Quantum Chemical and Dynamical Tools for Solving Photochemical Problems

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1 H H H H C 7 H H H H H H H 12 H Stoichiometry C6H6 Framework group D6H[3C2'(HC.CH)] Deg. of freedom 2 Full point group D6H NOp 24 Largest Abelian subgroup D2H NOp 8 Largest concise Abelian subgroup D2 NOp 4 Standard orientation: Center Atomic Atomic Coordinates (Angstroms) Number Number Type X Y Z Rotational constants (GHZ): Isotopes: C-12,C-12,C-12,C12,C-12,C-12,H-1,H-1,H-1,H-1,H-1,H-1 Leave Link 202 at Tue Oct 25 16:23: , MaxMem= cpu: 0.1 (Enter /r1/g98/l301.exe) Standard basis: 6-311G(d,p) (5D, 7F) There are 31 symmetry adapted basis functions of AG symmetry. There are 23 symmetry adapted basis functions of B1G symmetry. There are 7 symmetry adapted basis functions of B2G symmetry. There are 11 symmetry adapted basis functions of B3G symmetry. There are 7 symmetry adapted basis functions of AU symmetry. There are 11 symmetry adapted basis functions of B1U symmetry. There are 31 symmetry adapted basis functions of B2U symmetry. There are 23 symmetry adapted basis functions of B3U symmetry. Crude estimate of integral set expansion from redundant integrals= Integral buffers will be words long. Raffenetti 2 integral format. Two-electron integral symmetry is turned on. 144 basis functions 240 primitive gaussians 21 alpha electrons 21 beta electrons nuclear repulsion energy Hartrees. Leave Link 301 at Tue Oct 25 16:23: , MaxMem= cpu: 0.2 (Enter /r1/g98/l302.exe) 18 June One-electron integrals computed using PRISM. One-electron integral symmetry used in STVInt NBasis= 144 RedAO= T NBF= NBsUse= D-04 NBFU= Leave Link 302 at Tue Oct 25 16:23: , MaxMem= cpu: 8.1 (Enter /r1/g98/l303.exe) DipDrv: MaxL=1. Leave Link 303 at Tue Oct 25 16:23: , MaxMem= cpu: 3.8 (Enter /r1/g98/l401.exe) Projected INDO Guess. Initial guess orbital symmetries: Occupied (?A) (?A) (?A) (?A) (?A) (?A) (A1G) (E1U) (E1U) (E2G) (E2G) (A1G) (A2U) (B2U) (B1U) (E1U) (E1U) (E2G) (E2G) (E1G) (E1G) Virtual (E2U) (E2U) (B1U) (A1G) (E2G) (?B) (?B) (E2G) (E1U) (E1U) (B2G) (E2G) (E2G) (E1U) (E1U) (B1U) (A2G) (?B) Quantum Chemical and Dynamical Tools for Solving Photochemical Problems HΨ =E Ψ Workshop Inés Corral Institut für Chemie und Biochemie FREIE UNIVERSITÄT BERLIN 2007

2 Quantum Chemical and Dynamical tools for solving photochemical problems OUTLINE 1. Quantum Chemistry 2 2. Quantum Dynamics 18 June Quantum Control

3 Quantum Chemical and Dynamical tools for solving photochemical problems DNA isomerization Proton transfer Molecular rotors Singlet Oxygen Generators 18 June 20 07

4 Quantum Chemical and Dynamical tools for solving photochemical problems INFORMATION Quantum Chemistry π* OO σ * CO Vertical Excited Spectrum Structure of Stationary Points Conical Intersections Topology of the PES S 1 CI S 0 Minimum ΔE CI 18 June Minimum

5 Quantum Chemical and Dynamical tools for solving photochemical problems INFORMATION Quantum Dynamics Time-scale of the elementary processes Evolution of the molecular system in time Branching ratio and product distribution Absorption spectra 18 June 20 07

6 Quantum Chemical and Dynamical tools for solving photochemical problems Quantum Control INFORMATION Designing laser strategies for maximizing the production of 1 O 2 18 June 20 07

