Conical intersections in
quantum chemistry
Spiridoula Matsika
Temple University
Department of Chemistry
Philadelphia, PA
Nonadiabatic events in chemistry
and biology
Molecular Hamiltonian
i
e2/rij
j
Z e2 /ri
Z e2 /rj

Z Z e2 /R

1 2
1 2
1 2 1 2
H (r,R) 
 
 
i 
j 
2M
2M 
2m e
2m e
T
1 Z Z Z  Z  Z Z 





 T N  H e (r;R)
rij ri rj rj ri
R

The Born-Oppenheimer
(adiabatic) approximation
HT  T N  H e  

1 2
  H e (r;R)
2M
Separate the problem into an electronic and a nuclear part
T (r,R)   I (R)Ie (r;R)
H T T  E T T  (T N  H e ) I Ie  E T  I Ie
H  E 
e
e
I
e
I
Electronic
e
I
(T N  E Ie ) I  E T  I
Nuclear
When electronic states approach each other, more than one
of them should be included in the expansion
HT  T N  H e
Born-Huang expansion
(H T  E T )T  0
If the expansion is not truncated the
wavefunction is exact since the set Ie is
complete
(H e  E Ie )Ie  0
Na
T (r,R)    I (R)Ie (r;R)
I 1
1 2
1 N
e
T
  I  (E I  E )  I 
(2f IJ   J  k IJ  J )  0

2M
2M J 1
fIJ (R)   eI    eJ
k IJ (R)   eI  2  eJ
r
r

 eI He  eJ
EJ  EI
r
Derivative coupling
Two-state conical intersections
Two adiabatic potential energy surfaces cross (E=0).The
interstate coupling is large facilitating fast radiationless
transitions between the surfaces
Figure 4a
energy (a.u.)
0.015
0.01
0.005
0
-0.005
-0.01
-0.015
-0.2
-0.1
y (bohr)
0
0.1
0.2
-0.2
-0.1
0.1
0
x (bohr)
0.2
The Noncrossing Rule
The adiabatic eigenfunctions are expanded in terms of states I which are
diagonal to all other states. (the superscript e is dropped here)
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 
H ij   i H  j
E1,2  H  H 2  H122
The eigenvalues and
eigenfunctions are:
H11  H 22
H 
2
H  H 22
H  11
2
J. von Neumann and E. Wigner, Phys.Z 30, 467 (1929)
Atchity, Xantheas and Ruedenberg, J. Chem. Phys., 95, 1862, (1991)


1  cos 1  sin  2
2
2
2  sin
sin

cos
2

2



2
1  cos
H12
H 2 H122
H
H 2 H122

2
2
Degeneracy
E1,2  H  H 2  H122
E1  E 2 
H  H12  0

H11(R)=H22 (R)
H12 (R) =0
Dimensionality: Nint-2, where Nint is the number of internal
coordinates
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)

Evolution of conical intersections
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(R0 )  R
H12 (R)  0  H12 (R0 )  R
Then the conditions for degeneracy are
H(R 0 )  R  0
H12 (R 0 )  R  0
g  H
h  H12

For displacements x,y, along the branching plane
the Hamiltonian becomes (sx,sy are the projections
of H onto the branching plane) :
gx hy 
H  (sx x  sy y)I  

hy
gx


E1,2  sx x  sy y  (gx) 2  (hy) 2
Atchity, Xantheas and Ruedenberg, J. Chem. Phys., 95, 1862, (1991)
D. R. Yarkony, J. Phys. Chem. A, 105, 6277, (2001)
The Branching Plane
 (E I  E J )
R
H
h IJ   I
J
R
g IJ 
Tuning coordinate:
Coupling coordinate:
Seam coordinates
Figure 4a
energy (a.u.)

