Int. J. Chem. Sci.: 12(4), 2014, 1311-1318
ISSN 0972-768X
www.sadgurupublications.com


CORRELATION EFFECTS IN He-LIKE IONS: AN ANALYSIS OF THE GROUND STATE IN MOMENTUM SPACE

INAS M. AL-ROBAYI*


Laser Physics Department, College of Science for Women, Babylon University, HILLA, IRAQ

ABSTRACT

An electron correlation has been examined in detail for the 1s2 -state of He and comparisons are made with the ground state of Li+, Be2+ and B3+. An expression has been obtained for the partial Coulomb hole associated with any pair of occupied HF spin orbital for many electron systems. The required partitioning of the correlated second order density matrix was achieved here, up to and including the pair- correlation effects. Partial coulomb holes were determined in momentum space. The concept of partial coulomb holes and their collective presentation as surfaces has been demonstrated to be particularly informative. These surfaces enabled us to interpret correlation effects in momentum space providing a rationalization of the correlation mechanisms in each state.

Key words: Correlation effects, He-like ions, Momentum space.

INTRODUCTION

Two electron atoms or ions present an excellent testing ground for checking new calculation approaches and for studying photoelectron and other atomic processes. However, the Schrodinger equation for atoms/ions with more than one electron cannot be solved analytically. Approximation must be applied order in to solve the problem. A large variety of techniques were developed to obtain the nonrelativistic bound energies and wave functions for mentioned systems1,2.

The Hartree–Fock self-consistent field approximation, which is based on the idea that we can approximately describe an interacting Fermion system in terms of an effective single-particle model, remains the starting point and the major approach for quantitative electronic structure calculations. The correlation energy is defined as the energy error of the Hartree-Fock wave function, i.e., the difference between the Hartree- Fock limit energy and

*Author for correspondence; E-mail: Dr.Enas17@yahoo.com; Mo.: 009647811487602


exact solution of nonrelativistic Schrodinger equation3. Electron correlation has been examined within electron shells for a series of He-like systems in the ground state. Inter- shell description was obtained using a Haretree-Fock (HF) level (Weiss 1963)4 by partioning the second order density into its pairwise components. Correlation effects were then analyzed in terms of various partial Coulomb holes.

RESULTS AND DISCUSSION

Electron correlation has a direct influence on distribution function for inter particle separation r12. Following Coulson and Neilson5, the total Coulomb hole for any two electron m and n within an N-electron system described by uncorrelated wave function (HF) a Coulomb hole can be defined as –


?f (p12) = fcorr (p12) ? fHF (p12) Both functions satisfies the normalization conditions6:

…(1)

?
?f (p12) dp12 = 1
0


… (2)


Partial distribution function g (p12; p1) represents the probability of finding an inter electronic separation r12, when a test electron is located at distance r1 from the nucleus7.

The partial Coulomb hole, ? g (p12, p1, ?1), allows us to analyze a characteristics of the Coulomb hole, when a test electron, say particle 1, is located at specified radial distance r1 from the nucleus. When the system possesses a unique axis of symmetry, a partial Coulomb hole, ? g (p12, p1, ?1), may be defined. ? being measured relative to the symmetry axis. These functions are related to, ? f (p12) as follows8:

? ? ?

???g (p12; p1,?1) sin ?1d?1dp1 = ??g (p12;p1) dp1 = ?f (p12)
0 0 0

…(3)


For (i, j) labels, a pair of occupied spin orbitals ?i and ?j in the restricted HF description of the system, then the associated Coulomb hole can be written as9:



?f (P )

?
?? (x


, x ) dx dx


…(4)

ij mn = ?
0


ij m n

m n
drmn


Where ?(xn, xm) is the second order density matrix, which is already normalized to the number of electron pairs within the system.


D (p , p ) =

? (p , p ) p2p2 d?1 d?2


…(5)


ij 1 2

?? ij 1 2 1 2


where Dij (p1, p2) represent the two particle density distribution function and, d?k = sin?kd?kd?k where k = 1 or 2.

The one particle radial density distribution function is –


?
Dij (p1) = ? Dij (p1, p2) dp2
0


…(6)


Each inter particle separation function fij (pmn) is normalized to unity and integral of
? fij (pmn) against pmn is identically zero10.

Also, if the position of electron 1 is specified as r1, then we consider a related distribution function such as g (p12; p1).


g (p ; p ) = 8?2


p12p1 J1 …(7)


where

p p +p
12 12 1

J1 =

? p1dp1

? ?(p1, p2) p2dp2

…(8)

0 p12 ?p1

and

2 2

?(p1, p2) = ?1S (p1) ?1S

(p2)

…(9)


The influence of the correlation in partial distribution function g (p12; p1) of p12 and p1 for Z = 2, 3, 4 and 5 for K-shell are shown in Table 1. Fig. 1 shows g (p12; p1) surface for the S-symmetry states for He-like ions. Selected contour diagrams can be seen in Fig. 2. In HF approximation, each electrons move independently of each other, so this hypothesis neglecting the details of the electronic repulsion will reduce the HF results. The HF model indicates that the average angle between the electronic momentum vectors is 90o, the



location of the most probable distribution of g (p12, p1)

density can be estimated by using

“Pythagoras s theorem”. The

g (p12, p1) surface show that the maximum density is always

located at the diagonal such that between p1 and p2).

