Kantbp 3. 0: New Version of a program for Computing energy Levels, reflection and transmission matrices, and corresponding wave functions in the coupled-channel adiabatic approach Текст научной статьи по специальности «Математика»

UDC 517.958:530.145.6

KANTBP 3.0: New Version of a Program for Computing Energy Levels, Reflection and Transmission Matrices, and Corresponding Wave Functions in the Coupled-Channel

Adiabatic Approach

A. A. Gusev*, O. Chuluunbaatart, S. I. Vinitsky*, A. G. Abrashkevich*

* Joint Institute for Nuclear Research 6, Joliot-Curie, Dubna, Moscow region, Russia, 141980 ^ School of Mathematics and Computer Science National University of Mongolia, Mongolia 1 IBM Toronto Lab, 8200 Warden Avenue, Markham, ON L6G 1C7, Canada

Brief description of a FORTRAN 77 program for calculating energy values, refection and transmission matrices, and corresponding wave functions in a coupled-channel approximation of the adiabatic approach is presented. In this approach, a multidimensional Schrodinger equation is reduced to a system of the coupled second-order ordinary differential equations on a finite interval with the homogeneous boundary conditions of the third type at the left-and right-boundary points for continuous spectrum problem, or a set of first, second and third type boundary conditions for discrete spectrum problem. The resulting system of these equations containing the potential matrix elements and first-derivative coupling terms is solved using high-order accuracy approximations of the finite element method.

Key words and phrases: boundary value problem, multichannel scattering problem, finite element method, Kantorovich method.

1. Introduction

In this work we present a brief description of a KANTBP3 program for calculating with a required accuracy approximate eigensolutions of the continuum spectrum for systems of coupled differential equations on finite intervals of the variable z G [¿mm, ¿max] using a general homogeneous boundary condition of the third-type [1]. The third-type boundary conditions are formulated for problems under consideration by using known asymptotics for a set of linear independent asymptotic regular and irregular solutions in the open channels, and a set of linear independent regular asymptotic solutions in the closed channels, respectively [2]. These problems are solved by the finite element method [3,4]. This approach can be used in calculations of effects of electron screening on low-energy fusion cross sections, channeling processes, threshold phenomena in the formation and ionization of (anti)hydrogen-like atoms and ions in magnetic traps, scattering problem for quantum dots and quantum wires in magnetic field, potential scattering with confinement potentials, penetration through a two-dimensional fission barrier, tunneling from false vacuum of two interacted particles and three-dimensional tunneling of a diatomic molecule incident upon a potential barrier [2,5].

2. Statement of the Problem

In the Kantorovich method or close-coupling adiabatic approach, the multidimensional Schrodinger equation is reduced to a finite set of N ordinary second-order differential equations on the finite interval [zmin ,zmax] for the partial solution

Received 27th September, 2013. The work was supported partially by grants 13-602-02 JINR, 11-01-00523 and 13-01-00668 RFBR..

X(j)(z) — (;X[J\z),..., X%\z))T

(L-2E])x(%) = 0, L = -1-L-d+ ^d^ Q(Z). (1) v ' v ' zd-i dz dz w wdz z d-i d z v '

Here I, V(z) and Q(z) are the unit, symmetric and antisymmetric N x N matrices, respectively. We assume that V( z) and Q( z) matrices have the following asymptotic behaviour at large z = z± ^

( 2 Z ± N y( '± a{ 1 '±)

Va{ z±) — e, + + ^ -, Q%0( z±) — ^ ^V, (2)

V Z± J I=2 Z± I = 1 Z±

where ei < ... < e n are the threshold energy values.

In the present work, scattering problem is solved using the boundary conditions at

d 1 Z ¿min and ^ ¿max-

d$(z)

dz

/v( Zmin)$( ¿min^

d

— 2max)^( ¿max^ (3)

where '(z) is a unknown N x N matrix-function, $(z) — {x(^ (z)}Ij=L1 is the required

N xNa matrix-solution and Na is the number of open channels, Na = max2£^e- j ^ N. From this we obtain the quadratic functional at d =1 (similar to Eq. (5) in [3])

'max

E($,E, Zmin, ¿max) = J $T (z) (L - 2E I) $(z)d* — n($, E, Zmin, ¿max)-

'min

$ ( Zmax)G( ^max)^( ^max) + $ ( ^min)G( ^min)$( ^min^ (4)

where n($,E, Zmin, ^max) is the symmetric functional

'ma

П($, E, ^mim ^max) J

'min

d$T(z)d$(z) ,.T. ,

+$T(z)Q(z)d-d^) - Q(¿)$(¿) - 2E$T(z)G(z)

dz, (5)

and G( z) — '( z) - Q( z) is the N x N matrix-function which should be symmetric according to the conventual R-matrix theory.

