The general ansatz

This implies that when the ZORA ansatz is employed, the small components approach zero at the nuclei. The singular behaviour encountered in quasi-relativistic approaches based on kinetic-energy balance condition is therefore avoided. The ZORA method will be discussed in more detail in the next Section, and now we just note that the ZORA ansatz has a couple of desirable properties that shall be taken into consideration when the general ansatz function is constructed. 

In equation (6), we define f r) such that it has the property of approaching a constant value of one at very large distances from the nuclei, whereas at small distances it becomes zero. As seen in equation (4), for the kinetic-energy balance condition (KEBC), /(r) is independent of r and equal to one. In the ZORA approach, f r) approaches one at large distances when 2c j and becomes zero at the nuclei. 

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The present idea is to replace the ZORA ansatz, which already is an approximation to the energy-dependent elimination of the small component approach, with another but similar expression that relates the large and the small components. The general ansatz function should have the same shape as the ZORA function close to the nucleus. Its first derivative should also be reminiscent of that of the ZORA function. A general function f[r) that fulfills the desired asymptotic conditions for r —0 and for r -> < can for example consist of one exponential function or of a linear combination of a couple of exponential functions as... 

To simplify the expression for the effective Hamiltonian (11) and (12), one can separate out a constant of one from /(r) as (/(r) — 1) + 1, where /(r) — 1 is describing the ansatz difference between the kinetic-energy balance condition and the general ansatz. By using the identity... 

The quasi-relativistic Hamiltonians obtained using the general ansatz have usually a metric that include spin-orbit contributions. This can be a undesirable situation since in a perturbation study of spin-orbit effects, the addition of the spin-orbit coupling requires reorthogonalization of the orbitals [69]. However, as seen in equation (18), the relativistic correction term (V) consists of two contributions f and B. B is several orders of magnitude less significant than f. Furthermore, the B operator can also be separated into scalar relativistic and spin-orbit contributions. 

The generality of a simple power series ansatz and an open-ended formulation of the dispersion formulas facilitate an alternative approach to the calculation of dispersion curves for hyperpolarizabilities complementary to the point-wise calculation of the frequency-dependent property. In particular, if dispersion curves are needed over a wide range of frequencies and for several optical proccesses, the calculation of the dispersion coefficients can provide a cost-efficient alternative to repeated calculations for different optical proccesses and different frequencies. The open-ended formulation allows to investigate the convergence of the dispersion expansion and to reduce the truncation error to what is considered tolerable. 

The Wess-Zumino term in Eq. (11) guarantees the correct quantization of the soliton as a spin 1/2 object. Here we neglect the breaking of Lorentz symmetries, irrelevant to our discussion. The Euler-Lagrangian equations of motion for the classical, time independent, chiral field Uo(r) are highly non-linear partial differential equations. To simplify these equations Skyrme adopted the hedgehog ansatz which, suitably generalized for the three flavor case, reads [40] ... 

However, in the late 1960s, fir Cfzek and Josef Paldus introduced a some what different approach to the electron correlation problem instead of using a linear expansion of functions as in Eq. (13.3) they suggested an exponential ansatz of the general form [7-9]... 

One special topic for field propagation techniques in general is the minimization of the effect of the transversal boundaries. Uncared, they correspond to abrupt changes of the refractive index distribution, and back-reflections from the boundary into the computational domain do occur. After the obvious ansatz of absorbing BC, TBC " and PML indicate the major improvements so far, which eliminate the problem almost completely. 

First, we note that the determination of the exact many-particle operator U is equivalent to solving for the full interacting wavefunction ik. Consequently, some approximation must be made. The ansatz of Eq. (2) recalls perturbation theory, since (as contrasted with the most general variational approach) the target state is parameterized in terms of a reference iko- A perturbative construction of U is used in the effective valence shell Hamiltonian theory of Freed and the generalized Van Vleck theory of Kirtman. However, a more general way forward, which is not restricted to low order, is to determine U (and the associated amplitudes in A) directly. In our CT theory, we adopt the projection technique as used in coupled-cluster theory [17]. By projecting onto excited determinants, we obtain a set of nonlinear amplitude equations, namely,... 

