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哥伦布训练营|DeePMD——正极材料实战之性质计算篇
正极材料
正极材料
zhanglinshuang
发布于 2023-10-22
推荐镜像 :DeePMD-kit:2.2.1-cuda11.6-notebook
推荐机型 :c16_m32_cpu
赞 2
7
前言
实践
1 课前准备
数据下载和解压
lbg 的安装和登录
2 LiCoO2​体系结构优化
3 DP 对LiCoO2​体系电压平台描述
3 基于DP模型的大尺度、长时间MD模拟
4 低能空位构型的搜索
总结
参考

作者：梁文硕liangws@dp.tech, 张琳爽zhanglinshuang@dp.tech

推荐镜像： bohrium-notebook:2023-04-07

推荐资源：CPU

内容：📖 本文档为「能源学人——DeePMD用于电极材料前沿探索案例」课程教学文档。 本教程主要以电池正极材料LiCoO2体系为例，具体展示DP模型在热力学和动力学性质计算中的应用。 如您遇到任何问题，请联系 bohrium@dp.tech 。

注意：由于许可限制，部分内容以视频展示。你可以通过有效的许可在 Bohrium 上运行相关程序。

使用方式： 您可以在 Bohrium Notebook 上直接运行。你可以点击界面上方按钮 开始连接，选择 bohrium-notebook:2023-04-07 镜像及任意节点配置，稍等片刻即可运行。

共享协议：本作品采用知识共享署名-非商业性使用-相同方式共享 4.0 国际许可协议进行许可。

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前言

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🎯 欢迎阅读《哥伦布训练营 | DeePMD——正极材料实战之性质计算篇》

本文档旨在结合J. Phys. Chem. Lett. 2023, 14, 3677−3684 这篇于2023年发表的使用深度势能进行电池正极材料研究工作，介绍机器学习与分子动力学的结合如何发挥作用的。 接下来，本文档手把手带你复现文章中的电压数据、评估势函数对空位低能构型的搜索能力。

alt image.png

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📖 本文将直接使用已经构建好的适用于LiCoO2体系的深度神经网络势函数进行计算，而不再展示势函数构建的过程。
这是因为我们的重点在于展示DPMD方法在实际应用中的性能，而构建神经网络势函数的过程可以在其他相关文献中找到详细介绍。

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实践

1 课前准备

数据下载和解压

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[1]
!wget https://dp-public.oss-cn-beijing.aliyuncs.com/community/demo_LiCoO2.tar.gz
!tar -xzf demo_LiCoO2.tar.gz
--2023-07-14 12:53:18--  https://dp-public.oss-cn-beijing.aliyuncs.com/community/demo_LiCoO2.tar.gz
Resolving ga.dp.tech (ga.dp.tech)... 10.255.254.37, 10.255.254.7, 10.255.254.18
Connecting to ga.dp.tech (ga.dp.tech)|10.255.254.37|:8118... connected.
Proxy request sent, awaiting response... 200 OK
Length: 392891040 (375M) [application/gzip]
Saving to: ‘demo_LiCoO2.tar.gz’

demo_LiCoO2.tar.gz  100%[===================>] 374.69M  11.8MB/s    in 31s     

2023-07-14 12:53:51 (12.1 MB/s) - ‘demo_LiCoO2.tar.gz’ saved [392891040/392891040]
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lbg 的安装和登录

