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He and hydrogenoid ions

He and hydrogenoid ions. The one nucleus-electron system. topic. Mathematic required. Schrödinger for a hydrogenoid Orbital s Orbital p. Two prerequisites. Our world is 3D! We need to calculate integrals and derivatives in full space (3D).

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He and hydrogenoid ions

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  1. He and hydrogenoid ions The one nucleus-electron system

  2. topic • Mathematic required. • Schrödinger for a hydrogenoid • Orbital s • Orbital p

  3. Two prerequisites Our world is 3D! We need to calculate integrals and derivatives in full space (3D). A system of one atom has spherical symmetry. Spherical units are appropriate. Y= Y(r,q,f) rather than Y= Y(x,y,z) except that D was defined in cartesian units.

  4. Spherical units x = r sin q. cos f y = r sin q. sin f z = r cos q

  5. derivation dY/dr for fixed q and f

  6. derivation If we know making one derivation, we know how to make others, to make second derivatives, and the we know calculating the Laplacian, D

  7. Integration in space dV = r2sinqdrdqdf

  8. Integration limits r = ∞ f = 2p f = 0 r = 0 q = 0 q = p

  9. Integration in space

  10. Integration in space Integration over q, f and r gives V = 4/3 pr4 The integration over q and f gives the volume between two spheres of radii r and r+dr: dV = 4p r2dr Volume of a sphere Volume between two concentric spheres

  11. Dirac notationtriple integrals

  12. Spherical symmetry dV = 4 p r2 dR

  13. Radial density dP/dR = 4 p r2 Y*Y It is the density of probability of finding a particule (an electron) at a given distance from a center (nucleus) It is not the density of probability per volume dP/dV= Y*Y It is defined relative to a volume that increases with r. The unit of Y is L-3/2

  14. Schrödinger for a hydrogenoid (1 nucleus – 1 electron)The definition of an orbital atomic orbital:any function Ye(x,y,z) representing a stationary state of an atomic electron. Born-Oppenheimer approximation: decouplingthe motion of N and e Y(xN,yN,zN,xe,ye,ze)= Y(xN,yN,zN)Y(xe,ye,ze) mH=1846 me : When e- covers 1m H covers 2.4 cm, C 6.7 mm and Au 1.7 mm

  15. Schrödinger for a hydrogenoid (1 nucleus – 1 electron)

  16. Schrödinger for a hydrogenoid (1 nucleus – 1 electron)

  17. Schrödinger for a hydrogenoid (1 nucleus – 1 electron) We first look for solutions valid for large r

  18. Solution for large r Which of the 2 would you chose ?

  19. Solution Y = e-ar still valid close to the nucleus New To be set to zero leading to a condition on a : a quantification due to the potential Already set to zero by taking Ne-ar

  20. The quantification of a is a quantification on E This energy is negative. The electron is stable referred to the free electron

  21. Energy units 1 Rydberg = 21.8 10-19 Joules =14.14 105 J mole-1 1 Rydberg = 13.606 eV = 0.5 Hartree (atomic units) 1eV (charge for an electron under potential of 1 Volt) 1eV = 1.602 10-19 Joules = 96.5 KJoules mole-1 (→n = 8065.5 cm-1) 1eV = = 24.06 Kcal mole-1.

  22. Atomic units The energy unit is that of a dipole +/- e of length a0 It is the potential energy for H which is not the total energy for H (-1/2 a.u.) (E=T+V) Atomic units : h/2p =1 and 1/4pe0 =1 • The Schrödinger equation becomes simpler

  23. Normalization of Ne-ar From math textbooks The density of probability is maxima at the nucleus and decreases with the distance to the nucleus.

