User:Tamcd00dle/sandbox

KSSOLV
Original author(s)	Chao Yang, Juan C. Meza
Developer(s)	Lawrence Berkeley National Laboratory
Website	KSSOLV

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KSSOLV is a MATLAB toolbox^[1] for solving a class of nonlinear eigenvalue problems known as the Kohn-Sham equations^[2]. Developed by Chao Yang and Juan C. Meza from the Lawrence Berkeley National Laboratory, KSSOLV is designed using the objected oriented programming features available in MATLAB. It allows users to assemble an atomistic system and facilitates development and comparison of new numerical methods for solving the Kohn-Sham equations.

Background Information

Solving the Kohn-Sham equations is a problem that arises from electronic structure calculation, used to study the microscopic quantum mechanical properties of molecules, solids and other nanoscale materials. Through the density functional theory (DFT) formalism, the many-body Schrödinger equation can be reduced to a set of single-electron equations with fewer degrees of freedom.

The Kohn-Sham equations have the form

$H(\rho )\psi _{i}=\lambda _{i}\psi _{i},\ \ i=1,2,...,n_{e},$

where $n_{e}$ is the number of electrons in the system,

$\{\psi _{i}(r)\}$ ( $i=1,2,...,n_{e}$ ) is a set of single electron wavefunctions that are mutually orthonormal, $\rho (r)$ is the total electron charge defined by

$\rho (r)=\sum _{i=1}^{n_{e}}|\psi _{i}(r)|^{2},$

and

$\lambda _{1}\leq \lambda _{2}\leq ...\leq \lambda _{n_{e}}$

are the $n_{e}$ smallest eigenvalues of $H(\rho )$ , a linear operator that depends on $\rho$ . The operator $H$ is known as the Kohn-Sham Hamiltonian and defined by

$H(\rho )=-{\frac {1}{2}}\Delta +V_{ion}(r)+\rho \star {\frac {1}{|r|}}+V_{xc}(\rho ),$

where $\Delta$ is the Laplacian operator, $V_{ion}$ and $V_{xc}$ are known as the ionic and exchange-correlation potentials respectively^[2], and $\star$ denotes the convolution operator.

The Kohn-Sham equations are the the Euler-Lagrange equations for the total energy minimization problem

$\min E_{total}^{KS}(\psi _{i})$

${\mbox{s.t.}}$ $\psi _{i}^{\ast }\psi _{j}=\delta _{i,j},$

where

$E_{total}^{KS}={\frac {1}{2}}\sum _{i=1}^{n_{e}}\int _{\Omega }|\nabla \psi _{i}|^{2}dr+\int _{\Omega }\rho V_{ion}dr+{\frac {1}{2}}\int _{\Omega }\int _{\Omega }{\frac {\rho (r_{1})\rho (r_{2})}{|r_{1}-r_{2}|}}dr_{1}dr_{2}+E_{xc}(\rho ),$

$E_{xc}(\rho )$ is the exchange-correlation energy^[2].

The Kohn-Sham equations are different from the standard linear eigenvalue problem in that the matrix operators are functions of the eigenvectors to be computed. Due to this nonlinear coupling between the matrix operator and its eigenvector, the Kohn-Sham equations are more difficult to solve. Currently, the most widely used numerical method for solving the this type of problem is the Self Consistent Field (SCF) iteration^[3]^[4]. An alternative approach is designed to minimize the total energy directly. Both methods have been implemented in KSSOLV to allow users to compute the solution to the Kohn-Sham equations associated with various molecules and solids.

Several classes are created in KSSOLV to define both physical objects such as atoms and molecules and mathematical entities such as wavefunctions and Hamiltonians. These classes are defined in a hierarchical fashion so that a molecule object can be constructed from atom objects.

Setup

KSSOLV is built around the MATLAB language. The environmental variable KSSOLVPATH should be defined prior to running KSSOLV.

Test Problems

KSSOLV contains a number of test problems designed for experimentation:

setup file name	nocc	Description
c2h6_setup.m	7	an ethane molecule
co2_setup.m	8	a carbon dioxide molecule
h2o_setup.m	4	a water molecule
hnco_setup.m	8	an isocyanic acid molecule
qdot_setup.m	8	a 8-electron quantum dot confined by an external potential
si2h4_setup.m	6	a planar singlet silylene molecule
sibulk_setup.m	6	a silicon bulk system
sih4_setup.m	4	a silane molecule
ptnio_setup.m	43	a Pt₂Ni₆O molecule

Molecular Systems

A molecular system is defined by creating an Molecule object, using

 mol = Molecule ();

Properties of this object are specified using the set method associated with the object. For example,

 mol = set(mol, 'atomlist', alist);

Atoms can be specified by their chemical symbols or atomic numbers; for example, the silicon atom can be defined by

 a = Atom('Si')

or

 a = Atom(14);

The position of each atom in the atom list is specified by an n_a× 3 array; the ith row of this array gives the $x,y,z$ coordinates of the ith atom in the atom list. The unit of the coordinates should be in Bohr (1 Bohr = 0.5291772083 Angstrom).

