Chemistry 431. Lecture 28

Transcription

1 Chemistry 43 Lecture 28 Introduction to electrostatics Charge and dipole moment Polarizability van der Waal s forces Review of transition dipole NC State University

2 Review of Electrostatics The Coulombic force on charge j due to charge i is: F j = 4πε q i q j r r 2 ij ij The Coulombic force is additive. The combined force is a superposition. The force on charge k due to a number of charges with the index j is: F k = 4πε Σj k q j q k 2 r jk r jk The constant ε is the permittivity of vacuum. In MKS units the value is ε = x -2 C 2 N - m -2. In the cgs-esu unit system the permittivity of free space is /4π and the constant /4πε does not appear in the Coulomb force.

3 Electric Field The electric field is is the force per unit charge. The most precise statement is that it is the force per unit charge in the limit that the charge is infinitesimally small: E j = F j q When applied to the Coulomb force the electric field becomes: E k = 4πε Σj k r jk q j 2 r jk

4 Electrostatic Potential The electric field is the negative gradient of the scalar potential: E = φ The potential at a distance r from a charge is: φ = q 4πε r The electric field represents the force per unit charge. The potential is the work per unit charge. W 2 = φ(q 2 q ) In MKS units the potential has units of V where V = J/C.

5 Definition of a dipole moment A dipole is defined as a charge displaced through a distance. It is a vector quantity, i.e. it has direction: μ = q(d 2 d ) The units of dipole are Cm as well as Debye. Debye = 3.33 x -3 Cm One can also use units electron-angstroms. 4.8 Debye = ea The quantum mechanical definition of a dipole moment is Expectation value of the dipole operator er. μ = e all space Ψ r Ψdr

6 Potential and Field due to a Dipole The potential due to a dipole is: φ(r) = μ r 4πε r 3 The assumption in this equation is that the distance between the charge and dipole, r, is large relative to the separation of charges in the dipole, d, r >> d. The electric field due to a dipole is: E = 4πε 3(μ r)r r 5 μ r 3 Using the expression W = -μ. E we can calculate the interaction energy of two dipoles. W = μ μ 2 3(μ r)(μ 2 r) 4πε r 3 r 5

7 Example: Effect of Dipole Orientation Consider two dipoles, which have the orientations below that we can call aligned and head-to-tail Aligned: μ. μ 2 = μ 2, μ. r = μ r, μ 2. r = μ r, W = -2μ 2 /4πε r 3 Head-to-tail: μ. μ 2 = -μ 2, μ. r =,, μ 2. r =, W = -μ 2 /4πε r 3

8 Interaction of electric moments with the electric field The interaction of a collection of charges subjected to an electric field is given by: The picture is that of a charge interacting with the potential, the dipole interacting with a field, etc. An electric field can exert a force: or a torque: on a collection of charges. W = qφ μ E +... F = Σ q i E(r i ) i T = Σ r i q i E(r i ) i

9 Polarizability In the presence of an externally applied electric field the eipole moment of the molecule can also be expressed as an expansion in terms of moments: μ = μ α E + β :EE The leading term in this expansion is the permanent dipole moment, μ. The polarizability is a tensor whose components can be described as a follows: α xy = μ x E y Where the subscript refers to the fact that the derivative is Evaluated at zero field. The β tensor is called the hyperpolarizability and is third ranked tensor.

10 Polarizability as second rank tensor The dipole moment components each can depend on as many as three different polarizability components as described by the matrix: μ x μ y = μ z α xx α xy α xz α yx α yy α yz α zx α zy α zz E x E y E z If a molecule has a center of symmetry (e.g. CCl 4 ) then The polarizability is a scalar (i.e. the induced dipole moment Is always in the direction of the applied field). However, for non-centrosymmetric molecules components can be induced in other directions. The directions are often determined by the directions of chemical bonds, which may not be aligned with the field. This is the significance of the tensor.

11 Properties of the polarizability tensor Like the quadrupole moment, the polarizability can be made diagonal in the principle axes of the molecule. In the laboratory frame of reference the polarizability depends on the orientation of the molecule. The average polarizability is independent of orientation. It is given by the Trace, which is written Tr α. Tr α = 3 α xx + α yy + α zz The polarizability increases with the number of electrons in The molecule or with the volume of the charge distribution. Classically, for a molecule of radius a, α = 4πε a 3.

