This article was downloaded by: [Indian Instiute of Technology Mandi]On: 23 July 2014, At: 06:06Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Molecular Physics: An International Journal at theInterface Between Chemistry and PhysicsPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/tmph20

Multi-channel scattering problems: an analyticallysolvable modelDiwaker a & Aniruddha Chakraborty aa School of Basic Sciences , Indian Institute of Technology Mandi , Mandi , Himachal Pradesh175001 , IndiaAccepted author version posted online: 16 Mar 2012.Published online: 27 Apr 2012.

To cite this article: Diwaker & Aniruddha Chakraborty (2012) Multi-channel scattering problems: an analytically solvablemodel, Molecular Physics: An International Journal at the Interface Between Chemistry and Physics, 110:18, 2257-2267, DOI:10.1080/00268976.2012.674569

To link to this article: http://dx.doi.org/10.1080/00268976.2012.674569

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Molecular PhysicsVol. 110, No. 18, September 2012, 2257–2267

RESEARCH ARTICLE

Multi-channel scattering problems: an analytically solvable model

Diwaker and Aniruddha Chakraborty*

School of Basic Sciences, Indian Institute of Technology Mandi, Mandi, Himachal Pradesh 175001, India

(Received 29 January 2012; final version received 2 March 2012)

We have proposed a general method for finding an exact analytical solution for the multi-channel scatteringproblem in the presence of a delta function coupling. Our solution is quite general and is valid for any set ofpotentials, if the uncoupled diabatic potential has an exact solution. We have also discussed a few examples,where our method can easily be applied.

Keywords: quantum mechanics; scattering; multi-channel; two state; analytical model; Green’s function

1. Introduction

Nonadiabatic transition due to potential curve crossing

is one of the most important mechanisms to effectivelyinduce electronic transitions in collisions [1–14]. This is

a very interdisciplinary concept and appears in various

fields of physics, chemistry and even in biology

[2,3,5,7,15]. The theory of non-adiabatic transitions

dates back to 1932, when the pioneering works for

curve-crossing and non-crossing were published byLandau [16], Zener [17], Stuckelberg [18] along with

Rosen and Zener [19] respectively. Two categories in

general can be classified be classified for finding an

exact analytical solution of the curve crossing problem.

The first is that an exact analytical solution can be

obtained for the whole region of the variable (say x

here, see in the next section). For example, Osherovand Voronin solved the case where two diabatic

potentials are constant with exponential coupling

[20]. Zhu solved the case where two diabatic potentials

are exponential with exponential coupling [21]. In our

earlier publications we have reported analytical solu-

tion in those cases where two or more arbitrarypotentials are coupled by Dirac Delta interactions

[15,22–26]. The second is that an exact analytical

solution is only possible for the asymptotic region.

Then, physical quantities such as eigenvalues, scatter-

ing matrices can still be solved in an exact analytical

form, providing that the connection problem of theasymptotic solution is known. The Stokes phenom-

enon [27] of the asymptotic solution of the ordinary

differential equation provides a powerful tool to deal

with these kinds of problems [28–30]. Generalizing thereal variable to the complex variable and tracing theasymptotic solution around the complex plane, theconnection matrix which connects the asymptoticsolution in the complex plane can be expressed interms of Stokes constants. Recent work by Zhu andNakamura [30] found an exact analytical solution ofthe Stokes constants for the second-order ordinarydifferential equation with the coefficient function asthe fourth-order polynomial. In this way, exactanalytical solutions of scattering matrices wereobtained for the two state linear curve crossingproblem with constant coupling [31]. In this paper weconsider the case of two or more arbitrary diabaticpotentials with Dirac Delta couplings. The Dirac Deltacoupling model has the advantage that it can be exactlysolved [15,22–24] if the uncoupled diabatic potentialhas an exact solution.

2. Our model

We consider two diabatic curves, crossing each other.There is a coupling between the two curves, whichcauses transitions from one curve to another. Thistransition would occur in the vicinity of the crossingpoint. In particular, it will occur in a narrow range ofx, given by

V1ðxÞ � V2ðxÞ ’ V12ðxcÞ, ð1Þ

where x denotes the nuclear coordinate and xc is thecrossing point. V1 and V2 are determined by the shape

*Corresponding author. Email: aniruddha_oregon@yahoo.com

ISSN 0026–8976 print/ISSN 1362–3028 online

� 2012 Taylor & Francis

http://dx.doi.org/10.1080/00268976.2012.674569

http://www.tandfonline.com

Dow

nloa

ded

by [

Indi

an I

nstiu

te o

f T

echn

olog

y M

andi

] at

06:

06 2

3 Ju

ly 2

014

of the diabatic curves and V12 represents the couplingbetween them. Therefore it is interesting to analyse amodel where coupling is localized in space near xc.Thus we put

V12ðxÞ ¼ K0�ðx� xcÞ, ð2Þ

where K0 is a constant.

3. Formulation of multi-channel scattering problems

3.1. Two channel scattering problems

We start with a particle moving on any of the twodiabatic curves and the problem is to calculate theprobability of the particle to be still on that diabaticcurve after a time t. We write the probability amplitudefor the particle as

CðxÞ ¼ 1ðxÞ 2ðxÞ

� �, ð3Þ

where 1(x) and 2(x) are the probability amplitudesfor the two states. The Hamiltonian is given by

H ¼H11ðxÞ V12ðxÞV21ðxÞ H22ðxÞ

� �, ð4Þ

where H11(x), H22(x) and V12(x) are defined by

H11ðxÞ ¼ ��h2

2m

@2

@x2þ V1ðxÞ,

H22ðxÞ ¼ ��h2

2m

@2

@x2þ V2ðxÞ,

V12ðxÞ ¼ V21ðxÞ ¼ K0�ðx� xcÞ: ð5Þ

The above V1(x) and V2(x) are determined by theshape of that diabatic curve. V12(x) is a couplingfunction which we assume to be a Dirac delta function.The time-independent Schrodinger equation is writenin the matrix form

