Advanced techniques: Quantum treatment of nuclei and non-adiabatic - PowerPoint PPT Presentation

Advanced techniques: Quantum treatment of nuclei and non-adiabatic (surface hopping) approaches Mauro Boero Institut de Physique et Chimie des Matériaux de Strasbourg University of Strasbourg - CNRS, F-67034 Strasbourg, France and @Institute of Materials and Systems for Sustainability, Nagoya University - Oshiyama Group, Nagoya Japan 1

About Excited States • Photoactive molecules are also the target of potential technological applications in molecular optoelectronics, photocatalysis and photo-biochemistry • They involve electron excitations • Time-dependent DFT (TDDFT) has been proposed as a way to include electron excitation (see M. E. Casida, Recent Advances in Density Functional Methods , Vol. 1, ed. by Chong, D.P., World Scientific, Singapore, 1995) • Although TDDFT is computationally expensive… 2

About Excited States • Generally organic photoreactions involve mainly the first excited singlet state ( S 1 ) and the lowest triplet state ( T 1 ). Other excited states have a too short lifetime to be of real practical interest and can generally be neglected • see N. J. Turro, Modern Molecular Photochemistry , University Science Books, Mill Valley1991 A. Zewail* Femtochemistry: Ultrafast Dynamics of the Chemical Bond , World Scientific Series in 20 th Century Chemistry, Vol. 3, World Scientific, Singapore 1994 * 1999 Chemistry Nobel Prize 3

The “minimal” excited states • If a single valence electron is excited from the highest occupied (ground state) orbital a to the lowest unoccupied orbital b , four different determinants can be obtained according to the Pauli’s principle 4

The “minimal” excited states: Wavefunctions 5

The “minimal” excited states: Energies 6

ROKS : Excited states@KS • Instead of separate wavefunctions for the t and m states, it has been shown that it is possible to determine a single set of spin-restricted single-particle orbitals y i ( x ) for the states i = 1,…, N +1 in such a way that          m m t t ( x ) ( x ) ( x ) ( x ) ( x )     • A new DFT functional, the restricted open shell Kohn- Sham (ROKS) functional, can be written as    N 1         y       y y   ROKS KS KS 3 * H { ( x )} 2 E E d x ( x ) ( x ) i m t ij i j ij  i , j 1 I. Frank, J. Hutter, D. Marx, M. Parrinello, J. Chem. Phys . 108 , 4060 (1998) 7

Total energy functionals for excited states • The functionals with the superscript KS are Kohn-Sham total energy functionals with the difference reduced only to the exchange-correlation term         y  y         KS m m E { } E { } E E [ , ] E E   m i k i H xc eI II         y  y         KS t t E { } E { } E E [ , ] E E   t i k i H xc eI II 8

…and associated equations to solve (I) • The minimization of the functional H ROKS [ y i ( x )] with respect to the orbitals leads to two sets of Schrödinger-like equations, one for the doubly occupied orbitals…       1 ( y )      2  3    m  m   m  m d y V ( x ) v , v ,      eI xc xc 2 x - y        N 1 1 1          y   y t t t t v , v , ( x ) ( x )      xc xc i ij j  2 2  j 1   m , t  m , t   m , t  m , t E [ , ] E [ , ]        xc  xc v , v xc   xc   m , t m , t   9

…and associated equations to solve (II) • …and one for the singly occupied a and b states            1 1 ( y ) 1              y 2 3 m m t t d y V ( x ) v , v , ( x )         eI xc xc a  2 2 x - y 2       N 1    y ( x ) aj j  j 1            1 1 ( y ) 1              y 2 3 m m t t d y V ( x ) v , v , ( x )         eI xc xc a  2 2 x - y 2       N 1    y ( x ) aj j  j 1 10

…and does it actually work? 11

…well, maybe yes ! 12

Where can it be used ? • This approach has been used to study the isomerization and energy changes of the rhodopsin chromophore. • This is the photosensitive protein in the rod cells of the retina of vertebrates and the process of vision. • It, involves the photoisomerization ћ w as a response to the absorption of photons (in about 200 fs) and triggers a cascade of slower reactions that produce a specific biological signal ( C. Molteni et al . JACS 121 , 12177 (1999)). 13

Rhodopsin: Light-sensitive receptor protein responsible for vision 14

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Rhodopsin: ROKS simulated isomerization 16

Doing CPMD-like dynamics with more than one PES • If the ground state S 0 and the ROKS excited S 1 surface are two accessible states (e.g. photochemistry) it is possible to adopt a Tully scheme (J. C. Tully, J. Chem. Phys . 93 , 1061 (1990); ibid . 55 , 562 (1971)) • The electronic wavefunction for the whole system reads excit N    i E dt     a e j j j  j 0 Energy expectation Adiabatic state on S j value on S j 17

Doing CPMD-like dynamics with more than one PES S 1 (adiabatic) surfaces between which we want to hop S 0 F 1 F 0 ..and a j ? ˆ    E H j j j 18

Doing CPMD-like dynamics with more than one PES • a j are determined by the solution of the time-dependent Schrödinger equation  ˆ    H i  t • For the closed shell KS ground state S 0    0  * 0  0  * 0  0 1 1 n n • And for the excited ROKS S 1   1            1 * 1 1 * 1 1 * 1 * 1 1     1 1 1 n n 1 1 1 n n 1 2 • n = half the (even) number of electrons 19

Doing CPMD-like dynamics with more than one PES • These  0 and  1 are normalized on S 0 and S 1 , respectively but they are not orthogonal to each other • It is possible to define the quantities         S S S S S 1 ij i j ij 01 10 ii            D D D D 0 ij i j i j ij ji ii  t Non-adiabatic coupling matrix → easy to do: wfs velocities are directly available in CPMD 20

Doing CPMD-like dynamics with more than one PES   1    ˆ     a exp i E dt      * * H i • Solving for gives j j j  t  j 0   1 p p       a ia 1 S E E a D 1 a D S    0 1 0 1 1 01 0 10  2 S 1 p p   0 0   1 p        2 a a D 0 a D S ia S E E   1 0 10 1 01 1 0 1  2 S 1 p   1 Doltsinis & Marx, Phys. Rev. Lett . 88 , 166402 (2002) 21

Doing CPMD-like dynamics with more than one PES  i E dt  • Note that  p e j j ˆ       E H H H H E S j j j jj 01 10 0 • If the wavefunctions were eigenfunctions of the KS Hamiltonian, then | a 0 | 2 and | a 1 | 2 would be occupation numbers • …but they are not. So what ? Doltsinis & Marx, Phys. Rev. Lett . 88 , 166402 (2002) 22

Doing CPMD-like dynamics with more than one PES • Expand on an orthonormal auxiliary set of wfs  ’ j           d d b b b a p 0 0 1 1 0 0 1 1 j j j  d  2 2 d 1 true state population 0 1      c c j 0 j 0 1 j 1     c c Eigenvectors of      00  01 c c     H c i = E i S c i 0 1 c c     10 11 23

Doing CPMD-like dynamics with more than one PES • Hence, we get  2 E S E     E E E 1 0 > E 1 if E 0 < E 1 0 0 1  2 1 S     1 S       c c     0 1 0 1     • and the orthonormal auxiliary wavefunctions and occupations     S      0 1 0 0 1  2 1 S     2 2 2    2 * d b S b 2 S Re b b 2 2   2 d 1 S b 0 0 1 0 1 1 1 24