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
E is the total energy, is the wave function, and H is the Hamiltonian operator. [[tau]]i is the coordinates fo the ith particle.
where we have used atomic units:
. Also, orthonormal: 1s, 2s, etc.
Note cusp behavior as r --> 0: cancels 1/r term in H.
Multiplying by on the left and integrating over all space gives
Often written as
Diracn "bra-ket" notation.
the equality holding only for the exact solution (Variation Principle).
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.
Normalized by , this term is termed a Slater determinant. Often represented by the diagonal alone: .