Figure 1b
0.015
0.01
E (eV)
0.005
0
3
2
1
0
-1
-2
-3
E
-0.005
h
-0.01
-0.015
g
0.6
0.4
2.9
0.2
3
0
3.1
r (a.u.)
-0.2
3.2
3.3
seam
-0.4
3.4 -0.6
x (a.u.)
-0.2
g-h or branching plane
-0.1
y (bohr)
0
0.1
0.2
-0.2
-0.1
0.1
0
x (bohr)
0.2
topography
asymmetry
tilt
E  E0  sxx  sy y  g x  h y
2
2
2
2
Conical intersections are described in terms of the
characteristic parameters g,h,s
Three-state conical intersections
Based on the same non-crossing rule one can derive the dimensionality of a three-state
seam. Using a simple 3x3 matrix degeneracy between three eigenvalues is obtained if 5
conditions are satisfied. So the subspace where these conditions are satisfied is 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
J. von Neumann and E. Wigner, Phys.Z 30, 467 (1929)
J. Katriel and E. Davidson, Chem. Phys. Lett. 76, 259, (1980)
S. Matsika and D. R.Yarkony, J.Chem. Phys., 117, 6907, (2002)
Requirements of electronic structure
methods for conical intersections
•
•
•
•
•
Accurate excited states
Gradients for excited states
Describe coupling
Correct topography of conical intersections
Correct description at highly distorted geometries
(needed usually for S0/S1 conical intersections in closed
shell molecules but not necessary for all ci)
*
Excited states configurations
Ground state
Singly excited conf.
Doubly excited conf.
Configurations can be expressed as Slater determinants in
terms of molecular orbitals. Since in the nonrelativistic case
the eigenfunctions of the Hamiltoian are simultaneous
eigenfunctions of the spin operator it is useful to use
configuration state functions (CSFs)- spin adapted linear
combinations of Slater determinants, which are
eigenfunctiosn of S2
-
Singlet CSF
+
Triplet CSF

Or in a mathematical language:
Slater determinant:
1 (r1 )
1 1 (r2 )
SD
 (r1,r2 ,r3,r4 ) 
4 1 (r3 )
1 (r4 )
1(r1)
1(r2 )
1(r3 )
1 (r4 )
 2 (r1 )
 2 (r2 )
 2 (r3 )
 2 (r4 )
3 (r1)
3 (r2 )
3 (r3 )
3 (r4 )
Configuration state function:
1(r1)

1 1(r2 )
CSF
 (r1,r2 ,r3,r4 ) 
4 1(r3 )

1(r4 )
1 (r1 )
1 (r2 )
1 (r3 )
1 (r4 )
 2 (r1)
 2 (r2 )
 2 (r3 )
 2 (r4 )
3 (r1 ) 1(r1)
3 (r2 ) 1(r2 )

3 (r3 ) 1(r3 )
3 (r4 ) 1(r4 )
1 (r1 )
1 (r2 )
1 (r3 )
1 (r4 )
2 (r1)
2 (r2 )
2 (r3 )
2 (r4 )
 3 (r1 ) 

 3 (r2 ) 
 3 (r3 ) 

 3 (r4 ) 
•
•
Electronic structure methods for conical
intersections
Multireference methods satisfy these requirements
MCSCF and MRCI
I 
N CS F
c 
I
m
m
m1
H(R)c I (R)  E I (R)c I (R)
H ij (R)   i (r;R) H  j (r;R)
r
H(R 0 )  R  0
H(R) J
c
R
H(R) I
H(R) J
IJ
g (R)  c I
c  cJ
c
R
R
h (R)  c I
IJ
H12 (R 0 )  R  0
g  H
h  H12

Locate 2-state intersections
ji
Eij  g  R  0
h  R  0
ji
Locate 3-state intersections
Eij  g ij  R  0
Ekj  g kj  R  0
hij  R  hkj  R  hik  R  0
Additional geometrical constrains, Ki, , can be imposed. These conditions can be imposed
by finding an extremum of the Lagrangian

