Relativistic energy operator

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. 

Accounting for the properties of the seniority (quasispin) quantum numbers, we are able to express the matrix elements of the energy operator in terms of the corresponding quantities for the electronic configuration, for which this term has occurred for the first time (/f ajV L/S —  

This question is considered in detail in [102]. The utilization of the relationship between the CFP and the submatrix elements of the operators, composed of unit tensors, which was established in [105], allows one to find a large number of new expressions for the above-mentioned matrix elements. 

As was mentioned at the beginning of this chapter, the matrix element of the interaction between subshells may be expressed in terms of the CFP with one detached electron and two-electron matrix elements of the operator considered. The corresponding formula for jj coupling is as follows  

However, formulas of the kind (20.26) are rather inconvenient for calculations. Therefore, one has usually to insert explicit expressions for two-electron matrix elements, to perform, where it turns out to be possible, the summations necessary and to find finally the representation of the energy matrix element of the interaction between two subshells in the form of direct and exchange parts. Thus, for the electrostatic interaction we find 

Accounting for the properties of the seniority (quasispin) quantum numbers, we are able to express the matrix elements of the energy operator in terms of the corresponding quantities for the electronic configuration, for which this term has occurred for the first time (/, — /, a,d,L,S,). This question is considered in detail in [102], The utilization of the relationship between the CFP and the submatrix elements of the operators, composed of unit tensors, which was established in [105], allows one to find a large number of new expressions for the above-mentioned matrix elements. 

Here the radial integrals are defined according to (19.72), whereas their coefficients are equal to 

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It is necessary to underline that the dependence of the matrix elements of the relativistic energy operator on orbital quantum numbers is contained only in phase multipliers and radial integrals. There are only a few types of radial integrals and their coefficients, which considerably simplifies their calculation. Radial integrals have no derivatives of radial orbitals, whereas... 

Now we shall have to express operators for physical quantities in terms of irreducible tensors in the spaces of total angular momentum and quasispin. One-electron terms of relativistic energy operator (2.1) (formulas (2.2)-(2.4)) are expressed in terms of operators (23.69), (23.71)-(23.73) in a trivial way. With two-electron operators the procedure of deriving the pertinent relations is more complex. The relativistic counterpart of (18.50)... 

V. I. Sivcev, I. S. Kickin and Z. B. Rudzikas. Matrix Elements of Relativistic Energy Operator for Four Subshells of Equivalent Electrons (No 371-76 Dep) Non-Diagonal with Respect to Configurations Matrix Elements of Energy Operator for Four Subshells of Equivalent Electrons (No 2181-Dep76). Allunion Institute of Scientific and Technical Information, Moscow, 1976. 

A fully relativistic treatment of more than one particle has not yet been developed. For many particle systems it is assumed that each electron can be described by a Dirac operator (ca ir + p mc ) and the many-electron operator is a sum of such terms, in analogy with the kinetic energy in non-relativistic theory. Furthermore, potential energy operators are added to form a total operator equivalent to the Hamilton operator in non-relativistic theory. Since this approach gives results which agree with experiments, the assumptions appear justified. 

The Dirac operator incorporates relativistic effects for the kinetic energy. In order to describe atomic and molecular systems, the potential energy operator must also be modified. In non-relativistic theory the potential energy is given by the Coulomb operator. 

Let us, therefore, assume that the amplitude >ft x) describing a relativistic spin particle is an -component object. We are then looking for a hermitian operator H, the hamiltonian or energy operator, which is. linear in p and has the property that H2 = c2p2 + m2c4 = — 2c2V2 + m2c4. We also require H to be the infinitesimal operator for time translations, i.e., that... 

Generally, it is not required to retain all the terms in the resulting approximate Hamiltonian, except those operators which describe the actual physical processes involved in the problem. For example, in the absence of an external electromagnetic field, the non-relativistic energy calculations only requires... 

For a quasi-relativistic framework as relevant to chemistry (21), we may neglect the magnetic retardation between the electrons and the nuclei and therefore employ standard Coulombic interaction operators for the electrostatic interaction. The interaction between the electrons and the nuclei is not specified explicitly but we only describe the interactions by some external 4-potential. For the sake of brevity this 4-potential shall comprise all external contributions. Explicit expressions for the interaction between electrons and nuclei will be introduced at a later stage. Furthermore, we can neglect the relativistic nature of the kinetic energy of the nuclei and employ the non-relativistic kinetic energy operator denoted as hnuc(I),... 

By inserting the equations defining the kinetic energy operators and the pairwise interaction operators into Eq. (8) we obtain the Dirac-Coulomb-Breit Hamiltonian, which is in chemistry usually considered the fully relativistic reference Hamiltonian. 

As well as these additional terms there will also be changes to the Hamiltonian operator due to the relativistic change of electron mass with velocity. In ordinary optical spectroscopy the first two phenomena, (1) and (2), are the most important, leading to changes to the non-relativistic energy levels which are observable (effects (4) and (5) are important in n.m.r. and e.s.r. spectroscopy). 

As we shall see later on, for a large variety of atoms and ions the relativistic effects can be accounted for fairly precisely in the framework of the so-called Hartree-Fock-Pauli (HFP) approximation, as corrections of the order a2 (a = e1 /he is the fine structure constant and c stands for the velocity of light). Then, energy operator H will have the form... 

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