Approximating the Schrodinger Equation

2 Approximating the Schrodinger Equation. - With the approaches we have discussed so far one attempts to solve the exact Schrodinger equation as accurately as possible. All quantities are formulated in terms of the many-particle wavefunctions and, consequently, these approaches are called wave-function-based methods. Their main disadvantages are that the equations are highly complicated and, therefore, that the solutions become more and more approximate the more complex the system of interest is. Typically, the computational efforts scale as N4 for the Hartree-Fock approximation and up to N1 for the methods that include correlation effects. 

A completely different aspect is that the outcome of such calculations, i.e., the many-particle wavefunction, contains much more information than is required for the calculation of any observable. For the latter, one needs usually only the dependence of the wavefunction on one or two particles, whereas that on the coordinates of the N — 1 or N — 2 other particles is redundant. 

Gaspar7 argued that the potential on the right-hand side should be multiplied by j, and ultimately a more general approach with a scaling parameter a on the right-hand side was devised. This was the so-called Xoc method. Here, the Hartree-Fock equations (5) are consequently approximated as 

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The quantum mechanics methods in HyperChem differ in how they approximate the Schrodinger equation and how they compute potential energy. The ab initio method expands molecular orbitals into a linear combination of atomic orbitals (LCAO) and does not introduce any further approximation. 

HyperChem s semi-empirical calculations solve (approximately) the Schrodinger equation for this electronic Hamiltonian leading to an electronic wave function I eiecW for the electrons ... 

Using the BO approximation, the Schrodinger equation describing the time evolution of the nuclear wave function, can be written... 

Within the Born-Oppenheimer approximation, the Schrodinger equation for a whole molecular system can be divided into two equations. The electronic Schrodinger equation needs to be solved separately for each different (fixed) set of positions for the nuclei making up the system and gives the electronic wavefunction and the electronic... 

The END theory was proposed in 1988 [11] as a general approach to deal with time-dependent non-adiabatic processes in quantum chemistry. We have applied the END method to the study of time-dependent processes in energy loss [12-16]. The END method takes advantage of a coherent state representation of the molecular wave function. A quantum mechanical Lagrangian formulation is employed to approximate the Schrodinger equation, via the time-dependent variational principle, by a set of coupled first-order differential equations in time to describe the END. 

We consider a hydrogen-like atom, with nuclear charge Z, enclosed in a spherical well, of radius R, with an impenetrable wall. The nucleus is assumed fixed at the centre of the well and we note that, for finite R, it is not therefore possible to separate out the translational motion of the centre of mass of the system. Pupyshev [18] proved that, for the ground state, the energy is a minimum when the nucleus is at r = 0. In a non-relativistic approximation the Schrodinger equation for the electronic motion is1... 

Information about the wavefunction is obtained from the Schrodinger wave equation, which can be set up and solved either exactly or approximately the Schrodinger equation can be solved exactly only for a species containing a nucleus and only one electron (e.g. H, He ), i.e. a hydrogen-like system. 

Unfortunately, the molecular spectra based on the eigen-problem (36) are neither directly nor completely solved without specific atoms-in-molecule and/or symmetry constraints and approximation. As such, at the mono-electronic level of approximation, the Schrodinger equation (36) is rewritten under the so-called independent-electron problem ... 

The Schrodinger equation is not exactly soluble for polyelectronic systems due to the electron-electron interaction. In order to solve approximately the Schrodinger equation, different methods are available. The most traditional way is based on the Hartree-Fock (HF) method. An alternative treatment is based on the Density-Functional Theory (DFT). 

Information about the wavefunction is obtained from the Schrodinger wave equation, which can be set up and solved either exactly or approximately. The Schrodinger equation... 

Under adiabatic approximation, the Schrodinger equation solution of multi-particle system can be written as the product of a nuclear wave function R) and an electronic wave function The electron wave function > i(f,R is... 

One of the common schemes used to solve the Schrodinger equation assumes that the electrons can be approximated as independent particles that interact mainly with the nuclear charges and with an average potential from the other electrons. With this approximation, the Schrodinger equation becomes a set of independent one-electron equations, and the usual separation of variable method can be used. In this case, can be written as a product of functions of only one electron coordinates, Wt x,y,z), an approximation that is called the Hartree-Fock approximation or HF. 

The central-field approximation. The Schrodinger equation for the electrostatic interaction of N electrons in an atom with a nuclear charge Z is given by... 

In the real world, any system consists of a collection of atoms (nuclei and electrons). The whole energy of a system can only be accurately obtained through the resolution of the Schrodinger equation. It is quite customary to accept the so-called Born-Oppenheimer approximation based on the different mass scale of nuclei and electrons. Under this approximation the Schrodinger equation is divided in two parts ... 

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