H11ðxÞ V12ðxÞ

V21ðxÞ H22ðxÞ

� � 1ðxÞ

2ðxÞ

� �¼ E

1ðxÞ

2ðxÞ

� �: ð6Þ

This is equivalent to

H11ðxÞ 1ðxÞ þ K0�ðx� xcÞ 2ðxÞ ¼ E 1ðxÞ,

K0�ðx� xcÞ 1ðxÞ þH22ðxÞ 2ðxÞ ¼ E 2ðxÞ:ð7Þ

Integrating the above two equations from xc� � toxcþ � (where �! 0) we get the following twoboundary conditions

��h2

2m

d 1ðxÞ

dx

� �xcþ�xc��

þK0 2ðxcÞ ¼ 0,

��h2

2m

d 2ðxÞ

dx

� �xcþ�xc��

þK0 1ðxcÞ ¼ 0: ð8Þ

Also we have two more boundary conditions

1ðxc � �Þ ¼ 1ðxc þ �Þ,

2ðxc � �Þ ¼ 2ðxc þ �Þ: ð9Þ

Using the above four boundary conditions one can

derive the transition probability from one diabatic

potential to the other.

3.2. Three channel scattering problems

Here we start with a particle moving on any of the

three diabatic curves and the problem is to calculate

the probability that the particle will still be in that

diabatic curve after a time t. We write the probability

amplitude for the particle as

CðxÞ ¼

1ðxÞ

2ðxÞ

3ðxÞ

0B@

1CA, ð10Þ

where 1(x), 2(x) and 3(x) are the probability

amplitudes for the three states. The Hamiltonian

matrix of this system is given by

H ¼

H11ðxÞ V12ðxÞ V13ðxÞ

V21ðxÞ H22ðxÞ 0

V31ðxÞ 0 H33ðxÞ

0B@

1CA, ð11Þ

where H11(x), H22(x), H33(x), V12(x), V21(x), V31(x)

and V13(x) are defined by

H11ðxÞ ¼ ��h2

2m

@2

@x2þ V1ðxÞ,

H22ðxÞ ¼ ��h2

2m

@2

@x2þ V2ðxÞ,

H33ðxÞ ¼ ��h2

2m

@2

@x2þ V3ðxÞ,

V12ðxÞ ¼ V21ðxÞ ¼ K2�ðx� x2Þ,

V13ðxÞ ¼ V31ðxÞ ¼ K3�ðx� x3Þ: ð12Þ

In the above equations V1(x), V2(x) and V3(x) are

determined by the shape of the diabatic curve. The

time-independent Schrodinger equation for this pro-

blem is given by

H11ðxÞ K2�ðx� x2Þ K3�ðx� x3Þ

K2�ðx� x2Þ H22ðxÞ 0

K3�ðx� x3Þ 0 H33ðxÞ

0B@

1CA

1ðxÞ

2ðxÞ

3ðxÞ

0B@

1CA

¼ E

1ðxÞ

2ðxÞ

3ðxÞ

0B@

1CA: ð13Þ

2258 Diwaker and A. Chakraborty

Dow

nloa

ded

by [

Indi

an I

nstiu

te o

f T

echn

olog

y M

andi

] at

06:

06 2

3 Ju

ly 2

014

This matrix representation is equivalent to the

following three equations

H11ðxÞ 1ðxÞ þ K2�ðx� x2Þ 2ðxÞ

þ K3�ðx� x3Þ 3ðxÞ ¼ E 1ðxÞ,

K2�ðx� x2Þ 1ðxÞ þH22ðxÞ 2ðxÞ ¼ E 2ðxÞ,

K3�ðx� x3Þ 1ðxÞ þH33ðxÞ 3ðxÞ ¼ E 3ðxÞ:

ð14Þ

Now for simplicity, we consider the case where

K2¼K3¼K0 and x2¼x3¼xc. Then integrating the

above three equations from xc� � to xcþ � (where

�! 0) we get the following three boundary conditions

��h2

2m

d 1ðxÞ

dx

� �xcþ�xc��

þK0 2ðxcÞ þ K0 3ðxcÞ ¼ 0,

��h2

2m

d 2ðxÞ

dx

� �xcþ�xc��

þK0 1ðxcÞ ¼ 0,

��h2

2m

d 3ðxÞ

dx

� �xcþ�xc��

þK0 1ðxcÞ ¼ 0: ð15Þ

Also we have three more boundary conditions

1ðxc � �Þ ¼ 1ðxc þ �Þ,

2ðxc � �Þ ¼ 2ðxc þ �Þ,

3ðxc � �Þ ¼ 3ðxc þ �Þ: ð16Þ

Using the above six boundary conditions one can

derive analytical expressions for the transition prob-

ability from one diabatic potential to the other.