Lbg是一款用于资源管理、任务管理、项目监控的工具。

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[2]
!pip install lbg
#如果速度较慢，可以切换至清华源：
#!pip install -i https://pypi.tuna.tsinghua.edu.cn/simple/ lbg
Looking in indexes: https://pypi.tuna.tsinghua.edu.cn/simple
Requirement already satisfied: lbg in /opt/conda/lib/python3.8/site-packages (1.2.17)
Requirement already satisfied: packaging in /opt/conda/lib/python3.8/site-packages (from lbg) (23.0)
Requirement already satisfied: tqdm in /opt/conda/lib/python3.8/site-packages (from lbg) (4.64.1)
Requirement already satisfied: requests in /opt/conda/lib/python3.8/site-packages (from lbg) (2.28.2)
Requirement already satisfied: aliyun-python-sdk-core in /opt/conda/lib/python3.8/site-packages (from lbg) (2.13.36)
Requirement already satisfied: validators in /opt/conda/lib/python3.8/site-packages (from lbg) (0.20.0)
Requirement already satisfied: pytimeparse in /opt/conda/lib/python3.8/site-packages (from lbg) (1.1.8)
Requirement already satisfied: pandas in /opt/conda/lib/python3.8/site-packages (from lbg) (1.5.3)
Requirement already satisfied: pyhumps in /opt/conda/lib/python3.8/site-packages (from lbg) (3.8.0)
Requirement already satisfied: requests-toolbelt in /opt/conda/lib/python3.8/site-packages (from lbg) (0.10.1)
Requirement already satisfied: colorama in /opt/conda/lib/python3.8/site-packages (from lbg) (0.4.6)
Requirement already satisfied: aliyun-python-sdk-sts in /opt/conda/lib/python3.8/site-packages (from lbg) (3.1.0)
Requirement already satisfied: oss2 in /opt/conda/lib/python3.8/site-packages (from lbg) (2.16.0)
Requirement already satisfied: pyreadline3 in /opt/conda/lib/python3.8/site-packages (from lbg) (3.4.1)
Requirement already satisfied: argcomplete in /opt/conda/lib/python3.8/site-packages (from lbg) (2.0.0)
Requirement already satisfied: pyyaml in /opt/conda/lib/python3.8/site-packages (from lbg) (6.0)
Requirement already satisfied: pyreadline in /opt/conda/lib/python3.8/site-packages (from lbg) (2.1)
Requirement already satisfied: readchar in /opt/conda/lib/python3.8/site-packages (from lbg) (4.0.3)
Requirement already satisfied: aliyun-python-sdk-kms in /opt/conda/lib/python3.8/site-packages (from lbg) (2.16.0)
Requirement already satisfied: cryptography>=2.6.0 in /opt/conda/lib/python3.8/site-packages (from aliyun-python-sdk-core->lbg) (39.0.1)
Requirement already satisfied: jmespath<1.0.0,>=0.9.3 in /opt/conda/lib/python3.8/site-packages (from aliyun-python-sdk-core->lbg) (0.10.0)
Requirement already satisfied: six in /opt/conda/lib/python3.8/site-packages (from oss2->lbg) (1.16.0)
Requirement already satisfied: crcmod>=1.7 in /opt/conda/lib/python3.8/site-packages (from oss2->lbg) (1.7)
Requirement already satisfied: pycryptodome>=3.4.7 in /opt/conda/lib/python3.8/site-packages (from oss2->lbg) (3.17)
Requirement already satisfied: charset-normalizer<4,>=2 in /opt/conda/lib/python3.8/site-packages (from requests->lbg) (3.0.1)
Requirement already satisfied: urllib3<1.27,>=1.21.1 in /opt/conda/lib/python3.8/site-packages (from requests->lbg) (1.26.14)
Requirement already satisfied: certifi>=2017.4.17 in /opt/conda/lib/python3.8/site-packages (from requests->lbg) (2022.12.7)
Requirement already satisfied: idna<4,>=2.5 in /opt/conda/lib/python3.8/site-packages (from requests->lbg) (3.4)
Requirement already satisfied: python-dateutil>=2.8.1 in /opt/conda/lib/python3.8/site-packages (from pandas->lbg) (2.8.2)
Requirement already satisfied: pytz>=2020.1 in /opt/conda/lib/python3.8/site-packages (from pandas->lbg) (2022.7)
Requirement already satisfied: numpy>=1.20.3 in /opt/conda/lib/python3.8/site-packages (from pandas->lbg) (1.22.4)
Requirement already satisfied: setuptools>=41.0 in /opt/conda/lib/python3.8/site-packages (from readchar->lbg) (65.6.3)
Requirement already satisfied: decorator>=3.4.0 in /opt/conda/lib/python3.8/site-packages (from validators->lbg) (5.1.1)
Requirement already satisfied: cffi>=1.12 in /opt/conda/lib/python3.8/site-packages (from cryptography>=2.6.0->aliyun-python-sdk-core->lbg) (1.15.1)
Requirement already satisfied: pycparser in /opt/conda/lib/python3.8/site-packages (from cffi>=1.12->cryptography>=2.6.0->aliyun-python-sdk-core->lbg) (2.21)
WARNING: Running pip as the 'root' user can result in broken permissions and conflicting behaviour with the system package manager. It is recommended to use a virtual environment instead: https://pip.pypa.io/warnings/venv
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首次使用 Bohrium，需要先使用如下命令配置账号：'lbg config account'。然后输入你的Bohrium 邮箱和对应密码即可。