L (R,  , )= Ek + 1Eij + 2Eij + 3Hij + 4Hik + 5Hkj + iKi
•The algorithms are implemented in the COLUMBUS suite of programs.
M. R. Maana and D. R. Yarkony, J. Chem. Phys.,99,5251,(1993)
S. Matsika and D. R. Yarkony, J.Chem.Phys., 117, 6907, (2002)
H. Lischka et al., J. Chem. Phys.,120, 7322, (2004)
Derivative coupling
f IJ (R)  I (r;R)
I 


R
J (r;R)
r
N CSF
c 
I
m
m
m1
CI
fJI (R) fJI (R)


CSF
 j
c I
fJI (R)  c
  c J c iI  i

R

R
i, j
J
j
 j
1
H I
cJ
c   Di,JIj  i
EI  EJ
R
R
Similar to energy gradient

H I
c I
c I
H c  EI c 
c H
 EI

R
R
R
I
I
I
H I
J c
c
c  E I  E J  c
R
R
J
Geometric phase effect (Berry phase)
•
•
An adiabatic eigenfunction changes sign after traversing a closed loop
around a conical intersection
The geometric phase can be used for identification of conical
intersections. If the integral below is  the loop encloses a conical
intersection
f

IJ
(R)  dR  
Radiationless decay in DNA
bases
O
HN
O
N
HO
O
H
H
O
H
H
O
H
P
NH2
N
O-
N
-O
N
O
H
H
O
H
H
O
H
P
O-
N
• Implications in
photoinduced damage
in DNA
• Absorption of UV
radiation can lead to
photocarcinogenesis
• DNA and RNA bases
absorb UV light
• Bases have low
fluorescence quantum
yields and ultra-short
excited state lifetimes
• Fast radiationless
decay can prevent
further photodamage
O-
Daniels and Hauswirth, Science, 171, 675 (1971)
Kohler et al, Chem. Rev., 104, 1977 (2004)
Exploring the PES of excited states using
electronic structure methods
•State-averaged multiconfigurational self-consistent field (SA-MCSCF)
•Multireference configuration-interaction (MRCI)
•MRCI1: Single excitations from the CAS (~1 million CSFs)
•MRCI: Single excitations from CAS +  orbitals (~10-50 million CSFs)
•MRCI: Single and double excitations from CAS + single exc. from  orbitals
(100-400 million CSFs)
•Locate minima, transition states, and conical intersections
•Determine pathways that provide accessibility to conical intersections
Conical intersections in nucleic acid bases
Uracil
Cytosine
S. Matsika, J. Phys. Chem. A, 108, 7584, (2004)
K. A. Kistler and S. Matsika, J. Phys. Chem. A, 111, 2650, (2007)
K. A. Kistler and S. Matsika, Photochem. Photobiol.,83, 611, (2007)
Three-state intersections in
DNA/RNA bases
O
NH2
H 3C
NH
N
N
In pyrimidines:
N
H
O
N
H
O
N
H
NH2
C
In purines:
N
C
N
CH
HC
C
N
N
H
They connect different two-state conical intersections
O
Three-state conical intersections in nucleobases
Three state ci
Three state ci
Three-state conical intersections in cytosine
Multiple seams of three-state conical intersections were located for cytosine
and 5M2P, an analog with the same pyrimidinone ring.
7
6
S3
S2
S1
Energy (eV)
5
4
3
2
1
S0
0
-1
GS
ππ*/nNπ*/nOπ*
gs/ππ*/nNπ*
gs/ππ*/nOπ*
coupling vectors
The branching space for the S1-S2-S3
(*,n*,n*)(ci123) conical intersection.
coupling vectors
energy difference gradient vectors
The branching space for the S1-S2-S3
(*,*,n*)(ci123’) conical intersection
energy difference gradient vectors
The three-state ci012 are connected
to two state seam paths and finally
the Franck Condon region.
K. A. Kistler and S. Matsika, J. Chem. Phys. 128, 215102, (2008)
Phase effects and 3-state Conical Intersections
Different pairs of 2-state CIs exist around a 3-state CI.
Around a S0/S1/S2 CI there can be S0/S1 and S1/S2 CIs


