L2 Born‐Oppenheimer Approximation and Beyond Mario Barbatti A*Midex Chair Professor mario.barbatti@univ‐amu.fr Aix Marseille Université, Institut de Chimie Radicalaire LIGHT AND MOLECULES Adiabatic x diabatic x nonadiabatic 2 LIGHT AND MOLECULES From Greek diabatos: to be crossed or passed diabatic = with crossing a-diabatic = without crossing non-a-diabatic = with crossing!? 3 LIGHT AND MOLECULES without exchanging (cross) heat or energy with environment 4 LIGHT AND MOLECULES “A physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum.” Adiabatic theorem (Born and Fock, 1928). E k x In this example (adiabatic process), the spring constant k of a harmonic oscillator is slowly (adiabatically) changed. The system remains in the ground state, which is adjusted also smoothly to the new potential shape. Its state is always an eingenstate of the Hamiltonian at each time (“no crossing”). 5 LIGHT AND MOLECULES “A physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum.” Adiabatic theorem (Born and Fock, 1928). E k x In this example (diabatic process), the spring constant k of a harmonic oscillator is suddenly (diabatically) changed. The system remains in the original state, which is not a eingenstateof the new Hamiltonian. It is a superposition (“crossing”) of several eingenstates of the new Hamiltonian. 6 LIGHT AND MOLECULES “The nuclear vibration in a molecule is a slowly acting perturbation to the electronic Hamiltonian. Therefore, the electronic system remains in its instantaneous eigenstate if there is a gap between the eigenvalue and the rest of the Hamiltonian's spectrum.” This is another way to say that: The electrons see the nuclei instantaneously frozen 7 LIGHT AND MOLECULES adiabatic S2 S2 n* * diabatic Energy S1 n* S1 * Reaction coordinate 8 LIGHT AND MOLECULES Beyond Born-Oppenheimer I: Time-independent formulation 9 LIGHT AND MOLECULES H U 0 T – Kinetic energy nuclei H T H N N e He – potential energy terms (adiabatic basis) i which solves: HEeii 0 depends on the electronic coordinates r and i i r;R parametrically on the nuclear coordinates R. i k ik Since i is a complete basis, any function in the Hilbert space can be exactly written as a linear combination of i. 10 Ns LIGHT AND MOLECULES rR,; kk R rR k nuclear wave function k1 H TN H e H U 0 Multiply by i at left and integrate in the electronic coordinates Ns Ei U i i Tn k k 0 k 1 2 Nat 2 2 I 2 2 2 2 TN M AB AB 2AB A B 2 I 1 M I 2 Non-adiabatic coupling terms Prove it! N s 2 U TN Ei i M k i M k k i TN k 0 k 1 11 N s 2 U TN Ei i M k i M k k i TN k 0 k 1 If non-adiabatic coupling terms = 0 UTNii E 0 Nuclear vibrational problem. If Ei is expanded to the second order around the equilibrium position: 3Nat 3Nat 1 Ei 1/ 2 Ei Ei Req qk ql qk M k xk xk ,eq 2 k 1 l 1 qk ql eq it can be treated by normal mode analysis. 12 LIGHT AND MOLECULES Beyond Born-Oppenheimer II: Time-dependent formulation 13 LIGHT AND MOLECULES i H(r,R,t) 0 t T – Kinetic energy nuclei H T H N N e He – potential energy terms (adiabatic basis) i which solves: H e E i 0 depends on the electronic coordinates r and i i r;R parametrically on the nuclear coordinates R. Since i is a complete basis, any function in the Hilbert space can be exactly written as a linear combination of i. 14 Ns nuclear wave function r,R k R k r;R k k k1 H TN H e i H(r,R,t) 0 t Multiply by i at left and integrate in the electronic coordinates Prove it! N s i T E i T 0 N i i i k i N k k t k 1 t Time dependent Schrödinger equation for the nuclei 15 Nonadiabatic coupling terms N s i T E i T 0 N i i i k i N k k t k 1 t First suppose the couplings are null (adiabatic approximation): i TN Ei i 0 t Independent equations for each surface. 0(t) E0 0(t0) 0(t) time 16 LIGHT AND MOLECULES Classical limit of nuclear motion 17 LIGHT AND MOLECULES i i T E 0 Adiabatic approximation t N i i i (R,t) A(R,t)exp S(R,t) Write nuclear wave function in i polar form t S(R,t) Ldt' The phase (action) is the integral 0 of the Lagrangian 2 2 S S A Ei t I 2M I I 2M I A 2 S S Classical limit 0 Ei 0 t I 2M I Tully, Faraday Discuss. 110, 407 (1998) 18 Hamilton-Jacobi Equation S S 2 Ei 0 t I 2M I To solve the Hamilton-Jacobi equation for the action is totally equivalente to solve the Newton`s equations for the coordinates! Newton equation d 2R E M I i I dt In the classical limit, the solutions of the time dependent Schrödinger equation for the nuclei in the adiabatic approximation are equivalent to the solutions of the Newton`s equations. In which cases does this classical limit lose validity? 19 In which cases does this classical limit lose validity? LIGHT AND MOLECULES 1 adiabatic quantum terms ≠ 0 2 2 S S A Ei t I 2M I I 2M I A 2 nonadiabatic coupling terms ≠ 0 N s i T E i T 0 N i i i k i N k k t k 1 t 20 Non-adiabatic coupling terms N s i T E i T 0 N i i i k i N k k t k 1 t E1 1(t) ) 0 (a 2 x E0 0(t) x1 (a0) 21 LIGHT AND MOLECULES Ns rR,; kk R rR Born-Huang Model k1 rR,;ii R rR Adiabatic approximation HEeii 0 Born-Oppenheimer Approximation UTNii E 0 N s 2 U TN Ei i M k i M k k i TN k 0 k 1 Nonadiabatic coupling terms 22.