7 Quantum Chemical and Dynamical tools for solving photochemical problems 1. Quantum Chemistry H el Ψ el = E el Ψ el 18 June 20 07

8 H el Ψ el = E el Ψ el FullCI B3LYP MP2 CCSDT HF CASSCF Method

9 Basis set Hel Ψ el = E el Ψ el 5Z TZ 6-31G* cc-p-aug-cvdz DZ STO3-G

10 Hartree-Fock approximation H O H Electronic configuration: 10 e - (1s O ) 2 (σ OH ) 2 (σ OH ) 2 (η O ) 2 (η O ) 2 (σ * OH )0 (σ * OH ) 0 χ 2K χ r χ N+1 χ N=10 χ b χ a χ 2 χ 1 Equivalent to the MO approximation ψ i ( r ) α( ω) χ = ψ i ( x) i i k ψ i ( r ) β ( ω) k k χ = i 2 { } 2 { } 1 i= 1 k χ SPIN s ORBITALS SPATIAL ORBITALS = χ ( x ) χ ( x ) χ ( x ) χ i 1 j 1 k 1 1 χi( x2) χ j( x2) χk( x2) ψ ( χ1 χn ) = = N! SLATER DETERMINANT χ ( x ) χ ( x ) χ ( x ) i N j N k N 1 χ 2 χ χ χ i j N

11 In practice: Molecular orbital model the unknown spatial orbitals (MO) are described as a linear combination of a set of known basis functions ψ i = N μ = 1 c μiφ μ φμ one-electron basis function If the basis function contains the atomic orbitals MO=LCAO Next problem: How do we determine the unknown coefficients c μi? Variational principle= an approximate wavefunction has an energy which is above or equal to the exact energy, and the unknown coefficients are determined so that the total energy calculated from the many-electron wavefunction is minimized.

12 Basis sets 1 2 Slater-type orbitals φ φ e ξ r STO Gaussian-type orbitals e ξ r GTO 2 Closest to H atom solutions Less accurate than STO Fast integral calculation!! 3 SOLUTION: Linear combination of Gaussians L φ = μφ CGTO d p p= 1 contraction GTO p primitives L=length of contraction

13 Number of basis sets used Minimal basis set: using the minimum number of basis functions 1s 1s,1s 1s,1s,1s 1s,2s, 2p x,y,z 1s,2s, 2p x,y,z 1s, 2s, 2p x,y,z 1s,2s, 2p x,y,z 1s, 2s, 2p x,y,z 1s, 2s, 2p x,y,z DZ: doubling all basis functions TZ: contains 3 times as many basis functions as the minimal basis set

14 Any Extras.? Polarization functions: higher angular momentum functions*or(d,p) p orbitals are used for the polarization of s orbitals d orbitals are used for the polarization of p orbitals. H N C Diffuse functions: small exponents to describe weakly bound electrons+ Examples: anions, Hydrogen bonds

15 Type of basis sets Pople basis sets Split-valence: 3-21G, 6-31G, 6-311G,... C: 1s 2 2s 2 p 2 2s 2G 2s 1G (6s3p) [3s2p] 3G 2p 2G 2p 1G Dunning basis sets, ANO basis sets, Core correlated

16 Electronic correlation TWO APPROXIMATIONS: Our wavefunction consists on one single slater determinant Use of limited basis set in the orbital expansion MAIN DEFICIENCY: lack of correlation between motion of the electrons E corr = E exact -E HF-limit HF Improvement of correlation Full-CI SOLUTION: more than a single configuration!! Improvement basis set HF-limit Exact

17 Perturbation theory H=H 0 +λv can be solved exactly E=E 0 +λe 1 +λ 2 E 2 +λ 3 E 3 + Ψ=Ψ 0 +λψ 1 +λ 2 Ψ 2 +λ 3 Ψ 3 + perturbation is assumed to be small Ψt V Ψ 0 Ψ 1 = t E0 E t H 0 =ΣF i i H 0 Ψ s =E s Ψ s MP2, MP3, MP4. In principle, leads to good results, but perturbation must be small

