Quantum Mechanics

• Historical viewpoints

  The underlying physical laws necessary for mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.

  P. A. M. Dirac

  Nobody understands quantum mechanics.

  R. P. Feynman

• Wave/particle duality: double slit experiment. Wave equation for quantum particles. Classical limit for long wavelengths.

• Uncertainty principle:

  .

• Einstein didn't like it; various "interpretations".

• Schrödinger's cat. Tunnelling phenomena ( -decay). Hidden variables.

• Difference between physicists and chemists.

• Set of assumptions governing the dynamics of a subset of fundamental particles.

• Part physics, part mathematics, part philosophy.

• We shall ignore all the philosophy, and most of the mathematics.

• However, a serious study of quantum chemistry inevitably requires some mathematics.

• No derivations: proof by assertion.

The Schrödinger Equation

• Wave mechanical equation

  

  E is the total energy, is the wave function, and H is the Hamiltonian operator. [[tau]]i is the coordinates fo the ith particle.

• Classically, H = T + V, kinetic and potential energy, respectively.

• [[Psi]] contains all the information about the system.

• Since all the N particles have to be somewhere, we have the normalization condition:

  

• Probabilistic interpretation: gives the probability of finding the system with particle 1 at [[tau]]1, etc.

Hamiltonian Operator

• Kinetic energy: classically given as

  

  since p=mv.

• Quantum mechanically,

  

  so

  

  where we have used atomic units:

  

• Hence,

  .

• Consider the hydrogen atom: one electron and one fixed proton, Coulomb (1/r) potential.

The Hydrogen Atom

• T is expressed in terms of Cartesians, V is expressed in terms of internal scalar distance.

• From the symmetry of the system (spherically symmetric potential) we are better off using spherical coordinates .

• [[Psi]] then factorizes into radial and angular parts.

• The angular part is the spherical harmonic,

  

  like

  

  etc. Orthonormality:

  

• The radial part is a polynomial in r times

  

  . Also, orthonormal: 1s, 2s, etc.

  

  Note cusp behavior as r --> 0: cancels 1/r term in H.

figure 2

figure 3

The Hydrogen Atom

• Exact solution (basically by a suitable choice of coordinates).

• Near perfect agreement with experimental observations (not perfect because nonrelativistic).

• Gave enormous confidence in early days that quantum mechanics was fundamentally correct.

• So, beyond H? He, H2 ?

The Helium Atom

• Two electrons, fixed nucleus:

  

• Defies analytical solution (so far ---- probably a Nobel prize in it, though....)

• Term in is nonseparable: no product of functions of electrons 1 and 2 can be the exact solution.

• The naive guess: 1s(1)1s(2), spin-paired, is not an eigenfunction of H.

Variational Methods

• Return to the Schrödinger Equation

  

  Multiplying by on the left and integrating over all space gives

  

  Often written as

  

  Diracn "bra-ket" notation.

• Any guess at the wave function (say, ) obeys,

  

  the equality holding only for the exact solution (Variation Principle).

• Guess a trial wave function with some adjustable parameters, and then adjust them to minimize . This is the Variational Method.

The Helium Atom

• We know that a wave function of the form 1s² is not exact, but what is our best estimate? We can imagine that the hydrogen orbitals are too diffuse, because He has a higher charge than H. This could be accommodated by increasing in the radial function to two.

• Energy is 2.75 atomic units, compared to an experimental value of 2.90 atomic units - 5%, but this error is 94 kcal/mol!

• A better trial function (Hylleraas) would be

  

  where we see our original product of exponentials has been multiplied by polynomials in r1, r2, and r12. Six terms give an accuracy of 1 kcal/mol.

• But this only works for helium, or (in a modified form) H2. How do we deal with many-electron systems?

Spin and Antisymmetry

• Electrons have a spin which is quantized, up or down. Particles with half-integral spins (like electrons) are fermions. Fermion wave functions are antisymmetric with respect to particle interchange:

  

• Our simple helium wave functin was two 1s electrons, spin paired. Denoting spin-down by a bar, it was : neither symmetric nor antisymmetric with respect to particle interchange, which gives .

• The form is antisymmetric. It could be written as

  

  Normalized by , this term is termed a Slater determinant. Often represented by the diagonal alone: .

• The great advantage of the Slater determinant is that it can be generalized to any number of electrons.

Slater Determinants

• Consider three electrons in three arbitrary orbitals a, b, c. The Slater determinant |a(1)b(2)c(3)| can be exapanded as

  

• Note the compact notation: N! terms represented in an NxN array, or just as the elements of the diagonal.