Loop
f01
f12
f02
No CI
+
+
+
One (or odd #) S0/S1
+ One (or odd #) S1/S2
-
-
+
One (or odd #) S0/S1
+
-
-
Keat and Meating, J. Chem. Phys., 82, 5102, (1985)
Manolopoulos and Child, PRL, 82, 2223, (1999)
Han and Yarkony, J. Chem. Phys., 119, 11561, (2003)
Schuurman and Yarkony J. Chem. Phys., 124, 124109, (2006)
We will now explore the phases on loops
around the conical intersections that may or
may not enclose a second ci
K. A. Kistler and S. Matsika, J. Chem. Phys. 128, 215102, (2008)
Derivative coupling:
An
gh
f IJ (R) 
2g2 cos2   h 2 sin 2  

 constant
3.5


3
2.5
2
f
1.5
1
0.5
0
0
1
2
3
4

5
6
Plots of the derivative coupling along , f, the magnitude of f, and the energy of
states S0, S1, and S2 for a loop around a regular two-state ci away from a threestate ci.
CI
f10
An 
10
f
An 
10
f
|f20|
Energies along four branching plane loops around point D.
f01
f12
f02
No CI
+
+
+
One (or odd #) S0/S1
+ One (or odd #) S1/S2
-
-
+
One (or odd #) S0/S1
+
-
-
0.07
The magnitude of the derivative coupling is shown for four branching plane loops
around point D. f10, f20, f21, with respect to  are plotted.
Detail of c) above.
QM/MM for solvent effects
Including the solvent effects
Basics of QM/MM
H  HQM  HQM / MM  H MM

M: solvent charges
i: electrons
: nuclei
HQM / MM  H el  H vdw  H pol 
12
6 




qM
Z qM
 M
 M 
pol



  4M 


H
 
 
r
R
R
R
M
iM iM
M
M
 M   M  
A Hybrid QM/MM approach using ab
initio MRCI wavefunctions
• Quantum Mechanics:
– MCSCF/ MRCISD using COLUMBUS
• Molecular Mechanics/Dynamics
– TINKER or MOLDY
• Coupling between QM-MM regions
• Averaged solvent electrostatic potential
following the Martin et al. scheme
Martin et al. J. Chem. Phys. 116, (2002), 1613
General scheme
•Solve Schrodinger Eq. HQM=E -> q0
•Molecular dynamics -> average potential
•(HQM+HQM/MM)=E
• Compute energy and solute properties
( excitation energies)
Polarization
• Uncoupled method
– Do not change the partial charges of the solute during the MD simulation
• Polarization of solute:
– Partial charges (Mulliken):
–
MRCI
initial: C +0.1074, O -0.2414, H +.0670
–
final:
+0.141,
-0.393,
+0.126
–
MCSCF initial: +0.1549,
-0.2621,
+0.054
–
final:
+0.1954 ,
-0.4154,
+0.1115
• Polarization of the solvent:
General scheme
Until convergence
•Solve Schrodinger Eq. HQM=E -> q0
•Molecular dynamics -> average potential
•(HQM+HQM/MM)=E -> q
• Compute energy and solute properties
( excitation energies)
Average electrostatic potential
on a grid
f* Rvdw
f=0.8
S0=1.2 Å
S1=1.15 Å
4 shells
624 points
S0
S1
qi
Vj
q  A 1 b
m 
Grid A: inside the surface to calculate the
1 1 1 1 
Akj    
 


solvent induced electrostatic potential
r
r
r
r


ik
in  ij
in 
i1
q
Vj  
i
i
 1 1  0 Z 
bk    Vi  
r
rin 
rin 
i1  ik
m
rij
Grid B: outside to obtain q
Least squares fitting for q
Error in fitting: 150 cm-1

