Matsika_ConicalIntersectionsMCC - Conical Intersections...

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Unformatted text preview: Conical Intersections Conical Spiridoula Matsika The Born-Oppenheimer The approximation Energy TS Nuclear coordinate R Nuclear The study of chemical systems is The based on the separation of nuclear and electronic motion The potential energy surfaces The (PES) are generated by the solution of the electronic part of the Schrodinger equation. This solution gives an energy for every fixed position of the nuclei. When the energy is plotted as a function of geometries it generates the PES as a(3N-6) dimensional surface. Every electronic state has its own Every PES. PES. On this potential energy surface, On we can treat the motion of the nuclei classically or quantum mechanically mechanically Hamiltonian for molecules lsio n e pu cr Nu c-n u cti on elnu ca ttra lsio n l re pu . ki net ic E n ele Nu c kin etic H =T +V Ele ctr En The total Hamiltonian operator for a molecular system is the sum of The the kinetic energy operators (T) and potential energy operators (V) of kinetic potential all particles (nuclei and electrons). In atomic units the Hamiltonian is: all H tot (r, R) = T N + T e + V ee + V eN + V NN =å a -1 2 -1 2 Ña + å Ñi + å 2 Ma 2me i i = T N + H e (r;R) 1 årj > i ij åå a i Za +å ra i a Za Z b åR ab b >a H =T +H =å T N e a -1 2 Ña + H e (r;R) 2 Ma A ssuming that the motion of electrons and nuclei is separable, the Assuming Schrodinger equation is separated into an electronic and nuclear part. R and r are nuclear and electronic coordinates respectively. The total wavefunction ΨT is a product of electronic ΨIe and nuclear χI wavefunctions for an I state. ΨT (r, R) = c I (R) YIe (r;R) HT YT = E T YT e e I e I e I H Y =E Y Electronic eq. (T N + E Ie )c I = E T c I Nuclear eq. Nonadiabatic processes are facilitated by the close proximity of Nonadiabatic potential energy surfaces. When the potential energy surfaces approach each other the BO approximation breaks down. The rate for nonadiabatic transitions depends on the energy gap. Energy Avoided crossing Nuclear coordinate R When electronic states approach each other, more than one of them should be included in the expansion Born-Huang expansion Born-Huang I f the expansion is not truncated the wavefunction is exact since the set ΨIe is If complete. The total Schrodinger equation using the Born-Huang expansion becomes becomes 1 II (T + K + E Ie ) c I + m N N 1 (- 2f IJ ×Ñc J + K IJ c J ) = E T c I 2m J¹ I +å faIJ (R) = Y e Ña Y e I J k IJ (R) = Y e Ñ 2 Y e I J r r Derivative coupling: couples the couples different electronic states different Derivative coupling fIJ = YI Ñ YJ = YI ÑH YJ EJ - EI fIJ = - fJI fII = 0 For real wavefunctions YI Ñ 2 YJ = Ñ ×fIJ + fIJ ×fIJ The derivative coupling is inversely proportional to the energy The difference of the two electronic states. Thus the smaller the difference, the larger the coupling. If ∆ E= f is infinity. 0 E= What is a conical intersection intersection Two adiabatic potential energy surfaces cross. The interstate coupling is large facilitating fast radiationless transitions between the surfaces The Noncrossing Rule The adiabatic eigenfunctions are expanded in terms of ϕi The ψ1 = c11j 1 + c 21j 2 y 2 = c12j 1 + c 22j 2 The electronic Hamiltonian is built and diagonalized æH11 H =ç èH 21 e H12 ö ÷ H 22 ø H ij = j i H e j j DH = H11 - H 22 The eigenvalues and The eigenfunctions are: eigenfunctions 2 H11 + H 22 ± DH 2 + H12 E1,2 = 2 a a ψ1 = cos j 1 + sin j 2 2 2 a a y 2 = - sin j 1 + cos j 2 2 a H12 sin = 2 2 DH 2 + H12 cos a H -H = 11 2 22 2 2 DH + H12 2 In order for the eigenvalues to become degenerate: H11(R)= 22 (R) H H12 (R) = 0 Since two conditions are needed for the existence of a conical intersection the dimensionality is Nint-2, where Nint is thenumber of internal coordinates For diatomic molecules there is only one internal coordinate and so states of the same symmetry cannot cross (noncrossing rule). But polyatomic molecules have more internal coordinates and states of the same symmetry can cross. J. von Neum and E. Wigner, Phys.Z 30,467 (1929) ann Conical intersections and symmetry symmetry æH11 H =ç èH 21 e Symmetry required conical intersections, Jahn-Teller effect Symmetry Jahn-Teller • • • H12=0, H11=H22 by symmetry seam has dimension N of high symmetry Example: E state in H3 in D3h symmetry in Symmetry allowed conical intersections (between states of different Symmetry symmetry) • • • H12 ö ÷ H 22 ø H12=0 by symmetry