12 Van der Waal s Forces

13 /r 6 Interactions. Keesom - permanent dipole/permanent dipole 2 3 μ 2 μ 2 2 4πε 2kTr 6 2. Debye - permanent dipole/induced dipole α 2 2,μ 2 + α,2 μ 4πε 2r 6 3. London - induced dipole/induced dipole 3h 2 ν ν 2 ν + ν 2 α,α,2 4πε 2r 6

14 Dipolar Interactions The field around a dipole can be resolved into two components as shown in the Figure. The components are: E E^ E = 2μ 4πε E = μ 4πε The total field is: cos θ r 3 sin θ r 3 E = μ 4πε r 3 + 3cos 2 θ m

15 The Debye Term A permanent dipole on molecule will induce a dipole moment on molecule 2. μ 2 = α 2 E The total energy of the second dipole is: Φ 2 = 2 α 2E 2 Substituting for E and averaging over all orientations yields: Φ 2 = α 2 μ 2 2 4πε 2r 6 A similar equation can be derived for F to yield the Debye equation.

16 The Keesom Term The Keesom term arises from the interaction of two permanent dipoles. Here we consider the Debye term for the polarizability of a polar solvent. P = N A 3ε α + μ2 3kT Using a similar reasoning applied to the Debye term we can substitute in m 2 /3kT for a to obtain the Keesom term. 2 3 μ 2 μ 2 2 4πε 2kTr 6

17 London Interactions No permanent dipole is required for London forces to apply. The London equation for the attraction between two particles represents a quantum mechanical effect. The derivation uses a harmonic oscillator model. Consider a dipole-dipole interaction: ± 2μ2 4πε r =±2 e e 2 3 4πε r 3 since the definition of a dipole is: μ = e

18 Harmonic oscillator model Consider the electrons in a material as a harmonic oscillator. The nuclei represent the restoring force. The potential energy is given by: in which: Φ = K 2 K = e2 α Combining these various contributions we have: 2 Φ T = K ± 2 e e 2 4πε r 3

19 Harmonic oscillator model The energies of the harmonic oscillator are: in which: E = n +/2hν + n 2 + /2 hν 2 ν = ν 2α, ν 4πε r 3 2 = ν + 2α 4πε r 3 An illustration of the two induced dipoles for the London interaction is shown below: + - l + - l 2 r

20 Harmonic stabilization energy Taking the lowest energy harmonic oscillator state: E = h 2 ν + ν 2 Two independent oscillators in their ground state have energy: E = h 2 2ν The difference is the contribution of dispersion forces to the interaction energy: Φ = h 2 ν + ν 2 2ν

21 The London term Plugging in the frequencies obtained above and solving yields: Φ = hνα 2 24πε 2r 6 for identical molecules or: Φ = 3h 2 ν ν 2 ν + ν 2 α,α,2 4πε 2r 6 two different types of molecule and 2.

22 The van der Waal s parameter β The van der Waal s potential is the sum of the three terms derived: Φ = 4πε 2α 2 2,μ + 2μ 4 3kT hνα 2 = β, r 6 r 6 In this case it is derived for a pair of identical molecules of type. Thus, the parameter β is an interaction parameter for molecules of type that includes Keeson, Debye and London terms.

23 Protein folding energetics

24 Non-covalent forces in proteins What holds them together? Hydrogen bonds Salt-bridges Dipole-dipole interactions Hydrophobic effect Van der Waals forces What pulls them apart? Conformational Entropy

25 Dipole-Dipole Interactions Dipoles often line up in this manner. Example: α-helix

26 Main Chain Electrostatic Interactions Coulomb s Law: V = q q 2 /εr Example of a hydrogen bond -N-H.. O=C- Example of a Salt Bridge Main Chain Lysine Glutamate

27 Hydrogen bonding in water

28 Hydrophobic interactions

29 Contributions to ΔG - + -TΔS ΔH Internal Interactions Conformational Entropy -TΔS Hydrophobic Effect Net: ΔG Folding

30 Review of transitions

31 The Fermi Golden Rule can be used to calculate many types of transitions Transition H(t) dependence. Optical transitions Electric field 2. NMR transitions Magnetic field 3. Electron transfer Non-adiabaticity 4. Energy transfer Dipole-dipole 5. Atom transfer Non-adiabaticity 6. Internal conversion Non-adiabaticity 7. Intersystem crossing Spin-orbit coupling

32 Optical electromagnetic radiation permits transitions among electronic states Η t = μ E t where μ is the dipole operator and the dot represents the dot product. If the dipole μ is aligned with the electric vector E(t) then H(t) = - μe(t). If they are perpendicular then H(t) =. μ = er where e is the charge on an electron and r is the distance.

33 The time-dependent perturbation has the form of an time-varying electric field E t = E cos ωt where ω is the angular frequency. The electric field oscillation drives a polarization in an atom or molecule. A polarization is a coherent oscillation between two electronic states. The symmetry of the states must be correct in order for the polarization to be created. The orientation average and time average over the square of the field is [-μ. E(t)] 2 is μ 2 E 2 /6.