  24. Radial density of probability a0/Z is the most probable distance to the nucleus; it was found by Niels Bohr using a planetary model. The radial density close to zero refers to a dense volume but very small; far to zero, it corresponds to a large volume but an empty one

  25. Orbital 1s

  26. Average distance to the nucleus Distance: Operator r larger than a0/Z From math textbooks In an average value, the weight of heavy values dominates: (half+double)/2 = 1.25 > 1)

  27. Distances the nucleus

  28. Excited states We have obtained a solution using Y = e-ar; it corresponds to the ground state. There are other quantified levels still lower than E=0 (classical domain where the e is no more attached to the nucleus)o We can search for other spherical function Y= NnPn (r)e-ar where Pn(r) is a polynom of r of degree n-1

  29. Orbital ns Principal quantum number E2s = Z2 /4 E1s (H) Ens = Z2 /n2 E1s (H) Nodes: spheres for solution of equation Pn=O n-1 solutions Average distance

  30. Orbital 2s E2s = Z2 /4 E1s (H) A more diffuse orbital: One nodal surface separating two regions with opposite phases: the sphere for r=2a0/Z. Within this sphere the probability of finding the electron is only 5.4%. The radial density of probability is maximum for r=0.764a0/Z and r=5.246a0/Z. Between 4.426 et 7.246 the probability of finding the electron is 64%. = Average distance 5a0/Z

  31. Orbital 2s

  32. Radial distribution

  33. Resolution of Schrödinger equation in f, Solving the equation in q and f leads to define two other quantum numbers. They also can be defined using momentum instead of energy. For q: Angular Momentum (Secondary, Azimuthal) Quantum Number (l):  l = 0, ..., n-1. For f: Magnetic Quantum Number (ml):  ml = -l, ..., 0, ..., +l.

  34. Resolution of Schrödinger equation in f,the magnetic quantum number

  35. Quantum numbers • Principal Quantum Number (n):  n = 1, 2,3,4, …, ∞Specifies the energy of an electron and the size of the orbital (the distance from the nucleus of the peak in a radial probability distribution plot). All orbitals that have the same value of n are said to be in the same shell (level). • Angular Momentum (Secondary, Azimunthal) Quantum Number (l):  l = 0, ..., n-1. • Magnetic Quantum Number (m):  m = -l, ..., 0, ..., +l. • Spin Quantum Number (ms):  ms = +½ or -½.Specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions (sometimes called up and down).

  36. Name of orbitals The letter indicates the secondary Quantum number, l The number indicates the Principal Quantum number The index indicates the magnetic Quantum number

  37. Degenerate orbitalsE= Z2/n2 (E1sH) Depends only on the principal Quantum number 1 function 4 functions 9 functions 16 functions Combination of degenerate functions: still OK for hydrogenoids. New expressions (same number); real expressions; hybridization.

  38. Functions 2p

  39. symmetry of 2pZ Nodes: No node for the radial part (except 0 and ∞) cosq = 0 corresponds toq=p/2 : the xy plane or z/r=0 : the xy plane The 2pz orbital is antisymmetric relative to this plane cos(-q)=-cosq The z axis is a C∞ axis

  40. Directionality of 2pZ This is the product of a radial function (with no node) by an angular function cosq. It does not depend on f and has the z axis for symmetry axis. The angular contribution to the density of probability varies like cosq2 Within cones: Full space 2 cones: a diabolo

  41. Directionality of 2pZ This is the product of a radial function (with no node) by an angular function cosq. It does not depend on f and has the z axis for symmetry axis. The angular contribution to the density of probability varies like cosq2 Within cones: Probability is 87.5% in half of the space 22.5% in the other half

  42. Spatial representation of the angular part. Let us draw all the points M with the same contribution of the angular part to the density The angular part of the probability is OM = cosq2 All the M points belong to two spheres that touch at O

  43. Isodensities, isolevels

  44. 2p orbital

  45. 3p orbital

  46. The 2px and 2py orbitals are equivalent

  47. One electron equally distributed on the three 2p levels Y2 is proportional to x2/r2+ y2/r2+ z2/r2=1 and thus does not depend on r: spherical symmetry An orbital p has a direction, like a vector. A linear combination of 3 p orbitals, is another p orbital with a different axis: The choice of the x,y, z orbital is arbitrary

  48. orbitals • = N radial function angular function rl (polynom of degree n-l-r) n-l-r nodes l nodes

  49. d orbitals Clover, the forth lobe is the lucky one; clubs have three

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