【CP2K教程（三）】元动力学 （Metadynamics）与增强采样

1. Simple metadynamics simulation guide

2. 集合变量配位数函数

3. Biochemical systems metadynamics 

4. 自由能面绘图软件graph.sopt

5. cp2k和plumed联用简单案例

6. Tree diagram of keywords ralated to metadynamics

7. CP2K元动力学中获取重构势能面的方法

8. 基于restart文件继续执行元动力学

9. 元动力学测试题

Metadynamics is a computational method used in molecular simulations to enhance the sampling of free energy landscapes by adding a bias potential. It was developed to overcome the problem of slow conformational transitions and improve the exploration of rare events in complex systems.

Metadynamics was first introduced in 2002 by Michele Parrinello and Alessandro Laio. It builds upon previous methods such as umbrella sampling and adaptive biasing force, and is based on the idea of filling free energy basins with Gaussian hills to encourage the system to explore new areas of the configuration space.

Metadynamics allows researchers to study complex phenomena such as protein folding, protein-ligand binding, and phase transitions. It has also been applied to study chemical reactions and material properties. By enhancing the sampling of free energy landscapes, metadynamics enables the calculation of kinetic and thermodynamic properties that would be difficult to obtain otherwise.

In metadynamics, the external history-dependent bias potential is a function that is added to the potential energy of the system in order to drive it towards regions of phase space that have not been explored yet. The bias potential depends on the collective variables (CVs), which are coordinates that describe the system and are used to monitor its progress in exploring phase space. The bias potential is constructed so that it grows with time in regions of phase space that have not been visited frequently, effectively creating a "hill" in the potential energy landscape that drives the system towards unexplored regions. This results in an increased exploration of the phase space and the generation of a smooth free energy surface that describes the system's behavior. The external history-dependent bias potential is what makes metadynamics a powerful technique for exploring complex, high-dimensional systems.

The external history-dependent bias potential in metadynamics can be expressed by the following equation:

V_bias(s) = ∑ w_i exp(-||s-s_i||^2/2σ^2)

where V_bias is the bias potential, s is a vector of collective variables that describe the system, w_i is the height of the Gaussian hills added to the potential energy landscape, s_i is the position of the ith Gaussian hill in collective variable space, and σ is the width of the Gaussians. The summation is taken over all previous time steps of the simulation. The idea is to add up a series of Gaussians at previously visited regions of the collective variable space, effectively creating a repulsive bias that pushes the system away from those regions and towards unexplored regions. As a result, the system is encouraged to explore a larger portion of the phase space, which can lead to more accurate free energy calculations and a better understanding of the system's behavior.

CP2K can perform metadynamics simulations, which are a type of free energy calculation that use a history-dependent bias potential to explore the energy landscape of a system. You can set up a metadynamics calculation by adding the section MOTION / FREE_ENERGY / METADYN in your CP2K input file. In this section, you can specify the collective variables (CVs) on which to apply metadynamics, the frequency and height of the Gaussian hills, the friction term and the temperature for the extended Lagrangian scheme, and other options. You can also monitor the behavior of the CVs and the bias potential during the simulation.

1. Simple metadynamics simulation guide

Metadynamics is a powerful enhanced sampling technique that can be used to investigate complex systems, including those with multiple degrees of freedom. In metadynamics, a history-dependent bias is added to the potential energy of the system, which drives the system towards regions of high free energy and enhances the exploration of the configuration space.

One possible implementation of metadynamics using the coordination numbers as variables could be as follows:

1. Define the coordinates of the system: In this case, the coordination numbers can be used as collective variables (CVs) to represent the system. The coordination numbers could be calculated based on the distances between the atoms in the system and their neighbors.

2. Choose the Gaussian height and width: The height and width of the Gaussian hills used in the bias potential should be carefully chosen to achieve a good balance between exploration and convergence. A common choice is to use a height of 0.1-1.0 kJ/mol and a width of 0.1-0.5 Å.

3. Initialize the simulation: Start the simulation from an initial configuration and run it for a certain amount of time without the metadynamics bias. This allows the system to equilibrate and reach its initial state.

4. Add the metadynamics bias: At regular intervals (e.g., every 500-1000 time steps), add a Gaussian hill centered at the current value of the CVs to the potential energy of the system. Repeat this process until the simulation has run for a sufficient amount of time to generate a good sampling of the configuration space.

5. Analyze the results: The final state of the simulation can be analyzed to gain insights into the thermodynamics and kinetics of the system. The distribution of the CVs can be plotted and used to identify free energy basins and transition states.

This is a simple example of how metadynamics can be implemented using coordination numbers as variables, but many other variations and modifications are possible, depending on the specifics of the system being studied.

The important keywords related to metadynamics in CP2K.

METADYN	This section sets parameters to set up a calculation of metadynamics.
METAVAR	This section specify the nature of the collective variables.
FREE_ENERGY	Controls the calculation of free energy and free energy derivatives with different possible methods
FREE_ENERGY/METHOD	Defines the method to use to compute free energy.  Default value: METADYN
DO_HILLS	This keyword enables the spawning of the hills. Default .FALSE
NT_HILLS	Specify the maximum MD step interval between spawning two hills.  Default value: 30
WW	Specifies the height of the gaussian to spawn. Default 0.1
SCALE	Specifies the scale factor for the following collective variable. Alias names for this keyword: WIDTH
METAVAR/COLVAR	Specifies the colvar on which to apply metadynamics.
METADYN/PRINT	Controls the printing properties during an metadynamics run
METADYN/PRINT/COLVAR	Controls the printing of COLVAR summary information during metadynamics. When an extended Lagrangian use used, the files contain (in order): colvar value of the extended Lagrangian, instantaneous colvar value, force due to the harmonic term of the extended Lagrangian and the force due to the previously spawned hills, the force due to the walls, the velocities in the extended Lagrangian, the potential of the harmonic term of the Lagrangian, the potential energy of the hills, the potential energy of the walls and the temperature of the extended Lagrangian. When the extended Lagrangian is not used, all related fields are omitted.
COMMON_ITERATION_LEVELS	How many iterations levels should be written in the same file (no extra information about the actual iteration level is written to the file) Default value: 0
METADYN/PRINT/HILLS	Controls the printing of HILLS summary information during metadynamics. The file contains: instantaneous colvar value, width of the spawned gaussian and height of the gaussian. According the value of the EACH keyword this file may not be synchronized with the COLVAR file.  也就是说如果只有一个集合变量的话，输出的HILLS.metadynLog文件中会存在4列，第一列为时间，第二列为相应时刻添加gaussian hill时集合变量瞬时值，第三列为添加的gasuuian hill的宽度，对应SCALE关键词数值，第四列是添加的gaussian hill的高度，对应设WW关键词数值，该值默认为0.1。
MOTION / PRINT / RESTART	Controls the dumping of the restart file during runs. By default keeps a short history of three restarts. See also RESTART_HISTORY
MOTION / PRINT / RESTART_HISTORY	Dumps unique restart files during the run keeping all of them.Most useful if recovery is needed at a later point.
FORCE_EVAL / SUBSYS / COLVAR	This section specifies the nature of the collective variables.
SUBSYS / COLVAR / COORDINATION	Section to define the coordination number as a collective variable.
KINDS_FROM	Specify alternatively kinds of atoms building the coordination variable.
KINDS_TO	Specify alternatively kinds of atoms building the coordination variable.
R0	Specify the R0 parameter in the coordination function。 Default value: 3.00000000E+000 Default unit: [bohr] Alias n