34 Absorption of visible or ultraviolet radiation leads to electronic transitions σ Polarization of Radiation s s σ

35 Absorption of visible or ultraviolet radiation leads to electronic transitions σ Transition moment s s The change in nodal structure also implies a change in orbital angular momentum. σ

36 The interaction of electromagnetic radiation with a transition moment The electromagnetic wave has an angular momentum of. Therefore, an atom or molecule must have a change of in its orbital angular momentum to conserve this quantity. This can be seen for hydrogen atom: Electric vector of radiation l = l =

37 The Fermi Golden Rule for optical electronic transitions 2 π e Ψ q Ψ 2 2 E k = δω ω 2 6h 2 2 The rate constant is proportional to the square of the matrix element e< Ψ q Ψ 2 > times a delta function. The delta function is an energy matching function: δ(ω - ω 2 ) = if ω = ω 2 δ(ω - ω 2 ) = if ω ω 2.

38 A propagating wave of electromagnetic radiation of wavelength l has an oscillating electric dipole, E (magnetic dipole not shown) λ E

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49 The oscillating electric dipole, E, can induce an oscillating dipole in a molecule as the radiation passes through the sample λ E

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64 The oscillating electric dipole, E, can induce an oscillating dipole in a molecule as the radiation passes through the sample l v= R O R E = hc/λ v= R R O The type of induced oscillating dipole depends on λ. If λ corresponds to a vibrational energy gap, then radiation will be absorbed, and a molecular vibrational transition will result

65 The oscillating electric dipole, E, can induce an oscillating dipole in a molecule as the radiation passes through the sample λ LUMO HOMO E = hc/λ If λ corresponds to a electronic energy gap, then radiation will be absorbed, and an electron will be promoted to an unfilled MO

66 The absorption of light by molecules is is subject to several selection rules. From a group theory perspective, the basis of these selection rules is that the transition between two states a and b is electric dipole allowed if the electric dipole moment matrix element is non-zero, i.e., aμ b = * ψ a μψ b dτ where μ = μ x + μ y + μ z is the electric dipole moment operator which transforms in the same manner as the p- orbitals ψ a μ ψ b = ψ a (μ x + μ y + μ z ) ψ b, must contain the totally symmetric irrep or put another way, ψ a ψ b must transform as any one of μ x, μ y, μ z

67 Direct Products: The representation of the product of two representations is given by the product of the characters of the two representations. Verify that under C 2v symmetry A 2 B = B 2 C 2v E C 2 σ v σ' v A B - - A 2 B - - As can be seen above, the characters of A 2 B are those of the B 2 irrep.

68 Verify that A 2 B 2 = B, B 2 B = A 2 Also verify that the product of any non degenerate representation with itself is totally symmetric and the product of any representation with the totally symmetric representation yields the original representation Note that, A x B = B; while A x A = B x B = A g x u = u; while g x g = u x u =g.

69 Light can be depicted as mutually orthogonal oscillating electric and magnetic dipoles. In infrared and electronic absorption spectroscopies, light is said to be absorbed when the oscillating electric field component of light induces an electric dipole in a molecule. Electric vector of radiation l = l = For a hydrogen atom, we can view the electromagnetic radiation as mixing the s and 2p orbitals transiently.

70 Is the orbital transition d yz p x electric dipole allowed in C 2v symmetry? p x μ d yz b b b 2 b 2 = a a a 2 b b 2 = b 2 b a 2 None of the three components contains the a representation, so this transition is electric dipole forbidden A transition between two non-degenerate states will be allowed only if the direct product of the two state symmetries is the same irrep as one of the components of the electric dipole

71 How about an a b 2 orbital transition? μ b 2 b b 2 a a = a 2 a b 2 a = a 2 a b 2 Since m y makes the transition allowed, the transition is said to be "y-allowed" or "y-polarized" Remember the shortcut: a b 2 = b 2 which transforms as μ y Problem Indicate whether each of the following metal localized transitions is electric dipole allowed in PtCl (a) d xy p z (b) d yz d z 2 (c) d x 2 -y 2 p x,p y (d) p z s

72 Example: Myoglobin/Hemoglobin Heme spectroscopy Transition moment Franck-Condon active Vibronic coupling

73 Myoglobin Structure G helix Globular α-helical protein F helix A helix Heme B helix E helix H helix

74 The iron in heme is the binding site for oxygen and other diatomics Heme is iron protoporphyrin IX. Fe is found in Fe 2+ and Fe 3+ oxidation states. Diatomics bind to Fe 2+. Examples, CO, NO, O 2. O 2 is the physiologically relevant ligand, but it can oxidize iron and it is difficult to study directly. N N O O - O C Fe N N O O -

75 The porphine ring is an aromatic ring that has a fourfold symmetry axis The ring and metal can be considered separately. The ring has been succesfully modeled using the Gouterman four orbital model. In globins the iron is Fe(II) and can be either high spin or low spin. MbCO low spin Deoxy Mb - high spin N N N N

