Towards Closed-form Solutions to the Few-body Schrödinger Equation

Paul Abbott

School of Physics
The University of Western Australia
Australia
paul.c.abbott@uwa.edu.au

Introduction

Exact solutions to the few-body Schrödinger equation (SE) have been sought since 1926

Introduction

Exact solutions to the few-body Schrödinger equation (SE) have been sought since 1926

Closed-form solutions only known for variables-separable systems, such as H and (Subscript[H, 2]^(+))

Introduction

Exact solutions to the few-body Schrödinger equation (SE) have been sought since 1926

Closed-form solutions only known for variables-separable systems, such as H and (Subscript[H, 2]^(+))

For He power series solutions for Ψ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12])) in interparticle coordinates ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12])) do not exist

Introduction

Exact solutions to the few-body Schrödinger equation (SE) have been sought since 1926

Closed-form solutions only known for variables-separable systems, such as H and (Subscript[H, 2]^(+))

For He power series solutions for Ψ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12])) in interparticle coordinates ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12])) do not exist

Ψ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12])) is not analytic—there is no Taylor series about ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12]))==(0,0,0)

Introduction

Exact solutions to the few-body Schrödinger equation (SE) have been sought since 1926

Closed-form solutions only known for variables-separable systems, such as H and (Subscript[H, 2]^(+))

For He power series solutions for Ψ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12])) in interparticle coordinates ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12])) do not exist

Ψ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12])) is not analytic—there is no Taylor series about ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12]))==(0,0,0)

Ψ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12])) is finite and continuous for all ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12]))

Introduction

Exact solutions to the few-body Schrödinger equation (SE) have been sought since 1926

Closed-form solutions only known for variables-separable systems, such as H and (Subscript[H, 2]^(+))

For He power series solutions for Ψ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12])) in interparticle coordinates ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12])) do not exist

Ψ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12])) is not analytic—there is no Taylor series about ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12]))==(0,0,0)

Ψ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12])) is finite and continuous for all ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12]))

Steps towards exact solution provide insights about structure, especially at the Coulomb singular points—interparticle coalescences or cusps

Introduction

Exact solutions to the few-body Schrödinger equation (SE) have been sought since 1926

Closed-form solutions only known for variables-separable systems, such as H and (Subscript[H, 2]^(+))

For He power series solutions for Ψ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12])) in interparticle coordinates ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12])) do not exist

Ψ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12])) is not analytic—there is no Taylor series about ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12]))==(0,0,0)

Ψ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12])) is finite and continuous for all ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12]))

Steps towards exact solution provide insights about structure, especially at the Coulomb singular points—interparticle coalescences or cusps

⟹ better variational basis sets and more accurate values of ⟨E⟩ and ⟨(Subscript[r, 12]^n)⟩ (needed for muon catalyzed fusion)

Introduction

Exact solutions to the few-body Schrödinger equation (SE) have been sought since 1926

Closed-form solutions only known for variables-separable systems, such as H and (Subscript[H, 2]^(+))

For He power series solutions for Ψ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12])) in interparticle coordinates ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12])) do not exist

Ψ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12])) is not analytic—there is no Taylor series about ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12]))==(0,0,0)

Ψ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12])) is finite and continuous for all ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12]))

Steps towards exact solution provide insights about structure, especially at the Coulomb singular points—interparticle coalescences or cusps

⟹ better variational basis sets and more accurate values of ⟨E⟩ and ⟨(Subscript[r, 12]^n)⟩ (needed for muon catalyzed fusion)

With each step a clearer picture of the overall structure emerges

Hydrogen

Wavefunction

Wavefunction in atomic units:

(Subscript[Ψ, n,l.m])(r)∝(e^(-ρ/2)) (ρ^l) (Subscript[L, n-l-1]^(2 l+1))(ρ)(Subscript[Y, l]^m)(θ,φ)

where ρ=((2 Z r)/n).

Hydrogen

Radial wavefunction

Series[(E^(-ρ/2)) (ρ^l) (Subscript[L, n-l-1]^(2 l+1))(ρ),{ρ,0,2}]//FullSimplify

Hydrogen

Spherical harmonics

GraphicsGrid[Table[SphericalPlot3D[Evaluate[Tooltip[(Abs[(Subscript[Y, l]^m)(θ,φ)]),HoldForm[(Subscript[Y, l]^m)(θ,φ)]==(Subscript[Y, l]^m)(θ,φ)]],{θ,0,π},{φ,0,((3 π)/2)},ColorFunction->({x,y,z,θ,φ,r}|->Evaluate[Hue[((m φ)/(2 π))]]),ColorFunctionScaling->False,PlotStyle->Opacity[0.5],BoxRatios->1,Mesh->False,Boxed->True,Axes->None,PlotRange->All,ImageSize->200],{l,0,3},{m,0,l}]]

(Subscript[H, 2]^(+))

Coordinates

The electron is at distances (Subscript[r, A]) and (Subscript[r, B]) from the nuclear charges (Subscript[Z, A]) and (Subscript[Z, B]) located at points A and B along the z axis (z=∓R/2). Curves of constant ξ (ellipse) and η (hyperbola) are shown.

(Subscript[H, 2]^(+))

Equation

((1/2)(∇^2)+ℰ+((Subscript[Z, A])/(Subscript[r, A]))+((Subscript[Z, B])/(Subscript[r, B])))ψ(r)=0

(Subscript[H, 2]^(+))

Visualization

(Subscript[H, 2]^(+))

E(R)

As R->0, E(R) depends on R and log(R) (Byers-Brown and Steiner 1966, Abramov and Slavyanov 1978, Klaus 1983)

E(R)~-2+(2/3)(((2R))^2)-(2/3)(((2R))^3)+(22/135)(((2R))^4)-(2/9)(((2R))^5)log(R)+O((R^5))

Helium

Interparticle coordinates (IC):  (Subscript[r, 1]), (Subscript[r, 2]), (Subscript[r, 12])

Manipulate[Module[{r1=Norm[b-a],r2=Norm[c-a],r12=Norm[c-b]},
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Tooltip[Arrow[{a,b},0.15],Style[(Subscript[r, 1]),Large]],Tooltip[Arrow[{a,c},0.15],Style[(Subscript[r, 2]),Large]],Tooltip[Arrow[{b,c},0.15],Style[(Subscript[r, 12]),Large]],LightBlue,Disk[a,0.2],Disk[b,0.15],Disk[c,0.15]}, 
ImageSize->360,
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{s==NumberForm[r1+r2,{4,2}],p==NumberForm[(r12/(r1+r2)),{4,2}],q==NumberForm[((r1-r2)/r12),{4,2}]},
{s==NumberForm[r1+r2,{4,2}],p==NumberForm[(r12/(r1+r2)),{4,2}],τ==NumberForm[((r1-r2)/(r1+r2)),{4,2}]},{r==NumberForm[(Sqrt[(r1^2)+(r2^2)]),{4,2}],y==NumberForm[((2r1 r2)/((r1^2)+(r2^2))),{4,2}],Ω==NumberForm[(((r1^2)+(r2^2)-(r12^2))/(2r1 r2)),{4,2}]},
{r==NumberForm[(Sqrt[(r1^2)+(r2^2)]),{4,2}],α==NumberForm[ArcSin@((2r1 r2)/((r1^2)+(r2^2))),{4,2}],θ==NumberForm[ArcCos@(((r1^2)+(r2^2)-(r12^2))/(2r1 r2)),{4,2}]},
{r==NumberForm[(Sqrt[(r1^2)+(r2^2)]),{4,2}],α==NumberForm[ArcSin@((2r1 r2)/((r1^2)+(r2^2))),{4,2}],β==NumberForm[ArcCos@(((r1^2)+(r2^2)-(r12^2))/((r1^2)+(r2^2))),{4,2}]}}}]],
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Helium

Schrödinger Equation

SE for a two-electron atom with nuclear charge Z

ℋ Ψ(r)=-(1/2) ((Subscript[Δ, 1])+(Subscript[Δ, 2]))Ψ(r)+(-(Z/(Subscript[r, 1]))-(Z/(Subscript[r, 2]))+(1/(Subscript[r, 12])))Ψ(r)=ℰ Ψ(r)

ℋ is the Hamiltonian

Helium

Schrödinger Equation

SE for a two-electron atom with nuclear charge Z

ℋ Ψ(r)=-(1/2) ((Subscript[Δ, 1])+(Subscript[Δ, 2]))Ψ(r)+(-(Z/(Subscript[r, 1]))-(Z/(Subscript[r, 2]))+(1/(Subscript[r, 12])))Ψ(r)=ℰ Ψ(r)

ℋ is the Hamiltonian

𝒯 is the kinetic energy operator, homogenous of degree -2

Helium

Schrödinger Equation

SE for a two-electron atom with nuclear charge Z

ℋ Ψ(r)=-(1/2) ((Subscript[Δ, 1])+(Subscript[Δ, 2]))Ψ(r)+(-(Z/(Subscript[r, 1]))-(Z/(Subscript[r, 2]))+(1/(Subscript[r, 12])))Ψ(r)=ℰ Ψ(r)

ℋ is the Hamiltonian

𝒯 is the kinetic energy operator, homogenous of degree -2

𝒱 is the potential energy, homogenous of degree -1

Helium

Schrödinger Equation

SE for a two-electron atom with nuclear charge Z

ℋ Ψ(r)=-(1/2) ((Subscript[Δ, 1])+(Subscript[Δ, 2]))Ψ(r)+(-(Z/(Subscript[r, 1]))-(Z/(Subscript[r, 2]))+(1/(Subscript[r, 12])))Ψ(r)=ℰ Ψ(r)

ℋ is the Hamiltonian

𝒯 is the kinetic energy operator, homogenous of degree -2

𝒱 is the potential energy, homogenous of degree -1

ℰ is the energy eigenvalue, homogenous of degree 0

Helium

Laplacian Δ==(Subscript[Δ, 1])+(Subscript[Δ, 2]) in IC

Laplacian operator

Δ=f|->(1/(Subscript[r, 1]^2))(∂_(Subscript[r, 1]))((Subscript[r, 1]^2)(∂_(Subscript[r, 1]))f)+(1/(Subscript[r, 2]^2))(∂_(Subscript[r, 2]))((Subscript[r, 2]^2)(∂_(Subscript[r, 2]))f)+(2/(Subscript[r, 12]^2))(∂_(Subscript[r, 12]))((Subscript[r, 12]^2)(∂_(Subscript[r, 12]))f)+(((Subscript[r, 1]^2)+(Subscript[r, 12]^2)-(Subscript[r, 2]^2))/((Subscript[r, 1])(Subscript[r, 12])))((((∂^2)f)/(∂(Subscript[r, 1]) ∂(Subscript[r, 12]))))+(((Subscript[r, 2]^2)+(Subscript[r, 12]^2)-(Subscript[r, 1]^2))/((Subscript[r, 2])(Subscript[r, 12])))((((∂^2)f)/(∂(Subscript[r, 2]) ∂(Subscript[r, 12]))));

Hyperspherical harmonics

Δ[1]

Δ[(Subscript[r, 1]^2)+(Subscript[r, 2]^2)-(Subscript[r, 12]^2)]

Δ[(Subscript[r, 1]^2)-(Subscript[r, 2]^2)]

These solutions to Laplace's equation are not determined by the SE but by application of the boundary condition of normalizability, which is equivalent to finding the correct asymptotic form of the eigenfunction.

More complicated examples

Δ[((Subscript[r, 1]^2)+(Subscript[r, 2]^2)-(Subscript[r, 12]^2))log((Subscript[r, 1]^2)+(Subscript[r, 2]^2))]//Simplify

Δ[((Subscript[r, 1]^2)-(Subscript[r, 2]^2))log((Subscript[r, 1]^2)+(Subscript[r, 2]^2))]//Simplify

Helium

Potential in IC

𝒱(r1_,r2_,r12_):=-((Z/r1)+(Z/r2))+(1/r12)

𝒱(λ (Subscript[r, 1]),λ (Subscript[r, 2]),λ (Subscript[r, 12]))==(λ^(-1))𝒱((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12]))//Simplify

Helium

Power series expansion

Ψ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12]))==(∑_(i=0)^∞)(∑_(j=0)^∞)(∑_(m=0)^∞)(Subscript[c, i,j,m]) (Subscript[r, 1]^i) (Subscript[r, 2]^j) (Subscript[r, 12]^m)

Use k=i+j+m to denote the degree of the homogenous polynomials in IC and try to solve in order of increasing k=0,1,2,…

Helium

Power series expansion

Introducing k has three advantages:

ℰ, 𝒱, and 𝒯 are homogenous of degree 0, -1 and -2 respectively

Helium

Power series expansion

Introducing k has three advantages:

ℰ, 𝒱, and 𝒯 are homogenous of degree 0, -1 and -2 respectively

expansions in different coordinate systems can be compared simply by matching their k-lines

Helium

Power series expansion

Introducing k has three advantages:

ℰ, 𝒱, and 𝒯 are homogenous of degree 0, -1 and -2 respectively

expansions in different coordinate systems can be compared simply by matching their k-lines

recurrence relations are solved to find (Subscript[Ψ, k]) in order of increasing k ▶

Ψ==(Subscript[Ψ, 0])+(Subscript[Ψ, 1])+(Subscript[Ψ, 2])+⋯

(Subscript[ΔΨ, k])==2𝒱 (Subscript[Ψ, k-1])-2ℰ (Subscript[Ψ, k-2])

Helium

Schrödinger equation in IC

Schrödinger:=ψ|->Δ(ψ)-2 𝒱((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12]))ψ+2 ℰ ψ

Helium

Recurrence relation

Schrödinger[(Subscript[c, i,j,m]) (Subscript[r, 1]^i) (Subscript[r, 2]^j) (Subscript[r, 12]^m)]//Expand

ℛℛ[i_,j_,m_]=Collect[%/.a_ (Subscript[r, 1]^(i+x_.)) (Subscript[r, 2]^(j+y_.)) (Subscript[r, 12]^(m+z_.)):>(a/.{i->i-x,j->j-y,m->m-z}),(Subscript[c, ___]),Simplify]==0

This links k+2, k+1, and k only.

Helium

Boundary conditions

(Subscript[c, i_,j_,m_]):=0/;i<0∨j<0∨m<0

Helium

Solution of the Recurrence Relation

The recurrence relations are solved in order of increasing k

k=0

We are solving Laplace’s equation:

(Subscript[ΔΨ, 0])==0

The general solutions to this equation are hyperspherical harmonics

c/:(Subscript[c, 0,0,0])=1;

Ψ/:(Subscript[Ψ, 0])=1;

Helium

Solution of the Recurrence Relation

k=1

Table[ℛℛ[i,j,-1-i-j],{i,-1,0},{j,-1,0}]

(Subscript[k, 1])=Solve[%//Flatten,{(Subscript[c, 1,0,0]),(Subscript[c, 0,1,0]),(Subscript[c, 0,0,1])}]//First

Collect[(∑_(i=0)^1)(∑_(j=0)^(1-i))(∑_(m=0)^(1-i-j))(Subscript[c, i,j,m]) (Subscript[r, 1]^i) (Subscript[r, 2]^j) (Subscript[r, 12]^m)/.(Subscript[k, 1]),Z]

Ψ/:(Subscript[Ψ, 1])=-Z ((Subscript[r, 1])+(Subscript[r, 2]))+((Subscript[r, 12])/2);