Blueshift (cm-1) in formaldehyde
Exp:
~1700 cm-1
1A
2
MCSCF/PCM
595
CASSCF/ASEP
1470
MRCISD/COSMO
1532
CCSD/MM(TIP3P)
2139
CCSD/MM(SPCpol)
2803
MRSDCI
1500
MCSCF
1580
MRSDCI(no solute pol)
970
CASSCF/MC
CASSCF/MC (pol)
2610-2690
2540-2660
Exp.:
4.07 eV
MCSCF: 3.94 eV
MRCI:
4.01 eV
1A
1
CAS+MRCI(4,3) /aug-cc-pvtz
Z. Xu and S. Matsika, J. Phys. Chem. A,110, 12035, (2006)
Dipole moment shifts
Gas phase
MCSCF
1A : 2.27
1
1A : 1.19
2
MRCI
2.41
1.28
exp
2.33
1.57
Solution phase
/D
/D
Method
/D
/D
Method
2.98
0.54
MRCISD/COSMO
1.66
0.38
MRCISD/COSMO
3.00
0.65
CASSCF/ASEP
1.74
0.52
CASSCF/ASEP
3.377
1.269
CASSCF/MC
2.270
1.305
CC2/MM(TIP3P)
3.442
1.206
CC2/MM(TIP3P)
2.306
0.982
CASSCF/MC
3.597
1.187
CCSD/MM(TIP3P)
2.455
1.490
CC2/MM(SPCpol)
3.662
1.426
CC2/MM(SPCpol)
1.95
1.237
CCSD/MM(TIP3P)
3.816
1.406
CCSD/MM(SPCpol)
2.683
1.420
CCSD/MM(SPCpol)
2.91
0.64
MCSCF/MM
1.78
0.59
MCSCF/MM
3.09
0.68
MRCISD/MM
2.597
0.67
MRCISD/MM
Solvatochromic Shifts in
Uracil
S4(ππ*)
S3(nO’π*)
S2(ππ*)
6.57
6.31
5.79
4.80
-0.05 (=6.47 D)
+0.41 (=2.67 D)
S1 (nOπ*)
S0
(=4.30 D)
EOM-CCSD/MM: S1: +0.44 eV, S2: +0.07 eV (Epifanovsky et al., JPC A, (2008)
NH2
Solvatochromic shifts in Cytosine
N
8.00
7.00
+0.83 (=2.86 D)
Energy/ eV
nO*
5.29 nN*
5.14 *
5.00
4.46
4.00
4.06 4.31
3.90
+0.25
5.14(=5.29 D)
4.46 4.28
3.00
2.05
2.00
2.05
1.09
1.00
0.00
0.00
O
+0.56 (=2.93 D)
5.93
6.00
4.06
3.90
N
H
4.38 4.31 4.45 4.28
4.09
4.38
3.38
3.38
2.42
2.42
1.09
0.00
(=6.14
D)
EOM-CCSD/MM: S1: +0.25eV, S2: +0.57eV (Vallet and Kowalski, JCP,125,211101 )
4.
4.0
Conclusions
Two- and Three-state conical intersections are ubiquitous in polyatomic
molecules.
Multireference configuration interaction electronic structure theory methods can
be used to describe these features. Single reference methods are more
challenging and varios problems need to be overcome
Three-state conical intersections connect two-state seams
Three-state conical intersections can affect couplings and possibly the dynamics.
Phase changes can be used as signature for the presence of these conical
intersections.
A QM/MM method using MRCI wavefunctions can describe solvation effects on
excited states
Acknowledgments
Dr. ZongRong Xu
Dr. Dimitri Laikov
Kurt Kistler
Kandis Gilliard
Madiyha Muhammad
Benjamin Mejia
Elizabeth Mburu
Chris Kozak
Financial support by:
NSF CAREER Award
DOE
Temple University

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