{"id":233,"date":"2022-06-18T01:55:43","date_gmt":"2022-06-17T17:55:43","guid":{"rendered":"https:\/\/www.homechemer.com\/?p=233"},"modified":"2022-06-19T13:48:17","modified_gmt":"2022-06-19T05:48:17","slug":"%e5%9f%ba%e4%ba%8etensorflow%e5%8c%85%e6%96%b9%e4%be%bf%e5%9c%b0%e8%ae%a1%e7%ae%97wilsons-b%e7%9f%a9%e9%98%b5%e8%bf%9b%e8%a1%8c%e5%9d%90%e6%a0%87%e7%b3%bb%e8%bd%ac%e6%8d%a2","status":"publish","type":"post","link":"https:\/\/www.homechemer.com\/computchem\/4","title":{"rendered":"\u57fa\u4e8eTensorflow\u65b9\u4fbf\u5730\u8ba1\u7b97Wilson’s B\u77e9\u9635\u53ca\u5176\u5e7f\u4e49\u9006\u8fdb\u884c\u5750\u6807\u7cfb\u8f6c\u6362"},"content":{"rendered":"\n

\u57fa\u4e8eTensorflow\u65b9\u4fbf\u5730\u8ba1\u7b97Wilson’s B\u77e9\u9635\u53ca\u5176\u5e7f\u4e49\u9006\u8fdb\u884c\u5750\u6807\u7cfb\u8f6c\u6362
Calculating Wilson’s B Matrix and Its Generalized Inverse Conveniently Based on Tensorflow for Coordinate Systems Conversion<\/strong><\/p>\n\n\n\n

Jiawei Xu
Released: 2022-06-19 \/ Updated: 2022-06-19<\/p>\n\n\n\n

1. Wilson’s B\u77e9\u9635\u7684\u5b9a\u4e49<\/strong><\/p>\n\n\n\n

\u7b1b\u5361\u5c14\u5750\u6807\u7cfb\u5177\u6709\u76f4\u89c2\u3001\u6613\u4e8e\u6570\u5b66\u5904\u7406\u3001\u5b9a\u4e49\u552f\u4e00\u7b49\u4f18\u70b9\uff0c\u800c\u66f4\u5177\u6709\u5316\u5b66\u610f\u4e49\u7684\u5750\u6807\u7cfb\u662f\u5185\u5750\u6807\u6216\u5197\u4f59\u5185\u5750\u6807(redundant internal coordinates, RIC)\uff0c\u5408\u7406\u5b9a\u4e49\u7684\u5197\u4f59\u5185\u5750\u6807\u5728\u51e0\u4f55\u4f18\u5316\u7b49\u95ee\u9898\u4e0a\u5177\u6709\u660e\u663e\u7684\u4f18\u52bf\u3002\u56e0\u6b64\uff0c\u5728\u4ee5\u4e0a\u4e24\u7c7b\u5750\u6807\u7cfb\u95f4\u8fdb\u884c\u6570\u636e\u8f6c\u6362\u662f\u91cf\u5316\u8ba1\u7b97\u4e2d\u4e0d\u53ef\u907f\u514d\u7684\u95ee\u9898\u3002Wilson’s B\u77e9\u9635\u7ed9\u51fa\u4e86\u7b1b\u5361\u5c14\u5750\u6807\u7cfb\u548c\u5197\u4f59\u5185\u5750\u6807\u7cfb\u4e4b\u95f4\u7684\u8f6c\u6362\u5f62\u5f0f\uff0c\u5176\u77e9\u9635\u5143\u6709\u4ee5\u4e0b\u5f62\u5f0f\uff1a<\/p>\n\n\n\n

\\begin{equation}
B_{ij}=\\frac{\\partial q_i}{\\partial x_j}
\\label{eq1}
\\end{equation}<\/p>\n\n\n\n

