The Generator Coordinate Method

1 Introduction

The Hartree–Fock theory [1], at its limit, may provide about 98% of the energy of an atom or molecule. Still, we wish for better, not only in the search for testing Quantum Mechanics, but also because the “small” 2% error may have the magnitude of an ionization potential or an electronic transition.

Several alternatives are available if we endeavor to recover the missing portion of energy (correlation energy), such as many body perturbation theory (MBPT), density functional theory (DFT), and the variational configuration interaction (CI) method, often at the cost of nontrivial computational efforts.

Wheeler and collaborators [2], in the context of Nuclear Physics, showed in 1953–1957 that the limit in the variational procedure capacity itself was not reached. As we indicated in the Introduction (Chapter 1), the generator coordinate method (GCM) introduces an integral transform capable, in principle, of finding the best functional form for a given trial function through the Griffin–Hill–Wheeler (GHW) integral equation defined below.

2 Background for the Formulation of the Method

The GCM was introduced [2] in the field of Nuclear Physics. The proposition of Wheeler and collaborators was one of the first attempts to incorporate collective and single-particle nuclear motions into a single coherent quantum-mechanical formulation.

The α parameter, which plays an important role in the method, is initially introduced as a shape parameter of the nuclear liquid drop model, defining the size and shape of the drop.

Physics and chemistry have been interacting and feeding each other with questions and answers for centuries. However, the speed of interpenetration is variable. Thus, until the introduction of what we now call the generator coordinate Hartree–Fock (GCHF) method for atoms and molecules in 1986 [3] (see Chapter 4), the major part of the literature on the GCM dealt with the collective aspects of nuclei. It started with the classical paper of Hill and Wheeler [2a], which aimed at relating collective and single-particle aspects in the fission problem. The direct application to the bound state case was pioneered by Griffin and Wheeler [2b], who clearly recognized the GCM as a variational procedure. In Section 4 we review some of the literature in Nuclear Physics.

3 Formulation of the Method

Here, we shall follow closely the method presented by Griffin and Wheeler in 1957 [2b]. We search for a solution to the Schrödinger equation:

(x)?Ψ(x)=EΨ(x),

where H is the Hamiltonian operator, Ψ the eigenfunction, E the energy of the system, and x represents the space and spin coordinates. Then, a trial function Φ (x; α) is chosen, where α represents one or several generator coordinates. The trial function Φ may be an approximate solution of Equation (2.1), or the exact solution of a problem similar to Equation (2.1), or some other function appropriate for the case. Next, the integral transform function is built:

(x)=∫d?αf(α)?Φ(x;?α),

where f(α) is the weight function that needs to be determined. If the exact f(α) can be determined, then the integral transform in Equation (2.2) leads to the exact solution Ψ.

The energy functional E can now be written,

=∫dx?Ψ*(x)?H(x)?Ψ(x)/∫dx?Ψ*(x)?Ψ(x).

The use of Equation (2.2) in Equation (2.3) gives

=∫f*(α)?H(α,?β)?f?(β)?dαdβ/f*(α)?S(α,β)?f(β)?dαd?β,

with the energy H(α,β) and overlap S(α,β) kernels defined as:

(α,?β)=<Φ(x,α)|H|Φ(x,?β)>=∫dx?Φ*(x,?α)?H?Φ(x,β)

and

(α,?β)=<Φ(x,α)|Φ(x,?β)>=∫dx?Φ*(x,?α)?Φ(x,β),

respectively.

The generator wave function must now be chosen to make the integral an extreme value [see Equation (2.7)]:

=δE=∫dα?δf*?(α)?∫dβ?[H(α,?β)−ES(α,?β)]?f(β)+comp.conj.∫f*?(α)?S(α,?β)f(β)dα?dβ

The coefficients of δf*(α) and δf*(α) must vanish separately, because these are two linearly independent variations. Thus, one arrives at the generator wave equation (often called GHW equation):

[H(α,?β)−ES(α,?β)]?f(β)dβ=0.

The analytical solution of the GHW equation for many electron atoms and molecules (or many particle systems in general) is beyond present mathematical capabilities. Thus most applications have relied on either approximations, which is the case for nuclei, or discretization techniques, as in the case of atoms and molecules (see Chapter 5).

There seems to be only a few analytical solutions for the GCM, which we comment upon in the next chapter.

4 Applications in Nuclear Physics

Certainly the early and numerous applications of the GCM arose in the field of Nuclear Physics. From the very beginning, the Nuclear Physics community gave preference to the Gaussian overlap approximation (GOA) for the solution of the GHW equation.

In the GOA [4], the overlap kernel [Equation (2.6)] is replaced by a Gaussian function of the form

(α,?β)≅SGaussian?(α,?β)=exp{−12[α−βa(α¯)]},

where the width α is often chosen as a function of the average of the GCM generator coordinate α¯?=(α+β)/2.

In the seventies, attempts were made to mobilize the GCM for the scattering problem of complex particles as an alternative to the resonating group method [5]. Considerable literature for the bound state has been reviewed by Klein [6], Villars [7], Brink [8], Mihailovich and Rosina [9], Wong [10], and, with specific emphasis on the scattering aspects, Giraud et al. [11]. We also refer to a conference held in Belgium in 1975 summarizing this topic [12].

More recent revisions may be found in the book by Ring and Schuck [13] and the review by Bender et al. [4].

5 Some Alternative Proposals to the Generator Coordinate Method

In 1968, Somorjai and, later, Somorjai and Bishop [14] introduced an integral transform method, closely related to GCM, for atomic and molecular systems. Somorjai was aware of the work of Wheeler and collaborators, but was critical of the use of Gaussian distributions for solving the integral equation.

The procedure to solve the integral transform employed by Somorjai and Bishop was to choose a definite form for the weight function and then to use the upper and lower integration limits as variational parameters. In his first work on the subject [14a], Somorjai employed Hulthen functions to represent 1s orbitals following a proposition by Parr and Wave [15]. The two-parameter Hulthen function, Fα β (r), has the form

αβ=r−1[e−αr−e−βr].

The identity,

ɛβ(r)=r−1[e−αr−e−βr]=∫αβe−rxdx,

shows that Fαβ is a linear combination of an infinite number of screened 1s functions, with orbital exponents ranging continuously in a variationally optimized interval [α, β]. Each 1s orbital has the same weight. Further, Somorjai rewrites the finite integral transform [Equation (2.10)] as a general Laplace transform

(r)=∫0∞f(x)?e−rxdx,

thus recovering the form of the trial function of the GCM. In work to follow, Somorjai and other authors did not enter the GCM ansatz, but treated the function f(x) in Equation (2.12) as having adjustable parameters. An extensive review of this method was later organized by Bishop and Schneider [16]. Also, in this context, accurate correlated functions for two- and three-electron atomic systems were obtained by Thakkar and Smith [17].

While the former solutions were certainly original and innovative, they masked to some extent the full...