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Tutorials_Libra

Tutorials showcasing various capabilities of Libra

TOC

1. Rigid body

• 1.1. Construct & describe properties of rigid bodies

2. Integrators

• 2.1. Runge-Kutta 4-th order for Classical Mechanics
• 2.2. Runge-Kutta 4-th order for Quantum Mechanics

3. Linear algebra

• 3.1. operations with VECTOR, MATRIX, and CMATRIX data types
• 3.2. matrix functions, inversion, linear equations

4. Optimization

• 4.1. Gradient descent optimizer
• 4.2. line search using DIIS

5. Electronic structure calculations in Libra

• 5.1. Extended Huckel Theory, EHT

  • 5.1.1. Compact version
  • 5.1.2. Detailed version

• 5.2. Incomplete Neglect of Differential Overlap, INDO

  • 5.2.1. Compact version

6. Dynamics with Libra

• 6.1. Quantum-classical, trajectory methods

  • 6.1.1. Model, adiabatic MD
    • 6.1.1.1. NVE ensemble
    • 6.1.1.2. NVT ensemble
      • 6.1.1.2.1. 1 electronic state
  • 6.1.2. Model, common approach to adiabatic, Ehrenfest, and TSH
  • 6.1.3. Model, Ehrenfest recipes
  • 6.1.5. Atomistic, adiabatic dynamics, ground/excited
  • 6.1.6. Model, NBRA and non-NBRA
  • 6.1.7. Model, TSH with thermostat, quantum-vs-classical partitioning of DOFs, and constrining

• 6.2. Quantum-classical, neglect-of-back-reaction trajectory workflows

  • 6.2.1. step 1 with DFTB+
  • 6.2.2. step 1 with QE
  • 6.2.3. step 2 with QE
  • 6.2.4. step 3
    • 6.2.4.1. Compute single-particle NACs
    • 6.2.4.2. Compute many-body NACs
  • 6.2.5. step 4
    • 6.2.5.1. Initialze_data
    • 6.2.5.2. Dynamics

• 6.3. Hierarchical equations of motion, HEOM

  • 6.3.1. Computing population dynamics and lineshapes

• 6.4. DVR, on-the-grid wavepackets

  • 6.4.1. Gaussian wavepackets
    • 6.4.1.1. Computing matrix elements
  • 6.4.2. DVR basics
  • 6.4.3. SOFT dynamics in 1D and 2D
  • 6.4.4. More examples of DVR and dynamics

• 6.5. Fermi Golden Rule rates, FGR

7. Special functions

• 7.1. Sorting, matrix statistics
• 7.2. Fitting distributions to a superposition of Gaussians
• 7.3. Data statistics
• 7.4. Random numbers
  • 7.4.1. Basic random number generator, chaotic systems
  • 7.4.2. Random number generation using Metropolis Monte Carlo method

8. Model Hamiltonians

• 8.1. Define model Hamiltonians and plot PES
• 8.2. Define atomistic Hamiltonians and plot PES

9. Machine learning

• 9.1. Basics of the artificial neural networks (ANNs), Multilayer Perceptron (MLP)
• 9.2. Analytical derivatives of the ANNs
• 9.3. Advanced ANN training algorithms

10. Auxiliary functions and data types

• 10.1. Util functions and vectorized Libra types

11. Program-specific methods

• 11.1. ErgoSCF methods
  • 11.1.1. Basic methods
  • 11.1.2. Basic methods
• 11.2. QE methods
  • 11.2.1. pDOS
  • 11.2.2. MD
  • 11.2.3. Normal modes

12. Molecular builders

• 12.1. Crystal and QD builders

---

Use cases

1. Create a chemical system

• 5.1.1. |
• 12.1.

2. EHT calculations with Libra

• 5.1.1.

3. INDO calculations with Libra

• 5.2.1.

4. Compute .cube files from orbitals computed with Libra

• 5.1.1.
• 5.2.1.

5. Visualize the MOs from .cube files

• 5.1.1.
• 5.2.1.

6. Compute pDOS

• 5.1.1.
• 11.2.1.

7. Plot pDOS

• 5.1.1.
• 11.2.1.

8. Visualize MD trajectory with py3Dmol:

• 6.1.5.
• 6.2.1.
• 8.2.
• 11.2.2.

9. Construct the vibronic Hamiltonian from the QE MD calculations

• 6.2.3.

