From 39fc2feded38fb50f7b1c74dba66523010dcedba Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Ole=20Sch=C3=BCtt?= Date: Fri, 17 Nov 2023 12:38:09 +0100 Subject: [PATCH] Manual: Migrate howto:cdft from wiki (Orig. 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No previous +experience with CDFT simulations is required to complete this tutorial. However, a good +understanding of running DFT simulations with CP2K/QS is recommended before proceeding. + +This tutorial is divided as follows. First, a brief overview of the underlying theory behind CDFT +will be presented. Typical applications where CDFT simulations have been used will also be +highlighted. The CDFT implementation in CP2K will then be described in detail in the next section +with the aid of realistic example calculations. The last part of this tutorial covers how to +calculate properties involving multiple CDFT states. + +______________________________________________________________________ + +## CDFT in summary + +CDFT is a tool for constructing charge and/or spin localized states. Such localized states are +needed in a number of applications. These include for example the following + +- studying charge transfer phenomena and calculating electronic couplings (e.g. using the Marcus + theory approach) +- correcting spurious charge delocalization due to self-interaction error +- parametrizing model Hamiltonians (e.g. the Heisenberg spin Hamiltonian) + +A more exhaustive list of potential applications has been presented in this +[review article](https://dx.doi.org/10.1021/cr200148b). + +The charge and spin localized states are created by enforcing electron and spin density localization +within atom centered regions of space. The relevant theory has been derived by Wu and Van Voorhis in +a series of key papers: [paper 1](https://dx.doi.org/10.1103/PhysRevA.72.024502), +[paper 2](https://dx.doi.org/10.1063/1.2360263), [paper 3](https://dx.doi.org/10.1063/1.2360263). +Further useful references can be found in the aforementioned review article. The CDFT implementation +of CP2K has been throughly described in these two papers: +[paper 1](https://dx.doi.org/10.1021/acs.jctc.6b01085) and +[paper 2](https://dx.doi.org/10.1063/1.5038959). + +In this tutorial, only the main theoretical aspects needed to understand what is happening during a +CDFT simulation will be summarized. The charge/spin localized states can be generated by augmenting +the Kohn-Sham energy functional, $E_\mathrm{KS}$, by additional constraint potentials + +$$ +E_\mathrm{CDFT}[\rho, \vec{\lambda}] = \max_\vec{\lambda} \min_\rho \left( E_\mathrm{KS}[\rho] + \sum_c \lambda_c \left[\sum_{i = \uparrow, \downarrow} \int w_c^i(\mathbf{r})\rho^i(\mathbf{r})d\mathbf{r} - N_c \right] \right) +$$ + +where $\vec\lambda = [\lambda_1, \lambda_2, \cdots]^T$ are the constraint Lagrangian multipliers +("strength(s) of the constraint potential(s)"), $w^i(\mathbf{r})$ is an atom centered weight +function, and $N_c$ is the target value of the constraint. Multiple constraints can be included in a +CDFT simulation (the sum over $c$ above). The weight function is constructed as a normalized sum +over select constraint atoms $\mathcal{C}$ + +$$ +w^i(\mathbf{r}) = \frac{\sum_{j \in \mathcal{C}}c_jP_j(\mathbf{r})}{\sum_{j \in \mathcal{N}}P_j(\mathbf{r})} +$$ + +where $c_j$ are atomic coefficients which determine how each atom is included in the constraint +(more on this later), $P_j$ is the so-called cell function which determines the volume occupied by +atom $j$ according to some population analysis method, and $\mathcal{N}$ is the set of all atoms in +a system. Different types of constraints can be constructed by modifying the weight function +according to the following conventions + +- charge density constraint ($\rho^\uparrow + \rho^\downarrow$): $w^\uparrow = w^\downarrow = w$ +- magnetization density constraint ($\rho^\uparrow - \rho^\downarrow$): + $w^\uparrow = -w^\downarrow = w$ +- spin specific constraint ($\rho^{\uparrow/\downarrow}$): + $w^{\uparrow/\downarrow} = w, w^{\downarrow/\uparrow} = 0$ + +The Becke and Hirshfeld space partitioning schemes can be used as constraint weight functions in +CP2K. The main differences between these two constraints will be explained in a subsequent section. + +When CDFT is used in a molecular dynamics or a geometry optimization simulation, additional force +terms arising from the constraints are calculated + +$$ +\mathbf{F}_{c,i} = -\lambda_c \int \frac{\partial w(\mathbf{r})}{\partial \mathbf{R}_i}\rho(\mathbf{r})d\mathbf{r} +$$ + +The CDFT energy expression, $E_\mathrm{CDFT}$, is solved self-consistently using a two-tiered +approach: one external optimization loop for the constraints, and an inner loop to converge the +electronic structure. In practice, three SCF loops are needed to integrate CDFT with the OT method, +which uses its own outer loop to reset the OT preconditioner. This process has been schematically +illustrated in Figure 1. + +![Schematic of the CDFT SCF procedure](images/cdftscf.png){align=center} + +**Figure 1.