Conical Intersections Spiridoula Matsika
The Born-Oppenheimer approximation Energy TS Nuclear coordinate R ν The study of chemical systems is based on the separation of nuclear and electronic motion The potential energy surfaces (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 PES. On this potential energy surface, we can treat the motion of the nuclei classically or quantum mechanically
H = T + V H tot (r,r) = T N + T e + V ee + V en + V NN = 1 2 α + 2M α α Hamiltonian for molecules The total Hamiltonian operator for a molecular system is the sum of the kinetic energy operators (T) and potential energy operators (V) of all particles (nuclei and electrons). In atomic units the Hamiltonian is: Nuc kinetic En = T N + H e (r;r) 1 2 i + 2m e i Electr. kinetic En el-el repulsion el-nuc attraction i j>i 1 r ij Nuc-nuc repulsion Z α r αi + α i α β >α Z α Z β R αβ
H T = T N + H e = α Ψ T (r,r) = χ I (R)Ψ I e (r;r) 1 2 α + H e (r;r) 2M α Assuming that the motion of electrons and nuclei is separable, the Schrodinger equation is separated into an electronic and nuclear part. R and r are nuclear and electronic coordinates respectively. The total wavefunction Ψ T is a product of electronic Ψ e I and nuclear χ I wavefunctions for an I state. H T Ψ T = E T Ψ T H e Ψ I e = E I e Ψ I e Electronic eq. (T N + E I e )χ I = E T χ I Nuclear eq.
Nonadiabatic processes are facilitated by the close proximity of 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. Avoided crossing Energy Nuclear coordinate R
When electronic states approach each other, more than one of them should be included in the expansion Ψ T (r,r) = N a I =1 χ I (R)Ψ I e (r;r) (T N + 1 µ K II + E I e )χ I + + N J I 1 2µ ( 2f IJ χ J + K IJ Born-Huang expansion If the expansion is not truncated the wavefunction is exact since the set Ψ e I is complete. The total Schrodinger equation using the Born-Huang expansion becomes χ J ) = E T χ I f α IJ (R) = Ψ I e α Ψ J e k IJ (R) = Ψ I e 2 Ψ J e r r Derivative coupling: : couples the different electronic states
Derivative coupling f IJ = Ψ I Ψ J f IJ = f JI f II = 0 = Ψ I H Ψ J E J E I For real wavefunctions Ψ I 2 Ψ J = f IJ + f IJ f IJ The derivative coupling is inversely proportional to the energy difference of the two electronic states. Thus the smaller the difference, the larger the coupling. If ΔE=0 f is infinity.
What is a conical 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 ϕ i ψ 1 = c 11 ϕ 1 + c 21 ϕ 2 ψ 2 = c 12 ϕ 1 + c 22 ϕ 2 The electronic Hamiltonian is built and diagonalized H e = H 11 H 12 H 21 H 22 H ij = ϕ i H e ϕ j ΔH = H 11 H 22 The eigenvalues and eigenfunctions are: E 1,2 = H 11 + H 22 ± ΔH 2 + H 12 2 2 ψ 1 = cos α 2 ϕ 1 + sin α 2 ϕ 2 ψ 2 = sin α 2 ϕ 1 + cosα 2 ϕ 2 sin α 2 = H 12 ΔH 2 +H 12 2 cos α 2 = H 11 H 22 ΔH 2 +H 12 2
In order for the eigenvalues to become degenerate: H 11 (R)=H 22 (R) H 12 (R) =0 Since two conditions are needed for the existence of a conical intersection the dimensionality is N int -2, where N int is the number 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 Neumann and E. Wigner, Phys.Z 30,467 (1929)
Conical intersections and H e = H 11 H 12 H 21 H 22 symmetry Symmetry required conical intersections, Jahn-Teller effect H 12 =0, H 11 =H 22 by symmetry seam has dimension N of high symmetry Example: E state in H 3 in D3h symmetry Symmetry allowed conical intersections (between states of different symmetry) H 12 =0 by symmetry Seam has dimension N-1 Example: A 1 -B 2 degeneracy in C2v symmetry in H 2 +OH Accidental same-symmetry conical intersections Seam has dimension N-2
Example: X3 system branching coordinates α Q x R Q y r Seam coordinate Q s
Figure 4a Two internal coordinates lift the degeneracy linearly: g-h or branching plane energy (a.u.) 0.015 0.01 0.005 0-0.005-0.01-0.015 E g h -0.2-0.1 y (bohr) 0 0.1 0.2-0.2-0.1 0 x (bohr) 0.1 0.2 E (ev) N int -2 coordinates form the seam: points of conical intersections are connected continuously 3 2 1 0-1 -2-3 2.9 3 3.1 r (a.u.) 3.2 3.3 3.4-0.4-0.20-0.6 0.6 0.4 0.2 x (a.u.)
