Block Diagonal Hamiltonians

Most ah initio calculations use symmetry-adapted molecular orbitals. Under this scheme, the Hamiltonian matrix is block diagonal. This means that every molecular orbital will have the symmetry properties of one of the irreducible representations of the point group. No orbitals will be described by mixing dilferent irreducible representations. 

For configuration interaction calculations of double excitations or higher, it is possible to solve the Cl super-matrix for the 2nd root, 3rd root, 4th root, and so on. This is a very reliable way to obtain a high-quality wave function for the first few excited states. For higher excited states, CPU times become very large since more iterations are generally needed to converge the Cl calculation. This can be done also with MCSCF calculations. 

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Calculating the dipole intensities, with resonances included, requires one step in addition to the procedure described above. After the matrix representation of the effective dipole moment operator is calculated, in the same basis as was used to set up the block-diagonal Hamiltonian matrix, the similarity transformation that diagonalizes the Hamiltonian matrix is applied to the dipole moment matrix. The intensity of the transition... 

This is quite a different classification of states than that in Eq. (4.37). Consequently, the block-diagonal Hamiltonian matrix, introduced by the action of the Majorana operator, will contain the following polyads of... 

With this explicit form of the unitary matrix LI, we can calculate the block-diagonal Hamiltonian given by Eq. (11.15). Its components are... 

In addition, approximate decoupling schemes can be envisaged in order to arrive at the block-diagonal Hamiltonian of Eq. (11.15), which will be particularly valuable in cases with complicated expressions for the potential V. 

At this stage it is, however, not clear which of these parametrizations should be preferred over the others for application in decoupling procedures and whether they all yield identical block-diagonal Hamiltonians. Furthermore, the four possibilities given above to parametrize unitary transformations differ obviously in their radius of convergence Rc, which is equal to unity for the square root and the McWeeny form, whereas Rc = 2 for the Cayley parametrization, and Rc = co for the exponential form. 

Toward Well-Defined Analytic Block-Diagonal Hamiltonians 465 —, j Ep - mc + ApVAp + Ap(r-PpV(r-PpAp (1 + Pp Q)... 

Toward Well-Defined Analytic Block-Diagonal Hamiltonians... 

An expansion in terms of V, i.e., the Douglas-Kroll-Hess expansion, is the only valid analytic expansion technique for the Dirac Hamiltonian, where the final block-diagonal Hamiltonian is represented as a series of regular even terms of well-defined order in V, which are all given in closed form. For the derivation, the initial transformation step has necessarily to be chosen as the closed-form, analytical free-particle Foldy-Wouthuysen transformation defined by Eq. (11.35) in order to provide an odd term depending on the external potential that can then be diminished. We now address these issues in the next chapter. 

Chapter 11 introduced the basic principles for elimination-of-the-small-component protocols and noted that the Foldy Wouthuysen scheme applied to one-electron operators including scalar potentials yield ill-defined 1 /c-expansions of the desired block-diagonal Hamiltonian. In contrast, the Douglas Kroll-Hess transformation represents a unique and valid decoupling protocol for such Hamiltonians and is therefore investigated in detail in this chapter. 

The higher transformations can hardly be carried out manually and require an automatic, symbolic derivation. For practical purposes it would be desirable to calculate any high-order DKH Hamiltonian which approximates the exact block-diagonal Hamiltonian sufficiently closely. This is, however, not directly possible because of the nested character of the sequence of unitary transformations. Because of the linear dependence of Wi on V, an n-th order decoupled DKH Hamiltonian /idkhh requires an expansion of the inner Ui transformation up to the n-th power of Wi- That is, the higher the final order of the Hamiltonian shall be the further must the inner transformations have to be expanded. 

The dependence of the DKH Hamiltonians of fifth and higher order on the expansion coefficients in the parametrization of the unitary transformation in Eq. (11.57) clearly shows that the DKH expansion in Eq. (12.2) cannot be unique, in the sense that it is possible to have different expansions of the block-diagonal Hamiltonian in terms of the external potential. At first sight, this seems to be odd because one would expect to obtain a unique block-diagonal and unique Hamiltonian. At infinite order, all infinitely many different expansions with respect to the external potential V formally written in Eq. (11.57) as a single unique one necessarily yield the same diagonal Hamiltonian, i.e., the same spectrum. The individual block-diagonal operators may, however, still be different in accord with what has been said in section 4.3.3. 

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