ax

3. The Physical Scattering Asymptotic Forms

Matrix-solution ( z) = $( z) describing the incidence of the particle and its scattering, which has the asymptotic form "incident wave + outgoing waves" is

(z ^ ±ro) — <

X(+)(z)T, z> 0,

X(+)(z) + X(-)( z)Rv, z< 0,

X(-)( /) + X(+)( z)Rv, z > 0,

X(-)(z)Tv, z < 0,

v

(6)

where R^ and T are the reflection and transmission N0 x N0 matrices, v and v denote the initial direction of the particle motion along the z axis. Here the leading term of the asymptotic rectangle-matrix functions X(±) (z) has the form [2]

X(±\z) ^ p~l/2 exp z - ^ ln(2pjM))) Sij, (7)

Pj = y/2E - ej i = 1,...,N, j = 1,...,N0,

where Zj = Z+ at z > 0 and Zj = Z~ at z < 0. The matrix-solution (z,E) is normalized by

œ

J (z,E')®v(z,E)dz = 2n5(E' - E)SV,VI00, (8)

_œ

where I00 is the unit N0 x N0 matrix. Let us rewrite Eq. (6) in the matrix form at ^ and ^ -œ as

/*:(*+) *:(z+)\ = ( 0 X(_)(z+ )\ + ( 0 X(+)(z+)\ S V*:(*_) *^(z_)J VX(+)(*_) 0 ) + \X(_)(z_) 0 (9)

where the scattering matrix S

s=(R: R: ) ('o>

is composed of the reflection and transmission matrices.

In addition, it should be noted that functions X(±)(z) satisfy relations

Wr(Q(z); X(^(z), X(±)(z)) = ±2d00, Wr(Q(z); X(±)(z), X(±)(z)) = 0, (11) where

T

Wr(.; a(z), b(z)) = a7 (z) ( ^^ - •b(z) ) - ( ^^ - •a(z) ) b(z). (12)

'db(z) \ f da(z) V

bT -*(z)) -{"ZT -™(z))

This Wronskian is used to estimate a desirable accuracy of the above expansion.

Note, using a wronskian, we obtain the following properties of the reflection and transmission matrices:

T:T: + R:R: = I00 = T^: + R^:, T:R: + R:T: = 0 = R^: + T^:, (13)

T^ = T:, R^ = R:, Rr = R:. This means that the scattering matrix (10) is symmetric and unitary.

4. Test Desk

We consider the boundary problem (1)-(3) with parameters d =1, Z1 = Z2 = 0.1, m1 = 1, m2 = 3, s = 8, a:min = 0.1. This problem is followed from Kantorovich expansion of the 2D BVP described the tunneling problem of transmission of two ions

through repulsive barrier (for details, see [2])

( p2 p2 \ ( -gy - + x2 + Ui(xi) + ^2) - y y) = 0, (14)

where Ui(xi) = 22>i/^|x^|s + xmin are Coulomb-like barrier potentials, xi = s2y + six and x2 = s2y - s3x are Jacobi coordinates with si = mi/M, S3 = m2/M, s2 =

^toTTO2/M, S2 = \jM, M = mi + m2.

The required asymptotics of regular and irregular solutions given in [2]. The following values of numerical parameters and characters described in [1] have been used in the test run via the supplied input file SQRTBT.INP

&PARAS TITLE=' REFLECTION AND TRANSMISSION MATRICES IPTYPE=1,NROOT=1,MDIM=4,IDIM=1,NPOL=4, SHIFT= 4D0,IPRINT=1,IPRSTP=120,

NMESH=7,RMESH=-25D0,100D0,-6D0,100D0,6D0,100D0,25D0, NDIR=1, NDIL=4, NMDIL=0,THRSHL= 1.D0,3D0,5D0,7D0,IBOUND=8, FNOUT='KANTBP.LPR,,IOUT=7,POTEN=,ODPEVP.PTN,,IOUP=10, FMATR=,KANTBP.MAT,,IOUM=11,EVWFN=,KANTBP.WFN,,IOUF=0

&END

Boundary problem (14) and the corresponding matrix elements V(y), Q(y) have been solved by the ODPEVP program [6] on grids ilx{xmin, xmax} = {-xmin(64)xmax} with accuracy eps = 10-i0. Boundary points are xmax = -xmin = 8.1. All calculation details of this problem were written into file ODPEVP.LPR.

