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

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https://doi.org/10.22214/ijraset.2022.46641

September 2022

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

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 k(r, , ) is expressed in terms of ion-core electron density, Pl(r) of Hartree or Hartree-Fock one electron wave function k(r, , ) To develop the present equation, a new pseudo spherical function Yl(r) has been investigated which is helpful in determining many types of electron densities.

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

Keywords: radial part, pseudo potential, wave function, Hartree-Fock, electron densities I. INTRODUCTION In the free electron concepts of solids, Hartree and Hartree-Fock (HF) methodology and pseudo potential (PS) formalism have given new orientations. The potential usedin both cases is of central field type. In both cases the self-consistent field approximation is used where the effect of the interaction of a given electron with all others is replaced by some repulsive potential. Using these repulsive potentials the corresponding single particle Hamiltonian of HF methodology and PS methodology are given by –

   2 2 Ze 2    2 2  HHF      V (r )     VHF (r ) r  2m   2m     2 2 Ze 2    2 2     V (r )     VPS (r ) r  2m   2m 

HPS  

….. (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

 k  [T  H HF ]  k  Ek  k

HPS

k [T  H PS ]  k  Ek k

….. (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

   2 2 Ze 2   2m   r  VPS (r )  k  Ek  k     2  2m



….. (3)

  1   2     Ze 2  VPS (r )  k  Ek  k r  2   2  r r  r  r  r 

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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



Here,  



1   1 2  sin   sin    sin 2   2 

   2   2 2    Ze 2   2  VPS (r )  k  Ek  k  2 r r r  r  2m  r 



Put  = l(l+1)

  2   2m

   2 2  l (l  1)    ( E  V )  2   k  0 k r r r2   r    Ze 2   VPS (r )  r 

Where, V = 

   2   2m

 



  r

2



   2 2  l (l  1)       ( E  V )  k  0  2  k r r r 2    r 

  2m    2r  l (l  1)  r 2 2 ( Ek  V ) k  0 2 r r   

2 2  r y   2ry  {r  l (l  1)} y  0

Where  =

2m 2

…… (4)

( Ek  V ) & y= k



 am r m and solution y is of the form

Suppose, r2 =

m 0 

y=

b r

k

…… (5)

k

k 0

thus by some mathematical exercise, A solution of equation (4) is given by   Ck  Yl (r )  bo r  1   rk   k 1 k!(2   2) k 

Where,





1  1  ( 2l  1) 2  4a0 2



…… (6)

…… (7)

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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

and

Ck 

2  2

0......

0.....

0.....

a1

a1

2( 2   3)....

0.....

0......

a2

a2

a1

3( 2   4)....

0....

a3

.............................................................................................................

a n 1

Note: Solution (6) for

  2 

… (8)

............................................................................................................. a n2 a n 3 ... a1 ( n 1) (2   n)... a n 1 a n2

  1 



....



a1

an



1  1  (2l  1) 2  4a0 and 2



1  1  (2l  1) 2  4a0 Are independent, if 1   2 and 1   2 is not an integer. We have, 2

  Ck  Yl (r )  bo r  1   rk   k 1 k!(2   2)k 

In spherical co-ordinate

…. (9)

(r , , ) system the Hartree-Fock wave function is expressed as

1  k (r , , )  Pn,l (r ) y, l m ( , )  ms r

…. (10)

Whose radial function is given by

1 Yn,l (r )  Pn.l ( r ) r

….. (11)

Multiply and divide equation (9) by r

 Yl ( r )  bo

 Ck  r  1  rk  1   r  k 1 k!(2   2)k 

 from equation (11)   Ck  Pl (r )  bo r  1 1   rk   k 1 k!(2   2)k 

…. (12)

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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 Now we express the pseudo Schrödinger equation in spherical co-ordinate system

(r , , ) similar to Hartree-Fock wave function

as k(r, , )  r

l 2

Fn.l ( r ) L (( , )

…. (13)

 pseudo spherical radial function is given by

Y ( r )  r l 2 F ( r ) Now multiply and divide equation (9) by r =

bo

…. (14) l 2

r  l  2 r l 2

  Ck  rk  1    k 0 k!(2   2) k 

r r l 2

  Ck  1  rk     k 0 k!(2   2)k 

 Fl (r )  bo

…. (15)

B. Special Case If a  0 , then 

 l therefore equation (12) becomes

  Ck  Pl (r )  bo r l 1 1   rk   k 1 k!(2   2) k 

….. (16)

Similarly,   Ck  Fl (r )  bo r 2 1   rk   k 0 k!(2   2) k 

…. (17)

II. NUMERICAL COMPUTATION AND RESULTS 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) Fig.(1d) ,Variation of F0(r) for l=0 , as r increases then F0(r) varies from zero ,initially it is -ve then varies from –ve to 9.8107Å up to r = 8 Å and then varies exponentially.

2) Fig.(1e) F1(r) for l=1 as r increases from zero F1(r) varies from -2.46102Å to -7.1108Å up to r = 8 Å. The behavior shows that upto r= 4Å , F1(r) has variation almost constant and for higher values of r it shows exponential behavior.

3) Fig.(1f ), Variation of F2(r) for l=2 as r increases then F2(r) varies from -6.5101Å to -41010Å up to r = 8 ÅThe behavior shows that up to r= 4Å , F2(r) has variation almost constant and for higher values of r it shows exponential behavior .

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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

Fig.1 F n ,l(r) Versus ‘r’ REFERENCES [1] [2] [3] [4] [5] [6] [7]

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

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

W.Kohn and L.J.Sham, Phys.Rev.,A140,1133(1965) R.W.Shaw and W.A. Harrision,Phys.Rev. 163 ,604 (1967) G.N.Watson,Theory of Bessel functions,2nd Ed. Macmillan NewYork,1945 W.W.Piper,Phys.Rev.123,1281(1961) B.J.AustinV.Heine and L.J. Sham,Phys. Rev. 127,276(1962).

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