10. Read the vibronic Hamiltonian data files to obtain its properties

• 6.2.3.
• 6.2.4.1.
• 6.1.6.
• 6.1.7.

11. Compute the time-averaged nonadiabatic couplings of the vibronic Hamiltonian

• 6.2.3.
• 6.2.4.1.
• 6.2.4.2.

12. Manually construct a Slater Determinant basis

• 6.2.4.1.

13. Auto-generate a Slater Determinant basis

• 6.2.4.1.

14. Compute the energies and nonadiabatic couplings in the SD basis

• 6.2.4.1.
• 6.2.4.1.

15. Calculate population and coherence dynamics of a quantum system embedded in a bath

• 6.3.1.

16. Calculate absorbance spectral lineshapes of a quantum system embedded in a bath

• 6.3.1.

17. Construct and plot the Heller's wavepackets

• 6.4.1.1.

18. Compute the matrix elements of various operators with Heller's wavepackets

• 6.4.1.1.

19. Define diabatic abstract model Hamiltonian

• 8.1.
• 6.1.7.

20. Define adiabatic abstract model Hamiltonian

• 8.1.

21. Define adiabatic file-based model Hamiltonian

• 8.1.
• 6.1.6.

22. Plot 1D PES

• 8.1.
• 8.2.
• 6.1.6.
• 6.1.7.

23. Plot diabatic-to-adiabatic transformaitons vs. coordinate in 1D

• 8.1.

24. Plot 1D PES vs. time

• 8.1.
• 6.1.6.
• 6.1.7.

25. Define Libra/Psi4 intraface Hamiltonian

• 8.2.

26. Define Libra/DFTB+ intraface Hamiltonian

• 8.2.
• 6.1.5.
• 6.2.4.2.

27. Plot the PES of LiH at the EOM-CCSD/sto-3G level computed via interface of Libra with Psi4

• 8.2.

28. Plot the 1D PES of HFCO at the TD-DFTB level compute with interface of Libra with DFTB+

• 8.2.

29. Generate XYZ trajectory from a list of matrices

• 8.2.

30. Perform a ground state adiabatic MD with Libra

• 6.1.5.
• 6.1.6.

31. Perform an excited state adiabatic MD with Libra

• 6.1.5.
• 6.1.6.

32. Compute MD with DFTB+ via Libra

• 6.1.5.

33. Generate XYZ trajectory from HDF5 files

• 6.1.5.

34. Compute trajectory-averaged dephasing times

• 6.2.5.1

35. Compute trajectory-averaged energy gaps

• 6.2.5.1

36. Plot trajectory-averaged dephasing times

• 6.2.5.1

37. Fit the probability density of randomly distributed point with Gaussian density kernel functions

• 7.2.

38. Read the HDF5 files to setup Hamiltonians

• 6.1.6.

39. Read the HDF5 files to plot results of dynamical calculations

• 6.1.6.
• 6.1.7.

40. Compute nonadiabatic dynamics for atomistic systems with NBRA using Kohn-Sham states

• 6.2.5.1.

41. Plot the PES profiles with multidimensional model Hamiltonians

• 6.1.7.

42. MD and NAMD in the NVT ensemble

• 6.1.7.

43. Partitioning quantum and classical DOFs

• 6.1.7.

44. Coupling classical DOFs to thermostat

• 6.1.7.

45. Constraining DOFs in dynamics

• 6.1.7.

46. Numerically exact TD-SE

• 6.4.2.
• 6.4.3.
• 6.4.4.
• 6.4.4.
• 6.4.4.
• 6.4.4.

47. DVR calculations

• 6.4.2.
• 6.4.3.
• 6.4.4.
• 6.4.4.
• 6.4.4.
• 6.4.4.

48. Making animated gifs

• 6.4.3.

49. Integrating quantum Liouville's equation of motion

• 2.2.

50. Machine learning with MLP

• 9.1.
• 9.2.
• 9.3.

51. Artificial neural networks (ANN) and error Back Propagation algorithm

• 9.1.
• 9.3.

52. Derivatives of ANNs

• 9.2.

53. Convert Libra and Python data types

• 10.1.

54. Manipulate vectors (lists of data)

• 10.1.

55. Setup default values of Python dictionary

• 10.1.

56. Interfacing ErgoSCF and Libra

• 11.1.1.
• 11.1.2.

57. Sampling random numbers from common distributions

• 7.4.1.

