Born
-
Oppenheimer
Coupling Terms
as
Molecular Fields
Michael Baer
The Fritz Haber Research Center for Molecular
Dynamics, The Hebrew University of Jerusalem,
Jerusalem, Israel
2
Colleagues
–
Past & Present
Longstanding Collaborations
Prof. G.D. Billing (deceased)
Prof. A. Vibok (Debrecen, Hungary)
Prof. G.J. Halasz (Debrecen, Hungary)
Prof. R. Englman (Soreq, Israel)
Prof. A.M. Mebel (Intl. Univ. Miami, Fl
USA)
Prof. S. Adhikari (ITT, Guwahati, India)
Mr. B. Sarkar, (ITT, Guwahati, India)
Dr. T. Vertesi (Debrecen, Hungary)
Prof, D.J. Kouri, (Houston, TX, USA)
Prof. D.K. Hoffman (Ames, IA, USA)
Prof. R. Baer (Jerusalem, Israel)
Short Collaborations
Prof. A. Alijah (Coimbra, Portugal)
Dr. E. Bene (Debrecen, Hungary)
Dr. A. Yahalom (Ariel, Israel)
Dr. S. Hu (Houston, TX, USA)
Prof. A.J.C. Varandas (Coimbra,
Portugal)
Dr. Z.R. Xu (Coimbra, Portugal)
Dr. D. Charutz (Soreq, Israel)
Prof. R. Kosloff (Jerusalem, Israel)
Prof. J. Avery (Copenhagen, Denmark)
Prof. S.H. Lin (IAMS, Taipei, Taiwan)
Introduction
The Non
-
adiabatic Coupling Term
as a Physical Entity
4
The Non
-
adiabatic Coupling Term
(NACT)
,1,,
,,......
jk j k
j k N
q p
z z
=
= Ñ
æ ö
¶ ¶
÷
ç
Ñ =
÷
ç
÷
ç
¶ ¶
è ø
L
5
Four Coupled Potential Surfaces
2 2
Conical Int ersect ions for
t he C H
molecule
-
Halasz, Vibok, Baer
Chem. Phys. Lett.
413
, 226 (2005)
6
What is the Purpose of this
Lecture?
To understand the physical contents of the NACTs
By Definition a NACT is a vector but….?
We show that NACTs behave like fields
7
Contents
I. The Hilbert Space
II. Degeneracy Points as Poles
III. Vector Algebra to Form Two
-
state (Quantum) fields
IV.
Field Equations to Form Multi
-
State (Quantum) Fields
Chapter I
The Hilbert Space
9
Introduction
Consider a series of N
-
dimensional Hilbert
spaces
Resolution of unity:
(
)
1,2,3,...,
|
j e
j N
z
=
s s
(
)
(
)
1
ˆ
| |
N
j e j e
j
I
z z
=
=
å
s s s s
10
The connection between the Hilbert spaces of adjacent points is
described in terms of an NxN vectorial matrix:
is the electronic Born Oppenheimer NACT matrix
is an anti
-
symmetric matrix
The NACT
,1,,
,,......
jk j k
j k N
q p
z z
=
= Ñ
æ ö
¶ ¶
÷
ç
Ñ =
÷
ç
÷
ç
¶ ¶
è ø
L
11
Connecting Hilbert Spaces
Two Hilbert spaces at nearby points
:
Recalling the resolution of the unity (
and multiplying by
)
and
s
= + D
s s s
%
(
)
(
)
(
)
| |
ik i e k e
z z
= Ñ
s s s s s
(
)
(
)
(
)
(
)
| | | |
N
k e j e j e k e
j
z z z z
Ñ = Ñ
å
s s s s s s s s
(
)
(
)
(
)
| | |
k e k e k e
z z z
+ D = + D ×Ñ
s s s s s s s s
12
We obtain:
Solving the First order Differential
The Integration is along a prescribed contour
(
)
(
)
(
)
(
)
0
0 0
| | exp''|
e e
d
z z
é ù
= - ×
ò
ê ú
ë û
s
s
s s s s s s s
R. Baer, J. Chem. Phys.
117
,
405
(
2002
).