** Schematic of the CDFT SCF procedure. The constraint Lagrangians $\vec\lambda$ are +first optimized in the outer CDFT loop, their values are subsequently fixed, and the electron +density corresponding to these fixed values is solved like in traditional CP2K DFT simulations. The +control is then returned to the outer CDFT loop where convergence of the constraints is checked. +This iteration process is repeated until convergence is achieved or until the number of maximum CDFT +SCF steps is reached. The structure of the CDFT loop will be further described below. + +By definition, all constraints are satisfied when + +$$ +\vec c(\vec\lambda) = \left[ \sum_{i = \uparrow, \downarrow} \int w_1^i(\mathbf{r})\rho^i(\mathbf{r})d\mathbf{r} - N_1, \cdots \right]^T = \vec 0 +$$ + +The constraint Lagrangian multipliers $\vec\lambda$ can therefore be optimized by minimizing the +constraint error expression $\max |\vec c(\vec\lambda)|$ until the largest element decreases below a +threshold $\varepsilon$. Root-finding algorithms are used to optimize $\lambda$. For Newton and +quasi-Newton class optimizers, a new guess for $\vec\lambda$ at step $n$ is generated according to +the following iteration formula + +$$ +\vec\lambda_n = \vec\lambda_{n-1} - \alpha \mathbf{J}_n^{-1}\vec c(\vec\lambda_{n-1}) +$$ + +where $\alpha \in (0, 1]$ is the step size and $\mathbf{J}^{-1}$ is the inverse Jacobian matrix. The +step size $\alpha$ can be fixed or its value can be optimized with backtracking line search, where +the value of $\lambda$ is successively reduced if it decreases the constraint error function. The +Jacobian matrix is approximated with finite differences, e.g. using a first order forward difference +stencil, by perturbing each element of $\vec\lambda$ slightly and re-evaluating the value of +$\vec c$ self-consistently (an SCF energy optimization) + +$$ +\mathbf{J}_{ij} = \frac{\partial \vec c_i(\vec\lambda)}{\partial \lambda_j} \approx \frac{\vec c_i(\vec\lambda+\vec\delta_j)-\vec c_i(\vec\lambda)}{\left|\vec\delta_j\right|} +$$ + +where $\vec\delta_j$ is a small perturbation of the $j$th component of $\vec\lambda$. + +______________________________________________________________________ + +## Using the CDFT module + +The input section [](#CP2K_INPUT.FORCE_EVAL.DFT.QS.CDFT) is used to set up a CDFT simulation. A +brief description of this input section will be given in the next two subsections. Subsequently, +various aspects of running CDFT simulations will be explored through example calculations. + +### Defining CDFT SCF parameters + +Settings for the CDFT SCF loop are controlled by the input section +[](#CP2K_INPUT.FORCE_EVAL.DFT.QS.CDFT). An example of a typical CDFT input is given below. These +parameter selections should be suitable for most systems. + +``` +&QS + ... + ! CDFT loop settings + ! Please note that prior to CP2K version 7.0, + ! Becke constraints were separate from the CDFT section + &CDFT + TYPE_OF_CONSTRAINT BECKE + ! Compute CDFT charges? + ATOMIC_CHARGES TRUE + ! Constraint strength and target values + ! Give one value per constraint + STRENGTH ${BECKE_STR} + TARGET ${BECKE_TARGET} + ! Constraint definitions, each repetition defines a new constraint + &ATOM_GROUP + ATOMS 1 + COEFF 1 + CONSTRAINT_TYPE CHARGE + &END ATOM_GROUP + ! No constraint applied but calculate charges + &DUMMY_ATOMS + ATOMS 2 + &END DUMMY_ATOMS + ! CDFT convergence and optimizer settings + &OUTER_SCF ON + TYPE CDFT_CONSTRAINT + EXTRAPOLATION_ORDER 2 + MAX_SCF 10 + ! Convergence threshold + EPS_SCF 1.0E-3 + ! Optimizer selection: + ! Now Newton's method with backtracking line search + OPTIMIZER NEWTON_LS + ! Optimizer (initial) step size + STEP_SIZE -1.0 + ! Note that the section CDFT_OPT exists in CP2K version >= 6.1 + ! Remove section for CP2K version 5.1 (keywords are unchanged) + &CDFT_OPT ON + ! Line search settings + MAX_LS 5 + CONTINUE_LS + FACTOR_LS 0.5 + ! Finite difference settings for Jacobian matrix + JACOBIAN_STEP 1.0E-2 + JACOBIAN_FREQ 1 1 + JACOBIAN_TYPE FD1 + JACOBIAN_RESTART FALSE + &END CDFT_OPT + &END + ! Settigs specific to Becke constraints + &BECKE_CONSTRAINT + ... + &END BECKE_CONSTRAINT + ! Print information about CDFT calculation + &PROGRAM_RUN_INFO ON + &EACH + QS_SCF 1 + &END EACH + COMMON_ITERATION_LEVELS 2 + ADD_LAST NUMERIC + FILENAME ./${NAME} + &END PROGRAM_RUN_INFO + &END CDFT +&END QS +``` + +The structure of this input section is quite straightforward and consists of three parts: + +- Constraint definitions (type, which atoms to include, constraint target, etc) +- CDFT SCF loop settings (solver, convergence criterion, etc) +- Constraint weight function specific settings (Becke/Hirshfeld subsections) + +In the above example, a Becke constraint is selected using the keyword +[TYPE_OF_CONSTRAINT](#CP2K_INPUT.FORCE_EVAL.DFT.QS.CDFT.TYPE_OF_CONSTRAINT). The actual constraints +are defined using the section [ATOM_GROUP](#CP2K_INPUT.FORCE_EVAL.DFT.QS.CDFT.ATOM_GROUP). Each +repetition of this section defines a new constraint. The constraint atoms are selected with the +keyword [ATOMS](#CP2K_INPUT.FORCE_EVAL.DFT.QS.CDFT.ATOM_GROUP.ATOMS) and the keyword +[COEFF](#CP2K_INPUT.FORCE_EVAL.DFT.QS.CDFT.ATOM_GROUP.COEFF) determines how the atoms are summed up +to form the constraint. Usually all coefficients are set to +1, but mixing +1 and -1 coefficients +would define the constraint as the difference between two groups of atoms. The keywords +[TARGET](#CP2K_INPUT.FORCE_EVAL.DFT.QS.CDFT.TARGET) and +[STRENGTH](#CP2K_INPUT.FORCE_EVAL.DFT.QS.CDFT.STRENGTH) define the constraint target values and the +initial constraint strengths $\vec\lambda$, respectively. The constraint target value should be the +desired number of valence electrons on the constraint atoms, suitably multiplied by atomic +coefficients in case a relative constraint between two atom groups has been used. The constaint type +is selected with the keyword +[CONSTRAINT_TYPE](#CP2K_INPUT.FORCE_EVAL.DFT.QS.CDFT.ATOM_GROUP.CONSTRAINT_TYPE). It is also +possible to use fragment based constraints +[FRAGMENT_CONSTRAINT](#CP2K_INPUT.FORCE_EVAL.DFT.QS.CDFT.ATOM_GROUP.FRAGMENT_CONSTRAINT), in which +case the constraint target value is calculated from the superposition of isolated fragment densities +according to the scheme in Figure 2. + +![](images/cdft-fragment-constraint.png){align=center width=350px} + +**Figure 2.