Construction of Trial Wave Functions

Explain the principles underlying the construction of trial wave functions ... 

Clearly, computational resources are not yet to the point that brute force methods will suffice for high-precision calculations. Physical and chemical intuition play an important role in constructing appropriate trial wave functions and, despite the complexity and size of the basis sets used in these calculations, can improve the convergence of the basis significantly. It is important, therefore, to understand the nature of an atomic or molecular system in terms of its physical as well as its more formal mathematical properties. 

A final word about electron-impact ionization, which persists as one of the fundamentally unsolved problems in atomic and molecular physics. The central difficulty is constructing a trial wave function with the correct asymptotic form. In electron-impact ionization three charged particles (two electrons and one positive ion) separate in the asymptotic region. The correct asymptotic form is a solution of the three-body Coulomb problem, and is extremely complicated. Simply applying these boundary conditions is a difficult problem in itself, even for atoms. In the last few years significant progress has been made for the simplest case of a one- or two-electron atomic target. However, there is currently no practical ab initio approach to electron-impact ionization of molecules. 

At present, it is possible to achieve accuracy for two- and three-electron systems superior to that once obtained for two-electron atoms by Pekeris. This can be accomplished by dealing explicitly with the most difficult points of the wave function, as in the Fock basis, or implicitly by constructing sufficiently flexible trial functions through the use of multiple basis sets. In any case, such flexibility is required to deal with the differing character of the wave function at large and small length scales. Drake s[2,27] calculations on helium demonstrate how a double basis set can achieve this kind of flexibility, while Morgan and co-workers[23,31] have combined Fock and double basis sets in a relatively compact wave function to produce equally precise results. 

Among the classes of the trial wave functions, those employing the form of the linear combination of the functions taken from some predefined basis set lead to the most powerful technique known as the linear variational method. It is constructed as follows. First a set of M normalized functions dy, each satisfying the boundary conditions of the problem, is selected. The functions dy are called the basis functions of the problem. They must be chosen to be linearly independent. However we do not assume that the set of fdy is complete so that any T can be exactly represented as an expansion over it (in contrast with exact expansion eq. (1.36)) neither is it assumed that the functions of the basis set are orthogonal. A priori they do not have any relation to the Hamiltonian under study - only boundary conditions must be fulfilled. Then the trial wave function (D is taken as a linear combination of the basis functions dyy... 

The original Heitler-London calculation, being for two electrons, did not require any complicated spin and antisymmetrization considerations. It merely used the familiar rules that the spatial part of two-electron wave functions are symmetric in their coordinates for singlet states and antisymmetric for triplet states. Within a short time, however, Slater[10] had invented his determinantal method, and two approaches arose to deal with the twin problems of antisymmetrization and spin state generation. When one is constructing trial wave functions for variational calculations the question arises as to which of the two requirements is to be applied first, antisymmetrization or spin eigenfunction. 

The response equations are usually solved in some iterative manner, in which the explicit construction of Q is avoided, being replaced by the repeated construction of matrix-vector products of the form Q where v is some trial vector . In general, the solution of one set of response equations is considerably cheaper than the optimization of the wave function itself. Moreover, since the properties considered in this chapter involves at most three independent perturbations (corresponding to the three Cartesian components of the external field), the solution of the full set of equations needed for the evaluation of the molecular dipole-polarizability and magnetizability tensors is about as expensive as the calculation of the wave function in the first place. 

Gorecki and Byers Brown [14,15] proposed an approach based on the variational boundary perturbation theory. In this method the trial wave function for the confined system / is constructed as the product of the wave function for the free (unbounded) system /, times a non-singular cut off function /, to ensure fulfillment of the boundary condition /(ro, cp, 0) = 0. The cut-off function clearly vanishes at ro, /(ro) = 0... 

In the procedure outlined we have used a linear combination of atomic orbitals as the trial wave function. This procedure is known as the LCAO method or as the LCAO "MO method, the latter designation meaning that we have constructed a molecular orbital (MO) as a linear combination of atomic orbitals. The LCAO method is frequently employed,-but unless additional terms are added the agreement with experiment is never very close. 

The construction of a trial wave function ( JM) for a Al-electron atomic system starts with a set of one-electron basis ftmetions oi. The Al-electron configuration-state function

simultaneous eigenfunction of the angular momentum operators J and... 

Suppose a trial wave function for the description of a molecule is constructed having the form... 

The construction of the linear energy model starts from an explicitly normalized expression for the trial wave function. 

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