Variational principle solving Schrodinger equation

If the kinetic balance condition (5) is fulfilled then the spectrum of the L6vy-Leblond (and Schrodinger) equation is bounded from below. Then, in each case there exists the lowest value of E referred to as the ground state. In effect, this equation may be solved using the variational principle without any restrictions. On the contrary, the spectrum of the Dirac equation is unbounded from below. It contains the negative ( positronic ) continuum. Therefore the variational principle applied unconditionally would lead to the so called variational collapse [2,3,7]. The variational collapse maybe avoided by properly selecting the trial functions so that they fulfil the boundary conditions specific for the bound-state solutions [1]. 

But there is a more basic difficulty in the Hohenberg-Kohn formulation [19-21], which has to do with the fact that the functional iV-representability condition on the energy is not properly incorporated. This condition arises when the many-body problem is presented in terms of the reduced second-order density matrix in that case it takes the form of the JV-representability problem for the reduced 2-matrix [19, 22-24] (a problem that has not yet been solved). When this condition is not met, an energy functional is not in one-to-one correspondence with either the Schrodinger equation or its equivalent variational principle therefore, it can lead to energy values lower than the experimental ones. 

In principle, a laser control field e(/) could be designed with the evolution - reliably producing an acceptable value for (opf o xpi, where O is a chosen observable operator. This design problem may be best treated variationally, seeking an optimal control e(t) for this purpose[14,l5]. The practical implementation of quantum optimal control theory (OCT) poses challenging numerical tasks due to the need to repeatedly solve the Schrodinger equation, Eq. 

Price and Halley (PH) [136] and Halley, Johnson, Price and Schwalm (HJPS) [137] have described a different theory of electron overspill into the layer between the solvent and metal-ion cores at a metal-electrolyte interface in the absence of specific adsorption of ions. Previous authors avoided the use of Schrodinger s equation altogether by introducing trial functions for the electron density function n(x). In contrast Halley and co-workers (HQ [138-141] used the Kohn-Sham version [122] of the variational principle of Hohenberg and Kohn [121] in which n(x) was described in terms of wave functions obeying Hartree-like equations. An effective one-electron Schrodinger equation is solved... 

We can then solve the Schrodinger equation with the full Hamiltonian (Eq. (1.5)) by varying the coefficients c, so as to minimize the energy. If the summation is over an infinite set of these N-electron functions, i, , we will obtain the exact energy. If, as is more practical, some finite set of functions is used, the variational principle tells us that the energy so computed will be above the exact energy. 

As we have noted, electronic stracture techniques attempt to solve the Schrodinger equation. The traditional approach in quantum chemistry has been to use the Hartree Fock (HF) approximation, in which a determinantal, antisymmetrized wave function is optimized in accordance with the variational principle. The wave function is normally written as an expansion of atomic orbitals (the LCAO approximation). A major weakness of the HF method is that in its single... 

Except for a small number of intensively-studied examples, the Schrodinger equation for most problems of chemical interest cannot be solved exactly. The variational principle provides a guide for constructing the best possible approximate solutions of a specified functional form. Suppose that we seek an approximate solution for the ground state of a quantum system described by a Hamiltonian H. We presume that the Schrodinger equation... 

The problem is to get a computable expression for the ground state wave function without solving the Schrodinger equation for the many body hamiltonian of (1), obviously an impossible task for any non trivial system. As usual in many body problems, we can resort to the variational principle which states that the energy of any proper trial state ) will be greater or equal to... 

Now we have reduced the problem to solving the Schrodinger equation for just one unit cell (actually even a primitive cell in the cases where it is possible to reduce the unit cell even more), and for separated electronic and nuclear degrees of freedom. Still, as long as we have more than two electrons, the problem is unmanageable. Therefore we must make more approximations in order to get some way to solve the problem. One of the most common approximations is the Hartree-Fock method [8], in which the variational principle is used together with so called Slater determinants of electron orbitals to do calculations. One of the problems with this method is that you have to approximate the orbitals. 

The variation principle is used when the Schrodinger equation cannot be solved. However let us assume that we know the eigen function of the correct Hamiltonian. 

