6. Structural chemistry

Transcription

1 6. Structural chemistry 6.1. General considerations What does structure determination do? obtains of chemical formula determines molecular geometry, configuration, dynamical structure (molecular forces, vibrations, etc.) Grouping of the methods: spectroscopy: absorption or emission of electromagnetic radiation diffraction methods observation of ionized states (mass spectrometry and similar methods) There are lots of development on this field: improvement of methods (e.g. use of Furier techniques) lasers new and tricky setups Common basis of spectroscopical methods Event in all cases: interaction of matter and electromagnetic radiation What can happen during the interaction? 64

2 Arrangement of the measurement see figure on the blackboard The spectrum: relativ or absolute intensity as a function of the wave number (λ), frequency (ν) or wavenumber (ν ): frequency: ν, measured in [1/s], e.g. in Hz, MHz, but also in energy unit as ev wavenumber: ν measured in [1/m], e.g. in cm 1 wave length: λ measured in [m], e.g nm, Å Important relations: λ = c ν ν = 1 λ where c is the speed of light. Lambert-Beer law: I = I 0 exp( ε n l) where: I 0 is the intensity of the incoming light I is the intensity of the outcoming light ε: molar absorption coefficient (also called as extinction coefficient) n: concentration l: thickness of the sample This means that the intensity drops exponentially in the sampel. This law aplies only for small intensity and small thickness. Absorbance: A = ln( I 0 I ) = ε n l, which is proportional with the concentration I Transmittance: I 0 65

3 Common physical background: During the spectroscopical event there is a transition between energy levels: incresing the energy of the molecule by absorption, and lowering the energy by emission. It is seen that the the energy of the matter (molecule) changes exactly with the energy of photon: E 2 E 1 = E = hν Factors determining the spectrum a) energy levels nothing to be said about this b) selection rules The spectrum can be described by the tools of quantum mechanics. It can be derived that the probability of a transition between two energy levels are: P 1 2 = Ψ 1 ˆK Ψ2 dv 2 where Ψ 1, Ψ 2 : are the wave functions of the two sates ˆK: operator of the interaction between the molecule and the electromagnetic field. Usually it is approximated by the dipole moment. 66

4 The above integral is called transition integral. Transition occurs, if the transition integral does not vanish: it is allowed. If it vanishes, we say that the transition is forbidden. When is the transition allowed? This can be derived by, e.g. consideration of molecular symmetry. c) population of the energy levels The transition probability under b) is the same in both direction, i.e. absorption and induced emission has the same probability!! But: the population of the energy levels are different and given by the Boltzmandistribution: where N i = N 0 exp ( E i kt ) E i : relative energy of energy level i ( E i = E i E 0 ) k: Boltzman-constant T : temperature N 0 : population of the ground state The observed absorption is determined by the population and the transition probability at the same time The lasers LASER: Light Amplification of Stimulated Emission of Radiation Usually, the population of the lower states are significantly larger probability of absorption is larger than that of stimulated emission. However, by the so called population inversion (doing a long and intense irradiation) we can get the higher population for the higher states emission will be more likely. Properties of the lasers: monochromatic: the radiation includes only a narrow range of the electromagnetic spectrum coherence: the waves are in a common phase directed (small radial distribution) large intensity 67

5 Distinction of different spectroscopic methods Principle: the different motions in molecules can be distinguished by a good approximation: Ψ spatial (r, R) Ψ electron,spatial (r) Ψ nuclei,spatial (R) Ψ electron,spatial (r) Ψ vibration Ψ rotation Ψ translation }{{} Bohr Oppenheimer approximation where r and R are the coordinates of electrons and nuclei, respectively. If the above approximation for the wave function works, we can also approximate the energy as: E tér = E electron + E vibration + E rotation + E translation Additional degree of freedom is spin, which both electrons and nuclei have: E = E spatial + E electron spin + E nuclei spin The different contributions are of different size and therefore they will be obtained with electromagnetic radition belonging to different region of the electromagnetic spectrum can be investigated by different methods of spectroscopy 68

