Table of Contents
Changes to this chapter were last made on 25 March 2005.
A set of N one-electron Hamiltonians can be used to describe a system of N noninteracting particles. If exact solutions to a one-electron Hamiltonian are known, then the many-body problem for a fictitious system of independent electrons (noninteracting), with only single-electron interactions, will be trivial. A noninteracting many-electron Hamiltonian can be written as a sum over one-electron Hamiltonians
where the one-electron eigenvalue equation is given by
and we include the subscript "1" to emphasize that a single electron is being described. An example of a one-electron Hamiltonian is
where v(r) is an arbitrary external potential that involves only one-electron interactions, and is often assumed simply to be the electron-nucleus attraction energy,
The Hilbert space of a N-electron Hamiltonian, , is given by the direct product Hilbert space of N particles,
where is the Hilbert space of , the Hamiltonian operator of the ith electron. Many-body eigenstates written as a simple direct product of one-electron eigenkets
will form a set of exact solutions to the noninteracting many-body energy eigenvalue equation
where is given by Eq. (3.1). It should be emphasized that in the notation of Eq. (3.6), the subscripts label the states, not the electrons. A given electron is distinguished mathematically by its position in the sequence. In other words
We could have also written this as
is a 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 are neglected, then the direct product state Eq. (3.6) is a mathematically exact solution to the many-electron Hamiltonian in Eq. (3.7). This, however, is a good example of a mathematically exact solution giving a physically incorrect result. The problem is more serious than the neglection of the electron-electron interactions, which will be dealt with in the next chapter.
It is an empirical fact that a wavefunction describing a system of identical particles must be either symmetric or antisymmetric under the exchange of particle labels. It can also be shown, using relativistic quantum field theory, that systems of identical particles that have half-integral values of spin angular momentum (i.e., ) are antisymmetric under the exchange of particles, and those with integral values of spin (i.e., s = 1, 2, 3, 4,…) are symmetric under the exchange of particles. All known particles fall into one of these two categories and are referred to as fermions and bosons, respectively. As will be shown later, the antisymmetry property of a system of identical fermions leads to an important conclusion: two identical fermions cannot exist in the same quantum state. This is the Pauli exclusion principle, which Pauli originally introduced to explain the structure of atoms, but, in fact, all systems of identical fermions must obey it.
If we define the permutation operator as the operator that exchanges the ith particle with the jth particle, i.e.,
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 is applied to Eq. (3.12) a second time we find that
In other words
so the possible eigenvalues of are +1 and -1, as all ready indicated by Eq. (3.13). It is also easy to see that
Note that if is antisymmetric and , then . This can only be true if , and therefore two fermions cannot exist at the same point in space with the same value of spin.
If is an arbitrary operator in the subspace of the ith electron, then
where is the same operator in the subspace of the jth electron. In general, the order in which the permutation operator is applied is important. Expressed another way, this means that, in general,
If, however, the operator is symmetric, as would be the case if represents an observable, then
or two-electron operators
and the sum is over all the electrons in the system. The fact that the Hamiltonian operator is symmetric means that
and hence is a constant of the motion.
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.
In Section 2.2 we defined to be an orthonormal basis in the direct product space , where , is the single-particle Hilbert space, and is the single-particle spin vector space. Similarly, if is a base eigenket in the direct product Hilbert space of N particles, , and is a base eigenket in the direct product spin vector space of N particles
will be a base eigenket in the direct product space . The position-spin space representation of Eq. (3.26) is given by
The set of spin-orbitals used to construct the many-body wavefunction will have the form
where, following convention, we have used the notation
to denote the combined spatial and spin coordinates, and, as mentioned previously, the spatial part of the spin-orbital, , is referred to as an atomic orbital (AO) when applied to atoms and as a molecular orbital (MO) when applied to molecules.
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 , and
But the two sets of unrestricted spin-orbitals, and , are not orthogonal, i.e.,
where is the overlap matrix.
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 is antisymmetric. The simplest normalized antisymmetric N-particle state , constructed from single-particle states, and that satisfies the Pauli principle as required for fermions, is given by
is the 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, . To illustrate with the more specific example, consider a system of five electrons; among the 5! = 120 possible permutations we would have
and among the combinations that give zero are
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, , and the angled bracket is used for a normalized antisymmetrized N-particle state, . Angled brackets will still be used for a subspace (via a monomorphism [Geroch, p. 58]) of a N-dimensional direct product space, e.g., , as found in the matrix elements of an arbitrary two-body operator, . It should be emphasized that in Eq. (3.36), the subscripts label the states, not the electrons. A given electron is distinguished mathematically by its position in the sequence. In other words
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 to label the coordinate of the ith 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
where P is the parity of the permutation, and the factor is equal to +1 for an even number of permutations, and -1 for an odd number of permutations. The following shorthand notation is also commonly used for the Slater determinant
The Slater determinant is the simplest expansion of in terms of one-electron spin-orbitals that yields an antisymmetric wavefunction. One of the general properties of determinants is that if any of the rows or columns are the same, then the determinant is zero. If this rule is applied to the Slater determinant, it means that if any of the quantum states are the same (e.g., ), or if any of the electrons have the same coordinates (e.g., ), then at least two of the rows or columns will be identical and the determinant will be equal to zero. Hence, the Pauli principle is upheld by construction.
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 in terms of orthonormal one-electron spin-orbitals will only provide an approximate solution. is a 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 into an effective one-body operator. In the Hartree-Fock theory this effective one-body operator will remain nonlocal, but in Kohn-Sham theory will be transformed into a 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.[Back to Table of Contents] [Top of Chapter 3] [scienceelearning.org]