TEST RUN OUTPUT

PROBLEM: REFLECTION AND TRANSMISSION MATRICES

CONTROL INFORMATION

NUMBER OF DIFFERENTIAL EQUATIONS.....(MDIM ) = 4

NUMBER OF FINITE ELEMENTS........(NELEM ) = 300

NUMBER OF GRID POINTS..........(NGRID ) = 1201

ORDER OF SHAPE FUNCTIONS.........(NPOL ) = 4

ORDER OF GAUSS-LEGENDRE QUADRATURE. . . . (NGQ ) = 5

DIMENSION OF ENVELOPE SPACE ....... (IDIM ) = 1

BOUNDARY CONDITION CODE ......... (IBOUND) = 8

DOUBLE ENERGY SPECTRUM..........(SHIFT ) = 4.00000

SUBDIVISION OF RHO-REGION ON THE FINITE-ELEMENT GROUPS:

NO OF NUMBER OF BEGIN OF LENGTH OF GRID END OF GROUP ELEMENTS INTERVAL ELEMENT STEP INTERVAL

100 -25.000 0.19000 0.04750 -6.000

100 -6.000 0.12000 0.03000 6.000

100 6.000 0.19000 0.04750 25.000

TOTAL SYSTEM DATA

TOTAL NUMBER OF ALGEBRAIC EQUATIONS. . . . (NN ) = 4804

TOTAL NUMBER OF MATRIX ELEMENTS......(NWK) = 60010

MAXIMUM HALF BANDWIDTH..........(MK ) = 20

MEAN HALF BANDWIDTH..........(MMK) = 12

NDIM, MDIM= 4 4

CALCULATION OF WAVE FUNCTION WITH DIRECTION <--

NUMBER OF OPEN CHANNELS.........(NOPEN) =

VALUE OF I-TH MOMENTUM.........(I,QR ) =

VALUE OF I-TH MOMENTUM.........(I,QR ) =

I M P A R T: W R O N S K I A N

-2.00000 -.168196E-08 -.168196E-08 -2.00000

RE PART: RRMATRIX

-.194759 -.590855E-03 -.590855E-03 -.485377E-01

IM PART: RRMATRIX

-.124681 0.172716 0.172716 0.931470

RE PART: TT MATRIX

0.600459 -.317924E-01 0.317924E-01 -.276468

I M P A R T: TT M A T R I X

-.729781 0.150166 -.150166 0.134581E-01

Z REPART: FUNCTIONS

2

1 0.1732E+01

2 0.1000E+01

-25.0000 0.6664D+00 -.1165D+00

-13.6000 0.6802D+00 -.7978D-01

-3.6000 0.1490D-01 -.5461D-01

0.0000 -.8416D+00 0.7861D-01

3.6000 -.4115D+00 -.6691D-01

13.6000 0.5769D+00 -.6829D-01

25.0000 0.2716D+00 -.1259D+00

0.1531D+00 -.1120D+00 0.7601D-06

0.4223D-01 0.2431D+00 -.2701D-04

-.3718D-01 -.2780D+00 -.9230D-02 0.9335D-02 0.4446D+00 0.5115D-01

0.8351D-01 0.1351D+01 -.4630D-02

-.8088D-01 -.1298D+01 -.3777D-04 -.1631D+00 -.5370D+00 0.1506D-05

0.8680D-05 0.2445D-07 0.4751D-06 0.3867D-04 -.2948D-05 0.3128D-05 0.2404D-02 -.1299D-02 -.2425D-03 -.1732D-01 -.2247D-02 -.6850D-02 0.2048D-01 -.2890D-04 -.9308D-04 -.5932D-04 0.3999D-05 0.9632D-05 0.4406D-04 -.6915D-07 -.2284D-05

Z IMPART: FUNCTIONS

-25.0000 -13.6000 -3.6000 0.0000 3.6000 13.6000 25.0000

0.2735D+00 -.2428D+00 0.7372D+00 0.5262D+00 -.5284D+00 -.5507D+00 -.8982D+00

0.1055D-01 0.8603D-01 -.1083D+00 -.1487D+00 -.8131D-01 0.1129D+00 0.3837D-01

-.2391D-01 0.1506D+00 -.1518D+00 -.1846D-01 0.1780D+00 -.1559D+00 0.6149D-01

-.2560D+00 -.1425D+00 0.1221D+00 0.6235D+00 0.1380D+01 -.1335D+01 -.6546D+00

0.6563D-05 0.2784D-04 0.1107D-02 -.3508D-01 0.1289D-01 0.6059D-05 0.5103D-05

-.4645D-05 0. 0.5248D-04 0. -.5541D-02 -. -.4223D-03 -. 0.1938D-01 -. -.8405D-04 -. 0.4498D-04 -.

