Solving Schrodingers Equation

The first problem is the subject of modern quantum chemistry. Many efficient methods and computer programs are nowadays available to solve the electronic Schrodinger equation from first principles without using phenomenological or experimental input data (so-called ab initio methods Lowe 1978 Carsky and Urban 1980 Szabo and Ostlund 1982 Schmidtke 1987 Lawley 1987 Bruna and Peyerimhoff 1987 Werner 1987 Shepard 1987 see also Section 1.5). The potentials Vfe(Q) obtained in this way are the input for the solution of the nuclear Schrodinger equation. Solving (2.32) is the subject of spectroscopy (if the motion is bound) and... 

Let us consider a rectangular box (Fig. 4.4) with sides L and L2 and V = 0 inside and V = 00 outside. We first separate the variables x and y, which leads to the two 1-D Schrodinger equations (solved as shown above). 

Electrons are much lighter than nuclei and move much faster. Thus the electrons can adjust to the movement of the nuclei which means that the electronic states are at any moment essentially the same as if the nuclei were fixed. This is the basis of the Born-Oppenheimer approximation which assumes fixed nuclei. The wave function can be expressed as a product of an electronic wavefunction with the nuclei assumed fixed and a nuclear wavefunction describing the relative nuclear motion. Energy eigenvalues for the electronic Schrodinger equation, solved for different nuclear separations, form a potential, that is inserted into the nuclear Schrodinger equation together with the nuclear repulsion term. 

Balint-Kurti G G, Dixon R N and Marston C C 1992 Grid methods for solving the Schrodinger equation and time-dependent quantum dynamics of molecular photofragmentation and reactive scattering processes/of. Rev. Phys. Chem. 11 317—44... 

Upon solving the Schrodinger equation, the energy levels are , = + 1/2), where is related to the force... 

In the two-adiabatic-electronic-state Bom-Huang description of the total orbital wave function, we wish to solve the corresponding nuclear motion Schrodinger equation in the diabatic representation... 

De Raedt, H. Product formula algorithms for solving the time-dependent Schrodinger equation. Comput. Phys. Rep. 7 (1987) 1-72. 

I 1 11 Schrodinger equation can be solved exactly for only a few problems, such as the particle in a box, the harmonic oscillator, the particle on a ring, the particle on a sphere and the hydrogen atom, all of which are dealt with in introductory textbooks. A common feature of these problems is that it is necessary to impose certain requirements (often called boundary... 

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 ... 

For translational, rotational and vibrational motion the partition function Ccin be calculated using standard results obtained by solving the Schrodinger equation ... 

If the Hamiltonian operator contains the time variable explicitly, one must solve the time-dependent Schrodinger equation... 

The Hydrogenic atom problem forms the basis of much of our thinking about atomic structure. To solve the corresponding Schrodinger equation requires separation of the r, 0, and (j) variables... 

In summary, separation of variables has been used to solve the full r,0,( ) Schrodinger equation for one electron moving about a nucleus of charge Z. The 0 and (j) solutions are the spherical harmonics YL,m (0,(1>)- The bound-state radial solutions... 

In solving differential equations such as the Schrodinger equation involving two or more variables (e.g., equations that depend on three spatial coordinates x, y, and z or r, 0,... 

Statistical mechanics is the mathematical means to calculate the thermodynamic properties of bulk materials from a molecular description of the materials. Much of statistical mechanics is still at the paper-and-pencil stage of theory. Since quantum mechanicians cannot exactly solve the Schrodinger equation yet, statistical mechanicians do not really have even a starting point for a truly rigorous treatment. In spite of this limitation, some very useful results for bulk materials can be obtained. 

One of the advantages of this method is that it breaks the many-electron Schrodinger equation into many simpler one-electron equations. Each one-electron equation is solved to yield a single-electron wave function, called an orbital, and an energy, called an orbital energy. The orbital describes the behavior of an electron in the net field of all the other electrons. 

The relativistic Schrodinger equation is very difficult to solve because it requires that electrons be described by four component vectors, called spinnors. When this equation is used, numerical solution methods must be chosen. 

After solving the electronic Schrodinger equation (equation 4), to calculate a potential energy surface, you must add back nuclear-nuclear repulsions (equation 5). 

NDO calculations use the Hartree-Fock (HE) approximation to solve the Schrodinger equation. HE methods deal with several kinds of electron-electron interactions. By understanding these interactions, you can appreciate differences between the NDO methods and gain insight into why the NDO approximation works well or fails. 

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 ... 

This last equation is the nuclear Schrodinger equation describing the motion of nuclei. The electronic energy computed from solving the electronic Schrodinger equation (3) on page 163 plus the nuclear-nuclear interactions Vjjjj(R,R) provide a potential for nuclear motion, a Potential Energy Surface (PES). 

Rather than solve a Schrodinger equation with the Nuclear Hamiltonian (above), a common approximation is to assume that atoms are heavy enough so that classical mechanics is a good enough approximation. Motion of the particles on the potential surface, according to the laws of classical mechanics, is then the subject of classical trajectory analysis or molecular dynamics. These come about by replacing Equation (7) on page 164 with its classical equivalent ... 

Within the Born-Oppenheimer approximation discussed earlier, you can solve an electronic Schrodinger equation... 

Electrons are identical, and each term in this sum is essentially the same operator. You can then solve an independent-electron Schrodinger equation for a wave function /, describing an individual electron ... 

Now that you know the mathematical form, you can solve the independent-electron Schrodinger equation for the molecular orbitals. First substitute the LCAO form above into equation (47) on page 193, multiply on the left by and integrate to represent... 

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