Study of Radial Part of Pseudo wave Function using Pseudo-spherical Methodology in Condenser Matter by IJRASET

10 IX September 2022 https://doi.org/10.22214/ijraset.2022.46641

Study of Radial Part of Pseudo wave Function using Pseudo-spherical Methodology in Condenser Matter

K. G. Kolhe

Department of Physics, Sardar Vallabhbhai Patel Arts and Science College, Ainpur

Abstract: The radial part of Fl(r) of the pseudo Schrödinger equation has been developed in pseudo potential approach. Also the pseudo spherical function Fl(r), the radial part of pseudo wave function

(r,

is expressed in terms of ion-core electron density, Pl(r) of Hartree or Hartree Fock one electron wave function

To develop the present equation, a new pseudo spherical function

l(r) has been investigated which is helpful in determining many types of electron densities.

present study orients the study related to condenser matter as well as mathematical physics.

radial part, pseudo potential, wave function, Hartree Fock, electron densities

I. INTRODUCTION

Hartree Fock (HF) methodology and pseudo potential (PS) formalism have given new orientations.

In the free electron concepts of solids, Hartree

usedin both cases

both cases the self consistent field approximation is used where

replaced by some repulsive potential. Using these repulsive potentials

of the interaction of a given electron with all others

corresponding single

methodology are given by

k



)

k(r, , )

The

Keywords:

and

The potential

central field type. In

the effect

the

particle Hamiltonian of HF methodology and PS

HHF             2()2() 2 22 2 2 Vr m Vr r Ze m HF  HPS             2()2() 2 22 2 2 Vr m Vr r Ze m PS  ….. (1) Here, Hartree Fock potential VHF(r) consists of self consistent core potential and valance electron potential Ze2/r, while pseudo potential VPS(r) is the sum of same valance electron potential Ze2/r and electron ionic model interaction potential VR(r). The corresponding Schrödinger quantum mechanical equations are given by HHF kHFkkk THE   [] HPS kPSkkk THE   [] ….. (2) Where T is kinetic energy and Ek is energy Eigen value. A. Physical Developments In spherical co ordinate system, (r, , ), the pseudo Schrödinger equation is given by PSkkkVrE r Ze m        2() 2 2 2 ….. (3)  PSkkkVrE r Ze rr r mrr                       () 1 2 2 2 2 2 2

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465 465©IJRASET: All Rights are Reserved | SJ Impact Factor 7.538 | ISRA Journal Impact Factor 7.894 | Here,                2 2 2sin sin1 sin 1  PSkkkVrE r Ze mrrrr                    () 2 2 2 22 22 Put  = l(l+1)  22(1)()0 22 22                   kkEV r ll mrrr  Where, V =        () 2 Vr r Ze PS  0)(1)(2 2 22 22                             kkEV r ll mrrr   0)( 2 1)(2 2 2 2 2                   kkEV mllr r r r r   2{(1)}022 ryryrlly  …… (4) Where  = 2() 2 mEV k  & y= k Suppose, r 2 =   0m m m ar and solution y is of the form y =     0k k k br  …… (5) thus by some mathematical exercise, A solution of equation (4) is given by           1 !(22)()1 k k k lo r k Ck Yrbr   …… (6) Where,  0 2 21(21)4 1 la …… (7)

(12)

(9)

(10)

(11)

In spherical co ordinate (,,) r system the Hartree Fock wave function is expressed as knlms Prylm r r  (,,)1(),(,) ,  ….

Whose radial function is given by ()1() , Pr r Yrnlnl  …..

Multiply and divide equation (9) by r              1 1 !(22)()1 k k lo r kk Ck r Yrbr    from equation (11)             1 1 !(22)()1 k k lo r kk Ck Prbr   ….

COMPUTATION AND RESULTS

Fig.(1d)

is ve then varies from

up to r = 8 Å. The behavior shows that upto r= 4Å

Fig.(1e)

3) Fig.(1f

(r)

values of r it shows exponential behavior.

Variation of F2(r)

(r)

varies from

Å up to r = 8 ÅThe behavior shows that up to r= 4Å

higher values of r it shows exponential behavior .

International Journal for Research in Applied Science & Engineering Technology (IJRASET) ISSN: 2321 9653; IC Value: 45.98; SJ Impact Factor: 7.538 Volume 10 Issue IX Sep 2022 Available at www.ijraset.com 467 467©IJRASET: All Rights are Reserved | SJ Impact Factor 7.538 | ISRA Journal Impact Factor 7.894 | Now we express the pseudo Schrödinger equation in spherical co ordinate system (,,) r similar to Hartree Fock wave function as k(r, , ) ()((,)2  rFrL nl l …. (13)  pseudo spherical radial function is given by ()() 2 YrrFr l   …. (14) Now multiply and divide equation (9) by 2l r =           0 2 2 !(22) 1 k k k k l l o r k C r br              0 2 !(22)()1 k kk lol r kk C r Frbr   …. (15) B. Special Case If 0a , then l therefore equation (12) becomes             1 1 !(22)()1 k k k lk lo r k C Prbr  ….. (16) Similarly,           0 2 !(22)()1 k k k k lo r k C Frbr  …. (17) II. NUMERICAL

In order to see the study between pseudo spherical methodology and Hartree and Hartree Fock methodology, we have computed: Fl(r)using equation (15) for different values of l= 0,1 and 2 .We have taken the values of a0=1,a1=1.Computation results are shown in fig.(1) a,b,c,d,e & f. 1)

,Variation of F0(r) for l=0 , as r increases then F0(r) varies from zero ,initially it

ve to 9.8107Å up to r = 8 Å and then varies exponentially. 2)

F1(r) for l=1 as r increases from zero F1(r) varies from 2.46102Å to 7.1108Å

, F1

has variation almost constant and for higher

for

=2 as

increases then F2(r)

6.5101Å to 41010

, F2

has variation almost constant and for

ISSN: 2321 9653; IC Value: 45.98; SJ Impact Factor: 7.538

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Fig.1 F n ,l(r) Versus ‘r’ REFERENCES

[1] D.R. Hartree, The calculations of Atomic structures, John Wiley and Sons, Inc., Newyork 1957.p.58.

[2] W. A. Harrison, Phys. Rev. 131, 2433(1963), Pseudo potential theory of metals, Benjamin, Inc. Newyork,1966.

[3] W.Kohn and L.J.Sham, Phys.Rev.,A140,1133(1965)

[4] R.W.Shaw and W.A. Harrision,Phys.Rev. 163 ,604 (1967)

[5] G.N.Watson,Theory of Bessel functions,2nd Ed. Macmillan NewYork,1945

[6] W.W.Piper,Phys.Rev.123,1281(1961)

[7] B.J.AustinV.Heine and L.J. Sham,Phys. Rev. 127,276(1962).

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