p12 > p1 because < cos ?12 > = 0 where (?12 is the angle


Table 1: Momentum partial radial distribution function of He-like ion

Atomic number Z shell gHF(p12,p1) gCI(p12,p1)
2 K?K? 0.144647 0.145591
3 K?K? 0.123550 0.125593
4 K?K? 0.018233 0.019683
5 K?K? 0.015388 0.159330

The partial coulomb shift ?g (p12, p1) against (p12; p1) to study the correlation effects when the test electron be particle 1, with fixed magnitude p1; and the integral of partial coulomb shift against p1 is equal to ?fij (p12). Due to the s-symmetry of inter-shells, K? LB KBL?, the integration procedure in equation (3) and each fij (p12) is normalized to unity, and ?fij (p12) against p12 is equal to zero. The shapes of the partial coulomb shifts in Fig. 3 and 4 reflect the behaviors corresponding to g (p12, p1) surfaces, and contour diagrams. For K? KB, the HF distribution straddles the (p12 = p1) axis with a single peak located at p1? p1 and p12 ?

(2p2 )0.5, where p

being the most probable value of p , is deduced from the maximum of the

k k 1
radial density distribution function D(p1, p2) (see equation 5). This relationship between (p12, p1) value at gHF peak and mode of D(p1) follows that found in position space by Banyard and Al-Bayati. The ?g (p , p ) surface showed strong similarities with that for Li+-ion there is
reduction in the depth p1 < 0.4, and an increase in height of approximately (10%) when p1 >
0.4 (and for all Z). For inter-shells, it was found that, as in the K-shift radial correlation alone increases p12 whereas angular correlation decreases p12.

These opposing effects produce respective expansions and contraction in the normalized p12 distribution. ?g (p12, p1) arises, when the test electron is located in the L-shell. Therefore, the p12 variation should reflect the influence of correlation acting on K-shell electron. When p1 >1, the test electron is located in K-shell, and hence, the diagonal fluence in ?g (p12, p1) represent. For closed –shell system, it was found that both positive regions were equally obvious for each inter-shell; it was also shown that the results for k? L? were in excess of KB L?

































Fig. 1: The partial Coulomb holes g(p12; p1), as a surface diagrams for (A) He-atom,
(B) Li+ ion, (C) Be2+ ion, (D) B3+ ion. Momentum correlated description of each state, with expectation of 1s2 was obtained from
Hartree-Fock (HF) wave function
















Cont…

















Fig. 2: The partial Coulomb holes g(p12; p1), as a contour diagrams for (A) He-atom,
(B) Li+ ion, (C) Be2+ ion, (D) B3+ ion. Momentum correlated description of each state, with expectation of 1s2 was obtained from
Hartree-Fock (HF) wave function





Fig. 3: The partial Coulomb holes ? g(p12; p1), as a surface diagrams for (A) He-atom,
(B) Li+ ion, (C) Be2+ ion, (D) B3+ ion. Momentum correlated description of each state, with expectation of 1s2 was obtained from Hartree-Fock (HF) wave function





0.81

0.6

0.4

0.2

0

(A)










0 0.2 0.4 0.6 0.81
p1



0.812

0.6

0.4

0.2

0

(B)










0 0.2 0.4 0.6 0.81
p1





0.812

0.6

0.4

0.2

0

(C)










0 0.2 0.4 0.6 0.81
p1



0.812

0.6

0.4

0.2

0


(D)










0 0.2 0.4 0.6 0.81
p1


Fig. 4: The partial Coulomb holes ? g(p12; p1) , as a contour diagrams for (A) He-atom,
(B) Li+ ion, (C) Be2+ ion, (D) B3+ ion. Momentum correlated description of each state, with expectation of 1s2 was obtained from
Hartree-Fock (HF) wave function

CONCLUSIONS

From the present work, it was noted that as Z increases, partial distribution function decreases in momentum treatment for both approximations. The results of partial distribution function for configuration interaction is larger than that for Hartree-Fock because each electrons move independently of each other, so this hypothesis neglectes the details of the electronic repulsion, which will reduce the HF results.

REFERENCES

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3. Z. Huang and S. Kais, Chem. Phys. Lett., 413, 1 (2005).

4. A. W. Weiss, Phys. Rev., 122(6), 1826 (1961).

5. C. A. Coulson and A. H. Neilson, Proc. Phys. London, 78, 831 (1961).

6. K. E. Banyard and P. K. Youngman, J. Phys. B: At Mol. Phys., 20, 5585 (1987).

7. C. C. J. Roothaan, L. Sachs and A. W. Weiss, Rev. Modern Phys., 32(2), 186 (1960).

8. K. E. Banyard and P. K. Youngman, J. Phys. B: At Mol. Phys., 15, 854 (1982).

9. K. H. Al-Bayati and K. E. Banyard, J. Phys. B: At Mol. Phys., 20, 367 (1987).

10. L. S. Bartell and R. M. Gavin, J, Chem. Phys., 43(3), 856 (1965).



Accepted : 16.07.2014