Approximations Schrodinger

When the non-adiabatic coupling terms x and x 2 are considered negligibly small and dropped from Eq. (B.15), we get the uncoupled approximate Schrodinger equation... 

If we are in possession of solutions to the approximate Schrodinger equation (10), we can represent the wave function of a state by an expansion of the form... 

In solving the many-particle Schrodinger equation, it is desirable to choose the approximate potential V0(x) to be as close as possible to the actual potential V(x), since this leads to rapid convergence of the expansion (18). Goscinski showed [20, 21] that for atoms, the approximate Schrodinger equation (10) can be solved exactly provided that V0(x) is chosen to be the attractive Coulomb potential of the bare atomic nucleus ... 

Before discussing in detail the numerical results of our computational work, we describe the theoretical and computational context of the present calculations apart from deficiencies of models employed in the analysis of experimental data, we must be aware of the limitations of both theoretical models and the computational aspects. Regarding theory, even a single helium atom is unpredictable [14] purely mathematically from an initial point of two electrons, two neutrons and two protons. Accepting a narrower point of view neglecting internal nuclear structure, we have applied for our purpose well established software, specifically Dalton in a recent release 2.0 [9], that implements numerical calculations to solve approximately Schrodinger s temporally independent equation, thus involving wave mechanics rather than quantum... 

In order to be sure that a bond actually forms between two atoms linked by a molecular orbital it is necessary to calculate the energies of the molecular orbitals and then allocate electrons to them. In essence, the approximate Schrodinger equation... 

The quantum mechanical treatment of a hamionic oscillator is well known. Real vibrations are not hamionic, but the lowest few vibrational levels are often very well approximated as being hamionic, so that is a good place to start. The following description is similar to that found in many textbooks, such as McQuarrie (1983) [2]. The one-dimensional Schrodinger equation is... 

Peskin U and Steinberg M 1998 A temperature-dependent Schrodinger equation based on a time-dependent self consistent field approximation J. Chem. Phys. 109 704... 

The mixed, v t — % notation here has historic causes.) The Schrodinger equation is obtained from the nuclear Lagrangean by functionally deriving the latter with respect to t /. To get the exact form of the Schrodinger equation, we must let N in Eq. (95) to be equal to the dimension of the electronic Hilbert space (viz., 00), but we shall soon come to study approximations in which N is finite and even small (e.g., 2 or 3). The appropriate nuclear Lagrangean density is for an arbitrary electronic states... 

When the wave function is completely general and pennitted to vary in the entire Hilbert space the TDVP yields the time-dependent Schrodinger equation. However, when the possible wave function variations are in some way constrained, such as is the case for a wave function restricted to a particular functional form and represented in a finite basis, then the corresponding action generates a set of equations that approximate the time-dependent Schrodinger equation. 

In Section II, molecular dynamics within the BO approximation was introduced. As shown in Appendix A, the full nuclear Schrodinger equation is, however. 

The familiar BO approximation is obtained by ignoring the operators A completely. This results in the picture of the nuclei moving over the PES provided by the electrons, which are moving so as to instantaneously follow the nuclear motion. Another common level of approximation is to exclude the off-diagonal elements of this operator matrix. This is known as the Bom-Huang, or simply the adiabatic, approximation (see [250] for further details of the possible approximations and nomenclature associated with the nuclear Schrodinger equation). 

We will study the equations of motion that result from inserting all this in the full Schrodinger equation, Eq. (1). However, we would like to remind the reader that not the derivation of these equations of motion is the main topic here but the question of the quality of the underlying approximations. 

In making certain mathematical approximations to the Schrodinger equation, we can equate derived terms directly to experiment and replace dilTiciilL-to-calculate mathematical expressions with experimental values. In other situation s, we introduce a parameter for a mathematical expression and derive values for that parameter by fitting the results of globally calculated results to experiment. Quantum chemistry has developed two groups of researchers ... 

It was stated above that the Schrodinger equation cannot be solved exactly for any molecular systems. However, it is possible to solve the equation exactly for the simplest molecular species, Hj (and isotopically equivalent species such as ITD" ), when the motion of the electrons is decoupled from the motion of the nuclei in accordance with the Bom-Oppenheimer approximation. The masses of the nuclei are much greater than the masses of the electrons (the resting mass of the lightest nucleus, the proton, is 1836 times heavier than the resting mass of the electron). This means that the electrons can adjust almost instantaneously to any changes in the positions of the nuclei. The electronic wavefunction thus depends only on the positions of the nuclei and not on their momenta. Under the Bom-Oppenheimer approximation the total wavefunction for the molecule can be written in the following form ... 

Whereas the tight-binding approximation works well for certain types of solid, for other s. items it is often more useful to consider the valence electrons as free particles whose motion is modulated by the presence of the lattice. Our starting point here is the Schrodinger equation for a free particle in a one-dimensional, infinitely large box ... 

Approximation Methods Can be Used When Exact Solutions to the Schrodinger Equation Can Not be Eound. 

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