Undoubtedly, the Hohenberg-Kohn theorem has spurred much activity in density functional theory. In fact, most of the developments in this field are based on its tenets. Nevertheless, the approximate nature of all such developments, renders them functionally" non-jV-representable. This simple means that all approximate methods based on the Hohenberg-Kohn theorem are not in a one to one correspondence with either the Schrodinger equation or with the variational principle from which this equation ensues [21, 22], Thus, the specter of the 2-matrix N-representability problem creeps back in density functional theory. Unfortunately, the immanence of such a problem has not been adequately appreciated. It has been mistakenly assumed that this 2-matrix /V-representability condition in density matrix theory may be translocated into /V-representability conditions on the one-particle density [22], As the latter problem is trivially solved [23, 24], it has been concluded that /V-representability is of no account in the Hohenberg-Kohn-based versions of density functional theory. As discused in detail elsewhere [22], this is far from being the case. Hence, the lack of functional. /V-representability occurring in all these approximate versions, introduces a very serious defect and leads to erroneous results. 

The central field approximation and the simplifications which result from it allow one to construct a highly successful quantum-mechanical model for the AT-electron atom, by using Hartree s principle of the self-consistent field (SCF). In this method, one equation is obtained for each radial function, and the system is solved iteratively until convergence is obtained, which leaves the total energy stationary with respect to variations of all the functions (the variational principle ). The Hartree-Fock equations for an AT-electron system are equivalent to several one electron radial Schrodinger equations (see equation (2.2)), with terms which make the solution for one orbital dependent on all the others. In essence, the full AT-electron problem is approximated by a smaller number of coupled one-electron problems. This scheme is sometimes (somewhat inappropriately) referred to as a one-electron model in fact, the Hartree-Fock equations are a genuine AT-electron theory, but describe an independent particle system. 

This way to solve the Schrbdinger equation has obtained the name Full CI-FCI. It becomes an exact solution in the limit of a complete basis set. The Full Cl wave function is the best solution to the Schrodinger equation that can be obtained with a given basis set. This is true both for the ground state and for the excited states following the variation principle and McDonald s theorem [5]. The above formulation was made in terms of Slater determinants. It is, however, also possible to constmct spin-adapted functions, which are linear combinations of determinants with a given orbital occupation. We shall call such functions as Configuration State Functions (CSF). 

Since its eigenvalues correspond to the allowed energy states of a quantum-mechanical system, the time-independent Schrodinger equation plays an important role in the theoretical foundation of atomic and molecular spectroscopy. For cases of chemical interest, the equation is always easy to write down but impossible to solve exactly. Approximation techniques are needed for the application of quantum mechanics to atoms and molecules. The purpose of this subsection is to outline two distinct procedures—the variational principle and perturbation theory— that form the theoretical basis for most methods used to approximate solutions to the Schrodinger equation. Although some tangible connections are made with ideas of quantum chemistry and the independent-particle approximation, the presentation in the next two sections (and example problem) is intended to be entirely general so that the scope of applicability of these approaches is not underestimated by the reader. 

Calculations that employ the linear variational principle can be viewed as those that obtain the exact solution to an approximate problem. The problem is approximate because the basis necessarily chosen for practical calculations is not sufficiently flexible to describe the exact states of the quantum-mechanical system. Nevertheless, within this finite basis, the problem is indeed solved exactly the variational principle provides a recipe to obtain the best possible solution in the space spanned by the basis functions. In this section, a somewhat different approach is taken for obtaining approximate solutions to the Schrodinger equation. 

However, it is the Pauli principle which prevents us from simply ignoring the existence of electron spin altogether. The trial wavefunction must be antisymmetric with respect to the exchange of the coordinates (space-spin) of any two particles. Without this constraint the solutions of the many-electron Schrodinger equation would be wrong there are many more solutions of the Schrodinger equation than there aje antisymmetric solutions of that equation. Electron spin, at this level, simply ensures that the spatial part of the wavefunction behaves properly when the electrons coordinates are permuted. Thus, notwithstanding the manipulational convenience of the use of spin functions it would be attractive to be able to deal explicitly only with a spatial trial function and solve a spatial variational problem. 

The key idea behind solving the electronic Schrodinger equation is to start with an initial guess for the total electronic wave function with some freely adjustable parameters, and vary these parameters until the lowest energy is found. The variational principle then states that this energy is the ground-state energy. 

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