Non-variational method

B. Non-Variational Methods Sueh as MPPT/MBPT and CC do not Produee Upper Bounds, but Yield Size-Extensive Energies... 

We shall review next the time-dependent version of the CC theory of Monkhorst/56/ which also generates a linear response function closely corresponding to the CC-LRT response function. This formalism also thus falls in the category of non-variational method. To underline the similarity of this method with CC-LRT, we shall denote analogous entities by the same symbols and no confusion should arise if the context is remembered. 

Ahlrichs et al., 1985 Pulay and Saeb0, 1985), still has an edge over non-variational methods, but not as decisively as assumed earlier. 

For 7 < 1, no eigenvalue of this Hamiltonian can be found analytically and a variational method, like the one initiated by Faulkner [17], is used in most cases [13,26]. However, a non-variational method has also been used by the Kogan group in the late Soviet Union (see [7,8], and references therein). This method is facilitated by transforming Hamiltonian (5.6) using a deformed new coordinate frame X = 71/6x/a, Y = 71/6y/a, Z = 71/3z/a, where a =... 

A non-variational method of calculation has been used for the determination of eigenvalues of EM-donor Hamiltonian [31]. It is based on the finite boundedness method. A review of this method can be found in [1]. This... 

A non-variational method has also been used by [25] to determine the donor energy levels in uniaxial crystals, with an application to 4 //-SiC. It considers first a constant-energy ellipsoid with three different electron effective masses mi, my and m-z along three mutually orthogonal axes, which... 

Table 5.4. Calculated energies (meV) of the first odd-parity EM donor states in silicon for m = 0 and 1. The values of the last column are obtained by a non-variational method and the corresponding states are denoted by nPo for m = 0 and nP for m = 1...

The energies of the odd-parity states of donors in silicon calculated by variational and non-variational methods are given in Table 5.4. 

One can note the good agreement between the values obtained by the variational and non-variational methods. The difference between Faulkner s and Janzen et al. s values is only a matter of accuracy. 

The odd-parity acceptor states in germanium have been calculated varia-tionally [14,16]. As for silicon, the acceptor states in germanium have also been calculated by a non-variational method [36]. In this latter study, a screened Coulomb potential is used, but no correction is made for the acceptor-dependent central cell potential. The results of these calculations are given in Table 5.16. 

In order to illustrate how intra-orbit optimization of the energy may be accomplished by non-variational methods, let us consider some of the entries in Table 7. Let us assume that the orbit-generating wavefunction for orbit is W, which, according to Eq. (86) has the expansion coefficients (7 and yields the density pg(x). For the primitive orbital set A, the energy associated with this wavefunction is -14.538796 hartrees. Now, any displacement within orbit O must be accomplished by means of a local-scaling transformation. Consider that we carry out such a transformation between densities pg(x) and phf(x) and that by solving Eq. (37) we obtain the transformation function f(r). By means of Eq. (110), we can then transform the initial set A into a locally-scaled one from which the new wavefunction M HF can be constructed. Notice that because local-scaling transformations act only on the orbitals, the transformed wavefunction conserves the... 

Specific expressions for molecular properties can be developed for both variational and non-variational methods, the latter through the use of Lagrange s method of undetermined multipliers. 

NON-VARIATIONAL METHODS WITH SLATER DETERMINANTS (A4(g)) Coupled Cluster (CC) Method (40)... 

There are lots of exehange-eorrelation potentials in the literature. There is an impression that their authors worried most about theory/experiment agreement. We ean hardly admire this kind of seienee, but the alternative (i.e., the practiee of ab initio methods with the intact and holy Hamiltonian operator) has its own disadvantages. This is because finally we have to choose a given atomic basis set, and this influences the results. It is true that we have the variational principle at our disposal, and it is possible to tell which result is more accurate. But more and more often in quantum chemistry, we use some non-variational methods (cf. Ch ter 10). Besides, the Hamiltonian holiness disappears when the theory becomes relativistic (cf. Qiapter 3). 

Coupled Cluster A non-variational method of solving the Schrddinger equation with the wave function in the form of an exponential operator (to he determined) acting on the Hartree-Fock wave function. 

EOM-CC Equation-of-Motion Coupled Cluster A non-variational method of solving... 

Analytic gradient methods became widely used as a result of their implementation for closed-shell self-consistent field (SCF) wavefunctions by Pulay, who has reviewed the development of this topic. Since then, these methods have been extended to deal with all types of SCF wavefunctions, - as well as multi-configuration SCF (MC-SCF), - " configuration-interaction (Cl) wavefunctions, and various non-variational methods such as MoUer-Plesset (MP) perturbation theory - - and coupled-cluster (CC) techniques. - In short, it is possible to obtain analytic energy derivatives for virtually all the standard ab initio approaches. The main use of analytic gradient methods is, and will remain, the location of stationary points on a potential energy siuface, to obtain equilibrium and transition-state geometries. However, there is a specialized use in the calculation of quantities such as dipole derivatives. 

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