Determinantal wavefunction

The normalisation factor is assumed. It is often convenient to indicate the spin of each electron in the determinant this is done by writing a bar when the spin part is P (spin down) a function without a bar indicates an a spin (spin up). Thus, the following are all commonly used ways to write the Slater determinantal wavefunction for the beryllium atom (which has the electronic configuration ls 2s ) ... 

VVc can now see why the normalisation factor of the Slater determinantal wavefunction is I v/N . If each determinant contains N terms then the product of two Slater determinants, ldeU rminant][determinant], contains (N ) terms. However, if the spin orbitals form an oi lhonormal set then oidy products of identical terms from the determinant will be nonzero when integrated over all space. We Ccm illustrate this with the three-electron example, k ljiiiidering just the first two terms in the expansion we obtain the following ... 

Such a determinantal wavefunction is called a Slater determinant, after Slater (1929), and you should appreciate that a... 

The sums run over the occupied orbitals note that we have not made any reference to the LCAO approximation. The energy expression is correct for a determinantal wavefunction irrespective of whether the orbitals are of LCAO form or not. 

Suppose we have an HF determinantal wavefunction fi o constructed from singly occupied spin orbitals , (that is, a UHF wavefunction). Other... 

In the uncorrelated limit, where the many-electron Fock operator replaces the full electronic Hamiltonian, familiar objects of HF theory are recovered as special cases. N) becomes a HF, determinantal wavefunction for N electrons and N 1) states become the frozen-orbital wavefunctions that are invoked in Koopmans s theorem. Poles equal canonical orbital energies and DOs are identical to canonical orbitals. 

The method described previously is known as the unrestricted HF (UHF) method. It appears to be a logical choice. However, it does have one significant drawback the total single-determinantal wavefunction ... 

There is a technical point that deserves a closer discussion. This concerns the assignment of electron densities at the nucleus to the Is through 4s shells. First, it is realized that for a single-determinantal wavefunction such as HF or KS DFT, the electron density can be written in two alternative forms ... 

We may ask now, whether the same procedure may be applied to density-functional theory, just by replacing the Fock operator by the corresponding Kohn-Sham operator. To this end we have to look at the minimization of the total energy with respect to the density of a multi-determinantal wavefunction 4. We write the density as ... 

Further details of how to obtain the unitary matrix U are unimportant here. It suffices to recognize25 that any unitary transformation of CMOs leaves a determinantal wavefunction and density unchanged, and thus has no effect on the energy or other properties that could be calculated with this wavefunction or density. Thus, at the HF (or DFT) level we can rigorously write... 

The remaining step (d) in Fig. 4.106 (NLMO— MO) involves more complex tie-line patterns, but fortunately these are all superfluous and can be ignored as having no physical consequence This remark follows from the general (unitary) equivalence of MO determinantal wavefunctions formed either from NLMOs or... 

For small determinantal wavefunctions these statements are easily verified by explicit expansion the general proof rests on the fact that the determinant of a matrix product is equal to the product of the determinants of the matrices. 

For a single determinantal wavefunction built from orbitals labelled ( ),-, the quantity D(r) defined according to ... 

Suppose we have an HF determinantal wavefunction I o constructed from singly occupied spin orbitals 2. . fyn (that is, a UHF wavefunction). Other determinantal wavefunctions are derived from P0 by substitution of occupied spin orbitals by virtual spin orbitals. If we use indices A, B,. .. for the occupied spin orbitals and X, Y,... for the virtual spin orbitals, then we define complete single, double, triple,. .. substitution operators... 

The first term in brackets is the usual kinetic energy operator. The noninteracting reference system has the property that its one-determinantal wavefunction of the lowest N orbitals yields the exact density of the interacting system with external potential v(r) as a sum over densities of the occupied orbitals, that is, p(r) = Xl<)>,l2, and the corresponding exact energy E[p(r)]. The Kohn-Sham potential should account for all effects stemming from the electron-nuclear and electron-electron interactions. Not only does the Kohn-Sham potential contain the attractive potential v(r) of the nuclei and the classical Coulomb repulsion VCoul(r) within the electron density p(r), but it also accounts for all exchange and correlation effects, which have so to say been folded into a local potential vxc r) ... 

When forming the one-determinantal wavefunction with Kohn-Sham orbitals instead of Hartree-Fock orbitals, and taking the expectation value of the Hamiltonian, one obtains... 

Table 1 Correlation Corrections (eV) for Various Energy Components with Respect to Kohn-Sham and Hartree-Fock One-Determinantal Wavefunctions, for N2 at Three Internuclear Distances3...

In spite of the large differences in the individual energy terms, the total correlation energies of the HF and the KS determinantal wavefunctions are rather close, indicating that considerable cancellation occurs between the errors of opposite signs in the various energy components in the HF case. 

Due to the properties of determinants, a Slater determinantal wavefunction ° automatically fulfils the Pauli principle and takes care of the antisymmetric character of fermions. If written explicitly in terms of the single-particle orbitals,... 

For example, the ground state f 0 of the magnesium atom for which 12 electrons must be placed in the spin-orbitals lsOm% 2sOm% 2prn"s,3sOms is represented by the Slater determinantal wavefunction... 

In the independent particle picture, the ground state of helium is given by Is2 xSo. For this two-electron system it is always possible to write the Slater determinantal wavefunction as a product of space- and spin-functions with certain symmetries. In the present case of a singlet state, the spin function has to be... 

Table 3.1. Slater determinantal wavefunctions for the 2p2-electron configuration (see Sla60J).

For a calculation of this matrix element one first changes the order of orbitals in such a way that the two different orbitals in the determinantal wavefunctions are at the same positions. Since in the expansion of the continuum function into partial waves, equ. (3.5a), only S is allowed, one gets... 

The extension of a given determinantal wavefunction (called the parent wave-function, which in the simplest case can be just a one-electron spin-orbital) to include another non-equivalent electron (or even a group of non-equivalent electrons) is made with the help of vector-coupling or Clebsch-Gordan coefficients... 

---