\u5176\u4e2dq_i<\/span>\u548cx_j<\/span>\u5206\u522b\u4e3a\u5197\u4f59\u5185\u5750\u6807\u548c\u7b1b\u5361\u5c14\u5750\u6807\u3002\u5f53\u4e14\u4ec5\u5f53\u5197\u4f59\u5185\u5750\u6807\u7684\u4e2a\u6570\u4e3a3N<\/span>\u65f6\uff0cWilson’s B\u77e9\u9635\u4e3a\u4e00\u4e2a\u65b9\u9635\u3002\u7531\u5b9a\u4e49\u53ef\u77e5\uff0cWilson’s B\u77e9\u9635\u5b9e\u9645\u4e0a\u5c31\u662f\u7531\u4e24\u7ec4\u5750\u6807\u53d8\u91cf\u7ed9\u5b9a\u7684Jacobi\u77e9\u9635\u3002\u4e00\u822c\u6765\u8bf4\uff0c\u5bf9\u4e8e\u5750\u6807\u5fae\u5143\u6709\u4ee5\u4e0b\u8f6c\u6362\u6210\u7acb\uff1a<\/p>\n\n\n\n

\\begin{equation}
\\mathbf B\\delta \\mathbf x=\\delta \\mathbf q
\\label{eq2}
\\end{equation}<\/p>\n\n\n\n

\\begin{equation}
\\mathbf B^+\\delta \\mathbf q=\\delta \\mathbf x
\\label{eq3}
\\end{equation}<\/p>\n\n\n\n

\u5bf9\u4e8e\u5750\u6807\u7cfb\u4e2d\u7684\u77e2\u91cf\uff1a<\/p>\n\n\n\n

\\begin{equation}
\\mathbf g_x=\\mathbf B^{\\rm T}\\mathbf g_q
\\label{eq4}
\\end{equation}<\/p>\n\n\n\n

\\begin{equation}
\\mathbf g_q=(\\mathbf B^{\\rm T})^+\\mathbf g_x
\\label{eq5}
\\end{equation}<\/p>\n\n\n\n

Wilson’s B\u77e9\u9635\u7684\u5f62\u5f0f\u5f88\u7b80\u6d01\uff0c\u4f46\u5404\u5143\u7d20\u7684\u6c42\u89e3\u5341\u5206\u9ebb\u70e6\uff0c\u5c24\u5176\u662f\u4e8c\u9762\u89d2\u5bf912\u4e2a\u7b1b\u5361\u5c14\u5750\u6807\u7684\u89e3\u6790\u504f\u5bfc\u6570\u5f62\u5f0f\u6781\u4e3a\u590d\u6742\uff0c\u5177\u4f53\u6570\u5b66\u8868\u8fbe\u5f0f\u53ef\u53c2\u8003J. Chem. Phys.<\/em> 2002<\/strong>, 117, 9160<\/a>\u3002\u5373\u4f7f\u662f\u7528\u73b0\u6210\u7684\u89e3\u6790\u5f62\u5f0f\u7f16\u7a0b\uff0c\u4e5f\u9700\u8981\u82b1\u8d39\u4e0d\u5c11\u65f6\u95f4\u5199\u6210\u5e76\u68c0\u67e5\u4ee3\u7801\u3002\u4e0b\u9762\u5c06\u4ecb\u7ecdTensorflow\u81ea\u52a8\u6c42\u5bfc\u529f\u80fd\u7684\u57fa\u672c\u7528\u6cd5\uff0c\u5e76\u7528Tensorflow\u7ed3\u5408Python\u6c42\u89e3Wilson’s B\u77e9\u9635\uff0c\u8fdb\u4e00\u6b65\u5b9e\u73b0\u7ed3\u6784\u3001\u77e2\u91cf\u7b49\u6570\u636e\u5728\u4e0d\u540c\u5750\u6807\u7cfb\u4e4b\u95f4\u7684\u8f6c\u6362\u3002<\/p>\n\n\n\n

