Variational energy

In an orthonomial basis,. S. = 1 if k=j, and vanishes otherwise. The problem of finding the variational energy of the ground state then reduces to that of detemiining the smallest value of e that satisfies... 

The simultaneous optimization of the LCAO-MO and Cl coefficients performed within an MCSCF calculation is a quite formidable task. The variational energy functional is a quadratic function of the Cl coefficients, and so one can express the stationary conditions for these variables in the secular form ... 

Coupled cluster calculations give variational energies as long as the excitations are included successively. Thus, CCSD is variational, but CCD is not. CCD still tends to be a bit more accurate than CID. 

The variational energy principles of classical elasticity theory are used in Section 3.3.2 to determine upper and lower bounds on lamina moduli. However, that approach generally leads to bounds that might not be sufficiently close for practical use. In Section 3.3.3, all the principles of elasticity theory are invoked to determine the lamina moduli. Because of the resulting complexity of the problem, many advanced analytical techniques and numerical solution procedures are necessary to obtain solutions. However, the assumptions made in such analyses regarding the interaction between the fibers and the matrix are not entirely realistic. An interesting approach to more realistic fiber-matrix interaction, the contiguity approach, is examined in Section 3.3.4. The widely used Halpin-Tsai equations are displayed and discussed in Section 3.3.5. 

Every positive choice of a will give a variational energy higher than the true energy, and the best value of a occurs when... 

I am assuming that this particular electronic state is the lowest-energy one of that given spatial symmetry, and that the i/f s are orthonormal. The first assumption is a vital one, the second just makes the algebra a little easier. The aim of HF theory is to find the best form of the one-electron functions i/ a,. .., and to do this we minimize the variational energy... 

We satv earlier that the variational energy for a closed-shell state formed from electron configurations such as... 

It should be noted that the functions Xn need not necessarily form an orthonormal set The linearly independent coefficients c can be considered to be variable parameters that are determined by minimization of the variational energy, W. If the functions Xn are not orthonormal, Eq. (105) can be rewritten in the form... 

Show that the variational energies of a homonuclear diatomic molecule are given in the LCAO approximation by Eq. (137) and that the corresponding wavefiine-tions are as indicated in Eqs. (141) and (142). 

We conclude that L2a-Qb donor-donor interactions are generally ineffective at lowering the total variational energy,28 whereas L2a-f2b donor-acceptor interactions are universally stabilizing. Comparison of Fig. 3.2 (or Fig. 1.3) with Fig. 3.13 shows clearly how this fundamental difference arises from the Pauli restriction on orbital occupancies. 

Let us first inquire whether basic criteria for the validity of low-order perturbation theory are actually satisfied in the present case. As described in Section 1.4, the perturbative starting point is an idealized natural Lewis-structure wavefunction (t//,l )) of doubly occupied NBOs. The accuracy of this Lewis-type starting point may be assessed in terms of the percentage accuracy of the variational energy (E) or density (p(l ). as shown for each molecule in Table 3.20. 

In Table II we also compare our total variational energies with the energies obtained by Wolniewicz. In his calculations Wolniewicz employed an approach wherein the zeroth order the adiabatic approximation for the wave function was used (i.e., the wave function is a product of the ground-state electronic wave function and a vibrational wave function) and he calculated the nonadiabatic effects as corrections [107, 108]. In general the agreement between our results... 

Nonadiabatic Variational Energies for 15 States of the H2 Molecule with Zero Total Angular Momentum (the Ground Rotational States) Obtained with 3000 Basis Functions for Each State and Expectation Values of the Internuclear Distance and the Square of the Internuclear... 

Comparison of the Total Nonrelativistic Nonadiabatic Variational Energies, E (in a.u.), and Dissociation Energies, D (in cm ), of HD+ Vibrational States with Zero Total Angular Momentum, Obtained in Work [124] and the Corresponding Quantities Obtained by Moss [111] ... 

In Table IV we compare our variational energies with the best literamre values of Moss [112]. As one can see, the values agree very well. The agreement is consistently very good for all the states calculated. 

As one can see, for each set of parameters Z, k, L and d, one can select a in such a way that Eqs. (21) and (22) are fulfilled. Then, the resulting expectation values correspond to the variational minima and are equal to the appropriate exact eigenvalues of either Dirac or Schrodinger (or rather Levy-Leblond) Hamiltonian. However the corresponding functions do not fulfil the pertinent eigenvalue equations they are not eigenfunctions of these Hamiltonians. This example demonstrates that the value of the variational energy cannot be taken as a measure of the quality of the wavefunction, unless the appropriate relation between the components of the wavefunction is fulfilled [2]. 

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