- 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.

- 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.

- 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.

- 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

- 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 ?

- 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.

- 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*.

- We know that a wave function of the form 1s
^{2}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?

- 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.

- 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.