This preview shows page 1. Sign up to view the full content.
Unformatted text preview: Conical Intersections
Conical
Spiridoula Matsika The BornOppenheimer
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(3N6) 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
cn
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 BornHuang expansion
BornHuang
I f the expansion is not truncated the wavefunction is exact since the set ΨIe is
If
complete. The total Schrodinger equation using the BornHuang 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 Nint2, 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, JahnTeller effect
Symmetry
JahnTeller
•
•
• 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 N1
Seam
Example: A1B2 degeneracy in C2v symmetry in H2+OH Accidental samesymmetry conical intersections
• Seam has dimension N2 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:
gh 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) Nint2 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 Threestate 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 Nint5
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: Nint5, 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 spinorbit 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 Nint5 or
5
Nint3 C.A.Mead J.Chem.Phys., 70, 2276, (1979) ...
View
Full
Document
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
 Electron

Click to edit the document details