3403D-06 -.2162D-06 2187D-05 0.5597D-05 1592D-02 -.3799D-03 5965D-02 -.9388D-02 2320D-02 0.4662D-04 3894D-06 0.1222D-04 2851D-06 -.2320D-05

CALCULATION OF WAVE FUNCTION WITH DIRECTION -->

NUMBER OF OPEN CHANNELS.........(NOPEN) =

VALUE OF I-TH MOMENTUM.........(I,QR ) =

VALUE OF I-TH MOMENTUM.........(I,QR ) =

IM PART: WRONSKIAN

2.00000 -.168196E-08 -.168196E-08 2.00000

2

1 0.1732E+01

2 0.1000E+01

RE PART: RRMATRIX

-.194759 0.590855E-03 0.590855E-03 -.485377E-01

I M P A R T: RR M A T R I X

-.124681 -.172716 -.172716 0.931470

RE PART: TT MATRIX

0.600459 0.317924E-01 -.317924E-01 -.276468

I M P A R T: TT M A T R I X

-.729781 -.150166 0.150166 0.134581E-01

Z REPART: FUNCTIONS

-25.0000 0.2716D+00 0.1259D+00

-13.6000 0.5769D+00 0.6829D-01

-3.6000 -.4115D+00 0.6691D-01

0.0000 -.8416D+00 -.7861D-01

3.6000 0.1490D-01 0.5461D-01

13.6000 0.6802D+00 0.7978D-01

25.0000 0.6664D+00 0.1165D+00

0.1631D+00 -.5370D+00 0.1506D-05

0.8088D-01 -.1298D+01 -.3777D-04

-.8351D-01 0.1351D+01 -.4630D-02 -.9335D-02 0.4446D+00 0.5115D-01

0.3718D-01 -.2780D+00 -.9230D-02

-.4223D-01 0.2431D+00 -.2701D-04 -.1531D+00 -.1120D+00 0.7601D-06

-.4406D-04 0.6915D-07 -.2284D-05 0.5932D-04 -.3999D-05 0.9632D-05 -.2048D-01 0.2890D-04 -.9308D-04 0.1732D-01 0.2247D-02 -.6850D-02 -.2404D-02 0.1299D-02 -.2425D-03 -.3867D-04 0.2948D-05 0.3128D-05 -.8680D-05 -.2445D-07 0.4751D-06

