Femtochemistry: A theoretical overview Mario Barbatti III – Adiabatic approximation and non-adiabatic corrections This lecture can be downloaded at lecture3.ppt

2 Diabatic x adiabatic From Greek diabatos: to be crossed or passed, fordable diabatic = with crossing a-diabatic = without crossing non-a-diabatic= with crossing!?

3 Diabatic x adiabatic In thermodynamics without exchanging (cross) heat or energy with environment

4 Diabatic x adiabatic In quantum mechanics “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 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”). k

5 Diabatic x adiabatic In quantum mechanics “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). 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 eingenstate of the new Hamiltonian. It is a superposition (“crossing”) of several eingenstates of the new Hamiltonian. E x k

6 Diabatic x adiabatic In quantum chemistry “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 Beyond adiabatic approximation I: Time independent Time-independent formulation Since is a complete basis, any function in the Hilbert space can be exactly written as a linear combination of  i. T N – Kinetic energy nuclei H e – potential energy terms depends on the electronic coordinates r and parametrically on the nuclear coordinates R. which solves: (adiabatic basis)

8 nuclear wave function Multiply by  i at left and integrate in the electronic coordinates Non-adiabatic coupling terms Prove it!

9 Non-adiabatic coupling terms If non-adiabatic coupling terms = 0 Nuclear vibrational problem. If E i is expanded to the second order around the equilibrium position: it can be treated by normal mode analysis.

10 Beyond adiabatic approximation II: Time dependent (adiabatic representation) Time-dependent formulation T N – Kinetic energy nuclei H e – potential energy terms which solves: depends on the electronic coordinates r and parametrically on the nuclear coordinates R. Since is a complete basis, any function in the Hilbert space can be exactly written as a linear combination of  i. (adiabatic basis)

11 nuclear wave function Prove it! Multiply by  i at left and integrate in the electronic coordinates Time dependent Schrödinger equation for the nuclei

12 Non-adiabatic coupling terms First suppose the couplings are null (adiabatic approximation): Independent equations for each surface. E0E0 time 0(t)0(t) 0(t0)0(t0) 0(t)0(t)

13 Classical limit of the nuclear motion Write nuclear wave function in polar form The phase (action) is the integral of the Lagrangian Adiabatic approximation Classical limit Tully, Faraday Discuss. 110, 407 (1998)

14 Hamilton-Jacobi Equation To solve the Hamilton-Jacobi equation for the action is totally equivalente to solve the Newton`s equations for the coordinates! Newton‘s equations 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?

15 adiabatic quantum terms ≠ 01 non-adiabatic coupling terms ≠ 02 In which cases does this classical limit lose validity?

16 Non-adiabatic coupling terms x 1 (a 0 ) x 2 (a 0 ) E0E0 E1E1 1(t)1(t) 0(t)0(t)

17 Beyond adiabatic approximation III: Time dependent (general representation) which solves: (adiabatic basis) Before: We got: general orthonormal complete basis Now: We get: another non-adiabatic coupling term

18 A very important result: Nuclear kinetic energy operator This equation is the basis for the mixed quantum-classical methods such as Surface Hopping and Mean Field (Ehrenfest) dynamics.

19 For example, Surface Hopping For a fixed nuclear geometry, solve time- independent Schrödinger Eq. for electrons. Get the energy gradient.1 Use the energy gradient to update the nuclear geometry according to the Newton`s Eq. 2 For the new nuclear geometry (only!), solve the TDSE and correct classical solution by performing a hopping if necessary.3 Go back to step 1 and repeat the procedure until the end of the trajectory.4 Repeat procedure for a large number of trajectories to have the nuclear wave packet information.5

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21 I. Time-independent II. Time-dependent (adiabatic basis) III. Time-dependent (general basis) IV. Time-dependent (diabatic basis)

22 Coupling terms This term is often neglected in local approximations adiabatic This term is simplified to the Born-Oppenheimer energy if the electronic basis is adiabatic. Otherwise, it can be important. Using the chain rule, this term can be written as: v is the nuclear velocity. diabaticadiabatic The non-adiabatic coupling vector is by definition equal zero in an electronic diabatic basis. In adiabatic basis it is important close to degeneracies. It diverges at conical intersections.

23 non-adiabatic coupling terms In both cases, the derivatives are in nuclear coordinates.

24 Present situation of quantum chemistry methods Methods allowing for excited-state calculations:

25 They define the limit of validity of the adiabatic approximation and of the breakup of the Hilbert space into uncoupled subspaces, which is important for reducing the dimensionality of the problem to be treated. The coupling vectors allow to connect the Hilbert space from one set of nuclear coordinates R to another R +  R nearby, which is important for the time-dependent formulation of the problem. The coupling vectors define one of the directions of the branching space around the conical intersections, which is important for the localization of these points of degeneracy. Why are non-adiabatic coupling vectors important?

26 They define the limit of validity of the adiabatic approximation and of the breakup of the Hilbert space into uncoupled subspaces, which is important for reducing the dimensionality of the problem to be treated. Why are non-adiabatic coupling vectors important? E If h 2k ~ 0 (k ≠ 2), then state 2 can be treated alone (adiabatically). E If h ik ≠ 0 for i = 1, 2, k = 1, 2 and h ik ~ 0 for i = 1, 2 and k > 2, then states 1,2 for a uncoupled subspace and can be isolated treated.

27 The coupling vectors allow to connect the Hilbert space from one set of nuclear coordinates R to another R +  R nearby, which is important for the time-dependent formulation of the problem. Why are non-adiabatic coupling vectors important? R t Prove it! Hint: Expand the electronic wave function till the firts order and use the fact that the first derivative is a function of R and, therefore can be expanded in terms of R R+  R t t+  t

28 The coupling vectors define one of the directions of the branching space around the conical intersections, which is important for the localization of these points of degeneracy. Why are non-adiabatic coupling vectors important?

29 Relation between time derivative and spatial derivative couplings = 0 Computation of the coupling can be reduced to the computation of overlaps! Hammes-Schiffer and Tully, J. Chem. Phys. 101, 4657 (1994)

30 Pittner, Lischka, and Barbatti, Chem. Phys. 356, 147 (2009)

31 Implemented for MRCI, MCSCF, and TD-DFT Computational saving may be one-order higher For TD-DFT it is restricted to excited state crossings It can (in principle) be implemented for any method

32 Next lecture Non-crossing rule Conical intersections Contact This lecture can be downloaded at lecture3.ppt