Self-consistent procedure in DFT Construct effective potential Guess density n(r) Ver (r)=Vest (r)+Vu[n]+Vse[n] 0imtniatn Solve KS equation @(hm( v.rtb.m-. Calculate new density and Compare r)=∑w,r) no Self- yes Obtain output: consistent Total energy,Force, ? Eigenvalues,…
Self-consistent procedure in DFT
The mathematical problem How to search for the local minimum of a function f) The Hessian matrix is a square matrix of second-order partial derivatives of a f阀=a+脉+2B数=a+民-B民-。 scalar-valued function. Where B is the Hessian matrix 5 80f1 of 8r好 8r18r2 821 82n Bj Oxioxj 80f 8 80f 0r202x1 For a stationary point,one requires B= 8r好 8r20n g(= af 0派 =B(-) of of f 8xn 8x1 8zn Ox2 Bui g()= at the minimum the Hessian matrix must be additionally positive definite
The mathematical problem How to search for the local minimum of a function f(x) at the minimum the Hessian matrix must be additionally positive definite. B The Hessian matrix is a square matrix of second-order partial derivatives of a scalar-valued function. Where B is the Hessian matrix For a stationary point, one requires
Ionic relaxation Steepest descent 1.initial guess 最速下降法 初始点 J6,0)。 2.calculate the gradient g( 3.make a step into the direction of the steepest descent 2=-1/Tmax(B)g() 最小值 4.repeat step 2 and 3 until convergence is reached for functions with long steep valleys convergence can be very slow max min
Ionic relaxation 最速下降法
Ionic relaxation Newton Algorithm 牛顿迭代法 .start with an arbitrary start point .calculate the gradient g() multiply with the inverse of the Hessian matrix and perform a step 十 2=-B1() by inserting g()=B(),one immediately recognises that hence one can find the minimum in one step 0
Ionic relaxation 牛顿迭代法 Newton Algorithm
Ionic relaxation Full DIIS algorithm 1. steepest descent step from to(arrows correspond to gradients go andg) 2. gradient along indicated red line is now know.determine optimal position 迭代子空间中直接求逆 3. another steepest descent step formalong gopt=g() 4. caluateradinnowthe radient isknown in the entiredimensional space (linearity condition)and the function can be minimised exactly search in the space spanned by {gi=1,...,N}for the minimal gradient gpt=∑cdg. ak9 ax.I a+al=1 0 and calculate the corresponding position opt=∑c't 0 opt
Ionic relaxation 迭代子空间中直接求逆