6 6.2. Rotation of molecules microwave spectroscopy Physical background Event: The molecule as a rotating dipole interacts with the electromagnetic radiation. Therefore, the molecules with dipol moment will give rotational spectrum. Quantum mechanical description of rotation also results in quantized energy levels. The difference in the energy levels corresponds to microwave (MW) region of the electromagnetic radiation microwave spectroscopy Experimental details: source: klystron, the radiation is caused by vibrating electrons sample: gase phase, the cell is often several meter long information to be obtained: most accurate method for molecular geometry (bond length) Rotation of diatomic molecules The rotating molecule possesses moment of inertia: ( I B = m 1 r1 2 + m 2 r2 2 m1 m 2 ) 2 = (m 1 + m 2 ) + m2 1m 2 r 2 2 (m 1 + m 2 ) 2 m 1 m 2 (m 1 + m 2 ) (m 1 + m 2 ) 2 r 2 = m 1 m 2 m 1 + m 2 r 2 m red r 2 From the point of view of the mathematical treatment, it can be described as a single particle with mass m red rotating on an orbit of radius r (c.f. with the electron in hydrogen atom; it is simpler here since r is fixed). Classically: there is only kinetic energy: E = T = (1/2)Iω 2, where ω is the angular velocity (c.f. translational motion, where E = (1/2)mv 2 ). Alternatively, with angular momentum: (L = Iω): E = T = 1 2I L2 69

7 The corresponding Hamiltonian: Ĥ = 1 2I ˆL 2 i.e. it differs from ˆL 2 only by a constant eigenvalues will be the same but multipied by the constant. ˆL2 eigenvalues (see hydrogen atom): Λ = l(l + 1) h 2, l = 0, 1, 2,... The energy of rotation of diatomic molecules differs from this only by a factor 1 2I. The quantum number is traditionally denoted J. E rot J = 1 2I J(J + 1) h2 BJ(J + 1)hc J = 0, 1, 2,... with B being the rotational constant. Similarly to the z component of the angular momentum, the z component is given by a quantum number M: M = J, J + 1, 0, 1,..J and the energy is independent of this the rotational energy levels are (2J+1)-fold degenerate! From the formula of the energy above one can conclude that with increasing J the energy levels get further and further away! There is no zero point energy!!! Selection rules: µ 0, only molecules with dipole moment have rotational spectrum; J = ±1, transition only between neighboring states are possible. 70

8 A rotational spectrum of diatomic molecules consists of equidistante lines: The distribution of the intensity follows maximum curve: this is because the population of the energy levels, despite of the Boltzman distribution, grows first due to the increasing degeneracy (2J + 1). Informations from the rotational spectrum: Distance of the lines: rotational constant (B) moment of inertia (I) bond distance (r), or in generla, the molecular structure. A true rotational spectrum: Rotation of molecules with more than two atoms Basic concept: The moment of inertia is in this case a tensor of dimension 3x3 (I). The I tensor of the moment of inertia defines three main axes a, b, c and the corresponding three main moments of inertia (I a, I b, I c ). The spectrum can be described by this three numbers (or the corresponding rotational constants B a, B b and B c ). 71

9 In the main-axis system the Hamiltonian is given as the independent summ of the three components: Ĥ = 1 2I a ˆL2 a + 1 2I b ˆL2 b + 1 2I c ˆL2 c The molecules can be grouped according to the relative size of the main moments of inertia: spherical rotor: I a = I b = I c ; symmetric rotor: I a I b = I c ; asymmetric rotor: I a I b I c. Examples: spherical rotor: CX 4, but these does not posses dipole, therefore nor rotational spectrum. symmetric rotor: favorite of the MW-spectroscopists, easy to evaluate; e.g. NH 3, CH 3 X, C 6 H 6 (For the latter, however, no dipol moment!!) Summarized: The rotational (microwave - MW) spectroscopy results in the most accurate molecular structure for small és symmetric molecules in the gase phase. 72

