Perturbation Theory & Energy Levels in Quantum Mechanics: Rhodamine Dyes Case Study, Assignments of Physics

Download Perturbation Theory & Energy Levels in Quantum Mechanics: Rhodamine Dyes Case Study and more Assignments Physics in PDF only on Docsity! 1 Particle in a box An additional bit while we are discussing the particle in a box. What we have found in the particle in a box are “stationary solutions”. These are invariant in time – but there is no reason that these are the only allowed states in the box. They are only the only stationary solutions in the box. Remember that the eigenfunctions are a complete set (the sin and cos functions form a complete set – e.g. Fourier series). But Dirichlet’s theorem says that any function where ( ) ( )0 0Lψ ψ= = is also a solution. And we know that we can write any solution in terms of a weighted sum of all the eigenvalues: ( ) ( ) 2 sinn n nn n nf x c x c x L L πψ ⎛ ⎞= = ⎜ ⎟ ⎝ ⎠ ∑ ∑ . We can determine the coefficients, cis, in the usual way: ( ) ( ) ( ) ( )1 1m m n n n mn mn nx f x dx x c x dx c cψ ψ ψ δ ∞ ∞ ∞ ∞∗ ∗ = =−∞ −∞ = = =∑ ∑∫ ∫ . Actually the function ( )f x will in general not be stationary, and the general behavior of such a prepared state is: ( ) ( ) niE tn nnf x c x eψ −=∑ . This function is obviously time dependent. Also note that if we were able to prepare this state, and if there were no perturbation from the outside of the box, the behavior of the solution would oscillate forever according to the above solution. Of course, it would not be a simple sinusoidal oscillation. Homework: In homework problems you will calculate 2 2, , ,x x p p for a particle in a box, and compare it to what you would expect for the classical solution. Variational Principle: this is important when we talk about approximate MOs This principle says for any Hamiltonian and boundary conditions, the solution to the Schroedinger equation is the solution for which the minimum energy is found. That is, any other solution that may obey the boundary conditions will have an energy that is higher than the energy of the real solution (that is the solution of the Schroedinger equation). That is, say that we pick a function that obeys the boundary conditions, but is just a guess at the real solution – because the problem is too hard for us to get the real solution in an analytical form. Then say that we guess that a good estimate of the lowest energy wavefunction (the correct function is 0ψ with the correct energy E0) is φ . Then the variational principle says: 0 H d E E dφ φ φ τ φ φ τ ∗ ∗ = >∫ ∫ This gives us a way to get the best guess by minimizing the value E. This can be done by adjusting some parameters in the functional form of φ . 2 Take as an example the particle in a box and we try the solution 20 1 2a a x a xφ = + + . We have to obey the boundary conditions at x=o and x=L. At x=0 we get 0 0a = , and at x=L we get 21 2 1 20, or a L a L a a L+ = = − . Therefore, ( ) ( )22x a x xLφ = − . We can normalize this to give: ( ) ( )2530x x xLLφ = − ← guessed solution In this case there is nothing else to adjust to minimize it, so this is the best guess with the lowest energy for the guess we made. Note that the form of the function is quite reasonable considering what we already know from the last lecture about the correct solution. Now let’s calculate the energy using the rules we covered in the last lecture. ( ) ( ) ( )2 sinn x L n x Lψ π= , ← correct solution The energy of our guess is: ( ) ( ) ( ) ( ) ( ) ( ) ( ) 22 2 2 2 2 2 22 2 2 5 2 2 2 2 2 30 10 2 8 x E x V x x dx x dx m mx x x xL hx xL dx mL x mL φ φ φ φ π ∗ ∗⎛ ⎞ ∂⎛ ⎞ ∂= − + = −⎜ ⎟⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠ ∂ − ⎛ ⎞= − = ⎜ ⎟∂ ⎝ ⎠ ∫ ∫ ∫ The real value of the energy is: 2 2 2 2 2 2 02 2 22 8 8n n n h hE E mL mL mL π = = → = , where we realize that the lowest energy for the particle in a box is for the n=1 solution. Since 2 10 1.015 π ≅ the agreement is very good. But, as predicted, the energy of the calculated function is higher than the correct energy. We can now try to minimize it more by including more terms or changing the actually guess function. But the energy will always be larger. It is the same for the molecular orbital approximations, and we try to parameterize the trial functions so that we can find a minimum. For instance we could try ( ) ( ) ( )2 31 2 3 ...c x x L c x x L c x x Lφ = − + − + − + , find the conditions to obey the boundary conditions, evaluate E H d dφ φ φ τ φ φ τ ∗ ∗= ∫ ∫ , and minimize Eφ with respect to the cis. ------------------------------------------------------------------------------------------------------- Electron density: The electron density is calculated by: 2electron density k kk nψ=∑ , where nk is the number of electrons in the kth orbital. The electron density for hexatriene is ( ) ( )3 212 2 sin hexatriene elec density k L k x Lρ π = = ⋅∑ . Look at the distribution of the ( )2k xψ in the last lecture for the first three levels. 