Category:Electronic minimization: Difference between revisions

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The algorithms {{VASP}} offers for electronic minimisation (i.e., determining the electronic ground state) can be divided in two categories:
By '''electronic minimization''' we denote the process of determining the electronic ground state.
* Iterative matrix diagonalisation + density mixing, ''aka'' the "Self-Consistency Cycle" (SCC).
Within the context of Hohenberg-Kohn-Sham density functional theory, the ground state is that state of the system that minimizes the Kohn-Sham free energy:
* Direct optimisation of the orbitals.


=== The Self-Consistency Cycle ===
<span id="KohnShamFreeEnergy">
:<math>
F = \sum_n f_n \epsilon_n -E_{\rm H}\left[ \rho \right] +
E_{\rm xc} \left[ \rho \right] -\int V_{\rm xc}({\bf r})\rho({\bf r})d{\bf r} -
\sum_n \sigma S \left( \frac{\epsilon_n - \mu}{\sigma} \right)
</math>
</span>


# The SCC starts with an initial guess for the electronic density of the system under consideration: {{VASP}} uses the approximation of overlapping atomic charge densities. This density defines the initial Hamiltonian.
where the electronic density is given by:
# By means of iterative matrix diagonalisation techniques one obtains the ''N'' lowest lying eigenstates of the Hamiltonian (where ''N'' is of the order of the number of electrons in the unit cell). The iterative matrix diagonalisation algorithms implemented in {{VASP}} are the blocked-Davidson algorithm and the Residual Minimization Method with Direct Inversion in the Iterative Subspace (RMM-DIIS). Per default {{VASP}} uses the blocked-Davidson algorithm ({{TAG|ALGO}} = Normal).
# After the eigenstates and eigenvalues have been the determined with sufficient accuracy, they are used to to compute the total energy of the system, and to construct a new electronic density.
# In principle, this new density could be taken to define a new Hamiltonian. However, in order to arrive at a stable algorithm this new density is not used as is, but is mixed with the old density. By default {{VASP}} uses a Broyden mixer. The resulting density then defines the new Hamiltonian for the next round of iterative metrix diagonalisation (step 2).


Steps 2-4 are repeated until the change in the total energy from one cycle to the next drops below a specific threshold ({{TAG|EDIFF}}).
:<math>
\rho({\bf r})= \sum_n f_{n} |\psi_{n}({\bf r})|^2
</math>


Note that when starting from scratch ({{TAG|ISTART}} = 0), the self-consistency cycle procedure of {{VASP}} always begins with several ({{TAG|NELMDL}}) cycles where the density is kept fixed at the initial approximation (overlapping atomic charge densities).
and the Kohn-Sham orbitals and eigenenergies, <math>\{\psi_n, \epsilon_n \}</math> are solutions to the Kohn-Sham equations:
This ensures that the wave functions that are initialised with random numbers have converged to a something sensible before they are used to construct a new charge density.


For a more detailed description of the SCC have a look at: the [[Self-Consistency Cycle]].
:<math>
H \left[ \rho \right] | \psi_n \rangle = \epsilon_n S | \psi_n \rangle
</math>


=== Direct optimisation ===
under the constraint that the orbitals are ''S''-orthonormal:


Similar to the SCC procedure described above, when starting from scratch ({{TAG|ISTART}} = 0), the direct optimisation procedures in {{VASP}} always begin with several ({{TAG|NELMDL}}) self-consistency cycles where the density is kept fixed at the initial approximation (overlapping atomic charge densities).
:<math>
This ensures that the wave functions that are initialised with random numbers have converged to reasonable starting point for the subsequent direct optimisation.
\langle \psi_m | S | \psi_n \rangle = \delta_{mn}
</math>


The direct optimisation of the orbitals uses the gradient of the total energy with respect to the orbitals to move towards the ground state of the system: the orbitals are changed such that the total energy is lowered, using, ''e.g.'' the Conjugate Gradient Approximation, or Damped Molecular Dynamics.
The various algorithms for '''electronic minimization''' {{VASP}} offers, can be roughly divided into two categories:
* Iterative matrix diagonalisation + density mixing, ''aka'' the [[Self-consistency cycle|self-consistency cycle]] (SCC).
* [[Direct optimization of the orbitals]].
 
