3/1/2005 CHEM 408 - Sp05L8 - 1Vibrations & Vibrations & ThermochemistryThermochemistry• molecules store energy in various forms of nuclear motion:translation: {X, Y, Z}rotation: {θ, ϕ, χ} or {θ, ϕ}vibration: {Qi}• for N nuclei, the number of “degrees of freedom” fiassociated with these types of motion are: center of mass motionnon-linear linear“external”“internal”ftrans= 3frot= 0, 2, or 3 (for atoms, linear mols., or nonlinear mols.)fvib= 3N - frot- ftrans= 0, 3N-5, or 3N-6 θϕχRθϕχRR3/1/2005 CHEM 408 - Sp05L8 - 2Figure adapted form Atkins 16.33Molecular Vibrationsvvvx...)(!31)(!21)()()(333222+−⎟⎟⎠⎞⎜⎜⎝⎛∂∂+−⎟⎟⎠⎞⎜⎜⎝⎛∂∂+−⎟⎠⎞⎜⎝⎛∂∂+=exelexelexeleelelxxxExxxExxxExExEeee221)()()(exeelelxxkxExE −+≅exelxxEk⎟⎟⎠⎞⎜⎜⎝⎛∂∂≡22• general expansion:• harmonic approximation:mforce constantEelxxeharmonic fit3/1/2005 CHEM 408 - Sp05L8 - 3• classical trajectory:)2cos{)(δπν++= tAxtxeamplitude (phase)2/121⎟⎠⎞⎜⎝⎛=mkxπν221AkExvib=• frequency:• classical energy:Harmonic Oscillator (HO) Behavior)(xnψ)(21+=nhEvibν• quantal energy:• quantal state:Atkins, Fig. 12.22n = 0, 1, 2, …xEel(x)2||ψxzero-pointenergyνh21νh3/1/2005 CHEM 408 - Sp05L8 - 4Normal Modes of Vibration∑∑=>−−+≅vib vibfifijjejieiijeelelRRRRkRERE1))((21)()(rrr1r2r3θ2θ1θ3eRjielijRREkr⎟⎟⎠⎞⎜⎜⎝⎛∂∂∂=22nd derivative (or Hessian) matrix∑=+≅vibfiiielelQkEQE1221)0()(rr022rr=⎟⎟⎠⎞⎜⎜⎝⎛∂∂=QieliQEktransform• normal coordinates Q are the particular coordinates that remove cross terms in both potential (Eel) and kinetic energy; in these coordinates the (harmonic) vibrations can be viewed as a collection of uncoupled oscillators:∑=+=vibfiiivibnhE121)(νni= 0, 1, 2, …2/121⎟⎟⎠⎞⎜⎜⎝⎛=iiikµπν3/1/2005 CHEM 408 - Sp05L8 - 5• normal modes of CF2Oν1579 cm-1CF2bendν2619 cm-1CO wagν3775 cm-1umbrella def.ν4964 cm-1sym. stretchν51229 cm-1asym. stretchν61991 cm-1CO stretch3/1/2005 CHEM 408 - Sp05L8 - 6• For T>0K also add effects of thermal populations of trans. & rot. states∑==vibfiiZPEhE121ν• comparison of electronic structure energies to thermochemicaldata requires consideration of nuclear (trans, rot, vib) energies (even at 0 K)Thermal Energies & ThermochemistryA+BABREReexptD0Eel=De012)(21, rottransBrottrffTkE+≅population1T1< T2< T332)()( TEZPEETEnucelδ++≅• G03 calculates the thermal energy for an ideal gas system:TkTETHB+= )()(• enthalpy is:• the vibrational zero-point energy for a polyatomic molecule is:3/1/2005 CHEM 408 - Sp05L8 - 7G03 Frequency & Thermochemistry Output CF2O3/1/2005 CHEM 408 - Sp05L8 - 83/1/2005 CHEM 408 - Sp05L8 - 9xAtomic Charges Atomic Charges • the way in which e density is distributed within a molecule determines many aspects of intermolecular interactions and reactivityρ(x,y)e Density Contours of N2From: Bader in The Force Concept in Chemistry (van Nostrand, 1981) p.39.3/1/2005 CHEM 408 - Sp05L8 - 10• it is often convenient to try to summarize the charge density variations by assigning charges to atoms• there is no unique way to do so• 3 general approaches are:- orbital population methods:Mulliken, natural population analysis (NPA)- spatial partitioning of ρ“atoms in molecules” (AIM) - electrostatic potential (ESP) fittingMKS, CHELP, CHELPG3/1/2005 CHEM 408 - Sp05L8 - 11• Mulliken charges are obtained directly from the density matrix (MO coefficients) of an SCF calculation.The total number of electrons in a system is:1. Orbital Population Methods:∑∑∑∑∫∫>+===µµνµνµνµµµψρSPPrrdrrdNii2|)(|2)(fn basis MOrrrr)1()1(1νµµνϕϕ∗∫≡ rdSr∑=∗≡niiiCCP1 MO2σλλσ• dividing the overlap population equally between atoms on which µand νreside provides the Mulliken charges:populationin basis fn µµ-νoverlap population∑∑∑∈≠∈−−=AAAASPPZqµµνµνµνµµµ• easy to calculate, useful for qualitative comparisons, but basisset dependent and can be non-sensical3/1/2005 CHEM 408 - Sp05L8 - 122. Spatial Partitioning “AIM” :Fig. from LeachCONHHH• topological approach developed by Baderhttp://www.chemistry.mcmaster.ca/faculty/bader/aim/• more than just charges… see3/1/2005 CHEM 408 - Sp05L8 - 13Fig. from U. C. Singh & P. A. Kollman, J. Comput. Chem. 5, 129 (1984).• MKS grid based on 4 surfaces surrounding each atom at 1.4, 1.6, 1.8, 2.0 times van der Waals radius; all internal points discarded3. Electrostatic Potential (ESP) Fit Charges:ESP Contours (acetic acid)MKS Grid• MKS, CHELP(G) methods in G03 differ only on grid used• best choice for constructing MM
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