5 Structures of type A: There are 2N+2 electrons we have to place into the conjugated system; 2 for every double bond, and two on the neutral nitrogen. They will occupy the first N+1 molecular orbitals (of our particle in a box). Remember the wave functions and the energies of the particle in a box: ( ) ( ) ( )2 sinn x L n x Lψ π= and 2 2 28n n hE mL = L is the total distance from one end of the chain to the other through the pi-bonded network The A structure has two resonant structures that have identical structures; so the C-C bonds of the resonance structure all have 50% double bond character and 50% single bond character. We assume that the average C-C bond length is the same as in benzene (1.39 o A ). We also assume that the length of the box extends one bond length beyond each nitrogen (because of the non-bonded electrons). C H C H C N + N N-1 and C H C H CN + N N-1 So, ( ) ( ) o 2 1 1.39 AL N= + ⋅ . The first excited state should be from the transition from the highest occupied molecular orbital (with n=N+1) to the lowest unoccupied molecular orbital (with n=N+2). So the change in the energy will be: ( ) ( ) ( ) ( )( ) 22 2 2 16 22 2 2 3 2 1 10 8 8 1.39 2 2 h NhE N N x mL m N +⎡ ⎤Δ = + − + =⎣ ⎦ + . The absorption will take place at a wavelength of: ( )2 o2550 1 A 2 3 Nhc E N λ + = = Δ + See the next page for the comparison to actual data. After we get an expression for the oscillator strength from quantum considerations we will come back to these examples and give the comparison for that too. 6 The following table is taken from the literature on the series of compounds C and D (with different values of N). Wavelength of maximum absorbance for polymethene dyes N Observed o A Compound (C) Observed o A Compound (D) Calculated o A Equation 2 4250 - 3200 3 5600 - 4540 4 6500 5900 5800 5 7600 7100 7060 6 8700 8200 8330 7 9900 9300 9600 See K.F. Herzfeld and A.L. Skalar, Revs. Mod. Phys., 14, 294 (1942) and A.L Skalar, J. Chem. Phys., 10, 521 (1942). So we see that this very simple model is actually very good. Structure type B Now let’s look at dyes that are similar to structure B, which we give again here: Structure type B) C H C H C CC C N-2 ; see the table on next page. There is no nitrogen at the ends this time, and interestingly the model is not so good. But let’s look at it to see if we can figure out why the agreement is not so good. This is the central structure of the carotenoids, which are the molecules in plants that give the color to the leaves in the fall. The following table shows the absorption data for a series of carotenoids. The bonds are different for this case than for the molecules considered above. The double and single bonds alternate with the single bonds, and we do not have the resonance that we had before. Lets take the distances for the double and single bonds for butadiene, which are 1.35 o A and 1.46 o A respectively, and we assume that the bond structure representing the box extends a single bond beyond the terminal carbon atom. This will give a length of o 2.81( ) AL N= . Now we can calculate the wavelength of absorption to be ( )( ) o 22610 2 1 AN Nλ = + . The measured and calculated values are given in the following table. 