By Warren J Hehre

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This is termed the total electron density (or more simply the electron density), and corresponds to the electron density measured in an X-ray diffraction experiment. * This equation as well as multi-particle Schrödinger equation and all approximate equations which follow are given in so-called atomic units. This allows fundamental constants as well as the mass of the electron to be folded in. 22 Chapter 2 22 3/21/03, 11:45 AM Graphical representations of the electron density will be provided in Chapter 4, and connections drawn between electron density and both chemical bonding and overall molecular size and shape.

These become competitive and ultimately superior to conventional algorithms for very large molecules, where “pure” density functional procedures will actually be significantly faster than Hartree-Fock models. Socalled “hybrid” density functional models, such as the popular B3LYP model, make use of Hartree-Fock exchange terms. These do not benefit from multipole Coulomb methods and can never surpass Hartree-Fock models in computation speed. The second part of the calculation involves dealing with the exchange/ correlation functional.

32) Ψ = Ψ0 + λΨ(1) + λ2Ψ(2) + λ3Ψ(3) + ... (33) Substituting the expansions 31 to 33 into the Schrödinger equation and gathering terms in λn yields. ˆ 0Ψ0 = E(0)Ψ0 H (34a) (0) (1) ˆ Hˆ 0Ψ(1) + VΨ + E(1)Ψ0 0= E Ψ (34b) ˆ (1) = E(0) Ψ(2) + E(1) Ψ(1) + E(2) Ψ0 ˆ 0Ψ(2) + VΨ H (34c) Multiplying each of the equations 34 by Ψ0 and integrating over all space yields the following expression for the nth order (MPn) energy. E(0) = ∫ ... dτn Ψ0H (35a) E(1) = ∫ ... dτn Ψ0VΨ (35b) E(2) = ∫ ... dτn Ψ0VΨ (35c) In this framework, the Hartree-Fock energy is the sum of the zero and first-order Møller-Plesset energies.