Constrained DFT (cDFT)¶
Introduction¶
cDFT is a method for build charge/spin localized or diabatic states with a user-defined charge and/or spin state. As such cDFT, is a useful tool for widening the scope of ground-state DFT to excitation processes, correcting for self- interaction energy in current DFT functionals, excitation energy, and electron transfer as well as parametrizing model Hamiltonians, for example.
Theoretical Background¶
The general cDFT methodology is reviewed in 1 and the publication of the GPAW implementation is available in 2. In short, the cDFT works by specifying an additional constraining term to the KS functional. The role of this constraint is to enforce a user specified charge/spin state (\(N_c\)) on the chosen regions of atoms. The constrained regions are specified by a weight function \(w(\mathbf{r})\) and the strength of the constraining potential acting on the specified region is \(V_c\). With these definitions the new energy functional with the constraint is
where \(E^{KS}\) is the Kohn-Sham energy, \(c\) specifies the region, and \(\sigma\) is the spin variable. It is also seen that \(V_c\) is also a Lagrange multiplier. The modified energy functional leads to an additional external potential
This is just a sum of the usual KS potential and the constraining potential which is also used in the self-consistent calculation. The constraint is further enforced by introducing the convergence criteria
The \(V_c\) is self-consistently optimized so that the specified constraints are satisfied. Formally, this is written as
\(V_c\) is obtained from
In the end, one ends up with a modified electron/spin density localized on the chosen atoms.
Notes on using cDFT¶
The weight function¶
In the GPAW implementation a Hirschfeld partition scheme with atom-centered
Gaussian functions is used. These Gaussian have two tunable parameters: the
cut-off \(R_c\) and the width \(\mu\). If the constrained region cuts a covalent
bond, the results are sensitive to width parameter. There is no universal
guide for choosing the width in such cases. A sensible choice is to compute
match the Gaussian-Hirschfeld charges with e.g. Bader charges. The function
get_number_of_electrons_on_atoms()
helps in this
process.
- class gpaw.cdft.cdft.CDFT(calc, atoms, charge_regions=[], charges=None, spin_regions=[], spins=None, charge_coefs=None, spin_coefs=None, promolecular_constraint=False, txt='-', minimizer_options={'gtol': 0.01}, Rc={}, mu={'F': 0.7, 'Li': 0.5, 'O': 0.7, 'V': 0.5}, method='BFGS', forces='analytical', use_charge_difference=False, compute_forces=True, maxstep=100, tol=0.001, bounds=None, restart=False, hess=None)[source]¶
Constrained DFT calculator.
- calc: GPAW instance
DFT calculator object to be constrained.
- charge_regions: list of list of int
Atom indices of atoms in the different charge_regions.
- spin_regions: list of list of int
Atom indices of atoms in the different spin_regions.
- charges: list of float
constrained charges in the different charge_regions.
- spins: list of float
constrained spins in the different charge_regions. Value of 1 sets net magnetisation of one up/alpha electron
- charge_coefs: list of float
Initial values for charge constraint coefficients (eV).
- spin_coefs: list of float
Initial values for spin constraint coefficients (eV).
- promolecular_constraint: bool
Define charge and/or spin constraints from promolecular densities, see: dx.doi.org/10.1021/cr200148b Eq. 29-31 If true, user specified charges/spins are overwritten! The atoms (of Atoms object) specifying the charge/spin regions need to contain have correct charge/spin state! (atoms.set_initial_charges([atomic_charges]) and atoms.set_initial_magnetic_moments([moments]))
- txt: None or str or file descriptor
Log file. Default id ‘-’ meaning standard out. Use None for no output.
- minimizer_options: dict
options for scipy optimizers, see:scipy.optimize.minimize
- method: str
One of scipy optimizers, e.g., BFGS, CG
- forces: str
cDFT weight function contribution to forces ‘fd’ for finite difference or ‘analytical’
- difference: bool
If True, constrain charge difference between two regions Then charge_regions needs two regions and charges needs only one item which is the charge difference between the two regions, the first beign donor, the second acceptor
If False, each region is treated with the corresponding charge constraint
- compute_forces: bool
Should the forces be computed?
- restart: bool
starting from an old calculation from gpw
- hess: ‘2-point’, ‘3-point’, ‘cs’
scipy hessian approximation
- calculate(atoms, properties, system_changes)[source]¶
Do the calculation.
- properties: list of str
List of what needs to be calculated. Can be any combination of ‘energy’, ‘forces’, ‘stress’, ‘dipole’, ‘charges’, ‘magmom’ and ‘magmoms’.
- system_changes: list of str
List of what has changed since last calculation. Can be any combination of these six: ‘positions’, ‘numbers’, ‘cell’, ‘pbc’, ‘initial_charges’ and ‘initial_magmoms’.
Subclasses need to implement this, but can ignore properties and system_changes if they want. Calculated properties should be inserted into results dictionary like shown in this dummy example:
self.results = {'energy': 0.0, 'forces': np.zeros((len(atoms), 3)), 'stress': np.zeros(6), 'dipole': np.zeros(3), 'charges': np.zeros(len(atoms)), 'magmom': 0.0, 'magmoms': np.zeros(len(atoms))}
The subclass implementation should first call this implementation to set the atoms attribute and create any missing directories.
Optimizing \(V_c\)¶
Updating and optimizing the Lagrange multipliers \(V_c\) is achieved using
Scipy optimization routines. As it is easy to compute the gradient of the
energy wrt \(V_c\), gradient based optimizers are recommended. The best
performing optimizer seems to be the L-BFGS-B
. The accuracy of the
optimization is controlled mainly by the
minimizer_options={'gtol':0.01})
parameter which measurest the
error between the set and computed charge/spin value. A typical
gtol
value is around 0.01-0.1.
Choosing the constraint values and regions¶
Both charge and spin constraints can be specified. The charge constraints are
specified to neutral atoms: specifying charges = [1]
constrains
the first regions to have a net charge of +1. Note, that if the usual DFT
calculation yields e.g. a Fe ion with charge +2.5, specifying the charge
constraint +1 will result in +1, not an additional hole on Fe with a charge
state +3.5!
A constrained regions may contain several atoms and also several constrained regions can be specified. However, converging more than two regions is usually very difficult.
Tips for converging cDFT calculations¶
Unfortunaly, the cDFT sometimes exhibits poor convergence.
Choose a meaningful constraints and regions
Try to provide a good initial guess for the \(V_c\). In the actual calculation this initial guess given by the parameter
charge_coefs
orspin_coefs
.Use L-BFGS-B to set bounds for \(V_c\) by using
minimizer_options={'bounds':[min,max]})
.Converge the underlying DFT calculation well i.e. use tight convergence criteria.
Constructing diabatic Hamiltonians¶
One of the main uses for cDFT is constructing diabatic Hamiltonians which utilize cDFT states as the diabats. For instance, a 2x2 Hamiltonian matrix would have diagonal elements \(H_{11}\) and \(H_{22}\) as well as off-diagonals \(H_{12}\) and \(H_{21}\). \(H_{11}\) and \(H_{22}\) values are given directly by cDFT energies (not cDFT free energies). The off-diagonal coupling elements \(H_{12}\) and \(H_{21}\) are also needed and often utilized in e.g. Configuration Interaction-cDFT, in parametrizing model Hamiltonians or in computing non-adiabatic charge transfer rates. Note, that all parameters need to be computed at the same geometry.
The coupling elements are computed using the CouplingParameters class. There are several options and optional inputs. There are two main methods for computing coupling elements: the original cDFT approach from 1 and a more general overlap or Migliore method detailed in 3.
cDFT coupling constant¶
This method is the original approach in the context of cDFT. It works well
for simple system. For complex system it becomes quite sensitive to small
errors in the Lagrange multipliers and constraints, and the overlap method is
recommended instead. The inputs for the calculation are two cDFT objects with
different constraints. The coupling constant is computed using
get_coupling_term
.
Overlap coupling constant¶
This approach has been found to perform very well for a range of systems.
However, it has one major limitation: it can only be used if the two diabatic
states have different energies. In addition to the two cDFT states/objects
also a normal DFT calculator without any constraints is needed. The coupling
constant is computed using get_migliore_coupling
.