58. Sampling random numbers from arbitrary distributions

• 7.4.1.
• 7.4.2.

59. Computing data probability densities and cumulative distribution functions

• 7.3.
• 7.4.1.
• 7.4.2.
• 11.2.2.

60. Generating (deterministic) quasi-random numbers

• 7.4.1.
• 7.4.2.

61. Dynamical regimes and chaotic systems

• 7.4.1.

62. Wigner sampling

• 7.4.2.

63. Canonical and microcanonical enesemble sampling

• 7.4.2.

64. Analyzing MD trajectories

• 11.2.2.

65. Normal modes

• 11.2.3.

66. Constructing quantum dots

• 12.1.

67. Constructing periodic structures

• 12.1.

68. Automatically determining connectivity in complex structures

• 12.1.

69. Computing NACs using ErgoSCF/Libra

• 11.1.2.

70. Computing NACs using DFTB+/Libra

• 6.2.4.1.
• 6.2.4.2.

71. Computing NACs using CP2K/Libra

• 6.2.4.1.
• 6.2.4.2.

72. Computing single-particle (KS-DFT, HF, semiempirical) NACs

• 6.2.3.
• 6.2.4.1.
• 11.1.2.

73. Computing many-body (TD-DFT, TD-DFTB, CI) NACs

• 6.2.4.2.

74. Saving ANNs to files and creating ANNs from XML files

• 9.1.
• 9.3.

75. Rprop algorithm for ANN training

• 9.3.

76. Momentum algorithm in the ANN training

• 9.3.

77. Weight decay in the ANN training

• 9.3.

---

Functions

• liblibra::libconverters

  • Py2Cpp_int 10.1.
  • Py2Cpp_double 10.1.
  • Py2Cpp_complex 10.1.
  • Py2Cpp_string 10.1.
  • Py2Cpp_VECTOR 10.1.
  • Py2Cpp_MATRIX 10.1.
  • Py2Cpp_CMATRIX 10.1.
  • Cpp2Py 10.1.

• liblibra::libdyn

  • libgwp
    • gwp_coupling 6.4.1.1.
    • gwp_dipole 6.4.1.1.
    • gwp_kinetic 6.4.1.1.
    • gwp_overlap 6.4.1.1.
    • gwp_value 6.4.1.1.
  • libwfcgrid
    • compute_imapping6.4.2.
    • compute_mapping 6.4.2.

• liblibra::liblinalg

  • pop_submatrix 9.1.

• liblibra::libmontecarlo

  • metropolis_gau 7.4.2.

• liblibra::libspecialfunctions

  • randperm 9.1.

• liblibra::libqm_tools

  • charge_density 5.1.1.
  • compute_dos 5.1.1.

• libra_py

  • autoconnect

    • autoconnect 12.1.
    • find_undercoordinated_atoms 12.1.

  • build

    • make_xyz_mat 11.2.2.
    • read_xyz 12.1.
    • make_xyz 12.1.
    • read_xyz_crystal 12.1.
    • generate_replicas_xyz 12.1.
    • generate_replicas_xyz2 12.1.
    • crop_sphere_xyz 12.1.
    • crop_sphere_xyz2 12.1.
    • crop_sphere_xyz3 12.1.
    • add_atom_to_system 12.1.
    • add_atoms_to_system 12.1.

  • dynamics

    • heom
      • compute
        • run_dynamics 6.3.1.
    • tsh
      • compute
        • generic_recipe 6.1.6. | 6.1.7.
        • init_electronic_dyn_var 6.1.5.
        • init_nuclear_dyn_var 6.1.5. | 6.1.6. | 6.1.7.
        • run_dynamics 6.1.5.
      • plot
        • add_basis_transform 6.1.6.
        • add_coordinates_vs_t 6.1.6.
        • add_energies 6.1.6.
        • add_momentum_vs_t 6.1.6.
        • add_phase_space 6.1.6.
        • add_populations 6.1.6.
        • add_time_overlap_projectors 6.1.6.
        • add_trajectory_resolved_ham_property 6.1.6.
        • hdf2xyz 6.1.5.
        • plot_dyn 6.1.5.
        • plot_dynamics 6.1.6. | 6.1.7.
      • recipes
        • adiabatic_md_interfaces_params 6.1.5.