3.3. N channel scattering problems

Here we start with a particle moving on any of the N

diabatic curves and the problem is to calculate the

probability that the particle will still be in the diabatic

curve after a time t. We write the probability amplitude

for the particle as

CðxÞ ¼

1ðxÞ

2ðxÞ

3ðxÞ

:

:

:

NðxÞ

0BBBBBBBBBBB@

1CCCCCCCCCCCA, ð17Þ

where n(x) is the probability amplitude

for the nth state and n can take any value from 1

to N. The Hamiltonian matrix of this system is

given by

H ¼

H11ðxÞ V12ðxÞ V13ðxÞ : : V1NðxÞ

V21ðxÞ H22ðxÞ 0 0 0 0

V31ðxÞ 0 H33ðxÞ 0 0 0

: : : : : :

: : : : : :

VN1ðxÞ 0 0 0 0 HNNðxÞ

0BBBBBBBB@

1CCCCCCCCA,

ð18Þ

where Hnn(x), V1n(x) and Vn1(x) are defined by

HnnðxÞ ¼ ��h2

2m

@2

@x2þ VnðxÞ,

V1nðxÞ ¼ Vn1ðxÞ ¼ Kn�ðx� xnÞ: ð19Þ

The time-independent Schrodinger equation for this

problem is given by

H11ðxÞ K2�ðx�x2Þ K3�ðx�x3Þ : : KN�ðx�xNÞ

K2�ðx�x2Þ H22ðxÞ 0 0 0 0

K3�ðx�x3Þ 0 H33ðxÞ 0 0 0

: : : : : :

: : : : : :

KN�ðx�xNÞ 0 0 0 0 HNNðxÞ

0BBBBBBBB@

1CCCCCCCCA

�

1ðxÞ

2ðxÞ

3ðxÞ

:

:

:

NðxÞ

0BBBBBBBBBBB@

1CCCCCCCCCCCA¼E

1ðxÞ

2ðxÞ

3ðxÞ

:

:

:

NðxÞ

0BBBBBBBBBBB@

1CCCCCCCCCCCA: ð20Þ

This matrix representation is equivalent to the follow-

ing equations

H11ðxÞ 1ðxÞ þXn¼Nn¼2

Kn�ðx� xnÞ nðxÞ ¼ E 1ðxÞ,

Kn�ðx� xnÞ 1ðxÞ þHnnðxÞ nðxÞ ¼ E nðxÞ: ð21Þ

Now for simplicity, we consider the case where

K2¼K3¼ � � � ¼KN¼K0 and x2¼x3 � � � ¼ xN¼ xc.

Integrating the above equations from xc� � to xcþ �(where �! 0) we get the following N boundary

conditions

��h2

2m

d 1ðxÞ

dx

� �xcþ�xc��

þK0

Xn¼Nn¼2

nðxcÞ ¼ 0,

��h2

2m

d nðxÞ

dx

� �xcþ�xc��

þK0 1ðxcÞ ¼ 0: ð22Þ

Molecular Physics 2259

Dow

nloa

ded

by [

Indi

an I

nstiu

te o

f T

echn

olog

y M

andi

] at

06:

06 2

3 Ju

ly 2

014

Also we have N more boundary conditions

nðxc � �Þ ¼ nðxc þ �Þ ð23Þ

Using all 2N boundary conditions one can derive

analytical expressions for the transition probability

from one diabatic potential to the other.

4. Multi-channel scattering problems: Green’s

function method

4.1. Two channel scattering problems

We start with the time-independent Schrodinger

equation for a two state system

H11ðxÞ V12ðxÞ

V21ðxÞ H22ðxÞ

� � 1ðxÞ

2ðxÞ

� �¼ E

1ðxÞ

2ðxÞ

� �: ð24Þ

The above two equations can be written in the

following form

2ðxÞ ¼ V12ðxÞ�1 E�H11ðxÞ½ � 1ðxÞ,

2ðxÞ ¼ E�H22ðxÞ½ ��1V21ðxÞ 1ðxÞ: ð25Þ

Eliminating 2(x) from the above two equations we get

E�H11ðxÞ½ � 1ðxÞ � V12ðxÞ E�H22ðxÞ½ ��1

V21ðxÞ 1ðxÞ ¼ 0: ð26Þ

The above equations simplify considerably if V12(x)

and V21(x) are Dirac Delta functions at xc, which in

operator notation may be written as V¼K0S¼K0jxci

hxcj. The above equation now becomes

E�H11ðxÞ½ � � K20jxcihxcj E�H22ðxÞ½ �

�1jxcihxcj

� � 1ðxÞ ¼ 0: ð27Þ

This may be written as

H11ðxÞ þ K20�ðx� xcÞG

02ðxc, xc;EÞ

� 1ðxÞ ¼ E 1ðxÞ:

ð28Þ

The above equation can be written in the following

form

E�H11½ � 1ðxÞ ¼ K20�ðx� xcÞG

02ðxc, xc;EÞ 1ðxÞ, ð29Þ

where the right-hand side is considered as an inhomo-

geneous term. The general solution of this equation

can be written as

1ðxÞ ¼ 0ðxÞ þ K20G

01ðx,xc;EÞG

02ðxc, xc;EÞ 1ðxcÞ,

ð30Þ

where 0(x) is a solution of the homogeneousequations

E�H11½ � 0ðxÞ ¼ 0, E�H11½ �G01ðx, x

0;EÞ ¼ �ðx� x0Þ:

ð31Þ

In the above equations

1ðxÞ ¼

Z 1�1

dx0Gðx,x0;EÞ 0ðx0Þ,

0ðxÞ ¼

Z 1�1

dx0G01ðx, x

0;EÞ 0ðx0Þ: ð32Þ

So Equation (30) can be written asZ 1�1

dx0Gðx,x0;EÞ 0ðx0Þ ¼

Z1�1

dx0G01ðx,x

0;EÞ 0ðx0Þ

þK20G

01ðx,xc;EÞG

02ðxc,xc;EÞ

�

Z 1�1

dx0Gðxc,x0;EÞ 0ðx

0Þ:

ð33Þ

The solution in terms of Green’s function is as follows

Gðx, x0;EÞ ¼ G01ðx, x

0;EÞ þ K20G

01ðx, xc;EÞG

02ðxc, xc;EÞ

� Gðxc, x0;EÞ: ð34Þ

In the above equation we put x¼ xc

Gðxc,x0;EÞ ¼ G0

1ðxc, x0;EÞ þ K2

0G01ðxc, xc;EÞ

� G02ðxc, xc;EÞGðxc, x

0;EÞ: ð35Þ

After simplification, we get

Gðxc,x0;EÞ ¼

G01ðxc, x

0;EÞ

1� K20G

01ðxc, xc;EÞG

02ðxc, xc;EÞ

, ð36Þ

so that

Gðx, x0;EÞ ¼ G01ðx, x

0;EÞ

þK2

0G01ðx, xc;EÞG

02ðxc, xc;EÞG

01ðxc, x

0;EÞ

1� K20G

01ðxc,xc;EÞG

02ðxc, xc;EÞ

:

ð37Þ

Using this expression of G(x, x0;E) one can calculatethe wave function and from the wave function one caneasily calculate the transition probability from onediabatic potential to the other.