10 6.3. Vibration of molecules IR and Raman spectroscopy Physical background Event: The atoms building up the molecules are in constant vibration, their vibrational states are quantized. We can observe transition between these quantum states: directly: in the infra red (IR) region, which frequency corresponds to the energy differences between vibrational states. indirectly, when the frequency difference of the absorbed and emitted light corresponds to the energy differences between vibrational states (Raman effect, inelastic scattering) Experimental details of the IR measurement Classical arrangement (as in the general case, see figure above): source: IR lamp sample: film or solution, eventually solid monochromator: optical grid or prism detector: thermo-cell Sample rack and monochromator: salt of alkali metals (e.g. KI) since glas absorbes in IR. Details of the Raman measurement Effect: inelastic scattering; first excite to an electrically excited state, which will emit light and lead back to a vibrationally excited level of the ground electronic state. Generally, the left (Stokes) effect is used in Raman spectroscopy Experimental instruments: similar to the VIS and UV measurements (see later) 73

11 Vibrations of diatomic molecules Classical description Model: harmonic oscillator, which follows the Hook-law: F = kx, where F is the force, x is the displacement, k force constant of the spring. F = k(r r e ) where r e is the equilibrium distance between the two atoms, r is the actual distance (changes during vibration). The vibration of the diatomic molecule can be considered as a vibration of a particle of mass m red around its equilibrium position; the frequency of the vibration is given by: ν = 1 2π k m red i.e. it depends on the reduced mass (m red = m 1ṁ 2 m 1 +m 2 ) and the force constant (k). The energy of the system is continuous, it can have any vale, even 0! Quantum mechanical description in the harmonic approximation: Hamiltonian: Ĥ = ˆT + ˆV V = 1 2 kq2 (= F = kq) By solving the Schrödinger equation, we get the energy levels: E v = hν ( v + 1 ) 2 v = 0, 1, 2,... Thus the energy: quantized equidistant (E v+1 E v = hν) there is zero point energy ( 1hν) 2 74

12 Wave function: i.e. similar to the wave functions of the particle in the box model. (On the figure: ω = 2πν.) The figure also shows the distribution function ( Ψ 2 ), i.e. the probability of finding the vibrating particle. These can be compared with the classical probability of the same energy (the figure uses the letter m for the quantum number instead of v): It is seen that in the classical case the probability at q = 0 is the smallest since the particle runs here the fastest. In the quantum case, however, the largest probability is at q = 0. (This contradiction can be resolved if we do not compare states with the same energy, but the lowest energy state with the lowest energy state. In the classical case it has enery zero, so it does not move, i.e. the particle is at q = 0 which corresponds to a maximum of the quantized system.) from v = 2 the two cases hardly resemble each other, the maximum of the classical case is still at the turning point. in case of v = 10 it is seen already that classical and quantum distributions are similar (correspondence principle). 75

13 The form of the vibrational wave function has an important role in case of the fine structure of the UV spectrum (see later). Selection rules The transition probability between levels v and v in case of IR measurement: P v,v = Ψ v ˆµΨ v dv where ˆµ is the dipole operator. Skipping the details, this probability is not zero if dµ dq 0, i.e. the dipole moement changes during vibration v = v ± 1, i.e. the vibrational quantum number changes by 1. This means that in the spectrum of diatomic molecule only one line will appear and E = hν! 2 Please note that homo-nuclear diatomic molecules have no IR spectrum, since the dipole moment remains zero during the vibration. The transition probability between levels v and v in case of Raman measurement: P v,v = Ψ v ˆα Ψ v 2 where ˆα is the operator of polarizability. Skipping the details, this probability is not zero if dα dq 0, i.e. the polarizability changes along the coordinate v = v ± 1, i.e. the vibrational quantum number changes by 1. This means that in the Raman spectrum of diatomic molecule only one line will appear and E = hν! Please note that homo-nuclear diatomic molecules have Raman spectrum, since the polarizability changes if we elongate the bond. Illustration: Vibrational frequency of hidrogene halogenides HX HF HCl HBr HI ν (cm 1 ) Structural information: ν k, i.e. from the vibrational frequency we can get the force constant, which is representative for the strength of the bond. Above the strongest is the bond of the HF molecule. 76