2 LiCoO体系结构优化

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[3]
!ls /data/demo_LiCoO2/LiCoO2-opt/DP
LiCoO2.pb  POSCAR  POSCAR_optimized  energy  opt.py
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from IPython.display import Image
Image("/data/demo_LiCoO2/LiCoO2-opt/Structure/LiCoO2_structure.png",width=400,height=400)
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我们可以使用DP模型作为计算器，使用ASE程序，对上述构型进行结构优化。

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!cat /data/demo_LiCoO2/LiCoO2-opt/DP/opt.py
from ase.io import read, write
from ase.io import Trajectory
from ase.constraints import StrainFilter
from ase.optimize import BFGS
from deepmd.calculator import DP
import time

start_time = time.time()
calculator = DP(model="/data/demo_LiCoO2/LiCoO2-opt/DP/LiCoO2.pb")
stru = read('/data/demo_LiCoO2/LiCoO2-opt/DP/POSCAR', format='vasp')
stru.set_calculator(calculator)
sf = StrainFilter(stru)
dyn = BFGS(sf)
dyn.run(1e-5)
write('/data/demo_LiCoO2/LiCoO2-opt/DP/POSCAR_optimized', stru, format='vasp')
energy = stru.get_total_energy()
with open('/data/demo_LiCoO2/LiCoO2-opt/DP/energy', 'w') as fp:
    fp.write(str(energy))
print("Execution time: %s seconds" % (time.time() - start_time))
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[6]
!python /data/demo_LiCoO2/LiCoO2-opt/DP/opt.py
2023-07-14 12:55:03.945950: I tensorflow/core/platform/cpu_feature_guard.cc:193] This TensorFlow binary is optimized with oneAPI Deep Neural Network Library (oneDNN) to use the following CPU instructions in performance-critical operations:  AVX2 AVX512F FMA
To enable them in other operations, rebuild TensorFlow with the appropriate compiler flags.
2023-07-14 12:55:06.841700: W tensorflow/compiler/xla/stream_executor/platform/default/dso_loader.cc:64] Could not load dynamic library 'libnvinfer.so.7'; dlerror: libnvinfer.so.7: cannot open shared object file: No such file or directory; LD_LIBRARY_PATH: /usr/mpi/gcc/openmpi-4.1.0rc5/lib:/usr/local/nccl-rdma-sharp-plugins/lib:/usr/local/nvidia/lib64:/usr/local/cuda/lib64:/usr/local/nvidia/lib:/usr/local/nvidia/lib64
2023-07-14 12:55:06.842787: W tensorflow/compiler/xla/stream_executor/platform/default/dso_loader.cc:64] Could not load dynamic library 'libnvinfer_plugin.so.7'; dlerror: libnvinfer_plugin.so.7: cannot open shared object file: No such file or directory; LD_LIBRARY_PATH: /usr/mpi/gcc/openmpi-4.1.0rc5/lib:/usr/local/nccl-rdma-sharp-plugins/lib:/usr/local/nvidia/lib64:/usr/local/cuda/lib64:/usr/local/nvidia/lib:/usr/local/nvidia/lib64
2023-07-14 12:55:06.842824: W tensorflow/compiler/tf2tensorrt/utils/py_utils.cc:38] TF-TRT Warning: Cannot dlopen some TensorRT libraries. If you would like to use Nvidia GPU with TensorRT, please make sure the missing libraries mentioned above are installed properly.
WARNING:tensorflow:From /opt/conda/lib/python3.8/site-packages/tensorflow/python/compat/v2_compat.py:107: disable_resource_variables (from tensorflow.python.ops.variable_scope) is deprecated and will be removed in a future version.
Instructions for updating:
non-resource variables are not supported in the long term
WARNING:root:To get the best performance, it is recommended to adjust the number of threads by setting the environment variables OMP_NUM_THREADS, TF_INTRA_OP_PARALLELISM_THREADS, and TF_INTER_OP_PARALLELISM_THREADS.