2. Tensorflow\u5305\u7684\u5b89\u88c5\u548c\u81ea\u52a8\u6c42\u5bfc\u529f\u80fd\u7684\u7b80\u4ecb<\/strong><\/p>\n\n\n\n

\u7531\u4e8e\u4e0b\u9762\u5c06\u5728Python\u4e2d\u8c03\u7528Tensorflow\uff0c\u6545\u5047\u5b9a\u7535\u8111\u4e0a\u5df2\u7ecf\u5b89\u88c5\u4e86Python 3.0\u53ca\u4ee5\u4e0a\u7248\u672c\u3002\u6211\u4eec\u4f7f\u7528pip\u5b89\u88c5Tensorflow\uff0c\u8fd9\u662f\u6700\u4e3a\u4fbf\u6377\u7684\u65b9\u6cd5\u3002\u9996\u5148\u68c0\u67e5pip\u662f\u5426\u662f\u6700\u65b0\u7248\u672c\uff0c\u5982\u4e0d\u662f\u5219\u66f4\u65b0\u81f3\u6700\u65b0\u7248\u672c\uff0c\u7136\u540e\u76f4\u63a5\u5b89\u88c5\u5373\u53ef\u3002<\/p>\n\n\n\n

\npip install --upgrade pip\npip install tensorflow==2.6.0\n\n<\/code><\/pre>\n\n\n\n
\u5982\u679c\u901f\u5ea6\u8fc7\u6162\uff0c\u53ef\u4ee5\u5c1d\u8bd5\u6e05\u534e\u5927\u5b66\u7684\u955c\u50cf\u3002<\/p>\n\n\n\n
\npip install -i https:\/\/pypi.tuna.tsinghua.edu.cn\/simple tensorflow==2.6.0\n\n<\/code><\/pre>\n\n\n\n
\u4e0b\u9762\u7b80\u5355\u4ecb\u7ecd\u4e00\u4e0bTensorflow\u7684\u81ea\u52a8\u6c42\u5bfc\u529f\u80fd\u3002Tensorflow\u81ea\u52a8\u6c42\u5bfc\u7684\u539f\u7406\u5f88\u7b80\u5355\uff0c\u662f\u4e2d\u5b66\u6570\u5b66\u4e2d\u5c31\u5b66\u8fc7\u7684\u94fe\u5f0f\u6cd5\u5219\u3002Tensorflow\u5728\u6574\u4e2a\u8ba1\u7b97\u8fc7\u7a0b\u4e2d\u6784\u5efa\u4e00\u5f20\u6709\u5411\u56fe\uff0c\u5728\u6bcf\u4e2a\u8ba1\u7b97\u8282\u70b9\u4e0a\u6c42\u5f53\u524d\u8ba1\u7b97\u5f0f\u7684\u5bfc\u6570\uff0c\u5b8c\u6210\u6784\u56fe\u540e\u6839\u636e\u94fe\u5f0f\u6cd5\u5219\u5c31\u53ef\u4ee5\u6c42\u51fa\u6574\u4e2a\u8ba1\u7b97\u8fc7\u7a0b\u7684\u89e3\u6790\u5bfc\u6570\u3002\u867d\u7136\u4e0d\u80fd\u76f4\u63a5\u7ed9\u51fa\u89e3\u6790\u5f62\u5f0f\uff0c\u4f46\u5176\u7ed3\u679c\u5fc5\u7136\u662f\u548c\u89e3\u6790\u5bfc\u6570\u4e00\u81f4\u7684\u3002Tensorflow\u57fa\u4e8e\u6709\u5411\u56fe\u7684\u6846\u67b6\u4f7f\u5f97\u5b83\u6210\u4e3a\u4e86\u56fe\u795e\u7ecf\u7f51\u7edc(graph neural network, GNN)\u7b97\u6cd5\u4e2d\u6700\u91cd\u8981\u7684\u5de5\u5177\uff0c\u8fd1\u51e0\u5e74\u5728\u6df1\u5ea6\u5b66\u4e60\u7b49\u9886\u57df\u5e94\u7528\u5341\u5206\u5e7f\u6cdb\u3002Tensorflow\u81ea\u52a8\u6c42\u5bfc\u7684\u57fa\u672c\u8bed\u53e5\u5982\u4e0b\uff1a<\/p>\n\n\n\n
\nimport tensorflow as tf\nx=tf.Variable(initial_value=3.0)\nwith tf.GradientTape() as tape:\n    y=tf.square(x)\ngrad=tape.gradient(y,x)\nprint(grad)\n\n# tf.Tensor(6.0, shape=(), dtype=float32)\n\n<\/code><\/pre>\n\n\n\n
