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Unformatted text preview: Conical Intersections
Conical
Spiridoula Matsika The BornOppenheimer
approximation
Energy TS ν Nuclear coordinate R 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(3N6) 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 Hamiltonian for molecules ion uls on rep
uc
c n
Nu ttra
ca
elnu lsio
epu
el r
el cti n En
tic
ine
Ele
ctr
.k tic
ine
Nu ck H =T +V En The total Hamiltonian operator for a molecular system is the
sum of the kinetic energy operators (T) and potential energy
sum
kinetic
potential
operators (V) of all particles (nuclei and electrons). In atomic
units the Hamiltonian is: H tot (r, R) = T N + T e + V ee + V eN + V NN
Zα Z β
−1 2
−1 2
1
Zα
=∑
∇α + ∑
∇ i + ∑∑ − ∑∑ + ∑ ∑
α
α 2 Mα
i 2 me
i j > i rij
α
i ri
α β >α Rαβ
= T N + H e (r;R) −1 2
H =T +H =∑
∇ α + H e (r;R)
α 2 Mα
T N e 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
electronic coordinates respectively. The total wavefunction
electronic
wavefunction
ΨT is a product of electronic ΨIe and nuclear χI
wavefunctions for an I state. ΨT (r, R) = χ I (R) ΨIe (r;R)
H T ΨT = E T ΨT
e e
I e
I e
I H Ψ =E Ψ
€ Electronic eq.
Electronic eq (T N + E Ie ) χ I = E T χ I
Nuclear eq.
Nuclear eq Nonadiabatic processes are facilitated by the close
processes
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. Energy Avoided crossing Nuclear coordinate R When electronic states approach each other, more than one of them
should be included in the expansion
Na ΨT (r, R) = ∑ χ I (R) ΨIe (r;R) BornHuang expansion I =1 If the expansion is not truncated the wavefunction is exact since the set
is
ΨIee is e omplete. The total Schrodinger equation using the BornHuang
c
H ΨI = E Ie ΨIe
expansion becomes 1 II
(T + K + E Ie ) χ I +
µ
N N 1
(−2f IJ ⋅ ∇χ J + K IJ χ J ) = E T χ I
J ≠ I 2µ +∑ fαIJ (R) = Ψ e ∇ α Ψ e
I
J € k IJ (R) = Ψ e ∇ 2 Ψ e
I
J r
r Derivative coupling: couples the
different electronic states Derivative coupling
Derivative
fIJ = ΨI ∇ ΨJ = ΨI ∇H ΨJ
EJ − EI fIJ = −fJI
fII = 0 For real wavefunctions
For
wavefunctions ΨI ∇ 2 ΨJ = ∇ ⋅ fIJ + fIJ ⋅ fIJ The derivative coupling is inversely proportional to the
energy difference of the two electronic states. Thus the
the
smaller the difference, the larger the coupling. If ΔE=0 f is
infinity. What is a conical
What
intersection
Two adiabatic potential
energy surfaces cross.
The interstate coupling is
large facilitating fast
radiationless transitions
between the surfaces The Noncrossing Rule
Rule
The adiabatic eigenfunctions are expanded in terms of ϕi
The
ψ1 = c11ϕ1 + c 21ϕ 2
ψ 2 = c12ϕ1 + c 22ϕ 2 The electronic Hamiltonian is built and diagonalized
€ H11 H12 H = H 21 H 22 e H ij = ϕ i H e ϕ j
ΔH = H11 − H 22 The eigenvalues
The eigenvalues
and eigenfunctions
and eigenfunctions
€
are: 2
H11 + H 22 ± ΔH 2 + H12
E1,2 =
2 α
α
ψ1 = cos ϕ1 + sin ϕ 2
2
2
α
α
ψ 2 = − sin ϕ1 + cos ϕ 2
2
2
α
H12
sin =
2
2
ΔH 2 + H12
cos € α
H −H
= 11 2 22 2
2
ΔH + H12 In order for the eigenvalues to become degenerate:
In
to
H11(R)=H22 (R)
H12 (R) =0
Since two conditions are needed for the existence of a
conical intersection the dimensionality is Nint2, where Nint 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
Conical
symmetry H11 H12 H = H 21 H 22 e Symmetry required conical intersections, JahnTeller effect
Jahn
effect
• € •
• Symmetry allowed conical intersections (between states of different
symmetry)
•
•
• H12=0, H11=H22 by symmetry
by
seam has dimension N of high symmetry
Example: E state in H3 in D3h symmetry
in H12=0 by symmetry
Seam has dimension N1
Example: A1B2 degeneracy in C2v symmetry in H2+OH Accidental samesymmetry conical intersections
• Seam has dimension N2 Example: X3 system branching coordinates
α
R Qx Qy
r Seam coordinate
Qs Figure 4a E energy (a.u.) Two internal