  • models

    • Holstein
      • Holstein2 6.1.6. | 6.4.3.
    • Martens
      • model1 6.4.3.
      • model2 6.4.3.
    • Tully
      • chain_potential 6.1.7.

  • workflows

    • nbra
      • compute_hprime
        • compute_hprime_dia
        • hprime_py
      • compute_properties
        • compute_properties_onekpt
      • decoherence_times
        • decoherence_times2rates
        • energy_gaps
        • energy_gaps_ave 6.2.3.
        • decoherence_times
        • decoherence_times_ave_old
        • decoherence_times_ave 6.2.3.
      • lz
        • Belyaev_Lebedev
        • adjust_SD_probabilities
        • run
      • mapping
        • sd2indx
        • energy_arb
        • energy_mat_arb
        • orbs2spinorbs
        • ovlp_arb
        • ovlp_mat_arb 6.2.4.2.
      • qsh
        • compute_freqs
        • compute_qs_Hvib
        • run
      • step2
        • run 6.2.3.
      • step2_analysis
        • compute_oscillator_strengths
        • compute_spectrum
        • get_step2_mb_sp_properties
      • step2_dftb
        • do_step
        • do_ovlp
        • run_step2
        • run_step2_lz
      • step2_ergoscf
        • do_step
        • do_ovlp
        • clean
        • run_step2 11.1.2.
      • step2_many_body
        • curr_and_final_step_job
        • normalize_ci_coefficients
        • get_excitation_analysis_output
        • integrate_cube_set
        • compute_cube_ks_overlaps
        • reindex_cp2k_sd_states 6.2.4.2.
        • form_Hvib_real
        • run_step2_many_body
      • step3
        • get_step2_data 6.2.4.1. | 6.2.4.2.
        • print_SD_basis 6.2.4.1.
        • sort_SD_energies 6.2.4.1.
        • output_sorted_Hvibs 6.2.4.1.
        • build_SD_basis 6.2.4.1.
        • get_Lowdin
        • apply_normalization 6.2.4.2.
        • get_Lowdin_general
        • apply_orthonormalization_general 6.2.4.2.
        • make_cost_mat
        • apply_state_reordering 6.2.4.2.
        • do_phase_corr
        • apply_phase_correction
        • sac_matrices
        • scale_H_vib
        • compute_Hvib
        • run 6.2.4.1.
        • map_Hvib
        • pyxaid2libra
        • apply_state_reordering_general 6.2.4.2.
        • apply_phase_correction_general 6.2.4.2.
        • sort_unique_SD_basis 6.2.4.2.
      • step3_many_body
        • get_step2_mb_sp_properties 6.2.4.2.
        • compute_ci_energies_midpoint 6.2.4.2.
        • make_T_matrices 6.2.4.2.
        • run 6.2.4.2.
      • step4
        • get_Hvib
        • get_Hvib2 6.2.5.1.
        • traj_statistics
        • traj_statistics2
        • traj_statistics2_fast
        • printout
        • run 6.2.5.1.

  • data_conv

    • make_list 6.1.5.
    • matrix2list 6.1.5.
    • MATRIX2nparray 9.2. | 11.2.1. | 6.2.5.1.
    • nparray2CMATRIX 6.2.4.1.

  • data_outs

    • print_matrix 6.1.5. | 9.2. | 11.1.1. | 6.2.4.1.

  • data_read

    • get_data_sets 6.2.3.
    • read_2D_grid 6.4.3.

  • data_stat

    • cmat_stat2 6.2.3. | 6.2.4.1. | 6.2.4.2.

  • data_visualize

    • colors 7.2.
    • plot_map 6.4.1.1. | 6.4.3. | 9.2. | 9.3.
    • plot_map_nparray 6.2.5.1

  • DFTB_methods

    • read_dftb_output 8.2.
    • run_dftb_adi 8.2. | 6.1.5.
    • make_dftb_input 8.2.

  • dynamics_plotting

    • plot_pes_properties 8.1. | 8.2.
    • plot_surfaces 8.1. | 6.1.6. | 6.1.7. | 6.4.3.

  • ERGO_methods

    • get_mtx_matrices 11.1.1.
    • read_mo_restricted 11.1.1.
    • read_mo_unrestricted 11.1.1.
    • read_spectrum_restricted 11.1.1.
    • read_spectrum_unrestricted 11.1.1.

  • ft

    • ft26.3.1.