4.2. Three channel scattering problems

We start with the time-independent Schrodingerequation for a three state system, given by

H11ðxÞ V12ðxÞ V13ðxÞ

V21ðxÞ H22ðxÞ 0

V31ðxÞ 0 H33ðxÞ

0B@

1CA

1ðxÞ

2ðxÞ

3ðxÞ

0B@

1CA ¼ E

1ðxÞ

2ðxÞ

3ðxÞ

0B@

1CA:ð38Þ

2260 Diwaker and A. Chakraborty

Dow

nloa

ded

by [

Indi

an I

nstiu

te o

f T

echn

olog

y M

andi

] at

06:

06 2

3 Ju

ly 2

014

This matrix equation can be written in the following

form

H11ðxÞ � E½ � 1ðxÞ þ V12ðxÞ 2ðxÞ þ V13ðxÞ 3ðxÞ ¼ 0,

H22ðxÞ � E½ � 2ðxÞ þ V21ðxÞ 1ðxÞ ¼ 0,

H33ðxÞ � E½ � 3ðxÞ þ V31ðxÞ 1ðxÞ ¼ 0:

ð39Þ

The above equation after rearranging is given below

E�H11ðxÞ½ � 1ðxÞ � V12ðxÞ 2ðxÞ � V13ðxÞ 3ðxÞ ¼ 0,

ð40Þ

2ðxÞ ¼ E�H22ðxÞ½ ��1V21ðxÞ 1ðxÞ, ð41Þ

3ðxÞ ¼ E�H33ðxÞ½ ��1V31ðxÞ 1ðxÞ: ð42Þ

After eliminating both 2(x) and 3(x) from

Equation (40) we get�E�H11ðxÞ½ � � V12ðxÞ E�H22ðxÞ½ �

�1V21ðxÞ � V13ðxÞ

E�H33ðxÞ½ ��1V31ðxÞÞ 1ðxÞ ¼ 0: ð43Þ

The above equations are true for any general V12, V21,

V13 and V31. The above equations simplify consider-

ably if V12, V13, V31 and V21 are Dirac Delta functions,

which we write in operator notation as V12¼V21¼

K2S¼K2jx2ihx2j and V13¼V31¼K3S¼K3jx3ihx3j.

The above equation now becomes�E�H11ðxÞ½ � � K2

2�ðx� x2ÞG02ðx2, x2;EÞ � K2

3�ðx� x3Þ

� G02ðx3, x3;EÞ

� 1ðxÞ ¼ 0: ð44Þ

This may be written as

E�H12ðxÞ½ � � K23�ðx� x3ÞG

03ðx3,x3;EÞ

� � 1ðxÞ ¼ 0,

ð45Þ

where

H12ðxÞ ¼ H11ðxÞ þ K22�ðx� x2ÞG

02ðx2, x2;EÞ: ð46Þ

For H12(x), one can find the corresponding Green’s

function G12(x, x0;E) using the method as we have used

in the two-state case.

E�H11½ � 1ðxÞ ¼ K22�ðx� x2ÞG

02ðx2, x2;EÞ 1ðxÞ, ð47Þ

where the right-hand side is considered as an inhomo-

geneous term. The general solution of this equation

can be written as

1ðxÞ ¼ 0ðxÞ þ

Z 1�1

dxG01ðx, x

0;EÞK22�ðx

0 � x2Þ

� G02ðx2, x2;EÞ 1ðx

0Þ, ð48Þ

where 0(x) is a solution of the homogeneous equation

ðE�H11Þ 0ðxÞ ¼ 0, ð49Þ

where

ðE�H11ÞG01ðx,x

0;EÞ ¼ �ðx� x0Þ: ð50Þ

So

1ðxÞ ¼ 0ðxÞ þ K22G

01ðx,x2;EÞG

02ðx2,x2;EÞ 1ðx2Þ:

ð51Þ

In the above expression

1ðxÞ ¼

Z 1�1

dx0G12ðx, x0;EÞ 0ðx

0Þ,

0ðxÞ ¼

Z 1�1

dx0G01ðx, x

0;EÞ 0ðx0Þ: ð52Þ

So Equation (51) can be written asZ 1�1

dx0G12ðx, x0;EÞ 0ðx

0Þ

¼

Z 1�1

dx0G01ðx, x

0;EÞ 0ðx0Þ

þ K22G

01ðx, x2;EÞG

02ðx2, x2;EÞ

�

Z 1�1

dx0G12ðx2, x0;EÞ 0ðx

0Þ: ð53Þ

The solution in terms of Green’s function, extracted

from the last equation

G12ðx, x0;EÞ ¼ G0

1ðx, x0;EÞ þ K2

2G01ðx, x2;EÞ

� G02ðx2, x2;EÞG12ðx2, x

0;EÞ: ð54Þ

In the above equation we put x¼ x2 to get

G12ðx2, x0;EÞ ¼ G0

1ðx2, x0;EÞ þ K2

2G02ðx2, x2;EÞ

� G01ðx2, x2;EÞG12ðx2, x

0;EÞ: ð55Þ

So after simplification we get

G12ðx2, x0;EÞ ¼

G01ðx2, x

0;EÞ

1� K22G

01ðx2, x2;EÞG

02ðx2, x2;EÞ

,

ð56Þ

so that

G12ðx,x0;EÞ ¼G0

1ðx,x0;EÞ

þK2

2G01ðx,x2;EÞG

02ðx2,x2;EÞG

01ðx2,x

0;EÞ

1�K22G

01ðx2,x2;EÞG

02ðx2,x2;EÞ

:

ð57Þ

Molecular Physics 2261

Dow

nloa

ded

by [

Indi

an I

nstiu

te o

f T

echn

olog

y M

andi

] at

06:

06 2

3 Ju

ly 2

014

Now we will incorporate the effect of a third state

which is coupled to the first state only, i.e. we will solve

Equation (46) in terms of Green’s function.

