In Section 2.2 we defined   { | χ 〉 }   to be an orthonormal basis in the direct product space   ℋ ⊗ 𝒮 , where   | χ 〉 ≡ | ψ 〉 ⊗ | σ 〉 ,   ℋ   is the single-particle Hilbert space, and  𝒮   is the single-particle spin vector space. Similarly, if   | ψ 1 〉 ⊗ | ψ 2 〉 ⊗ ⋯ ⊗ | ψ N 〉   is a base eigenket in the direct product Hilbert space of N particles,   ℋ 1 ⊗ 2 ⊗ ⋯ ⊗ N ,  and   | s 1 ⁢ m 1 〉 ⊗ | s 2 ⁢ m 2 〉 ⊗ ⋯ ⊗ | s N ⁢ m N 〉   is a base eigenket in the direct product spin vector space of N particles

then

will be a base eigenket in the direct product space   ℋ 1 ⊗ 2 ⊗ ⋯ ⊗ N ⊗ 𝒮 1 ⊗ 2 ⊗ ⋯ ⊗ N . The position-spin space representation of Eq. (3.26) is given by

The set of spin-orbitals   { χ i ⁢ ( x ) }   used to construct the many-body wavefunction   Ψ ⁢ ( x 1 , x 2 , … , x N )   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,   ψ i ⁢ ( r i ) ,  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   ψ i α ( r ) ≠ ψ i β ( r ) ,   and

But the two sets of unrestricted spin-orbitals,   { ψ i α }   and   { ψ i β } ,   are not orthogonal, i.e.,

where   S ij αβ   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   Ψ   is antisymmetric. The simplest normalized antisymmetric N-particle state   | Ψ 〉 ,  constructed from single-particle states, and that satisfies the Pauli principle as required for fermions, is given by

ε ijk⋯  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,   ε 12345 ⁢ ⋯ . 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   5 5 - 5 != 3005   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,   | χ 1 ⁢ χ 2 ⁢ ⋯χ N ) = | χ 1 〉 ⊗ | χ 2 〉 ⊗ ⋯ ⊗ | χ N 〉 ,  and the angled bracket is used for a normalized antisymmetrized N-particle state,   | Ψ 〉 = | χ 1 ⁢ χ 2 ⁢ ⋯χ N 〉 . Angled brackets will still be used for a subspace (via a monomorphism [Geroch, p. 58]) of a N-dimensional direct product space, e.g.,   | χ 1 ⁢ χ 2 〉 = | χ 1 〉 ⊗ | χ 2 〉 ,  as found in the matrix elements of an arbitrary two-body operator,   〈 φ i ⁢ φ j | V ^ ij | ϕ i ⁢ ϕ j 〉 . 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   i = x i   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   ( - 1 ) P   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  Ψ  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.,   χ 2 = χ 5 ),   or if any of the electrons have the same coordinates (e.g.,   x 3 = x 6 ),   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 Ψ in terms of orthonormal one-electron spin-orbitals χ i   will only provide an approximate solution. V ^ ee   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   V ^ ee   into an effective one-body operator. In the Hartree-Fock theory this effective one-body operator will remain nonlocal, but in Kohn-Sham theory   V ^ ee   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.

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