2023-07-14 12:55:10.665666: W tensorflow/compiler/xla/stream_executor/platform/default/dso_loader.cc:64] Could not load dynamic library 'libcuda.so.1'; dlerror: /usr/lib/x86_64-linux-gnu/libcuda.so.1: file too short; LD_LIBRARY_PATH: /usr/mpi/gcc/openmpi-4.1.0rc5/lib:/usr/local/nccl-rdma-sharp-plugins/lib:/usr/local/nvidia/lib64:/usr/local/cuda/lib64:/usr/local/nvidia/lib:/usr/local/nvidia/lib64
2023-07-14 12:55:10.665736: W tensorflow/compiler/xla/stream_executor/cuda/cuda_driver.cc:265] failed call to cuInit: UNKNOWN ERROR (303)
WARNING:tensorflow:From /opt/conda/lib/python3.8/site-packages/deepmd/utils/batch_size.py:61: is_gpu_available (from tensorflow.python.framework.test_util) is deprecated and will be removed in a future version.
Instructions for updating:
Use `tf.config.list_physical_devices('GPU')` instead.
WARNING:tensorflow:From /opt/conda/lib/python3.8/site-packages/deepmd/utils/batch_size.py:61: is_gpu_available (from tensorflow.python.framework.test_util) is deprecated and will be removed in a future version.
Instructions for updating:
Use `tf.config.list_physical_devices('GPU')` instead.
WARNING:deepmd.utils.batch_size:You can use the environment variable DP_INFER_BATCH_SIZE tocontrol the inference batch size (nframes * natoms). The default value is 1024.
      Step     Time          Energy         fmax
BFGS:    0 12:55:11     -276.076977       13.9439
BFGS:    1 12:55:11     -243.941126      232.3193
BFGS:    2 12:55:12     -275.882182       26.7527
BFGS:    3 12:55:12     -268.527002       98.5378
BFGS:    4 12:55:12     -276.099741       14.2600
BFGS:    5 12:55:12     -263.858882      127.5108
BFGS:    6 12:55:12     -276.082801       10.1510
BFGS:    7 12:55:12     -274.918630       25.0773
BFGS:    8 12:55:12     -276.200384        1.8851
BFGS:    9 12:55:12     -276.137437        8.3510
BFGS:   10 12:55:12     -276.206263        1.0258
BFGS:   11 12:55:12     -276.205628        0.7183
BFGS:   12 12:55:13     -276.206874        0.1333
BFGS:   13 12:55:13     -276.206885        0.1421
BFGS:   14 12:55:13     -276.206893        0.0012
BFGS:   15 12:55:13     -276.206893        0.0003
BFGS:   16 12:55:13     -276.206893        0.0000
BFGS:   17 12:55:13     -276.206893        0.0000
Execution time: 3.2218334674835205 seconds
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我们将DFT和DP数据汇总到/data/demo_LiCoO2/LiCoO2-opt/plot/plot_data ，作图对比DFT和DP的计算精度和计算时长：

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[7]
!cat /data/demo_LiCoO2/LiCoO2-opt/plot/plot_data
name energy time  
DFT  -279.9 1121
DPMD -276.2 3
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[8]
import numpy as np
import matplotlib.pyplot as plt

with open("/data/demo_LiCoO2/LiCoO2-opt/plot/plot_data", "r") as f:
lines = f.readlines()

data = [line.strip().split() for line in lines[1:]]
names = [item[0] for item in data]
energies = [float(item[1]) for item in data]
times = [float(item[2]) for item in data]

bar_width = 0.35

index = np.arange(len(names))

fig, ax = plt.subplots(1, 2, figsize=(6, 3))
ax[0].bar(index, energies, bar_width, label="Energy")
ax[0].set_xlabel("Methods")
ax[0].set_ylabel("Energy (eV)")
ax[0].set_xticks(index)
ax[0].set_xticklabels(names)
ax[0].set_title("Energy Comparison")

ax[1].bar(index, times, bar_width, label="Time")
ax[1].set_xlabel("Methods")
ax[1].set_ylabel("Time (s)")
ax[1].set_xticks(index)
ax[1].set_xticklabels(names)
ax[1].set_title("Time Comparison")

plt.tight_layout()
plt.show()
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可以看出，DP模型在能量精度方面与DFT方法表现一致，但计算成本更低。这意味着DP模型，为研究和优化LiCoO等材料的性能提供了更高效的途径。