\u9996\u5148\u7528tf.Variable<\/mark>\u58f0\u660e\u88ab\u6c42\u5bfc\u7684\u53d8\u91cf\uff0c\u5e76\u5bf9\u5176\u8d4b\u503cinitial_value=3.0<\/mark>\u3002\u7136\u540e\u7528with tf.GradientTape() as tape:<\/mark>\u4e00\u884c\u5f15\u5bfc\u4e00\u4e2a\u8ba1\u7b97\u6a21\u5757\uff0c\u5728\u8be5\u6a21\u5757\u4e2d\u7684\u6240\u6709\u8ba1\u7b97\u8fc7\u7a0b\u4f1a\u88ab\u7528\u4e8e\u6784\u56fe\uff0c\u5e76\u8bb0\u5f55\u4e3atape<\/mark>\u3002\u5bf9\u4e8e\u4e0a\u9762\u7684\u8fd9\u4e2a\u4f8b\u5b50\uff0c\u6a21\u5757\u4e2d\u8ba1\u7b97\u4e86y=x^2<\/span>\u3002\u5728\u6a21\u5757\u5916\u8fdb\u884c\u6c42\u5bfc\uff0ctape.gradient(y,x)<\/mark>\u7528\u4e8e\u6307\u5b9a\u6c42\u6a21\u5757\u4e2d\u7684\u51fd\u6570y<\/mark>\u5bf9\u53d8\u91cfx<\/mark>\u7684\u5bfc\u6570\u3002\u7a0b\u5e8f\u8f93\u51fa\u7ed3\u679c6.0\uff0c\u5e76\u7ed9\u51fa\u53d8\u91cf\u5f62\u72b6\u548c\u7c7b\u578b\u3002<\/p>\n\n\n\n
\u4e5f\u53ef\u4ee5\u58f0\u660e\u591a\u4e2a\u53d8\u91cf\u6c42\u591a\u5143\u51fd\u6570\u7684\u504f\u5bfc\u6570\uff0c\u6b64\u65f6\u4e00\u5b9a\u8981\u5199persistent=True<\/mark>\uff1a<\/p>\n\n\n\n
\nimport tensorflow as tf\nx=tf.Variable(initial_value=2.0)\ny=tf.Variable(initial_value=3.0)\nwith tf.GradientTape(persistent=True) as tape:\n    z=tf.square(x)+tf.square(y)\ngrad1=tape.gradient(z,x)\ngrad2=tape.gradient(z,y)\ngrad3=tape.gradient(x,y)\nprint(grad1,grad2,grad3,sep='\\n')\n\n# tf.Tensor(4.0, shape=(), dtype=float32)\n# tf.Tensor(6.0, shape=(), dtype=float32)\n# None (No output since dx\/dy does not exist!)\n\n<\/code><\/pre>\n\n\n\n
3. \u4ee3\u7801\u5b9e\u73b0<\/strong><\/p>\n\n\n\n
3.1 \u4ea7\u751f\u5185\u5750\u6807\u7cfb<\/strong><\/p>\n\n\n\n
\u4e3a\u4e86\u65b9\u4fbf\u6bd4\u8f83\u4e0d\u540c\u7ed3\u6784\uff0c\u9700\u8981\u6709\u4e00\u5957\u552f\u4e00\u786e\u5b9a\u7684\u5185\u5750\u6807\u4ea7\u751f\u65b9\u6cd5\u3002\u5bf9\u4e8e\u66f4\u4e00\u822c\u7684\u5197\u4f59\u5185\u5750\u6807\u7684\u60c5\u51b5\uff08\u5f53\u5197\u4f59\u5185\u5750\u6807\u6ca1\u6709\u5197\u4f59\u81ea\u7531\u5ea6\u65f6\uff0c\u5373\u9000\u5316\u4e3a\u5185\u5750\u6807\uff09\uff0c\u4e5f\u4e00\u6837\u9700\u8981\u89c4\u5b9a\u552f\u4e00\u786e\u5b9a\u7684\u4ea7\u751f\u65b9\u6cd5\u3002\u8fd9\u91cc\u6211\u4eec\u89c4\u5b9a\uff0c\u5bf9\u7ed3\u6784\u4e2d\u5e8f\u53f7\u76f8\u90bb\u7684\u4e24\u4e2a\u539f\u5b50\u5b9a\u4e49\u952e\u957f\uff0c\u76f8\u90bb\u7684\u4e09\u4e2a\u539f\u5b50\u5b9a\u4e49\u952e\u89d2\uff0c\u76f8\u90bb\u7684\u56db\u4e2a\u539f\u5b50\u5b9a\u4e49\u4e8c\u9762\u89d2\u3002\u6545\u4e00\u5171\u6709(k-1)<\/span>\u4e2a\u952e\u957f\u3001(k-2)<\/span>\u4e2a\u952e\u89d2\u548c(k-3)<\/span>\u4e2a\u4e8c\u9762\u89d2\u3002\u4ee3\u7801\u5b9e\u73b0\u5982\u4e0b\uff1a<\/p>\n\n\n\n
