Direct variational method

A direct variational method was used in Refs. 23 and 24 to go beyond the Condon approximation. Functions of the type... 

Use of Equation (1) in numerical work requires a means of generating x(r, r i(o) as well as the average charge density. Direct variational methods are not applicable to the expression for E itself, due to use of the virial theorem. However, both pc(r) and x(r, r ico) (39-42, 109-112) are computable with density-functional methods, thus permitting individual computations of E from Eq. (1) and investigations of the effects of various approximations for x(r, r ico). Within coupled-cluster theory, x(r, r ico) can be generated directly (53) from the definition in Eq. (3) then Eq. (1) yields the coupled-cluster energy in a new form, as an expectation value. 

The weak interaction region can be defined as one for which the total electron density is approximately equal to the sum of the densities of the separate interacting particles. Whether one uses a direct variational method to calculate the energy or a perturbation expansion it is found that good results are only obtained if the wavefunctions for the interacting particles give accurate values for these atomic densities. 

This part introduces variational principles relevant to the quantum mechanics of bound stationary states. Chapter 4 covers well-known variational theory that underlies modern computational methodology for electronic states of atoms and molecules. Extension to condensed matter is deferred until Part III, since continuum theory is part of the formal basis of the multiple scattering theory that has been developed for applications in this subfield. Chapter 5 develops the variational theory that underlies independent-electron models, now widely used to transcend the practical limitations of direct variational methods for large systems. This is extended in Chapter 6 to time-dependent variational theory in the context of independent-electron models, including linear-response theory and its relationship to excitation energies. 

Direct Variational Methods for Complex Resonance Energies... 

On the other hand, Marin and Cruz [16-18] used the direct variational method or Rayleigh-Ritz method with a trial function of the same form used by Gorecki and Byers Brown (Equation (25)). Marin and Cruz chose hydrogenic functions with a variational parameter as the exponent for the functions

[Pg.133]

In a first report, Marin and Cruz [16] studied the helium atom confined in an impenetrable spherical box where they used the direct variational method to optimize the energy value. The Hamiltonian for a spherically confined helium atom within a hard box is given by... 

The approximate solution of Euler-Lagrange equation (3.11) can be found with the help of direct variational method, as it has been proposed in Refs. [14-16]. 

The application of direct variational method for solution of Euler-Lagrange equation is similar to that in Sect. 3.2.2.1. It ailows to obtain the free energy in the form of polarization power series with the coefficients dependent on average particles radius and the parameters of Euler-Lagrange equation (3.45). In particular, surface polarization Pd in the boundary conditions leads to appearance of built-in field Ecyi(R), which can be written as... 

The application of direct variational method to Eq. (3.51) yields, similarly to the previous case of cylindrical particles, the free energy function with renormalized coefficients... 

Solving Euler-Lagrange equations by direct variational method with their subsequent substitution into Eq. (3.56), one can obtain ... 

Direct Variational Method for Schrodinger Equation Solution... 

The direct variational method has been used to solve Schrodinger equation (4.2) with respect to Eqs. (4.3), (4.4), and (4.5). Hydrogen-like atom wave functions 2p and 3p have been taken as trial functions for ground and exited states respectively. Two-fermion wave functions were written in conventional form as a product of coordinate part and symmetric or asymmetric spin part for triplet (spin S = 1) or singlet (spin E = 0) state. The energy level positions 23 (S = 0), 33 (S = 0),... 

Application of the direct variational method [8, 78] for the Euler-Lagrange Eq. (4.20) leads to the conventional form of the free energy with renormalized coefficients... 

Using the expressions for strains from Ref. [11] and direct variational method [8, 78, 91], the spontaneous polar ation P3 (T, R), magnetization M(T, R) and antiferromagnetic order parameter L(T, R) averaged over the wire radius R can be written in the form... 

Morozovska, A.N., Eliseev, E.A., Glinchuk, M.D. Ferroelectricity enhancement in confined nanorods direct variational method. Rhys. Rev. B 73,214106 (2006)... 

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