(
)
s
s
(
)
(
)
(
)
| |
e e
z z
Ñ = -
s s s s s
13
Closed Contour
For a closed contour (loop)
In loops: return to the same state (up to phase)
(
)
(
)
(
)
(
)
0 0 0
| | exp |
e e
d
z z
G
= Ã - ×
ò
s s s s s s s
Ñ
(
)
(
)
(
)
0 0 0
| | exp |
j e j j e
i
z q z
=
s s s s s
s
i
s
f
A. Alijah and M. Baer, Chem. Phys. Lett.
319
,
489
(
2000
)
14
Quantization Conditions
A group of N states forms a
Hilbert space
(in a
given region)
if and only if the
D
-
matrices
are
diagonal
along any
contour
:
(
)
(
)
(
)
(
)
,{1,}
exp exp;
jk j jk
jk
j k N
d i
q d
G
=
é ù
G = - × =
ò
ë û
D s s
Ñ
(
)
(
)
(
)
exp
d
G
G = Ã - ×
ò
D s s
Ñ
15
Quantization Conditions
In the case of real eigenfunctions:
In case of two states (N=
2
):
the D
-
matrix can be written as:
(
)
jk jk
d
G = ±
D
A. Alijah and M. Baer, Chem. Phys. Lett.
319,
489(2000)
12
æ ö
÷
ç
÷
ç
÷
ç
÷
÷
ç
è ø
(
)
(
)
(
)
(
)
(
)
cos sin
sin cos
a a
a a
æ ö
G G
÷
ç
÷
ç
G =
÷
ç
÷
ç
- G G
÷
ç
è ø
D
16
Bohr
-
Sommerfeld Quantization
Condition
(
) is the Topological (Berry) phase
:
The
2
2
D
(
)
-
matrix
becomes
diagonal
if:
21
( )
d
a
G
G = ×
ò
s
Ñ
(
)
;0,1,2,...
n n
a p
G = = ± ±
17
Hilbert subspace
We assume that
breaks up into
blocks
12 13 1N
12 23 2N
13 23 3N
1N 2N 3N
N+1N+2 N+1N+3
N+1N+2 N+2N+3
N+1N+3 N+2N+3
-
-
O( )
-
- - - -
-
- - -
- - -
- - -
O( )
- - - - -
- - - - -
- - - - -
0
τ τ τ
τ 0 τ τ
τ τ 0 τ
0
τ τ τ 0
0
τ τ
τ 0 τ
τ τ 0
0
0
0
τ
18
How to detect NACTs?
Halasz, Vibok, Baer,
CPL
413
,
226
(
2005
)
2 2
The C H Molecule
19
NH
2
:
2
-
state
results
Vibok, Halasz,
Suhai,Hoffman,
Kouri, Baer,
J. Chem. Phys.
124
,
024312
(
2006
)
20
NH
2
:
3
-
state
results
Vibok, Halasz, Suhai,
Hoffman, Kouri, Baer,
JCP
124
,
024312
(
2006
)
21
H+H
2
3
-
states
Halasz, Vibok, Mebel, Baer
JCP
118
,
3052
(
2003
)
22
The Curl Equation
The
eigenfunctions
of a Hilbert space satisfy, for any (p,q) tensorial
component, the
equality
:
This
equality
is termed as the
Curl Condition
If the NACT
-
matrix breaks up into
blocks
the
Curl
Condition
is fulfilled for each Block
,0
p q q p
q p
pq p q
p q
= -
¶ ¶
é ù
= - - =
ë û
¶ ¶
F
1442 443
M. Baer, Chem. Phys. Lett.
35
,
112
(
1975
)
23
The Curl Equation (continued)
Abelian
vs
non
-
Abelian
variables
Commutation Relation
The
-
浡m物r猬渠来g敲慬Ⱐ
non
-
Abelian
A
2
2
-
浡m物r猠
Abelian
12
æ ö
÷
ç
÷
ç
÷
ç
÷
÷
ç
è ø
p q q p
τ τ τ τ
24
The
curl condition
for Two
-
state System:
Here:
This case is Abelian and therefore:
In polar coordinates
12
æ ö
÷
ç
÷
ç
÷
ç
÷
÷
ç
è ø
12
12
12
,0 F 0
y
x
xy
x y
y x
¶
¶
é ù
= Þ = - =
ë û
¶ ¶
12 12
12 12
12
1
F 0 0
q q
q
q q q
j j
j
j j
æ ö
¶ ¶
¶ ¶
÷
ç
= - = Þ - =
÷
ç
÷
ç
¶ ¶ ¶ ¶
è ø
q
j
CI
25
The Divergence Equation
The
eigenfunctions
of a Hilbert space form also
a Divergence equation
where
In polar coordinates
:
(
)
Div
Ñ× - ×
q q
q q q
j
j
¶
¶
Ñ× + +
¶ ¶
(
)
2
i j
ij
z z
º Ñ
q
j
CI
26
Div
⡳⤠景爠a
呷T
-
却慴S卹S瑥t
In Cartesian Coordinates
The
2
)
matrix elements are:
In contrast to
-
浡瑲m砠敬敭敮瑳t⁴桥礠慲攠
獣慬a牳
12
12
Div
y
x
x y
¶
¶
= +
¶ ¶
(
)
(
)
(
)
2
12
2 2
2
11 22 12
= Div
= = -
Chapter II
Degeneracy Points
As
Poles
28
The Degeneracy Point and its
Close Vicinity
At the vicinity of DP the
corresponding
NACT behaves like
a
Pole
: it is singular and decays like (
1
/q)
At the vicinity of DP the
corresponding
NACT possesses
an
angular
component.
