The paper (J.Y.Hsu, PRL 2003) gives a new derivation of Density Functional Theory from a cluster expansion by truncating the higher-order correlations in one and only one term in the kinetic energy. The new formulation admits excited states and allows self-consistent calculation of the exchange correlation e¤ect without any ad hoc assumptions.
The wave function ª is chosen as the product of a single-electron wave function
©, and an N-body correlation function,
ª(¡) = ¦Ni=1©(ri)U(¡) (9)
¡ is the N-particle phase space point equivalent to the expression (r1; r2; :::; rn).
The exchange symmetry is imposed on U and on the indistinguishable particles so that each electron is described by the same ©. This gives the density function as follows:
n(r) =
Z
¦Ni=2d¿ij©(ri)j2jU(r1; r2; :::; rN)©(r)j2 ´ M(r)j©(r)j2 ´ jª0(r)j2; (10)
where The derivation of the density functional theory (DFT) from the cluster expansion corrects the spurious self-interaction energy in the classical DFT, admits the excited states, and has a self-consistent exchange correlation e¤ect.
Excited States (Singlet and Triplet States)
Triplet States Singlet States
For a particle j in an arbitrary orbital, the generalized equation is
"ªj(r) =¡1 ª0 constant to minimize the total energy, subject to particale conservation
N =
Z
d¿1d¿2jº(r1; r2)j2½(r1)½(r2)´
Z
d¿1d¿2jª(r1; r2)j2: (14) We reduce to the simpli…ed ¯ = 0 version
"ªs(¡!r1) =¡1
Example for excited Be(2 1S,1 2S,1 2P)
"ª2S(r1) =¡1
6 NUMERICAL METHODS 6.1 Finite Element Method
6.1.1 Introduction
The Finite Element Method is a numerical method which can be used for the accurate solution of complex engineering problems. The method was …rst devel-oped in 1956 for the analysis of aircraft structural problems. Thereafter, within a decade, the potentialities of the method for the solution of di¤erent types of applied science and engineering problems were recognized. Over the years, the
…nite element technique has been so well established that today it is considered to be one of the best methods for solving a wide variety of practical problems e¢ciently. In fact the method has become one of the active research areas for ap-plied mathematicians. One of the main reasons for the popularity of the method in di¤erent …elds of engineering is that once a general computer program is writ-ten, it can be used for the solution of any problem simply by changing the input data.
Many problems that …nd out its approximate numerical solution to predict the response of physical system subjected to the external in‡uences arise in many ar-eas of engineering, science, and applied mathematics. The Finite Element Method which is a computer-aid mathematical technique is just a powerful method for obtaining approximate numerical solution. Up to now,applications to date have occurred principally in the areas of solid mechanics, heat transfer, ‡uid mechanics, and electromagnetism. New areas of application are continually being discovered, recent ones include solid-state physics and quantum mechanics.
6.1.2 Procedure of Finite Element Method
Discretization The discretization of the domain into subregions is the …rst step in FEM. We choose the tetrahedral element as our basic element shape to partition the domain.
tetrahedral element
By the software (HyperMesh), we can partition the domain.
The meshes of 3D computation domain( r=20 Bohr)
Shape Functions The typical 3-D linear tetrahedral element trial solution can be written
U(r; ®) =e X4 j=1
®jNj(x; y; z) The special coordinates are introduced de…ned by
x = L1x1+ L2x2+ L3x3+ L4x4 (8.a) y = L1y1+ L2y2+ L3y3+ L4y4 (8.b) z = L1z1+ L2z2+ L3z3 + L4z4 (8.c)
1 = L1+ L2+ L3+ L4 (8.d)
Sloving Eq(8) gives Li = ai+ bix + ciy + diz
The linear shape functions for the linear element are simply Ni = Li; Nj = Lj; Nk = Lk; Nl = Ll
where Vc represents the volume of the tetrahedron.
6.1.3 Weighted residual approach
Variational approach requires a knowledge of a variational problem( functional to be extremized or made stationary ) for the given problem. Usually we encounter problems for which the variational principles are not known. Weighted residual approach in the …nite element method can be applied even in these cases.We can derive directly from the governing di¤erential equations of the problem without any need of knowing the ’functional’ by weighted residual approach.
For a particle J of one nucleus system in an orbital, the governing equation is written as
where ªJ is the density function of a a particle J in an orbital that is limited to two electrons with spin polarization to satisfy the Pauli exclusion principle, r and
R is the coordinate from the zero point of the computing space, Z is the number of positive charge of the nucleus, the subscript I means the kind of nucleus .
STEP 1.
Using a trial solution U(r; ®) to approximate the density function ª(r), wee
…rst write down the residual equation for each orbital J : RJ(r; ®) =¡1
2 52U(r; ®)e ¡ Z
rU (r; ®) + ¦e JU(r; ®)e ¡ "U(r; ®)e (19) The typical 3-D element trial solution is always written in the general form U(r; ®) =e Pnj=1®jNj(r) , and one residual equation is
Z Z Z
RJ(r; ®)Ni(r)dV = 0 ; i = 1; 2; :::; n (20) where it integrates over one element and n is the number of nodes in an element.
=)
where nx, ny and nz are the direction vectors of the outward unit normal to the element boundary. The R.H.S. of above Equation is the boundary term contains the ‡ux must vanish from the system equations for the eigenproblems , so Equation (22) is rewritten as follows:
1Z Z Z @Ni(r)@U(r; ®)e
+ @Ni(r)@U (r; ®)e
+@Ni(r)@U (r; ®)e
dV (23)
STEP 3.
Equation (23) is rearranged to satisfy the matrix solver as follows :
f" [M] + [N]g f®g = 0 (24)
To transfer the integrals of Equation(24) into a form appropriate for numerical evaluation.
r dV (Gauss Cubic integration) (30)
=Vc
where N is the total numbers of the Gauss points, wk is the weighting factor and the subscript k means that the related values on the Gauss point .
Numerical integration formula of Gauss for tetrahedral element [Zienkiewicz and Taylor, 20 [C] =¡
Z Z Z
¦jNi(r)Nj(r)dV (31) STEP 5.
We obtain the term ¦J from solving Poisson’s equation
52¦J(r) =¡4¼jªJ(r)j2 (32) Boundary Condition :
=) Deriving the element equations complete by above steps. The Conjugate Gradient Method can slovehAi[X] =hBi, and it means the ¦j term is known now. We get [C]ij =¡R R R ¦jNi(r)Nj(r)dV to utilize Gauss Cubic integration. All elemental matrix equations are assembled to be algebra system that are then solved by Jacobi-Davidson matrix solver.