G13ðx,x0;EÞ¼G12ðx,x

0;EÞ

þK2

2G12ðx,x3;EÞG02ðx3,x3;EÞG12ðx3,x

0;EÞ

1�K22G12ðx3,x3;EÞG

02ðx3,x3;EÞ

:

ð58Þ

In the above expression, we have Green’s function for

the three-state scattering problem using a delta

function coupling model. Using the above expression

one can calculate the wave function and from the wave

function one can easily calculate the transition prob-

ability from one diabatic potential to the other.

4.3. N channel scattering problems

We start with the time-independent Schrodinger

equation for a N state system, given by

H11ðxÞ V12ðxÞ V13ðxÞ : : V1NðxÞ

V21ðxÞ H22ðxÞ 0 0 0 0

V31ðxÞ 0 H33ðxÞ 0 0 0

: : : : : :

: : : : : :

VN1ðxÞ 0 0 : : HNNðxÞ

0BBBBBBBB@

1CCCCCCCCA

1ðxÞ

2ðxÞ

3ðxÞ:

:

NðxÞ

0BBBBBB@

1CCCCCCA

¼E

1ðxÞ

2ðxÞ

3ðxÞ

:

:

NðxÞ

0BBBBBBBB@

1CCCCCCCCA: ð59Þ

This matrix equation can be written in the following

form

H11ðxÞ � E½ � 1ðxÞ þXn¼Nn¼2

V1nðxÞ nðxÞ ¼ 0,

HnnðxÞ � E½ � nðxÞ þ Vn1ðxÞ 1ðxÞ ¼ 0: ð60Þ

The above equations after rearrangement are given

below as

E�H11ðxÞ½ � 1ðxÞ �XNn¼2

V1nðxÞ nðxÞ ¼ 0, ð61Þ

nðxÞ ¼ E�HnnðxÞ½ ��1Vn1ðxÞ 1ðxÞ: ð62Þ

After eliminating n(x) from Equation (61) we get

E�H11ðxÞ½ � 1ðxÞ �XNn¼2

V1nðxÞ E�HnnðxÞ½ ��1

Vn1ðxÞ 1ðxÞ ¼ 0: ð63Þ

The above equations are true for any general V1n and

Vn1. The above equation simplifies considerably if V1n

and Vn1 are Dirac Delta functions, which we write in

operator notation as V1n¼Vn1¼KnS¼Knjxnihxnj. The

above equation now becomes

E�H11ðxÞ½ ��XNn¼2

K2n�ðx�xnÞG

0nðxn,xn;EÞ

! 1ðxÞ ¼ 0:

ð64Þ

This may be written as

E�H12ðxÞ½ ��XNn¼3

K2n�ðx�xnÞG

0nðxn,xn;EÞ

! 1ðxÞ ¼ 0,

ð65Þ

where

H12ðxÞ ¼ H11ðxÞ þ K22�ðx� x2ÞG

02ðx2, x2;EÞ: ð66Þ

For H12(x), one can find the corresponding Green’s

function G12(x, x0;E) using the method as we have used

in the two-state case. Similarly we can define H13(x),

so that

E�H13ðxÞ½ ��XNn¼4

K2n�ðx�xnÞG

0nðxn,xn;EÞ

! 1ðxÞ ¼ 0,

ð67Þ

where

H13ðxÞ ¼ H12ðxÞ þ K23�ðx� x3ÞG

03ðx3, x3;EÞ: ð68Þ

For H13(x), one can find the corresponding Green’s

function G13(x, x0;E) using the method as we have used

in the two-state case. This is possible because we have

already derived analytical expression for G12(x, x0;E).

So if the analytical expression of G12(x, x0;E) is known,

it is possible to derive an analytical expression of G13(x,

x0;E) using the method we have discussed for the two

channel case. Using the same analogy, an analytical

expression of G1n(x, x0;E) (for any ‘n’) can be derived, if

we know the analytical expression for G1[n� 1](x, x0;E).

So we can define H1(N� 1)(x), in the following equation

E�H1½N�1�ðxÞ�

�K2N�ðx�xNÞG

0NðxN,xN;EÞ

� � 1ðxÞ ¼ 0,

ð69Þ

2262 Diwaker and A. Chakraborty

Dow

nloa

ded

by [

Indi

an I

nstiu

te o

f T

echn

olog

y M

andi

] at

06:

06 2

3 Ju

ly 2

014

where

H1½N�1�ðxÞ ¼ H11ðxÞ þXN�1n¼2

K2n�ðx� xnÞG

0nðxn, xn;EÞ:

ð70Þ

In this case, one can find the Green’s function

G1N(x,x0;E) using the method as we have already

used for two-channel case.

G1Nðx,x0;EÞ

¼G1½N�1�ðx,x0;EÞ

þK2

2G1½N�1�ðx,xN;EÞG0NðxN,xN;EÞG1½N�1�ðxN,x

0;EÞ

1�K22G1½N�1�ðxN,xN;EÞG0

NðxN,xN;EÞ:

ð71Þ

So it is possible to derive an analytical expression for

G1N(x, x0;E) and using G1N(x, x0;E), one can calculate

a wave function explicitly and from the wave function

one can easily calculate the transition probability from

one diabatic potential to all other diabatic potentials.