  • gaussian_kernel_algorithm

    • compute_apriory_prob_densities_1D 7.2.
    • compute_widths_1D 7.2.
    • gaussian_density_estimator_1D 7.2.
    • gaussian_kernel_algorithm_iteration_1D 7.2.

  • hpc_utils

    • distribute 6.2.3.

  • LoadMolecule

    • Load_Molecule 5.1.1.

  • LoadPT

    • Load_PT 5.1.1.

  • normal_modes

    • get_xyz2 11.2.3.

  • pdos

    • libra_pdos 5.1.1.
    • QE_pdos 11.2.1.

  • psi4_methods

    • run_psi4_adi 8.2.

  • QE_methods

    • out2inp 6.2.3.
    • read_md_data 11.2.2.
    • read_md_data_cell 11.2.2.
    • get_QE_normal_modes 11.2.3.

  • scan

    • coords2xyz 8.2.
    • make_path_xyz2 8.2.

• liblibra::libintegrators

  • RK4 2.1. | 2.2.

• liblibra::libutil

  • allocate_1D 10.1.
  • allocate_2D 10.1.
  • allocate_3D 10.1.
  • show_vector 10.1.
  • is_in_vector 10.1.
  • is_repeating 10.1.
  • delta 10.1.
  • split_line 10.1.
  • is_equal 10.1.
  • is_included 10.1.
  • is_present 10.1.
  • check_input 10.1.

---

Classes and class members

• liblibra::libann

  • NeuralNetwork 9.1. | 9.2. | 9.3.
    • Nlayers 9.1.
    • Npe 9.1.
    • W 9.1.
    • dW 9.1.
    • dWold 9.1.
    • grad_w 9.1.
    • grad_w_old 9.1.
    • B 9.1.
    • dB 9.1.
    • dBold 9.1.
    • grad_b 9.1.
    • grad_b_old 9.1.
    • propagate 9.1. | 9.2. | 9.3.
    • derivatives 9.2.
    • back_propagate 9.1.
    • init_weights_biases_uniform 9.1. | 9.2. | 9.3.
    • init_weights_biases_normal 9.1.
    • train 9.1. | 9.2. | 9.3.
    • load 9.1.
    • save 9.1. | 9.3.
    • error 9.1.

• liblibra::libcontrol_parameters::Control_Parameters 5.1.1.

• liblibra::libconverters

  • StringList 10.1.
  • StringMap 10.1.
  • StringDoubleMap 10.1.
  • StringIntMap 10.1.
  • StringVDoubleMap 10.1.

• liblibra::libchemobjects

  • libchemsys::System

    • System 12.1.
    • Number_of_atoms 5.1.1.
    • get_xyz 5.1.1. | 12.1.
    • init_box 12.1.
    • print_ent 12.1.

  • libmol

    • AtomList 10.1.
    • GroupList 10.1.
    • MoleculeList 10.1.

  • libuniverse::Universe 5.1.1.

• liblibra::libdata

  • DATA 7.3. | 7.4.2.
    • Data 7.3.
    • ave 7.3.
    • var 7.3.
    • sd 7.3.
    • se 7.3.
    • mse 7.3.
    • mae 7.3.
    • rmse 7.3.
    • min_val 7.3.
    • min_indx 7.3.
    • max_val 7.3.
    • max_indx 7.3.
    • scale_factor 7.3.
    • shift_amount 7.3.
    • LinearTransformData 7.3.
    • invLinearTransformData 7.3.
    • ScaleData 7.3.
    • ShiftData 7.3.
    • NormalizeData 7.3.
    • Calculate_Estimators 7.3. | 11.2.2.
    • Calculate_MiniMax 7.3. | 11.2.2.
    • Calculate_Distribution 7.3. | 7.4.1. | 7.4.2. | 11.2.2.
  • DATAList 10.1.

• liblibra::libdyn

  • libelectronic
    • ElectronicList 10.1.
  • libnuclear
    • NuclearList 10.1.
  • libthermostat
    • ThermostatList 10.1.

• liblibra::libforcefield

  • Angle_RecordList 10.1.
  • Bond_RecordList 10.1.
  • Dihedral_RecordList 10.1.
  • Fragment_RecordList 10.1.

• `liblibra::libhamiltonian

  • libhamiltonian_atomistic

    • libhamiltonian_qm
      • listHamiltonian_QM
        • compute_scf 5.1.1.
        • get_electronic_structure 5.1.1.
        • basis_ao 5.1.1.
        • atom_to_ao_map 5.1.1.
      • listHamiltonian_QMList 10.1.
      • listHamiltonian_QMMap 10.1.
    • Hamiltonian_AtomisticList 10.1.

