Local Vibrational Modes: A new Tool to Describe the Electronic Structure of Molecules Dieter Cremer and Wenli Zou Department of Chemistry, SMU Dallas, Texas 75275, USA E-mail: [email protected] http://smu.edu/catco/ SWTCC, Texas A&M, Friday, October 26, 2012 How to use computational methods for increasing the power of experimental methods ? Design of CalEx methods Calculational “Grafting” of Experimental Data Precise description of electronic structure and bonding with experimental means Tools X-ray NMR MS VibSpec UV PES ED IR UPS MW Raman XPS Auger geometry shifts fragm. bond excited IPs SSCC pattern strength states EAs charge distribution How to connect to bond strength and electronic structure? Modern Vibrational Spectroscopy High Performance Vibrational Spectroscopy Inrared Raman Near IR and Far IR Fourier transform Raman Fourier transform IR (FTIR) Raman Optical Active Spectr. (ROA) Two-dimensional IR Femtosecond Stimulated Nonlinear 2D IR Raman Spect. Transmission IR Coherent anti-Stokes Raman Spectroscopy (CARS) Diffuse Reflectance IR Spect. (DRIFTS) Resonance Raman Spectroscopy Reflection-Absorption IR Surface Enhanced Spectr. (RAIRS) Raman Spectr. (SERS) Tip Enhanced Multiple Internal Raman Spectr. (TERS) Reflection Spectr. Vibrational Spectroscopy as a generally applicable tool for the description of Chemical Bonding Molecular vibrations probe the strength of chemical bonds. Vibrational frequency and vibrational force constant of a stretching vibration can be related to the bond strength, i.e. to the Intrinsic Bond Dissociation Energy (IBDE) of a bond. Normal vibrational modes are always delocalized Individual bond stretching modes cannot be identified! Normal vibrational modes are always delocalized because of Mode-mode coupling i) electronic coupling: non-diagonal force constants ii) mass coupling: depends on ratio of masses, direction of mode vectors; closeness of eigen values Fermi resonances Closeness of frequency and overtone identical symmetries Nickeltetracarbonyl O C O C Ni C O C O 9 atoms 3 x 9 - 6 = 21 modes 8 stretches 5 + 4 x 2 bends Normal vibrational modes of Ni(CO)4: Exp. frequencies Normal mode vibrations are decomposed into local modes Local Vibrational Modes Adiabatic Internal Coordinate Modes: AICoMs • The local vibrational modes are the true equivalent of the normal vibrational modes. • For a given set of internal coordinates, there is only one set of local modes, which is directly related to the normal vibrational modes. How to get the normal vibrational modes? Basic Equations of Vibrational Spectroscopy Cartesian Coordinate Space Internal Coordinate Space x q -1 f lµ = λµ M lµ F dµ = λµ G dµ 2 2 2 q † x λµ = 4π c ωµ Fnm = cn f cm x f : Force constant matrix F: F:orce constant matrix M : Mass matrix G: Wilson's G-matrix lµ : Normal mode vector dµ: Normal mode vector fx L = M L Λ Fq D = G-1 D Λ lµ = C dµ Matrix C transforms the normal mode vector from internal to Cartesian coordinate space. How to get the Local Modes? Diagonalization of force constant matrix: suppression of electronic coupling Then, we suppress mass-coupling Electronic coupling is suppressed by finding the normal mode vectors. However, kinematic (mass) coupling) cannot be eliminated when obtaining the normal modes. Mass-Decoupling massless approximation points of mass zero Mass-Decoupling massless approximation points of mass zero Local Modes in the Harmonic Approximation Assume that the vibrational problem has been solved, the potential energy V and an internal coordinate qn (x1, x2, ...., x3Kn) qn can be expressed as function of Nvib normal mode coordinates Qµ, and the corresponding force constants kµ: Nvib 1 2 V(Q) = Σ kµ Qµ (1) 2 µ=1 Nvib qn(Q) = Σ Dnµ Qµ (2) µ=1 Matrix D collects the column vectors dµ, which represent the normal modes µ in internal coordinate space. Leading the local mode by qn* and relaxing all other internal coordinates qm (Leading parameter principle): qn(Q) = qn* (3) ∂ [ V(Q) - λ (qn(Q) - qn*) ] = 0 (4) ∂ Qµ (n) Dnµ Qµ = λ (5) kµ 1 λ = qn* (6) Nvib D 2 Σ nµ µ=1 kµ (n) 0 Qµ = Qµn qn* (7) The local modes are extracted out of the normal modes expressed in internal coordinates Elements µ of mode an associated with internal coordinate qn Dnµ kµ Q (a ) = an n µ Nvib 2 Dnν Σ kν ν=1 Dnµ contribution of displacement qn to normal mode µ kµ force constant of normal mode µ Nvib = 3N - 6 Properties of Local Vibrational Modes K = L†fx L AICoMs adiabatic internal coordinate modes