Selecting a particular method of '''electronic minimization''' is done by means of the {{TAG|ALGO}} (or {{TAG|IALGO}}) tag.
 
== Self-consistency cycle ==
 
# The SCC starts with an initial guess for the electronic density of the system. In particular, {{VASP}} uses the approximation of overlapping atomic charge densities. This density defines the initial Hamiltonian.
# By means of iterative matrix-diagonalization techniques, one obtains the {{TAG|NBANDS}} lowest lying eigenstates of the Hamiltonian. The iterative matrix-diagonalization algorithms implemented in {{VASP}} are the [[Blocked-Davidson algorithm|blocked-Davidson algorithm]] and the [[RMM-DIIS|residual-minimization method with direct inversion in the iterative subspace (RMM-DIIS)]]. Per default {{VASP}} uses the blocked-Davidson algorithm ({{TAG|ALGO}} = Normal).
# After the eigenstates and eigenvalues have been determined with sufficient accuracy, they are used in order to compute the total energy of the system and to construct a new electronic density.
# In principle, this new density could be taken to define a new Hamiltonian. However, in order to obtain a stable algorithm, this new density is not used as is but is mixed with the old density. By default {{VASP}} uses a Broyden mixer. The resulting density then defines the new Hamiltonian for the next round of iterative matrix diagonalization (step 2).
 
Steps 2-4 are repeated until the change in the total energy from one cycle to the next drops below a specific threshold set by {{TAG|EDIFF}}.
 
Note that when starting from scratch ({{TAG|ISTART}} = 0), the SCC procedure of {{VASP}} always begins with several ({{TAG|NELMDL}}) cycles where the density is kept fixed at the initial approximation, i.e., overlapping atomic charge densities.
This ensures that the wavefunctions that are initialized with random numbers have converged to something sensible before they are used to construct a new charge density.
 
For a more detailed description of the SCC have a look at: the [[Self-consistency cycle|self-consistency cycle]].
 
== Direct optimization ==
 
Similar to the SCC procedure described above, when starting from scratch ({{TAG|ISTART}} = 0), the direct optimization procedures in {{VASP}} always begin with several ({{TAG|NELMDL}}) self-consistency cycles where the density is kept fixed at the initial approximation (overlapping atomic charge densities).
This ensures that the wavefunctions that are initialized with random numbers have converged to a reasonable starting point for the subsequent direct optimization.
 
The direct optimization of the orbitals uses the gradient of the total energy with respect to the orbitals to move towards the ground state of the system: the orbitals are changed such that the total energy is lowered, using, e.g., the conjugate-gradient approximation, or damped molecular dynamics.


After every change of the orbitals, the total energy and electronic density are recomputed.
After every change of the orbitals, the total energy and electronic density are recomputed.
Per default, the electronic density is constructed directly from the orbitals at each step along the way, without any density mixing.
Per default, the electronic density is constructed directly from the orbitals at each step along the way, without any density mixing.
Optionally, though, density mixing may be used to stabilise these optimisation procedures when charge sloshing occurs.
Optionally, though, density mixing may be used to stabilize these optimization procedures when [[charge sloshing]] occurs.


Similar to the SCC described above, the direct optimisation of the orbitals stops when the change of the total energy drops below {{TAG|EDIFF}}.
As for the SCC described above, the direct optimization of the orbitals stops when the change of the total energy drops below {{TAG|EDIFF}}.


