Spinor upper-component

In Eq. (168), the first, magnetic-field term admixes different components of the spinors both in the continuity equation and in the Hamilton-Jacobi equation. However, with the z axis chosen as the direction of H, the magnetic-field temi does not contain phases and does not mix component amplitudes. Therefore, there is no contribution from this term in the continuity equations and no amplitude mixing in the Hamilton-Jacobi equations. The second, electric-field term is nondiagonal between the large and small spinor components, which fact reduces its magnitude by a further small factor of 0 particle velocityjc). This term is therefore of the same small order 0(l/c ), as those terms in the second line in Eqs. (164) and (166) that refer to the upper components. 

The lower 2-spinor components, f3 = —1, are related to the upper components by the kinetic matching relation. 

This is an equation for a 4-component spinor field o> analogous to the Dirac 4-component spinor with an upper component lower component xoi both of which are two-component spinors. 

The eigenfunctions of the effective Hamiltonian are the upper components ifk of the Dirac spinors, but L,- is not hermitean. 

Rather than start from a single nonrelativistic eigenfunction (po and improve it towards the upper component

Dirac spinor ijj, we now start from a d-dimensional model space of nonrelativistic functions (f = 1,... d) and we search for an effective Hamiltonian L as a. matrix representation in this model space, such that the eigenvalues of L are equal to a subset of eigenvalues of the Dirac operator. 

This equation shows us explicitly how the operator k, when staying on the right side, acts on the upper component spinor. The second trick will indicate how we can shift the k operator to the right side. For this purpose let us consider an identity fulfilled for any operators... 

There are, however, two features which should be reanalysed the upper component spinor was not normalised and the last term in the expansion of the Dirac Hamiltonian is not Hermitian. Thus we introduce a renormalised upper component spinor... 

Now, using the expression of the lower component spinor in terms of the upper component spinor... 

The renormalised upper component spinor satisfies the time-dependent Dirac equation in the form... 

The equivalence of the lOTC method to the four-component Dirac approach has been documented by calculations of spin orbital energies in several papers [18,20, 63]. The unitary transformation does not affect the energy eigenspectrum, though it reduces the four-component bi-spinors to two-component spinor solutions. Due to this fact the two-component methods are frequently addressed as being quasi-relativistic and it is assumed that some information is lost. It can be demonstrated [22] that the two-component lOTC wave function which is the upper component of the unitarly transformed four-component Dirac spinor I ... 

The two-component wave function is called a Pauli spinor. We will show in a later chapter that for this basis the Pauli matrices form a representation of the spin operators such that ha = 2s. With the conventional choice of basis, that is, the eigenfunctions of ff, the upper component represents the part of the wave function with spin projection nts = j, or a spin, and the lower component the part with rus = -, or p spin. The primitive a and p spin functions are represented by the vectors... 

What is the significance of the four components We noted above that in nonrelativistic quantum mechanics we can introduce the spin by replacing the mechanical momentum nr by a nr and making the wave function a two-component vector, or 2-spinor, where the upper component corresponds to spin j and the lower component corresponds to spin -i. The same concept applies to the Dirac wave function— components 1 and 3 correspond to spin and components 2 and 4 correspond to spin — i, and the wave function is called a 4-spinor. 

The no-pair DCB Hamiltonian (6) is used as a starting point for variational or many-body relativistic calculations [9], The procedure is similar to the nonrelativistic case, with the Hartree-Fock orbitals replaced by the four-component Dirac-Fock-Breit (DFB) functions. The spherical symmetry of atoms leads to the separation of the one-electron equation into radial and spin-angular parts [10], The radial four-spinor has the so-called large component the upper two places and the small component Q, in the lower two. The quantum number k (with k =j+ 1/2) comes from the spin-angular equation, and n is the principal quantum number, which counts the solutions of the radial equation with the same k. Defining... 

The Breit Hamiltonian operates on sixteen-component spinor functions which contain fourtypes of function, designated A L. 1- fiu- fu, which represent upper and lower... 

The upper two-component vector iftfL is called the large-component spinor, and the lower small-component spinor. In the REL4D program, we use two-component (large- and small-component) atomic spinors ([Pg.160]

In this notation the presence of two upper and two lower components of the four-component Dirac spinor fa is emphasized. For solutions with positive energy and weak potentials, the latter is suppressed by a factor 1 /c2 with respect to the former, and therefore commonly dubbed the small component fa, as opposed to the large component fa. While a Hamiltonian for a many-electron system like an atom or a molecule requires an electron interaction term (in the simplest form we add the Coulomb interaction and obtain the Dirac-Coulomb-Breit Hamiltonian see Chapter 2), we focus here on the one-electron operator and discuss how it may be transformed to two components in order to integrate out the degrees of freedom of the charge-conjugated particle, which we do not want to consider explicitly. 

A different approach is chosen when the screening of nuclear potential due to the electrons is incorporated in /z . Transformation to the eigenspinor basis is then only possible after the DHF equation is solved which makes it more difficult to isolate the spin-orbit coupling parts of the Hamiltonian. Still, it is also in this case possible to define a scalar relativistic formalism if the so-called restricted kinetic balance scheme is used to relate the upper and lower component expansion sets. The modified Dirac formalism of Dyall [24] formalizes this procedure and makes it possible to identify and eliminate the spin-orbit coupling terms in the selfconsistent field calculations. The resulting 4-spinors remain complex functions, but the matrix elements of the DCB Hamiltonian exhibit the non-relativistic symmetry and algebra. 

Since this chapter is meant to focus on 4-component type methods I will give some attention to the recent developments that reduce the time spent in evaluation of matrix elements over g. It not trivial to reduce the amount of work associated with the separate upper and lower component basis sets since the norm of the small component wave function may be rather large for heavy atoms. One possibility is to use the locality of the small components of the wave function to replace long-range interactions by a classical interaction [27]. If we distinguish between the upper (large) two and lower (small) two components of the basis 4-spinors... 

We proceed somewhat similarly as in the theory of effective Hamiltonians (see the Appendix). However, we do, of course, not consider the matrix representation of the Dirac operator in a given basis, but take directly the matrix form of D in terms of the upper and lower spinor components. 

This equation plays a central role in relativistic quantum mechanics, far beyond the concept of a FW transformation. One can actually show that the upper and lower components (p and y of an exact 4-component spinor tp are related as [50, 12, 13]... 

The basic reason for the divergences observed at the end of the previous subsection is that there is a fundamental difference between the upper large) component g> of the Dirac bispinor, and the 2-component FW spinor... 

---