14 Anharmonicity For the real molecules the harmonic approximation is not quite valid, the potential differs from the harmonic one: This can be described the best by a so called Morse-potencial: V (q) = D e (1 exp( αq)) 2 where D e is the dissociation energy (measured from the minimum of the curve), α is a constant. Consequence: the energy levels are not equidistant they get more dens with growing energy v = ±1 will not be true strictly in the spectrum so called overtones will aslo apear at 2ν, 3ν. This effect is, however, very weak, the vibrational spectrum can approximated by the harmonic picture in most cases: there will be only some weak lines beside the strong ones corresponding to the harmonic approximation Vibrations of molecules with more than two atoms Internal coordinates The number of internal degrees of freedom for a molecule with N atoms is 3N 6 (in case of linear molecules 3N 5). The use of internal coordinates is better than the Cartesian coordinates (3N), and also chemical intuition can be used. Main types: bond stretching bond bending torsion out-of-plane 77

15 Examples: water: three coordinates are necessary, r 1, r 2, α ammonia: 6 coordinates are necessary, there are different choices, e.g. r 1, r 2, r 3, α 1, α 2, α 3, but the description of inversion would require the out-of-plane coordinate. ethylene: 12 coordinates are necessary, one possible choice is: five stretching, four bending (note that two of the possible six bendings are redundant), τ 3126 torsion, two out-of-plane (CH 2 groups). Normal coordinates The theory gives a set of coordinates in which the 3N-6 (3N-5) degrees of freedom get independent, the Hamiltonian reads: Ĥ = i ĥ(q i ) Here Q i stands for the normal coordinates. Normal coordinates are built from combination internal coordinates, therefore they extend over the whole molecule. The energy is the sum of the energy of the 3N-6 oscillators: E(v 1, v 2,..., v n ) = n hν i (v i + 1 i 2 ) (50) with ν i being the frequency of the ith normal coordinate, with the corresponding vibrational frequency of v i. The ZPE is: E ZP E = h 2 i ν i. Selection rules: Without derivations: v i = ±1 és v j i = 0, i.e. only one oscillator can be excited by one photon; dµ dq i 0 (IR) or dα dq i 0 (Raman). 78

16 Considering anharmonicity, beside the overtones, in the spectrum so called combination bends will also appear, which are the results of simultaneous excitation of more than one mode. Thus, the spectrum is dominated by 0 1 transition of symmetry-allowed normal modes. These are called fundamental bands. Due to anharmonicity, also overtones and combination bends will apear. As an example here I show the IR and Raman spektra of toluene: Typical normal vibrations are shown on the following figure: 79

17 Practical application of vibrational spectroscopy identification of the molecule assignation of typical groups quantitative characterization Identification Without theoretical characterization, the so called finger-print regions are compared with databases. The fingerprint region is the low frequency (< 500 cm 1 ) part of the spectrum, which mostly corresponds to deformation of the molecular backbone. These are very sensitive to the molecular environment, thus very characteristic for individual molecules. Assignation of typical groups Although the normal vibrations extend over the whole molecule, these very often dominated by some local internal coordinates. Such vibration will apear in the same region of the spectrum even if the molecular environment is different. Mostly stretching vibration can be used for this purpose. The following figure demonstrate this and shows the regions characteristic for different functional groups (you should remember these!). 80

