In this exercise, you will perform ab initio molecular dynamics using Second Generation Car-Parrinello (SGCP) molecular dynamics.

Please cite [[ https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.98.066401 | Phys. Rev. Lett. 98, 066401  ]], if you use this method.

Published work using SGCP method:

[[https://pubs.acs.org/doi/10.1021/acs.jpcc.8b05933 | J. Phys. Chem. C 2018, 122, 42, 24068–24076]]

[[ https://doi.org/10.1021/acs.jpclett.0c01025 |J. Phys. Chem. Lett. 2020, 11, 9, 3724–3730]]

====1. Introduction====
Second Generation CPMD (2ndG CPMD) is a molecular dynamics method that combines the efficiency of Car-Parrinello MD (CPMD) with the accuracy of Born-Oppenheimer MD (BOMD). It avoids fully self-consistent field (SCF) optimizations at each time step while enabling larger integration steps and maintaining accuracy close to BOMD.


**Goal:** Retain the efficiency of CPMD while achieving BOMD-level accuracy.

- **Efficiency**:  Large time steps ; No full SCF loops  

- **Accuracy**:  Forces nearly indistinguishable from BOMD  

- **Stability**:  Effective for systems with vanishing band gaps  

- **Error Control**:  Controlled deviation from BO surface using adaptive correction

====2. Comparison with CPMD and BOMD====

| Feature                          | CPMD          | BOMD         | SGCP                |
| SCF at each step                 | No          | Yes        |  Partially (predictor-corrector) |
| Time step                        | Small (~0.1 fs) | Large (~1 fs) | Large (~1–2 fs)           |
| Conserved quantity preservation  | Excellent     | Reasonable   | Excellent                 |
| On Born-Oppenheimer surface      | Slightly above| Yes          | Very close                |
| Works for small-gap systems      | Poor          | Good         | Good                      |

====3. ASPC Method====
ASPC Method: Always Stable Predictor Corrector

ASPC is a **Gear-type integrator** for electronic wavefunctions:

Predictor:

\[
C_p(t_n) = \sum_{m=1}^{K} (-1)^{m+1} \cdot m \cdot B_m \cdot P_S(t_{n-m})
\]

where:
- \( B_m \): Kolafa predictor coefficients
- \( P_S \): projection onto the overlap matrix \( S \)

Corrector:

\[
C(t_n) = \omega \cdot \min[C_p(t_n)] + (1 - \omega) \cdot C_p(t_n), \quad \omega = \frac{K}{2K - 1}
\]

Langevin Dynamics & Dissipation Compensation

Because ASPC introduces small dissipation, **Langevin-type equations** are used to stabilize the dynamics:

\[
M_I \ddot{R}_I = F_\text{BO} - (\gamma_D + \gamma_L)\dot{R}_I + \Xi_I
\]

- \( \gamma_D \): implicit friction from ASPC  
- \( \gamma_L \): Langevin thermostat  
- \( \Xi_I \): Langevin random noise



====4. How to Set Up in CP2K====


| Parameter              | Purpose                                  | Notes                                      |
| EXTRAPOLATION_ORDER  | Higher gives better predictor            | 1–4 typical, 0 for metallic is more stable                               |
| MAX_SCF_HIST        | Controls SCF correction                  | ≥2 helps smoother convergence              |
| STEPSIZE             | Time step in fs                          | ~0.5–2 fs depending on system                |
| PRECONDITIONER       | Affects SCF convergence                  | `FULL_SINGLE_INVERSE` slightly better      |
| NOISY_GAMMA (γ_D)    | ASPC dissipation compensation            | Adjust to control drift in T and energy    |
| GAMMA (γ_L)          | Langevin thermostat strength             | Set to 0 for dissipation-only integration  |

1. ASPC Extrapolation

  &FORCE_EVAL
    &DFT
       &QS
        EXTRAPOLATION ASPC
        EXTRAPOLATION_ORDER 0 # Higher gives better corrector 
       &END QS
       &SCF
        MAX_SCF_HIST 2
       &END SCF
    &END DFT
  &END FORCE_EVAL

2. Langevin Thermostat

  &MOTION
    &MD
      ENSEMBLE LANGEVIN
      &LANGEVIN
        GAMMA 0.005         ! γ_L
        NOISY_GAMMA 4.0E-4  ! γ_D
      &END LANGEVIN
    &END MD
  &END MOTION

3. Atom-Specific γ_D (Optional)

  &THERMAL_REGION
    DO_LANGEVIN_DEFAULT TRUE
    &DEFINE_REGION
      TEMPERATURE 500
      NOISY_GAMMA_REGION 4.E-4
      LIST 577..745
    &END DEFINE_REGION
  &END THERMAL_REGION