Relativistic corrections matrices

Table 6 Matrix Dirac-Hartree-Fock (Edhf) and Hartree-Fock (Ehf) energies calculated using BERTHA. The Gaussian exponential parameters are those of the non-relativistic sets derived by van Duijenveldt and tabulated in Poirier et al [36]. Thejirst-order molecular Breit energy, Eb, v as calculated using methods described in [12] relativistic corrections to Ehf collected in the column labelled E energies are in atomic units.

Table 7 Estimates of total relativistic correction, E , and the first-order Breit energy correction,, obtained by combining the atomic or ionic contributions indicated by the second column. They may be compared with the values of the total relativistic correction, Er. and the first-order Breit interaction, Es, obtained directlyfrom matrix Dirac-Elartree-Fock and Elartree-Fock calculations of the molecular structure using BERTEIA [12], Only the results of the 13s7p2d atom-centred basis sets for Er and Eb are quoted. All energies in atomic units.

Eq. (4.3) with K = 0 may serve as starting point to find the relativistic corrections to the length form of the transition operator. The expansion of the matrix element (4.3) up to terms of order v2/c2 for the case of the... 

We discussed in detail the properties of the matrix elements of the electrostatic energy operator for shell lN. The corresponding expressions for the remaining two-electron operators may be found in a similar way, therefore, here we shall present only final results. For the case of relativistic corrections H2, H and H s to the Coulomb energy (formulas (19.8), (19.11) and (19.12), respectively) we have... 

The expression for quantity g(Rs) in (19.52) follows directly from (5.40). Thus, for p- and d-shells we have simple algebraic formulas for the coefficients of radial integrals (actually, for matrix elements) of all relativistic corrections of the order a2 to the Coulomb energy. For the /-shell such a formula exists only in the case of the orbit-orbit interaction. For the almost filled shell we find... 

Corresponding expressions for matrix elements of relativistic corrections to the electrostatic energy and to two-electron parts of magnetic interactions are rather cumbersome and are not presented here. They may be found in [14]. In Chapter 27 we describe the simplified method of taking them into account. 

The general definition of the electron transition probability is given by (4.1). More concrete expressions for the probabilities of electric and magnetic multipole transitions with regard to non-relativistic operators and wave functions are presented by formulas (4.10), (4.11) and (4.15). Their relativistic counterparts are defined by (4.3), (4.4) and (4.8). They all are expressed in terms of the squared matrix elements of the respective electron transition operators. There are also presented in Chapter 4 the expressions for electric dipole transition probabilities, when the corresponding operator accounts for the relativistic corrections of order a2. If the wave functions are characterized by the quantum numbers LJ, L J, then the right sides of the formulas for transition probabilities must be divided by the multiplier 2J + 1. 

The third form of the Efc-transition operator contains even two-particle terms [77], therefore its matrix elements are expressed in terms of the corresponding two-electron quantities. Unfortunately, these matrix elements are very cumbersome, and therefore are of little use. For the special case of transitions in shell lN they were established in [171]. Submatrix elements of /c-transitions with relativistic corrections (4.18)-(4.20) and (4.22) are considered in detail for one-electron and many-electron configurations in [172],... 

It can be shown that det C = 1 (Problem 2.13.1). The norm of C, or trace of C, or sum of its diagonal terms, is 2 + 2y. Since det C = 1, we can consider the Lorentz transformation matrix X like the four-dimensional analog of the Eulerian rotation in 3-space. We now seek quantities that are "covariant with the Lorentz transformation"—that is, are "relativistically correct". We next define in this new four-space a few essential quantities ... 

The Fock operator f and the one-particle density matrix 7 commute, i.e. have common eigenfunctions. This allows an iterative construction of 7 from the eigenstates of f. The leading relativistic corrections for the Dirac-Coulomb operator are ... 

The matrix elements of the relativistically corrected Hamiltonian, to be used in the DKH calculations, should result from a variation of the total energy expression - for consistency and to assist in the calculation of energy gradients. Variation of E vith respect to the density matrix F yields a contribution to the DKS Hamiltonian, see Eqs. (29) and (30) ... 

Since most quantum chemical calculations apply a linear combination of atom-centered basis functions, we may employ these basis functions to construct atomic projectors as is done in charge and spin population analysis [760]. For those Hamiltonian matrix blocks for which relativistic corrections are important, we need to derive a relativistic expression to evaluate them. The locality is then exploited in basis-function space. While it is clear that heavy-atom diagonal blocks of an operator matrix will require a relativistic description, the treatment of heavy-atom off-diagonal blocks depends on their contribution to physical observables. 

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