Seam has dimension N-1 Seam Example: A1-B2 degeneracy in C2v symmetry in H2+OH Accidental same-symmetry conical intersections • Seam has dimension N-2 Example: X3 system a 1 e e X Y branching coordinates α R Q x Q y r Seam coordinate Q s Figure 4a energy (a.u.) Two internal coordinates lift the degeneracy linearly: g-h or branching plane E 0.015 0.01 0.005 0 h ­0.005 ­0.01 ­0.015 g ­0.2 ­0.1 y (bohr) 0 0.1 0.2 ­0.2 0.1 0 x (bohr) ­0.1 Figure 1b E (eV) Nint-2 coordinates form the seam: points of conical intersections are connected continuously 3 2 1 0 -1 -2 -3 0.6 0.4 2.9 0.2 3 0 3.1 r (a.u.) -0.2 3.2 3.3 -0.4 3.4 -0.6 x (a.u.) The Branching Plane The Hamiltonian matrix elements are expanded in a Taylor series The expansion around the conical intersection expansion H (R) = H (R0 ) + Ñ H (R0 ) ×dR DH (R) = 0 + ÑDH (R0 ) ×dR H12 (R) = 0 + ÑH12 (R0 ) ×dR Then the conditions for degeneracy are ∇∆H (R0 ) ×dR = 0 ÑH12 (R0 ) ×dR = 0 g = ÑDH h = ÑH12 æ gx hy ö H = ( sx x + sy y )I + ç ÷ hy - gx ø è e E1,2 = sx x + sy y ± ( gx ) 2 + ( hy ) 2 Topography of a conical Topography intersection intersection asymmetry Conical intersections are described in terms of the characteristic parameters g,h,s tilt Geometric phase effect (Berry Geometric phase) phase) I f the angle α changes from α to α + π: 2 If æ aö æ aö ψ1 = çcos ÷ 1 + çsin ÷ 2 j j è è 2ø 2ø æ aö æ aö y 2 = - çsin ÷ 1 + çcos ÷ j j è 2ø è 2ø y 1 (a + 2p ) = - y 1 (a ) y 2 (a + 2p ) = - y 2 (a ) 2 The electronic wavefunction is doubled valued, so a phase has to be The added so that the total wavefunction is single valued added ΨT = e iA ( R )y ( R; r) c ( R) The geometric phase effect can be used for the identification of The conical intersections. If the line integral of the derivative coupling around a loop is equal to π Adiabatic and Diabatic represenation represenation Adiabatic representation uses the eigenfunctions Adiabatic of the electronic hamiltonian. The derivative coupling then is present in the total Schrodinger equation Diabatic representation is a transformation from Diabatic the adiabatic which makes the derivative coupling vanish. Off diagonal matrix elements appear. Better for dynamics since matrix elements are scalar but the derivative coupling is a vector. vector. Strickly diabatic bases don’t exist. Only Strickly quasidiabatic where f is very small. quasidiabatic Practically g and h are taken from ab initio Practically wavefunctions expanded in a CSF basis wavefunctions ΨIe = N CSF å I c my m m =1 Tuning, coupling vectors gIJ(R)= gI(R) - gJ(R) L ocating the minimum energy point on the seam of conical intersections Projected gradient technique: Projected M. J. Baerpack, M. Robe and H.B. Schlegel M. Chem. Phys. Lett. 223, 269, (1994) Chem. 223 Lagrange multiplier technique: M. R. Manaa and D. R. Yarkony, J. Chem. M. Phys., 99, 5251, (1993) Phys., 99 5251, L ocate conical intersections using lagrange multipliers: Additional geometrical constrains, Ki, , can be imposed. These conditions can be imposed by finding an extremum of the Lagrangian. L (R, ξ ,λ )= Ek + ξ1∆ Eij+ ξ2Hij + ∑λiKi Branching vectors for OH+ OH g 6 h 6 O 5 O 5 4 4 H 3 H 3 2 )0 Y(a 2 1 1 O 0 O 0 H -1 H -1 -2 -2 -4 -3 -2 -1 0 X(a0) 1 2 3 -4 4 -3 -2 -1 0 X(a0) 1 2 3 4 Routing effect: Routing E OH(A)+ OH(X) Figure 4a energy (a.u.) Quenching to OH(X) + OH(X) 0.015 0.01 0.005 g 0 ­0.005 ­0.01 h ­0.015 ­0.2 ­0.1 y (bohr) 0 0.1 0.2 ­0.2 Reaction to H2O+ O ­0.1 0 x (bohr) 0.1 0.2 Three-state conical intersections Three state conical intersections can exist between three states of the sam sym etry e m Three in a system with Nint degress of freedom in a subspace of dimension Nint-5 in -5 æ 11 H ç H = çH12 ç èH13 H12 H 22 H 23 H13 ö ÷ H 23 ÷ ÷ H 33 ø H11(R)= 22 (R)=H33 H H12 (R) =H13 (R) =H23 (R) = 0 Dimensionality: Nint-5, where Nint is thenumber of internal coordinates J . von Neumann and E. Wigner, Phys.Z 30,467 (1929) Conditions for a conical intersection including the spin-orbit interaction including Ψ1 Ψ2 TΨ1 TΨ2 In general 5 conditions need to be In satisfied. satisfied. Re(H12)=0 Im(H12)=0 Re(H1T2)=0, satisfied in Cs symmetry symmetry H11=H22 Im(H1T2)=0, satisfied in Cs symmetry symmetry The dimension of the seam is Nint-5 or -5 Nint-3 C.A.Mead J.Chem.Phys., 70, 2276, (1979) ...
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This note was uploaded on 12/07/2011 for the course CHEM 350 taught by Professor Duanejohnson during the Summer '06 term at University of Illinois, Urbana Champaign.

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