  • libhamiltonian_extern

    • Hamiltonian_ExternList 10.1.

  • libhamiltonian_generic

    • HamiltonianList 10.1.
    • nHamiltonianList 10.1. -libhamiltonian_model
    • Hamiltonian_ModelList 10.1.

  • Electronic_Structure

    • Nelec 5.1.1.
    • Norb 5.1.1.
    • get_bands_alp 5.1.1.
    • get_bands_bet 5.1.1.
    • get_occ_alp 5.1.1.
    • get_occ_bet 5.1.1.

• liblibra::libqobjects

  • PromitiveGList 10.1.
  • AOList 10.1.
  • SDList 10.1.
  • PWList 10.1.

• liblibra::librandom

  • Random
    • Random 7.4.1. | 7.4.2.
    • uniform 7.4.1. | 7.3. | 7.4.2.
    • p_uniform 7.4.1.
    • normal 7.4.1. | 7.2. | 7.3. | 9.1. | 7.4.2.
    • p_normal 7.4.1.
    • exponential 7.4.1.
    • exponential 7.4.1.
    • poiss1 7.4.1.
    • poiss2 7.4.1.
    • p_poiss 7.4.1.
    • gamma 7.4.1.
    • p_gamma 7.4.1.

• liblibra::liblinalg

  • CMATRIX
    • real 11.1.1
    • H 11.1.1
    • show_matrix 11.1.1
  • intList 10.1.
  • floatList 10.1.
  • doubleList 10.1.
  • complexList 10.1.
  • intList2 10.1.
  • floatList2 10.1.
  • doubleList2 10.1.
  • complexList2 10.1.
  • intList3 10.1.
  • floatList3 10.1.
  • doubleList3 10.1.
  • complexList3 10.1.
  • intMap 10.1.
  • floatMap 10.1.
  • doubleMap 10.1.
  • complexMap 10.1.
  • MATRIXList 10.1.
  • MATRIXMap 10.1.
  • CMATRIXList 10.1.
  • CMATRIXMap 10.1.
  • MATRIX3x3List 10.1.
  • MATRIX3x3Map 10.1.
  • QUATERNIONList 10.1.
  • QUATERNIONMap 10.1.
  • VECTORList 10.1.
  • VECTORMap 10.1.

• libra_py

  • data_savers
    • hdf5_saver
      • set_compression_level 8.1.
      • add_dataset 8.1.
      • save_matrix 8.1.
      • hdf5_saver 8.1.

• liblibra::libdyn::libwfcgrid2

  • Wfcgrid2 6.4.2.
    • nstates 6.4.2.
    • ndof 6.4.2.
    • Npts 6.4.2.
    • npts 6.4.2.
    • rmin 6.4.2.
    • rmax 6.4.2.
    • dr 6.4.2.
    • kmin 6.4.2.
    • dk 6.4.2.
    • gmap 6.4.2.
    • imap 6.4.2.
    • PSI_dia 6.4.2.
    • reciPSI_dia 6.4.2.
    • PSI_adi 6.4.2.
    • reciPSI_adi 6.4.2.
    • Hdia 6.4.2.
    • U 6.4.2.
    • add_wfc_Gau 6.4.2.
    • add_wfc_HO 6.4.2.
    • add_wfc_ARB 6.4.2.
    • norm 6.4.2.
    • e_kin 6.4.2.
    • e_pot 6.4.2.
    • e_tot 6.4.2.
    • get_pow_q 6.4.2.
    • get_pow_p 6.4.2.
    • get_den_mat 6.4.2.
    • get_pops 6.4.2. | 6.4.3.
    • update_propagator_H 6.4.2.
    • update_propagator_K 6.4.2.
    • SOFT_propagate 6.4.2.
    • update_real 6.4.3.
    • update_reciprocal 6.4.2.
    • normalize 6.4.2.
    • update_Hamiltonian 6.4.2.
    • update_adiabatic 6.4.2.
    • update_diabatic 6.4.3.
    • print_wfc_1D 6.4.3.
    • print_reci_wfc_1D 6.4.3.
    • print_wfc_2D 6.4.3.
    • print_reci_wfc_2D 6.4.3.