QMol_DFT_SCF_Anderson

Anderson's mising scheme for DFT ground-state calculations.

Description

Use QMol_DFT_SCF_Anderson to solve DFT self-consistent field (SCF) nonlinear eigen-state problem

$$ {\hat{\mathcal{H}} }_{{\mathrm{D}\mathrm{F}\mathrm{T}}} \left\lbrack \lbrace \phi_k \rbrace_k \right\rbrack ~\phi_k (x)=E_k ~\phi_k (x)~~{\mathrm{f}\mathrm{o}\mathrm{r}}~~k=1,2,\ldots~~~~~~(1) $$

where ${\hat{\mathcal{H}} }_{{\mathrm{D}\mathrm{F}\mathrm{T}}}$ is the DFT Hamiltonian operator and $\lbrace \phi_k \rbrace_k$ are the Kohn-Sham orbitals with energies $\lbrace E_k \rbrace_k$ . QMol_DFT_SCF_Anderson uses an Anderson's mixing scheme [Anderson 1965] in SCF iterations, as described in [Johnson 1988], for either the one-body density or the Kohn-Sham potential. QMol_DFT_SCF_Anderson is a handle class.

Class properties

Anderson's mixing

The QMol_DFT_SCF_Anderson class defines the following public get-access properties; each can be changed using the set method:

tolerance (tol)

Convergence tolerance [ nonnegative scalar (default 1e-10) ]

maxIterations (maxit)

Maximum number of SCF iterations [ nonnegative integer (default 100) ]

mixing (mix)

Mixing coefficient [ 0 to 1 scalar (default 0.5) ]

• Too large mixing coefficient may render the SCF unstable and prevent convergence.
• Too small mixing coefficient makes the SCF convergence slower.
• Optimal mixing coefficient is system dependent and should be adapted accordingly.

mixingMode (mode)

What to mix in the SCF iterations [ density (default) | potential ]

• density mixes the one-body density while potential mixes the Kohn-Sham potential. In both cases, only the explicit part of the potential is being mixed in the SCF iterations and any implicit exchange-correlation functional is not being mixed.

convergenceTest (test)

Convergence criterion [ 'density' (default) | 'eigenvalues' ]

• With the 'density' criterion, the result is considered converged when the ${\mathcal{L}}^2$ norm difference in the one-body density between two consecutive SCF iterations is smaller than tolerance*mixing.
• With the 'eigenvalues' criterion, the result is considered converged when the sum in the absolute-value difference between the DFT Hamiltonian operator eigenvalues between two consecutive SCF iterations is smaller than tolerance*mixing.

display (disp)

Whether to display the SCF iteration progress [ true (default) | false ]

Other properties

To facilitate simulations, QMol_DFT_SCF_Anderson defines a handful of additional transient properties. These cannot be edited with the set method.

isInitialized (isInit)

Whether the DFT-model object is properly initialized. This is used throughout the QMol-grid package to check that the DFT-model object holds meaningful information and is ready for use.

Class methods

Creation

constructor

Create an Anderson's mixing scheme object with empty class properties.

obj = QMol_DFT_SCF_Anderson;

Create an Anderson's mixing scheme object with the name properties set to the specified value. Several name-value pairs can be specified consecutively. Suitable name is any of Anderson's mixing and is case insensitive.

obj = QMol_DFT_SCF_Anderson(name1,value1);
obj = QMol_DFT_SCF_Anderson(name1,value1,name2,value2,___);

Changing class properties

set

Update the name properties of an Anderson's mixing scheme object to the specified value. Several name-value pairs can be specified consecutively. Suitable name is any of Anderson's mixing and is case insensitive.

obj.set(name1,value1);
obj.set(name1,value1,name2,value2,___);

This is the common name-value pair assignment method used throughout the QMol-grid package. The set method also reset the class. After running, the set property updates the isInitialized flag to a false value.

reset

Reset the object by deleting/re-initializing all run-time properties of the class and updating the isInitialized flag to false.

obj.reset;

• This is the common reset method available to all classes throughout the QMol-grid package.

clear

Clear all class properties

obj.clear;

Clear a specific set of the class properties. Suitable name is any of Anderson's mixing and is case insensitive.

obj.clear(name1,name2,___);

This is the common clear method available to all classes throughout the QMol-grid package. The clear method also reset the class.

Initializing the object

initialize

Initialize a QMol_DFT_SCF_Anderson object and set the isInitialized flag to true.

obj.initialize;

Run-time documentation

showDocumentation

Display the run-time documentation for the specific configuration of a QMol_DFT_SCF_Anderson object, wich must have been initialized beforehand.

ref = obj.showDocumentation;

• The output ref is a cell vector containing the list of references to be included in the bibliography.

SCF computation

solveSCF

Compute the solution of the SCF problem of Eq. (1) with

obj.solveSCF(DFT);

• DFT is the DFT-model handle object, i.e., QMol_DFT_spinPol or QMol_DFT_spinRes, that describes the DFT Hamiltonian operator of Eq. (1).
• To avoid any mismatch in internal properties, solveSCF first reset the object and (re)initializes the DFT before performing the SCF iterations.
• The first step of the SCF iterations uses linear mixing (with the mixing coefficient); the following iterations use Anderson's mixing.
• For grid-based DFT models, the SCF iterations use the QMol_DFT_eigs eigen-state solver with default properties; for basis set models, it uses the QMol_DFT_eig_basis solver (again with default properties).
• The result Kohn-Sham orbitals are stored in the input DFT object (in DFT.orbital).

Specify the eigen-state solver to use in the SCF iterations with

obj.solveSCF(DFT,ES);

• This enables non-default properties for the eigen-state solver object ES.

Specify the initial condition to start the SCF iterations with

obj.solveSCF(DFT,[],IC);    % Default eigen-state solver
obj.solveSCF(DFT,ES,IC);    % supplied eigen-state solver

• IC = [] (or omitted) starts the SCF iterations from scratch, from the equivalent of the eigen-states of the external potential only (no Hartree or exchange-correlation contribution). This is the default and most general option but my result in slow (or failed) convergence.
• IC = 'DFT' starts the SCF iterations from the one-body density from the input DFT object (DFT.getDensity).
• IC = rho, where rho is a compatible one-body density object, starts SCF iterations from the supplied one-body density (the total charge in rho may not match the DFT total charge, in which case the input density is rescaled to the proper total charge).
• IC = Vks, where Vks is a compatible Kohn-Sham potential object, starts SCF iterations from the supplied Kohn-Sham potential.
• The type of initial condition may not match the mixingMode. For instance, one my supply a starting one-body density while mixing the Kohn-Sham potential in the SCF iterations.

Examples

Examples of SCF calculations for spin-polarized and spin-restricted DFT models can be found in the QMol_DFT_spinPol and QMol_DFT_spinRes documentation pages, respectively.

Test suite

Run the test suite for the class in normal or summary mode respectively with

QMol_test.test('DFT_SCF_Anderson');
QMol_test.test('-summary','DFT_SCF_Anderson');

References

[Anderson 1965] D.G. Anderson, "Iterative procedures for nonlinear integral equations," Journal of the ACM 12, 547 (1965).

[Johnson 1988] D. D. Johnson, "Modified Broyden''s method for accelerating convergence in self-consistent calculations," Physical Review B 38, 12807 (1988).

Notes

• QMol_DFT_SCF_Anderson was introduced in version 01.00.