Variational ground-state energy

Nonadiabatic Variational Ground-State Energies of the EiH , EiH, EiD , and EiD Molecules Obtained with Different Basis Set Sizes [123] ... 

Figure 1. Variational ground state energy of a Z = 90 hydrogen-like atom obtained from the Dirac-Pauli equation as a function of a (abscissa) and (5 (ordinate) while a = b = s (left figure) and as a function of a (abscissa) and b while a = (5 = Z (right figure). The arrows are proportional to the gradient of . The saddle points correspond to the exact eigenvalues of the Dirac Hamiltonian.

The exact wavefunction corresponds to a = b = s and a = (3 = Z. The variational ground state energy of Z = 90 hydrogen-like ion in the Dirac-Pauli... 

Figure 8. The two-electron atom Variational ground-state energy (a), first derivative (b), and second derivative (c) as a function of X for N — 6,7,..., 13.

Wlien first proposed, density llinctional theory was not widely accepted in the chemistry conununity. The theory is not rigorous in the sense that it is not clear how to improve the estimates for the ground-state energies. For wavefiinction-based methods, one can include more Slater detenuinants as in a configuration interaction approach. As the wavellmctions improve via the variational theorem, the energy is lowered. In density fiinctional theory, there is no... 

To begin a more general approach to molecular orbital theory, we shall describe a variational solution of the prototypical problem found in most elementary physical chemistry textbooks the ground-state energy of a particle in a box (McQuanie, 1983) The particle in a one-dimensional box has an exact solution... 

Part I of the paper develops an exact variational principle for the ground-state energy, in which the density (r) is the variable function (i.e. the one allowed to vary). The authors introduce a universal functional F[n(r)] which applies to all electronic systems in their ground states no matter what the external potential is. This functional is used to develop a variational principle. 

March NH (1993) The Ground-State Energy of Atomic and Molecular Ions and Its Variation with the Number of Electrons. 80 71-86... 

The variation method gives an approximation to the ground-state energy Eq (the lowest eigenvalue of the Hamiltonian operator H) for a system whose time-independent Schrodinger equation is... 

In many applications of quantum mechanics to chemical systems, a knowledge of the ground-state energy is sufficient. The method is based on the variation theorem-, if 0 is any normalized, well-behaved function of the same variables as and satisfies the same boundary conditions as then the quantity = (p H (l)) is always greater than or equal to the ground-state energy Eq... 

As a simple application of the variation method to determine the ground-state energy, we consider a particle in a one-dimensional box. The Schrodinger equation for this system and its exact solution are presented in Section 2.5. The ground-state eigenfunction is shown in Figure 2.2 and is observed to have no nodes and to vanish at x = 0 and x = a. As a trial function 0 we select 0 = x(a — x), 0 X a... 

According to the variation theorem, the lowest root g o is an upper bound to the ground-state energy Eq Eo So- The other roots may be shown to be upper bounds for the excited-state energy levels... 

In this section we examine the ground-state energy of the helium atom by means of both perturbation theory and the variation method. We may then compare the accuracy of the two procedures. 

As a normalized trial function 0 for the determination of the ground-state energy by the variation method, we select the unperturbed eigenfunction r2) of the perturbation treatment, except that we replace the atomic number Zby a parameter Z ... 

Coming back to the variational principle, the strategy for finding the ground state energy and wave function should be clear by now we need to minimize the functional E[ F] by searching through all acceptable N-electron wave functions. Acceptable means in this context that the trial functions must fulfill certain requirements which ensure that these func-... 

Stated in still other words this means that for any trial density p(r) - which satisfies the necessary boundary conditions such as p( ) - 0, J p( ) dr = N, and which is associated with some external potential Vext - the energy obtained from the functional given in equation (4-6) represents an upper bound to the true ground state energy E0. E0 results if and only if the exact ground state density is inserted into equation (4-8). The proof of the inequality (4-11) is simple since it makes use of the variational principle established for wave functions as detailed in Chapter 1. We recall that any trial density p(r) defines its own Hamiltonian H and hence its own wave function. This wave function can now be taken as the trial wave function for the Hamiltonian generated from the true external potential Vext. Thus, we arrive at... 

Let us summarize what we have shown so far. First, all properties of a system defined by an external potential Vext are determined by the ground state density. In particular the ground state energy associated with a density p is available through the functional J P(f )VNedr + I)ikIp] Second, this functional attains its minimum value with respect to all allowed densities if and only if the input density is the true ground state density, i. e for p(r) = p0(r). Of course, the applicability of this variational recipe is limited to the ground... 

This functional satisfy a variational principle [3,5] EVo o[p0, m0 < EVo>Bo[p, m. EVoJ3o [p0, hi0] denote the ground state energy with density p0(r), and magnetization m0(r) of a particular system characterized by the external fields (v0(r), Bt>(r)). One of the main differences between the spin-restricted and spin-polarized cases is that the one-to-one relation between the external potential and the density cannot be extrapolated to the set of quantities (v0(r), B0(r)) and (p0(r), m0(r)) [3]. 

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