Spin-down orbitals

Here a is an adjustable parameter, usually determined by comparing Hartree-Fock and Xa atomic calculations. In the spin-unrestricted version, the spin-up and spin-down orbitals are distinct, so that in general the resulting wavefunction is not a spin eigenfunction. 

Due to the spin polarization effect, the magnetic orbitals can be difficult to identify from a spin-unrestricted calculation. Since the total energy of a Kohn—Sham determinant is invariant under unitary transformations between the spin-up orbitals among each other and spin-down orbitals among each other, one can arrange each spin-up orbital to overlap at most with each spin-down orbital on the basis of the corresponding orbital transformation (COT) (88—90). Then, the molecular orbitals (MOs) are ordered into pairs of maximum similarity between spin-up and spin-down orbitals and can be separated into three groups (i) the MOs with spatial overlap close to one (doubly occupied MOs),... 

Solve the Schrodinger equation (1) for the spin-up and spin-down orbitals with the potentials of step (1)... 

In the Xa scattered wave approximation, the exchange potential for spin-up electrons may be different from that for spin-down electrons. In particular, when unpaired electrons are present, the exchange potentials, and hence the spin-up and spin-down orbitals and their energy levels, are different. Thus, MO calculations are performed using a spin-unrestricted formalism so that separate orbital energy levels are given for spin-up (a) and spin-down (p) electrons. 

Fig. 4.25. Molecular-orbital diagrams for the MnO, " , MnO,.", and MnO clusters Oi, symmetry) obtained using the MS-SCF-Ya method. Orbital energies have been scaled relative to the nonbonding lt,g, and 6t, orbital energies. The numbers of electrons in the highest occupied molecular orbitals are shown by the arrow symbols, and the a and p symbols refer to spin-up and spin-down orbitals, respectively (after Sherman, 1984 reproduced with the publisher s permission).

Fig. 2. (a) The modulus squared of the overlap between the spin-up and spin-down orbitals in a spin-unrestricted Xa calculation on H2 the Coulson-Fischer point (b) The spin-polarized triplet, spin-restricted (R.Jfa), spin-unrestricted UA"a), energy-projected singlet (EQl)and self-consistently exchange-projected singlet (E02) Xa solutions for as a function of internuclear separation. 

The minus sign in the spin-down orbital is included to make the wavefunction odd under time reversal). The imaginary phase factor between the two contributions to each orbital results in the FON density... 

We can see that the time-reversed state corresponding to the original atomic orbital is a spin-down orbital with reversed angular momentum. Notice that if magnetic fields are present, time-reversal symmetry is lost. 

Fig. 11.7 Energy levels of MPc and MP (M = Fe, Co, a = spin-down orbital, orbital) [188]...

In the above procedure, a number of new orbitals are introduced, which are orthogonal to 5. Further, 1% ) are as many as the spin-down orbitals. Thus, a subspace 5 " of 5 is left without a basis. For this subspace, one can construct a basis of IXi), i > I, which are orthogonal to 5 and S . 

In practice, the construction of these orbitals is done by using linear combinations of the HF orbitals c ) (r). For this purpose, name the spatial parts of the spin-down orbitals by ( )J(r) = It is also considered that the orbitals are real as in the case of a complete set of orbitals, appropriate linear combinations can be taken so that real orbitals are obtained. 

HK theorems and KS equations can be extended to the spin-polarized systems where the electron- density components pa T),Pi r) for spin-up and spin-down orbitals differ i.e. the spin-density p (r) = Pc t) — pp r) is nonzero. 

Now, minimization is performed with respect to the coefficients while the basis functions are held fixed. The expansion is only exact in the limit of a mathematically complete basis set < >(x), which is impossible to obtain in practice. Thus, the results depend on the size and nature of the employed basis functions, but there is a well-defined basis set limit. Since the BO-operator is spin free, it is customary to let the orbitals be eigenfunctions of the single-electron spin-operator by choosing them to be either spin-up or spin-down orbitals. This leads to the spin-unrestricted (UHF) method a = a,P) ... 

For a closed-shell system, the spin-up and spin-down Fock operators are equal and the spin-orbitals are obtained in pairs of equal shape and energy. Instead of dividing the set of orbitals into spin-up and spin-down orbitals, it is also possible to pursue a division into closed-shell and open-shell orbitals. This leads to the restricted open-shell HF (ROHF) method [23-25]. The formalism for this method is slightly more involved than the UHF formalism, but the general ideas are identical. The ROHF wavefunction is an eigenfunction of the total spin squared (5 ) operator, while the UHF wavefunction does not have this feature. The energy of the UHF wavefunction, on the other hand, is lower than that of the ROHF wavefunc-... 

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