势能面的交叉

Avoided crossing
Nuclear coordinate R
Energy
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)
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.
T (r,R) I (R)Ie (r;R)
HT T ET T
H eIe
E
e I
Ie
Electronic eq.
(T N
E
e I
)
I
ET 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.
• 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
Born-Huang expansion
I 1
If the expansion is not truncated the wavefunction is exact since
tBhHoerens-eHIetuaEIneIegis ecIe xopmapnlseioten.
The total Schrodinger becomes
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.
Geometric phase effect (Berry
phase)
If the angle changes from to +2:
1
cos 2
1
sin
2
2
2
sin
2
1
c
os 2
2
1( 2 ) 1() 2( 2 ) 2()
The electronic wavefunction is doubled valued, so a
H TV
H tot (r,R) T N T e V ee V eN V NN
1 2M
2
i
1 2me
2 i
i
1
r ji ij
Z
Z Z
i ri
R
T N H e (r;R)
H T
TN He
1 2M
2
H
e
(r;R)
Assuming that the motion of electrons and nuclei is
seam: points of conical
-2 -3
intersections are connected 2.9
continuously
0.6
0.4
0.2
3
3.1
0 -0.2
x (a.u.)
r (a.u.) 3.2
3.3
-0.4 3.4 -0.6
The Branching Plane
The Hamiltonian matrix elements are expanded in a Taylor series expansion around the conical intersection
势能面交叉
Conical Intersections
The Born-Oppenheimer approximation
Energy
TS Nuclear coordinate R
The study of chemical systems is based on the separation of nuclear and electronic motion
H(R) H(R0) H(R0) R H(R) 0 H(R0) R H12(R) 0 H12(R0) R
Then the conditions for degeneracy are
H(R0) R 0 H12(R0) R 0
g H h H12
He
(sx x
sy
y)I
gx hy
phase has to be added so that the total wavefunction is
single valued
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
J. von Neumann and E. Wigner, Phys.Z 30,467 (1929)
Conical intersections and
symmetry
He HH1211
H12 H 22
Symmetry required conical intersections, Jahn-Teller effect
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.
• H12=0, H11=H22 by symmetry • seam has dimension N of high symmetry
• Example: E state in H3 in D3h symmetry
Symmetry allowed conical intersections (between states of different symmetry)
Example: X3 system
branching coordinates
R
Q
x
Qy
r
Seam coordinate
Q
s
Two internal coordinates lift the degeneracy linearly: g-h or branching plane
energy (a.u.)
hy gx
E1,2 sx x sy y (gx)2 (hy)2
Topography of a conical intersection
asymmetry
tilt
E E0 sxx syy g2x2 h2y2
Conical intersections are described in terms of the characteristic parameters g,h,s
equation
using the
(T N
1 K II
E
e I
)
I
N
JI
1 (2f IJ
2
J
K IJJ
)
ET I
fIJ (R) Biblioteka e Ie Jr
Derivative coupling: couples
k IJ (R) e 2e
the different electronic states
I
Jr
Derivative coupling
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: