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Transcript
⢠to describe molecules one need quantum mechanics; ⢠we need to develop methods which can give more and more accurate solutions to the Schrödinger equation ⢠we also need approximate methods which support chemical intuition without expensive calculations. In practice: one has to approximate Ψ. Chemists are very good at this! 2.3.3. The Hamilton operator The Hamilton operator of the system (HÌ) consists of the sum of the kinetic (TÌ ) and the potential energy (VÌ ) operators. HÌ = TÌ + VÌ (4) the form of TÌ is the same for all systems, while the potential energy represents the molecule by including the interactions between the electrons and nuclei. 2.3.4. State function In quantum mechanics the state of the system is represented by the wave function (or state function) which depends on the coordinates of the particles: Ψ = Ψ(x, y, z) = Ψ(r) (5) Ψ = Ψ(x1 , y1 , z1 , x2 , y2 , z2 , ..., xn , yn , zn ) = Ψ(r1 , r2 , ..., rn ) (6) or in case of n particles: The wave function has no physical meaning, but its square, the so called probability density can be given a probability interpretation: Ψâ (x0 , y 0 , z 0 ) · Ψ(x0 , y 0 , z 0 )dx dy dz (7) is the probability of finding a particle at point (x0 , y 0 , z 0 ) (more precisely in the infinitesimal proximity). Shorter notation: Ψâ Ψdv or |Ψ|2 dv We have to chose the wave function normalized, otherwise the valószÃnűség of finding the particle in the whole space would not be one: Z Z Z Ψâ · Ψ dx dy dz = 1 14 (8)
