Variational wavefunction

By carefully adjusting the kind of variational wavefunction used, it is possible to circumvent size-extensivity problems for selected species. For example, a CI calculation on Bc2 using all CSFs that can be formed by placing the four valence electrons into the orbitals 2ag, 2ay, 3 <5g, 3 a, 1 Tty, and 1 Tig can yield an energy equal to twice that of the Be... 

The analysis of the variational wavefunctions clearly shows admixtures of valence and Rydberg characters in many states, either at the orbital level or at the Cl level. We will not discuss this point here, but will focuse on transition energies. 

Perhaps the simplest and most cost-effective way of treating relativistic contributions in an all-electron framework is the first-order perturbation theory of the one-electron Darwin and mass-velocity operators [46, 47]. For variational wavefunctions, these contributions can be evaluated very efficiently as expectation values of one-electron operators. 

The major drawback for employing the Car-Parrinello approach in dynamics simulations is that since a variational wavefunction is required, the electronic energy should in principle be minimized before the forces on the atoms are calculated. This greatly increases the amount of computer time required at each step of the simulation. Furthermore, the energies calculated with the electronic structure methods currently used in this approach are not exceptionally accurate. For example, it is well established that potential energy barriers, which are of importance to chemical reactivity, often require sophisticated methods to be accurately determined. Nonetheless, the Tirst-principles calculation of the forces during the dynamics is an appealing idea, and will continue to be developed as computer resources expand. 

For any variational wavefunction which is not near the Hartree-Fock limit the Brillouin theorem is irrelevant, and even for those of Hartree-Fock accuracy low-lying important excited states may invalidate the conclusions drawn from it. The statement that values of one-electron properties are expected to be good because of the Brillouin theorem should therefore be regarded with caution. 

Ab initio modem valence bond theory, in its spin-coupled valence bond (SCVB) form, has proved very successful for accurate computations on ground and excited states of molecular systems. The compactness of the resulting wavefunctions allows direct and clear interpretation of correlated electronic structure. We concentrate in the present account on recent developments, typically involving the optimization of virtual orbitals via an approximate energy expression. These virtuals lead to higher accuracy for the final variational wavefunctions, but with even more compact functions. Particular attention is paid here to applications of the methodology to studies of intermolecular forces. 

The chosen variational wavefunction is the d-wave superconducting wave-function with projection to remove double occupation ... 

AFM ground state. An unasked question is At what x does the chosen RVB variational wavefunction have a lower energy than a wavefunction describing AFM at appropriate wave-vectors Q(jc) They also acknowledge that their approach is no help in understanding the universal normal state properties for compositions near those for the highest Tc. I will return to both these points first, let us compare the claims made with the experiments. 

Two popular modem VB approaches, GVB-PP-SO and SC, use fully-variational wavefunctions including a single orbital product. In the GVB-PP-SO wavefunction this single orbital product is combined with a single perfect-pairing (PP) spin function which in the Rumer spin 3-4,. .. , (JV-2S-1)-(N-2S)) ... 

Using the same form of an optimized variational wavefunction... 

Non-variational wavefunctions, in which there is no optimisation of parameters, also exist. One approach is Moller-Plesset (MP) perturbation theory which, like Cl methods, is employed most often to improve upon a previously determined HF wavefunction. Fuller details of all these approaches may be found in the monograph by Szabo and Ostlund [40]. 

This theorem is also valid for many variational wavefunctions, e.g. for the Hartree-Fock one, if complete basis sets are used. As only the one-electron part of the Hamiltonian depends on the nuclear coordinates, H is a one-electron operator, and the evaluation of the Hellmann-Feynman forces is simple. Because of this simplicity, there have been a number of early suggestions to use the Hellmann-Feynman forces for the study of potential surfaces. These attempts met with little success, and the discussion below will show the reason for this. It is perhaps fair to say that the main value of the Hellmann-Feynman theorem for geometrical derivatives is in the insight it provides, and that numerical applications do not appear promising. For other types of perturbations, e.g. for weak external fields, the theorem is widely used, however. For a survey, see a recent book (Deb, 1981). 

By carefully adjusting the variational wavefunction used, it is possible to circumvent size-extensivity... 

This first-order approximation to the true exp(i2) 0> then leads us to consider the variational wavefunction... 

The identity of expressions (25) and (32) for exact eigenfunctions of a model Hamiltonian and optimum variational wavefunctions [65] can be proven by the extra-diagonal hypervirial relationship... 

These questions will be discussed separately, starting with the first one, and showing that, within the conventional CO formulation of the magnetic response [58,67], exact cancellation takes place only in the case of optimal variational wavefunctions which satisfy hypervirial constraints [65]. 

To understand the difference between the spurious gauge terms (i), which are removed in the ideal case of optimal variational wavefunctions [65], and terms (ii), which account for the essential origin dependence of the property, let us first discuss die origin dependence of the quadrupole polarizability of magnetic susceptibility (22) within the conventional common-origin representation. 

Some of the above reasons for preferring the energy derivative over the expectation value will only hold for variational wavefunctions. However, Diercksen et have argued that this technique is more suitable with MBPT methods as well. The techniques developed in analytic derivative methods can also be applied to the calculation of MBPT properties. In Moller-Plesset theory (the simplest form of MBPT), the zeroth-order wavefunction is SCF and the Hamiltonian is partitioned so that... 

As variational wavefunctions of the ground and excited states we choose harmonic oscillator wavefunctions... 

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