For more details on the direct optimisation algorithms please read: [[Direct optimisation of the orbitals]].
For more details on the direct optimization algorithms, please read: [[Direct optimization of the orbitals|direct optimization of the orbitals]].
<!--
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*Electronic minimization: {{TAG|Algorithms used in VASP to calculate the electronic groundstate}}.
*Electronic minimization: {{TAG|Algorithms used in VASP to calculate the electronic groundstate}}.
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*A description to obtain band decomposed charge densities is given here: {{TAG|Band decomposed charge densities}}.
*A description to obtain band decomposed charge densities is given here: {{TAG|Band decomposed charge densities}}.
*k-point projection scheme: {{TAG|LKPROJ}}.
*k-point projection scheme: {{TAG|LKPROJ}}.
----
-->


[[Category:VASP|Electronic minimization]]
[[Category:VASP|Electronic minimization]]

Latest revision as of 09:49, 17 October 2024

By electronic minimization we denote the process of determining the electronic ground state. Within the context of Hohenberg-Kohn-Sham density functional theory, the ground state is that state of the system that minimizes the Kohn-Sham free energy:

where the electronic density is given by:

and the Kohn-Sham orbitals and eigenenergies, are solutions to the Kohn-Sham equations:

under the constraint that the orbitals are S-orthonormal:

The various algorithms for electronic minimization VASP offers, can be roughly divided into two categories:

Selecting a particular method of electronic minimization is done by means of the ALGO (or IALGO) tag.

Self-consistency cycle

  1. The SCC starts with an initial guess for the electronic density of the system. In particular, VASP uses the approximation of overlapping atomic charge densities. This density defines the initial Hamiltonian.
  2. By means of iterative matrix-diagonalization techniques, one obtains the NBANDS lowest lying eigenstates of the Hamiltonian. The iterative matrix-diagonalization algorithms implemented in VASP are the blocked-Davidson algorithm and the residual-minimization method with direct inversion in the iterative subspace (RMM-DIIS). Per default VASP uses the blocked-Davidson algorithm (ALGO = Normal).
  3. After the eigenstates and eigenvalues have been determined with sufficient accuracy, they are used in order to compute the total energy of the system and to construct a new electronic density.
  4. In principle, this new density could be taken to define a new Hamiltonian. However, in order to obtain a stable algorithm, this new density is not used as is but is mixed with the old density. By default VASP uses a Broyden mixer. The resulting density then defines the new Hamiltonian for the next round of iterative matrix diagonalization (step 2).

Steps 2-4 are repeated until the change in the total energy from one cycle to the next drops below a specific threshold set by EDIFF.

Note that when starting from scratch (ISTART = 0), the SCC procedure of VASP always begins with several (NELMDL) cycles where the density is kept fixed at the initial approximation, i.e., overlapping atomic charge densities. This ensures that the wavefunctions that are initialized with random numbers have converged to something sensible before they are used to construct a new charge density.

For a more detailed description of the SCC have a look at: the self-consistency cycle.

Direct optimization

Similar to the SCC procedure described above, when starting from scratch (ISTART = 0), the direct optimization procedures in VASP always begin with several (NELMDL) self-consistency cycles where the density is kept fixed at the initial approximation (overlapping atomic charge densities). This ensures that the wavefunctions that are initialized with random numbers have converged to a reasonable starting point for the subsequent direct optimization.

The direct optimization of the orbitals uses the gradient of the total energy with respect to the orbitals to move towards the ground state of the system: the orbitals are changed such that the total energy is lowered, using, e.g., the conjugate-gradient approximation, or damped molecular dynamics.

After every change of the orbitals, the total energy and electronic density are recomputed. Per default, the electronic density is constructed directly from the orbitals at each step along the way, without any density mixing. Optionally, though, density mixing may be used to stabilize these optimization procedures when charge sloshing occurs.

As for the SCC described above, the direct optimization of the orbitals stops when the change of the total energy drops below EDIFF.

For more details on the direct optimization algorithms, please read: direct optimization of the orbitals.