Berlin Lecture - The Fritz Haber Center for Molecular Research

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
.