Block diagonalization of the electronic

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. 

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. 

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 ... 

The discussion so far has focused on the block-diagonalization of the one-electron Dirac-like Fock operator / of Eq. (11.1). One may argue that the transformed Fock operator finally provides transformed molecular spinors to enter Slater determinants. Then, the total electronic wave function is constructed from two-component spinors only. Nonetheless, we need to investigate how the total unitary transformation valid for N molecular spinors. 

The unitary transformation required for the block diagonalization of the relativistic Fock operator can be obtained in one step if a matrix representation of the Fock operator is available this is achieved by the so-called eXact-2-Component (X2C) approach [725-728,731-734]. An important characteristic of the X2C approach is its noniterative construction of the key operator X of Eq. (11.2). In this noniterative construction scheme, the matrix operator X is obtained from the electronic eigenvectors of the relativistic (modified) Roothaan Eq. (14.13),... 

Exercise 3.12 Show that the off-diagonal blocks and of the electronic Hessian matrix and the off-diagonal blocks and of the overlap matrix vanish if one chooses the state transfer operators, Eq. (3.165), as operators h . ... 

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 ... 

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... 

It follows that the only possible values for la + Ip are S A and the computation of vibronic levels can be carried out for each K block separately. Matrix elements of the electronic operator diagonal with respect to the electronic basis [first of Eqs. (60)], and the matrix elements of T are diagonal with respect to the quantum number I = la + Ip. The off-diagonal elements of [second and third of Eqs. (60)] connect the basis functions with I — la + Ip and I — l + l — l 2A. 

The Natural Atomic Orbitals for atom A in the molecular environment may be defined as those which diagonalize the block, NAOs for atom B as those which diagonalize the D block etc. These NAOs will in general not be orthogonal, and the orbital oecupation numbers will therefore not sum to the total number of electrons. To achieve a well-defined division of the electrons, the orbitals should be orthogonalized. 

The RDMs for atoms and molecules have a special structure from the spin of the electrons. To each spatial orbital, we associate a spin of either a or f. Because the two spins are orthogonal upon integration of the N-particle density matrix, only RDM blocks where the net spin of the upper indices equals the net spin of the lower indices do not vanish. Hence a p-RDM is block diagonal with (p -f 1) nonzero blocks. Specifically, the 1-RDM has two nonzero blocks, an a-block and a -block ... 

The electrostatic interactions in a molecule are determined by the structure of the density matrix D. In constructing D from an atomic orbital or atomic spinor basis, we incorporate a lot of redundant information in the Gaussian overlap charge distributions, since most of the electron density is concentrated near the nuclei. One should therefore try to transform D into a block diagonal form in which each dense block corresponds to a one-centre density so that the... 

In particular, we could use the grills to relabel the atoms of the EM so that the he-matrix appears in block-diagonal form, the number of blocks being the number of molecules in the EM. Observe that, if there are no rows/columns without non-zero entries, representing unbonded cores without free electrons, e.g., H , Li , He, Ne, etc.) in that case, the number of rows/columns without non-zero entries will be on a separate grill. 

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. 

The interaction with the Coulomb potential of the electrons appears with the nuclear potential in the blocks that are diagonal in both the spin and the components, and the exchange contributions are spread between all blocks. 

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