1 MTLE-6120: Advanced Electronic Properties of Materials Atoms, many electron theories and the periodic table Contents: ◮ Hydrogen atom: quantum numbers and orbitals ◮ Many-electron systems and the density-functional (DFT) perspective ◮ Electronic configuration examples Reading: ◮ Kasap: 3.7 - 3.8
2 Hydrogenic atom ◮ Single electron with a nucleus of charge + Ze , where Z is the atomic number ◮ Z = 1 is hydrogen, Z = 2 is a He + ion, Z = 3 is Li 2+ etc. ◮ Schrodinger equation − � 2 ∇ 2 ψ ( � − Ze 2 r ) 4 πǫ 0 rψ ( � r ) = Eψ ( � r ) 2 m separable in spherical coordinates resulting in eigenfunctions ψ nlm ( � r ) = R nl ( r ) Y lm ( θ, φ ) and eigen-energies me 4 0 � 2 · Z 2 n 2 = − Z 2 2 n 2 E h = − Z 2 Ryd ≈ − Z 2 E nlm = − n 2 (13 . 6 eV ) 32 π 2 ǫ 2 n 2
3 Atomic quantum numbers ◮ For the box, we has n x , n y and n z ◮ Now in spherical coordinates, so correspond to r , θ and φ ◮ Principal quantum number n = 1 , 2 , . . . is for the radial r direction ◮ Angular quantum number l = 0 , 1 , 2 , . . . , n − 1 is for the θ direction ◮ Azimuthal quantum number m l = − l, − l + 1 , . . . , + l is for the φ direction ◮ But energy E nlm l ∝ n − 2 only depends on n ◮ States of various l and m l at same n are ‘degenerate’ i.e. have same energy
4 Radial wavefunctions ◮ Radial functions of the form � − Zr � · r l · p ( l ) R nl ( r ) ∝ exp n − l − 1 ( r ) na 0 1s 2s where a 0 = 4 πǫ 0 � 2 / ( me 2 ) ≈ 0 . 529 ˚ A 3s is the Bohr radius ◮ Typical radial extent ∼ na 0 /Z ◮ Polynomial degree n − l − 1 : first n of given l has no nodes, next has one node etc. ◮ Remember l = 0 , 1 , 2 , 3 denoted by s , p , d , f 2p 3p ◮ 1s has no nodes, 2s has 1 node etc. ◮ 2p has no nodes, 3p has 1 node etc.
5 Angular wavefunctions ◮ Spherical harmonics Y lm l ( θ, φ ) = P m l (cos θ ) e im l φ l ◮ Characteristic orbital shapes used in chemistry (typically Re Y lm and Im Y lm ) ◮ l controls number of lobes ◮ m l controls number in xy -plane ◮ All m l related by spherical symmetry
6 Electronic configuration of atoms ◮ Pauli exclusion principle: one electron per state (Fermi-Dirac statistics) ◮ Spin: m s = ± 1 / 2 (2 states) ◮ Azimuthal: m = − l, − l + 1 , . . . , + l ( 2 l + 1 states) ◮ Per n and l : 2(2 l + 1) states ◮ Periodic table by orbital being filled ( Z range): 1s (1-2) 2s (3-4) 2p (5-10) 3s (11-12) 3p (13-18) 4s (19-20) 3d (21-30) 4p (31-36) 5s (37-38) 4d (39-48) 5p (49-54) 6s (55-56) 4f (57-70) 5d (71-80) 7p (81-86) 7s (87-88) 5f (89-102)
7 The size of atoms ◮ Orbital size ∼ na 0 /Z ◮ Hydrogen atom Z = 1 , n = 1 : size ∼ a 0 ≈ 0 . 53 ˚ A ◮ Sodium atom Z = 11 , n = 3 : size ∼ 3 a 0 / 11 ≈ 0 . 14 ˚ A ◮ Platinum atom Z = 78 , n = 6 : size ∼ 6 a 0 / 78 ≈ 0 . 04 ˚ A ◮ What’s wrong? ◮ Hydrogenic orbitals are for one electron systems only! ◮ When more than one electron, electron-electron repulsion matters ◮ Effective charge seen by outer electrons is approximately that of nucleus + inner electrons