At the vicinity of DP the
radial
component of the
corresponding
NACT is negligible small.
At the vicinity of DP related to a
given
NACT the effect of
all
other
NACTs is negligible
small
29
The Epstein Theorem
The Epstein Theorem States that
(at every point
in CS):
Degeneracy Points (DP)
can only be
formed if
j e k
jk
k j
H
u u
z z
Ñ
=
-
1
k j
= ±
30
The Epstein Theorem at
Degeneracy Points
At the
vicinity of a
DP
we consider the
angular
component
:
At the vicinity of a
DP
we expand for q~
0
1
1
1
1 1
e
j j
jj
j j
H
q q u u
j
z z
j
±
±
±
¶
¶
=
-
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
1
0
0
1
1 0 1
0
lim
lim
,
,
m m
j j j
q
m m
j j j
q
u q u q O q
u q u q O q
j j l j
j j l j
+
®
+
± ±
®
» + +
» + +
q
j
CI
31
The
Pole
and the
Quantization
For the
DP
to be a
Pole
:
In such a case:
Recalling Quantization (Berry Phase):
(
)
(
)
1
1
0
lim
m m
j e j j
q
H q O q
z z h j
j
+
±
®
¶
» +
¶
(
)
(
)
(
)
(
)
(
)
1 1
0
1
lim
,
j
jj jj
q
j j
q f
j
h j
j j
l j l j
± ±
®
±
= =
-
(
)
(
)
2
1 1 1
0
( )
jj jj jj
f d n
p
a j j p
± ± ±
G = = G
ò
32
Stokes Theorem
The (Abelian)
2
㈠
䍵牬r䍯湤楴n潮
:
There is an unresolved issue at q=
0
: We have to
define
F
at q=
0
to fulfill
Stokes
Theorem.
Stokes theorem Asserts that
:
1
1
1
1
0
jj
qjj
q jj
q q
j
j
j
+
+
+
æ ö
¶
¶
÷
ç
= - =
÷
ç
÷
ç
¶ ¶
è ø
F
M. Maer, Chem. Phys. Lett.
349
,
149
(
2001
)
Vertesi, Vibok, Halasz, Yahalom, Englman and Baer, J. Phys. Chem. A
107
,
7189
(
2003
)
q
j
CI
S
[
]
1 1
(
)
jj jj
d d
s
s
s
s
+ +
G
Q = × = ×
òò ò
F n s
Ò Ñ
33
Stokes Theorem (Cont.)
The line integral Fulfills
Bohr
-
Sommerfeld
Quantization law
F
has to be extended to have a value at the
DP
point:
To fulfill the Stokes Theorem must obey
(
)
1 1 1
( )
2
q jj q jj jj
q
f
q
j j
d
p j
+ + +
× Þ × +
F n F n
%
(
)
(
)
(
)
2
1 1 1
0
1
;
jj jj jj
f d n f
p
j
j j j j
p
+ + +
= ± º
ò
% %
[
]
2
1 1
0
( | )
jj jj
d d n
p
j
s
j j p
+ +
G
× = G = ±
ò ò
s
Ñ
( )
f
j
%
34
Close vicinity of
DP
(Example: solving
the Curl Equation)
Having the
Curl equation
We derive the
angular
component of
:
Thus
near a
DP
(
q
0
):
(
)
(
)
1
1
1
1
2
jj
qjj
jj
q
f
q q q
j
d
p j
j
+
+
+
æ ö
¶
¶
÷
ç
- =
÷
ç
÷
ç
¶ ¶
è ø
(
)
(
)
(
)
1
1 1
0
,'
,
q
qjj
jj jj
q
q dq f
j
j
j p j
j
+
+ +
¶
¢
- =
¶
ò
(
)
(
)
jj 1
,0,0
jj
f q
p j
+
Will be taken as
boundary values
around
each
DP
35
36
The Angular NACT for H+H
2
37
Summary
(what did we achieve so far?)
The
Curl
-
Div
Equations fulfilled by the NACTs
can be applied to derive the related
fields
(just
like the
Maxwell
Equations are applied to
calculate the electro
-
magnetic fields)
The
boundary conditions
needed to calculate
these fields are formed at the close vicinity of the
DP
s and are obtained from Ab
-
initio calculations
using given packages (MOLPRO).
