Commit d2577dec by Jens Jørgen Mortensen

### Merge branch 'cdft_tutorial' into 'master'

Cdft tutorial

See merge request !615
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 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='He2+_initial_%3.2f.txt' % distance, 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='He2+_initial_%3.2f.cdft' % distance, # cDFT output file minimizer_options={'gtol': 0.01}) # tolerance for cdft # Get cdft energy sys.set_calculator(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='He2+_final_%3.2f.txt' % distance, 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='He2+_final_%3.2f.cdft' % distance, minimizer_options={'gtol': 0.01}) sys.set_calculator(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
 .. module:: gpaw.cdft .. _constrained_DFT: ====================== 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 [#cdft1]_ and the publication of the GPAW implementation is available in [#cdft2]_. 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 .. math:: F[n(\mathbf{r}), V_c] = E^{KS}[n] + \sum_c V_c \sum_{\sigma}\left(\int d\mathbf{r}w_c^{\sigma}(\mathbf{r}) n^{\sigma}(\mathbf{r})-N_c\right) 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 .. math:: v_{\rm eff}^{\sigma}=\dfrac{\delta E^{KS}[n(\mathbf{r})]} {\delta n(\mathbf{r})} + \sum_c V_cw_c^{\sigma}(\mathbf{r}) 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 .. math:: C \geq \bigg\lvert \sum_{\sigma} \int {\rm d}\mathbf{r} w_c^{\sigma}(\mathbf{r})n^{\sigma}(\mathbf{r}) - N_c \bigg\rvert \, ,\forall\, c The V_c is self-consistently optimized so that the specified constraints are satisfied. Formally, this is written as .. math:: F\left[V_{c}\right]= \min _{n} \max _{\left\{V_{c}\right\}} \left[E^{\mathrm{KS}}[n(\mathbf{r})]+ \sum_{c} V_{c}\left(\int \mathrm{d} \mathbf{r} w_{c}(\mathbf{r}) n(\mathbf{r})-N_{c}\right)\right] V_c is obtained from .. math:: \frac{\mathrm{d} F}{\mathrm{d} V_{c}} = \int \mathrm{d} \mathbf{r} w_{c}(\mathbf{r}) n(\mathbf{r})-N_{c}=0 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 :meth:~gpaw.cdft.cdft.CDFT.get_number_of_electrons_on_atoms helps in this process. .. autoclass:: gpaw.cdft.cdft.CDFT :members: 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 :literal:L-BFGS-B. The accuracy of the optimization is controlled mainly by the :literal:minimizer_options={'gtol':0.01}) parameter which measurest the error between the set and computed charge/spin value. A typical :literal: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 :literal: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. 1. Choose a meaningful constraints and regions 2. Try to provide a good initial guess for the V_c. In the actual calculation this initial guess given by the parameter :literal:charge_coefs or :literal:spin_coefs. 3. Use L-BFGS-B to set bounds for V_c by using :literal:minimizer_options={'bounds':[min,max]}). 4. 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 [#cdft1]_ and a more general overlap or Migliore method detailed in [#cdft3]_. 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 :literal: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 :literal:get_migliore_coupling. Additional comments ------------------- In [#cdft2]_ 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 :literal: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. .. literalinclude:: He2.py 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 ========== .. [#cdft1] B. Kaduk, T. Kowalczyk, T. Van Voorhis, Constrained Density Functional Theory _, *Chem. Rev.*, **112** 321–370 (2012) .. [#cdft2] 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) .. [#cdft3] 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)