1. PHYSICAL CHEMISTRY - ADVANCED MATERIALS H - Atom The Hydrogen Atom Z Y X + e-e- rere rprp r Coordinates Systems R z y x + e-e- r r     spherical polar coordinates z=r cos  y=rsin  sin  x=rsin  cos  z=r cos  y=rsin  sin  x=rsin  cos  center of mass or barycenter reduced mass

2. PHYSICAL CHEMISTRY - ADVANCED MATERIALS H - Atom RotationRotation + e-e- If m p >>m e :  ≈ m e r ≈ r e If m p >>m e :  ≈ m e r ≈ r e moment of inertia: I=mr 2 angular momentum: kinetic energy: T=L 2 /(2I) moment of inertia: I=mr 2 angular momentum: kinetic energy: T=L 2 /(2I) tangential component: radial component: angular velocity tangential component: radial component: angular velocity  

3. PHYSICAL CHEMISTRY - ADVANCED MATERIALS H - Atom angular momentum: vectorial representation meme meme quantum mechanics: Analogies: Translation Rotation m I v  p=mv L=I  Analogies: Translation Rotation m I v  p=mv L=I 

4. PHYSICAL CHEMISTRY - ADVANCED MATERIALS H - Atom Coordinates Transformation spherical polar coordinates cartesian coordinates z y x + e-e- r r    

5. PHYSICAL CHEMISTRY - ADVANCED MATERIALS H - Atom coulombic potential: U=-1/r Z Y X 1-Dr=|x|1-Dr=|x| 2-D r=|√x 2 + y 2 | 2-D 3-D r=|√x 2 + y 2 +z 2 | 3-D

6. PHYSICAL CHEMISTRY - ADVANCED MATERIALS H - Atom Equivalent to two ordinary (not partial) differential equations: Space: f(x) TIme: f(t) Space: X(x) Time: T(t)

7. PHYSICAL CHEMISTRY - ADVANCED MATERIALS H - Atom Schrödinger equation: free particle kinetic, rotational, coulombic

8. PHYSICAL CHEMISTRY - ADVANCED MATERIALS H - Atom Schrödinger equation: radial, angular wavefunctions

9. PHYSICAL CHEMISTRY - ADVANCED MATERIALS H - Atom Wavefunctions (solutions)

10. PHYSICAL CHEMISTRY - ADVANCED MATERIALS H - Atom Quantum numbers: Principal: n: 1,2,3,…….. Angular: l: 0,1,2,…(n-1) Magnetic: m: +l,(l-1)…0….-l Principal: n: 1,2,3,…….. Angular: l: 0,1,2,…(n-1) Magnetic: m: +l,(l-1)…0….-l Energy:Energy: Magnitude of the angular momentum: z component of the angular momentum:

11. PHYSICAL CHEMISTRY - ADVANCED MATERIALS H - Atom Effective potential eigenvalueseigenvalues The effective potential energy of an electron in the hydrogen atom. When the electron has zero orbital angular momentum, the effective potential energy is the Coulombic potential energy. When the electron has nonzero orbital angular momentum, the centrifugal effect gives rise to a positive contribution which is very large close to the nucleus. We can expect the l = 0 and l  0 wavefunctions to be very different near the nucleus.

12. PHYSICAL CHEMISTRY - ADVANCED MATERIALS H - Atom Radial wavefunctions

13. PHYSICAL CHEMISTRY - ADVANCED MATERIALS H - Atom Atomic orbitals What is an atomic orbital? Orbitals and orbits When the a planet moves around the sun, you can plot a definite path for it which is called an orbit. A simple view of the atom looks similar and you may have pictured the electrons as orbiting around the nucleus. The truth is different, and electrons in fact inhabit regions of space known as orbitals. Orbits and orbitals sound similar, but they have quite different meanings. It is essential that you understand the difference between them. The impossibility of drawing orbits for electrons To plot a path for something you need to know exactly where the object is and be able to work out exactly where it's going to be an instant later. You can't do this for electrons. The Heisenberg Uncertainty Principle says - loosely - that you can't know with certainty both where an electron is and where it's going next. (What it actually says is that it is impossible to define with absolute precision, at the same time, both the position and the momentum of an electron.) That makes it impossible to plot an orbit for an electron around a nucleus. Is this a big problem? No. If something is impossible, you have to accept it and find a way around it. What is an atomic orbital? Orbitals and orbits When the a planet moves around the sun, you can plot a definite path for it which is called an orbit. A simple view of the atom looks similar and you may have pictured the electrons as orbiting around the nucleus. The truth is different, and electrons in fact inhabit regions of space known as orbitals. Orbits and orbitals sound similar, but they have quite different meanings. It is essential that you understand the difference between them. The impossibility of drawing orbits for electrons To plot a path for something you need to know exactly where the object is and be able to work out exactly where it's going to be an instant later. You can't do this for electrons. The Heisenberg Uncertainty Principle says - loosely - that you can't know with certainty both where an electron is and where it's going next. (What it actually says is that it is impossible to define with absolute precision, at the same time, both the position and the momentum of an electron.) That makes it impossible to plot an orbit for an electron around a nucleus. Is this a big problem? No. If something is impossible, you have to accept it and find a way around it.

14. PHYSICAL CHEMISTRY - ADVANCED MATERIALS H - Atom It is not possible to determine the exact location of an electron in an atom. However, the probability of finding an electron at a given position can be calculated. The higher the probability of finding an electron at a given position, the larger the electron density at that position. An atomic orbitalorbital is derived using the mathematical tools of quantum mechanics, is a representation of the three-dimensional volume (i.e., the region in space) in which an electron is most likely to be found, and electron CANNOT be observed experimentally (electron density can, however, be observed experimentally).

15. PHYSICAL CHEMISTRY - ADVANCED MATERIALS H - Atom s (l=0)p (l=1)d (l=2)f (l=3) n=1 n=2 n=3 n=4 n=5... n=6...

16. PHYSICAL CHEMISTRY - ADVANCED MATERIALS H - Atom http://winter.group.shef.ac.uk/orbitron/ Some interesting Websites…. http://www.orbitals.com/orb/ http://www.falstad.com/qmatom/ http://bouman.chem.georgetown.edu/atomorbs/aovis.html http://electron6.phys.utk.edu/qm2/modules/m1-3/hydrogen.htm http://galileo.phys.virginia.edu/classes/751.mf1i.fall02/HydrogenAtom.htm