Chapter III
Vector Algebra to Form
Two
-
State Quantum Fields
39
Vector Algebra for
Abelian
Systems
Expression for a single
ci
as seen from a given
origin
Where:
1
(,) ( ) sin( )
(,) ( ) cos( )
τ
τ
q j j j
j
j j j
j
q f
q
q
q f
q
j
j j j j
j j j j
2 2
0 0 0 0
0 0
( cos cos ) ( sin sin )
cos cos
cos
j j j j j
j j
j
j
q q q q q
q q
q
j j j j
j j
j
40
For Several
DP
s
The NACT field formed at N
DP
s:
1
1
1
(,) ( ) sin( )
1
(,) ( ) cos( )
τ
τ
N
q j j j
j
j
N
j j j
j
j
q f
q
q q f
q
j
j j j j
j j j j
J. Avery, M. Baer and G.D. Billing, Molec. Phys.
100
,
1011
(
2002
)
41
NaH
2
:
Abelian
NACTs at Four DP
A. Vibok, T. Vertesi, E. Bene, G.J. Halasz and M. Baer, J.
Phys. Chem. A.
108
,
8590
(
2004
)
42
Ab
-
initio
vs.
Vector
Algebra
(NaH
2
) as
calculated along
four different contours
A.
Vibok, T. Vertesi, E. Bene, G.J. Halasz
and M. Baer, J. Phys. Chem. A.
108
,
8590
(
2004
)
Vibok, Vertesi,Bene,Halasz,
Baer, J. Phys. Chem. A
108
,
8590
(
2004
)
Chapter IV
Field Equations to Form
Multi
-
State Quantum Fields
44
Field Theory to derive NACTs
The Born
-
Oppenheimer NACTs are
:
Vector
-
fields
formed
by
sources
located at
DP
s.
Treated theoretically (and numerically) employing
field theory
.
Methodology:
Calculate Abelian NACTs,
jj+
1
formed, by states j and j+
1
, due to a single
DP
along a small contour surrounding this (j,j+
1
)
DP
(e.g. MOLPRO).
At larger regions, due to the existence of
DP
s
formed by
other states,
the system
becomes non
-
Abelian
The N
丠
-
浡m物砠桩捨晵汦楬汳l瑨攠湯n
-
A扥b楡渠
䍵牬
-
䕱畡E楯i
慮a瑨攠湯n
-
A扥汩慮b
䑩
-
䕱畡E楯渠
楳i潢瑡楮敤e批獯汶楮i瑨敳攠敱e慴楯湳n
This
calculated
matrix contains the non
-
Abelian
Quantum Fields
.
45
The Curl Equation for a
3
-
state System
23 13
12
Curl
τ τ τ
13 12
23
Curl
τ τ τ
12 23
13
Curl
τ τ τ
Curl
τ τ τ
M. Baer, A. M. Mebel and G.D. Billing, Int. J. Quant. Chem.
90
,
1577
(
2002
)
46
A
3
-
state system
(
Ab
-
initio
evidence for the Curl Eq.
)
:
12 13
12 23
13 23
0
τ τ
τ 0 τ
τ τ 0
τ
13
13
12 23 12 23
q
q q
q
j
j j
j
τ
τ
τ τ τ τ
Example: Forming Curl
13
in
two different ways:
(1)
By differentiation of
13
(2)
By forming the vector
product
Vertesi, Vibok, Halasz,
Baer, J. Chem. Phys.
120
,
8420
(
2004
)
47
The Div
-
Equation
for a
3
-
state System
(2)
2
τ τ τ
(2)
12 12 23 13
Div
τ τ τ τ
(2)
23 23 13 12
Div
τ τ τ τ
(2)
13 13 12 23
Div
τ τ τ τ
Vertesi,
Vibok, Halasz, M. Baer, J. Chem. Phys.
121
,
4000
(
2004
)
48
The Poisson Equations
Curl
-
Div Equations for each one of the NACTs
:
Decoupling of the two components
1
F (,)
C
q
q q
q q
j
j
j
τ
τ
D
1
F (,)
q
q q
q q
j
j
j
τ
τ
2 2
2 2 2
1 1
F (,)
τ τ τ
q
q q
q q
j j j
j
j
j
2 2
2 2 2
1 1
F (,)
τ τ τ
q q q
q
q
q q
q q
j
j
Vertesi, Vibok, Halasz, Baer, J. Chem. Phys.
121
,
4000
(
2004
)
49
Molecular Fields for
the H+H
2
System
50
51
Summary
We showed that NACTs are created at
degeneracy points
and their spatial distribution
can be derived by solving
Maxwell Equations
.
Consequently the NACTs are fields and we
suggest to call them
Quantum Fields
.
It is not clear if these fields are related to the
electromagnetic fields but, if so, they should be
termed as
Weak Electro
-
Magnetic Fields
.
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