\ndef genintval(geom,radunit=False):\n    bond,angle,dihedral=[],[],[]\n    N=len(geom)\/3\n    for i in range(N):\n        if i==0:\n            pass\n        else:\n            coord1=[geom[3*i-3],geom[3*i-2],geom[3*i-1]]\n            coord2=[geom[3*i],geom[3*i+1],geom[3*i+2]]\n            bond.append(distance(coord1,coord2))\n    for i in range(N):\n        if i<=1:\n            pass\n        else:\n            coord1=[geom[3*i-6],geom[3*i-5],geom[3*i-4]]\n            coord2=[geom[3*i-3],geom[3*i-2],geom[3*i-1]]\n            coord3=[geom[3*i],geom[3*i+1],geom[3*i+2]]\n            angle.append(angle(coord1,coord2,coord3,radunit))\n    for i in range(N):\n        if i<=2:\n            pass\n        else:\n            coord1=[geom[3*i-9],geom[3*i-8],geom[3*i-7]]\n            coord2=[geom[3*i-6],geom[3*i-5],geom[3*i-4]]\n            coord3=[geom[3*i-3],geom[3*i-2],geom[3*i-1]]\n            coord4=[geom[3*i],geom[3*i+1],geom[3*i+2]]\n            dihedral.append(dihedral(coord1,coord2,coord3,coord4,radunit))\n    intcoord=bond+angle+dihedral\n    return intcoord,bond,angle,dihedral\n\n<\/code><\/pre>\n\n\n\n
\u8be5\u51fd\u6570\u7684\u8f93\u5165geom<\/mark>\u4e3a\u7b1b\u5361\u5c14\u5750\u6807\u7cfb\u4e0b\u7684\u7ed3\u6784\uff0c\u53ef\u4ee5\u65b9\u4fbf\u5730\u4ece\u5404\u79cd\u91cf\u5316\u7a0b\u5e8f\u7684\u8f93\u51fa\u6587\u4ef6\u4e2d\u8bfb\u53d6\uff0c\u6392\u6210list\u578b\u3002\u8fd9\u91cc\u6d89\u53ca\u5230\u4e86\u5185\u5750\u6807\u7684\u6c42\u7b97\u3002\u952e\u957f\u952e\u89d2\u662f\u6bd4\u8f83\u65b9\u4fbf\u7684\uff0c\u4e8c\u9762\u89d2\u7684\u4e00\u79cd\u53ef\u884c\u7684\u6c42\u7b97\u65b9\u6cd5\u5982\u4e0b\uff0cradunit<\/mark>\u7528\u4e8e\u63a7\u5236\u8ba1\u7b97\u7ed3\u679c\u4f7f\u7528\u4f55\u79cd\u5355\u4f4d\u5236\u3002\u7ecf\u5178\u7684\u534a\u5e73\u9762\u6cd5\u5411\u91cf\u6c42\u4e8c\u9762\u89d2\u7684\u5173\u952e\u95ee\u9898\u5728\u4e8e\u5224\u65ad\u662f\u9510\u89d2\u8fd8\u662f\u949d\u89d2\uff0c\u8fd9\u91cc\u7a0b\u5e8f\u5b9e\u73b0\u7684\u66ff\u4ee3\u65b9\u6cd5\u662f\u5148\u6c42\u534a\u5e73\u9762\u4e0a\u4e24\u70b9\u5bf9\u4ea4\u7ebf\u5782\u8db3\u5230\u8fd9\u4e24\u4e2a\u70b9\u7684\u5411\u91cf\uff0c\u518d\u6c42\u5411\u91cf\u5939\u89d2\uff0c\u7ed5\u5f00\u4e86\u8fd9\u4e00\u95ee\u9898\u3002<\/p>\n\n\n\n
\ndef dotprod(vec1,vec2):\n    dotprod=0.0\n    for i in range(len(vec1)):\n        dotprod+=(vec1[i]*vec2[i])\n    return dotprod\ndef norm(vec):\n    norm=0.0\n    for i in range(len(vec)):\n        norm+=(vec[i])**2\n    norm=math.sqrt(norm)\n    return norm\ndef vecdiff(vec1,vec2):\n    vecdiff=[]\n    for i in range(len(vec1)):\n        vecdiff.append(vec1[i]-vec2[i])\n    return vecdiff\ndef vecangle(vec1,vec2,radunit=False):\n    dp=dotprod(vec1,vec2)\n    norm1=norm(vec1)\n    norm2=norm(vec2)\n    angle=math.acos(dp\/(norm1*norm2))\n    