3 DP 对LiCoO体系电压平台描述

LiCoO作为一种主要的正极材料，在其脱嵌过程中电压变化的计算尤为重要，以指导其在锂离子电池中的实际应用。

通过构建不同脱锂量下的多个构型，并通过DFT和DP对上述构型进行计算，绘制得到的凸包图如下图所示：

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from IPython.display import Image
Image("/data/demo_LiCoO2/Voltage/Convexhull/Convexhull.png",width=500,height=400)
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通过分析凸包图（Convexhull），我们提取得到了不同脱嵌量时稳定的LixCoO结构，并提取相关数据绘制DFT和DP的电压平台。

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[10]
import matplotlib.pyplot as plt

# import plot_utils
fig = plt.figure(figsize=(5, 4)) # Set the figure size
axes = fig.add_subplot()

x_data_new_1 = [1, 0.83, 0.50, 0.25, 0.00]

x_data_new_2 = [1, 0.75, 0.58, 0.50, 0.25, 0.00]

y_data_new_3 = [4.353977, 4.353977, 4.398701, 4.592491402, 4.689742834]
y_data_new_4 = [4.39, 4.39, 4.39, 4.500926933, 4.567768068, 4.633619805]
axes.step(x_data_new_1, y_data_new_3, color='red', label='DFT+U')
axes.step(x_data_new_2, y_data_new_4, color='black', label='DP')
axes.set_xlabel('x in Li$_{x}$CoO$_{2}$')
axes.set_ylabel('Voltage')
axes.legend()

plt.show()
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文献[1]构建的LiCoODP势函数和DFT在计算LiCoO脱嵌过程电压变化方面的结果具有很高的一致性。

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3 基于DP模型的大尺度、长时间MD模拟

在锂离子电池材料中，锂离子在充放电过程中的迁移能力是非常关键的。为了准确模拟这一过程，需要在长时间尺度和较大晶胞的条件下进行计算。然而，传统的AIMD方法往往无法胜任这样的任务。相较之下，DPMD模型可以有效地实现这一目标，为锂离子电池材料的研究提供了有力的支持。

输入文件 in.lammps LiCoO2.pb Li0.5CoO2.lmp param.json

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[11]
!cat /data/demo_LiCoO2/Diffusion/1000k/in.lammps
echo both
variable        pid world 0

# --------------------- VARIABLES-------------------------
variable        THERMO_FREQ     equal 10
variable        DUMP_FREQ       equal 100
variable        TEMP            equal 1000.000000
variable        TEMP2           equal 1000.000000
variable        PRES            equal 1.000000
variable        TAU_T           equal 0.100000
variable        TAU_P           equal 1.000000
variable        Seed 		    equal 32564
variable        msdxli          equal c_1[1]
variable        msdyli          equal c_1[2]
variable        msdzli          equal c_1[3]
variable        msdli           equal c_1[4]
variable        msdxCo          equal c_2[1]
variable        msdyCo          equal c_2[2]
variable        msdzCo          equal c_2[3]
variable        msdCo           equal c_2[4]
variable        msdxO           equal c_3[1]
variable        msdyO           equal c_3[2]
variable        msdzO           equal c_3[3]
variable        msdO            equal c_3[4]
variable        istep           equal step

# ---------------------- INITIALIZAITION ------------------
units           metal
boundary        p p p
atom_style      atomic
# --------------------- ATOM DEFINITION ------------------
box             tilt large 
read_data       Li0.5CoO2.input
mass            1 6.941
mass            2 58.933
mass		    3 16

