MOLECULES. ENERGY LEVELS electronic vibrational rotational

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Carmella Davidson
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1 MOLECULES BONDS Ionic: closed shell (+) or open shell (-) Covalent: both open shells neutral ( share e) Other (skip): van der Waals (He-He) Hydrogen bonds (in DNA, proteins, etc) ENERGY LEVELS electronic vibrational rotational

2 Ionic Bonds - NaCl First approximation. Both atoms are ions (no electron sharing) and bond due to EM force between two charged bodies E/4piepso =.4 ev*nm, separation=.36 nm 5.9 ev Bond Na bond Cl Atom valence ionization ~effz radius Na 3s 5. ev.8.7 nm Cl 3s 3p 5 3 ev.9.7 nm Ar 3s 3p 6 6 ev nm E 3.6 Z n eff Ar more tightly bound than Cl. But Cl - looks like Ar and more tightly bound then neutral Cl Has effective Z ~ 3 E( Cl ) E( Cl need : (5. 3.8) ev ) 3.8eV Na Cl Na Cl

3 3 terms NaCl potential vs R Pauli repulsion at small R shielding of nuclei becomes incomplete electrons wave functions overlap and electrons forced to higher states (p3s) 3

4 Ionic Bonds KCl - SKIP another example. What is the energy required to dissociate a KCl molecule into K atom and Cl atom given Ionization energy K = 4.34 ev electron affinity Cl = 3.8 ev separation distance R =.79 angstroms attractive EM potential for.79 A = -5.6 ev takes 5.6 ev to separate K+ and Cl- ions to infinity takes 3.8 ev to pull electron from Cl- to form neutral Cl gain 4.34 ev when K+ captures electron and forms neutral K = 4.64 ev = energy required 4

5 Ionic vs Covalent As R >>.5 (size of p orbit), there is little overlap in the electron wave function between the Na and Cl ions mostly ionic bond 94% ionic and 6% covalent (DH makes up numbers) look at HFl molecule H ionization energy = 3.6 ev Fl electron affinity = 3.4 ev need. ev in electrostatic energy e.4evnm R. 4nm U.eV 4 as the size of filled p in Fl is about.5 nm and the nominal s in an H atom is.5 nm, the electrons are attached to both atoms covalent bond % ionic and 9% covalent (DH made up numbers) the nuclei will start to not be shielded from each other some ++ repulsion 5

6 Covalent Bonds - Diatomic Molecules assume all valence electrons are shared often S= lowest energy but not always (Oxygen is S=) if both atoms are the same then y same if switch atom() and atom() - -- electron densities around each atom are the same (even sort of holds if different atoms like CO) H(s) <-- very far apart ---> H(s) close together H( s )H( s ) electron wavefunctions overlap - shared two energy levels (S=,) which have y (,) y (,) E R=infinity (atoms) s*s bands Vib and rot 6

7 Covalent Bonds - Hydrogen even if only electron, bond is covalent look first at ionized diatomic H H have repulsive potential between protons depends on R = p-p separation (about. nm) e 4 R V pp but then have attractive energy V e for the electron. It will depend on R (and one calculates R by looking at the minimum of V e + V pp ) V e lowest energy states 7

8 Covalent Bonds - Hydrogen guess at a 3D solution for the wave function. Must not depend on vs for two atoms. Only one electron and, are locations of two protons y (,) y (,) (, spatial ) at large separation just two H atoms y S ( H) y (,) e A( e r / a two possibilities: symmetric and antisymmetric when the separation becomes small r r / a e r r / a ) y y y symmetric p p y antisymmetric large separation small separation 8

9 Covalent Bonds - Hydrogen+ symmetric wave function has lower energy less curvature. node vs nodes (compare to particle in a box) also greater shielding of protons from each other as higher probability for the electron to be between the two protons (antisymmetric goes to at midpoint) can extrapolate to R= --- symmetric becomes a S state of He and antisymmetric (with wavefunction= at origin) becomes a P state total E V pp E e determine this as a function of R internuclear separation. Find there is a minimum for symmetric but not for antisymmetric covalent bond 9

10 Covalent Bonds - Hydrogen+ Mostly skip E-

11 Neutral Hydrogen Molecule H 4.7eV H H more tightly bound with electrons. Have: additional shielding of protons (lower E) H e-e repulsion (higher E).7eV end up: R=.7 nm (compared to about.9 nm with single electron) the size of a H atom is about.5 nm and so the s wavefunctions of the atoms are overlapping and need to use Fermi-Dirac statistics Pauli exclusion and a totally antisymmetric wavefunction if if S S y spin spin y ( e, e) y space y spin y ( e, e ) y sym, y H space antisym, y H antisym space sym