if radunit:\n        pass\n    else:\n        angle=180.0*angle\/math.pi\n    return angle\ndef dotpreptoline(coord1,coord2,coord3):\n    vec12=vecdiff(coord1,coord2)\n    vec23=vecdiff(coord2,coord3)\n    coeffvec1=[vec23[1],-vec23[0],0.0]\n    coeffvec2=[0.0,-vec23[2],vec23[1]]\n    coeffvec3=[-vec23[0],-vec23[1],-vec23[2]]\n    A=numpy.array([coeffvec1,coeffvec2,coeffvec3])\n    b=numpy.array([coord3[0]*vec23[1]-coord3[1]*vec23[0],coord3[2]*vec23[1]-coord3[1]*vec23[2],-coord1[0]*vec23[0]-coord1[1]*vec23[1]-coord1[2]*vec23[2]])\n    foot1at23=numpy.linalg.solve(A,b)\n    distance1to23=distance(foot1at23,coord1)\n    return distance1to23,foot1at23\ndef dihedral(coord1,coord2,coord3,coord4,radunit=False):\n    dis1to23,foot1at23=dotpreptoline(coord1,coord2,coord3)\n    dis4to23,foot4at23=dotpreptoline(coord4,coord2,coord3)\n    dihedral=vecangle(vecdiff(coord1,foot1at23),vecdiff(coord4,foot4at23),radunit)\n    return dihedral\n\n<\/code><\/pre>\n\n\n\n
3.2 Tensorflow\u81ea\u52a8\u6c42\u5bfc\u8ba1\u7b97Wilson’s B\u77e9\u9635\u5143<\/strong><\/p>\n\n\n\n
\u4e0b\u9762\u7684\u51fd\u6570\u4f7f\u7528Tensorflow\u8ddf\u8e2a\u8ba1\u7b97\u5185\u5750\u6807\u6570\u503c\u65f6\u7684\u6570\u636e\u6d41\u8ba1\u7b97Wilson’s B\u77e9\u9635\u5143\uff0c\u5728\u540c\u4e00\u4e2a\u51fd\u6570\u91cc\u8ba1\u7b97\u5185\u5750\u6807\u548cWilson’s B\u77e9\u9635\u3002\u4e0e\u76f4\u63a5\u5229\u7528\u89e3\u6790\u5f0f\u6c42\u89e3\u7684\u4f20\u7edf\u65b9\u6cd5\u76f8\u6bd4\uff0cTensorflow\u5728\u4ee3\u7801\u4e0a\u660e\u663e\u66f4\u4e3a\u7b80\u6d01\u548c\u76f4\u89c2\uff0c\u4e14\u80fd\u591f\u652f\u6301GPU\u52a0\u901f\uff0c\u5bf9\u4e8e\u5904\u7406\u5927\u5c3a\u5ea6\u4f53\u7cfb\uff0c\u6216\u8005\u5904\u7406\u590d\u6742\u6570\u636e\uff08\u5982Hessian\u77e9\u9635\u7b49\uff09\u7684\u5750\u6807\u7cfb\u8f6c\u6362\u65f6\u6709\u663e\u8457\u7684\u4f18\u52bf\u3002<\/p>\n\n\n\n
\ndef icm(geom,radunit=True):\n    dim=len(geom)\n    N=int(dim\/3)\n    bond12,angle123,dihedral1234=[],[],[]\n    for i in range(len(geom)):\n        geom[i]=geom[i]\/bohrradius\n    icm=numpy.zeros((dim-6,dim))\n    for i in range(N-1):\n        coord1,coord2=[],[]\n        for j in range(3):\n            coord1.append(tf.Variable(initial_value=geom[3*i+j]))\n            coord2.append(tf.Variable(initial_value=geom[3*i+j+3]))\n        with tf.GradientTape(persistent=True) as bondcalc:\n            R12=distance(coord1,coord2)\n            bond12.append(R12)\n        for j in range(3):\n            icm[i,3*i+j]=bondcalc.gradient(R12,coord1[j])\n            icm[i,3*i+j+3]=bondcalc.gradient(R12,coord2[j])\n        if i<N-2:\n            coord3=[]\n            