《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

幻灯片7Key points about PES5.Higher order saddle points have g =O and more than onenegative Hessian eigenvalue;Can always lead to lower energy first order saddle point;..These typically do not have chemical significance;It is usually our job to identify the critical points (minima and6.transitionstatesonaPES);In liquids, PES is much more flat and lightly corrugated;..Statisticalmechanicsbecomesmoresimportant;7.TherearemultiplePES'sforanyatomconfiguration,corresponding to different electronic states;Sometimes these states can interact, intersect, giving.avoided crossings,conical intersections...Lead to morecomplicated dynamical behaviorVibrational Analysis in Gaussianhttp://gaussian.com/vib/5、更高阶的鞍点g=0，多于一个H的负本征值。能够走向能量更低的一阶鞍点。它们一般没有化学意义。6、计算化学中，通常需要确认PES上的关键点。在液体中，PES更加平，带有轻微的褶皱，统计力学就变得更重要了。7、不同电子态对应不同的势能面。PES之间也会作用，交叉，产生规避交叉或者锥交叉，导致BOA崩溃和其他复杂的动力学行为。Vibrational Analysis in Gaussianhttp://gaussian.com/vib/The roots of the eigenvalues (Hessian)are the fundamentalfrequencies of themolecule

幻灯片 7 Key points about PES 7 5. Higher order saddle points have g = 0 and more than one negative Hessian eigenvalue; • Can always lead to lower energy first order saddle point; • These typically do not have chemical significance; 6. It is usually our job to identify the critical points (minima and transition states on a PES); • In liquids, PES is much more flat and lightly corrugated; • Statistical mechanics becomes mores important; 7. There are multiple PES’s for any atom configuration, corresponding to different electronic states; • Sometimes these states can interact, intersect, giving avoided crossings, conical intersections.Lead to more complicated dynamical behavior Vibrational Analysis in Gaussian http://gaussian.com/vib/ 5、更高阶的鞍点g=0，多于一个H的负本征值。能够走向能量更低的一阶鞍 点。它们一般没有化学意义。 6、计算化学中，通常需要确认PES上的关键点。在液体中，PES更加平，带 有轻微的褶皱，统计力学就变得更重要了。 7、不同电子态对应不同的势能面。PES之间也会作用，交叉，产生规避交 叉或者锥交叉，导致BOA崩溃和其他复杂的动力学行为。 Vibrational Analysis in Gaussian http://gaussian.com/vib/ The roots of the eigenvalues (Hessian) are the fundamental frequencies of the molecule

幻灯片8EnergygradientsIt is helpful to calculate first and second derivativesof Epes with respect tonuclear positions qi;SZZeEs(R...Rw)- Eet)1poRaFirst term is tougherEaH)( ((中）-（da[0.]Tca计算能量对于原子核坐标的一阶和二价导数很重要第二项容易，第一项较难，其一阶导数可以写为三项

幻灯片 8 Energy gradients 8 • It is helpful to calculate first and second derivatives of EPES with respect to nuclear positions qi ; • First term is tougher 计算能量对于原子核坐标的一阶和二价导数很重要 第二项容易，第一项较难，其一阶导数可以写为三项

幻灯片9Hellmann-FeynmantheoremThe Hellmann-Feynman theorem relates the derivativeofthetotalenergywithrespecttoaparameter,totheexpectation valueof thederivativeof theHamiltonianwith respect to that same parameter5一-1（会（8）（））（）会）-B（芸）+日（川芸）+（叫号（）B号（0（m）-(μ))总能量对于任何一个连续参数的导数dE/d2=，即假设不依赖于入，dy/d*y=0。此时剩余的项就是原子核受到的H-F力

幻灯片 9 Hellmann-Feynman theorem 9 • The Hellmann–Feynman theorem relates the derivative of the total energy with respect to a parameter, to the expectation value of the derivative of the Hamiltonian with respect to that same parameter 总能量对于任何一个连续参数的导数dE/dλ= ，即假设ψ不依赖于λ， dψ/dλ*ψ=0。 此时剩余的项就是原子核受到的H-F力

幻灯片10PulayforcesThemiddletermwouldbeeasy,sincetheonlypartofHthatdepends onthenuclearpositions istheone-electronelectron-nucleusattractionpiece;Ifourbasissetdepends upon the ionic positions,suchasatomic centered Gaussians,and if we have arapproximate eigenstate ,for example from usinganincompletebasisset,thenwemustkeepall3termsinthegeneral expression and the other derivatives in thegeneralexpressionwillcontributeso-calledPulayforces;NotethatPulayforceswillvanishinthelimitofacomplete basis set (i.e., the exact Hartree-FockwavefunctionortheHFlimit),butthatthis isneverrealizedinpractice,orifpositionindependentbasisfunctions, such as plane-waves, are used.P. Pulay, Mol. Phys. 19, 197 (1969根据H-F原理，只有中间项，中间项中依赖原子坐标的只有单电子算符里面的电子-原子核吸引项。但是，如果我们用的是基组是GTO，即依赖于原子的位置的函数，而且如果基组是不完全基组，那么我们必须保留所有的三项导数。“AnalyticalDerivativeMethodsinQuantumChemistryAdv.Chem.Phys:AbinitioMethodsinQuantumChemistryPart2,1987,69,241-286记得GTO里的Zeta吧，这个参数跟表示电子波函数距离原子核的衰减速度。还有里面的变量xyz都是以原子核为原点的，不同的GTO的原点（原子核位置）不同。这样造成每个原子轨道、分子轨道都依赖于原子坐标，因此当轨道对于原子坐标求导，不能再忽略。prb,2000,61,16207One requires that the basis set is such that the extra term is identically zero. Thisrequirement,known as Hurley's condition,is satisfied if the basisfunctions do notdependonq(e.g.,whentheparametersqarenuclearpositionvectors;suchsetsareknownasfloatingsets"inquantumchemistry;insolidstateapplications,thecommonlyusedplanewavesaresuchaset).Theconditionisalsosatisfied ifthederivatives ofthebasisfunctions with respecttogarethemselvespartof thebasisset.Insuchacase,theoriginal HFtheorem issatisfied.然而对于GTO，如果用的是完备基组，即H-Flimit，那么因为GTO的导数是另一个GTO，dGTO/dr=c*r*GTO。若是完备基，求导的结果还是某个GTO，多了个常数，最后这些高斯基函数的系数互相抵消（ForceinSCFtheories.chemphyslett,80,2,340），使求导的三项又回归为一项。Mathematically，thederivativesoftheGaussianfunction canberepresented usingHermitefunctions.Then-th