Helium

Exponentiated solution

Solution so far

(Subscript[Ψ, 0])+(Subscript[Ψ, 1])=1-Z ((Subscript[r, 1])+(Subscript[r, 2]))+((Subscript[r, 12])/2)

This solution is fine as (Subscript[r, i])->0 for i=1,2,12. However, it diverges as (Subscript[r, i])->∞

Exercise

Consider the exponentiated version of (Subscript[Ψ, 0])+(Subscript[Ψ, 1]):

ψ==(e^(-Z ((Subscript[r, 1])+(Subscript[r, 2]))+(Subscript[r, 12])/2))==1-Z ((Subscript[r, 1])+(Subscript[r, 2]))+((Subscript[r, 12])/2)+⋯

For Z≥1 show that ψ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12])) is finite as (Subscript[r, i])->∞ for i=1,2,12

The local energy (Subscript[E, local]) is defined by ((ℋ ψ)/ψ). Compute (Subscript[E, local]) and show that it is finite

Hints ▶

Note the triangularity condition: 0≤|(Subscript[r, 1])-(Subscript[r, 2])|≤(Subscript[r, 12])≤(Subscript[r, 1])+(Subscript[r, 2])<∞.

Expression such as (((Subscript[r, 1]^2)+(Subscript[r, 12]^2)-(Subscript[r, 2]^2))/((Subscript[r, 1])(Subscript[r, 12])))≡2cos((Subscript[θ, 1,12])) are finite

Helium

Non-analyticity

It is easily shown that the IC expansion leads to an inconsistent recurrence relation for the (Subscript[c, i,j,m]) for k=2

k=2

ℛℛ[-1,0,1]/.(Subscript[k, 1])

ℛℛ[1,0,-1]/.(Subscript[k, 1])

Reduce[{%%,%,Z>0}]

Helium

Early attempts

The following attempts suffered from the same problem—their solutions were not finite and continuous throughout configuration space, including at the electron-electron and electron-nuclear cusps:

Bartlett (1937)

Hyleraas (1956-60)

Pluvinage (1950)

Kinoshita (1957)

Kinoshita introduced a modified version of elliptic coordinates and (effectively) considered negative powers of the IC which violates the boundary conditions.

Helium

Fock Expansion

Helium

Fock Expansion

FE includes powers of logarithmic functions of the hyperradius, r=((((Subscript[r, 1]^2)+(Subscript[r, 2]^2)))^(1/2))

Helium

Fock Expansion

FE includes powers of logarithmic functions of the hyperradius, r=((((Subscript[r, 1]^2)+(Subscript[r, 2]^2)))^(1/2))

FE can be viewed as a generalization of Fuch's Theorem to partial differential equations

Helium

Fock Expansion

FE includes powers of logarithmic functions of the hyperradius, r=((((Subscript[r, 1]^2)+(Subscript[r, 2]^2)))^(1/2))

FE can be viewed as a generalization of Fuch's Theorem to partial differential equations

FE has been generalized to Z>2 and to molecular systems

Helium

Fock Expansion

In IC, the Fock Expansion (FE) reads

Ψ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12]))==(∑_(i=0)^∞)(∑_(j=0)^∞)(∑_(m=0)^∞)(∑_(p=0)^(⌊(k/2)⌋))(Subscript[c, i,j,m,p])(r^k) (((((Subscript[r, 1])/r)))^i)(((((Subscript[r, 2])/r)))^j) (((((Subscript[r, 12])/r)))^m) (log^p)(r),

where r=((((Subscript[r, 1]^2)+(Subscript[r, 2]^2)))^(1/2)) is the hyperradius

Notes ▶

log(r) is homogenous of degree zero

Although r log(r) is finite as r->0 it is not in the FE because action of the Laplacian Δ results in log(r), which is infinite as r->0

|(Subscript[r, 1])-(Subscript[r, 2])|≤(Subscript[r, 12])≤(Subscript[r, 1])+(Subscript[r, 2])≤r

