Variational Bounds

We now have a matrix form of the Dirac equation which tends to the right nonrelativistic limits, but we have no guarantee that the solutions of these equations are well-behaved. In particular, we need to ensure that the finite basis expansion does not produce solutions corresponding to energies below the positive energy electronic space, despite all our efforts to cast the equations in a form that should prevent this from occurring (chapter 5 and chapter 8). 

The tool we use to analyze this problem is the Rayleigh quotient corresponding to (11.12), 

This condition was first formulated by Lee and McLean (1982a, 1982b) and Grant (1982). 

We want to know if this quotient is bounded from below for what we expect to be bound-state energies, 0 E -2m c. Obviously we are looking for a bound that is greater than —2mc if we do not find one it is likely that there is no bound. 

The Rayleigh quotient is a function of E because of the elimination of the small component. In order to find a lower bound, we need to remove E and identify a positive term in the quotient, so that we can create an inequality. 

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Variational bounds for problems in diffusion and reaction (with W. Strieder). J. Inst. Maths Appl. 8, 328-334 (1971). 

Communications on the theory of diffusion and reaction-VIII Variational bounds on the effectiveness factor (with S. Rester). Chem. Eng. Sci 27, 347-360 (1972). 

Drachman, R.J. (1968). Variational bounds in positron-atom scattering. Phys. Rev. 173 190-201. 

In bound-state calculations, the Rayleigh-Ritz or Schrodinger variational principle provides both an upper bound to an exact energy and a stationary property that determines free parameters in the wave function. In scattering theory, the energy is specified in advance. Variational principles are used to determine the wave function but do not generally provide variational bounds. A variational functional is made stationary by choice of variational parameters, but the sign of the residual error is not determined. Because there is no well-defined bounded quantity, there is no simple absolute standard of comparison between different variational trial functions. The present discussion will develop a stationary estimate of the multichannel A -matrix. Because this matrix is real and symmetric for open channels, it provides the most... 

A further advance occurred when Chesnavich et al. (1980) applied variational transition state theory (Chesnavich and Bowers 1982 Garrett and Truhlar 1979a,b,c,d Horiuti 1938 Keck 1967 Wigner 1937) to calculate the thermal rate coefficient for capture in a noncentral field. Under the assumptions that a classical mechanical treatment is valid and that the reactants are in equilibrium, this treatment provides an upper bound to the true rate coefficient. The upper bound was then compared to calculations by the classical trajectory method (Bunker 1971 Porter and Raff 1976 Raff and Thompson 1985 Truhlar and Muckerman 1979) of the true thermal rate coefficient for capture on the ion-dipole potential energy surface and to experimental data (Bohme 1979) on thermal ion-polar molecule rate coefficients. The results showed that the variational bound, the trajectory results, and the experimental upper bound were all in excellent agreement. Some time later, Su and Chesnavich (Su 1985 Su and Chesnavich 1982) parameterized the collision rate coefficient by using trajectory calculations. 

Another set of configurations which it is sometimes useful to add to each spincoupling is based on the restricted Cl (RCI) ideas of Goddard and coworkers (e.g. see [21]). We have found that on some problems the variational bound is... 

There are only few rigorous results on critical phenomena for many-body systems, and most of them are variational bounds. As an example, for iV-electron atoms, experimental results [61] and numerical calculations [62] rule out the stability of doubly charged atomic negative ions in the gas phase. However, a rigorous proof exists only for the instability of doubly charged hydrogen negative ions [63]. 

Exact values of critical exponents are more difficult to obtain, because variational bounds do not give estimations of the exponents. Then the result presented by M. Hoffmann-Ostenhof et al. [64] for the two-electron atom in the infinite mass approximation is the only result we know for /V-body problems with N > 1. They proved that there exists a minimum (critical) charge where the ground state degenerates with the continuum, there is a normalized wave function at the critical charge, and the critical exponent of the energy is a = 1. 

