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Molecules Chapter 10 November 09 Modern Physics Outline Bonding mechanisms Molecular Rotation and Vibration Molecular Spectra Electron Sharing and the Covalent Bond Molecular orbital picture November 09 Modern Physics Force between atoms The attraction and bonding of atoms is electrostatic in original - electrons and protons attract each other. The attraction is a result of a dynamical correlation of the positions of electrons relative to the nuclei and is subtle - different models are used to understand the effect in different systems. November 09 Modern Physics Attraction of electrons p+ e- e- The far electron is closer to the repelling electron than to the attracting proton so repelled. e- p+ e- The far electron is farther from the repelling electron than from the attracting proton so attracted. November 09 Modern Physics Induced dipole moment p+ e- e- If the near electron is in a fast circular orbit, the time average force vanishes. e- p+ e- If the near electron is on average pushed away from the proton, the time average force is attractive and the hydrogen atom has an induced (dipole) polarization. November 09 Modern Physics Induced dipole interactions p+ e- e- p+ Two neutral atoms can co-induce dipole resulting in a net attraction. The net force is in this case called the van der Waals force. November 09 Modern Physics Field of an electric dipole e- p+ r At distance r along the x-axis of a charge -e at r=0 and charge +e at r=d (dipole moment p=ed), the electric field strength is inversely proportional to the third power of distance from the dipole November 09 Modern Physics Van der Waals potential e- p+ r If the dipole moment p induced on an atom is proportional to the inducing electric field strength (p=constant* E), the interaction energy between two dipoles is inversely proportional to the 6th power of distance - effectively rather short ranged. November 09 Modern Physics Interatomic effective potential The total potential energy includes the repulsive energy of interaction (1/r2)between the protons and attractive induced dipole-dipole energy (~1/r6) yielding a minimum at a finite “equilibrium separation.” November 09 Modern Physics Ionic dissociation Bring a neutral sodium and a neutral chlorine atom into contact. Then pull them apart. The sodium may lose its relatively loosely bound 3s outer electron to the chlorine which when neutral has a closed shell minus one 2p5 outer electron configuration. (The energy to remove an electron from Na is 5.1 eV and the attachment energy or electron affinity of chlorine is 3.7 eV.) The up-shot is the pair of Na+ and Cl - ions. The attractive force between such ions ~1/r2 at large distances. November 09 Modern Physics Na-Cl Total energy versus the nuclear separation for Na+ and Cl- ions. The energy required to separate the NaCl molecule into neutral atoms of Na and Cl is the dissociation energy, 4.2 eV. November 09 Modern Physics Electron Sharing An ionic bond is formed when one atom essentially steals an electron from a different atom, the two ions then attracting each other. A covalent bond is formed when electrons involved in bonding are more equally shared. In H2, the two electrons on equal footing orbit both protons and both tend to be between the protons - more of this later. November 09 Modern Physics Carbon The electron configuration in carbon (1s22s22p2) favors formation of 4 covalent bonds with say hydrogen to form tetrahedral methane. November 09 Modern Physics Hydrogen bonding In some circumstances, hydrogen can donate an electron to two high electron affinity atoms. November 09 Modern Physics Quantum excitations of molecules In its lowest energy state, a molecule is comprised of nuclei and core electrons at “fixed” equilibrium positions and a cloud of shared electrons. Electronic excitations involve promoting electrons to excited states of the complex 3-d potential associated with the nuclei . The minimum electronic excitation energy is a few eV. At lower available energies the electrons are frozen in the ground state. November 09 Modern Physics Non-electronic excitations Assuming the binding electrons are frozen in the lowest energy state for nuclei of fixed relative position, the entire molecule may still absorb energy internally in collective (rigid) rotational motion. With or without rotation present, the relative positions of the nuclei may change with the binding electrons constantly adjusting adiabatically to remain in the instantaneous electronic ground state - nuclear motion is slow compared to electronic motion. Vibrational motions of the nuclear positions are approximately harmonic about the (stable) equilibrium positions. November 09 Modern Physics External and internal kinetic energy We can separate the kinetic energy associated with the motion of the entire molecule from the internal motion: November 09 Modern Physics Rotational kinetic energy For rigid body rotational motion at fixed angular frequency about direction n, the internal kinetic energy may be written in terms of the moment of inertia I and angular momentum L: I and L refer to n. November 09 Modern Physics Quantization of rotational energy The orientation of a free rigid body is described by an angular wave function similar to that which describes the angular position of an electron in a spherically symmetric potential. The angular momentum and energy a rigid body are quantized. November 09 Modern Physics Diatomic molecule Note that the mass in a molecule is dominated by the pointlike nuclear masses and the rotational nuclear energy is small compared to electronic rotational energy: November 09 Modern Physics Example November 09 Modern Physics Diatomic molecule rotations Allowed rotational energies of a diatomic molecule November 09 Modern Physics Rotational transitions Electromagnetic transitions between rotational states correspond to emission and absorption of microwave (IR) frequencies, not optical frequencies. November 09 Modern Physics Diatomic vibration The potential energy of a diatomic molecule versus atomic separation. Oscillation about the equilibrium separation R0 has effective spring constant: November 09 Modern Physics Recall quantum oscillator November 09 Modern Physics Diatom vibrational states Allowed vibrational energies of a diatomic molecule, where ω is the fundamental frequency of vibration given by ω = √K/ µ . Note that the spacings between adjacent vibrational levels are equal. November 09 Modern Physics Vibrational data Note that (ground state) vibrational frequencies are larger than (low excitation) rotational frequencies. November 09 Modern Physics Nonlinear vibrations Potential energy U(r) around R0 is harmonic, but rises sharply as the atoms are brought closer together. The separation between adjacent levels decreases with increasing energy. Morse potential November 09 Modern Physics Combined rotational-vibrational spectrum The rotation– vibration levels for a typical molecule. Note that the vibrational levels are separated by much larger energies so that a complete rotational spectrum can be associated with each vibrational level. November 09 Modern Physics Transitions (a) Absorptive transitions between the v = 0 and v = 1 vibrational states of a diatomic molecule obey the selection rule Δ l = ± 1 and fall into two sequences: those for which Δ l = + 1 and those for which Δ l = - 1. The transition energies are given by Equation 11.14. (b) Expected lines in the optical absorption spectrum of a molecule. The lines on the right side of center correspond to transitions in which l changes by +1, and the lines to the left of center correspond to transitions for which l changes by -1. These same lines appear in the emission spectrum. November 09 Modern Physics HCL example The absorption spectrum of the HCl molecule. Each line is split into a doublet because chlorine has two isotopes, 35Cl and 37Cl, which have different nuclear masses. November 09 Modern Physics Electronic-rot-vibration spectrum If an electron is excited out of the (molecular) state (requires optical frequency excitation), the spring constant, the equilibrium separation, and the moment of inertia are slightly changed. The excited molecule exhibits a slightly different rotational-vibrational spectrum. November 09 Modern Physics Raman scattering An incoming photon with energy E scatters from a molecule and emerges with reduced energy E ‘. The energy lost by the photon increases the rotational energy of the molecule in accordance with the selection rule Δ l= 2. The energy loss translates into a change in photon frequency, the Raman shift, that can be used to probe molecular structure. November 09 Modern Physics Flourescence A photon with energy E is absorbed, in the process exciting a higher vibrational state of the molecule. This excess energy is lost to collisions with neighboring molecules. The molecule returns to its original state in step 3 by emitting a photon with reduced energy E’. In phosphorescence, the final transition is forbidden by selection rules, resulting in delayed photon emission. November 09 Modern Physics Simplest molecule The hydrogen molecular ion H2+. The lone electron is attracted to both protons by the electrostatic force between opposite charges. The equilibrium separation |R| of the protons in H2+ is about 0.1 nm. November 09 Modern Physics H2+ For R=0, the electron is bound in a hydrogen like orbital for Z=2. The ground state electronic binding energy is Z2=4 times that of hydrogen or -54 eV. For R=infinity, the electron is on one or the other proton and the energy is -13.6 eV. The total energy is the sum of the electronic energy and the repulsive energy of interaction between the protons. November 09 Modern Physics Electronic