# --------------------- FORCE FIELDS ---------------------
pair_style      deepmd   LiCoO2.pb
pair_coeff      * *
# --------------------- MD SETTINGS ----------------------
neighbor        1.0 bin
timestep        0.001
thermo          ${THERMO_FREQ}
thermo_style    custom step ke pe etotal enthalpy temp press vol lx ly lz xy xz yz pxx pyy pzz pxy pxz pyz
restart         ${DUMP_FREQ} restart.lmp.${pid} restart2.lmp.${pid}

dump            0 all custom ${DUMP_FREQ} dump_${DUMP_FREQ}fs.equi id element  x y z
dump_modify     0 sort id element Li Co O
velocity 	    all create ${TEMP} ${Seed}
velocity        all zero linear

group 		    liatoms type 1
compute         1 liatoms msd com yes
fix             4 liatoms print 1 "${istep} ${msdxli} ${msdyli} ${msdzli} ${msdli}" screen no file msd-li.dat

group           coatoms type 2
compute		    2 coatoms msd com yes
fix             5 coatoms print 1 "${istep} ${msdxCo} ${msdyCo} ${msdzCo} ${msdCo}" screen no file msd-co.dat

group           oatoms type 3
compute         3 oatoms msd com yes
fix             6 oatoms print 1 "${istep} ${msdxO} ${msdyO} ${msdzO} ${msdO}" screen no file msd-o.dat

# --------------------- RUN ------------------------------

fix             7 all npt temp ${TEMP} ${TEMP2} ${TAU_T} aniso ${PRES} ${PRES} ${TAU_P}
fix             mzero all momentum 10 linear 1 1 1
run             500000 #500 ps
unfix           7


write_data      data.final
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from IPython.display import Image
Image("/data/demo_LiCoO2/Diffusion/structure/supercell-756atoms.png",width=600,height=400)
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执行提交任务命令:

cd /data/demo_LiCoO2/Diffusion/1000k
lbg job submit -i job.json -p ./
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!cat /data/demo_LiCoO2/Diffusion/1000k/job.json
{
    "job_name":"1000k-msd",
    "command": "lmp -in in.start.lmp",
    "log_file": "log",
    "backward_file": [],
    "program_id": 15359,
    "job_group_id": "",
    "platform": "ali",
    "job_type": "container",
    "machine_type": "c8_m32_1 * NVIDIA V100",
    "image_name": "registry.dp.tech/dptech/deepmd-kit:2.2.1-cuda11.6",
    "result": "./"
}
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输出文件包括dump_100fs.equi log.lammps msd-li.dat 等

我们以msd-li.dat为例展示均方位移的输出格式:

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[14]
!cat /data/demo_LiCoO2/Diffusion/1000k/msd-li.dat | head -n 20
# Fix print output for fix 4
0 8.85642451463404e-31 1.05181454029468e-30 2.62953635073671e-31 2.20041062683176e-30
1 0.000116344458795415 0.000116963042388933 9.55739035090524e-05 0.0003288814046934
2 0.000464318945173845 0.000466784343643858 0.000380367050750345 0.00131147033956805
3 0.00104035139671962 0.00104544222445041 0.000848039051181463 0.00293383267235149
4 0.00183760483456243 0.00184481784544355 0.00148702984492018 0.00516945252492616
5 0.00284584814044392 0.00285247390772897 0.00228042533296386 0.00797874738113675
6 0.00405175279157205 0.00405186796364002 0.00320635038403628 0.0113099711392484
7 0.00543913690958795 0.00542271615451357 0.00423864221934627 0.0151004952834478
8 0.00698948386776909 0.00694145158105126 0.00534777243828375 0.0192787078871041
9 0.00868237651918469 0.00858184113920391 0.00650151301706877 0.0237657306754574
10 0.0104958904149312 0.0103154271342036 0.00766602313231568 0.0284773406814505
11 0.0124067912991325 0.0121125214659588 0.00880682480625211 0.0333261375713435
12 0.0143907465753516 0.0139428168584287 0.0098897270138642 0.0382232904476445
13 0.0164227275652906 0.0157758293597513 0.0108815819228767 0.0430801388479186
14 0.0184776396142197 0.0175815279550812 0.0117518061971564 0.0478109737664572
15 0.020530808394137 0.0193305624704187 0.0124739595168928 0.0523353303814485
16 0.0225583793817308 0.0209953367826327 0.0130268758515714 0.0565805920159348
17 0.0245376120970886 0.0225508865994023 0.0133952781816941 0.0604837768781849
18 0.0264476612937954 0.0239753580618043 0.0135704323753613 0.0639934517309609
cat: write error: Broken pipe
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MSD计算公式：

