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Transcript
Molecules Atoms can combine in various ways to form stable molecules. Atoms pair up into molecules when there is an energy gain (I.e. their energy is lowered; the atoms become more stable). Two views: 1. A molecule is a stable arrangements of a set of nuclei and electrons. The exact arrangement is determined by the electromagnetic interactions and quantum mechanics. “Set up the Schrödinger equation for all particles (nuclei + electrons) and solve.” 2. A molecule is a stable structure formed by two or more atoms. “Atoms in a molecule mostly preserve their identity.” Molecular bonding affects the outermost shell of electrons only (“the valence electrons”); often in quite dramatic ways. Here we typically speak of molecular orbitals instead of distinct atomic orbitals. Lower lying core-orbitals/subshells are almost exactly like those in the atom. Molecular Bonding - Overview A balance between attractive and repulsive forces between atoms. Attractive Forces: •Coulomb forces between oppositely charged ions. •Covalent bonds (courtesy of quantum mechanics). •Coulomb interactions between dipoles. •Induced dipole-dipole interactions. RAB Repulsive Forces: RAB •Coulomb forces between non-opposite charged ions. •Covalent antibonds (“antibonding molecular orbitals”). •Repulsion between nuclei. •Exchange Force/Pauli Exclusion Principle. The balance between attractive and repulsive potential often creates a net-potential with an energy minimum. The energy minimum defines the classical (!) equilibrium separation or bond length. The depth of the potential well defines the classical (!) molecular binding energy or dissociation energy. (! see vibrational quantization, zero-point energy for the more accurate quantum mechanical definitions). Molecules - Ionic Bonding Ionic bonding results, when the Coulomb attraction between oppositely charged ions is larger than the energy required to form ions from the neutral elements. This occurs when small ionization energies in one atom are paired with large electron affinities in another atom. Ionization: Na + 5.1 eV # Na+ + eCl + e- # Cl- + 3.8 eV " Ionized elements. +1.3 eV " Neutral elements. 0 eV Repulsive forces at short distances due to: (1) Repulsion between the nuclear charges; less shielding of the ZNa and ZCl with decreasing R. (2) Exchange forces - more electrons compete for less space. -3.6 eV The Simplest Molecule - H2+ Two hydrogen nuclei (Z=1) and one electron. Time independent Schrödinger Equation: The Born-Oppenheimer Approximation - H2+ “Electrons move much faster than the nuclei. From the timescale perspective of the electrons, the nuclei remain essentially fixed. From the timescale perspective of the nuclei, the electronic motion is averaged out.” This allows us to separate electronic and nuclear motion into two separate quantum problems: Starting with the time-independent Schrödinger equation from the previous slide: Assuming static nuclei, separation gives rise to an electronic Schrödinger equation: Can solve this with nuclear positions given. Obviously, different solutions for different nuclear positions. Molecular potential energy curve/surface: … and a nuclear Schrödinger equation: Describes vibrational and rotational motion of the molecule H2+ Correlation Diagram Interpolation between two extremes: RHH=0: He+ combined atom limit (Z=2). RHH=$: H+ + H separated atom limit (Z=1). RHH Energies at limits given by Bohr-equation for one-electron atoms: 2 2 n=4: -3.4 eV n=3: -6.0 eV En= -13.6 eV (Z /n ) n=2: -3.4 eV (8%) n=2: -13.6 eV (4%) n=1: -13.6 eV (2%) Note: This diagram only shows the electronic energy. The Coulomb repulsion between the two nuclei is not included. n=1: -54.4 eV (1%) RHH / bohrs Combined Atom Limit He+ Towards separated Atom Limit H+ + H H2+ Potential Energy Curves Potential energy curve for the two lowest electronic states of H2+ with nuclear repulsion included. Correlation Diagram without nuclear repulsion &- &+ &&+ H2+ - Molecular Orbitals &+ = &1s,A + &1s,B &- = &1s,A - &1s,B &+ Higher electron density between atoms. Lower electron density between atoms. Lower energy (more stable), because the electrons are more located where the nuclear potential is attractive (negative). Higher energy (less stable) Bonding molecular orbital Antibonding molecular orbital. H2-Molecule - Two electrons ! We need to make sure that our two-electron eigenfunction is anti-symmetric w.r.t. electron exchange. This is achieved by combining a symmetric space-eigenfunction with an antisymmetric spin-eigenfunction ' or vice versa. Remember from earlier lecture that triplet’s (“parallel spins”) have symmetric and singlets “(antiparallel spins”) have antisymmetric spin functions. [ &+(1)&-(2) + &-(1)&+(2) ]S 'A(1,2) ! singlet [ &+(1)&-(2) - &-(1)&+(2) ]A 'S(1,2) ! triplet [ &+(1)&+(2) + &+(1)&+(2) ]S 'A(1,2) ! singlet With both electrons in the same one-electron eigenfunction &+, an antisymmetric space function can not be constructed; so a triplet state is not possible for this configuration.