zH,molecule112Molecules1Ra- e.g. H2Y(r,r,R.,R,)-energyEandwavefunctionincludingallfourparticlesMolecular Schrodinger equation-22++2RV2RaβM2m=EY(r,..,In,R,..,Rn)
H2 molecule • Molecules – e.g. H2 – energy E and wavefunction including all four particles • Molecular Schrödinger equation 7
Hamiltonianforamoleculee?Z-h2-h2electronsnucleielectronsnucleielectronsnuclei77H=BZ2ZZN72me2marisYABrjAii>jA>Bkinetic energy of the electronskinetic energy of the nucleielectrostatic interaction between the electrons andthe nucleielectrostatic interaction between the electronselectrostatic interaction between the nuclei8
Hamiltonian for a molecule • kinetic energy of the electrons • kinetic energy of the nuclei • electrostatic interaction between the electrons and the nuclei • electrostatic interaction between the electrons • electrostatic interaction between the nuclei nuclei A B AB A B electrons i j ij nuclei A iA A electrons i A nuclei A A i electrons i e r e Z Z r e r e Z m m 2 2 2 2 2 2 2 2 2 ˆ H 8
Born-OppenheimerapproximationM>> melectronnuclei The nuclei move more slowly than the electrons;.Thenucleiinstantaneouslywillappearimmobile-Theelectronandnuclearmotionscanthusbeapproximately decoupled;》separation of variablesY(ri. ,,R,..R) Ye(ri. ,,R..R.)u(R..,R.)O
Born-Oppenheimer approximation • Mnuclei >> melectron – The nuclei move more slowly than the electrons; • The nuclei instantaneously will appear immobile; – The electron and nuclear motions can thus be approximately decoupled; »separation of variables 9
TheelectronicSchrodingerequationDepends only parametrically on the locations of thenuclei[2+2岁]Yele(r.. ,; R,., R) - Eee Ylee (ri., ,; R,..Rn)Z.e>27Orbital approximation, again!Y(.r)中(r),(r)Applying variational principle,again![i, + Ccouomb[p] + Uexchane[ + Uonehtio [ ] ; fiw; = );Ucoml[pl = JPCdr, p(r)=Z;(r)1210
The electronic Schrödinger equation • Depends only parametrically on the locations of the nuclei 10 • Orbital approximation, again! • Applying variational principle, again!
Potentialenergysurface(PEs). Defines a potential energy surface (adiabatic surface)+2Z.ZnEpes(R,..,Rv) Ele + Eachpoint onthePES isasolutiontoelectronicSchrodingerequationNucleican bethought to“"travel"onAthis PES(R)·3N degree offreedomWaystodescribenucleimotion:1.quantum mechanically, e.g.to capture"tunneling." Expensive andspecialized.2. classical particles rolling along the PEs. Essence of classical ab initiomoleculardynamics.3.focus on locating“critical points" along PES, like stable minima("molecules")and saddle points ("transition states"). Least expensive and11most common
Potential energy surface (PES) • Defines a potential energy surface (adiabatic surface) 11 Each point on the PES is a solution to electronic Schrodinger equation Ways to describe nuclei motion: 1. quantum mechanically, e.g. to capture “tunneling.” Expensive and specialized. 2. classical particles rolling along the PES. Essence of classical ab initio molecular dynamics. 3. focus on locating “critical points” along PES, like stable minima (“molecules”) and saddle points (“transition states”). Least expensive and most common. • Nuclei can be thought to “travel” on this PES • 3N degree of freedom