《分子电子结构》研究生课程教学资源（Electronic Structure of Molecules）Lecture 8 is about optimization, including the algorithms and software operation

幻灯片2LasttimePotentialenergysurfaces1.Potentialenergycurves/surfaces2.Born-Oppenheimer Approximation3.Propertiesfromenergyderivatives4. Exploring PESs1. Rigid scan2. Relaxed scan上次课程讨论了势能面，介绍了势能面和势能曲线，BOA，能量求导与分子性质，演示了扫描势能面的方法

幻灯片 2 Last time 2 Potential energy surfaces 1. Potential energy curves/surfaces 2. Born-Oppenheimer Approximation 3. Properties from energy derivatives 4. Exploring PESs 1. Rigid scan 2. Relaxed scan 上次课程讨论了势能面，介绍了势能面和势能曲线，BOA，能量求导与分子性 质，演示了扫描势能面的方法

幻灯片3Contents1. Geometryoptimization2.Vibrationalnormal mode3. Frequency calculation4.IRandRaman5. Thermochemistry1、几何优化；2、简正振动；3、振动频率；4、IR和拉曼

幻灯片 3 Contents 3 1. Geometry optimization 2. Vibrational normal mode 3. Frequency calculation 4. IR and Raman 5. Thermochemistry 1、几何优化； 2、简正振动； 3、振动频率； 4、IR和拉曼

幻灯片4Potentialenergy surfaceSecond Order Saddle PointTransition Structure ATransitionStructureBMinimum forProductAMinimumforProduct B10.5Second Order0.5Saddle Point/Valley-Ridge0.5X-0.5Inflection PointMinimumforReactantAbstractaxes,hypersurface (3N-6dimensions),multiplestates (S,, S.-), multiple pathways linking minima我们再回顾一下PES的主要特点除了能够简化为两个变量的体系，PES的坐标无实际定量意义。可以称为反应坐标。超曲面，曲面上极大值极小值，鞍点；多个面：多条反应路径连接极小值

幻灯片 4 Potential energy surface 4 Abstract axes, hypersurface (3N-6 dimensions), multiple states (S0 , S1.), multiple pathways linking minima 我们再回顾一下PES的主要特点 除了能够简化为两个变量的体系，PES的坐标无实际定量意义。可以称为反应坐 标。 超曲面，曲面上极大值极小值，鞍点；多个面；多条反应路径连接极小值

幻灯片5B-Oapproximation&energyderivativesSadfle PoinnaonGradient (forces)F=-g(R...)t4frrDmduetseconWarfateP'ETES0EEHessiangi28.04CE地Born-Oppenheimerapproximation: decoupleCnonH:the motion of electrons from nucleiWtotal =electronic XPnaclearO'ErEs'ErsCEr+22eCn,g.Epes(R...R)=ECg,og.g2RBOA和E导数1、PES是BOA的直接结果，即电子和原子核运动坐标分离2、BOA背后是电子与原子核质量以及速度的巨大差异的结果。3、SP能量是电子能+原子核库伦排斥能=高斯最后报告的HF能量。4、能量对原子坐标的一阶（1x3N的向量）和二阶导数（3Nx3N），此处未展开为xyz

幻灯片 5 B-O approximation & energy derivatives 5 Gradient (forces) Hessian F = −g Born–Oppenheimer approximation: decouple the motion of electrons from nuclei BOA和E导数 1、PES是BOA的直接结果，即电子和原子核运动坐标分离 2、BOA背后是电子与原子核质量以及速度的巨大差异的结果。 3、SP能量是电子能+原子核库伦排斥能= 高斯最后报告的HF能量。 4、能量对原子坐标的一阶（1x3N的向量）和二阶导数（3Nx3N），此处未展 开为xyz

幻灯片6Keypoints about PES1. Hessian matrix is real and symmetric, can be diagonalized to giveeignevalues and eigenvectors;Eigenvectors give "natural' directions along PES;The harmonic vibrational/normal modes;.Eigenvalues indicate curvature in that direction;2. Minimum on multidimensional PES has gradient vector g = 0 and allpositive Hessian eigenvalues;3.First order saddle point, or transition state, has g =O and one andonly one negative Hessian eigenvalue;:Physically,one unique direction that leads downhill in energy;·Alwayscorrespondstothelowestenergypointconnectingtwominima;4. Minimum energy pathway (MEP)or intrinsic reaction coordinate (IRC)issteepestdescentpathway (inmass-weightedcoordinates)fromsaddle point to nearby minima.Patha marble with infinite inertia wouldfollow.1、H矩阵是实数、对称、且可以对角化的。对角化后产生本征值和本征向量。本征向量描述PES上原子的集体运动，即简谐正则振动（简正振动）：本征值代表简正振动的势能曲面的曲率。2、PES上极小值的特征g=0，h>0。3、一级鞍点，TS，g=0，1且1个负h的本征值。物理图像，只有这个本征向量的方向导致能量降低，是连接两个极小值的能量最低点。4、MEP、IRC是从鞍点到附近极小值的最速下降通道（质量权重过的坐标）。即无限惯性的石球可以沿其滚动

幻灯片 6 Key points about PES 6 1. Hessian matrix is real and symmetric, can be diagonalized to give eignevalues and eigenvectors; • Eigenvectors give “natural” directions along PES; • The harmonic vibrational/normal modes; • Eigenvaluesindicate curvature in that direction; 2. Minimum on multidimensional PES has gradient vector g = 0 and all positive Hessian eigenvalues; 3. First order saddle point, or transition state, has g = 0 and one and only one negative Hessian eigenvalue; • Physically, one unique direction that leads downhill in energy; • Always corresponds to the lowest energy point connecting two minima; 4. Minimum energy pathway (MEP) or intrinsic reaction coordinate (IRC) is steepest descent pathway (in mass-weighted coordinates) from saddle point to nearby minima. • Path a marble with infinite inertia would follow. 1、H矩阵是实数、对称、且可以对角化的。 对角化后产生本征值和本征向量。 本征向量描述PES上原子的集体运动，即简谐正则振动（简正振动）；本征值代 表简正振动的势能曲面的曲率。 2、PES上极小值的特征g=0，h>0。 3、一级鞍点，TS，g=0，1 且1个负h的本征值。物理图像，只有这个本征 向量的方向导致能量降低，是连接两个极小值的能量最低点。 4、MEP、IRC是从鞍点到附近极小值的最速下降通道（质量权重过的坐 标）。即无限惯性的石球可以沿其滚动