幻灯片 10 Pulay forces 10 • The middle term would be easy, since the only part of H that depends on the nuclear positions is the one-electron electron-nucleus attraction piece; • If our basis set depends upon the ionic positions, such as atomic centered Gaussians, and if we have an approximate eigenstate ψ, for example from using an incomplete basis set, then we must keep all 3 terms in the general expression and the other derivatives in the general expression will contribute so-called Pulay forces; • Note that Pulay forces will vanish in the limit of a complete basis set (i.e., the exact Hartree-Fock wavefunction or the HF limit), but that this is never realized in practice, or if position independent basis functions, such as plane-waves, are used. P. Pulay, Mol. Phys. 19, 197 (1969) 根据H-F原理，只有中间项，中间项中依赖原子坐标的只有单电子算符里面的电 子-原子核吸引项。 但是，如果我们用的是基组是GTO，即依赖于原子的位置的函数，而且如果基 组是不完全基组，那么我们必须保留所有的三项导数。“Analytical Derivative Methods in Quantum Chemistry” Adv. Chem. Phys: Ab initio Methods in Quantum Chemistry Part 2, 1987, 69, 241-286 记得GTO里的Zeta吧，这个参数跟表示电子波函数距离原子核的衰减速度。还有 里面的变量xyz都是以原子核为原点的，不同的GTO的原点（原子核位置）不 同。这样造成每个原子轨道、分子轨道都依赖于原子坐标，因此当轨道对于原 子坐标求导，不能再忽略。 prb, 2000, 61, 16207 One requires that the basis set is such that the extra term is identically zero. This requirement, known as Hurley’s condition, is satisfied if the basis functions do not depend on q (e.g., when the parameters q are nuclear position vectors; such sets are known as ‘‘floating sets’’ in quantum chemistry; in solid state applications, the commonly used plane waves are such a set). The condition is also satisfied if the derivatives of the basis functions with respect to q are themselves part of the basis set. In such a case, the original HF theorem is satisfied. 然而对于GTO，如果用的是完备基组，即H-F limit，那么因为GTO的导数是另一 个GTO， dGTO/dr = c*r*GTO。若是完备基，求导的结果还是某个GTO，多了个 常数，最后这些高斯基函数的系数互相抵消（Force in SCF theories. chem phys lett,80,2,340），使求导的三项又回归为一项。 Mathematically, the derivatives of the Gaussian function can be represented using Hermite functions. The n-th

derivativeoftheGaussianistheGaussianfunctionitselfmultipliedbythen-thHermitepolynomial,uptoscale.对于平面波基组，由于其不在原子上，所以与原子核坐标无关。所以只需要计算H-F力即可，不需要PulayForces。然而其中又会出现“pulaystress"问题。主要原因是平面波基组不完备，导致在对不同体积计算时的时候出现误差（slab片层模型中的真空层会影响计算结果）(http://cms.mpi.univie.ac.at/vasp/vasp/Volume_vs_energy_volume_relaxations_Pulay_Stress.html因为在vasp优化cellshape、volume的计算中基组保持不变，volume或者cellshape改变都会影响计算结果。增加energycutoff或者在保持cutoff不变时，改变volume，最后用equationofstate拟合最后的能量或者体积。其实就是对不同体积的计算改变了基组的数量。）

derivative of the Gaussian is the Gaussian function itself multiplied by the n-th Hermite polynomial, up to scale. 对于平面波基组，由于其不在原子上，所以与原子核坐标无关。所以只需要计 算H-F力即可，不需要Pulay Forces。 然而其中又会出现“pulay stress”问题。主要原因是平面波基组不完备，导致在 对不同体积计算时的时候出现误差（slab片层模型中的真空层会影响计算结 果） （ http://cms.mpi.univie.ac.at/vasp/vasp/Volume_vs_energy_volume_relaxations_P ulay_Stress.html 因为在vasp优化cell shape、 volume的计算中基组保持不变， volume或者cell shape改变都会影响计算结果。增加energy cutoff或者在保持 cutoff不变时，改变volume，最后用equation of state 拟合最后的能量或者体 积。其实就是对不同体积的计算改变了基组的数量。）