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.

F( r ) = Σ _j   C _j   S _nl ( α _j ,r) Y _l,m ( r ) ,

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.