Development note | |
---|---|

Changes to this chapter were last made on 25 March 2005. |

where the one-electron eigenvalue equation is given by

will form a set of exact solutions to the noninteracting many-body energy eigenvalue equation

where

We could have also written this as

to emphasize that each one-electron Hamiltonian actually has a set of solutions,

Conspicuously absent from the Hamiltonian constructed from Eq. (3.2), (3.3), and (3.4) is the term describing the electron-electron interactions,

*nonlocal*
two-electron operator that prevents a product of one-electron states from being an
exact solution to the many-body Schrödinger equation. If the electron-electron interactions

then symmetrization requirement of systems of identical particles can be written as

where the *plus* sign corresponds to a *symmetric* wavefunction and the
*minus* sign corresponds to an *antisymmetric* wavefunction. If

so the possible eigenvalues of

If
*i*th electron, then

It is easy to see that Eqs. (3.20) and (3.21) are true from Eq. (3.17) in the case that

and hence

The direct product eigenstates given by Eq. (3.6), and its position space representation in Eq. (3.10), are unphysical because they don't include spin degrees of freedom and they lack the symmetry properties required of quantum mechanical particles. The direct product eigenket in Eq. (3.6) will be symmetric if all the single-electron eigenkets are the same, but in general it has no specific symmetry properties. However, such states can be used to construct symmetric and antisymmetric states, but first we must include the spin degrees of freedom.

[Back to Table of Contents] [Top of Chapter 4] [scienceelearning.org]

In Section 2.2 we defined
*N* particles,
*N* particles

will be a base eigenket in the direct product space

where, following convention, we have used the notation

The spin-orbitals in Eq. (3.28) are *restricted* spin-orbitals; they are *restricted* in the sense that the spatial part of the spin-orbital is the same for both the spin-up electron and the spin-down electron. In general, restricted spin-orbitals can be written as

On the other hand, the spatial part of an *unrestricted* spin-orbital can be different for the spin-up and spin-down electrons; *unrestricted* spin-orbitals can be written as [Szabo, p. 105]

where

But the two sets of *unrestricted* spin-orbitals,

where

Although the direct product eigenstates given by Eq. (3.26), and its position space representation in Eq. (3.27), are exact solutions to the noninteracting many-body Schrödinger equation, they will not describe a system of fermions unless
*antisymmetric*.
The simplest normalized *antisymmetric* *N*-particle state
*Pauli principle* as required for fermions, is given by

*N*-dimensional generalization of the
*Levi-Civita tensor*, which is defined to be zero if any of the indices are the same, +1
for even permutations of the indices, and -1 for odd permutations. Cyclic permutations, for example, require an even number of permutations
to return the tensor to its original, ordered form,
*5! = 120* possible permutations we would have

and among the

It should also be mentioned, that in the notation of Eq. (3.36) we have used a curved bracket for a simple *N*-particle state with no specific symmetry properties,
*antisymmetrized* *N*-particle state,
*subspace* (via a *monomorphism* [Geroch, p. 58])
of a *N*-dimensional direct product space, e.g.,

In the position-spin space representation, the electrons are identified mathematically by their coordinate labels, and in the position representation of Eq. (3.36) [or Eq. (3.40)] one could either perform the sum over the permutations of the state labels, or over the permutations of the electron coordinates. An antisymmetric *N*-particle wavefunction representation of the
state in Eq. (3.36) can be constructed from the spin-orbitals by using the *Slater determinant*.
The Slater determinant is the position-space representation of Eq. (3.36); it can be written as

where we have used the short-hand notation
*i*th electron.
For the special case of a *restricted close-shell* Slater determinant, the bar notation of
Eqs. (2.31) and (2.32) can be used to write the determinant as

The Slater determinant can also be written as a sum over permutations

and sometimes the antisymmetric sum over permutations in Eq. (3.43) appears in the literature as

In Chapter 5, we will discuss the contribution of electron-electron interactions to the Hamiltonian. When electron-electron interactions are included, the Slater determinant expansion of
*nonlocal*
two-electron operator that prevents a single Slater determinant from being an
exact solution to the many-body Schrödinger equation. In Chapter 5 and Chapter 6 we will discuss
the Hartree-Fock method and density-functional theory approaches to solving the many-body problem; the Rayleigh-Ritz energy minimization principle will result in a transformation of the two-body operator
*nonlocal*, but in Kohn-Sham theory
*local* one-body operator.
It is important to keep in mind that
even for the ground state, the single Slater determinant will be an exact solution
for systems of independent electrons (*noninteracting* with each other), with only single-electron interactions
[Szabo, p. 130]. This would include systems in which a many-body
interaction has been transformed into an effective one-body interaction.
In Chapter 5, approximate solutions to the Hartree-Fock equations will be found by expanding the spatial part of the spin-orbitals with a set of basis functions, and a similar approach will be used to solve the Kohn-Sham equations in Chapter 6.