Alternatively, the relative position of each atom in a unit (super)cell (of a periodic lattice) can first be specified by three lattice vectors c₁, c₂, and c₃. If the relative displacement of the atom along c_i is denoted by $\xi _{i}$ , then the absolute $x,y,z$ coordinates of the atom correspond to the three components of the row vector

$r=\xi _{1}\mathbf {c} _{1}^{T}+\xi _{2}\mathbf {c} _{2}^{T}+\xi _{3}\mathbf {c} _{3}^{T}=(\xi _{1}\xi _{2}\xi _{3})C$ ,

where $C=\left({\begin{array}{c}\mathbf {c} _{1}^{T}\\\mathbf {c} _{2}^{T}\\\mathbf {c} _{3}^{T}\\\end{array}}\right)$ .

The 3 × 3 matrix C defines a unit supercell in KSSOLV. It can be used to set the 'supercell' attribute of a Molecule object. For example,

 mol = set(mol,'supercell',10*eye(3));

sets the unit supercell of mol to a 10×10×10 cube.

In order to discretize the Kohn-Sham equations with an appropriate number of planewave basis functions, a kinetic energy cutoff E_cut must be specified through

mol = set(mol,’ecut’,cutoff_value);

where cutoff value is a user-supplied number in the unit of Rydberg (Ryd).

The Silane molecule might be set up as follows:

%
% 1. construct atoms
%
a1 = Atom('Si');
a2 = Atom('H');
alist = [a1; a2; a2; a2; a2];
%
% 2. set up primitive cell (Bravais lattice)
%
C = 10*eye(3);
%
% 3. define the coordinates the atoms
%
coefs = [
 0.0     0.0      0.0
 0.161   0.161    0.161
-0.161  -0.161    0.161
 0.161  -0.161   -0.161
-0.161   0.161   -0.161
];
xyzmat = coefs*C';
%
% 4. Configure the molecule
%
mol = Molecule();
mol = set(mol,'Blattice',C);
mol = set(mol,'atomlist',alist);
mol = set(mol,'xyzlist' ,xyzmat);
mol = set(mol,'ecut', 25);  % kinetic energy cut off
mol = set(mol,'name','SiH4');

Solving the Kohn-Sham Equations

Once an Molecule object is properly set up, computing the electron wavefunctions associated with the minimum total energy of the molecular system can be performed in KSSOLV by calling one of the available functions.

function	Description
scf.m	Self consistent field iteration
chebyscf.m	Chebyshev filtered self consistent field iteration
dcm.m	direct constrained minimization (DCM)^[5]
trdcm.m	direct constrained minimization with flexible trust region parameters^[5]
trdcm1.m	direct constrained minimization with a fixed trust region parameter^[5]

Visualization

The electron charge density, total potential and single particle wavefunctions can be visualized using either the MATLAB’s isosurface rendering function isosurface or the volume rendering function written by Joe Conti (see External Links), which is also included in KSSOLV.


The computed charge density of a silane molecule.	A volume rendering of the computed charge density of a silane molecule.

References

^ C. Yang, J. C. Meza, B. Lee, and L-W. Wang. KSSOLV: A MATLAB toolbox for solving the Khon-Sham equations. ACM Trans. Math. Soft., to appear, 2009.
^ ^a ^b ^c W. Kohn and L. J. Sham. Self-consistent equations including exchange and correlation effects. Phys. Rev., 140., (4A): A1133-A11388, 1965.
^ C. Yang, J. C. Meza, and L. W. Wang. A trust region direct constrained minimization algorithm for the Kohn-Sham equation. SIAM J. Sci. Comp., 29(5):1854-1875, 2007.
^ P. Pulay. Convergence acceleration of iterative sequences: The case of SCF iteration. Chemical Physics Letters, 73(2):393-398, 1980.
^ ^a ^b ^c C. Yang, J. C. Meza, and L. W. Wang. A constrained optimization algorithm for total energy minimization in electronic structure calculation. Journal of Computational Physics, 217:709-721, 2005.

External links

[1] C. Yang, J. C. Meza, B. Lee, and L-W. Wang. KSSOLV: A MATLAB toolbox for solving the Khon-Sham equations. ACM Trans. Math. Soft., to appear, 2009.

[:0-2] W. Kohn and L. J. Sham. Self-consistent equations including exchange and correlation effects. Phys. Rev., 140., (4A): A1133-A11388, 1965.

[3] C. Yang, J. C. Meza, and L. W. Wang. A trust region direct constrained minimization algorithm for the Kohn-Sham equation. SIAM J. Sci. Comp., 29(5):1854-1875, 2007.

[4] P. Pulay. Convergence acceleration of iterative sequences: The case of SCF iteration. Chemical Physics Letters, 73(2):393-398, 1980.

[:1-5] C. Yang, J. C. Meza, and L. W. Wang. A constrained optimization algorithm for total energy minimization in electronic structure calculation. Journal of Computational Physics, 217:709-721, 2005.

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