The resulting orbitals are then used in a standard nonorthogonal Cl expansion (single and double vertical excitations) in order to relax the spincoupling coefficients and to find a variational bound for the energy. We refer to such an expansion as an SCVB wavefunction. In order to improve the description of one-electron properties, we may choose to include, at very little additional cost, configurations with double occupancy of the occupied and/or virtual orbitals. 

Although the well-known variational bound property of classical TST does not exist in the PI-QTST formulation [44], analytical... 

J.F. Perkins, Variational-bound method for autoionization states, Phys. Rev. 178 (1969)... 

The translation-invariant decomposition (9.38) was first written by Post [95] and was rediscovered independently in refs. [85,86], The result (9.39) clearly constitutes an improvement with respect to the previous inequality (9.18) because the constituent mass in is decreased by a factor 3/4, and therefore the energy E. is algebraically increased. For an attractive power-law potential e(/3)r, this provides a factor (4/3). A numerical comparison is shown in table 9.1, where are listed the naive lower limit (9.18), the improved lower limit (9.39), the exact energy obtained by a hyperspherical expansion, and the variational bound derived from a Gaussian trial wave function. It is worth noticing that the new lower limit (9.39) becomes exact in the case of the harmonic oscillator. This is true for an arbitrary number A of bosons and the harmonic oscillator is the only potential for which the inequality is saturated. A beautiful proof of this property has been given by Wu and is included in ref. [ ]. 

In coupled cluster (CC) approaches, which are also size consistent and generally not variationally bound, instead of including all configurations to a particular order as... 

A subsequent relation proved by Hohenberg and Kotin [7] indicated a way to proceed. They proved that an approximate density function, po,approx, when subjected to the (unknown) procedure that relates the exact po to the exact Eq, must yield an energy higher than the exact Eq Eq,approx > Eq, so a variational bound exists. Note that the unknown process referred to here is one that assumes Vext ( ) to be the same for the analysis of po and po,approx, which means that the same nuclear framework applies in both cases. 

If a procedure were known for finding E from p, then the existence of a variational bound would allow a variational procedure analogous to what we have applied earlier. One would start with a trial p, calculate its energy, and vary p to locate the p that gives the lowest energy. 

In particular, in the early days of relativistic quantum chemistry, attempts to solve the DHF equations in a basis set expansion sometimes led to unexpected results. One of the problems was that some calculations did not tend to the correct nonrelativistic limit. Subsequent investigations revealed that this was caused by inconsistencies in the choice of basis set for the small-component space, and some basic principles of basis-set selection for relativistic calculations were established. The variational stability of the DHF equations in a finite basis has also been a subject of debate. As we show in this chapter, it is possible to establish lower variational bounds, thus ensuring that the iterative solution of the DHF equations does not collapse. 

The Breit-Pauli Hamiltonian describes a number of contributions that can be considered small compared to the nonrelativistic Hamiltonian given in O Eq. 11.2. These contributions are thus ideally suited to be treated by perturbation theory, at least as long as we do not consider the heaviest atoms of the periodic table, where the relativistic effects become substantial and the lack of a variational bound for the Breit-Pauli Hamiltonian makes any perturbation approach fail. For the energy of molecules consisting of light atoms, the relativistic effects can to a first approximation often be treated considering only the mass-velocity (O Eq. 11.9) and one-electron Darwin (O Eq. 11.20) terms. 

This observation is in accordance with a supposedly rigorous variational bound for the diffusion constant, which states that the Kirkwood value is an upper bound. A much stronger effect (w 30%) was found for the intrinsic viscosity [rf, which measures the polymer contribution to the viscosity p of the solution, relative to the viscosity rjs of the pure solvent. It is experimentally defined as... 

The first results concern the comparison between the Kirkwood formula for the diffusion constant and the diffusion constant obtained from pre-averaged BD simulations. Within the error bars, the numbers are identical. However, for fluctuating hydrodynamics a diffusion constant systematically above the Kirkwood value was found, at variance with the variational bound. Since the data seem to be rather accurate, this might be an indication that something is wrong with the rigorous bound. For a discussion, see Ref. 38. These questions must be regarded as completely unresolved today. 

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