** Using a fragment based CDFT constraint. The system is first divided into two fragments +with atomic positions fixed in the same configuration as in the full system. The electron and spin +densities of the fragment systems are then saved to cube files and subsequently used as input files +for the CDFT calculation, where the constraint target value is calculated from the superimposed +fragment densities. + +The OUTER_SCF section within the CDFT section defines settings for the CDFT SCF loop. The keyword +[EPS_SCF](#CP2K_INPUT.FORCE_EVAL.DFT.QS.CDFT.OUTER_SCF.EPS_SCF) defines the CDFT constraint +convergence threshold $\varepsilon$ and +[OPTIMIZER](#CP2K_INPUT.FORCE_EVAL.DFT.QS.CDFT.OUTER_SCF.OPTIMIZER) selects the CDFT optimizer. +Using Newton or quasi-Newton optimizers (Broyden methods) is recommended for most applications. +These optimizers accept additional control settings that define how the Jacobian matrix is +calculated (keywords `JACOBIAN_*`) and how to optimize the step size $\alpha$ (keywords `*_LS`). +These keywords are available in the +[CDFT_OPT](#CP2K_INPUT.FORCE_EVAL.DFT.QS.CDFT.OUTER_SCF.CDFT_OPT) section. MD simulations with a +single constraint might benefit from using the bisect optimizer, which avoids building the Jacobian +matrix, in case a considerable amount of the total time per MD step is spent in building the +Jacobian. Notice, however, that the frequency of Jacobian rebuilds +[JACOBIAN_FREQ](#CP2K_INPUT.FORCE_EVAL.DFT.QS.CDFT.OUTER_SCF.CDFT_OPT.JACOBIAN_FREQ) can be +controlled on a per MD step and per CDFT SCF step basis. The Broyden optimizers require less +frequent rebuilds of the Jacobian matrix because the matrix is +[rank-one updated](https://en.wikipedia.org/wiki/Broyden%27s_method) every iteration, although the +stability of the method with respect to the rebuild frequency needs to be carefully studied. + +Above, for instance, the Jacobian is explicitly calculated every CDFT SCF iteration and MD step by +perturbing each constraint Lagragian using a first order forward difference stencil with a step size +of $10^{-2}$. The Newton step size is optimized with backtracking line search using the update +formula $\alpha_n = 0.5*\alpha_{n-1}$ for a maximum of 5 steps as long as the CDFT constraint error +decreases. + +### Available constraints + +The CDFT module in CP2K currently supports using Becke or Hirshfeld based constraints. The main +aspects of these weight functions and their use as CDFT constraints will be explained in this +section. Weight function specific settings are defined in the sections +[BECKE_CONSTRAINT](#CP2K_INPUT.FORCE_EVAL.DFT.QS.CDFT.BECKE_CONSTRAINT) +[HIRSHFELD_CONSTRAINT](#CP2K_INPUT.FORCE_EVAL.DFT.QS.CDFT.HIRSHFELD_CONSTRAINT). If you want to +visualize the weight functions to e.g. see the effects of using different parameters, you can use +the section [WEIGHT_FUNCTION](#CP2K_INPUT.FORCE_EVAL.DFT.QS.CDFT.PROGRAM_RUN_INFO.WEIGHT_FUNCTION) +to print the CDFT weight function to a cube file. + +#### Becke constraints + +The Becke density partitioning method can be considered as a smoothed Voronoi scheme. In Voronoi +partitioning, the volume occupied by each atom is the set of real space grid points $\mathbf{r}$ +which are closer to that particular atom than to any other atom in the system. An example Voronoi +diagram is given below in Figure 3. The line segments in this figure define real space points which +are equidistant from two atoms, while vertices correspond to grid points which are equidistant from +three or more atoms. The Becke cell function $P_i$ is overlayed on top of the Voronoi diagram and it +decays smoothly from 1 to 0 across the Voronoi polyhedron boundary. Using a smooth density +partitioning function improves numerical stability in simulations. + +![](images/cdft-becke-atomicsize.png){align=center width=400px} + +![](images/cdft-becke.png){align=center width=400px} + +**Figure 3.** Comparison of the Voronoi (lines) and Becke partitioning (contours) schemes. At left, +the Becke partitioning is performed without atomic size information. At right, the size of the red +atom is 30 % larger than the black atoms, and the contours of the red atom extend farther than +without atomic size adjustments. + +The Voronoi and, by extension, the Becke partitioning methods treat each element equally. This leads +to unphysical partial charges in most systems. For example, the Becke scheme predicts a positive +charge on oxygen and a negative charge on hydrogen in water (see examples for input files). This +problem can be remedied by accounting for atomic radii during the partitioning. This behavior is +activated by the keyword +[ADJUST_SIZE](#CP2K_INPUT.FORCE_EVAL.DFT.QS.CDFT.BECKE_CONSTRAINT.ADJUST_SIZE) and the atomic radii +are defined with the keyword +[ATOMIC_RADII](#CP2K_INPUT.FORCE_EVAL.DFT.QS.CDFT.BECKE_CONSTRAINT.ATOMIC_RADII). The atomic radii +should be set to values that reflect the system under simulation, e.g. using additive covalent radii +for covalent molecules