How to make the Born-Oppenheimer approximation exact: A fresh look at potential energy surfaces and Berry phases E.K.U. Gross Max-Planck Institute of Microstructure Physics Halle (Saale)
Process of vision Light-induced isomeriztion
"Triad molecule": Candidate for photovoltaic applications C.A. Rozzi et al, Nature Communications 4, 1602 (2013) S.M. Falke et al, Science 344, 1001 (2014) TDDFT propagation with clamped nuclei
"Triad molecule": Candidate for photovoltaic applications C.A. Rozzi et al, Nature Communications 4, 1602 (2013) S.M. Falke et al, Science 344, 1001 (2014) Moving nuclei
Hamiltonian for the complete system of N e electrons with coordinates r r r R R R and N n nuclei with coordinates 1 N 1 N e n ˆ ˆ ˆ ˆ ˆ ˆ H T ( R ) W ( R ) T ( r ) W ( r ) V ( R , r ) n nn e ee en N 2 N 2 N Z Z 1 n e n ˆ ˆ ˆ i with T T W n e nn 2 M 2 m 2 R R 1 i 1 , N N N 1 1 Z e e n ˆ ˆ W V ee en 2 r r r R j , k j 1 1 j k j j k Stationary Schrödinger equation ˆ H r , R E r , R
Hamiltonian for the complete system of N e electrons with coordinates r r r R R R and N n nuclei with coordinates 1 N 1 N e n ˆ ˆ ˆ ˆ ˆ ˆ H T ( R ) W ( R ) T ( r ) W ( r ) V ( R , r ) n nn e ee en N 2 N 2 N Z Z 1 n e n ˆ ˆ ˆ i with T T W n e nn 2 M 2 m 2 R R 1 i 1 , N N N 1 1 Z e e n ˆ ˆ W V ee en 2 r r r R j , k j 1 1 j k j j k Time-dependent Schrödinger equation i r , R , t H r , R V r , R , t r , R , t laser t N N e n V r , R , t r Z R E f t cos t laser j j 1 1
Born-Oppenheimer approximation solve ˆ ˆ ˆ ˆ Φ Φ ext BO BO BO T ( r ) W ( r ) V ( r ) V ( r , R ) r r R e ee e en R R for each fixed nuclear configuration R . Make adiabatic ansatz for the complete molecular wave function: Ψ χ BO BO BO r , R r R R and find best χ BO by minimizing < Ψ BO | H | Ψ BO > w.r.t. χ BO :
Born-Oppenheimer approximation solve ˆ ˆ ˆ ˆ Φ Φ ext BO BO BO T ( r ) W ( r ) V ( r ) V ( r , R ) r r R e ee e en R R for each fixed nuclear configuration R . Make adiabatic ansatz for the complete molecular wave function: Ψ χ BO BO BO r , R r R R and find best χ BO by minimizing < Ψ BO | H | Ψ BO > w.r.t. χ BO :
Nuclear equation 1 ˆ ˆ ˆ ext T ( R ) W ( R ) V ( R ) (-i ) BO BO A R R n nn n υ M ˆ Φ Φ χ E χ BO * BO BO BO r T R r d r R R R n R Berry connection Φ Φ BO BO * BO A R r (-i ) r d r υ υ R R γ BO BO is a geometric phase C A R d R C In this context, potential energy surfaces and the vector potential BO BO R A R follow from an APPROXIMATION (the BO approximation).
Nuclear equation 1 ˆ ˆ ˆ ext T ( R ) W ( R ) V ( R ) (-i ) BO BO A R R n nn n υ M ˆ Φ Φ χ E χ BO * BO BO BO r T R r d r R R R n R Berry connection Φ Φ BO BO * BO A R r (-i ) r d r υ υ R R γ BO BO is a geometric phase C A R d R C In this context, potential energy surfaces and the vector potential BO BO R A R follow from an APPROXIMATION (the BO approximation).
Geometric Phases Concept of geometric phase: Discovered by S. Pancharatnam (1956) Proc. Indian Acad. Sci. A 44 : 247 – 262. In the context of quantum mechanics: Michael V. Berry (1984) Proc. Royal Society 392 (1802), 45 – 57.
Whenever the Hamiltonian of a quantum system depends on a vector of parameters, R, the Berry phase is defined as: where the line integral is along a closed loop, C, in parameter space. A non-vanishing value of only appears when C encircles some non-analyticity.
Standard representation of the full TD wave function Expand full molecular wave function in complete set of BO states: Ψ r,R,t Φ r χ R,t BO R,J J J and insert expansion in the full Schrödinger equation → standard . Φ BO r non-adiabatic coupling terms from T n acting on R , J
Plug Born-Huang expansion in full TDSE: i R,t T R,t R R,t t k n k k k 2 BO BO i i R,t R,k R R,j R j M j NAC-1 2 BO 2 BO R,t R,k R R,j j 2M j NAC-2 The dynamics is "non-adiabatic" when the NAC terms cannot be neglected
Φ r BO 1,R BO E R 1 BO E R Φ r BO 0 0,R Ψ χ R Φ r χ R Φ B O BO r ,R r , t ,t , t 0 0 0 0,R 01 1,R When only few BO-PES are important, the BO expansion gives a perfectly clear picture of the dynamics
Example: NaI femtochemistry Na + + I - Na + I
Example: NaI femtochemistry Na + + I - Na + I emitted neutral Na atoms
Effect of tuning pump wavelength (exciting to different points on excited surface) λ pump /nm 300 311 321 Different periods indicative of anharmonic potential 339 T.S. Rose, M.J. Rosker, A. Zewail, JCP 91 , 7415 (1989)
For larger systems one would like to (one has to) treat the nuclei classically.
Trajectory-based quantum dynamics
For larger systems one would like to (one has to) treat the nuclei classically. But what ’ s the classical force when the nuclear wave packet splits??
For larger systems one would like to (one has to) treat the nuclei classically. But what ’ s the classical force when the nuclear wave packet splits??
For larger systems one would like to (one has to) treat the nuclei classically. But what’s the classical force when the nuclear wave packet splits?? There is only one correct answer!
Outline • Show that the factorisation Ψ χ r , R r R R can be made exact • Concept of exact PES and exact Berry phase • Concept of exact and unique time-dependent PES • Mixed quantum-classical treatment
THANKS Ali Abedi Axel Schild Federica Agostini Yasumitsu Suzuki Seung Kyu Min Neepa Maitra (Hunter College, CUNY) Ryan Requist Nikitas Gidopoulo (Durham University, UK)
Theorem I The exact solutions of ˆ H r , R E r , R can be written in the form Ψ r,R r χ R R 2 Φ for each fixed where d r r 1 R . R N.I. Gidopoulos, E.K.U. Gross, Phil. Trans. R. Soc. 372, 20130059 (2014)
Proof of Theorem I: Ψ r,R Given the exact electron-nuclear wavefuncion 2 iS R R : e dr r,R Choose: S R with some real-valued funcion Φ r : Ψ r,R / χ R R 2 Φ Then, by construction, d r r 1 R
Proof of Theorem I: Ψ r,R Given the exact electron-nuclear wavefuncion 2 iS R R : e dr r,R Choose: S R with some real-valued funcion Φ r : Ψ r,R / χ R R 2 Φ Then, by construction, d r r 1 R Note: If we want (R) to be smooth, S(R) may be discontinuous
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