18 The vibrational spectroscopy can be used for identification of groups of atoms. Qualitative characterization From the vibrational frequencies, with mathematical tools, one can obtain the force constants, which are characteristic for the strength of the bonds. One measurement is usually not enough to obtain all force constants, therefore izotopomers are used, which give independent data. Here theoretical methods have a very important role. 81

19 6.4. Electronspectroscopy Two different methods: UV-VIS spectroscopy: M + hν M PES, XPS (ESCA): M + hν M + + e UV-VIS ultraviolet-visible spectroscopy Experimental details: Regions: visible UV VUV (Vacum UV) nm nm < 200 nm The instrument is simple: source: optic: detector: wolfram lamp, hydrogen lamp, etc. quartz photocell Physical background: transition between two electronic levels electron spectrum Theory: everything, what we have learned about electronic structure of molecules Selection rules: Transition probability between state i and j is given here also be the transition integral: where Ψ is the total wave function: Ψ i ˆµ Ψ j dv Ψ = Ψ vib Ψ el = Ψ vib Ψ spatial Ψ spin therefore Ψ i ˆµ Ψ j dv = Ψ vibr i Ψ vibr j dv Ψ spin i Ψ spin j dv Ψ spatial i ˆµ Ψ spatial j dv Ψ vibr i Ψ vibr j dv: called as the Frank-Condon factor (see later). Ψ spin i Ψ spin j dv: transition only between states of the same multiplicity is possible. Ψ spatial i ˆµ Ψ spatial j dv: determined by molecular symmetry Example: benzene HOMO-LUMO excitation (HOMO: highest occupied MO, LUMO: Lovest unoccupied MO) 82

20 Excitation with π electron configuration: (a 2u ) 2 (e 1g ) 4 (a 2u ) 2 (e 1g ) 3 (e 2u ) 1 With states: 1 A 1g B 2u, B 1u, E 1u (singlet or triplet) The transition can be classified as: symmetry allowed symemetry forbidden spin allowed 1 E 1u 1 B 2u, 1 B 1u spin forbidden 3 E 1u 3 B 2u, 3 B 1u The spectrum: excitation wave number (Å) extinction coefficient 1 A 1g 1 E 1u B 1u B 2u UV spectrum of benzene: 83

21 The triplet states can not be seen at all spin prohibition is much stronger than symmetry! Vibrational fine structure, Frank-Condon prinziple The vibrational levels superimpose on the electronic transitions: These are called vibronic transitions. E = E el + E vibr Question: what are the relative intensities of the the vibronic transitions? Transition integral: dv Ψ spatial i ˆµ Ψ spatial j dv Ψ vibr i,v Ψ vibr j,v 84

22 The second term is the same for all vibrational levels, therefore it is influenced only by the first term: the individual transition probabilities are given by the overlap of the vibrational wave functions in the ground and excited state. Qualitatively: the motion of the nuclei are slow compered to the motion of electrons the so called vertical excitations will be the most probable, i.e. transition to vibrational levels where the probability of the nuclear arrangement of the ground state is large. In more detail for the diatomic molecules 2 : a) The potential energy surface does not change too much (e.g. excitation from lone pair) See figure on the blackboard In this case the vibrational wave function is almost the same in the two electronic state, therefore v = 0 v = 0 transition will be the most probable. b) The potential energy surface of the two electronic states differ considerably, also the equilibrium bond length differ: See figure on the blackboard In this case some other but the first vibrational wave function will have larger overlap, the corresponding vibronic band will be the most intense, the lines left and right will have decreasing intensity (so called progressions will appear. If the resolution of the spectrum is not large, we can see only the envelop of the individual lines. c) Dissociation See figure on the blackboard The spectrum becomes continuous above the dissociation energy, since the energy of the fragments can take any value (translation). 2 The graphical representation of the vibrational wave function is seen in subsection 6.3.2, page