or Shannon's ionic radii for ionic compounds. An example on how atomic size +adjustments affect the Becke cell functions has been visualized above in Figure 3 at right, where +the size of the red atom is set to a value 30 % larger than the black atoms causing the red atom's +contours to extend farther than without atomic size adjustments. + +The algorithmic implementation of the Becke density partitioning method has been detailed +[here](https://dx.doi.org/10.1021/acs.jctc.6b01085). In brief, this involves iterating over each +atom pair permutation $\{\mathbf{R}_i, \mathbf{R}_j\}, j\neq i$ at every real space grid point +$\mathbf{r}$. This leads to a poor scaling with respect to the system size (cell size and planewave +cutoff) and the number of atoms within the system, and is particularly troublesome for solvated +system simulations. The computational cost of the Becke method can be considerably decreased by +noting that only the grid points within a cutoff distance $R_{cutoff}$ +([CUTOFF_TYPE](#CP2K_INPUT.FORCE_EVAL.DFT.QS.CDFT.BECKE_CONSTRAINT.CUTOFF_TYPE)) of atoms involved +in constraints actually need to be considered. The other grid points can be efficiently screened +with constraint atom centered spherical Gaussian functions, activated by the keyword +[CAVITY_CONFINE](#CP2K_INPUT.FORCE_EVAL.DFT.QS.CDFT.BECKE_CONSTRAINT.CAVITY_CONFINE) and controlled +by other keywords of the form `CAVITY_*`. The exact details of this confinement scheme and why it +can be used are explained in the implementation paper. + +An example of a Becke constraint input section is given below. This choice of parameters should be +reasonable for most systems, ignoring the atomic radii and constraint definitions which are system +dependent. Decreasing the partitioning cutoff might be useful for solvated system MD simulations, +but extensive testing is always necessary before starting production simulations. + +```none +&CDFT + ... + &BECKE_CONSTRAINT + ! Take atomic radii into account? + ADJUST_SIZE FALSE + ATOMIC_RADII 0.63 0.32 + ! Cutoff scheme + CUTOFF_TYPE ELEMENT + ELEMENT_CUTOFF 6.0 + ! Perform Becke partitioning only within the space + ! spanned by constraint atom centered spherical Gaussians + ! (reduces cost for solvated systems) + CAVITY_CONFINE TRUE + CAVITY_SHAPE VDW + EPS_CAVITY 1.0E-7 + IN_MEMORY TRUE + SHOULD_SKIP TRUE +&END CDFT +``` + +#### Hirshfeld constraints + +Hirshfeld constraints are cheaper to construct than Becke constraints in large systems because +Hirshfeld constraints are essentially just weighted sums of spherical Gaussian functions. The +keywords [SHAPE_FUNCTION](#CP2K_INPUT.FORCE_EVAL.DFT.QS.CDFT.HIRSHFELD_CONSTRAINT.SHAPE_FUNCTION) +and [GAUSSIAN_SHAPE](#CP2K_INPUT.FORCE_EVAL.DFT.QS.CDFT.HIRSHFELD_CONSTRAINT.GAUSSIAN_SHAPE) define +which type of Hirshfeld constraint to apply to the system, consistent with the options for printing +of Hirshfeld atomic charges (see +[SHAPE_FUNCTION](#CP2K_INPUT.FORCE_EVAL.DFT.PRINT.HIRSHFELD.SHAPE_FUNCTION)). + +The shape function keyword accepts two values: Gaussian or Density. + +- The first choice implies that the CDFT weight function for each atom is a single Gaussian function + whose radius is controlled by the GAUSSIAN_SHAPE keyword. By default, tabulated covalent radii are + used as the radii of the Gaussian, but it is also possible to select van der Waals radii or to + define custom radii. +- The latter choice implies that the atomic weight function are constructed from isolated atomic + densities which are expanded in terms of multiple spherical Gaussians. This choice avoids the + introduction of any empirical parameters, and generally provides a more robust description of + atomic charges than either Becke or Gaussian based Hirshfeld charge partitioning. + +### Selected examples + +(example-zn-dimer-cation)= + +#### Zn dimer cation + +In this example, we will perform two CDFT simulations for the Zn dimer cation $\mathrm{Zn}_2^+$. As +the distance $R$ between the two atoms is increased, the excess charge in the system should localize +onto one of the Zn atoms forming $\mathrm{Zn}^+ + \mathrm{Zn}$. With standard GGA and hybrid +functionals, such as PBE, PBE0, BLYP or B3LYP, this is however not the case. The excess charge will +instead be equally shared among the two atoms $\mathrm{Zn}^{0.5+} + \mathrm{Zn}^{0.5+}$ regardless +of separation. We can force the charge to localize onto one of these atoms with CDFT. + +You can download the input files from +[here](https://www.cp2k.org/_media/:howto:cdft-tutorial-zn.zip). Unzip the folder and execute the +file `energy.bash` (use flag `-h` for usage instructions) to run a standard DFT calculation, +followed by two CDFT simulations where the first Zn atom is constrained to charges +1 and 0, +respectively. The script will also run a mixed CDFT calculation, which will be analyzed in a +[subsequent section](#example-coupling-zn-cation-dimer). Modify paths to basis sets and other CP2K +data files in the file `dft-common-params.inc` in case they are in non-standard paths relative to +the CP2K binary. The calculations will take a while to run. In the meanwhile, you can + +- Check the calculated partial charges in the standard DFT output file +- Study the script file to understand how it works +- Study the generated output files from the CDFT simulations + +Assuming the CDFT simulations have finished, the following files were generated + +- `*.out` + - This is the standard CP2K output file and now also