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