Determinantal eigenfunctions

For tables and the construction of determinantal eigenfunctions of atoms see, for example. Slater. ... 

Our next problem is then to combine the determinantal eigenfunctions, the D functions, into new trial eigenfunctions which are eigenfunctions of M, S, Mg, and Sg. We shall first show that these trial eigenfunctions are already eigenfunctions of Mg and Sg. Let us take a general determinantal eigenfunction... 

Such a "general form of wave function is easily written explicitly for each set of values of N, S, and MS- Any appropriate form of approximate wave functions, like determinantal functions composed of one-electron functions ( molecular spin orbitals ), the "bond eigenfunctions" used in the valence bond approach, and so on, is shown to fulfil this requirement. 

It should be noted that in the case of the helium isoelectronic series (singlet GS of a two-electron system), because only one orbital is sufficient to construct the determinantal wave function and to obtain the density as n(r) — 2[t)r(r)], the minimization over n in Eq. (64) is equivalent to the minimization over ip in Eq. (28). Therefore the HF-KS and HF equations and their eigenfunctions are the same, ipir) = and consequently AT n =... 

In summary, to obtain a many-electron wave function of the single determinantal form [equation (A.12)] which will give the lowest electronic energy [equation (A.14) or (A.27)], one must use one-electron wave functions (orbitals) which are eigenfunctions of the one-electron Fock operator according to equation (A.42). There are many, possibly an infinite number of, solutions to equation (A.42). We need the lowest Ne of them, one for each electron, for equation (A. 12) [or (A.27)]. When the Ne MOs of lowest energy satisfy equation (A.42), then Eq=Ehf [equation (A.27)] and o= hf [equation (A.12)]. 

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. 

Methods based upon Slater determinantal functions (SDF). When we take this approach, we are, in effect, applying the antisymmetrization requirement first. Only if the orbitals are all doubly occupied among the spin orbitals is the SDF automatically, at the outset, an eigenfunction of the total spin. In all other cases further manipulations are necessary to obtain an eigenfunction of the spin, and these are written as sums of SDFs. 

The determinantal functions must be linearly independent and eigenfunctions of the spin operators S2 and Sz, and preferably they belong to a specified row of a specified irreducible representation of the symmetry group of the molecule [10, 11]. Definite spin states can be obtained by applying a spin projection operator to the spin-orbital product defining a configuration [12]. Suppose d>0 to be the solution of the Hartree-Fock equation. From functions of the same symmetry as d>0 one can build a wave function d>,... 

To consider that it is and not the form of each individual m.o. that ultimately matters is a consequence of the determinantal form of the wavefunc-tion. In fact, determinants remain unchanged when the respective elements are subject to some specific operations (unitary transformations) as was already illustrated on page 89 (Problem 5.2). For example, the eigenfunction (Eq. (8.15)) is not changed if we replace each m.o. (row i) by Xi= + Such mathematical alterations inside the determinant do not correspond to any physical change, because the eigenfunction S remains unchanged. 

The resulting wave functions are eigenfunctions that transform as irreducible representations of the molecular point group. Thus, existing ab initio molecular structure codes can be used to calculate these LCAO-MO determinantal wave functions. 

That is, any one-determinantal wave function is an eigenfunction of operator Nj with the eigenvalue nj. Operator Nj is said to be an occupation number operator. 

We have found that the eigenvalue of operator N is the total number of electrons in the system under consideration. Single determinantal wave functions are eigenfunctions of N. This operator is called the particle number operator namely, it is the operator of the total number of electrons in the system. 

While the solution of the Hiickel model is mathematically very simple (it consists of the diagonalization of the matrix of the one-electron integrals h ), the diagonalization of the Hubbard Hamiltonian is not always straightforward (Whaley Falicov 1987). One-determinantal wave functions can be calculated analogously to the Hartree-Fock theory by an iterative procedure this corresponds to averaging the two-electron term. Such wave functions are only approximations to the exact solution of the Hubbard Hamiltonian whose exact eigenfunctions can be expressed only as linear combinations of determinants. 

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