Balance kinetic

From Eqs. (5.136) and (5.137) in chapter 5 we know that the kinetic balance condition relates the small (lower) and large (upper) 2-spinors. In the stationary case, in which the time-dependence of the Dirac equation drops out and the total energy E if a one-electron state comes in as a separation constant according to (ih) d/df — e (cf. Eqs. (6.5)-(6.7)), we may write Eq. (5.136) as 

The kinetic balance condition is a consequence of the 2x2 superstructure of the one-electron Hamiltonian and must therefore be fulfilled by any basis set expansion of the molecular spinors. 

In 1967, Kim [510] formulated relativistic SCF equations by expanding the unknown spinors into a set of known basis functions. However, he did not take special precautions so that the relation between the upper and lower two 

In the numerical solution of the SCF orbital equations kinetic balance restrictions are not required, as this condition will be satisfied exactly. However, in the numerical solution of MCSCF equations for purely correlating orbitals difficulties may arise if the orbital energy e, becomes too negative [230,246,473,553]. Here it is suggested to use projection operators to eliminate the functions which correspond to the negative continuum. 

Elimination (or isolation) of the small component provides a useful basis for discussion of the properties of the Dirae equation. In particular, in this section we want to develop the relationship between the large- and small-component basis sets. We write the matrix 2-spinor Dirac equation in the form of two coupled matrix equations, 

V is negative definite for most atomic and molecular potentials. Assuming that we are looking for solutions above the negative-energy continuum, where E —2mc, it is permissible to invert the term in brackets in the seeond of these equations to get 

We now eliminate the small-component expansion coefficients, a, from the first equation to obtain 

This is an equation for the expansion coefficients of the large component only, and corresponds to solving the equations by standard matrix partitioning techniques. For actual calculations we would gain nothing, because we also have to find the expansion coefficients for the small component, but for purposes of analysis this form turns out to be convenient. In order to display the dependence on c more clearly, we can expand the inverse operator in the equation above by using the matrix relation 

Here we have collected terms of order c on the left-hand side of the equation, and terms of order c and smaller on the right. 

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The presence of the momentum operator means that the small component basis set must contain functions which are derivatives of the large basis set. The use of kinetic balance ensures that the relativistie solution smoothly reduees to the non-relativistic wave function as c is increased. 

This is not the case for the restricted kinetic balance scheme... 

Table 4. The isotropic indirect spin-spin coupling constant of calculated at various levels of theory. LL refers to the Levy-Leblond Hamiltonian, std refers to a full relativistic calculation using restricted (RKB) or unrestricted (UKB) kinetic balance, spf refers to calculations based on a spin-free relativistic Hamiltonian. Columns F, G and whether quaternion imaginary parts are deleted (0) or not (1) from the regular Fock matrix F prior to one-index transformation, from the two-electron Fock matrix G...

Instead of a two-component equation as in the non-relativistic case, for fully relativistic calculations one has to solve a four-component equation. Conceptually, fully relativistic calculations are no more complicated than non-relativistic calculations, hut they are computationally demanding, in particular, for correlated molecular relativistic calculations. Unless taken care of at the outset, spurious solutions can occur in variational four-component relativistic calculations. In practice, this problem is handled by employing kinetically balanced basis sets. The kinetic balance relation is... 

The radial functions Pmi r) and Qn ir) may be obtained by numerical integration [16,17] or by expansion in a basis (for recent reviews see [18,19]). Since the Dirac Hamiltonian is not bound from below, failure to observe correct boundary conditions leads to variational collapse [20,21], where admixture of negative-energy solutions may yield energies much below experimental. To avoid this failure, the basis sets used for expanding the large and small components must maintain kinetic balance [22,23]. 

Ishikawa and coworkers [15,24] have shown that G-spinors, with orbitals spanned in Gaussian-type functions (GIF) chosen according to (14), satisfy kinetic balance for finite c values if the nucleus is modeled as a uniformly-charged sphere. 

An application of the variational principle to an unbounded from below Dirac-Coulomb eigenvalue problem, requires imposing upon the trial function certain conditions. Among these the most important are the symmetry properties, the asymptotic behaviour and the relations between the large and the small components of the wavefunction related to the so called kinetic balance [1,2,3]. In practical calculations an exact fulfilment of these conditions may be difficult or even impossible. Therefore a number of minimax principles [4-7] have been formulated in order to allow for some less restricted choice of the trial functions. There exist in the literature many either purely intuitive or derived from computational experience, rules which are commonly used as a guidance in generating basis sets for variational relativistic calculations. 

If the kinetic balance condition (5) is fulfilled then the spectrum of the L6vy-Leblond (and Schrodinger) equation is bounded from below. Then, in each case there exists the lowest value of E referred to as the ground state. In effect, this equation may be solved using the variational principle without any restrictions. On the contrary, the spectrum of the Dirac equation is unbounded from below. It contains the negative ( positronic ) continuum. Therefore the variational principle applied unconditionally would lead to the so called variational collapse [2,3,7]. The variational collapse maybe avoided by properly selecting the trial functions so that they fulfil the boundary conditions specific for the bound-state solutions [1]. 

Abstract. An application of the Rayleigh-Ritz variational method to solving the Dirac-Coulomb equation, although resulted in many successful implementations, is far from being trivial and there are still many unresolved questions. Usually, the variational principle is applied to this equation in the standard, Dirac-Pauli, representation. All observables derived from the Dirac equation are invariant with respect to the choice of the representation (i.e. to a similarity transformation in the four-dimensional spinor space). However, in order to control the behavior of the variational energy, the trial functions are subjected to several conditions, as for example the kinetic balance condition. These conditions are usually representation-dependent. The aim of this work is an analysis of some consequences of this dependence. 

The pronounced dependence of the crossover potential on chemical sensitization, and particularly on the gaseous environment, shows that the crossover does not represent a division between dyes that can cause the appearance of conduction electrons in the silver halide and dyes that cannot. Instead, it represents an energy level determined by a kinetic balance between the formation and loss of photoelectrons and/or silver in the silver halide, a kinetic balance between sensitization and desensitization (259,265). One cancels the other and the net formation of latent image is zero. The actions of oxy-gen/moisture and of mobile holes are important sources of desensitization. 

This chapter gave an overview of how to simplify complex processes sufficiently to allow the use of analytical models for their analysis and optimization. These models are based on mass, momentum, energy and kinetic balance equations, with simplified constitutive models. At one point, as the complexity and the depth of these models increases by introducing more realistic geometries and conditions, the problems will no longer have an analytical solution, and in many cases become non-linear. This requires the use of numerical techniques which will be covered in the third part of this book, and for the student of polymer processing, perhaps in a more advanced course. 

In order to describe a signal by this method we will first use the classical approach. At the beginning we will ascertain how either probability density Pb(9, multipole moments ipq of the excited state 6, entering into the fluorescence intensity expressions (2.23) or (2.24), are connected to the corresponding magnitudes pa(9, ground state a. The respective kinetic balance equation for probability density and its stationary solution, assuming that the conditions supposed to hold in Eq. (3.4) are in force, is very simple indeed ... 

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