Chem3D 8.0 Manual - CambridgeSoft

AdministratorQuantum mechanical methods describe moleculesin terms of explicit interactions between electronsand nuclei. Both ab initio and semiempiricalmethods are based on the following principles:• Nuclei and electrons are distinguished fromeach other.• Electron-electron (usually averaged) andelectron-nuclear interactions are explicit.• Interactions are governed by nuclear andelectron charges (i.e. potential energy) andelectron motions.• Interactions determine the spatial distributionof nuclei and electrons and their energies.• Quantum mechanical methods are concernedwith approximate solutions to Schrödinger’swave equation.ΗΨ = ΕΨ• The Hamiltonian operator, H, containsinformation describing the electrons and nucleiin a system. The electronic wave function, Ψ,describes the state of the electrons in terms oftheir motion and position. E is the energyassociated with the particular state of theelectron.NOTE: The Schrödinger equation is aneigenequation, where the “H” operator, theHamiltonian, operates on the wave function toreturn the same wave function and a constant.The wave function is called an eigenfunction,and the constant, an eigenvalue.function. This function is the square of the wavefunction, and when properly normalized,describes the probability of finding an electronin that state.∫ Ψ 2 (r)dr = 1where r = radius (x, y and z)• There are many solutions to this probabilityfunction. These solutions are called atomicorbitals, and their energies, orbital energies.• For a molecule with many electrons and nucleithe aim is to be able to describe molecularorbitals and energies in as analogous a fashionto the original Schrödinger equation aspossible.Approximations to the HamiltonianThe first approximation made is known as the Born-Oppenheimer approximation, which allows separatetreatment of the electronic and nuclear energies. Dueto the large mass difference between an electron anda nucleus, a nucleus moves so much more slowlythan an electron that it can be regarded as motionlessrelative to the electron. In effect, this approximationconsiders electrons to be moving with respect to afixed nucleus. This allows the electronic energy tobe described separately from nuclear energy by anelectronic Hamiltonian, which can be solved at anyset of nuclear coordinates. The electronic version ofthe Schrödinger equation is:• Exact solutions to the Schrödinger equation arepossible only for the simplest 1 electron-1nucleus system. These solutions, however,yield the basis for all of quantum mechanics.• The solutions describe a set of allowable statesfor an electron. The observable quantity forthese states is described as a probabilityH elecΨ elec= E elecΨ elecAnother approximation assumes that electrons actindependently of one another, or, more accurately,that each electron is influenced by an average fieldcreated by all other electrons and nuclei. Eachelectron in its own orbital is unimpeded by itsneighbors.130 • Chapter 9: Computation Concepts CambridgeSoftQuantum Mechanics Theory in Brief