7 N Obs. Maximum A o Calculated A o 1 1600 870 2 2170 2080 3 2600 3360 4 3020 4650 5 3460 5940 6 3690 7230 11 4510 13700 12 4750 15000 15 5040 18900 This data is from N.S. Bayliss, J. Chem. Phys., 16, 287 (1948). Obviously this is not a good representation, especially for the longer lengths. We will see later that the oscillator strengths according to this simple model are not so bad, compared to the measured values. Possible reason for the disagreement: H. Kuhn has given a possible reason for the lack of agreement between the measured values and the calculated values of the wavelengths for the carotenoids. It is interesting because it shows how subtle changes in the structures. The resonance properties of the dyes can cause major changes in the spectroscopic properties of the dyes. Essentially it is because there are no simple resonance structures with equivalent energies that make the bottom of the potential well flat. Any alternate resonance structure of the carotenoid structures has a higher energy than the lowest one shown above. We already pointed out that the alternating double and single bonds are assumed to have different lengths, and therefore when the electrons are in the area of the single bonds they will have a higher energy (the potential well floor at that point will be higher). Thus there is a “perturbation” of the Hamiltonian that Kuhn assumed to be of the form ( ) ( ) ( )( ) ( ) ( )( ) 0 0 constant sin 2 constant sin 2 1.35 1.35 1.48 sb sb dbV x V x V x π π = + − + = + − + Calculations show that the energy levels become divided into groups of N (where N is the number of double bonds), such that the energy difference between the levels in each group of N becomes smaller than in the case of a flat potential, and the energy difference between the different groups of energy levels (that is, between the group 1 to N, and the group N-1 to 2N) becomes larger. The first transition will be from the highest occupied to the lowest unoccupied level, and this is between the lowest group and the second group, and this energy difference becomes greater due to the corrugated sinusoidal potential. Kuhn found out that if V0 is about 2 electron volts, the calculated values agree well with experiment. Actually, the value of V0 to give the best agreement decreases slightly as the length of the molecule increases (2.55 eV for N=2, 2.03 eV for N=8, and 1.89 eV for N=17). Actually this, although it seems at first to be sort of arbitrary, agrees qualitatively with sophisticated calculations showing that the alternating bonds become more similar as the length of the conjugated system increases. There will be a homework problem dealing with this simple model. 10 Let’s see how this looks like for two levels: 0 01 2and E E , where 0 0 1 2E E< . How do they affect each other though a perturbation? 2 120 1 1 0 0 1 2 2 210 2 2 0 0 2 1 ' ' H E E E E H E E E E + − + − Because 212' 0H > , and 0 0 1 2E E< , thus 0 1 1E E< and 0 2 2E E> . That is the mixing tends to make the energy levels repel each other! Note how symmetry and selection rules enters here. ---------------------------------------------------------------------------------------------------------- Let’s go back to the particle in a box, and “do” a perturbation. The box has walls at x=0 and x=L. Say 1' VH x L = . We use the known solutions of a particle in a one-dimensional box: 2 2 28n n hE mL = and ( ) ( ) ( )0 2 sinn x L n x Lψ π= . We just calculate the matrix element: ( ) ( ) ( ) ( )10' 2 sin 2 sin L km VH L n x L x L n x L dx L π π⎛ ⎞= ⎜ ⎟ ⎝ ⎠∫ 1 1' 2mm H V= So, every level is raised by the same amount. ( ) ( ) ( ) ( )1 2 22 ' 0 when is even, or k,m both even or both odd. 2 1 1' when is odd, or k=odd and m=even, or visa versa km km H k m VH k m k m k mπ = − ⎡ ⎤ = − −⎢ ⎥ + −⎢ ⎥⎣ ⎦ Using this equation, as a homework you will find whether the electron spends more time in the left side or the right side of the box, when the electron is in the lowest state. Only use the first three eigenfunctions and show that this is enough. 11 Absorption Intensity Molar absoptivities may be very large for strongly absorbing chromophores (>10,000) and very small if absorption is weak (10 to 100). The magnitude of ε reflects both the size of the chromophore and the probability that light of a given wavelength will be absorbed when it strikes the chromophore. A general equation stating this relationship may be written as follows: ε = 0.87 • 1020 Ρ • a Ρ is the transition probability ( 0 to 1 ) & a is the chromophore area in cm2 ) The factors that influence transition probabilities are complex, and are treated by what spectroscopists refer to as "Selection Rules". A rigorous discussion of selection rules is beyond the scope of this text, but one obvious factor is the overlap of the orbitals involved in the electronic excitation. This is nicely illustrated by the two common transitions of an isolated carbonyl group. The n __> π* transition is lower in energy (λmax=290 nm) than the