contains all the output from the CDFT SCF + iterations. Each iteration step starts a new DFT energy optimization using new values of the + constraint Lagrangian multipliers $\vec{\lambda}$, as generated by the selected CDFT optimizer. + - An example output from the end of one the CDFT simulations is provided below. + - Information about the CDFT SCF iteration process and the constraints and their convergence is + printed alongside the usual CP2K SCF iteration information. + - With CDFT optimizers that support backtracking line search, the density optimization process at + each CDFT SCF step is restarted from the optimized constraint strength and density obtained + previously during line search. Consequently, when the line search was successful (i.e. the + energy optimization converged), the density optimization should converge terminate in exactly 1 + step each CDFT SCF iteration (refer to the output files, as this should now be the case). +- `*-LineSearch.out` + - The progress of the optimization of the Newton step size $\alpha$ using backtracking line search + is reported in this file. Observe that the step size is halved on each iteration if the CDFT + constraint error ("Deviation from target" in the output) decreases. +- `*.cdftLog` + - CDFT parameters (atomic radii, constraint definitions and cutoffs) and Becke partial charges are + printed in these files. +- `*-JacobianInfo.out` + - The output from the calculation of the Jacobian matrix $\mathbf{J}$ is reported in this file. +- `*.inverseJacobian` + - This is a restart file for the inverse Jacobian matrix. + +Using the above list, study the generated output files to understand how the CDFT SCF loop is +integrated with the standard CP2K DFT SCF process. + +```none + SCF WAVEFUNCTION OPTIMIZATION + + ----------------------------------- OT --------------------------------------- + Minimizer : DIIS : direct inversion + in the iterative subspace + using 7 DIIS vectors + safer DIIS on + Preconditioner : FULL_ALL : diagonalization, state selective + Precond_solver : DEFAULT + stepsize : 0.15000000 energy_gap : 0.08000000 + ortho_irac : CHOL irac_degree : 4 + max_irac : 50 eps_irac : 0.10000E-09 + eps_irac_switch: 0.10000E-01 eps_irac_quick_exit: 0.10000E-04 + on_the_fly_loc : F + ----------------------------------- OT --------------------------------------- + + Step Update method Time Convergence Total energy Change + ------------------------------------------------------------------------------ +qs_ot_get_orbitals_ref 0: ||P-I||= 0.10493E-10, ortho_irac = POLY +qs_ot_ref_poly 1: quick exit! +qs_ot_get_orbitals_ref 0: ||P-I||= 0.11959E-12, ortho_irac = POLY +qs_ot_ref_poly 1: quick exit! + 1 OT DIIS 0.15E+00 2.5 0.00000022 -120.6126709217 -1.21E+02 + + *** SCF run converged in 1 steps *** + + + Electronic density on regular grids: -22.9999999253 0.0000000747 + Core density on regular grids: 24.0000000000 -0.0000000000 + Total charge density on r-space grids: 1.0000000746 + Total charge density g-space grids: 1.0000000746 + + Overlap energy of the core charge distribution: 0.00000000000000 + Self energy of the core charge distribution: -159.30058829583706 + Core Hamiltonian energy: 50.63132167014943 + Hartree energy: 5.66516329750697 + Exchange-correlation energy: -17.60803221234604 + Dispersion energy: -0.00058223840320 + + Total energy: -120.61267092172808 + + outer SCF iter = 1 RMS gradient = 0.22E-06 energy = -120.6126709217 + outer SCF loop converged in 1 iterations or 1 steps + + + CDFT SCF iter = 5 RMS gradient = 0.13E-03 energy = -120.6126709217 + CDFT SCF loop converged in 5 iterations or 37 steps + + + --------------------- Becke constraint information --------------------- + Atomic group : 1 + Type of constraint : Charge density constraint + Target value of constraint : 11.000000000000 + Current value of constraint : 11.000126158558 + Deviation from target : 1.262E-04 + Strength of constraint : 0.371415167271 + ------------------------------------------------------------------------ +``` + +#### Charge transfer energy in water dimer + +In this example, we will calculate the charge transfer energy, $-\Delta E_\mathrm{CT}$, of water +dimer. This quantity is conveniently defined in CDFT as (see +[here](https://dx.doi.org/10.1039/C6CP07475H)) + +$$ +-\Delta E_\mathrm{CT} = E_\mathrm{CDFT}-E_\mathrm{DFT} +$$ + +where $E_\mathrm{DFT}$ is the DFT total energy of the system, and $E_\mathrm{CDFT}$ is the CDFT +energy of the system when charge transfer between the two molecules is prevented. + +We will calculate the charge transfer energy with four different constraints: default Becke +constraint, Becke constraint with atomic size adjustments using covalent radii from +[this publication](https://dx.doi.org/10.1002/chem.200800987), and a fragment based Becke constraint +with and without the same atomic size adjustments. The input files can be downloaded from +[here](https://www.cp2k.org/_media/:howto:cdft-tutorial-water.zip). Execute the file `energy.bash` +to run all the simulations: standard DFT simulations for the full and two fragment systems, and the +aforementioned CDFT simulations with different constraints. Modify the include (`.inc`) file if +necessary as you would have before. Study the input files while the calculations are running. + +After the calculations have finished, answer the following questions + +- Compare the partial charges of unconstrained PBE water as predicted by the Becke population + analysis method with and without