Close encounters: The influence of electron coalescence on atomic wavefunctions.

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1 Close encounters: The influence of electron coalescence on atomic wavefunctions. David G. Ellis The University of Toledo Department of Physics and Astronomy April 2, 2003 (1) Atomic Structure and Spectra. What we are tryingto do. (2) Electron Correlation. One wavefunction for N electrons. (3) Cato s Cusp and Hund s Rule. It s important to consider close encounters. (4) A Proposed Solution. Try to add a new basis function to an already large set. 1

2 The startingmulti-configuration wavefunction Begin with a good approximate wavefunction for N electrons: Ψ 0 = γ C γφ γ The sum is over a number of configuration state functions (CSF). Each CSF Φ γ is an antisymmetric function of the electron coordinates, specified by the label γ, which includes not only the electron configuration {(n 1 l 1 ) w 1, (n 2 l 2 ) w 2, (n Ng l Ng ) w Ng}, but also the angular momentum coupling conditions and other quantities such as fractional parentage coefficients needed to specify the CSF. Φ γ (L, S, J, M) = α 1 α N α 1 α N γ,l,s,j,m Φ(α 1 α N ) Here Φ(α 1 α N ) is a Slater determinant, i.e. an antisymmetrized direct product of N single-particle Pauli wavefunctions (spin-orbitals): with of course Φ(α 1 α N )=A φ α1 (1) φ α2 (2) φ αn (N), φ α ( r, σ) = 1 r P nl(r)y lm (θ, φ)χ µ (σ) and α = {n, l, m, µ}.

3 The enhancement function for N electrons Given the best available orbital-based solution Ψ 0, we propose to construct an enhanced wavefunction Ψ=Ψ 0 +Ψ where Ψ is an antisymmetric N-electron function which contains explicitly correlated two-electron functions. We want Ψ to be zero or very small everywhere except where any two electrons come close together, and to be orthogonal to each CSF in the original solution Ψ 0. We write an antisymmetrized function as: Ψ = A { Ψ res (1, 2, N 2) Lr,S r,ψ ij(n } 1,N) Lp,S p Our initial trial form for the two-particle function is L,S,J,M ψ ij(1, 2) L,S = ψ L,M( r 1, r 2 ) φ(r 1,r 2,s) where s = r 12 is the inter-electron distance, and ψ L,M is the simplest possible function to give the electron pair the right total angular momentum. We can guarantee orthogonality by removing all the lower partial waves from our two-electron function, for example by the following definitions: with φ λ (r 1,r 2,s)=φ(s) λ 1 Γ k (r 1,r 2 )= 2k k=0 Γ k (r 1,r 2 ) P k (cos θ 12 ), φ(s) P k (cos θ 12 ) d cos θ 12.

4 Finding the change in energy We want: E = Ψ H Ψ E 0 We have: H = ij H ij Ψ=Ψ 0 + α Ψ H = E 0 +2α Ψ 0 H Ψ + α 2 Ψ H Ψ 1+α 2 Usingthe MCFP expansion we get: Ψ 0 H Ψ = ij C ij ψij (1, 2) H 12 ψ p(1, 2) So now the problem is reduced to doing a two-electron integral. But: ψ ij (1, 2) involves r 1,θ 1,φ 1,r 2,θ 2,φ 2 ψ p(1, 2) involves α, β, γ, r 1,r 2,θ 1 2

5 Calculation of Ψ( 1 L) H Φ( 1 L) (1) H is a symmetric spin-independent scalar two-body operator, for example the non-relativistic hamiltonian. (2) Φ( 1 L) is a two-electron spin-singlet state: Φ LM ( r 1, r 2 )= K ( 1) (M+K) φ LK (r 1,r 2,θ)D L MK(α, β, γ) Here θ is the angle between the vectors r 1, r 2, the Euler angles α, β, γ specify the orientation of the plane containing the two vectors, and D L MK is the Wigner D-function. (3) Ψ( 1 L) is a two-electron spin-singlet state represented by an orbital-based LS-coupled function defined by Ψ l1 l 2 LM( r 1, r 2 )= m 1,m 2 l 1,m 1 ; l 2,m 2 L, M ψ l1 m 1 ( r 1 )ψ l2 m 2 ( r 2 ) Now we can transform this also to the variables (r 1,r 2,θ,α,β,γ): Ψ l1 l 2 LM( r 1, r 2 )= K with the definition F LK (θ) = ( 1) M+K F LK (θ)r l1 (r 1 )R l2 (r 2 )D L MK(α, β, γ) m 1 m 2 l 1 m 1,l 2 m 2 LK Y l1 m 1 (π/2, θ/2)y l2 m 2 (π/2,θ/2)

