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Wave function of helium-like ions

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Abstract

The determination of the wave functions can be considered to be important problem for the theoretical study of some inelastic processes. This information is needed for description of interaction of atom and ion with charge particles and photons. The wave functions are needed for calculations of one-electron (ionization, single excitation, charge transfer) and two-electron (double ionization, double excitation, transfer ionization) cross sections in areas beam-foil spectroscopy, electron or multiply charged ion impact processes, photon - atom interaction, etc. A convenient form of the wave functions to describe an initial and final state of target and projectile is needed to calculate the cross sections of these processes.
The purpose of this project is to collect the analytical wave functions of atoms and ions. The project calculations are based on variational method and wave function expansion in a linear combination of hydrogen - like functions with effective charges.

Keywords

wave function of atoms, wave function of ions, cross section calculations, helium-like, variational method, Hartree - Fock, excited states, ionization, single excitation, double ionization, double excitation

Notation

Z - nuclear charge of the ion;
Ne – number of electrons of atom or ion;
Z ion = Z - Ne - ion charge;
r j - an electron position in the atom or ion (j=1,... Ne);
r = ( r 1 , r 2 ... r Ne );
Ψ( r , L,M,S) = Σ g j Φ j ( r , L,M,S) - 2S+1 L   wave function of atom or ion ;
S, L, M - spin, orbital moment and its projection;
Energy=E N - eigenvalue of Hamiltonian ;
H = Σ j [ -0.5 Δ j - Z/ r j + Σ i 1/ / r j - r i / ] - atomic Hamiltonian ;
Φ j ( r , L,M,S) - a basic function;
g j - a relative weight of a basic function;
C(l1,m1,l2,m2,L,M) - a Clebsch - Gordan coefficient;
φ n,l,m ( α , r j ) = R n,l (α , r j ) * Y l,m ( r j ) - hydrogenic function with charge α
R n,l (α , r j ) - radial part of a hydrogenic function with charge α ;
F( r j ) - Hartree - Fock wave function
S n,l (α , r j ) = (2 α) n+1/2   [(2n)!] -1/2   r j n-1   exp( - α r j ) - Slater basic function with charge α ;
Y l,m ( r j ) - a spherical harmonic.
ORB.ENERGY - Orbital energy in atomic units.
NORM = < F( r j ) / F( r j ) >
< R > = < F( r j ) / r j / F( r j ) >
< R2 > = < F( r j ) / r j 2 / F( r j ) >
<1/R> = < F( r j ) / 1/r j / F( r j ) >
<1/R**2> = < F( r j ) / 1/r j 2 / F( r j ) >

Variational method

Let N symbol describes unknown state 2S+1 L   of atom or ion. E N is the energy of N state. The wave functions of the 2S+1 L   states with energy E n < E N (n=1,...N-1) are known and the conditions
< Ψ n ( r , L,M,S) \ Ψ m ( r , L,M,S) > = δ n,m       (1)
are assumed to be satisfied. The unknown wave functions Ψ N ( r , L,M,S) can be written in the form [1,2]
Ψ N ( r , L,M,S) = χ( r , L,M,S) -   Σ n   Ψ n ( r , L,M,S)   < χ( r , L,M,S) / Ψ n ( r , L,M,S) > ,      (2)
where χ( r , L,M,S) is function with varied parameters. This function is determined by the expansion
χ( r , L,M,S) = Σ j=1,J     g' j Φ j ( r , L,M,S) ,       (3)
where Φ j ( r , L,M,S) is a basic function and g' j is a relative weight of a basic function in (3). A basic function Φ j ( r , L,M,S) can be presented by combination of hydrogenic function φ n,l,m ( α , r j ) with charge α. The eigenvalue
E N = < Ψ N \ H \ Ψ N >       (4)
of Hamiltonian H is the function of J-1 relative weight parameters g' j and of J* Ne charges α and of 2J * Ne integer parameters of hydrogenic function (n,l). These values are determined from the minimum of the energy (4). As the wave functions Ψ n ( r , L,M,S) in (2) are also determined by functions Φ j ( r , L,M,S), the equation (2) can be written in the form
Ψ( r , L,M,S) = Σ g j Φ j ( r , L,M,S) .       (5)

There are the additional conditions for the adequate description of some states.
1.A wave function of doubly exsited state (nln'l') 2S+1 L (n ≥ 2, n'≥2) of helium - like ion is orthogonal to all wave functions Ψ ( r , (1sn″L) 2S+1 L)
< Ψ ( r ,(nln'l') 2S+1 L) \ Ψ ( r , (1sn″L) 2S+1 L)> = 0   for all n″.     (6)
These conditions are satisfied if the special combination of hydrogenic function
φ' ns ( α , r j ) = { φ ns ( α , r j ) - <φ 1s (Z, r') / φ ns ( α , r')> φ 1s ( Z, r j ) } * [ 1 - /<φ 1s ( α , r') / φ ns ( α , r')> / 2 ] -1/2       (7)
is used in (3) [1,2].
2. The wave function for a state with S=1 takes into account a Pauli's exclusion principle
Φ( r 1 , r 2 , L,M,S=1) = Σ m1,m2 C(l1,m1,l2,m2,L,M) { φ n1,l1,m1 ( α1, r 1 ) φ n2,l2,m2 (α2, r 2 ) - φ n2,l2,m2 (α2, r 1 ) φ n1,l1,m1 (α1, r 2 ) } * 2 -1/2 ,       (8)
where
< φ n1,l1,m1 ( α1, r ) / φ n2,l2,m2 (α2, r ) > = 0 .       (9)



Approximation of Hartree - Fock wave function by Slater basic functions.

A numerical solution of Hartree - Fock equation χ(r) is approximated by wave function

F( r ) = Σ j   C j   S nl ( α j ,r) Y l,m ( r ) ,

where coefficients α j and C j   are defined from the minimum of the functional

I = ∫ dr { χ(r) - Σ j   C j   S nl ( α j ,r)} 2 .



This work started at Department of Physics of Moscow State University in 1998 - 2000 was supported by the program "Russian Universities - Fundamental Investigations", grant no. 98-1-5247. Now the work is continued in Laboratory of Atomic Collisions of MSU SINP.



References
[1] Novikov N. V. and Senashenko V.S.Description of (2s2)1S, (2s2p)1P, (2p2)1D and (1snl)1L (n ≤ 6, l ≤ 2) - states of the helium atom by the variational method. // Optics and Spectroscopy Vol. 86, No. 3, 1999 , p. 371-377.
Abstract
[2] Novikov N. V. and Senashenko V.S. Description of (2s2)1S, (2s2p)1P, (2p2)1D- states of the helium-like ions by the variational method. // Vestnik Moscow University, ser 3. Physics. Astronomy. No. 6, 2000 , p.37-40. Full text
[3] Novikov N.V. New Method of the Approximation of Hartree-Fock Wave Functions. // International Journal of Mathematics and Computational Science V. 1 (2015) Issue 2, P.55-58 Full text