其中，表示粒子的总数，表示粒子在时刻的位置，表示粒子在初始时刻的位置。

扩散系数（D）计算公式：

这里，表示时间，单位为秒。在实际计算中，我们通常使用MSD随时间的线性拟合斜率除以6得到扩散系数。

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[15]
import numpy as np
import matplotlib.pyplot as plt

def plot_msd_and_get_diffusion(temp):
data = np.loadtxt(f'/data/demo_LiCoO2/Diffusion/{temp}k/msd-li.dat', skiprows=1)

time = data[:, 0]
msd1 = data[:, 4]

plt.plot(time/1000, msd1, label=f'Li+ at {temp} K', lw=1) # 1fs = 1/1000ps

slope1, residuals1 = np.polyfit(time, msd1, 1)

Diff = slope1 / 6 * 1e-5 # D = 1/6 * slope; 1 A^2/fs = 1e-5 m^2/s

return Diff

temperatures = [800,1000,1200]
Diff = []

for i, temp in enumerate(temperatures):
diff = plot_msd_and_get_diffusion(temp)
Diff.append(diff)

data = np.column_stack((temperatures, Diff))
np.savetxt('/data/demo_LiCoO2/Diffusion/diffusion.txt', data, header='Temperature (K)\tDiffusion Coefficient (m^2/s)')

plt.xlabel('time (ps)')
plt.ylabel('MSD (Å^2)')
plt.legend()
plt.savefig('/data/demo_LiCoO2/Diffusion/msd.png', dpi=300)
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[16]
!cat /data/demo_LiCoO2/Diffusion/diffusion.txt
# Temperature (K)	Diffusion Coefficient (m^2/s)
8.000000000000000000e+02 1.512969348071375718e-10
1.000000000000000000e+03 7.521645281666237925e-10
1.200000000000000000e+03 2.103456264103633486e-09
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根据计算结果，我们得到了不同温度下锂离子的传导率。拟合Arrhenius公式，计算离子的激活能。

其中：

  • ：锂离子扩散系数
  • ：温度为绝对零度时的扩散系数
  • ：激活能，表示达到反应所需的能量阈值
  • ：气体常数，数值约为 8.314 J/(mol·K)
  • ：绝对温度（单位：开尔文，K）
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[18]
import numpy as np
import matplotlib.pyplot as plt

data = np.loadtxt('/data/demo_LiCoO2/Diffusion/diffusion.txt')
temperatures = data[:,0] # K
diffusion_coefficients = data[:,1] # m^2/s
R = 8.617333262145e-5 # Ideal gas constant,eV/K

ln_diffusion_coefficients = np.log(diffusion_coefficients)

slope, intercept = np.polyfit(1/temperatures, ln_diffusion_coefficients, 1)
activation_energy = slope * R
plt.figure(figsize=(5, 4))
plt.scatter(1/temperatures, ln_diffusion_coefficients, color='blue',label='Calculation data')
plt.plot(1/temperatures, slope * 1/temperatures + intercept, color='red', label='Linear fitting')
plt.xlabel('1 / T (K⁻¹)')
plt.ylabel('ln(D) m$^2$/s')
plt.legend()
plt.savefig('/data/demo_LiCoO2/Diffusion/activation_energy.png')

print(activation_energy, "eV")
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4 低能空位构型的搜索

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我们将DP模型应用于LiCoO中不同脱嵌量下的空位结构的搜索。

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我们基于LiCoO的初始构型，生成了和的结构，并对其进行DFT计算和DP计算。

输出结果总结在下述文档中:

