Local Vibrational Modes: a New Tool to Describe the Electronic Structure of Molecules

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?

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

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

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

k (CC) = 56.866 r(CC)-6.330 15 a

k(CC)

R^2 = 0.993

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. stretch

sym. stretch

red: stretches bend green: bends O1 2H H3 Coupling Frequencies

Water Light -heavy -light

Coupling Rules

small coupling

Equal masses Heavy - light - heavy

large coupling

largest coupling the mode vectors have to have parallel components; if they are orthogonal they do not couple Local Mode

Intensities Infrared Intensity of Normal Vibrational Modes Infrared Intensity of Local Vibrational Modes Infrared Intensity of Local Vibrational Modes

a I a I Molecule Parameter In µ µ Molecule Parameter In µ µ [km/mol] [km/mol] [km/mol] [km/mol] Internal Coordinate set 1

H-O-H H-O 23.4868 3 40.8595 H-O-D H-O 23.4868 3 24.7171 H-O 23.4868 2 3.2361 D-O 14.9527 2 11.2932 H-O-H 69.1712 1 69.5078 H-O-D 59.8634 1 59.5745

Internal Coordinate set 2

H-O-H H-O 23.4868 3 40.8595 H-O-D H-O 23.4868 3 24.7171 H-O 23.4868 2 3.2361 D-O 14.9527 2 11.2932 H...H 1 69.5078 H...D 1 59.5745 Investigation of the

Hydrogen bond Normal vibrational modes of the water dimer: Exp. frequencies

red: stretches green: bends light blue: torsions dark blue: HB stretch

Normal mode vibrations are decomposed into local modes Local modes Normal Modes

avoided crossing

Adiabatic connection scheme of the water dimer: Exp. frequencies Local modes Normal Modes

avoided crossing

switch of mode character

Adiabatic connection scheme of the water dimer: Exp. frequencies

lower range Adiabatic connection scheme including IR intensities HB stretch

local mode intensities

normal mode intensities

HB stretch Investigation of H-bonded complexes

M. Freindorf, E. Kraka, and D. Cremer, Int. J. Quant. Chem. 112, 3174-3187 H-Bonding X-H ... Y 1.2

1.1 n = 1 F-H 1 0.9 X, Y = N, O, F 0.8

0.7 O ... H 0.6 n = 0.5 O-H 0.5 ...... - 0.4 F H F N ... H Bond Order n(XH)

0.3 N-H

0.2 F ... H n = 0.521 k 0.291 0.1 a

0 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

Adiabatic Force Constants ka(XH) [mdyne/Å] H-Bonding X-H ... Y 0.4 X, Y = N, O, F 9 5 19 100 20 6 10 0.3 11 3 4 22 12 17 13 21 28 24 27 14 0.2 8 5 18 16 31 O...H Bond Order n(XH) O-H

0.1 N ... H n = 0.521 k 0.291 a N-H

F ... H 0 0 0.05 0.1 0.15 0.2

Adiabatic Force Constants ka(XH) [mdyne/Å] H-Bonding X-H ... Y

1.05 X, Y = N, O, F

1 F-H

0.291 25 n = 0.521 ka 23 0.95 7 O-H 5 24 13 11 6 14 4 21 3 ... 20 9 O H 0.9 12 2 O-H Bond Order n(XH) 19 N ... H 0.85 N-H 8 F-H 0.8 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

Adiabatic Force Constants ka(XH) [mdyne/Å]

M. Freindorf, E. Kraka, and D. Cremer, Int. J. Quant. Chem. 112, 3174-3187 Applications: Investigation of

H-bonding Dihydrogen bonding (borazene, etc.) Halogen Bonding Agostic / anagostic Bonding Extremely weak / strong Bonding Pnicogen Bonding Multiple bonds in TM complexes Gallium multiple bonds Bond pseudorotation Adiabatic puckering for aromaticity H H H H H H H H H H H H

H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H

H H H H H H H H H H H H H H H H H H H H

H H H H H H H

HH H Development and Application of AICoM Local Modes

http://smu.edu/catco/ E. Kraka and D. Cremer, Reviews Acc. Chem. Res., 43, 591-601 (2010).

D. Cremer and E. Kraka, E. Kraka, J.A. Larsson, and D. Cremer, Curr. Org. Chem, 14, 1524-1560 (2010) Computational Spectroscopy, 105-150 (2010).

D. Cremer, J. A. Larsson, and E. Kraka, Theory Theoretical and Computational Chemistry, Volume 5, Theoretical Organic Chemistry, C. Parkanyi, Edt., Elsevier, Amsterdam, 1998, p.259. Z. Konkoli, J. A. Larsson and D. Cremer, Int. J. Quant. Chem. 67, 11 (1998). Z. Konkoli and D. Cremer, Z. Konkoli and D. Cremer, Int. J. Quant. Chem. 67, 1 (1998). Int. J. Quant. Chem. 67, 29 (1998). Z. Konkoli, J. A. Larsson and D. Cremer, Int. J. Quant. Chem. 67, 41 (1998). E. Kraka and D. Cremer Applications Chem. Phys. Chem. 10, 686-98 (2009). J. Oomens, E. Kraka, M.K. Nguyen, T. H. Morton J. Phys. Chem., 112, 10774-83 (2008) J.A. Larsson and D. Cremer, E. Kraka and D. Cremer, J. Mol. Struct. 485, 385 (1999). J. Phys. Org. Chem. 15, 431 (2002). D. Cremer, A. N. Wu, A. Larsson, and E. Kraka, J. Mol. Model. 6, 396 (2000).

Reacting Molecules H. Joo, E. Kraka, W. Quapp, and D. Cremer Mol. Phys., 105, 2697 – 2717, 2007. Z. Konkoli, E. Kraka, and D. Cremer, D. Cremer, A. A. Wu, and E. Kraka, J. Phys. Chem. A 101, 1742 (1997). E. Kraka, A. Wu, and D. Cremer, Phys. Chem. Chem. Phys. 3, 674 (2001). J. Phys. Chem. A 107, 9008 (2003). Measurements Computations

Vibrational Spectroscopy

CalEx Normal Modes Method Local Modes Intensities Intensities Atomic charges Bond moments force constants frequencies

Bond strength (Stability)

static dynamic

Equilibrium Reaction Acknowledgments

Prof. Elfi Kraka (SMU) Dr. Wenli Zou (SMU) Robert Kalescky (SMU) Dr. Marek Freindorf (SMU) Thomas Sexton (SMU) Dani Setiawan Dr. Andreas Larsson (Tyndall, Cork, Ireland)

Prof. Thomas R. Morton (Riverside, California) NSF Grant CHE 1152357