76 The four orbital model is used to represent the highest occupied and lowest unoccupied molecular orbitals of porphyrins The two highest occupied orbitals (a u,a 2u ) are nearly equal in energy. The e g orbitals are equal in energy. Transitions occur from: a u e g and a 2u e g. M e g π a 2u π a u π

77 The transitions from ground state π orbitals a u and a 2u to excited state π* orbitals e g can mix by configuration interaction Both excited state configurations are E u so they can mix. Two electronic transitions are observed. One is very strong (B or Soret) and the other is weak (Q). The transition moments are: M M 2 e g π B band R s = M + M 2 Q band r s = M -M 2 a 2u π a u π

78 Porphine orbitals e g e g a u a 2u

79 Four orbital model of metalloporphyrin spectra There are four excited state configurations possible in D 4h symmetry. These are denoted B (strong) and Q(weak). B y = 2 a 2ue gy + a u e gx Q y = 2 a 2ue gy a u e gx B x = 2 a 2ue gx + a u e gy Q x = 2 a 2ue gx a u e gy

80 The transition moment for absorption The absorption probability amplitude for a Franck-Condon active transition is: e<g σ B> E B E G hω iγ B Here i and f represent individual vibrational levels in each electronic manifold. The polarization can be σ = x, y, or z. Remember that the ground state is totally symmetric (filled shell molecules). Here it is A g.

81 A g A 2g B g B 2g E g A u A 2u E u E 2 2 Character table for D 4h point group 2C 4 (z) The absorption probability amplitude for a Franck-Condon - - active transition - - is: - - B u B 2u C C' C'' i S σ h σ v - - 2σ d - - linears, rotations R z (R x, R y ) z (x, y) quadratic x2 -y2 xy x2 +y2, z2 (xz, yz)

82 A g A 2g B g B 2g E g A u A 2u E u E 2 2 Character table for D 4h point group 2C 4 (z) The absorption probability amplitude for a Franck-Condon - - active transition - - is: - - B u B 2u C C' C'' i S σ h σ v - - 2σ d - - linears, rotations R z (R x, R y ) z (x, y) quadratic x2 -y2 xy x2 +y2, z2 (xz, yz)

83 A g A 2g B g B 2g E g A u A 2u E u E 2 2 Character table for D 4h point group 2C 4 (z) The absorption probability amplitude for a Franck-Condon - - active transition - - is: - - B u B 2u C C' C'' i S σ h σ v - - 2σ d - - linears, rotations R z (R x, R y ) z (x, y) quadratic x2 -y2 xy x2 +y2, z2 (xz, yz)

84 A g A 2g B g B 2g E g A u A 2u E u E 2 2 Character table for D 4h point group 2C 4 (z) The absorption probability amplitude for a Franck-Condon - - active transition - - is: - - B u B 2u C C' C'' i S σ h σ v - - 2σ d - - linears, rotations R z (R x, R y ) z (x, y) quadratic x2 -y2 xy x2 +y2, z2 (xz, yz)

85 Mixing of the excited state configurations There are two transitions that both have E u symmetry. Thus, they can add constructively and destructively. B y = 2 a 2ue gy + a u e gx Q y = 2 a 2ue gy a u e gx B x = 2 a 2ue gx + a u e gy Q x = 2 a 2ue gx a u e gy Constructive (allowed) Destructive (forbidden) Constructive (allowed) Destructive (forbidden)

86 Vibrational modes that couple the states are determined by the direct product Γ coupling = E u E u which we can determine from the character table. E 2C 4 (z) C 2 2C' 2 2C'' 2 i 2S 4 σ h 2σ v 2σ d linears, rotations quadratic E u (x, y) Γ c This reducible representation can be decomposed into four irreps: A g + A 2g + B g + B 2g

87 Magnetic Circular Dichroism

88 The Perimeter Model The porphine ring has D 4h symmetry. The aromatic ring has 8 electrons. The p system approximates circular electron path. N N -5 Δm=9-4 5 Δm= 4 N N -3 3 Φ = 2π eimφ -2-2 m =

89 MCD spectra Δε ν = A f ν ν + B + C k B T f ν Δε m ν εν = A μ B D f ν ν f ΔL z =2 A D Franzen, JPC Accepted

90 MbCO MCD spectra follow the PM B Q The spectra are A-term MCD as shown by the derivatives of the absorption spectrum (red). The (Q MCD) = 9 x (B MCD). Franzen, JPC Accepted

91 Deoxy MCD spectra are anomalous B Q C-term 4 time larger than MbCO! A-term but with vibronic structure

92 MCD spectra: Vibronic coupling in the Perimeter Model Franzen, JPC Accepted

93 MCD spectra: Vibronic coupling in the Perimeter Model Metal Porphyrin Vibronic Distortions