23 Practical UV spectroscopy Importance: quantitativ analysis structural chemistry: detection of chromophores chromophores: are molecular fragments (group, bond, etc.) which can be excited easily. According to the chromophores the following types of excitation can be distinguished: 1. π π Chromophore: double bond or π-system in the molecule excitation from binding π orbital to antibonding π orbital ethylene butadiene hexatriene octatetraene λ(å) so called batochrome shift: λ increases Any explanation? Yes, with the energy levels of the particle in the box they decrease with the increasing size of the box ( E n n n2 n 2 ) L 2 2. n π chromophore lone pair 3. σ σ, n σ, stb 4. charge transfer excitation excitation from an orbital localized on one side of the molecule to a one which is localized to an other side there will be positive charge at one and negative charge on the other end, therefore big dipole moment. Example: C 6 H 6 I 2 C 6 H 6+ I 2 Such transitions have large intensity because of the large change in dipole moment! 5. transition metal complexes Relaxation of excited states According to quantum mechanics, the excited states are only quasi-stationary, i.e. they can not leave for ever (finite life time). Possible routes: without radiation: with a complicated mechanism the energy transfers to other degrees of freedom (e.g. vibration, rotation the sample gets warm) with radiation fluorescence immediately (nano-seconds time scale) phosphorescence timely delayed (seconds or even minutes time scale) 86

24 Fluorescence It can be seen on the figure: λ out λ in absorption shows the vibrations of the excited state emission shows the vibrations of the ground state Phosphorescence What is the reason for the delay? 87

25 Since triplet singlet transition is forbidden, it can proceed only with low probability, i.e. slow. The change in wave number is much larger than in case of fluorescence PES photoelectron spectroscopy and ESCA principle: electromagnetic radiation ejects an electron from the molecule: M + hν M + + e The kinetic energy of the electron (T electron ) will be measured, and the ionization energy (IE) can be calculated using the frequency of the radiation (ν). hν = E ion E }{{ molecule } IE +T electron 88

26 In practice: UPS: UV light is used, the valence electrons are ejected characteristic for the molecule XPS: X-ray is used: electrons of the core (1s) are ejected will be characteristic for the atoms of the molecule. Photoelectron spectroscopy Source: helium lamp, so called lines I and II ( ev). What will be seen in the spectrum? the molecule is usually in its ground state (both electronic and vibrational) several state of ion can be produced the spectrum consist of several lines. different vibrational states of the ion can be reached vibrational fine structure (as in case of UV) Example: photoelectron spectrum of the CO 2 molecule: We can see: there are three bands without vibrational structure these are ionization from lone pair orbitals one band has strong vibrational structure ionization from a bonding orbital. Explanation is very similar to that of the UV spectroscopy. The overlap of the vibrational wave functions of the molecule and the ion will determine the intensity of the vibrational bands. If we eject electron form a lone pair, the bonding of the molecule does not change much, vibration will be the same. On the other hand, ejecting an electron form a bonding orbitals, results in a big change in bindings and consequently in vibrational modes. Information from the photoelectric spectrum: ionization energy vibrational modes of the ion Application: mostly systems with lone pairs (the spectrum is simpler): 89

27 transition metal complexes organic molecules with heterocyclic (aromatic) fragments ESCA Electronspectroscopy for chemical analysis Excitation by X-ray radiation which ejects electrons from inner orbitals (mostly 1s) Inner shells: are with good approximation identical to atomic orbitals (see e.g. the orbitals of water) the ionization energy will be characteristic for the atom small dependence of the chemical environment the ionization energy of the atoms in different environment will be slightly different: chemical shift. Example: ethylpropionate: ESCA spectrum: Notes to the spectrum: the bands corresponding to O and C atoms differ considerably (540 and 290 ev, respectively) the signal of the O atom splits into two: there are two types of O atoms in this molecule the signal of the C atom splits into three: there are three types of C atoms in this molecule 90