Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
The Helium Atom The Helium Atom poses a problem that (like all other atomic and molecular systems) can only be solved through approximations – there exists no closed analytical solution. A spherical (“central-force”) problem similar to the hydrogen atom There exist two electrons and one nucleus (with z = 2) There are three interaction terms (two attractive and one repulsive) that all depend on the relative position of the electrons and the nucleus All three potentials are given by the Coulomb potential between two point charges. If Schrödinger’s equation describes the entire system, then the wave function will describe the particle waves of both electrons and the nucleus. The Schrödinger Equation for the He Atom is ̂ * + ̂ ̂ ̂ Kinetic Energy ̂ ̂ ̂ Electron nucleus attraction ̂ ̂ ̂ Electron electron repulsion ̂ To solve the SE, use separation of variables ̂ ̂ ̂ It can be shown that no analytical solution to this problem can be found. The product of the one-electron wave functions doesn’t solve Schrödinger’s equation! The problem is the electron-electron interaction term ̂ Approach #1: Brute force method – neglect of electron repulsion We neglect the term that gives us the headache and write ̂ ̂ Of course in this case there is no dependence of one electrons motion on the other electron’s motion As analogy with the hydrogen case the energy eigenvalues are ( ) For the lowest state of Helium (Z = 2, n1 = n2 = 1) the energy eigenvalues is ( ) This is a whole lot lower than the actual value of –2.904 Eh ! Our approximation is rather poor! Approach #2: Perturbation Theory When trying to solve any complicated problem, that resembles a simpler problem one can introduce the offensive term in the Hamiltonian as a “perturbation” to the “0th-order” Hamiltonian operator We can get an arbitrarily accurate wave function for state n by writing the true wave function as an expansion Here the parameters are numbers with < 1 that increase with the degree of perturbation. Consequently 2 is very close to zero. Now, for simplicity let’s only retain the 1st-order correction terms. Solving the equation will produce that So that This is a lot closer to the exact value of E=-2.904 Eh, but this time it is too high. Problem: the Hamiltonian is exact, but the wave function (and therefore the energy) only contains the first-order correction term. One can increase the accuracy by introducing higher terms. Approach #3: The Variational Method Any trial wave function being operated on by the exact Hamiltonian will always give a ground state energy value above the true energy eigenvalue of the system. | ̂| 〉 〈 〉 This is useful, since the only thing left to do is to guess a good wave function . The best guess will have the lowest energy! the one-electron 1s wave functions in atomic units for distance are 〈 ⁄ √ Using Z’ which is the average nuclear charge that the electrons see. This value may be lower than Z = 2, since there is a change that the other electron is “in front of” the nucleus and “shields” the charge. Using the variational method we’re looking for a value of Z’ for which the energy is minimal therefore The “effective charge” that an electron sees is therefore considerably less than “2”. Shielding is important!. Using this value so which is now MUCH BETTER since it is already very close to E=2.904Eh Molecular Orbital Theory: from atoms to molecules For a system with many nuclei (index a) and electrons (index i) This is the exact but unsolvable Schrödinger equation for ANY molecule The Born-Oppenheimer approximation Electrons’ motion is much faster than nuclear motion when solving S.E. we can assume that the atoms stand still find eigenvalues for electronic wave function at a fixed geometry move atoms a little and freeze them again find more energies at the new geometry repeat for The SE before B.O. A. is: After the B.O. A. the electronic part of the equation is The nuclear kinetic part is set to zero and the potential energy arising from internuclear repulsion is just added as a parameter (energies are additive) Linear Combination of Atomic Orbitals (LCAO) How can the molecular wave functions be obtained? Example H2 : linear combination of two H-atom wave functions The bonding 1 MO is symmetric with respect to inversion at the center-point even (german: even = “gerade”, so the subscript “g”) The antibonding 2* MO is antisymmetric with respect to inversion at the center point odd (german: odd = “ungerade”, so the subscript “u”) Molecular orbitals from p-atomic orbitals The MO diagram for a homonuclear diatomic molecule General Methods Ab initio methods: that is a Latin word means from the beginning, the approximations made are usually mathematical approximation, such as using a simpler functional form for a function. These methods are very accurate but needs a lot of time and high PC resources; an example of those methods is called HF method. Semi-empirical methods: these are many types of methods defer from each other according to the type of approximation which in these methods are approximating some integrals needed in calculation by experimental values, also that methods use only the valance electrons in molecules, an example of that methods is called PM3 method. These methods are more or less accurate but needs no time or PC resources.