The need for symmetry restrictions

Let us assume that 0) is an approximate state associated with the electronic Hamiltonian H and that 0) possesses definite spin and spatial symmetries. If we wish to determine a set of parameters Kpq in exp(— ) that transforms the approximate state 0 into a new state 0) with the same spin and spatial symmetries 

As discussed in Section 4.4, the conservation of spin and spatial symmetries in the optimized wave function is then guaranteed only if the symmetry restrictions are explicitly imposed on the parametrization. In other words, the spin and spatial symmetries are conserved only if k contains only those spin-orbital excitation operators that transform as the totally symmetric representation of the Hamiltonian H. If nontotally symmetric rotations are allowed, then 0 will not have the symmetry of 0) and the symmetries of 0) and 0) will be different. 

When a perturbation V is applied to the system, the Hamiltonian becomes H + V. The allowed variations then become those that transform according to the totally symmetric representation of H + V rather than of H. For example, if we consider the ground state of the oxygen molecule and if V corresponds to an electric field perpendicular to the intemuclear axis, then the allowed spin-orbital excitations are represented by spin-conserving orbital excitation operators that are totally symmetric in the C2v point group. In other words, the allowed variations are described by operators Ep where the direct product of the irreducible representations of the orbitals (t p and (p is totally symmetric in C2v 

These examples should suffice to illustrate the need for symmetry-constrained spin-orbital rotations. Let us summarize the most common situations where restricted rotations are required  

Real and imaginary rotations Field-free nonrelativistic wave functions may be constructed from real orbitals and only real rotations are needed for optimizing the wave functions. On the other hand, in the presence of perturbations such as an external magnetic field, imaginary rotations are required to describe the perturbed state. 

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