Towards the Hartree Method

Transcription

1 Towards the Hartree Method Recall from Lecture 11: Schrödinger Equation for Helium rewritten in simple abstract form as follows, where the subscript of H and V indicate which electrons these terms apply to. The V 12 (e-e Coulomb) term prevents us from separating!(1,2) into a product of one electron functions!(1)!(2). As a way out, we made the (zeroth-order) approximation to completely neglect V 12 ; that is, we set V 12 =0. Can we do better? ( whilst preserving the ability to separate!)

2 Hartree s Method Approximates the instantaneous interaction between two moving electrons (as captured by V 12 ) with the average interaction experienced by an electron due the presence of the other. This replaces V 12 with a term V 1 +V 2, a sum of two effective potentials. Because these are one-index terms, we can separate!(1,2)=! 1 (1)! 2 (2) Similar for other atoms (Z) with more electrons V 1 +V 2 + V Z There are variations to the method; typically, the effective potential is taken to be the same for all electrons; V eff (r) is computed from the probability distribution "(r). In a spherical symmetric atom, the effective potential and "(r) are spherically symmetric. Thus, the relevant quantities are: Radial effective potential V(r) and radial distribution function P(r).

3 What does the potential look like? (electronic shielding of the nuclear charge) Consider an electron near the surface of an atom (image on the left), and near the nucleus (image on the right). Total Coulomb potential experienced by the electron For: For: Near the nucleus, the repulsive interactions with other electrons tend to cancel (on average!). Thus, our electron experiences the full nuclear Coulomb force/potential. Near the surface, the repulsive interactions with other electrons do not cancel; there is a net repulsive force in the direction away from the nucleus. In effect, the nuclear Coulomb attraction is weakened or shielded by the other electrons. Its often convenient to think in terms of an effective nuclear charge Z(r) instead of the effective potential.

4 Hartree s Self-Consistent Solution

5 Results: Atomic Shells Eigenfunctions of the same n have their highest density at roughly the same radial distance. In a sense, atoms are like onions, with quantum number n defining a layer of electrons around the nucleus. These layers are called shells. Electrons in a shell experience similar shielding of the nuclear charge. Here (Argon): Z n=1 #16 Z n=2 #8 Z n=3 #3 n=1 (K) n=2 (L) n=3 (M) n=1 (K) n=2 (L) n=3 (M)

6 Helium - Low Energy Electronic States With e-e Coulomb interaction, the eigenenergies are no longer degenerate with quantum number l (as in the one-electron atoms). This is because the average e-e distance changes with different l values, leading to different strenghts of the Coulomb e-e interaction. Examine average distance (triangle) in radial distribution function: Electron 1 (n=1) Electron 2 (n=2) $ Less e-e-repulsion $ More e-e-repulsion

7 Periodic Table The chemical properties of the elements are a periodic function of the atomic number Z. Hartree theory provides a quantum mechanical explanation for the periodic table. Recalling that electron energies in Hartree theory are determined by quantum numbers n and l, the structure of the table is determined by the energetic ordering of the outermost filled subshell.

8 Spectroscopic Notation Use simplified spectroscopic notation for (n,l) pairs of quantum numbers of an electron. n is given as a number, together with a lowercase letter representing l where s stands for l=0, p for l=1, d for l=2, f for l=3, g for l=4, e.g. (n=1,l=0) $ 1s (n=2,l=1) $ 2p (n=3,l=2) $ 3d Use this notation to describe single electrons. Describe a set of electrons by stringing together several single-electron symbols. Use superscripts to indicate multiple occupancy of the same (n, l) subshell; e.g. 1s 2 2s 2 2p 3 (This is called a configuration ) Remember the maximum occupancy of a subshell is 2(2l+1). This follows from the exclusion principle and is determined by the number of unique combinations of m S and m l quantum numbers for a given n and l.

9 Ordering of Outermost Levels

10 Change of Ordering with Z