Block diagonalization of the electronic Hamiltonian

Pacher, T., Cederbaum, L.S., and Koppel, H. (1988). Approximately diabatic states from block diagonalization of the electronic Hamiltonian, J. Chem. Phys. 89, 7367-7381. 

Four-component methods are computationally expensive since one has to deal with small-component integrals. Therefore, various two-component methods in which small-component degrees of freedom are removed have flourished in the literature. We focus the present discussion on the X2c theory at one-electron level (X2c-le). The X2c-le scheme consists of a one-step block diagonalization of the Dirac Hamiltonian in its matrix representation via a Foldy-Wouthuysen-type matrix unitary transformation ... 

Most procedures, but not all of them, imply the preliminary determination of the adiabatic wavefuactions, which is.fe trivial way of achieving the block diagonalization of //ei by means of the standard methods of ab initio theoretical chemistry then, a rotation or projection of the adiabatic states yields the quasi-diabatic ones. In all cases, the central point is to obtain the 5 block H of the electronic Hamiltonian matrix. The diagonalization of H returns the adiabatic energies Ek and the eigenvectors Cg, which express the adiabatic states in the quasi-diabatic basis ... 

We see from the way in which the effective rotational Hamiltonian is constructed that it is naturally expressed in terms of the angular momentum operator N. In the scientific literature, however, it is frequently written in terms of the vector R (which represents the rotational angular momentum of the nuclei) rather than N. While R = N — L occurs in the fundamental Hamiltonian (7.71), its use in the effective Hamiltonian is not satisfactory because R has matrix elements (due to L) which connect different electronic states and so is not block diagonal in the electronic states. In practice, authors who claim to be using R in their formulations usually ignore any matrix elements which they find inconvenient such as those of Lx and Ly. We shall return to this point in more detail later in this chapter. 

In the present work we will focus mainly on the infinite order two-component method, lOTC. However, some comparison between the lOTC and DKHn methods will be also presented. So far the discussion has been focus on the block-diagonalization of the one-electron Dirac Hamiltonian. For the N electron system a Hamiltonian may be written as the sum of the one-electron transformed Dirac Hamiltonian plus the Coulomb electron-electron interaction and it is commonly used form of the relativistic Hamiltonian. 

All of the above procedures provide an approximate block diagonalization of the starting 4-component one-electron Hamiltonian from which a 2-component Hamiltonian may be extracted... 

The classes of two-electron integrals required for the construction of the matrix elements are therefore different for each block diagonal, and the use of time-reversal properties to classify the integrals provides an efficiency in the construction of the Hamiltonian matrix. 

Hamiltonian With these operators the two off-diagonal blocks E and E of the electronic Hessian matrix and S and of the overlap matrix become zero [see Exercise 3.12] and the propagator in Eq. (3.159) can be written as... 

Symmetry tools are used to combine these M objects into M new objects each of which belongs to a specific symmetry of the point group. Because the hamiltonian (electronic in the m.o. case and vibration/rotation in the latter case) commutes with the symmetry operations of the point group, the matrix representation of H within the symmetry adapted basis will be "block diagonal". That is, objects of different symmetry will not interact only interactions among those of the same symmetry need be considered. 

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