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1 He and hydrogenoid ions The one nucleus-electron system

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

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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. r rather than x,y,z except that was defined in cartesian units.

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

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5 derivation d /dr for fixed and

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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,

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7 Integration in space dV = r 2 sin drd d

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8 Integration limits r = 0 r = ∞

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9 Integration in space

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10 Integration in space Integration over , and r gives V = 4/3 r 4 The integration over and gives the volume between two spheres of radii r and r+dr: dV = 4 r 2 dr Volume of a sphere Volume between two concentric spheres

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11 Dirac notation triple integrals

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12 Spherical symmetry dV = 4 r 2 dR

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13 Radial density dP/dR = 4 r 2 * 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= It is defined relative to a volume that increases with r. The unit of is L -3/2

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14 Schrödinger for a hydrogenoid (1 nucleus – 1 electron) The definition of an orbital atomic orbital: any function e (x,y,z) representing a stationary state of an atomic electron. Born-Oppenheimer approximation: decoupling the motion of N and e (x N,y N,z N,x e,y e,z e )= (x N,y N,z N ) (x e,y e,z e ) m H =1846 m e : When e - covers 1m H covers 2.4 cm, C 6.7 mm and Au 1.7 mm

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

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

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

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

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

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

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21 Energy units 1 Rydberg = 21.8 10 -19 Joules =14.14 10 5 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 (→ = 8065.5 cm -1 ) 1eV = = 24.06 Kcal mole -1.

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22 Atomic units The energy unit is that of a dipole +/- e of length a 0 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/2 =1 and 1/4 0 =1 –The Schrödinger equation becomes simpler lengthchargemassenergy a 0, Bohr radius e, electron charge 1.602 10 -19 C m, electron mass 9.11 10 -41 Kg Hartree

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

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24 Radial density of probability a 0 /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

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25 Orbital 1s

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26 Average distance to the nucleus In an average value, the weight of heavy values dominates: (half+double)/2 = 1.25 > 1) larger than a 0 /Z Distance: Operator r From math textbooks

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27 Distances the nucleus

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28 Excited states We have obtained a solution using e - r ; 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 N n P n (r)e - r where P n (r) is a polynom of r of degree n-1

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29 Orbital ns E ns = Z 2 /n 2 E 1s (H) E 2s = Z 2 /4 E 1s (H) Nodes: spheres for solution of equation P n =O n-1 solutions Average distance Principal quantum number

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30 Orbital 2s = E 2s = Z 2 /4 E 1s (H) Average distance 5 a 0 /Z A more diffuse orbital: One nodal surface separating two regions with opposite phases: the sphere for r=2a 0 /Z. Within this sphere the probability of finding the electron is only 5.4%. The radial density of probability is maximum for r=0.764a 0 /Z and r=5.246a 0 /Z. Between 4.426 et 7.246 the probability of finding the electron is 64%.

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31 Orbital 2s

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32 Radial distribution

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

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34 Resolution of Schrödinger equation in ,the magnetic quantum number

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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 (m s ): m s = +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).

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36 Name of orbitals 1s 2s 2p (2p +1, 2p 0, 2p -1 ) 3s 3p (3p +1, 3p 0, 3p -1 ) 3d (3p +2, 3p +1, 3p 0, 3p -1 3p -2 ) 4s 4p (4p +1, 4p 0, 4p -1 ) 4d (4d +2, 4d +1, 4d 0, 4d -1 4d -2 ) 4f (4f +3, 4f +2, 4f +1, 4f 0, 4f -1 4f -2, 4f -3 The letter indicates the secondary Quantum number, l The index indicates the magnetic Quantum number The number indicates the Principal Quantum number

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37 Degenerate orbitals E= Z 2 /n 2 (E 1sH ) 1s 2s 2p (2p +1, 2p 0, 2p -1 ) 3s 3p (3p +1, 3p 0, 3p -1 ) 3d (3p +2, 3p +1, 3p 0, 3p -1 3p -2 ) 4s 4p (4p +1, 4p 0, 4p -1 ) 4d (4d +2, 4d +1, 4d 0, 4d -1 4d -2 ) 4f (4f +3, 4f +2, 4f +1, 4f 0, 4f -1 4f -2, 4f -3 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.

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38 Functions 2p

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

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

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41 angle Density inside The half cone (1-cos( l ) 3 )/2 Part of the volume (1-cos l )/2 15° 0.049 0.017 30°0.1750.067 45°0.3230.146 60°0.4370.25 75°0.4910.37 90°0.5

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42 Directionality of 2p Z This is the product of a radial function (with no node) by an angular function cos . It does not depend on and has the z axis for symmetry axis. The angular contribution to the density of probability varies like cos 2 Within cones: angle Density inside The half cone (cos l ) 3 -1)/2 Part of the volume (cos l -1)/2 15° 0.049 0.017 30°0.1750.067 45°0.3230.146 60°0.4370.25 75°0.4910.37 90°0.5 Probability is 87.5% in half of the space 22.5% in the other half

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43 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 = cos 2 All the M points belong to two spheres that touch at O

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44 Isodensities, isolevels

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45 2p orbital

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46 3p orbital

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47 The 2p x and 2p y orbitals are equivalent

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48 One electron equally distributed on the three 2p levels 2 is proportional to x 2 /r 2 + y 2 /r 2 + z 2 /r 2 =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

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49 orbitals = N radial function angular function r l (polynom of degree n-l-r) n-l-r nodes l nodes

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

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52 3d orbitals

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53 Compare the average radius of 1s for the hydrogenoides whose nuclei are H and Pb. Pb 207 82 Make comments on the 1s orbital the atom Pb?

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54 Name these orbitals Spherical coordinates x = r sin cos ; y = r sin sin ; z = r cos

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55 Paramagnetism Is an atom with an odd number of electron necessarily diamagnetic? Is an atom with an even number of electron necessarily paramagnetic? What is the (l, m l ) values for Lithium? Is Li dia or para? Why?

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56 Summary

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