coordinates lift the
degeneracy linearly:
gh or branching
plane 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) Nint2 coordinates form the
seam: points of conical
intersections are connected
continuously 3
2
1
0
1
2
3 2.9 0.2
3 3.1
r (a.u.) 3.2 0.2
3.3 3.4 0.6 0.4 0 0.4 0.6 x (a.u.) 0.2 The Branching Plane
The
The Hamiltonian matrix elements are expanded in a Taylor
series expansion around the conical intersection
H (R) = H (R0 ) + ∇ H (R0 ) ⋅ δR
ΔH (R) = 0 + ∇ΔH (R0 ) ⋅ δR
H12 (R) = 0 + ∇H12 (R0 ) ⋅ δR Then the conditions for degeneracy are
€ ∇ΔH (R0 ) ⋅ δR = 0
∇H12 (R0 ) ⋅ δR = 0
g = ∇ΔH
h = ∇H12 gx hy H = ( sx x + sy y )I + hy −gx e E1,2 = sx x + sy y ± ( gx ) 2 + ( hy ) 2 € Topography of a conical
intersection
asymmetry tilt E± = E0 + sx x + sy y ± g 2 x 2 + h2 y 2
Conical intersections are described in terms of the
characteristic parameters g,h,s Geometric phase effect (Berry
phase) If the angle α changes from α to α +2π:
If α α
ψ1 = cos ϕ1 + sin ϕ 2 2
2 α α
ψ 2 = −sin ϕ1 + cos ϕ 2 2 2
ψ1 (α + 2π ) = −ψ1 (α )
ψ 2 (α + 2π ) = −ψ 2 (α ) € The electronic wavefunction is doubled valued, so a phase
is
has to be added so that the total wavefunction is single
has
is
valued
ΨT = 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
coupling Adiabatic and Diabatic
Diabatic
represenation Adiabatic representation uses the eigenfunctions
Adiabatic
eigenfunctions
of the electronic hamiltonian. The derivative
of
hamiltonian
coupling then is present in the total Schrodinger
equation
Diabatic representation is a transformation from
Diabatic representation
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 exist. Only
Strickly
bases
quasidiabatic where f is very small.
quasidiabatic where is Practically g and h are taken from ab initio
Practically
ab
wavefunctions expanded in a CSF basis
wavefunctions expanded
N CSF ΨIe = I
c mψ m
∑
m =1 [H (R) − E (R)]c (R) = 0
e I I Tuning, coupling vectors
€ ∂H (R) J
h (R ) = c ( R x )
c (R x )
∂Rα
IJ
α I † ∂ H (R ) I
g (R ) = c ( R x )
c (R x )
∂Rα
I
α I † gIJ(R)= gI(R)  gJ(R) Locating the minimum energy point
on the seam of conical intersections Projected gradient technique: M. J. Baerpack, M. Robe and H.B. Schlegel
M. Baerpack
Chem. Phys. Lett. 223, 269, (1994)
Phys. Lett 223 Lagrange multiplier technique: M. R. Manaa and D. R. Yarkony, J. Chem.
M.
Yarkony
Chem
Phys., 99, 5251, (1993)
., 99 Locate conical intersections using
lagrange multipliers:
ji Δ Eij + g δ R = 0
ji h ⋅ δR = 0
Additional geometrical constrains, Ki, , can be imposed. These conditions can be imposed
by finding an extremum of the Lagrangian.
L (R, ξ ,λ )= Ek + ξ1ΔEij+ ξ2Hij + ∑λiKi Branching vectors for OH+OH
g 6 O 5 O 5 4 4 H 2 1 H 3 Y(a0) 3 Y(a0) h 6 2 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:
E OH(A)+OH(X) Figure 4a Quenching to
OH(X)+OH(X) energy (a.u.) 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 0.1
0
x (bohr) 0.2 Threestate conical intersections
Three state conical intersections can exist between three states of the same symmetry
Three
in a system with Nint degress of freedom in a subspace of dimension Nint5
5
in H11 H = H12 H13 H12
H 22
H 23 H13 H 23 H 33 H11(R)=H22 (R)= H33
H12 (R) = H13 (R) = H23 (R) =0
Dimensionality: Nint5, where Nint is the number of internal
coordinates
J. von Neumann and E. Wigner, Phys.Z 30,467 (1929) Conditions for a conical intersection
Conditions
including the spinorbit interaction
Ψ1 Ψ 2 H1 1 H1 2 H* 2 H2 2
1 TΨ1 0
−H 1T 2
H1 1
H1 2 T Ψ2 H1T2 0 *
H1 2 H2 2 In general 5 conditions need to be
satisfied. H11=H22
Re(H12)=0
Im(H12)=0
Re(H1T2)=0, satisfied in Cs symmetry
symmetry
Im(H1T2)=0, satisfied in Cs symmetry
symmetry The dimension of the seam is Nint5
or Nint3 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.
 Summer '06
 DuaneJohnson
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