ABSTRACT
The unrestricted broken symmetry (UBS) approach formulated by Noodleman1 is focused on the determination of the energies of pure spin states on the basis of either unrestricted Hartree-Fock (UHF) or unrestricted density functional theory (UDFT) calculations. This was an obvious response to the requirements for magnetic properties of molecular metal dimers to be described. The energy of pure spin states with different total spin S for systems with an even number of magnetic electrons is subordinated to a simple relation
E S E S S JPS PS( ) ( ) ( )= − +0 1 (2.1)
where EPS(0) presents the energy of the pure-spin singlet state, while J is the Heisenberg exchange integral, or magnetic coupling constant. A possibility to obtain the constant, which is the main characteristic of the magnetic behavior of the molecular compounds, was the main goal of the approach. Within the framework of the UBS approach, J can be determined as1
J E E S
=
−( ) ( )max max
0 2
(2.2)
Here EB(0) and EB(Smax) are the energy of singlet and Smax spin states obtained by either UHF or UDFT schemes using |ψB〉 (see Equation 1.1). Since any Smax spin state is presented by a single conœguration (see discussion of conœguration V in regard to Figure 1.1), the |ψB〉-based calculation always provides the pure spin state in this case. For n pairs of magnetic electrons Smax = n. According to Noodleman,1 the energy of a pure-spin singlet state is expressed as
E E S JPS B( ) ( ) max0 0= + (2.3)
while the energies of higher spin states, following Equations 2.1 and 2.3, are determined as
E S E S S S JPS B( ) ( ) ( ) max= − + −[ ]0 1 (2.4) Here
J J JF AF= + (2.5)
and
J n
a b r
=
∑1 12 12, (2.6)
J n
S a h b a b r
i i i i i ii i= ( ) +
(2.7)
JF and JAF are ferromagnetic (positive) and antiferromagnetic (negative) coupling constants, S-ai –biis the overlap of the nonorthogonal magnetic orbitals –ai = ai + cibi and
– bi = bi + ciai (see Equation 1.1), h is one-electron
Hamiltonian, and r12 determines the distance between electrons of atoms A and B in the dimer.1 The sign of magnetic coupling J indicates which kind of electronic interaction, exchange or Coulombic, dominates.