Additional comments¶
In 2 the coupling constants were computed by computing all-electron
wave functions on a grid. However, this is quite slow and much faster
implementation utilizing only pseudo wave functions and atomic corrections
has been added. For the tested cases both give equally good values for the
coupling constant. Hence, it is recommended to use the pseudo wave functions
which is set by AE=False
.
The quantities needed for computing the coupling constants can be parallellized only over the grid.
Example of hole transfer in \(He_2^+\)¶
Both the cDFT calculation and extraction of the coupling element calculation are demonstrated for the simple \(He_2^+\) system.
from ase import Atoms
from gpaw import GPAW, FermiDirac, Davidson, Mixer
from gpaw.cdft.cdft import CDFT
from gpaw.cdft.cdft_coupling import CouplingParameters
# Set up the system
distance = 2.5
sys = Atoms('He2', positions=([0., 0., 0.], [0., 0., distance]))
sys.center(3)
sys.set_pbc(False)
sys.set_initial_magnetic_moments([0.5, 0.5])
# Calculator for the initial state
calc_a = GPAW(
h=0.2,
mode='fd',
basis='dzp',
charge=1,
xc='PBE',
symmetry='off',
occupations=FermiDirac(0., fixmagmom=True),
eigensolver=Davidson(3),
spinpol=True, # only spin-polarized calculations are supported
nbands=4,
mixer=Mixer(beta=0.25, nmaxold=3, weight=100.0),
txt=f'He2+_initial_{distance:3.2f}.txt',
convergence={
'eigenstates': 1.0e-4,
'density': 1.0e-1,
'energy': 1e-1,
'bands': 4})
# Set initial state cdft
cdft_a = CDFT(
calc=calc_a,
atoms=sys,
charge_regions=[[0]], # choose atom 0 as the constrained region
charges=[1], # constrain +1 charge
charge_coefs=[2.7], # initial guess for Vc
method='L-BFGS-B', # Vc optimization method
txt=f'He2+_initial_{distance:3.2f}.cdft', # cDFT output file
minimizer_options={'gtol': 0.01}) # tolerance for cdft
# Get cdft energy
sys.calc = cdft_a
sys.get_potential_energy()
# the same for the final state
calc_b = GPAW(h=0.2,
mode='fd',
basis='dzp',
charge=1,
xc='PBE',
symmetry='off',
occupations=FermiDirac(0., fixmagmom=True),
eigensolver=Davidson(3),
spinpol=True,
nbands=4,
mixer=Mixer(beta=0.25, nmaxold=3, weight=100.0),
txt=f'He2+_final_{distance:3.2f}.txt',
convergence={
'eigenstates': 1.0e-4,
'density': 1.0e-1,
'energy': 1e-1,
'bands': 4})
cdft_b = CDFT(
calc=calc_b,
atoms=sys,
charge_regions=[[1]], # choose atom 1
charges=[1], # constrained charge +1
charge_coefs=[2.7],
method='L-BFGS-B',
txt=f'He2+_final_{distance:3.2f}.cdft',
minimizer_options={'gtol': 0.01})
sys.calc = cdft_b
sys.get_potential_energy()
# Now for the coupling parameter
coupling = CouplingParameters(cdft_a, cdft_b, AE=False) # use pseudo orbitals
H12 = coupling.get_coupling_term() # use original cDFT method
The most important cDFT results are found in the .cdft files. For instance, the errors, iterations, and cDFT parameters are shown. Also, the energy can be found in this file. The most relevant energy is the Final cDFT Energy (not the free energy).
References¶
- 1(1,2)
B. Kaduk, T. Kowalczyk, T. Van Voorhis, Constrained Density Functional Theory, Chem. Rev., 112 321–370 (2012)
- 2(1,2)
M. Melander, E. Jońsson, J.J. Mortensen, T. Vegge,J.M. Garcia-Lastra, Implementation of Constrained DFT for Computing Charge Transfer Rates within the Projector Augmented Wave Method, J. Chem. Theory Comput., 12, 5367−5378 (2016)
- 3
A. Migliore, Nonorthogonality Problem and Effective Electronic Coupling Calculation: Application to Charge Transfer in π-Stacks Relevant to Biochemistry and Molecular Electronics, J. Chem. Theory Comput., 7, 1712-1725 (2011)