5. Exact analytical solutions: three simple cases

5.1. Constant potential case

In region 1 ( x5 xc), the time-independent Schrodinger

equation for the first potential is given by

��h2

2m

@2 1ðxÞ

@x2þ V1 1ðxÞ ¼ E 1ðxÞ: ð72Þ

The above equation has the following solution

1ðxÞ ¼ A expðik1xÞ þ B expð�ik1xÞ, ð73Þ

where k1¼ [(2m/�h2)(E�V1)]1/2. In region 2 (x4 xc),

the time-independent Schrodinger equation for the first

potential is given by

��h2

2m

@2 1ðxÞ

@x2þ V1 1ðxÞ ¼ E 1ðxÞ: ð74Þ

The physically acceptable solution of the above

equation is given by

1ðxÞ ¼ C expðik1xÞ: ð75Þ

In region 1 ( x5 xc), the time-independent Schrodinger

equation for the second potential is given by

��h2

2m

@2 2ðxÞ

@x2þ V2 2ðxÞ ¼ E 2ðxÞ: ð76Þ

The physically acceptable solution is given by

2ðxÞ ¼ D expð�ik2xÞ, ð77Þ

where k2¼ [(2m/�h2)(E�V2)]1/2. In region 2 (x4 xc),

the time-independent Schrodinger equation for the

second potential is given by

��h2

2m

@2 2ðxÞ

@x2þ V2 2ðxÞ ¼ E 2ðxÞ: ð78Þ

The physically acceptable solution is given by

2ðxÞ ¼ F expðik2xÞ: ð79Þ

Here we put xc¼ 0. Now using four boundary

conditions we calculate

F

A

2¼ imK0k1�h

2

ðk1k2�h4 þm2K20Þ

2

ð80Þ

and

D

A

2¼ imK0k1�h

2

ðk1k2�h4 þm2K20Þ

2

: ð81Þ

So, the transition probability is given by

T ¼ 2k2k1

imK0k1�h2

ðk1k2�h4 þm2K20Þ

2

: ð82Þ

In our numerical calculation we use atomic units so

that �h¼ 1. In atomic units, we set V1(x)¼ 0, V2(x)¼ 5,

K0¼ 1.0 and m¼ 1.0. The result of our calculation is

shown in Figure 1.

5.2. Linear potential case

The time-independent Schrodinger equation for the

case where a linear potential coupled to another linear

p

Figure 1. The plot of transition probability from oneconstant potential to another constant potential as a functionof energy of incident particle (K0¼ 1.0).

Molecular Physics 2263

Dow

nloa

ded

by [

Indi

an I

nstiu

te o

f T

echn

olog

y M

andi

] at

06:

06 2

3 Ju

ly 2

014

potential through a Dirac delta interaction is given

below (see Figure 2).

� �h2

2m@2

@x2þp1x K0�ðx�xcÞ

K0�ðx�xcÞ ��h2

2m@2

@x2�p2x

! 1ðxÞ

2ðxÞ

� �¼E

1ðxÞ

2ðxÞ

� �:

ð83Þ

Equation (83) can be split into two equations

��h2

2m

@2

@x2þ p1x

� � 1ðxÞ þ K0�ðxÞ 2ðxÞ ¼ E 1ðxÞ,

��h2

2m

@2

@x2� p2x

� � 2ðxÞ þ K0�ðxÞ 1ðxÞ ¼ E 2ðxÞ:

ð84Þ

In our calculation, we use p1¼ p2¼ 1. The time-

independent Schrodinger equation for the first diabatic

potential is given below,

��h2

2m

@2

@x2þ x

� � 1ðxÞ ¼ E 1ðxÞ: ð85Þ

In region 1 (x5 xc) the physically acceptable solution

is given below

1ðxÞ ¼ ðAþ BÞAi 21=3ð�Eþ xÞ

� þ iðA� BÞBi 2

1=3ð�Eþ xÞ�

: ð86Þ

Here, Ai[z] and Bi[z] represent the Airy functions. In

the above expression A denotes the probability

amplitude for motion along the positive direction and

B denotes the probability amplitude for motion along

the negative direction [15]. The physically acceptable

solution in region 2 (x4 xc) is given by

1ðxÞ ¼ CAi 21=3ð�Eþ xÞ

� : ð87Þ

In this region the net flux is zero [24].

The time-independent Schrodinger equation for the

second diabatic potential is given below

��h2

2m

@2

@x2� x

� � 2ðxÞ ¼ E 2ðxÞ: ð88Þ

In region1 (x5 xc) the physically acceptable solution is

2ðxÞ ¼ DAi 21=3ð�E� xÞ

� : ð89Þ

In this region the net flux is zero [15]. In region 2

(x4 xc) the physically acceptable solution is

2ðxÞ ¼ F Ai 21=3ð�E� xÞ

� � iBi 2

1=3ð�E� xÞ� � �

:

ð90Þ

Using the four boundary conditions mentioned in

Section 3.1 (here we put xc¼ 0), we have derived an

analytical expression for the transition probability

from one diabatic potential to the other diabatic

potential and the final expression is given below:

T ¼NL

DL

2, ð91Þ

where

NL ¼ 4� 21=3Ai �21=3E

� 2�Ai �2

1=3E�

B0i �21=3E

� � A0i �2

1=3E�

Bi �21=3E

� �ð92Þ

and

DL¼4Ai �21=3E

� 4�8iAi �2

1=3E� 3

Bi �21=3E

� þ22=3A0i �2

1=3E� 2

Bi �21=3E

� 2þAi �2

1=3E� 2

� 22=3B0i �21=3E

� 2�4Bi �2

1=3E� 2h i

�25=3Ai �21=3E

� A0i �2

1=3E�

Bi �21=3E

� B0i �2

1=3E�

:

ð93Þ

In our numerical calculation we set p1¼ 1, p2¼ 1,

K0¼ 1 and m¼ 1 in atomic units. The result of our

calculation is shown in Figure 3.