atomic size adjustments (Hint: the CDFT calculations were + restarted from the DFT wavefunction) +- How much charge is transferred between the two water molecules according to the different + constraint methods? Note that net charges for fragment based constraints are not reported with + respect to the core charge, but they can be recovered by post-processing the reported absolute + populations. Instead, each atomic charge is referenced to the number of electrons per atom in the + system where the isolated densities are superimposed. +- Compare the calculated charge transfer energies to the + [reference value 1.7 mHa](https://dx.doi.org/10.1039/C6CP07475H), calculated at the + PBE0/def-QZVP/CDFT level of theory using a different code and constraint. Which is closest to the + reference value? Why? (Hint: Look at previous question) + +#### Zn dimer cation with Hirshfeld constraints + +This simulation requires CP2K version 7.0 or later. + +This tutorial is exactly the same as the Zn dimer example above but using Hirshfeld partitioning +based constraints instead of Becke constraints. You can find the input files +[here](https://www.cp2k.org/_media/:howto:cdft-tutorial-hirshfeld.zip). + +It might be instructive to visualize how the Becke and Hirshfeld weight function schemes differ, in +particular, how the methods assign a volume to each atom in the system. You can activate the section +[WEIGHT_FUNCTION](#CP2K_INPUT.FORCE_EVAL.DFT.QS.CDFT.PROGRAM_RUN_INFO.WEIGHT_FUNCTION) to output the +weight function as a cube file which you can visualize with e.g. VMD. Feel free to modify the water +tutorial above to look at the differences between Becke and Hirshfeld constraints in a system with +different chemical elements. + +______________________________________________________________________ + +## Using the mixed CDFT module + +Additional properties can be calculated from the interactions between CDFT states. In CP2K, these +types of simulations are called mixed CDFT simulations because the module leverages the +[MIXED FORCE_EVAL](#CP2K_INPUT.FORCE_EVAL.MIXED) type to efficiently treat multiple CDFT states in +parallel. Mixed CDFT calculations are useful in a number of applications including + +- Calculating charge transfer kinetics parameters +- Performing configuration interaction calculations within the basis of CDFT states + +In this part of the tutorial, the theoretical basis for mixed CDFT will first be established. The +quantities accessible through such simulations will also be introduced. The structure of a mixed +CDFT input file will then be discussed. The tutorial is concluded with a walk through of an example +calculation. + +### Theoretical basis + +The theoretical concepts related to mixed CDFT calculations are best introduced through an example. +Consider the following one electron transfer processs X^- + Y -> X + Y^-. Denote the +initial and final states of this reaction as A and B, respectively. Now, according to the Marcus +theory of electron transfer, the charge transfer rate of this reaction is given by the rate equation + +$$ +k_\mathrm{ab}=\frac{2\pi}{\hbar}\frac{\left<\left|\mathbf{H}_\mathrm{ab}\right|^2\right>_T}{\sqrt{4\pi k_bT\xi}}\exp\left(-\frac{(\xi+\Delta A)^2}{4\pi k_bT\xi} \right) +$$ + +where $\Delta A$ is the reaction free energy, $\xi$ is the solvent reorganization energy, and +$\left|\mathbf{H}_\mathrm{ab}\right|$ is the electronic coupling. The first two quantities can be +obtained from free energy simulations as discussed e.g. in +[here](https://dx.doi.org/10.1021/acs.jctc.6b01085). The electronic coupling is rigorously defined +as the interaction energy between wavefunctions $\Psi$ representing the two reaction states + +$$ +\mathbf{H}_\mathrm{ab} = \left<\Psi_\mathrm{a}\left| \mathcal{H}\right|\Psi_\mathrm{b}\right> +$$ + +where $\mathcal{H}$ is the many-electron Hamiltonian operator. The usefulness of the electronic +coupling quantity is not limited to the Marcus rate equation, but it also a central quantity in +other charge transfer theories as well as in CDFT based +[configuration interaction](https://dx.doi.org/10.1063/1.2800022). + +The true, interacting many-electron wavefunctions or the Hamiltonian are not available in CDFT +simulations. The electronic coupling is instead approximated using the CDFT surrogates + +$$ +\mathbf{H}_\mathrm{AB} \approx \left<\Phi_\mathrm{A}\left| \mathcal{H}_\mathrm{KS}\right|\Phi_\mathrm{B}\right> = E_\mathrm{B}S_\mathrm{AB}-\sum_c \lambda_c^\mathrm{B} \mathbf{W}_c^\mathrm{AB} \\ +\mathbf{H}_\mathrm{BA} \approx \left<\Phi_\mathrm{B}\left| \mathcal{H}_\mathrm{KS}\right|\Phi_\mathrm{A}\right> = E_\mathrm{A}S_\mathrm{BA}-\sum_c \lambda_c^\mathrm{A} \mathbf{W}_c^\mathrm{BA} +$$ + +where $\Phi$ are the CDFT Kohn-Sham determinants, +$S_\mathrm{AB}= \left<\Phi_\mathrm{A}|\Phi_\mathrm{B}\right>$, $E_\mathrm{I}$ is the CDFT energy of +state $\mathrm{I}$, and $\mathbf{W}_c^\mathrm{AB}$ are the weight function matrices defined by + +$$ +\mathbf{W}_c^\mathrm{AB} = \left<\Phi_\mathrm{A}\left| w_c^\mathrm{B}(\mathbf{r}) \right|\Phi_\mathrm{B}\right> +$$ + +In the above expressions, capital subscripts have been used to emphasize the fact that the CDFT +determinants are in general nonorthogonal. The electronic couplings and overlaps are collected into +matrices $\mathbf{H}$ and $\mathbf{S}$, respectively. The off-diagonal elements of $\mathbf{H}$ are +not symmetric. The matrix is converted to symmetric form by setting + +$$ +\mathbf{H'}_\mathrm{AB}=\frac{\mathbf{H}_\mathrm{AB}+\mathbf{H}_\mathrm{BA}}{2} = \frac{E_\mathrm{A}+E_\mathrm{B}}{2}S_\mathrm{AB}-\sum_c \left<\Phi_\mathrm{A}\left|\frac{\lambda_c^\mathrm{A}w_c^\mathrm{A}(\mathbf{r})+\lambda_c^\mathrm{B}w_c^\mathrm{B}(\mathbf{r})}{2} \right|\Phi_\mathrm{B}\right> +$$ + +and setting $\mathbf{H'}_\mathrm{BA}=\mathbf{H'}_\mathrm{AB}$. + +The resulting matrix $\mathbf{H'}$ is then orthogonalized to yield the final electronic coupling. +The following orthogonalization methods are available: + +- Rotate CDFT states to eigenstates of the weight matrix $\mathbf{W}$. This is the default behavior + for systems with only one constraint that is identically defined across all CDFT states, not + applicable otherwise. +- Löwdin's symmetrical orthogonalization + $\mathbf{H} = \mathbf{S}^{-1/2}\mathbf{H}'\mathbf{S}^{-1/2}$. This is the default behavior for + systems with multiple constraints and can always be activated with the keyword + [LOWDIN](#CP2K_INPUT.FORCE_EVAL.MIXED.MIXED_CDFT.LOWDIN). +- The so-called wavefunction overlap method where the ground state Kohn-Sham solution is represented + as the linear combination of CDFT states, see keyword + [WFN_OVERLAP](#CP2K_INPUT.FORCE_EVAL.MIXED.MIXED_CDFT.WFN_OVERLAP). + +### Structure of input file + +Mixed CDFT calculations are activated through the [MIXED FORCE_EVAL](#CP2K_INPUT.FORCE_EVAL.MIXED) +section by setting [MIXING_TYPE](#CP2K_INPUT.FORCE_EVAL.MIXED.MIXING_TYPE) to MIXED_CDFT and +providing an appropriate [MIXED_CDFT](#CP2K_INPUT.FORCE_EVAL.MIXED.MIXED_CDFT) input section. The +individual CDFT states involved in a mixed CDFT calculation should correspond to different +localizations of charge and/or spin in the same system. The constraint definitions do not have to be +identical (i.e. defined using the same sets of atoms) in all states as long as the number of +constraints is the same. + +The CDFT states are included as their own [FORCE_EVAL](#CP2K_INPUT.FORCE_EVAL) sections. It is +highly recommended that the CDFT states are first converged in separate simulations, the +[FORCE_EVAL](#CP2K_INPUT.FORCE_EVAL) sections from these simulations are then copy-pasted into the +mixed CDFT input file, and the mixed CDFT method is used as post-processing analysis tool. The +converged wavefunctions and constraint strengths $\vec\lambda$ should be supplied as restart +quantities for the mixed CDFT calculation. Using templates and the CP2K `@include` and `@set` +directives is strongly encouraged to keep the mixed CDFT input tidy, see the example input file +below + +```none +&FORCE_EVAL + METHOD MIXED + &MIXED + MIXING_TYPE MIXED_CDFT + NGROUPS 1 + &MIXED_CDFT + ! Calculate mixed CDFT properties every COUPLING step + COUPLING 1 + ! Settings determining how forces are mixed + FORCE_STATES 1 2 + LAMBDA 1.0 + ! Orthogonalize CDFT states using Lowdin's method + ! in addition to standard method + LOWDIN TRUE + ! Configuration interaction? + CI FALSE + ! Turn on printing + &PRINT + &PROGRAM_RUN_INFO ON + &END + &END PRINT + &END MIXED_CDFT + &END MIXED + @include subsys.inc +&END FORCE_EVAL +# Zn+ Zn +&FORCE_EVAL + @SET WFN_FILE ${WFN_FILE_1} + @SET RESTART ${RESTART_1} + @SET NAME ${PROJECT_NAME}-state1 + @SET BECKE_TARGET ${BECKE_TARGET_1} + @SET BECKE_STR ${BECKE_STR_1} + METHOD QS + @include ${DFT_FILE} +&END FORCE_EVAL +# Zn Zn+ +&FORCE_EVAL + @SET WFN_FILE ${WFN_FILE_2} + @SET RESTART ${RESTART_2} + @SET NAME ${PROJECT_NAME}-state2 + @SET BECKE_TARGET ${BECKE_TARGET_2} + @SET BECKE_STR ${BECKE_STR_2} + METHOD QS + @include ${DFT_FILE} +&END FORCE_EVAL +``` + +In the above example input file, a common file `${DFT_FILE}` is used as a template for the +[DFT](#CP2K_INPUT.FORCE_EVAL.DFT) subsection. The CDFT state specific constraint settings and the +wavefunction filename are passed through variables. The electronic coupling is calculated using the +default weight function matrix and Löwdin orthogonalization methods. If molecular dynamics were +performed with the above input file, the forces would be mixed according to linear mixing scheme +$F += \\lambda F_1 + (1-\\lambda) +F_2$, where the states $F_i$ are selected with the keyword +[FORCE_STATES](#CP2K_INPUT.FORCE_EVAL.MIXED.MIXED_CDFT.FORCE_STATES) and the mixing parameter +\$\\lambda\$ with [LAMBDA](#CP2K_INPUT.FORCE_EVAL.MIXED.MIXED_CDFT.LAMBDA). No configuration +interaction calculation is performed. The [MIXED_CDFT](#CP2K_INPUT.FORCE_EVAL.MIXED.MIXED_CDFT) +section accepts some additional keywords, which have been described in the manual. + +The keyword [NGROUPS](#CP2K_INPUT.FORCE_EVAL.MIXED.NGROUPS) is set to 1, which implies that the two +CDFT states are treated sequentially utilizing the full set of $N$ MPI processes for the simulation. +The CDFT weight function and its gradients are copied from state to state if the constraints +definitions are identical in each CDFT state, because construction of these terms might be expensive +in large systems. + +The keyword [NGROUPS](#CP2K_INPUT.FORCE_EVAL.MIXED.NGROUPS) could also be set to 2 or a larger value +if using more than 2 CDFT states. In this case, each CDFT state is solved in parallel using +$N/N_\mathrm{groups}$ processors. This will likely reduce the wall clock time of your simulation at +the expense of more computing resources. However, you should note that the weight function and its +gradients are computed separately for each state instead of copied from state to state (if +possible). This can be costly for large solvated systems. + +A