Introduction
==================================

KSSOLV2.0 :cite:`KSSOLV2` is a MATLAB toolbox for performing first-principles density functional theory (DFT) ground- and excited- state calculations
for molecules and solids within plane-wave basis sets under periodic boundary conditions.

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**Ground state calculation**
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The ground state calculation is based on the Kohn-Sham density functional 
theory.  The main equations to be solved are the Kohn-Sham equations of the form

.. math:: H(\rho) \psi_i({\bf r}) = \varepsilon_i \psi_i({\bf r}), 
   :label: kseig

where :math:`\psi_i`, :math:`i=1,2,...n_e`, with :math:`n_e` being the number 
of electrons, are orthonormal quasi-electron orbitals, and 
:math:`\varepsilon_i` are the corresponding quasi-electron energies. 
The function :math:`\rho({\bf r})` is the electron density defined to be

.. math:: \rho({\bf r}) = \sum_{i=1}^{n_e} \left| \psi_i({\bf r}) \right|^2. 

In the so called local density approximation (LDA) or general gradient approximation (GGA), the Kohn-Sham Hamiltonian :math:`H(\rho)` has the form

.. math:: H(\rho) = -\nabla^2 + \int \frac{\rho({\bf r}')}{\left| {\bf r}-{\bf r}' \right|} d{\bf r}' + v_{xc}(\rho({\bf r})) + v_{ext}({\bf r}),
   :label: ksham

where the first term in the above equation is associated with the kinectic 
energy of the electrons, the second term corresponds to the electro-static 
repulsion among electrons, the third terms is the so called 
exchange-correlation potential that accounts for other many-body properties
of the system, and the last term represents the ionic potential contributed
by nuclei. There are a number of expressions for :math:`v_{xc}` since
the exact form of the exchange-correlation potential is unknown. These 
expressions also depend on the way :math:`v_{ext}` is approximated. By 
treating inner electrons as part of an ionic core, an approach known
as the *pseudopotential* method, we obtain :math:`v_{ext}` that is easier to 
compute.

The Kohn-Sham equations form a nonlinear eigenvalue problem in which 
the Kohn-Sham Hamiltonian to be diagonalized is a function of the 
electron density :math:`\rho({\bf r})`, which is in turn a function 
of the eigenfunction :math:`\psi_i`'s to be computed.

These equations are the first order necessary condition of the 
contrained minimization problem:

.. math:: \min_{\langle\psi_i,\psi_j\rangle = \delta_{i,j}} E_{total}(\{\psi_j\}) \equiv \frac{1}{2}\sum_{i=1}^{n_e} \int \left|\nabla \psi_i ({\bf r}) \right|^2 dr + E_{ion} + E_{coul}(\rho) + E_{xc}(\rho), 
   :label: ksetot

where 

.. math:: E_{ion} = \int \rho({\bf r}) v_{ext} ({\bf r}) dr,

.. math:: E_{coul} =  \frac{1}{2}\int \int \frac{\rho(r)\rho(r')}{\left|r - r'\right|} dr dr'.

Therefore, the Kohn-Sham nonlinear eigenvalue problem can also be solved
as a constrained optimization problem.