energy spectrum versus separation [1 bohr (a0) = 0.529 Å.] At R = 0, the levels are those of the united atom (ion) He+, 1s, 2s,2p,.. At R = ∞ the levels are those of neutral H. The degeneracy of the various levels (excluding spin) given by the numbers in parentheses is twice that of hydrogen as there are two protons. November 09 Modern Physics Electronic wave functions We expect if the electron is on one atom, it will tunnel to the other and back. The two atomic states will mix to produce two energy eigenstates, one symmetric, one antisymmetric. At large R, we describe the electron approximately as a combination of localized atomic orbital waves. November 09 Modern Physics Symmetric wavefunction November 09 Modern Physics Symmetric probability density Note large probability for electron between the protons. November 09 Modern Physics Antisymmetric wavefunction The antisymmetric wavefunction vanishes between the protons and the energy is different. November 09 Modern Physics Antisymmetric probability density Note a) small probability for electron between the protons, and b) higher derivatives and more confinement implies higher energy. November 09 Modern Physics Estimation of electronic energy If the wave function is only approximately a solution to Schrodinger’s equation, the final expression approximates the energy. November 09 Modern Physics Estimation of electronic energy With hydrogenic wave solutions, the result for the integral is: where R is units of the Bohr radius. For large R November 09 Modern Physics H2+ energy The antisymmetric wavefunction yields positive molecular energy (not bonding even excluding pp repulsion). The symmetric wavefunction yields negative molecular energy (bonding). The predicted bond length occurs at the point of stable equilibrium, around R = 2.5 bohrs. The predicted bond energy is about09 1.77 eV. November Modern Physics H2+ energy compared to data Predict R = 2.5 bohrs= 0.132 nm. Observe 0.106 nm. Predict bond energy (energy to separate H and H+ )= 1.77 eV. Observe 2.65 eV. Refined calculation technique: Assume wave is symmetric but not simply a sum of atomic orbitals but distorted. Characterize distortion by analytical factor with parameters. Calculate E in terms of these parameters. Minimize result as a function of the parameters and separation simultaneously. It can be proved this procedure will produce better results and it does. November 09 Modern Physics Now on to H2 The diatomic neutral hydrogen molecule contains two electrons. First approximation: Treat the two electrons as independent and assume both electrons occupy the symmetric bonding molecular orbital with opposite spins to be consistent with the exclusion principle. The case of one in bonding one in antibonding with parallel spins is less bound. 2nd approximation: Include e-e repulsion and recalculate the energy and minimum energy equilibrium separation. November 09 Modern Physics Molecular Hydrogen: H2 Total molecular energy for the bonding and antibonding orbitals of H2. The bond energy for H2 is 4.5 eV (not quite twice that of H2+)and the bond length is 0.074 nm (less than 0.1 nm of H2+). Since the energy of the antibonding orbital exceeds that of the isolated H atoms, no stable molecule can be formed in this state. November 09 Modern Physics Sigma bonds The electronic configuration of N is 1s22s22p3 and the three outer p-state (L=1) electrons bond. We can think of them in pairs. The axially symmetric states can overlap to form an axially symmetric (sigma type) bond. Formation of a sigma bond in N2 from the overlap of the 2pz orbitals on adjacent N atoms. November 09 Modern Physics Sigma bond The symmetric bonding wave for two electrons formed from the pz states concentrates the electron pair between the atoms. November 09 Modern Physics Pi bond Formation of a pi bond by overlap of the 2px orbitals on adjacent N atoms. A similar bond is formed by overlap of the 2py orbitals. November 09 Modern Physics Complexity In heteronuclear diatoms (eg HF), the bonds are not symmetric and can be ionic in character if one atom is significantly more “attractive.” In complex molecules, electrons are distributed over more than two atoms and concepts from condensed matter (solids) are applicable. Condensed matter is the subject of the next chapter. November 09 Modern Physics Important lessons Molecular orbitals can be crudely understood as linear combinations of atomic orbitals. Weak bonds may be qualitatively understood as a case of tunneling between atomic bound states. In the simplest bond, an electron is localized and shared equally between two atoms in an antisymmetric wavefunction. Two electrons may share that wave state provided their spins are antiparallel. As in atoms, the Coulombic energies in junction with the exclusion principle favor spin pair formation and the absence of paramagnetism. November 09 Modern Physics