Helium

Fock Expansion

For the ground state of helium, solutions for k=0,1,2 are known ▶

Ψ((Subscript[r, 1]),(Subscript[r, 2]),(Subscript[r, 12]))==1-Z ((Subscript[r, 1])+(Subscript[r, 2]))+((Subscript[r, 12])/2)-((π-2)/(3π))Z (Subscript[Y, 2,1])log(r)-(Z/3)(Subscript[Y, 2,1])log((((Subscript[r, 1])+(Subscript[r, 2])+(Subscript[r, 12]))/r))
+(Z/12)(Subscript[Y, 2,0]) (2log((Subscript[r, 1])-(Subscript[r, 2])+(Subscript[r, 12]))-((Subscript[s, 1])-((Subscript[s, 2])/π)))
+(Z/(6π))(Subscript[r, 12]) (((2(r^2)-(Subscript[r, 12]^2)))^(1/2))(2(sin^(-1))(y Ω)+π)
-(E/6)(r^2)+((4(Z^2)+1)/12)(Subscript[r, 12]^2)-((2Z)/3)(Subscript[r, 12]) ((Subscript[r, 1])+(Subscript[r, 2]))+((Z (3Z+1))/3)(Subscript[r, 1])(Subscript[r, 2])+(Subscript[a, 2,1])(Subscript[Y, 2,1])/;(Subscript[r, 1])>(Subscript[r, 2])

Definitions ▶

(Subscript[s, 1])==log((Subscript[r, 12]) (((2(r^2)-(Subscript[r, 12]^2)))^(1/2))+(Subscript[r, 1]^2)-(Subscript[r, 2]^2))

(Subscript[s, 2])==α log(((1+Ω)/(1-Ω)))-β  (log((Subscript[r, 12]) (((2(r^2)-(Subscript[r, 12]^2)))^(1/2))+(Subscript[r, 1]^2)-(Subscript[r, 2]^2))-log((Subscript[r, 12]) (((2(r^2)-(Subscript[r, 12]^2)))^(1/2))-(Subscript[r, 1]^2)+(Subscript[r, 2]^2)))
+2 (ℒ(((α-β)/2))-ℒ(((α+β)/2))+ℒ(((π-α+β)/2))-ℒ(((π-α-β)/2)))

ℒ(x):=-(∫_0^x)log((Abs[cos(t)]))dt⟺ℒ''(x)==tan(x)

Coordinates ▶

(Subscript[Y, 2,0])==(Subscript[r, 1]^2)-(Subscript[r, 2]^2)

(Subscript[Y, 2,1])==(Subscript[r, 1]^2)+(Subscript[r, 2]^2)+(Subscript[r, 12]^2)==(Subscript[r, 12])sin(α)cos(θ)

r==((((Subscript[r, 1]^2)+(Subscript[r, 2]^2)))^(1/2))

y==sin(α)==((2(Subscript[r, 1])(Subscript[r, 2]))/((Subscript[r, 1]^2)+(Subscript[r, 2]^2)))

Ω==cos(θ)

β==(sin^(-1))(y Ω)

Helium

Fock Expansion

Physical solutions

For a formal solution to become a physical solution it must be capable of satisfying the boundary conditions

In Fock's expansion, Kato's cusp conditions, and the exponential ansatz, Myers, Umrigar, Sethna, and Morgan (1991) showed that the FE up to k=2 removed second-order cusps

Proofs of convergence

Macek (1968)

Leray (1982, 83, 84)

Morgan (1986)

Helium

Coordinate Systems

Much work has been done to find the “optimal” coordinate system for solving the SE.