Compliance Constants Derived from the inverse of the force constant matrix Decius, 1953, 1963 Compliance Constants Potential energy V expressed in terms of generalized displacement forces (rather than displacements 1 1 V(g) = g† C g rather than V(q) = q† F q 2 2 q vector of displacement coordinates g = F q g vector of displacement forces -1 q = F g = C g F force constant matrix C compliance matrix Γ = C = F-1 Cii: displacement of internal coordinate qi under the impact of a unit force with all other forces are being relaxed (Cij largely reduced) Large (small) displacement weak (strong) bond Decius, 1953, 1963 Compliance constants are not useful + are rarely used • Calculation is too expensive • They are often inaccurate • They cannot be related to a vibrational mode • They do not lead to a frequency or intensity • Meaning of the off-diagonal elements ? • Odd description of bond strength Local Mode Force Constants and Compliance Constants Fq = C†fx C C = L D-1 † x (n) -1 + -1 K = L f L ka = (dnK dn ) W. Zou, R. Kalescky, E. Kraka, and D. Cremer, J. Chem. Phys. 137, 084114 (2012) Compliance constants are not useful and superfluous • Calculation is too expensive • They are often inaccurate • They cannot be related to a vibrational mode • They do not lead to a frequency or intensity • Meaning of the off-diagonal elements ? • Odd description of bond strength What are stretching frequency and force constant of bond a-b of naphthalene? How do they compare to those of bonds b-c and c-d? a b d c Napthalene: Mode composition given in % Napthalene The 48 local modes are pure without any contamination How to relate local to normal modes ? Solution W. Zou, R. Kalescky, E. Kraka, and D. Cremer, J. Chem. Phys. 137, 084114 (2012) The C9-C10 stretching mode contributes to 5 different modes in the range 750 - 1580 cm-1 local modes normal modes 2 3 1 Napthalene Adiabatic Connection Schemes Napthalene: Bond lengths do not always reflect the bond strength H H peri-repulsion 1.364 1.364 1.415 1.418 1.415 1.418 1.421 1.421 H H competing benzene rings 3 4 2 1 ? delocalized 10 π system Napthalene: Determination of a bond order Badger Rule for Polyatomic Molecules op 38 1 4 42 18 17 43 39 ip 15 16 F O C C 2 3 14 N ip 1 2 3 40 41 op 22 23 24 op 6 N Si 44 5 Cl 8 O N 7 19 47 ip 45 c 4 5 6 20 21 t 46 t 25 26 27 9 11 N 12 O 10 C N 51 c S 24 25 7 8 9 48 50 op op t 26 19 22 23 52 15 17 13 F 3 C 28 29 c 49 30 ip 1 14 2 18 O ip 16 O 53 28 N 54 56 10 11 12 2 29 2 27 3 31 O 30 3 31 55 32 20 23 57 22 5 32 4 O C N 6 33 21 2 op N O S 13 14 15 60 3 O O O 1 62 2 c 33 4 Cl 5 35 37 26 7 27 28 3 C O 34 36 61 38 59 58 63 8 9 25 24 t 34 35 36 16 17 18 ax64 31 O 36 66 34 29 32 13 O eq ip ip t 65 10 11 12 37 38 33 30 c 35 39 op op 37 40 19 20 21 For the description of Chemical Bonds 4π2µ ν2 = k electronic effect Stretching Force Constants ν2 = k / (4π2µ ) are more useful than Stretching Frequencies electronic and mass effect Force constant - bond length relationship For diatomics AB, Badger’s Rule, 1934 3 ke ( re – dij) = const dij depends on the positions of atoms A and B in the periodic table effective bond length Generalization requires local force constants ! p -1/p ke ( re – d) = const; re = a ke + b with 2 ≤ p ≤ 6, similar equations for ωe Badger Relationships for 120 Diatomic Molecules 25 Experimental Values for AB 20 k[1-1] k[2-1] 15 k[2-2] [mdyne/Å] k k[3-1] k[3-2] 10 k[3-3] constant Force 5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 Bond length r [Å] y = 14.251x-7.456 r2 = 1.000 y = 56.866x-6.330 r2 = 0.993 Periody = 10.957xi - Period-8.183 rj2 =atom-atom 0.994 y = 53.974x bonding-6.730 r2 = 0.997 Experimental AICoMs -8.824 2 y = 7.916x r = 0.993 y = 50.801x-7.153 r2 = 0.979 25 CN k(BH) k(CH) 20 2-2 k(NH) k(OH) k(CC) 15 2-1 k(CN) CO CC k(CO) 10 k(CF) OH CH BH 5 NH CF Adiabatic Stretching Force Constant k [mdyne/Å] 0 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 Bond Length r [Å] Extending the Badger Rule from Diatomics to Polyatomics 20 k (CC) = 56.866 r(CC)-6.330 15 a k(CC) 10 5 R^2 = 0.993 0 Adiabatic Stretching Force Constant k(CC) [mdyne/Å] 1.2 1.3 1.4 1.5 1.6 Bond Length r(CC) [Å] 25 p 20 ke (re – dij) = const -p ke = const (re – dij) 15 CC, CN, CO, CF 10 5 with 2 ≤ p ≤ 6 Adiabatic Stretching Force Constant [mdyne/Å] 0 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 Effective Bond Length R [Å] Rules for Mode-Mode Coupling Adiabatic Connection Scheme and Coupling Frequencies local modes normal modes asym.