Hsu. Energy is in units of eV.
Atom Z exp G H GF
Be 4 399 390 387.5745747
B+ 5 663 662 649.4520744
C+2 6 991 989 979.4498507
Table 4.6 Hydrogen atom. The energies of the electron obtained by
three-dimensional FEDFT program compared with experimental ones and numerical ones obtained by one-dimensional FEDFT. Energy is in units of eV.
Atom Z exp n=11d n=21d n=13d n=23d
H 1 13.6 13.60002372 3.400007348 13.6006394 3.399937991
Table 4.7 Helium atom. The energies of the electron obtained by three-dimensional FEDFT program compared with experimental ones and numerical ones obtained by Hsu and one-dimensional FEDFT. Energy is in units of eV.
Atom Z exp n=1 H n=11d n=13d
He 2 79 78 77.10271 78.34722
Table B.1 Numerical integration formula of Gauss Quadrature for tetrahedral element [Zienkiewicz and Taylor, 2000].
Fig. 1.1 Sketch of the multiscale and physical processes in a DC-magnetron sputtering chamber.
Fig. 1.2 The analysis of vapor deposition spans both a wide length and time scale.
Overlapping modeling methods are beginning to allow an increasingly rigorous multiscale treatment [Ohno et al., 1999; Olson, 1997].
Fig. 1.3 The publications about DFT [Friedrich].
Fig. 1.4 STOs & GTOs [Friedrich]
Fig. 2.1 The proof of Hohenberg & Kohn second theorem
Fig. 3.1 The flow chart of the FEDFT.
Fig. 3.2 The flow chart of the J-D solver.
Fig. 4.1 One-dimensional meshes with 4 elements and 5 nodes.
Fig. 4.2 The tetrahedral element.
Fig. 4.3 The surface meshes of three-dimensional computational domain with different view points. (r=10 Bohr radii, θ=30°, φ=30°)
Fig. 4.4 The probabilities of finding the electron in a hydrogen atom at a distance between r and r + dr from the nucleus for the 1s and 2s state obtained by
one-dimensional FEDFT compared with exact solutions.
Fig. 4.5 The probabilities of finding the electron in a He at a distance between r and r + dr from the nucleus for the 1s and 2s state obtained by one-dimensional FEDFT.
Fig. 4.6 The probabilities of finding the electron in a Li+ at a distance between r and r + dr from the nucleus for the 1s and 2s state obtained by one-dimensional FEDFT.
Fig. 4.7 The probabilities of finding the electron in a Be+2 at a distance between r and r + dr from the nucleus for the 1s and 2s state obtained by one-dimensional FEDFT.
Fig. 4.8 The probabilities of finding the electron in a B+3 at a distance between r and r + dr from the nucleus for the 1s and 2s state obtained by one-dimensional FEDFT.
Fig. 4.9 The probabilities of finding the electron in a C+4 at a distance between r and r + dr from the nucleus for the 1s and 2s state obtained by one-dimensional FEDFT.
Fig. 4.10 The probabilities of finding the electron in helium-like atoms at a distance between r and r + dr from the nucleus for the 1s state obtained by one-dimensional FEDFT.
Fig. 4.11 The probabilities of finding the electron in helium-like atoms at a distance between r and r + dr from the nucleus for the 2s state obtained by one-dimensional FEDFT.
Fig. 4.12 The probabilities of finding the electron in a Be at a distance between r and r + dr from the nucleus for the 1s and 2s state obtained by one-dimensional FEDFT.
Fig. 4.13 The probabilities of finding the electron in a B+ at a distance between r and r + dr from the nucleus for the 1s and 2s state obtained by one-dimensional FEDFT.
Fig. 4.14 The probabilities of finding the electron in a C+2 at a distance between r and r + dr from the nucleus for the 1s and 2s state obtained by one-dimensional FEDFT.
Fig. 4.15 The probabilities of finding the electron in beryllium-like atoms at a distance between r and r + dr from the nucleus for the 1s and 2s state obtained by one-dimensional FEDFT.
Fig. 4.16 Photographic representation of the electron probability-density distribution for 1s and 2s states. These may be regard as sectional views of the distribution in a plane containing the polar axis, which is vertical and in the plane of the paper [Arthur, 1995].
Fig. 4.17 The 1s state convergence of residual with iterations for hydrogen atom by one-dimensional and three-dimensional FEDFT.
Fig. 4.18 The 2s state convergence of residual with iterations for hydrogen atom by one-dimensional and three-dimensional FEDFT.
Fig. 4.19 The probabilities of finding the electron in a hydrogen atom at a distance between r and r + dr from the nucleus for the 1s and 2s state obtained by
three-dimensional FEDFT compared with exact solutions.
Fig. 4.20 The probability of finding the electron in a helium atom at a distance between r and r + dr from the nucleus for the 1s state obtained by three-dimensional FEDFT compared with one-dimensional FEDFT.
Fig. B.1 The coordinates of point P described by four edge nodes in the tetrahedral element [Zienkiewicz and Taylor, 2000].