Z IMPART: FUNCTIONS

-25.0000 -.8982D+00 -.3837D-01 -.6149D-01 -.6546D+00 0.5103D-05 -.4498D-04 0.2851D-06 -.2320D-05

-13.6000 -.5507D+00 -.1129D+00 0.1559D+00 -.1335D+01 0.6059D-05 0.8405D-04 0.3894D-06 0.1222D-04

-3.6000 -.5284D+00 0.8131D-01 -.1780D+00 0.1380D+01 0.1289D-01 -.1938D-01 0.2320D-02 0.4662D-04

0.0000 0.5262D+00 0.1487D+00 0.1846D-01 0.6235D+00 -.3508D-01 0.4223D-03 0.5965D-02 -.9388D-02

3.6000 0.7372D+00 0.1083D+00 0.1518D+00 0.1221D+00 0.1107D-02 0.5541D-02 0.1592D-02 -.3799D-03

13.6000 -.2428D+00 -.8603D-01 -.1506D+00 -.1425D+00 0.2784D-04 -.5248D-04 -.2187D-05 0.5597D-05

25.0000 0.2735D+00 -.1055D-01 0.2391D-01 -.2560D+00 0.6563D-05 0.4645D-05 -.3403D-06 -.2162D-06

C H E C K P R O P E R T I E S

CHECK |RR_<-|~2 + |TT_<-|~2

1.00000 0.242339E-09 0.242339E-09 1.00000

CHECK |RR_->|~2 + |TT_->|~2

1.00000 -.407011E-09 -.407011E-09 1.00000

RE PART: TT_->~1 * RR_<- + RR_->~1 * TT_<-

0.185469E-09 0.420999E-09 -.476236E-09 0.157399E-09

I M PART: TT_->~1 * RR_<- + RR_->~1 * TT_<-

0.219235E-11 -.125379E-09 -.197244E-09 0.129723E-10

RE PART: RR_<-~T - RR_<-

0.00000 -.185546E-09 0.185546E-09 0.00000

I M PART: RR_<-~T - RR_<-

0.00000 0.356981E-09 -.356981E-09 0.00000

RE PART: RR_->~T - RR_->

0.00000 0.103188E-09 -.103188E-09 0.00000

I M PART: RR_->~T - RR_->

0.00000 -.533526E-09 0.533526E-09 0.00000

RE PART: TT_->~T - TT_<-

0.231348E-10 0.847061E-10 -.952086E-13 0.142655E-10

I M PART: TT_->~T - TT_<-

0.186473E-11 0.452252E-09 0.511466E-09 -.116038E-10

References

1. A Program Package for Solution of Two-Dimensional Discrete and Continuum Spectra Boundary-Value Problems in Kantorovich (Adiabatic) Approach / O. Chuluun-baatar, A. A. Gusev, S. I. Vinitsky, A. G. Abrashkevich // JINR Lib. — 2013. — http://wwwinfo.jinr.ru/programs/jinrlib/kantbp/indexe.html.

2. Symbolic-Numerical Algorithms to Solve the Quantum Tunneling Problem for a Coupled Pair of Ions / A. A. Gusev, S. I. Vinitsky, O. Chuluunbaatar et al. // Lecture Notes in Computer Science. — 2011. — Vol. 6885. — Pp. 175-191.

3. KANTBP: A Program for Computing Energy Levels, Reaction Matrix and Radial Wave Functions in the Coupled-Channel Hyperspherical Adiabatic Approach / O. Chuluunbaatar, A. A. Gusev, A. G. Abrashkevich et al. // Comput. Phys. Commun. — 2007. — Vol. 177. — Pp. 649-675.

4. KANTBP 2.0: New Version of a Program for Computing Energy Levels, Reaction Matrix and Radial Wave Functions in the Coupled-Channel Hyperspherical Adia-batic Approach / O. Chuluunbaatar, A. A. Gusev, S. I. Vinitsky, A. G. Abrashke-vich // Comput. Phys. Commun. — 2008. — Vol. 179. — Pp. 685-693.

5. Symbolic-Numerical Algorithm for Generating Cluster Eigenfunctions: Tunneling of Clusters Through Repulsive Barriers / A. A. Gusev, S. I. Vinitsky, O. Chulu-unbaatar et al. // Lecture Notes in Computer Science. — 2013. — Vol. 8136. — Pp. 427-442.

6. ODPEVP: A program for Computing Eigenvalues and Eigenfunctions and Their First Derivatives with Respect to the Parameter of the Parametric Self-Adjoined Sturm-Liouville Problem / O. Chuluunbaatar, A. A. Gusev, S. I. Vinitsky, A. G. Abrashkevich // Comput. Phys. Commun. — 2009. — Vol. 180, No 8. — Pp. 1358-1375.

УДК 517.958:530.145.6

KANTBP 3.0: новая версия программы для вычисления энергетических уровней, матриц амплитуд отражения и прохождения и соответствующих волновых функций в адиабатическом подходе со связанными каналами

А. А. Гусев*, О. Чулуунбаатар^, С. И. Виницкий*, А. Г. Абрашкевич*

* Объединённый институт ядерных исследований ул. Жолио-Кюри, д. 6, г.Дубна, Московская область, Россия, 141980 ^ Факультет математики и компьютерных наук Монгольский государственный университет, Монголия * Лаборатории IBM в Торонто 8200 Ворден авеню, Маркхэм, ON L6G 1C7, Канада

Представлено краткое описание программ на языке Фортран 77 для вычисления энергетических уровней, матриц амплитуд отражения и прохождения и соответствующих волновых функций в адиабатическом подходе со связанными каналами. В этом подходе многомерное уравнение Шрёдингера сводится к системе связанных обыкновенных дифференциальных уравнений второго порядка на конечном интервале с однородными граничными условиями третьего рода на левой и правой граничных точках для задачи непрерывного спектра или набора граничных условий первого, второго и третьего рода для задачи дискретного спектра. Полученная система уравнений, содержащая матричные потенциалы, а также связанная слагаемыми, содержащими первые производные, решается в приближении высокого порядка точности методом конечных элементов.

Ключевые слова: краевая задача, многоканальная задача рассеяния, метод конечных элементов, метод Канторовича.