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[19]
!cat /data/demo_LiCoO2/Vacancy-opt/data/energy-vac-3.txt |head -n 10
!cat /data/demo_LiCoO2/Vacancy-opt/data/energy-vac-9.txt |head -n 10
45 	0	     	-.24508752E+03	   -255.47650945156326
45 	1	     	-.24622006E+03	   -255.94926231529158
45 	2	     	-.24595753E+03	   -255.9107038163852
45 	3	     	-.24595679E+03	   -255.91419784563394
45 	4	     	-.24655206E+03	   -256.53936125243933
45 	5	     	-.24664783E+03	   -256.59951533233277
45 	6	     	-.24621900E+03	   -255.94893680322923
45 	7	     	-.24664790E+03	   -256.5985602153588
45 	8	     	-.24681387E+03	   -256.70201657520926
39 	0	   	-0.196256107445E+03	-202.03445795499707
39 	1	   	-.21093893E+03	   	-216.60215580718597
39 	2	   	-.21071719E+03	   	-216.00456367651793
39 	3	   	-.21105025E+03	   	-216.59512932266978
39 	4	   	-.21088330E+03	   	-216.5721124205345
39 	5	   	-.21062566E+03	   	-216.53997244707477
39 	6	   	-.21094058E+03	   	-216.36702942136822
39 	7	   	-.21035044E+03	   	-216.03239314524544
39 	8	   	-.21150228E+03	   	-216.56808433936908
39 	9	   	-.21090383E+03	   	-216.54783461302847
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为了直观得对比DP的预测效果，我们统计了DP和DFT各自预测的最低空位结构的构型编号如下：

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[20]
import pandas as pd
import matplotlib.pyplot as plt
import openpyxl

file_path = "/data/demo_LiCoO2/Vacancy-opt/data/energy-vac-all.txt"
data = pd.read_csv(file_path, sep="\\s+", header=None, names=["Atoms", "ID", "DFT_Energy", "DP_Energy"])

grouped_data = data.groupby("Atoms")

sorted_data = {}

for name, group in grouped_data:
sorted_group = group.sort_values("DFT_Energy")
sorted_data[name] = sorted_group["ID"].tolist()

min_dpmd_index = sorted_group["DP_Energy"].idxmin()
print(f"For {name} atoms:")
print(f"DFT min energy ID: {sorted_group.at[min_dpmd_index, 'ID']}")
print(f"DPMD min energy ID: {sorted_group.at[min_dpmd_index, 'ID']}")

output_file = "/data/demo_LiCoO2/Vacancy-opt/data/energy_comparison.xlsx"

with pd.ExcelWriter(output_file, engine='openpyxl') as writer:
for name, ids in sorted_data.items():
df = pd.DataFrame(ids, columns=["ID"])
df.to_excel(writer, sheet_name=f"{name}_atoms", index=False)

fig, axes = plt.subplots(1, 2, figsize=(6, 4))

for ax, (name, group) in zip(axes, grouped_data):
if name == 39:
label_name = "Li$_3$Co$_{12}$O$_{24}$"
ax.set_ylim(-5.6, -4.8)
elif name == 45:
label_name = "Li$_9$Co$_{12}$O$_{24}$"
ax.set_ylim(-5.8, -5.2)
else:
label_name = f"{name} atoms"

ax.scatter(group["ID"], group["DFT_Energy"] / name, label=f"{label_name} DFT")
ax.scatter(group["ID"], group["DP_Energy"] / name, label=f"{label_name} DP")
ax.set_xlabel("ID")
ax.set_ylabel("Energy (eV/atom)")
ax.legend(loc='upper right')
ax.set_title(f"{label_name}")

plt.tight_layout()
plt.show()
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[21]
from IPython.display import Image
Image("/data/demo_LiCoO2/Vacancy-opt/data/minimize-structure.png",width=800,height=600)
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由图可知，DP对空位结构的绝对能量预测结果与DFT数据存在一定偏差，但是能量变化趋势完全一致。

总结

DP模型具有较高的计算精度和较小的计算成本，基于DP模型可以实现大尺度、长时间的MD模拟，是电池等材料体系研究的有力工具，将极大地拓宽研究领域的边界，推动电池等领域相关科学问题的解决，促进新发现的产生。

参考

[1] Hu T, Dai F, Zhou G, et al. Unraveling the Dynamic Correlations between Transition Metal Migrations and the Oxygen Dimer Formation in the Layered LiCoO2 Cathode[J]. 2023.

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