The Branching Plane The Hamiltonian matrix elements are expanded in a Taylor series expansion around the conical intersection H(R) = H(R 0 ) + H(R 0 ) δr ΔH(R) = 0 + ΔH(R 0 ) δr H 12 (R) = 0 + H 12 (R 0 ) δr Then the conditions for degeneracy are ΔH(R 0 ) δr = 0 H 12 (R 0 ) δr = 0 g = ΔH h = H 12 H e = (s x x + s y y)i + gx hy hy gx E 1,2 = s x x + s y y ± (gx) 2 + (hy) 2
Topography of a conical intersection asymmetry tilt E ± = E 0 + s x x + s y y ± g 2 x 2 + h 2 y 2 Conical intersections are described in terms of the characteristic parameters g,h,s
Geometric phase effect (Berry If the angle α changes from α to α +2π: ψ 1 = cos α 2 ϕ + sin α 1 2 ϕ 2 ψ 2 = sin α 2 ϕ + cos α 1 2 ϕ 2 ψ 1 (α + 2π) = ψ 1 (α) ψ 2 (α + 2π) = ψ 2 (α) phase) The electronic wavefunction is doubled valued, so a phase has to be added so that the total wavefunction is single valued Ψ T = e ia(r ) ψ(r;r)χ(r) The geometric phase effect can be used for the identification of conical intersections. If the line integral of the derivative coupling around a loop is equal to π
Adiabatic and Diabatic represenation Adiabatic representation uses the eigenfunctions of the electronic hamiltonian.. The derivative coupling then is present in the total Schrodinger equation Diabatic representation is a transformation from 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. Strickly diabatic bases don t t exist. Only quasidiabatic where f is very small.
Practically g and h are taken from ab initio wavefunctions expanded in a CSF basis Ψ I e = N CSF m=1 c m I ψ m [ H e (R) E I (R)]c I (R) = 0 Tuning, coupling vectors h α IJ (R) = c I (R x ) H(R) R α c J (R x ) g αi (R) = c I (R x ) H(R) R α c I (R x ) g IJ (R)= g I (R) - g J (R)
Locating the minimum energy point on the seam of conical intersections Projected gradient technique: M. J. Baerpack,, M. Robe and H.B. Schlegel Chem.. Phys. Lett. 223,, 269, (1994) Lagrange multiplier technique: M. R. Manaa and D. R. Yarkony, J. Chem. Phys., 99,, 5251, (1993)
Locate conical intersections using lagrange multipliers: ΔE ij + g ji δr = 0 h ji δr = 0 Additional geometrical constrains, K i,, can be imposed. These conditions can be imposed by finding an extremum of the Lagrangian. L (R, ξ,λ )= E k + ξ 1 ΔE ij + ξ 2 H ij + λ i K i
Branching vectors for OH+OH g h O O H H O O H H
Routing effect: E OH(A)+OH(X) Figure 4a Quenching to OH(X)+OH(X) g h Reaction to H 2 O+O
Three-state conical intersections Three state conical intersections can exist between three states of the same symmetry in a system with N int degress of freedom in a subspace of dimension N int -5 H 11 H 12 H 13 H = H 12 H 22 H 23 H 13 H 23 H 33 H 11 (R)=H 22 (R)= H 33 H 12 (R) = H 13 (R) = H 23 (R) =0 Dimensionality: N int -5, where N int is the number of internal coordinates J. von Neumann and E. Wigner, Phys.Z 30,467 (1929)
Conditions for a conical intersection including the spin-orbit interaction Ψ 1 Ψ 2 TΨ 1 TΨ 2 H 11 H 12 0 H 1T2 * H 12 H 22 H 1T 2 0 * H 11 H 12 H 12 H 22 In general 5 conditions need to be satisfied. H 11 =H 22 Re(H 12 )=0 Im(H 12 )=0 Re(H )=0, 1T satisfied in C 2 s symmetry Im(H )=0, 1T satisfied in C s symmetry 1T 2 The dimension of the seam is N int -5 or N int -3 C.A.Mead J.Chem.Phys., 70, 2276, (1979)