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

LasttimePotentialenergy surfaces1. Potential energy curves/surfaces2. Born-Oppenheimer Approximation3. Properties from energy derivatives4. Exploring PESs1. Rigid scan2. Relaxed scan

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

Contents1. Geometry optimization2.Vibrationalnormalmode3. Frequency calculation4. IR and Raman5. Thermochemistry3

Contents 3 1. Geometry optimization 2. Vibrational normal mode 3. Frequency calculation 4. IR and Raman 5. Thermochemistry

PotentialenergysurfaceSecondOrderSaddlePointTransitionStructureATransitionStructureBMinimumforProductAMinimumforProductB0-0.5Second Order。0.5SaddlePoint0Valley-Ridge0.5-0.5InflectionPointMinimumforReactant-1Abstract axes, hypersurface (3N-6 dimensions), multiplestates (So, Si...), multiple pathways linking minima4

Potential energy surface 4 Abstract axes, hypersurface (3N-6 dimensions), multiple states (S0 , S1.), multiple pathways linking minima

B-Oapproximation&energyderivativesOEPESSecondOrderSaddlePointTransitionStructureAodTransitionStructureBGradient (forces)OEPESMinimumforF=-gg(R...RN)=od2ProductA:MinimumforProductBOEPES05aqnSecondOrderg0.5SaddlePointValley-Ridge0.5-0.5Inflection.PointMinimumforReactantOEPESOEPESO'EPESHessianOq?og,og2OgoanOEPESEPESO'EPESBorn-Oppenheimerapproximation:decoupleH=aq?ogonthemotionofelectronsfromnucleicog.:.total=electronicXnuclearOEPESO"EPESO"EPESNNEepes(R... Ry)=Eee+22odhgoannoRapα-1β-α+15

B-O approximation & energy derivatives 5 Gradient (forces) Hessian F = −g Born–Oppenheimer approximation: decouple the motion of electrons from nuclei

KeypointsaboutPES1. Hessian matrix is real and symmetric, can be diagonalized to giveeignevalues and eigenvectors;Eigenvectorsgive"natural"directions alongPES;Theharmonicvibrational/normalmodes;:Eigenvalues indicate curvature in that direction;2. Minimum on multidimensional PES has gradient vector g =O and allpositive Hessian eigenvalues;3. First order saddle point, or transition state, has g = 0 and one andonlyone negative Hessianeigenvalue;Physically, one unique direction that leads downhill in energy;:Always corresponds to the lowest energy point connecting twominima;4. Minimum energy pathway (MEP) or intrinsic reaction coordinate (IRC)is steepestdescentpathway (inmass-weightedcoordinates)fromsaddle point to nearby minima..Pathamarblewithinfiniteinertia wouldfollow.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; • Eigenvalues indicate 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

KeypointsaboutPES5. 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;6. It is usually our job to identify the critical points (minima andtransition states on a PES);In liguids, PES is much moreflat 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, givingavoided crossings, conical intersections....Lead to morecomplicateddynamicalbehaviorVibrational Analysis inGaussianhttp://gaussian.com/vib/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/

EnergygradientsIt is helpful to calculate first and second derivativesof Epes with respect to nuclear positions q;;N22Z.Z.Epes(R,.,R) = Eelec +)Ropα-1B=α+1FirsttermistougheraH0EaleaaaHIFYY12HFHFHFHHFHFodiodiodioqiag8

Energy gradients 8 • It is helpful to calculate first and second derivatives of EPES with respect to nuclear positions qi ; • First term is tougher

Hellmann-FeynmantheoremThe Hellmann-Feynman theorem relates the derivativeof the total energy with respect to a parameter, to theexpectation value of the derivative of the Hamiltonianwith respect tothat same parameterddEx(H/山)dxdadbxFd)ddxdbdbdHEEaddddHFadHda9

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

PulayforcesThe middle term would be easy, since the only part of Hthat depends on the nuclear positions is the one-electronelectron-nucleus attraction piece;If our basis set depends upon the ionic positions, such asatomic centered Gaussians, and if we have anapproximate eigenstate , for example from using anincomplete basis set, thenwe mustkeepall3terms inthegeneral expression and the other derivatives in thegeneral expression will contribute so-called Pulay forces;Note that Pulayforces will vanish in the limit of acomplete basis set (i.e., the exact Hartree-FockwavefunctionortheHFlimit),butthatthisisneverrealized in practice, or if position independent basisfunctions, such as plane-waves, are used.10P.Pulay,Mol.Phys.19,197 (1969)

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)

Second-derivatives of energyCan be evaluated analytically, through solution of"Coupled Perturbed Hartree-Fock"equations;More common for correlated methods and DFT touse finite differences;Select the point of interest (too expensive tocompute over the entire PES)A minimum, i.e. that g = 0;Assume its Epes is locally harmonicCalculate the first-derivative at some smalldisplacements h awayfromthepoint of interestJansen,chp11.6.1CoupledPerturbedHartree-Fock11

Second-derivatives of energy 11 • Can be evaluated analytically, through solution of “Coupled Perturbed Hartree-Fock” equations; • More common for correlated methods and DFT to use finite differences; • Select the point of interest (too expensive to compute over the entire PES) • A minimum, i.e. that g = 0; • Assume its EPES is locally harmonic; • Calculate the first-derivative at some small displacements h away from the point of interest Jansen, chp 11.6.1 Coupled Perturbed Hartree–Fock