5.3. Exponential potential case

The time-independent Schrodinger equation for the

case where an exponential potential coupled to another

exponential potential through a Dirac delta interaction

is given below (see Figure 4).

� �h2

2m@2

@x2þ V0 expðaxÞ K0�ðx� xcÞ

K0�ðx� xcÞ � �h2

2m@2

@x2þ V0e

�ax

! 1ðxÞ

2ðxÞ

� �

¼ E 1ðxÞ

2ðxÞ

� �: ð94Þ

Figure 2. Schematic diagram of the two-state problem,where one linear potential is coupled to another linearpotential in the diabatic representation.

2264 Diwaker and A. Chakraborty

Dow

nloa

ded

by [

Indi

an I

nstiu

te o

f T

echn

olog

y M

andi

] at

06:

06 2

3 Ju

ly 2

014

Equation (94) can be split into the following two

equations

��h2

2m

@2

@x2þV0 expðaxÞ

� � 1ðxÞþK0�ðx�xcÞ 2ðxÞ ¼E 1ðxÞ,

��h2

2m

@2

@x2þV0 expð�axÞ

� � 2ðxÞþK0�ðx�xcÞ 1ðxÞ ¼E 2ðxÞ:

ð95Þ

In our calculation we took V0¼ 1.0 and a¼ 1. The

time-independent Schrodinger equation for the first

diabatic potential is given below

��h2

2m

@2

@x2þ V0 expðaxÞ

� � 1ðxÞ ¼ E 1ðxÞ: ð96Þ

In the region x5 xc, the solution of the above equationis given below

1ðxÞ ¼ AIð2ið2EÞ1=2Þ 2½2 expðxÞ�

1=2� �

þ BIð�2ið2EÞ1=2Þ 2½2 expðxÞ�

1=2� �

: ð97Þ

Here In(z) represents the modified Bessel function ofthe first kind. In the above expression A denotes theprobability amplitude for motion along the positivedirection and B denotes the probability amplitude formotion along the negative direction [15]. In the region,where x4 xc, the physically acceptable solution is

1ðxÞ ¼ CKð�2ið2EÞ1=2Þ 2½2 expðxÞ�

1=2� �

: ð98Þ

Here Kn(z) represents the modified Bessel function ofthe second kind. In this region the net flux is zero [15].

For the second diabatic potential, the time-inde-pendent Schrodinger equation is given below

��h2

2m

@2

@x2þ V0 expð�axÞ

� � 2ðxÞ ¼ E 2ðxÞ: ð99Þ

In the region, where x4 xc, the physically acceptablesolution is

2ðxÞ ¼ FIð�2ið2EÞ1=2Þ 2½2 expð�xÞ�

1=2� �

: ð100Þ

In the region, where x5 xc, the physically acceptablesolution is

2ðxÞ ¼ DKð�2ið2EÞ1=2Þ 2½2 expðxÞ�

1=2� �

: ð101Þ

In this region the net flux is zero [15]. Using fourboundary conditions as discussed in Section 3.1 (withxc¼ 0), we have derived an analytical expression forthe transition probability from one exponential poten-tial to the other exponential potential, and is given by

T ¼ 1�NE

DE

2, ð102Þ

where

NE ¼ �Ið2:83iE1=2Þð2:83Þ

Ið�2:83iE1=2Þð2:83Þ� Kð�2:83iE1=2Þð2:83Þ

� ð½Ið�2:83iE1=2�1Þð2:83Þ þ I1�ð2:83iE1=2Þð2:83Þ�

� Ið2:83iE1=2Þð2:83ÞðIð�2:83iE1=2Þð2:83Þ½Kð�2:83iE1=2�1Þ

� ð2:83Þ þ K1�ð2:83iE1=2Þð2:83Þ�

þ ½Ið�2:83iE1=2�1Þð2:83Þ þ I1�ð2:83iE1=2Þð2:83Þ�

� Kð�2:83iE1=2Þð2:83ÞÞ � Ið�2:83iE1=2Þð2:83Þ

� ½Ið2:83iE1=2�1Þð2:83Þ þ Ið2:83iE1=2þ1Þð2:83Þ�

� ðIð�2:83iE1=2Þð2:83Þ½Kð�2:83iE1=2�1Þð2:83Þ

þ K1�ð2:83iE1=2Þð2:83Þ� þ ½Ið�2:83iE1=2�1Þð2:83Þ

þ I1�ð2:83iE1=2Þð2:83Þ�Kð�2:83iE1=2Þð2:83ÞÞÞ

c

Figure 4. Schematic diagram of the two-state problem,where one exponential potential is coupled to anotherexponential potential in the diabatic representation.

p

Figure 3. The plot of transition probability from one linearpotential to another linear potential, as a function of energyof incident particle (K0¼ 1.0).

