Hamiltonian Dirac matrix

The Dirac Hamiltonian matrix h can be written in terms of x sub-matrices... 

Another major two-component approximation is the Foldy-Wouthuysen transformation (Foldy and Wouthuysen 1950), which makes the large-component and small-component submatrices of the Dirac Hamiltonian matrix, Hd, linear independent by a unitary transformation such as... 

The different techniques utilized in the non-relativistic case were applied to this problem, becoming more involved (the presence of negative energy states is one of the reasons). The most popular procedures employed are the Kirznits operator conmutator expansion [16,17], or the h expansion of the Wigner-Kirkwood density matrix [18], which is performed starting from the Dirac hamiltonian for a mean field and does not include exchange. By means of these procedures the relativistic kinetic energy density results ... 

A matrix element Hpq of the many-electron Dirac Hamiltonian H can be expanded as... 

Here, at is the Dirac a-matrix, A(r, t) is the vector potential of the field (divA = 0), and Ho is the Hamiltonian of an isolated atom whose energy in the initial state of the reaction is denoted by Ef. H0 i) = E i). Equation (5) defines the effective two-photon operator Q(2 ui,u> ) which has the dimension of L3 and is a straightforward relativistic generalization of its nonrelativistic counterpart that has been first introduced in [28] to describe the process of two-photon absorption. Generally, the matrix elements of u>,u> ) can be expressed explicitly as... 

We will start by setting up a simple 2x2 matrix that (without interaction) displays perfect symmetry between the particle and its antiparticle image. Note that it is well known that the Klein-Gordon and the Dirac equation can be written formally as a standard self-adjoint secular problem (see e.g. [11,12]), based on the simple Hamiltonian matrix (in mass units)... 

The Dirac operator Ho is interpreted as the operator corresponding to the energy of a free particle, which fits well to its role as the generator of the time evolution. As a consequence of the anti-commutation relations (5), the square of the Dirac Hamiltonian is a diagonal matrix. It is simply given by... 

There are other reasons why methods based on Dirac Hamiltonians have been unpopular with quantum chemists. Dirac theory is relatively unfamiliar, and the field is not well served with textbooks that treat the topic with the needs of quantum chemists in mind. Matrix self-consistent-field equations are usually derived from variational arguments, and as a result of the debates on variational collapse and continuum dissolution , many people believe that such derivations are invalid for relativistic problems. Most implementations of the Dirac formalism have made no attempt to exploit the rich internal structure of Dirac... 

All other operators can now be expressed as sums of products of matrix elements (complex numbers) with these creation or annihilation operators. The Dirac Hamiltonian becomes... 

Having defined our starting point, the second quantized no-pair Hamiltonian, we may now take a closer look at the relations between the matrix elements. For future convenience we will also change the notation of these matrix elements slightly. Due to hermiticity of the Dirac Hamiltonian and the Coulomb-Breit operator we have... 

The transformation of the Dirac Hamiltonian to two-component form is accompanied by a corresponding reduction of the wavefunction. As discussed in detail in section 2, the four-component Dirac spinor will have only two nonvanishing components, as soon as the complete decoupling of the electronic and positronic degrees of freedom is achieved, and can thus be used as a two-component spinor. This feature can be exploited to calculate expectation values of operators in an efficient manner. However, this procedure requires that some precautions need to be taken care of with respect to the representation of the operators, i.e., their transition from the original (4 x 4)-matrix representation (often referred to as the Dirac picture) to a suitable two-component Pauli repre-... 

As seen in equation (26), the quasi-relativistic Hamiltonian and the operators describing the difference between the exact Dirac Hamiltonian and the quasi-relativistic one are now explicitly separated and the direct perturbation theory method can be applied. In the direct perturbation theory approach, the metric is also affected by the perturbation [12]. Note that the interaction matrix is block diagonal at the lORA level of theory, whereas the coupling between the upper and the lower components still appears in the metric. 

The natural solution of the operator Eq. 4.38 would be in the momentum space due to the presence of momentum p which replaces the standard configuration space formulation by a Eourier transform. The success of the Douglas-Kroll-Hess and related approximations is mostly due to excellent idea of Bernd Hess [53,54] to replace the explicit Eourier transformation by some basis set (discrete momentum representation) where momentum p is diagonal. This is a crucial step since the unitary transformation U of the Dirac Hamiltonian can easily be accomplished within every quantum chemical basis set program, where the matrix representation of the nonrelativistic kinetic energy T = j2m is already available. Consequently, all DKH operator equations could be converted into their matrix formulation and they can be solved by standard algebraic techniques [13]. 

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 ... 

DHF calculations on molecules using finite basis sets require considerably more computational effort than the corresponding nonrelativistic calculations and cause several problems due to the presence of the Dirac one-particle operator. It is therefore desirable to find (approximate) relativistic Hamiltonians for many-electron systems which are not plagued by unboundedness from below and therefore do not cause problems like the variational collapse at the self-consistent field level or the Brown-Ravenhall disease at the configuration interaction level. It is also desirable to find forms in which the quality of a matrix representation of the kinetic energy is more stable than for the Dirac Hamiltonian, i.e., forms which are not affected by the finite basis set disease . 

The proportionality constant k is usually assigned a value of approximately 1.75. The overlap matrix elements Sy are calculated with respect to a set of two component basis functions with lsjm) quantization. The radial parts were chosen to be one or two Slater functions yielding (r ) (k=-l,0,1,2) expectation values as close as possible (Lohr and Jia 1986) to the Dirac-Hartree-Fock or Hartree-Fock results tabulated by Desclaux (1973) for the relativistic and nonrelativistic case, respectively. The diagonal Hamiltonian matrix elements Hu were set equal to the corresponding orbital energies from Desclaux s tables. Due to the use of a two-component lsjm) basis set the matrices H and S are generally complex and of dimension 2nx2n, when is the number of spatial orbitals. 

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