Unrestricted determinants

Another significant advantage is that DFT methods based on unrestricted determinants (analogously to UHF, Section 3.7) for open-shell systems are not very prone to spin contamination , i.e. (S ), is normally close to S S + 1) (see also Sections 4.4 and 11.5.3). This is basicly a consequence of electron correlation being included in the determinantal wave function ... 

The simplest level of the END approach employs a single spin unrestricted determinant... 

The performance of various DFT methods resembles the FIF results. A restricted type determinant leads to an incorrect dissociation, while an unrestricted determinant has the energetically correct dissociation limit. Figure 11.12 shows the performance of restricted and unrestricted type determinants with the BLYP and B3LYP functionals. 

We have described the spin of a single electron by the two spin functions a(this section we will discuss spin in more detail and consider the spin states of many-electron systems. We will describe restricted Slater determinants that are formed from spin orbitals whose spatial parts are restricted to be the same for a and p spins (i.e., xi = il iP ), Restricted determinants, except in special cases, are not eigenfunctions of the total electron spin operator. However, by taking appropriate linear combinations of such determinants we can form spin-adapted configurations which are proper eigenfunctions. Finally, we will describe unrestricted determinants, which are formed from spin orbitals that have different spatial parts for different spins (i.e., fjS ). 

Unrestricted determinants are formed from unrestricted spin orbitals. Unrestricted spin orbitals have different spatial orbitals for different spins. Given a set of K orthonormal spatial orbitals, ... 

Unrestricted determinants are not eigenfunctions of 6. Moreover, they cannot be spin-adapted by combining a small number of unrestricted determinants as is the case for restricted determinants. Thus the UHF ground state (2.26S) of the Li atom is not a pure doublet as is the RHF ground state (2.264). Nevertheless, unrestricted wave functions are commonly used as a first approximation to doublet and triplet states. 

Figure 2.15 An unrestricted determinant that is approximately a doublet.

In spite of spin contamination, an unrestricted determinant is often used as a first approximation to the wave function for doublets and triplets because unrestricted wave functions have lower energies than the corresponding restricted wave functions. 

The next step concerns the calculation of the energy of a collection of spin unrestricted determinants with different occupations and relate their energies to the electronic structure parameters in order to calculate the X parameter and in this way obtain a measure for the biquadratic interaction strength from methods like DFT (Fig. 5.17). 

The Spin Projection approach can be generalized to any spin system. The energy of any intermediate spin state S can be expressed as a function of the energy of the higher multeplicity spin state as and of an unrestricted determinant corresponding to Mg = S, E(Mg) as ... 

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