Molecular Physics 2265

Dow

nloa

ded

by [

Indi

an I

nstiu

te o

f T

echn

olog

y M

andi

] at

06:

06 2

3 Ju

ly 2

014

and

DE¼ Ið�2:83E1=2Þð2:83ÞðIð�2:83E1=2Þð2:83Þ2ð�Kð�2:83iE1=2�1Þð2:83Þ

2

�2Kð1�2:83iE1=2Þð2:83ÞKð�2:83iE1=2�1Þð2:83Þ

�Kð1�2:83iE1=2Þð2:83Þ2þ0:08Kð�2:83iE1=2Þð2:83Þ

2Þ

þðIð�2:83E1=2�1Þð2:83Þþ Ið1�2:83iE1=2Þð2:83ÞÞ

� Ið�2:83E1=2Þð2:83Þð�2Kð�2:83E1=2�1Þð2:83Þ

�2Kð1�2:83iE1=2Þð2:83ÞÞKð�2:83iE1=2Þð2:83Þ

þð�Ið�2:83E1=2�1Þð2:83Þ2�2Ið1�2:83iE1=2Þð2:83Þ

� Ið�2:83E1=2�1Þð2:83Þ� Ið1�2:83iE1=2Þð2:83Þ2Þ

�Kð�2:83E1=2Þð2:83Þ2Þ

In our numerical calculation we set K0¼ 1 and m¼ 1 in

atomic units. The result of our calculation is shown in

Figure 5.

6. Conclusions

We have proposed a general method for finding an

exact analytical solution for the multi-channel quan-

tum scattering problem in the presence of a delta

function coupling. Our solution is quite general and is

valid for any potential. We have also extended our

model to deal with general one-dimensional multi-

channel scattering problems. The same procedure is

also applicable to the case where S is a non-local

operator and may be represented by S� j f i K0 hgj,

where f and g are arbitrary acceptable functions.

Choosing both of them to be Gaussian should be an

improvement over the delta function coupling model. S

may even be a linear combination of such operators.

Acknowledgements

One of the authors (A.C.) thanks Prof. K.L. Sebastian forsuggesting this very interesting problem. A.C. would like tothank Prof. M.S. Child for his kind interest, suggestions andencouragements.

References

[1] H. Nakamura, Int. Rev. Phys. Chem. 10, 123

(1991).[2] E.E. Nikitin and S.Ia. Umanskii, in Theory of Slow

Atomic Collisions, edited by M. Bayer and C.Y. Ng

(Springer, Berlin, 1984).[3] M.S. Child, Molecular Collision Theory (Dover,

Mineola, NY, 1996).[4] E.S. Medvedev and V.I. Osherov, Radiationless

Transitions in Polyatomic Molecules (Springer,

New York, 1994).

[5] E.E. Nikitin, Annu. Rev. Phys. Chem. 50, 1 (1999).[6] H. Nakamura, Nonadiabatic Transition: Concepts, Basic

Theories and Applications (World Scientific, Singapore,

2002).

[7] H. Nakamura, in Theory, Advances in Chemical Physics,

edited by M. Bayer and C.Y. Ng (John Wiley and Sons,

New York, 1992).[8] S.S. Shaik and P.C. Hiberty, in Theoretical Models of

Chemical Bonding, Part 4, edited by Z.B. Maksic

(Springer-Verlag, Berlin, 1991), Vol. 82.[9] B. Imanishi and W. von Oertzen, Phys. Rep. 155, 29

(1987).[10] A. Thiel, J. Phys. G 16, 867 (1990).[11] A. Yoshimori and M. Tsukada, in Dynamic Processes

and Ordering on Solid Surfaces, edited by A. Yoshimori

and M. Tsukada (Springer-Verlag, Berlin, 1985).[12] R. Engleman, Non-Radiative Decay of Ions and

Molecules in Solids (North-Holland, Amsterdam,

1979).[13] N. Mataga, in Electron Transfer in Inorganic, Organic

and Biological Systems, Advances in Chemistry,

edited by J.R. Bolton, N. Mataga, and G. Mclendon

(American Chemical Society, Washington, DC, 1991),

p. 228.[14] D. Devault, Quantum Mechanical Tunneling in

Biological Systems (Cambridge University Press,

Cambridge, 1984).[15] A. Chakraborty, in preparation (2012).[16] L.D. Landau, Phys. Z. Sowjetunion 2, 46 (1932).

[17] C. Zener, Proc. Roy. Soc. A 137, 696 (1932).[18] E.C.G. Stuckelberg, Helv. Phys. Acta 5, 369 (1932).[19] N. Rosen and C. Zener, Phys. Rev. 40, 502 (1932).

[20] V.I. Osherov and A.I. Veronin, Phys. Rev. A 49, 265

(1994).

[21] C. Zhu, J. Phys. A 29, 1293 (1996).[22] A. Chakraborty, Mol. Phys. 107, 165 (2009).[23] A. Chakraborty, Mol. Phys. 107, 2459 (2009).

[24] A. Chakraborty, Mol. Phys. 109, 429 (2011).

p

Figure 5. The plot of transition probability from oneexponential potential to the other as a function of energyof incident particle (K0¼ 1.0).

2266 Diwaker and A. Chakraborty

Dow

nloa

ded

by [

Indi

an I

nstiu

te o

f T

echn

olog

y M

andi

] at

06:

06 2

3 Ju

ly 2

014

[25] A. Chakraborty, Ph.D. thesis, Indian Institute ofScience, India, 2004.

[26] A. Chakraborty, Nano Devices, 2D Electron Solvationand Curve Crossing Problems: Theoretical ModelInvestigations (Lambert Academic Publishing,Germany, 2010).

[27] G.G. Stokes, Trans. Camb. Phil. Soc. 10, 105 (1864).

[28] J. Heading, An Introduction to Phase-Integral Methods(Methuen, London, 1962).

[29] F.L. Hinton, J. Math. Phys. 20, 2036 (1979).[30] C. Zhu and H. Nakamura, J. Math. Phys. 33, 2697

(1992).[31] C. Zhu, H. Nakamura, N. Re and Z. Aquilanti, J.

Chem. Phys. 97, 1892 (1992).

Molecular Physics 2267

Dow

nloa

ded

by [

Indi

an I

nstiu

te o

f T

echn

olog

y M

andi

] at

06:

06 2

3 Ju

ly 2

014