Introduction to the octopus code - TDDFT.org

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8 EigenSolverInitTolerance = 1.0e-6 EigenSolverFinalToleranceIteration = 7 EigenSolverFinalTolerance = 1.0e-6 EigenSolverMaxIter = 50 [The values given are the default ones. If you find that the convergence is too slow, you can try increasing the tolerance thresholds. Note that these variables do not only apply to the calculation of unoccupied states; they also apply to the eigenproblem at each one of the SCF step during the calculation of the ground state]. Note also that you can perform a first run setting a poor convergence criterion, and afterwards re-start the run with a more stringent criterion, making sure that you re-read the previously badly converged wave functions by setting: FromScratch = no This will restart the unoccupied states calculation from the point where the previous calculation ended. [This also holds for any other run mode; the FromScratch variable instructs the code to try and restart a previous calculation, or not.] This is important as using the default value of EigenSolverMaxIter, it is very likely that your unoccupied states will not converge (the program will tell, in the standard output, whether or not the eigenstates are converged or not). You may increase the value of EigenSolverMaxIter and restart. The distribution of KS eigenvalues (especially the differences between occupied and unoccupied ones) provides us with a first impression about the possible excitations of the system. However, it is only a very crude approximation; in the following we will learn how to improve on it. Q6. How do the eigenvalue differences differ from the experimental excitation energies provided, e.g., by an absorption spectrum 4. Determine the optimal time-step for the propagation of the TDKS equations While the time-independent Kohn-Sham equations constitute a boundary value problem, the solution of the time-dependent Kohn-Sham equations is an initial value problem. At t = t 0 the system is in some initial state described by the Kohn- Sham orbitals ϕ i (⃗r, t 0 ). In most cases the initial state will be the ground state of the system (i.e., ϕ i (⃗r, t 0 ) will be the solution of the ground-state Kohn-Sham equations). To solve the TDKS equations amounts to propagate this initial state until some final time, t f . The time-dependent Kohn-Sham equations can be rewritten in the integral form where the time-evolution operator, Û, is defined by [ ϕ i (⃗r, t f ) = Û(t f, t 0 )ϕ i (⃗r, t 0 ) , (8) Û(t ′ , t) = ˆT exp −i ∫ t ′ t dτ ĤKS(τ) ] . (9)
section 2. First session 9 Note that ĤKS is explicitly time-dependent due to the Hartree and xc potentials. It is therefore important to retain the time-ordering product ˆT, in the definition of the operator Û. The exponential in expression (9) is clearly too complex to be applied directly, and needs to be approximated in some suitable manner. octopus provides several approximation methods: the split-operator, Suzuki-Trotter (a higher order split-operator method), Enforced Time-Reversal Symmetry, Approximated Enforced Time-Reversal Symmetry, Exponential Midpoint Rule, and Magnus Expansion methods. The method used in the propagation is chosen putting in the input file the keyword TDEvolutionMethod = xxx where xxx is either split, suzuki-trotter, etrs, aetrs, exp mid, or magnus. Some of these methods require also a numerical approximation to the exponential of the Hamiltonian. The methods supplied by octopus with the input variable TDExponentialMethod = xxx are the split-operator, Suzuki-Trotter, Lanczos, Chebyshev methods and a standard n-th order expansion of the exponential. The value of the above variable is either split, suzuki-trotter, lanczos, chebyshev, or standard. You can look in the manual for some information about these methods. Other octopus variables that are relevant for the propagation are, obviously, the duration of the propagation (TDMaximumIter, which is the number of time iterations that the code will compute) and the time-step for the propagation (TDTimeStep). If we had infinite speed computers the latter could be chosen as small as numerical accuracy would allow. But, in the real world, we have to compromise and use a large time-step. However, the discretization of the evolution operator imposes an upper limit on the allowed time-step. This upper limit depends on the grid-spacing, becoming smaller for finer grids. You will now determine the relation between the maximum time-step that allows for an energy-conserving propagation and the gridspacing. To do this, you will have to perform several TDDFT runs for each grid spacing with different time-steps, and monitor the evolution of the system and the variation of the total energy. You will notice that an energy-conserving well-behaved propagation will occur only below a certain ∆t max . To start a simple TDDFT run with octopus you replace CalculationMode = gs with CalculationMode = td in the input file and include, e.g., the following lines
Page 1 and 2: Practical session octopus I-II Intr
Page 3 and 4: section 1. Introduction 3 The time-
Page 5 and 6: section 2. First session 5 http://w
Page 7: section 2. First session 7 valence
Page 11 and 12: section 3. Second session 11 Tasks
Page 13 and 14: section 3. Second session 13 Q4. Co
Page 15: REFERENCES, COMMENTS, ETC. 15 Refer

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