for j in range(3):\n                coord3.append(tf.Variable(initial_value=geom[3*i+j+6]))\n            with tf.GradientTape(persistent=True) as anglecalc:\n                A123=angle(coord1,coord2,coord3,radunit)\n                angle123.append(A123)\n            for j in range(3):\n                icm[N-1+i,3*i+j]=anglecalc.gradient(A123,coord1[j])\n                icm[N-1+i,3*i+j+3]=anglecalc.gradient(A123,coord2[j])\n                icm[N-1+i,3*i+j+6]=anglecalc.gradient(A123,coord3[j])\n            if i<N-3:\n                coord4=[]\n                for j in range(3):\n                    coord4.append(tf.Variable(initial_value=geom[3*i+j+9]))\n                with tf.GradientTape(persistent=True) as dihedralcalc:\n                    D1234=dihedral(coord1,coord2,coord3,coord4,radunit)\n                    dihedral1234.append(D1234)\n                for j in range(3):\n                    icm[2*N-3+i,3*i+j]=dihedralcalc.gradient(D1234,coord1[j])\n                    icm[2*N-3+i,3*i+j+3]=dihedralcalc.gradient(D1234,coord2[j])\n                    icm[2*N-3+i,3*i+j+6]=dihedralcalc.gradient(D1234,coord3[j])\n                    icm[2*N-3+i,3*i+j+9]=dihedralcalc.gradient(D1234,coord4[j])\n    intcoord=bond12+angle123+dihedral1234\n    return icm,intcoord\n\n<\/code><\/pre>\n\n\n\n
3.3 \u5947\u5f02\u503c\u5206\u89e3\u6cd5\u6c42Wilson’s B\u77e9\u9635\u7684Penros\u5e7f\u4e49\u9006<\/strong><\/p>\n\n\n\n
\u4e0d\u6ee1\u79e9\u7684\u77e9\u9635\u4e0d\u53ef\u9006\uff0c\u957f\u65b9\u77e9\u9635\u4e5f\u65e0\u6cd5\u4f7f\u7528\u65b9\u9635\u6c42\u9006\u7684\u65b9\u6cd5\u3002\u7531\u4e8eWilson’s B\u77e9\u9635\u662f\u4e0d\u6ee1\u79e9\u7684\uff0c\u4e14\u4e0d\u4e00\u5b9a\u662f\u65b9\u9635\uff0c\u5373\u5197\u4f59\u5185\u5750\u6807\u5b9a\u4e49\u7684\u4e2a\u6570\u4e0d\u4e00\u5b9a\u7b49\u4e8e3N<\/span>\uff0c\u6545\u6b64\u5904\u4f7f\u7528\u5e7f\u4e49\u9006(generalized inverse)\u7684\u6982\u5ff5\u7ed9\u51fa\u5e7f\u4e49\u9006\u77e9\u9635\\mathbf A^+<\/span>\u3002\u5e7f\u4e49\u9006\u7684\u6c42\u6cd5\u6709\u597d\u51e0\u79cd\uff0c\u8fd9\u91cc\u4f7f\u7528\u5947\u5f02\u503c\u5206\u89e3\u6cd5(singular value decomposition, SVD)\u3002\u5947\u5f02\u503c\u5206\u89e3\u662f\u4e00\u79cd\u56e0\u5b50\u5206\u89e3\u8fd0\u7b97, \u5c06\u4e00\u4e2a\u77e9\u9635\u5206\u89e3\u4e3a3\u4e2a\u77e9\u9635\u7684\u4e58\u79ef\u3002\u5176\u4e2d, \u5947\u5f02\u503c\u77e9\u9635\\mathbf \\Sigma<\/span>\u662f\u5bf9\u89d2\u9635\uff0c\\mathbf U<\/span>\u548c\\mathbf V<\/span>\u662f\u4e24\u4e2a\u6b63\u4ea4\u9635\u3002<\/p>\n\n\n\n
\\begin{equation}
  \\mathbf A=\\mathbf U\\mathbf \\Sigma \\mathbf V
  \\label{eq6}
