Berlin Lecture - The Fritz Haber Center for Molecular Research

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16 Νοε 2013 (πριν από 3 χρόνια και 9 μήνες)

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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


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



-
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
  
  

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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
.