Conical Intersections
Spiridoula Matsika
The Born-Oppenheimer
approximation
Energy


TS


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(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
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:
H T V
H tot (r,R)  T N  T e  V ee  V eN  V NN
Z Z 
1 2
1 2
1
Z

  
 i       
 2M
i 2m e
i ji rij
 i ri
   R
 T N  H e (r;R)
1 2
H T H 
  H e (r;R)
 2M
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 coordinates respectively. The total wavefunction
T is a product of electronic Ie and nuclear I
wavefunctions for an I state.
T (r,R)   I (R)Ie (r;R)
H T T  E T T
H  E 

e
e
I
e
I
e
I
Electronic eq.
(T N  E Ie )  I  E T  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.
Energy
Avoided crossing
Nuclear coordinate R
When electronic states approach each other, more than one of them
should be included in the expansion
Na
T (r,R)    I (R)Ie (r;R)
Born-Huang expansion
I 1
If the expansion is not truncated the wavefunction is exact since the set
Ieeis ecomplete.
e
e The total Schrodinger equation using the Born-Huang
H I  E Ibecomes
I
expansion
(T 
N
1

K II  E Ie )  I 
N

1
(2f IJ   J  K IJ  J )  E T  I
J I 2
fIJ (R)   eI    eJ

k IJ (R)   eI  2  eJ
r
r
Derivative coupling: couples the
different electronic states
Derivative coupling
fIJ  I  J 
I H J
EJ  EI
fIJ  fJI
fII  0
For real wavefunctions
I  2 J    fIJ  fIJ  fIJ
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 E=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 i
1  c111  c 21 2
2  c121  c 22 2
The electronic Hamiltonian is built and diagonalized

H11 H12 
H  

H 21 H 22 
e
H ij   i H e  j
H  H11  H 22
H11  H 22  H 2  H122
E1,2 
2
The eigenvalues
and eigenfunctions

are:


1  cos 1  sin  2
2
 2  sin
sin

cos
2

2



2
2

1  cos  2
2
H12
H 2 H122
H11  H 22
H 2 H122
In order for the eigenvalues to become degenerate:
H11(R)=H22 (R)
H12 (R) =0
Since two conditions are needed for the existence of a
conical intersection the dimensionality is Nint-2, 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
symmetry
H11 H12 
H  

H
H
 21
22 
e

Symmetry required conical intersections, Jahn-Teller effect
•

•
•

Symmetry allowed conical intersections (between states of different
symmetry)
•
•
•

H12=0, H11=H22 by symmetry
seam has dimension N of high symmetry
Example: E state in H3 in D3h symmetry
H12=0 by symmetry
Seam has dimension N-1
Example: A1-B2 degeneracy in C2v symmetry in H2+OH
Accidental same-symmetry conical intersections
•
Seam has dimension N-2
Example: X3 system
branching coordinates

R
Qx
Qy
r
Seam coordinate
Qs
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 expansion around the conical intersection
H(R)  H(R 0 )  H(R 0 )  R
H(R)  0  H(R 0 )  R
H12 (R)  0  H12 (R 0 )  R
Then the conditions for degeneracy are
H(R0 )  R  0
H12 (R0 )  R  0
g  H
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  sxx  sy y  g 2x 2  h2y 2
Conical intersections are described in terms of the
characteristic parameters g,h,s
Geometric phase effect (Berry
phase)
If the angle  changes from  to  +2:
     
1  cos 1  sin  2

2   2 
     
 2  sin 1  cos  2
 2   2 
1 (  2 )  1( )
 2 (  2 )   2 ( )
The electronic wavefunction is doubled valued, so a phase
has to be added so that the total wavefunction is single
valued
T  e iA(R ) (R;r) (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 
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 exist. Only
quasidiabatic where f is very small.

Practically g and h are taken from ab initio
wavefunctions expanded in a CSF basis
Ie 
N CSF
I
c
 m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 )
R
IJ
I
†
H(R) I
g (R)  c (R x )
c (R x )
R
I
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
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:
ji
Eij  g  R  0
h  R  0
ji
Additional geometrical constrains, Ki, , can be imposed. These conditions can be imposed
by finding an extremum of the Lagrangian.
L (R,  , )= Ek + 1Eij+ 2Hij + iKi
g
6
h
6
O
5
O
5
4
4
3
H
H
3
Y(a0)
0
Branching vectors for OH+OH
2
1
2
1
O
O
0
0
-1
H
-1
-2
H
-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
energy (a.u.)
Quenching to
OH(X)+OH(X)
0.015
0.01
0.005
0
g
-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
Three-state conical intersections
Three state conical intersections can exist between three states of the same symmetry
in a system with Nint degress of freedom in a subspace of dimension Nint-5
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: Nint-5, where Nint 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
1 2
H 11
H *12




H12
H 22
T1
0
H 1T 2
H 11
H12
T2
H1T2 
0 

*
H 12 

H 22 

In general 5 conditions need to be
satisfied.






H11=H22
Re(H12)=0
Im(H12)=0
Re(H1T2)=0, satisfied in Cs symmetry
Im(H1T2)=0, satisfied in Cs symmetry
The dimension of the seam is Nint-5
or Nint-3
C.A.Mead J.Chem.Phys., 70, 2276, (1979)