Report

Lecture 23 Born-Oppenheimer approximation (c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has been developed and made available online by work supported jointly by University of Illinois, the National Science Foundation under Grant CHE-1118616 (CAREER), and the Camille & Henry Dreyfus Foundation, Inc. through the Camille Dreyfus Teacher-Scholar program. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the sponsoring agencies. The Born-Oppenheimer approximation For polyatomic molecules, we use approximate separation of variables between the nuclear and electronic variables. We first solve the electronic Schrödinger equation, in which nuclei are held fixed. Next, we solve the nuclear Schrödinger equation. The BO approximation introduces important chemical concepts such as potential energy surfaces, equilibrium geometries, binding energies, etc. The molecular Hamiltonian The molecular Hamiltonian is … n N 2 2 N n Z Z e Z e Hˆ = -å Ñ2e - å Ñ 2N + å + å I J - åå I i I i=1 2me I =1 2mN i< j 4pe 0 rij I <J 4pe 0 rIJ I i 4pe 0 rIi 2 i Kinetic (electron) n 2 e2 N I Kinetic (nuclei) Coulomb (e – e) Coulomb (n – n) Coulomb (e – n) There is no hope that we can solve the Schrödinger equation with this Hamiltonian exactly. Can we separate variables approximately? Separation of the Hamiltonian He(r) n N 2 2 N n Z Z e Z e Hˆ = -å Ñ2e - å Ñ 2N + å + å I J - åå I i I i=1 2me I =1 2mN i< j 4pe 0 rij I <J 4pe 0 rIJ I i 4pe 0 rIi 2 i Kinetic (electron) n 2 e2 N I Kinetic (nuclei) Coulomb (e – e) Hn(R) Coulomb (n – n) Coulomb (e – n) Separation of the wave function The first step of (approximate) separation of variables: assumption of product form of wave function Y(r,R) = Y e (r)Y n (R) Electronic coordinates Electronic wave function Nuclear coordinates Nuclear wave function Separation of the wave function The second step: substitution into the Schrödinger equation. resists separation éë H e (r) + H n (R) ùû Y e (r)Y n (R) = H e ( Y eY n ) + Y e ( H nY n ) = EY e (r)Y n (R) n n N n 2 Z e Hˆ e = - å Ñ 2e + å - åå I i i=1 2me i< j 4pe 0 rij I i 4pe 0 rIi 2 e2 i N Hˆ n = - å 2 I =1 2mN Ñ 2N I N I Z I Z J e2 +å I <J 4pe 0 rIJ involves also nuclear coordinates Separation of the wave function … we must demote some variables to parameters. parameters R are held fixed éë H e (r;R) + H n (R) ùû Y e (r;R)Y n (R) = Y n ( H eY e ) + Y e ( H nY n ) = EY e (r;R)Y n (R) n n N n 2 Z e Hˆ e = - å Ñ 2e + å - åå I i i=1 2me i< j 4pe 0 rij I i 4pe 0 rIi 2 e2 i N Hˆ n = - å 2 I =1 2mN Ñ 2N I N I Z I Z J e2 +å I <J 4pe 0 rIJ nuclear coords. are considered parameters Parameters versus variables Parameters are arguments of a function with which no differentiation or integration is performed. For example, the electron mass is a parameter. Parameters are held fixed. Variables are arguments of a function with which differentiation or integration is performed. For example, electron coordinates in the hydrogenic Schrödinger equation are variables. Variables do vary. Separation of the wave function The third step: divide the whole equation by wave function. 1 1 H eY e ) + H nY n ) = E ( ( Ye Yn ( ) Ee R Electronic structure Nuclear dynamics Separation achieved En Hˆ e ( r;R ) Y e ( r;R ) = Ee ( R ) Y e ( r;R ) { } Hˆ n ( R ) Y n ( R ) = E - Ee ( R ) Y n ( R ) The Born-Oppenheimer approximation The BO approximation breaks the original problem into two smaller problems that must be solved in sequence: Electronic structure (nuclei held fixed) 2 ü 2 n N n ìï n 2 Z e e ï 2 I Ñe + å - åå í- å ý Y e ( r;R ) = Ee ( R ) Y e ( r;R ) i i< j 4pe 0 rij I i 4pe 0 rIi ï ïî i=1 2mei þ Nuclear dynamics 2 2 N ìï N üï ZI ZJ e 2 ÑN + å + Ee ( R ) ý Y n ( R ) = EY n ( R ) í- å I 2m I =1 I <J 4pe 0 rIJ NI ïî ïþ Potential energy surface (PES) The Born-Oppenheimer approximation In the electronic Schrödinger equation, nuclear coordinates are parameters and their kinetic energy operator does not act on them; the nuclei are held fixed. This is justified by that a nucleus is 1800+ times heavier than an electron and sluggish. The electronic Schrödinger equation must be solved for various nuclear positions, forming a part of the potential energy surface. The PES is the effective potential the nuclei feel and a part of the Hamiltonian for the nuclear Schrödinger equation. Chemical concepts from BO Electronic structure and nuclear dynamics (molecular vibration and rotation). Translational motion is separable exactly. Potential energy (hyper)surfaces and curves – effective potentials that nuclei feel. One PES for each electronic state. Equilibrium structure, binding energy, vibrational energy levels, rotational energy levels, Franck-Condon factors (see later lectures), nonadiabatic transition. Chemical concepts from BO Dynamical degrees of freedom n electrons + N nuclei: 3(n+N) dynamical degrees of freedom. Electronic structure: 3n dynamical DOF. Nuclear dynamics: 3N dynamical DOF. Translational DOF: 3. Rotational DOF: 3 (nonlinear) or 2 (linear). Vibrational DOF: 3N−6 (nonlinear) or 3N−5 (linear). Dynamical degrees of freedom all 3n+3N Born-Oppenheimer approximation electronic 3n nuclear 3N Exact separation translational 3 relative 3N−3 Rigid rotor approximation rotational 3 or 2 vibrational 3N−6 or 3N−5 Summary The BO is the approximate separation of variables. It is one of the most accurate approximations in chemistry. This leads to solving (1) electronic structure with clamped nuclei and then (2) nuclear J. Robert Oppenheimer dynamics with the PES from (1). Public-domain image This is justified by nuclear mass >> electron mass. It is the basis of many chemistry concepts.