Born-OppenheimerCoupling 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¶ ¶ ÷çÑ = ÷ç ÷ç¶ ¶è ø

L

5

Four Coupled Potential Surfaces

2 2

Conical Intersections for

the 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 Nz =s s

( ) ( )1

ˆ | |N

j e j ej

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¶ ¶ ÷çÑ = ÷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= +Ds s s%

( ) ( ) ( ) | |ik i e k ez z= Ñs s s s s

( ) ( ) ( ) ( )| | | |N

k e j e j e k ej

z z z zÑ = Ñås s s s s s s s

( ) ( ) ( )| | |k e k e k ez z z+D = +D ×Ñs 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 edz zé ù= - ×òê úë ûs

ss s s s s s s

R. Baer, J. Chem. Phys. 117, 405 (2002).

( )s s

( ) ( ) ( )| |e ez zÑ = -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 edz zG

=Ã - ×òs s s s s s sÑ

( ) ( ) ( )0 0 0| | exp |j e j j eiz q z=s s s s s

sisf

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 jkjkj k Nd iq d

G=é ùG = - × =òë ûD s sÑ

( ) ( )( )exp dG

G =Ã - ×òD s sÑ

15

Quantization Conditions In the case of real eigenfunctions:

In case of two states (N=2):

the D-matrix can be written as:

( )jk jkdG = ±D

A. Alijah and M. Baer, Chem. Phys. Lett. 319,489(2000)

12

æ ö÷ç ÷ç ÷ç ÷÷çè ø

( )( ) ( )

( ) ( )

cos sin

sin cos

a a

a a

æ öG G ÷ç ÷çG = ÷ç ÷ç- G G ÷çè øD

16

Bohr-Sommerfeld Quantization Condition

() is the Topological (Berry) phase:

The 22 D()-matrix becomes diagonal if:

21( ) daG

G = ×ò sÑ

( ) ; 0, 1, 2,...n na pG = = ± ±

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 ττ 0 τ ττ τ 0 τ

0τ τ τ 0

0 τ ττ 0 ττ τ 0

00

0

τ

18

How to detect NACTs?

Halasz, Vibok, Baer, CPL 413,226(2005)

2 2The C H Molecule

19

NH2: 2-state resultsVibok, Halasz, Suhai,Hoffman, Kouri, Baer,J. Chem. Phys. 124, 024312 (2006)

20

NH2: 3-state results Vibok, Halasz, Suhai,Hoffman, Kouri, Baer,JCP 124, 024312 (2006)

21

H+H2 3-states

Halasz, Vibok, Mebel, BaerJCP 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

, 0p q q p

q ppq p qp q

= -

¶ ¶ é ù= - - =ë û¶ ¶F

1442443

M. Baer, Chem. Phys. Lett. 35, 112 (1975)

23

The Curl Equation (continued)

Abelian vs non-Abelian variables Commutation Relation

The -matrix is, in general, non-Abelian

A 22 -matrix is Abelian

12

æ ö÷ç ÷ç ÷ç ÷÷çè ø

p q q pτ 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 ÷÷çè ø

1212

12, 0 F 0yxxyx y y x

¶¶é ù= Þ = - =ë û ¶ ¶

12 1212 1212

1F 0 0q qq q q q

j jj j j

æ ö¶ ¶¶ ¶÷ç= - = Þ - =÷ç ÷ç ¶ ¶ ¶ ¶è ø

q

CI

25

The Divergence Equation

The eigenfunctions of a Hilbert space form also a Divergence equation

where

In polar coordinates:

( ) DivÑ× - ×

q q

q q qj

j

¶ ¶Ñ × + +

¶ ¶

( ) 2

i jijz zº Ñ

q

CI

26

Div(s) for a Two-State System

In Cartesian Coordinates

The 2) matrix elements are:

In contrast to -matrix elements, they are scalars

1212Div yx

x y

¶¶= +

¶ ¶

( )

( ) ( )