Helium

Coordinate Systems

Elliptic coordinates

Hyleraas (1928) defined “elliptic” coordinates: s=(Subscript[r, 1])+(Subscript[r, 2]), t=(Subscript[r, 1])-(Subscript[r, 2]), u=(Subscript[r, 12])

FE reads

Ψ(s,t,u)==∑(Subscript[c, i,j,k,p]) ((((u/s)))^i)((((t/u)))^j) (s^k) (log^p)(r)

Used by Freund, Huxtable and Morgan (1984) to obtain variational energies for the ground state of Helium to ~(10^(-13))

Ground state energy is -2.9037243770340… au

Helium

Coordinate Systems

Perimetric coordinates

x==-(Subscript[r, 1])+(Subscript[r, 2])+(Subscript[r, 12]), y==(Subscript[r, 1])-(Subscript[r, 2])+(Subscript[r, 12]), z==(Subscript[r, 1])+(Subscript[r, 2])-(Subscript[r, 12])

Used by Coolidge (1937), Pekeris (1959), and Barnett (1995)

One advantage is simplification of matrix elements:

⟨f|ℒ|g⟩=2(π^2) (∫_0^∞)(∫_0^∞)(∫_0^∞)(SuperStar[f])ℒ gdxdydz

Instead of the FE one can use expand Ψ in an orthogonal basis of Laguerre polynomials in perimetric coordinates and log terms are not required

With ϵ=(Sqrt[-ℰ]) and

u==ϵ (-(Subscript[r, 1])+(Subscript[r, 2])+(Subscript[r, 12])), v==ϵ ((Subscript[r, 1])-(Subscript[r, 2])+(Subscript[r, 12])), w==2ϵ ((Subscript[r, 1])+(Subscript[r, 2])-(Subscript[r, 12]))

Ψ(u,v,w)==(∑_(a,b,c))(Subscript[A, a,b,c])(Subscript[L, a])(u)(Subscript[L, b])(v)(Subscript[L, c])(w)

Disadvantage is slow convergence

Helium

Coordinate Systems

Spherical polar coordinates

(Subscript[r, 1]),(Subscript[r, 2]),Ω=cos(θ)

Expansion reads

Ψ((Subscript[r, 1]),(Subscript[r, 2]),Ω)==(∑_l)(Subscript[f, l])((Subscript[r, 1]),(Subscript[r, 2])) (LegendreP[l, Ω])

where log behaviour is in (Subscript[f, l])((Subscript[r, 1]),(Subscript[r, 2]))

Used by Hylleraas (1960), Newman (1973), Davis (1981), Pluvinage (1982) and Gottschalk (1987)

Advantage is orthogonality of Legendre polynomials (LegendreP[l, cos(θ)])

Disadvantage is resulting slowly convergent infinite sums over (LegendreP[l, cos(θ)])

Helium

Coordinate Systems

Hyperspherical coordinates

r,sin(α)==((2 (Subscript[r, 1]) (Subscript[r, 2]))/(r^2)),Ω==cos(θ)

Used by Fock (1954-58), Ermolaev (1959-61), and Abbott (1986)

FE reads

Ψ(r,α,θ)==(∑_(k=0)^∞)(∑_(p=0)^(⌊(k/2)⌋))(∑_(n,l))(Subscript[c, k,p,n,l])(r^k) (log^p)(r)(Subscript[Y, n,l])(α,θ)

Advantages ▶

Hypersherical harmonics (Subscript[Y, n,l])(α,θ) are a a complete orthonormal basis for functions ψ(α,θ)

Very easy to solve FE

Disadvantages ▶

Slowly convergent series

Extremely difficult to re-sum and simplify

Conclusions

The Fock expansion is ubiquitous and unavoidable for atomic and molecular problems

Solutions the FE are relatively complicated but many simplification techniques have been explored and many discarded (at least for the goal of exact solutions)

Some promising techniques remain, especially analytic expansions in (Subscript[r, 12]) and perturbation series

The greatly improved power and availability of computer algebra systems (CAS) should assist solving the FE for higher k

Many problems remain ▶

The k=3 line contains sets of complicated elliptic integrals

Resumming series is tedious, but can be automated using CAS

A simple and accurate scheme for determining the arbitrary coefficients is yet to be found

Expressions for the asymptotic expansions as r->∞, which determine the arbitrary coefficients, are very complicated