12 Neutral Hydrogen Molecule the antisymmetric space has y when r = r gives: lower electron probability between protons less shielding higher energy in this case (and in most cases) have covalent bond when electrons are paired with antiparallel spin S= S= E E e E e V pp R pp S=

13 Covalent p Bonding bonding in N and O (and sort of in molecules with C and other atoms) depend on p orbital shapes 3 different 3p wave functions. Along x y and z directions. (If have sp 3 then along 4 legs of tetrahedron) For covalent bonds with identical atoms, symmetry requires that both atoms have the same electron distributions. So y will have xx,yy,zz combinations only. Let x be the axis of the molecule and y,z be perpendicular. y,z give same energy eigenvalues and so can have mixing f symmetry 3

14 Covalent p Bonding just based on symmetry between the atoms electron distributions, can sketch out the bond and antibonding wave functions. If electrons are between nuclei lower energy BOND YY,ZZ ANTIBOND N N N N BOND XX ANTIBOND p N N N N 4

15 Covalent p Bonding give energy levels which are then filled energy ordering changes between O and N as different electron distributions usually think of covalent bonds as S= but have S= for O N O x P anti x y z y z x P bond y z y z x S antibond S bond S

16 Energy Levels for given electronic states (s,3p, etc S=, S=) determine effective V(R) and see if a minimum (bound state) exists. Transitions between at higher energy, visible/uv as NOT V(r) potential, Sch. Eq. Not separable into (THETA,PHI) parts L not eigenfunction, L not good eigenvalue but often phi symmetry L z m good will then have H.O. vibrations around minimum, and rotations V y y electronic y rotational R=nuclear separation y vibrational 6

17 Rotations Go to center-of-mass frame for two atoms M M R write down Schrod. Eq. For rotational energy R c. m. R M E rot with spherically symmetric in cm frame. Rotationally confined to sphere r=r y L L H rot z rot y Y rot rm r( r m L y I (, f) ) M R rot r r m L I m I R m m m E r rot spherical int y eger rot eger ( V harmonics int v ) 7

18 Rotations As know angular momentum eigenvalues E rot E E L I r E r( r) I r I spacing r( r ) ( r ) r r 3 I use NaCl as example. R =.4 nm ( c ) I c R 3.9 (97eV fm) MeV (.4 fm).5ev kt =.5 ev for T=3. Easy to be in an excited state and relative amount is another way to measure temperature Eta really hbar It is 97 ev*nm and,4 nm 8

19 Rotation example. That is an hbar find # # n n r # in level r # in level quantum states r quantum states (r ) e E r( r ) I r( r) / IkT e e E E r / kt / kt # states r n r which level is most highly populated? n r r r prob (r ) Ikt IkT.5 ( ).5 E / kt.5/ e r NaCl at T=3 9

20 Molecular Vibrations E tot R separation minimum in separation distance can be approximated as a parabola let v=vibrational quantum number =,,... E vib ( v Etot C R R NaCl : h ) h R.4eV curvature of HCl C : h parabola.4ev

21 Rotational-Vibrational Energy Levels The # of rotational levels only fixed by the top of the finite energy well different vibrational levels can overlap E v r ( v ) C r( r ) a often just called rotational-vibrational band spacing (between vibrational levels and rotational levels) will vary as move to the top of the well. The electron distribution changes and so average separation changes. Well non-symmetric at room T, most molecules in lowest vib. State n e e n # states # states h / kt.35/.5 7 well is finite limited number of vibrational states (~4 for some before dissociation)

22 See water vibrational levels on web page link Absorption in atmospher e---in IR primarily from water

23 Photon Scattering Both light passing through a gas and a technique in M b ~M a M a out if photon is at a resonant frequency. Then M b is an excited state and the outgoing photon energy depends on the details of the energy levels. If photon not at a resonance, M b is virtual. But electric-dipole selection rules hold at each vertex r each vertex for both r r v v ( Rayleigh) or, ( Raman) 3

Molecular Physics. Attraction between the ions causes the chemical bond.

Molecular Physics. Attraction between the ions causes the chemical bond. Molecular Physics A molecule is a stable configuration of electron(s) and more than one nucleus. Two types of bonds: covalent and ionic (two extremes of same process) Covalent Bond Electron is in a molecular