18 Configuration interaction Ψ = c o ψ o + ra c r a ψ r a single, S a r Orbitals not re-optimized Complete expansion: FullCI Truncated CI: CIS, CISD, CISDT, + rs rs cab ψ ab + a r < < b s double, D a r b s H 2 O 6-31G* 10 electrons 38 spinorbitals SD!!!!!! Size-consistency problem! Quadratic CI: QCISD, QCISDT... Coupled Cluster: CCSD, CCSDT χ 2K χ s χ r χ N χ b χ a χ 2 χ 1

19 Multiconfigurational methods When a single Lewis structure is not enough.. excited states diradicals often transition states Coefficients in front of the determinants are optimized MOs used for the construction of the determinants are also optimized CASSCF (Complete Active Space Self Consistent Field) Select electrons and orbitals involved in the chemical process CASSCF(n, m) n number of electrons m is the number of orbitals

20 CASSCF (Complete Active Space Self Consistent Field) CAS All excitations Dynamic correlation: Second Order Perturbation Theory CASPT2

21 Density functional Theory (DFT) In Hartree-Fock Ψ(x 1,x 2,..x N ) Exact Hamiltonian, but approx. wavefunction. ELECTRON DENSITY Important Saving: The wavefunction depends on 3N spatial coordinates + Spin The electron density depends just on 3 spatial variables Principles of DFT The ground state density of the system is enough to obtain the energy and other properties. Variational principle: for any trial density ρ the energy will represent an upper bond to the ground state energy E 0

22 Exact wavefunction, but unknown Hamiltonian!!! Looking for an approximate Exchange Correlation functional LDA F XC = f X + f Exchange Correlation C + VWN (Vosko, Wilk and Nusair) GGA B-P86 (Becke88 and Perdew86) Hybrid Functionals f XC =c HF f X HF+c GGA f XC GGA B3-LYP (Becke 3 parameter functional-lee Yang Paar) Excited States Time-Dependent Density Functional theory

23 Quantum Chemical and Dynamical tools for solving photochemical problems 2. Quantum Dynamics 18 June 20 07

24 Goal: Follow the evolution molecular system in real time The speed of atomic motion: 1000m/s t = s = 100 fs Chemical changes involve distances of Angstroms Femtosecond laser camera

25 Time dependent Schrödinger equation i Ψ () ˆ tot t = H() t Ψ tot () t t el Ψ ({ r },{ R }, t) = ψ ({ r };{ R }) ψ ({ R }, t) tot i A el i A nuc A el Ht ˆ() = Hˆ μ Et ()

26 1. Obtention of an initial wavefunction Solution of the time independent nuclear Schrödinger equation Numerically: FGH (Fourrier Grid Hamiltonian) 2.Propagation: solution of the TDSE Wavepacket: coherent superposition of stationary states

27 Quantum Chemical and Dynamical tools for solving photochemical problems 3. Quantum Control 18 June 20 07

28 Conventional chemical control Temperature Concentration Catalysis Pressure

29 Mode-selective chemistry + hν Fails in many cases since bonds are not independent!! energy flows between them: IVR

30 Brumer-Shapiro phase control Tannor-Kosloff-Rice pump-dump control Control through the manipulation of the amplitude and the phase parameters CPL 126 (1986) 541. JCP 85 (1986) 5805.

31 Judson-Rabitz Optimal Control Theory Laser Pulse-shape Modified E(t) E(t) Control system (molecule) Learning algorithm Products Detector Objective Teaching laser to control molecules Rabitz, PRL 68 (1992) 1500.

32 Quantum Chemical and Dynamical tools for solving photochemical problems Bibliography Introduction to computational chemistry. Frank Jensen. Wiley Modern Quantum Chemistry. Szabo and Ostlund. Dover Introduction to quantum mechanics. A time dependent perspective. Tannor. University Science Books. 18 June 20 07