\\end{equation}<\/p>\n\n\n\n
\u8fd9\u6837\uff0c\\mathbf A<\/span>\u7684Penros\u5e7f\u4e49\u9006\u53ef\u4ee5\u7531\\eqref{eq7}\u7ed9\u51fa\uff1a<\/p>\n\n\n\n
\\begin{equation}
  \\mathbf A^+=\\mathbf V^{\\rm T}\\mathbf \\Sigma^{-1} \\mathbf U^{\\rm T}
  \\label{eq7}
\\end{equation}<\/p>\n\n\n\n
\u5bf9\u4e8e\u957f\u65b9\u77e9\u9635\uff0c\u901a\u8fc7\u589e\u52a0\u5143\u7d20\u5168\u4e3a0\u7684\u884c\u6216\u5217\u8f6c\u6362\u4e3a\u65b9\u9635\u540e\uff0c\u53ef\u4ee5\u4f9d\u7167\u4e0a\u8ff0\u65b9\u6cd5\u6c42\u5e7f\u4e49\u9006\uff0c\u8fd9\u5728Numpy\u4e2d\u6709\u73b0\u6210\u7684\u51fd\u6570\u3002<\/p>\n\n\n\n
\nimport numpy as np\npinv=np.linalg.pinv(mat)\n\n<\/code><\/pre>\n\n\n\n
\u6c42\u5f97Wilson’s B\u77e9\u9635\u7684\u5e7f\u4e49\u9006\u6216\u8f6c\u7f6e\u5e7f\u4e49\u9006\u540e\uff0c\u5373\u53ef\u6839\u636e\\eqref{eq2}~\\eqref{eq5}\u8fdb\u884c\u5750\u6807\u53d8\u6362\u3002<\/p>\n\n\n\n
<\/p>\n\n\n\n
<\/p>\n\n\n\n
<\/p>\n\n\n\n
<\/p>\n\n\n\n
<\/p>\n\n\n\n
<\/p>\n\n\n\n
* Xuliang Wang from Institute of Computing Technology, Chinese Academy of Sciences provided valuable advice for this work.<\/p>\n","protected":false},"excerpt":{"rendered":"
\u57fa\u4e8eTensorflow\u65b9\u4fbf\u5730\u8ba1\u7b97Wilson&# …<\/p>\n","protected":false},"author":9,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"spay_email":"","footnotes":""},"categories":[5],"tags":[],"class_list":["post-233","post","type-post","status-publish","format-standard","hentry","category-computchem"],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/www.homechemer.com\/index.php?rest_route=\/wp\/v2\/posts\/233","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.homechemer.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.homechemer.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.homechemer.com\/index.php?rest_route=\/wp\/v2\/users\/9"}],"replies":[{"embeddable":true,"href":"https:\/\/www.homechemer.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=233"}],"version-history":[{"count":66,"href":"https:\/\/www.homechemer.com\/index.php?rest_route=\/wp\/v2\/posts\/233\/revisions"}],"predecessor-version":[{"id":1620,"href":"https:\/\/www.homechemer.com\/index.php?rest_route=\/wp\/v2\/posts\/233\/revisions\/1620"}],"wp:attachment":[{"href":"https:\/\/www.homechemer.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=233"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.homechemer.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=233"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.homechemer.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=233"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}