Operator orbital rotation

In the self-consistent field linear response method [25,46,48] also known as random phase approximation (RPA) [49] or first order polarization propagator approximation [25,46], which is equivalent to the coupled Hartree-Fock theory [50], the reference state is approximated by the Hartree-Fock self-consistent field wavefunction < scf) and the set of operators /i j consists of single excitation and de-excitation operators with respect to orbital rotation operators [51],... 

This shows that all of the matrix elements of the partitioned orbital Hessian matrix involving redundant orbital rotation operators vanish. In particular the diagonal elements involving such operators vanish. This diagonal element relation, along with the gradient relation of Eq. (249), has been used by the author to identify redundant variables during the iterative MCSCF optimiz-... 

In the above expression 0) is the reference determinant, whereas exp[—ic(t)] is the (unitary) orbital-rotation operator, being the exponential of the anti-Hermitian operator... 

The K operator must be anti-Hermitian in order for the orbital-rotation operator to be unitary. The advantage of this parameterization is that it preserves... 

It is often convenient to express this in terms of general excitation operators which act on the Hartree-Fock determinant. The excitation level is indicated by the subscript i and /x refers to a particular operator of this general excitation level. The whole set of excitation operators of level i is often collected in a column vector denoted by hi- Alternatively, one often expresses excitation operators of a particular level in terms of the single excitation or orbital rotation operators where the subscript ai then refers to the involved virtual and occupied orbitals. 

Pi t) is the time-dependent orbital rotation operator, defined in Eq. (11.37) with the time-dependent orbital rotation parameters Kpq t) and Kpq t), which are equal to the... 

For a closed-shell RHF state, the orbital-rotation operator k in equation (116) takes the form... 

An alternative parametrization of the singles manifold is obtained by employing the orthogonal orbital-rotation operator... 

The variational conditions require the GBT to be satisfied for the occupied-virtual orbital-rotation operators (13.8.22). For an optimized OCC state, these conditions are trivially satisfied also for the occupied-occupied and virtual-virtual orbital-rotation operators. Consider the occupied-occupied operator a]aj — a]ai. Invoking the BCH expansion, this operator may be rewritten as... 

In short, for the OCC optimization, the same orbital-rotation operators are redundant as for the optimization of the single-determinant reference state RF). 

The exponential parametrization of a unitary operator is independent in the sense that there are no restrictions on the allowed values of the numerical parameters in the operator - any choice of numerical parameters gives rise to a bona fide unitary operator. In many situations, however, we would like to carry out restricted spin-orbital and orbital rotations in order to preserve, for example, the spin symmetries of the electronic state. Such constrained transformations are also considered in this chapter, which contains an analysis of the symmetry properties of unitary orbital-rotation operators in second quantization. We begin, however, our exposition of spin-orbital and orbital rotations in second quantization with a discussion of unitary matrices and matrix exponentials. 

In contrast to the spin-orbital rotation operator in (3.3.4), we have in (3.3.19) and (3.3.21) introduced irreducible tensor components in spin space and at the same time made a clear distinction between real and imaginary rotational parameters. This procedure makes it easy to select the components of ic that are necessary for a particular task. For example, in optimizations of real orbitals, we may neglect all complex rotations and rotations that mix spin states as well as the complex phase factors. The orbital-rotation operator then reduces to... 

Show that, in this eigenvector basis, the orbital-rotation operator and its first-order variation may be expressed as ... 

Show that the first derivatives of the orbital-rotation operator are given by... 

Inserting these expansions in (3E.5.1), we obtain an expression for the orbital-rotation operator in terms of the transformed elementary operators... 

For more general orbital-rotation operators, see Section 3.3.3. If we also wish to conserve the spatial symmetry of the CSF, we must retain in the orbital-rotation operator (10.1.9) only those excitation operators that transform as the totally symmetric irreducible representation of the molecular point group. For Abelian point groups, this is accomplished by summing over only those pairs pq where p and q transform as the same irreducible representation. 

As it stands, the orbital-rotation operator in (10.1.9) contains parameters that mix all classes of MOs among one another - the inactive orbitals, the active orbitals and the virtual orbitals. These parameters are all necessary in order to carry out a general rotation of the individual MOs. For a general transformation of the wave function, however, not all of these parameters are needed those parameters that are not needed for a general transformation of the wave function are referred to as redundant. We shall here use the term redundant in a more specialized sense, applying it to those parameters Kpq that are not needed for a general first-order transformation of the wave function — CSF). A parameter Kp is thus said to be redundant if the corresponding operator in (10.1.9) satisfies the condition... 

By group-theoretical arguments, it may be shown that, if the set of redundant orbital-rotation operators (10.1.10) constitute a group, then their elimination from the orbital-rotation operator (10.1.9) will not affect our description of the wave function to any order in - that is. any state that can be reached with the full set of redundant and nonredundant parameters in (10.1.8) can then also be reached with the set of nonredundant parameters. 

The unitary matrix exp(—ic ) transforms the inactive orbitals among themselves, the active orbitals among themselves, and the virtual orbitals among themselves. For the optimization of the wave function, it is convenient to set the redundant parameters equal to zero in each iteration since this choice of k simplifies the orbital-rotation operator ic. In Exercise 10.2, it is shown that, for a closed-shell state, the choice of k = 0 leads to transformed MOs that (in the least-squares sense) are as similar as possible to the original MOs - that is, to the transformed MOs that have the largest possible overlap with the original ones. 

---