Each of these approaches to the density functional theory can be generalized to 2-density functional theory. In Section I, we mentioned the commonly considered generalization of the Weisacker ansatz, namely. 

Looking at Eqs. (20) and (23), one realizes that there are many other choices that reduce to the exact FEG limit. For example, taking Eq. (24) into accoimt, one can introduce a much more general ansatz for the DM1,... 

We shall now employ the SU Ansatz for the wave operator U, Eq. (4), with the cluster operator T i) having the same general form as in the SR case, namely... 

In this section we describe the general approach to constructing conformally invariant ansatzes applicable to any (linear or nonlinear) system of partial differential equations, on whose solution set a linear covariant representation of the conformal group 0(1,3) is realized. Since the majority of the equations of the relativistic physics, including the Klein-Gordon-Fock, Maxwell, massless Dirac, and Yang-Mills equations, respect this requirement, they can be handled within the framework of this approach. 

Summarizing, we conclude that the problem of construction of P(l,3)-invariant ansatzes reduces to finding solutions of linear systems of first-order partial differential equations that are integrated by rather standard methods of the general theory of partial differential equations. 

Note that in contrast to the case of the nonlinear Dirac equation, it is not possible to construct the general solutions of the reduced systems (59)-(61). For this reason, we give whenever possible their particular solutions, obtained by reduction of systems of equations in question by the number of components of the dependent function. Let us emphasize that the miraculous efficiency of the t Hooft ansatz [5] for the Yang-Mills equations is a consequence of the fact that it reduces the system of 12 differential equations to a single conformally invariant wave equation. 

Consequently, to describe all the ansatzes of the form (53),(54) reducing the Yang-Mills equations to a system of ordinary differential equations, one has to construct the general solution of the overdetermined system of partial differential equations (54),(86). Let us emphasize that system (54),(86) is compatible since the ansatzes for the Yang-Mills field ( ) invariant under the three-parameter subgroups of the Poincare group satisfy equations (54),(86) with some specific choice of the functions F, F2, , 7Mv, [35]. 

Equations 2.86 and 2.90 are equivalent these are often taken as the starting point for the theory of spectral moments and line shapes. For the treatment of binary systems, one may start with the Schrodinger expression when dealing with many-body systems, the correlation function formalism is generally the preferred ansatz. 

The equations are written for the specific reaction of NH3 formation as a fully general approach would be unwieldy. The modification of the approach to other reactions is not trivial but could be done following the outline below. Another application of the formalism to a very complex reaction system (CO + O2 on a Pt/Sn disordered catalyst) is demonstrated, as well as the generality of the stochastic ansatz [28],... 

As already discussed at the end of Section 2.2.3, we derived a universal superposition principle from a complex symmetric ansatz arriving at a Klein-Gordon-like equation relevant for the theory of special relativity. This approach, which posits a secular-like operator equation in terms of energy and momenta, was adjoined with a conjugate formal operator representation in terms of time and position. As it will be seen, this provides a viable extension to the general theory [7, 82]. We will hence recover Einstein s laws of relativity as construed from the overall global superposition, demonstrating in addition the independent choice of a classical and/or a quantum representation. In this way, decoherence to classical reality seems always possible provided that appropriate operator realizations are made. 

Ll(t)Pii(-t) is evaluated at t = 0 in the field off case and hence is zero, but in the field on case one must evaluate O/SpO JexpC-tLgJcos . = (3/3p0 )cos0.(-t) for t i 0 which does not vanish. The ansatz used which reduces its consequences to a simple and useful form is that the effect of an increment in initial momentum p0 in the ensemble average is produced by the increments in 0 (t) at time t which are in the initial direction 0.(0). With this and some manipulation of the averages, one finally obtains as the generalization of Benoit s equations... 

Regarding the problems of the electronic structure of molecular systems, we notice that in the past, the importance of the qualitative concepts and explanations has been stressed many times. In this context, V.A. Fock [3,4] discussed the (basically metaphysical) problem of interrelation between exact solution and approximate explanation . His point was that any approximation (more precisely, the general form of the trial electron wave function i.e. an Ansatz used for it) sets the system of qualitative concepts (restricted number of variables), which can only be used for interpreting the calculation results and for describing the experiments. A characteristic example... 

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