8 Many-electron Schrodinger equation ◮ So far, we discussed wavefunction ψ ( � r ) satisfying − � 2 2 m ∇ 2 ψ + V ( � r ) ψ = Eψ which is strictly a one-electron theory only. ◮ For N electrons, need to keep track of all N electronic coordinates with a wavefunction ψ ( � r 1 ,� r 2 , . . . ,� r N ) ◮ Corresponding Schrodinger equation with e-e interactions: − � 2 e 2 � � � ∇ 2 r i ψ + V ( � r i ) ψ + r j | ψ = Eψ � 2 m 4 πǫ 0 | � r i − � i i i � = j � �� � � �� � � �� � e-nuc Kinetic e-e which is impossible to solve exactly beyond special N = 2 cases
9 Many-electron non-interacting case ◮ Without e-e interactions: − � 2 � � ∇ 2 r i ψ + V ( � r i ) ψ = Eψ � 2 m i i � �� � � �� � e-nuc Kinetic which is separable in each � r i . ◮ Therefore solution must be consist of products ψ ( � r 1 ,� r 2 , . . . ,� r N ) ∼ φ 1 ( � r 1 ) φ 2 ( � r 2 ) · · · φ N ( � r N ) ◮ Each ‘orbital’ φ i : N = 1 Schrodinger equation with orbital energy ε i ◮ Total energy E = � i ε i ◮ Strictly, fermionic wavefunctions need to be antisymmetric ⇒ ψ = det[ φ i ( � r j )] (Slater determinant)
10 Kohn-Sham density functional theory (DFT) ◮ A single-particle theory − � 2 2 m ∇ 2 φ i ( � r ) + V KS ( � r ) ψ = ε i φ i ( � r ) in an effective potential V KS ( � r ) ◮ V KS ( � r ) = V ( � r ) + contribution from electron density n ( � r ) ◮ Total energy E = � i ε i + contribution from electron density n ( � r ) r ) | 2 made self-consistent r ) = � ◮ Electron density n ( � i | φ i ( � ◮ DFT works surpisingly well even for strongly interacting electrons ◮ When DFT does not work, material called ‘strongly-correlated’!
11 Atoms revisited ◮ Orbital energies are those from effective potential (not hydrogenic) ◮ For spherical atoms, still degenerate in m and m s , but not in l ◮ At same n , energy increases with l ◮ In particular, energy of ( n + 1) s < ( n − 1) f < n d < ( n + 1) p ⇒ 1s (1-2) 2s (3-4) 2p (5-10) 3s (11-12) 3p (13-18) 4s (19-20) 3d (21-30) 4p (31-36) 5s (37-38) 4d (39-48) 5p (49-54) 6s (55-56) 4f (57-70) 5d (71-80) 7p (81-86) 7s (87-88) 5f (89-102)
12 Electronic configuration example: Sc Z = 21 , Configuration: 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 1 3d 4s 3p 3s 2p 2s 1s
13 Electronic configuration example: V Z = 23 , Configuration: 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 3 3d 4s 3p 3s 2p 2s 1s
14 Electronic configuration example: Cr Z = 24 , Configuration: 1s 2 2s 2 2p 6 3s 2 3p 6 3d 5 4s 1 3d 4s 3d 3p 3s 2p 2s 1s
15 Electronic configuration example: Ni Z = 28 , Configuration: 1s 2 2s 2 2p 6 3s 2 3p 6 3d 8 4s 2 3d 4s 3p 3s 2p 2s 1s
16 Electronic configuration example: Cu Z = 29 , Configuration: 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 4s 1 4s 3d 3p 3s 2p 2s 1s
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