212

2 2 211 22 12

=Div

= = -

Chapter II

Degeneracy PointsAs

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 kjk

k j

H

u u

z zÑ=

-

1k 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

11

1 1e

j j

jjj j

H

q q u uj

z zj ±

±±

¶¶=-

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

100

11 0 10

lim

lim

,

,

m mj j jq

m mj j jq

u q u q O q

u q u q O q

j j l j

j j l j

+

®

+± ±®

» + +

» + +

q

CI

31

The Pole and the Quantization

For the DP to be a Pole:

In such a case:

Recalling Quantization (Berry Phase):

( ) ( )110

lim m mj e j jq

H q O qz z h jj

+±®

¶» +

¶

( )( )

( ) ( )( ) 1 10

1

lim , jjj jjq

j j

q fj

h jj j

l j l j± ±®±

= =-

( ) ( )2

1 1 10

( )jj jj jjf d np

a j j p± ± ±G = = Gò

32

Stokes Theorem

The (Abelian) 22 Curl Condition:

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 11

10jj qjj

q jj q qj

j j+ +

+

æ ö¶ ¶ ÷ç= - =÷ç ÷ç ¶ ¶è øF

M. Maer, Chem. Phys. Lett. 349,149 (2001)Vertesi, Vibok, Halasz, Yahalom, Englman and Baer, J. Phys. Chem. A 107, 7189 (2003)

q

CI

[ ]

1 1( ) jj jjd ds

s

ss + +

G

Q = × = ×òò òF n sÒ 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

( )2q jj q jj jj

qf

qj j

dp j+ + +× Þ × +F n F n %

( ) ( ) ( )2

1 1 10

1;jj jj jjf d n f

p

jj j j jp+ + += ± ºò % %

[ ]

2

1 1

0

( | )jj jjd d np

j

s

j j p+ +

G

× = G = ±ò òsÑ

( )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 (q0):

( )( ) 1 1

1

12jj qjj

jj

qf

q q qj d

p jj

+ ++

æ ö¶ ¶ ÷ç - =÷ç ÷ç ¶ ¶è ø

( )( )

( )

11 1

0

, ',

qqjj

jj jj

qq dq fj

jj p j

j+

+ +

¶¢- =

¶ò

( )( ) jj 1 ,0,0jjf qp j+

Will be taken as boundary values around each DP

35

36

The Angular NACT for H+H2

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 DPs 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 jj

j j jj

q fq

qq f

q

2 2

0 0 0 0

0 0

( cos cos ) ( sin sin )

cos coscos

j j j j j

j j

j

j

q q q q q

q q

q

40

For Several DPs

The NACT field formed at N DPs:

1

1

1( , ) ( ) sin( )

1( , ) ( ) cos( )

τ

τ

N

q j j jj j

N

j j jj j

q fq

q q fq

J. Avery, M. Baer and G.D. Billing, Molec. Phys. 100, 1011 (2002)

41

NaH2: 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 (NaH2) as calculated alongfour 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 DPs.

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 DPs formed by other states, the system becomes non-Abelian

The NN -matrix which fulfills the non-Abelian Curl-Equation and the non-Abelian Div-Equation is obtained by solving these equations.

This calculated matrix contains the non-Abelian Quantum Fields.

45

The Curl Equation for a 3-state System

23 1312Curl

τ τ τ

13 1223Curl

τ τ τ

12 2313Curl

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

1313

12 23 12 23

q

q q

q

ττ

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) 2τ τ τ

(2)12 12 23 13

Div τ τ τ τ(2)

23 23 13 12Div 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

1F ( , )C

q q qq q

ττ

D

1F ( , )q q q

q q

ττ

2 2

2 2 2

1 1F ( , )

τ τ τq

q qq q

2 2

2 2 2

1 1F ( , )

τ τ τq q q

qq

q qq q

Vertesi, Vibok, Halasz, Baer, J. Chem. Phys. 121,4000 (2004)

49

Molecular Fields for the H+H2 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.