《分子电子结构》研究生课程教学资源（Electronic Structure of Molecules）8-GeomOpt

TheNewton-RaphsonmethodThe two above only use/require information about g;If we also knew something about the second-derivative of the energy, H, that should speed upoptimization even further;In one dimension, if function was quadratic, cang(q)exactly write (dE/dq'=O):q=qH(q)If not quadratic, make Taylor expansion of energy in qup to second-order, arrive at same expression forpredicting q';Apply iteratively;In well-behaved cases this algorithm rapidly convergesto localminimumhttps://en.wikipedia.org/wiki/Newton%27smethod17

The Newton-Raphson method 17 • The two above only use/require information about g; • If we also knew something about the second￾derivative of the energy, H, that should speed up optimization even further; • In one dimension, if function was quadratic, can exactly write (dE/dq’=0): • If not quadratic, make Taylor expansion of energy in q up to second-order, arrive at same expression for predicting q′; • Apply iteratively; • In well-behaved cases this algorithm rapidly converges to local minimum https://en.wikipedia.org/wiki/Newton%27s_method

TheNewton-Raphsonmethodfor finding a minimumTaylorexpandaboutq:dE1d2EE(g)~E(g)Tdq2dd'EdEdE0dq'dqdq?dE/dqgg=qHd'EIdqq'qCangeneralize this approach to multiple dimensions;Makesecond-orderTaylorexpansionof energyabout R,can showOptimal R'= R -H-1(R)g(R)Again can perform line-search in direction R'- R if you want;Apply iteratively;The Newton-Raphson method works really well;18

The Newton-Raphson method for finding a minimum 18 • Taylor expand about q: • Can generalize this approach to multiple dimensions; R′ = R −H−1(R)g(R) • Make second-order Taylor expansion of energy about R, can show optimal • Again can perform line-search in direction R′− R if you want; • Apply iteratively; • The Newton-Raphson method works really well;