Variational collapse equation

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]. 

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... 

The matrix form of the atomic Dirac-Hartree-Fock (DHF) equations was presented by Kim [37,95], who used a basis set of modified radial Slater-type functions, without the benefit of a balancing presciption for the small component set. A further presentation of the atomic equations was made by Kagawa [96], who generalized Kim s work to open shells and discussed matrix element evaluation. An extension to include the low-ffequency form of the Breit interaction self-consistently in an S-spinor basis was presented by Quiney [97], who demonstrated that this did not produce variational collapse. Our presentation of the DHFB method is based on [97-99]. 

The terms etc. in (10) represent the one-body mean-field potential, which approximates the two-electron interaction in the Hamiltonian, as is the practice in SCF schemes. In the DFB equations this interaction includes the Breit term (4) in addition to the electron repulsion l/rjj. The radial functions Pn ( ) and Qn/c( ) may be obtained by mmierical integration [20,21] or by expansion in a basis (for more details see recent reviews [22,23]). Since the Dirac Hamiltonian is not bound from below, failure to observe correct boundary conditions leads to variational collapse [24,25], 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 [26,27]. In the nonrelativistic limit (c oo), the small component is related to the large component by [24]... 

We could not show here the results of solving the relativistic Dirac-Coulomb equation. The FC method can be extended to the case of the Dirac-Coulomb equation with only a small modification [36]. It is important to use the inverse Dirac-Coulomb equation to circumvent the variational collapse problem which often appears in the relativistic calculations [37]. 

The requirement that the wave function should be stationary with respect to a variation in the orbitals, results in an equation that is formally the same as in non-relativistic theory, FC = SCe (eq. (3.51)). Flowever, the presence of solutions for the positronic states means that the desired solution is no longer the global minimum (Figure 8.1), and care must be taken that the procedure does not lead to variational collapse. The choice of basis set is an essential component in preventing this. Since practical calculations necessarily use basis sets that are far from complete, the large and small component basis sets must be properly balanced. The large component corresponds to the normal non-relativistic wave function, and has similar basis set requirements. The small component basis set is chosen to obey the kinetic balance condition, which follows from (8.15). 

P. Falsaperla, G. Fonte, J. Z. Chen. Two methods for solving the Dirac equation without variational collapse. Phys. Reo. A, 56(2) (1997) 1240-1248. 

The fuUy relativistic LCAO method for soUds, based on the DKS scheme in the LDA approximation was represented in [541]. The basis set consists of the numerical-type orbitals constructed by solving the DKS equations for atoms. This choice of basis set allows the spurious mixing of negative-energy states known as variational collapse to be overcome. Furthermore, the basis functions transform smoothly to those in the nonrelativistic limit if one increases the speed of light gradually in a hypothetical way. 

Figure 5.11a shows mercury porosimetry data for a silica xerogel with mixed behavior the material first shrinks (full circles) up to a critical pressure Per beyond which its small, uncollapsed pores are intruded (open circles). The critical pressure is identified by a sudden change of slope of the volume variation curve. If the sample is depressurized before reaching Per (crosses), then indeed no mercury uptake is detected in the sample. The pore volume distribution in Fig. 5.11b can be computed using either Eqs. 5.4 or 5.5 depending on the considered pressure domain. At P=Pc both equations are valid so that the mechanical constant C - necessary for the hierarchical collapse equation - can be obtained conveniently (Pirard et ah, 1998) ... 

What this means for mean-field theory is that the lowest electron eigenvalue of the one-particle matrix that we are diagonalizing can never fall below the lowest eigenvalue of the positive-positive block of the matrix in any one iteration, and therefore there is no problem with variational collapse in a self-consistent field procedure, provided that the set of states in which we have formed the matrix represent the solutions of some one-particle Dirac equation. Failure to ensure a proper representation in a finite basis has been the occasion of problems that appear to exhibit variational collapse. Further discussion of this issue will be postponed to chapter 11, which covers finite basis methods. 

The developments of this section show that for energy solutions in the domain of interest to us, the Rayleigh quotient is bounded below, and there is therefore no danger of variational collapse when solving the Dirac equation in a kinetically balanced finite basis. For the Dirac-Hartree-Fock equations, the only addition is the electron-electron interaction, which is positive and therefore will not contribute to a variational collapse. 

Dirac equation. This method of eliminating the small component is not a procedure that leads to a simplification. It does, however, have some motivation, both physical and practical. First, it projects out the negative-energy states, and leaves a Hamiltonian that may have a variational lower bound, avoiding the potential problem of variational collapse. Second, it removes from explicit consideration the small component, and with the use of the Dirac relation (4.14) it yields a one-component operator that can be used in nonrelativistic computer programs. 

One useful trick in solving complex kinetic models is called the steady-state approximation. The differential equations for the chemical reaction networks have to be solved in time to understand the variation of the concentrations of the species with time, which is particularly important if the molecular cloud that you are investigating is beginning to collapse. Multiple, coupled differentials can be solved numerically in a fairly straightforward way limited really only by computer power. However, it is useful to consider a time after the reactions have started at which the concentrations of all of the species have settled down and are no longer changing rapidly. This happy equilibrium state of affairs may never happen during the collapse of the cloud but it is a simple approximation to implement and a place to start the analysis. 

Calibration of FAGE1 from a static reactor (a Teflon film bag that collapses as sample is withdrawn) has been reported (78). In static decay, HO reacts with a tracer T that has a loss that can be measured by an independent technique T necessarily has no sinks other than HO reaction (see Table I) and no sources within the reactor. From equation 17, the instantaneous HO concentration is calculated from the instantaneous slope of a plot of ln[T] versus time. The presence of other reagents may be necessary to ensure sufficient HO however, the mechanisms by which HO is generated and lost are of no concern, because the loss of the tracer by reaction with whatever HO is present is what is observed. Turbulent transport must keep the reactor s contents well mixed so that the analytically measured HO concentration is representative of the volume-averaged HO concentration reflected by the tracer consumption. If the HO concentration is constant, the random error in [HO] calculated from the tracer decay slope can be obtained from the slope uncertainty of a least squares fit. Systematic error would arise from uncertainties in the rate constant for the T + HO reaction, but several tracers may be employed concurrently. In general, HO may be nonconstant in the reactor, so its concentration variation must be separated from noise associated with the [T] measurement, which must therefore be determined separately. 

From the characteristics of our reactivity curves (presented later), we selected the random pore model developed by Bhatia and Perlmutter as the model can represent the behaviour of a system that shows a maximum in the reactivity curve as well as that of a system that shows no maximum. The maximum arises from two opposing effects the growth of the reaction surface associated with the growing pores and the loss of surface as pores progressively collapse at their intersections (coalescence). In the kinetically controlled regime, the model equations derived for the reaction surface variation (S/S ) with conversion and conversion-time behaviour are given by ... 

This experiment confirms the good identification of successive mechanisms responsible for the volume variation. It confirms also that equation (2) proposed to interpret a mercury porosimetry curve when the sample collapses leads to a pore size distribution identical to which obtained from Washburn equation when mercury intrudes the pores. 

With the Coulomb and exchange parts of the MP discussed so far, the core-like solutions of the valence Fock equation would still fall below the energy of the desired valence-like solutions. In order to prevent the valence-orbitals collapsing into the core during a variational treatment and to retain an Aufbau principle for the valence electron system, the core-orbitals are moved to higher energies by means of a shift operator... 

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