Download The Helium Atom

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.