6 The two-electron hamiltonian The non-relativistic hamiltonian for two electrons in the field of a fixed nucleus of charge Z can be written Hnr = H rad + Hang Z Z + 1. r 1 r 2 r 12 H rad = 1 2 2r 1 r1 2 r r 2 r2 r 2 Hang = 1 2r1 2 L r2 L 2 2. We must rewrite Hang in terms of the coordinates (α, β, γ, θ). First we define the sum and difference of the angular momenta L = L1 + L 2 and = L 1 L 2, and the components in the rotated frame L. The shift operators are L ± = L x ±il y and L ± = L x±il y. These have the following action on the Wigner D-functions: L 2 DMK(α, L β, γ) = L(L +1)DMK(α, L β, γ) L z DMK(α, L β, γ) = M DMK(α, L β, γ) L ± DMK(α, L β, γ) = C LM ± DM±1,K(α, L β, γ) L z DMK(α, L β, γ) = K DMK(α, L β, γ) L ± DMK(α, L β, γ) = C LK ± DM,K±1(α, L β, γ) The result for the angular part of the kinetic energy operator is Hang = r r2 (L ) r1 2 1 r2 L 2 = 4 csc θ θ sin θ θ + 2 sin2 θ ( L 2 sin 2 L z 2) + cos θ θ sin 2 θ L = i θ + cos θ L z i ( L 2 + L 2) 2sinθ 4sinθ ( L L 2)

7 The integrations which must be done To write the necessary integrals for matrix elements using Euler angles (α, β, γ) and internal coordinates (r 1,r 2,θ), define d 3 r 1 d 3 r 2 = dτ dω dω = 2π 0 dτ = 0 dα π 2π 0 sinβdβ dγ 0 r 2 1dr 1 0 r 2 2dr 2 π 0 sinθdθ. Now we can use the orthonormality integral for the Wigner functions 1 8π 2 D L M K (α, β, γ) D L MK(α, β, γ) dω = δ L Lδ M Mδ K K 2L +1 to write our interaction matrix element in terms of Rl1 (r 1 )R l2 (r 2 )H (L) (r 1,r 2,θ)φ λ (r 1,r 2,θ)dτ where we have defined H (L) (r 1,r 2,θ) K K F LK D L MK H nr DMKdω L F0L LK Here Hnr is the nonrelativistic hamiltonian operator for the twoelectron system. It contains derivatives which act on the remaining variables (r 1,r 2,θ) in the enhancement function φ λ (r 1,r 2,θ), in addition to the derivatives with respect to the Euler angles, which we work out analytically.

8 Finally the crucial function is H (L) (r 1,r 2,θ)= 2 1 [l1 l 2 ]/[L] l 1 0,l 2 0 L0 P l1 (cos θ) H rad + U 1 2g + Dθ 2 L(L +1) + 24 πi L(L +1)/3( 1) l 1 [ ] 1/2 l 1 0,l 2 0 L0 4 Fl 10 1 l 1 (θ)(2g + g )(cot θ +2 θ ) + π( 1)L+1 [l2 ] 24 (2L +3)! l 2 0,L0 J0 5 (2L 2)! J FJl 20 1 (θ)(2g + cot 2 θ + g ) + π 4 ( 1)L [L]L(L +1) [l 2 ] 2 LL L 1 1 J [ 2g+ (1 + 2 csc 2 θ) g ] l 1 l 1 1 LLl 2 L L 2 J l 2 0,L0 J0 L L 2 J csc θ [ F 22 Jl 1 (θ)(2g + cot θ ig )+F 2 2 Jl 1 (θ)(2g + cot θ + ig ) ]. Here we have used the definitions U = Z/r 1 Z/r 2 +(N 1)/r 12 g ± = 1/r 2 1 ± 1/r 2 2 Note that H rad and D 2 θ contain derivatives which act on φ λ (r 1,r 2,θ) and which must be worked out analytically, and then the threedimensional integral dτ must be done numerically.

9 Current and future work (1) Extend to pairs with L > 0. This is what I m working on now. The equations that must be programmed are much more complicated. (2) Make the integrals more efficient. The three-dimensional integrals are very time-consuming. (3) Make the method relativistic. The whole thing needs to be redone using Dirac wavefunctions. (4) Transition probabilities. We don t want just wavefunctions, we need to be able to calculate transition probabilities and other atomic properties. This will involve working out more three-dimensional integrals.