special run type is available for `NGROUPS 2` if the keyword +[PARALLEL_BUILD](#CP2K_INPUT.FORCE_EVAL.MIXED.MIXED_CDFT.PARALLEL_BUILD) is activated. In this case, +the CDFT weight function and gradients are first built in parallel on $N$ MPI processes, which are +subsequently copied onto the two MPI processor groups of size $N/2$ which solve the CDFT states in +parallel. This operating mode is limited to two CDFT states with one identically defined total +charge density constraint. This operating mode is an advanced feature which might be useful for +large scale MD simulations to save computational wallclock time at the expense of higher CPU core +usage. The operating mode should only be used in conjuction with +[dynamic load balancing](#CP2K_INPUT.FORCE_EVAL.MIXED.MIXED_CDFT.DLB) if possible, and will likely +require tweaking the load balancing parameters. + +(example-coupling-zn-cation-dimer)= + +### Example: Electronic coupling of Zn cation dimer + +In this example, we will calculate the electronic coupling for the reaction +$\mathrm{Zn}^+ +\mathrm{Zn} \rightleftharpoons \mathrm{Zn}+ \mathrm{Zn}^+$. The initial and final +states of this reaction were already converged with CDFT in a +[previous section of this tutorial](#example-zn-dimer-cation). That part of the tutorial must be +completed before proceeding. The full input files, in particular the mixed CDFT input file +`energy_mixed.inp`, were also given in that section. + +The converged CDFT states are used as input for the mixed CDFT calculation. The calculation does not +take long to run as a result. The mixed CDFT input file uses template files to keep the input tidy, +as was discussed in the previous section. Find and study the corresponding section in the +`energy.bash` script file to see how variables in the main mixed CDFT template `energy_mixed.inp` +are initialized. + +A number of files are generated by the mixed CDFT calculation. The main output from the calculation +can be found in the file `Zn-5A-mixed-cdft.out`. The relevant part of the output is included below. +The mixed CDFT analysis is printed after the header lines `MIXED_CDFT|`. For each unique CDFT state +permutation $\{i,j\}, i$ and +$|\mathrm{HH^+}>$ as the basis. + +We can use CDFT states as the basis of a configuration interaction (CI) simulation to correct for +SIE in this system. As the figure above shows, CDFT-CI using the PBE functional is able to reproduce +the exact dissociation profile. You can read up on the theory behind CDFT-CI simulations from the +references given at the start of this tutorial. Very briefly, CDFT-CI simulations involve +representing the system's wavefunction as a linear combination of multiple CDFT states where the +charge/spin density is constrained differently in different states. The CI expansion coefficients +and energies are then obtained by solving a generalized eigenvalue equation where the effective +Hamiltonian matrix describes how the CDFT states interact with each other. + +In this tutorial, you will reproduce the DFT and CDFT results from the figure above. You can find +the input files [here](https://www.cp2k.org/_media/:howto:cdft-tutorial-h2.zip). The reference data +used to plot Figure 4 are also included in the zip-folder. Please note that the reference results +were obtained with a larger basis set and planewave cutoff as well as tighter convergence criteria +than the settings you will be using in this tutorial. + +- Start by examining the simulation script `energy.bash`. This tutorial involves a rather large + number of simulations so running them will take a while. You can use the flag `-x` to separately + run the different types of simulations (DFT, CDFT, CDFT-CI) needed in this tutorial. +- While the CDFT simulations are running, look at the results from the DFT simulations with PBE. Can + you figure out the reason why PBE predicts an unphysical dissociation profile? (Hint. Compute the + partial charges). +- Inspect the output files produced by the CDFT-CI simulations once they are done. Find the output + from the CDFT-CI module in the main output files. Look at the CI expansion coefficients in terms + of the CDFT states. How would you characterize the CI wavefunction? Is the result sensible? What + about the atomic partial charges in different CDFT states as a function of distance? +- Plot the CDFT-CI and DFT dissociation profiles with your favorite plotting tool. Use the + Hartree-Fock data from the provided data file as a reference. You can produce a similar data file + from your simulations by calling `energy.bash` with the flag `-x results`. diff --git a/docs/methods/dft.md b/docs/methods/dft.md index f418f10a53..823a3b95a5 100644 --- a/docs/methods/dft.md +++ b/docs/methods/dft.md @@ -9,7 +9,7 @@ gpw gapw hfx lrigpw -Constrained DFT +cdft ls-dft basis_sets pseudopotentials diff --git a/docs/methods/images/cdft-becke-atomicsize.png b/docs/methods/images/cdft-becke-atomicsize.png new file mode 100644 index 0000000000000000000000000000000000000000..65b4823124f075650953c21b19abcec85dd13555 GIT binary patch literal 197392 zcma&O1z1(<(>}aF5fu@GP*O!fKw3H#6_FBuCr^!zv2tp|F@SZ$^;CmqmUhK&e z@J_`?tzP)gaof8RPfo(i?c~!p@bf9FhpM&+!a|FF;W&K}aez1Z?Cz`BDOeiXIqKRN zAm+MuR%VuVW+r-991Lu1O)M?0aImnmu;033Y-eZ1&&v8=zrbQ?W5kL~$VMQ@6-46R zohMF_bAygfPwe|@mR;M$KGnU9kH5x-gR6A?%KclSS&nZrcKU8SBe7J%GA>V*l6$TSM)n)x=Z zH{D+euI^F$bWGc!PyXjs>A^>NmV}H<@>8+HJYz(Yp-ucg)2&;9*L&w30s<;y?lzwN 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