π __> π* transition (λmax=180 nm), but the ε of the former is a thousand times smaller than the latter. The spatial distribution of these orbitals suggests why this is so. As illustrated in the following diagram, the n-orbitals do not overlap at all well with the π* orbital, so the probability of this excitation is small. The π __> π* transition, on the other hand, involves orbitals that have significant overlap, and the probability is near 1.0. a* hw AE cee eee eee ee eee nontnnnige seeece eee sence sees ee eeeeeeeeeeeeeee nx” —®)—_®—_ a —}———_« cc my ee a*g —————@— a°3 hy see ce cee ece cesses estes eeeeee cece seeernnrtnnnened ABs ooo seeeeeeeeeeeeseeees a+ —&—_)— _ a —_®—___—__ &2 E221 My cac-c—e at xt, at a xy —————— 54 > hv Brrr rrr rrr renee erences wenmnomnoteccos son cclan BE sone s con ecn secon sce seas o +I uy —e—e— m3 83 ——_®—_®—-_ §2 ———_®—_®— 2 2 1 2 ®t CSc cS o-cS 12  15 Some interesting Dye References • The entries are organized alphabetically by first author. • Most of these are journal articles; some are books. • The source of this information is a database maintained by a research group using EndNote Plus 2.2 (see TM, http://www.niles.com/). CRC Handbook of Triarylmethane and Xanthene Dyes M. A. Ali, et al. 1990 Examination of Temperature Effects on the Lasing Characteristics of Rhodamine CW Dye Lasers Applied Optics F. López Arbeloa, et al. 1989 Influence of the molecular structure and the nature of the solvent on the absorption and fluorescence characteristics of rhodamines Chem Phys. F. López Arbeloa, et al. 1990 Molecular structure effects on the lasing properties of rhodamines J. Photochem. Photobiol. A: Chem. F. López Arbeloa, et al. 1989 Fluorescence self-quenching of the molecular forms of rhodamine B in aqueous and ethanolic solutions J. Lumin. F. López Arbeloa, et al. 1988 On the Aggregation of Rhodamine B in Ethanol I. López Arbeloa 1980 Fluorescence Quantum Yield Evaluation: Corrrections for Re-Absorption and Re- Emission J. Photochem. I. López Arbeloa, et al. 1982 Dimeric States of Rhodamine B I. Lopez Arbeloa, et al. 1981 Molecular Forms of Rhodamine B I. López Arbeloa, et al. 1986 Solvent effects on the photophysics of the molecular forms of rhodamine B. Internal conversion mechanism Chem. Phys. Letters I. López Arbeloa, et al. 1986 Solvent effects on the photophysics of the molecular forms of rhodamine B. Solvation models and spectroscopic parameters Chem. Phys. Letters T. B. Babaev, et al. 1970 Optical Characteristics of Monomers and Dimers of Dyes in Solution C. Bojarski, et al. 1977 Anti-Stokes Fluorescence in Rhodamine Acta Phys. Chem. 16 Solutions C. Bojarski, et al. 1978 Concentrational Quenching of Anti-Stokes Fluorescence in the Solutions of Rhodamin 6G C. Bojarski, et al. 1975 Resonance Quenching of Anti-Stokes Luminescence from Rhodamine B in Water Solutions Z. Naturforsch., A N. A. Borisevich, et al. 1964 On the Anti-Sokes Fluorescence of Molecules Optics and Spectroscopy A. Bujko, et al. 1983 Investigation of Anti-Stokes Fluorescence in Mixed Rhodamine 6G-Malachite Green Systems F. Cichos, et al. 1997 Solvation Dynamics in Mixtures of Polar and Nonpolar Solvents J. Phys. Chem. R. Fraser Code, et al. 1984 Electric-Field-Induced Modulation of Fluorescence from Rhodamine 610 Dye in a Thin Plastic Film Chem. Phys. V. V. Danilov, et al. 1973 Anti-Stokes Excitation of Luminescence of Dyes by High-Power Radiation F. J. Duarte 1995 Tunable Lasers Handbook Academic Press L. E. Erickson 1972 On Anti-Stokes Luminescence From Rhodamine 6G in Ethanol Solutions Tsuneo Fujii, et al. 1990 Absorption and fluorescence spectra of rhodamine B molecules encapsulated in silica gel networks and their thermal stability J. Photochem. Photobiol. A: Chem. D. M. Gakamsky, et al. 1992 Wavelength-Dependent Rotation of Dye Molecules in a Polar Solution Journal of Fluorescence Mariette E. Gál, et al. 1973 Derivation and Interpretation of the Spectra of Aggregates J. Chem. Soc. Farad. T. II J. Gardecki, et al. 1995 Ultrafast Measurements of the Dynamics of Solvation in Polar and Non-Dipolar Solvents Journal of Molecular Liquids Keith Goodling, et al. 1994 Luminescent Characterization of Sodium Dodecyl Sulfate Micellar Solution Properties J. Chem. Educ. P. R. Hammond 1979 Self-Absorption of Molecular Fluorescence, the Design of Equipment for Measurement of Fluorescence Decay, and the Decay Times of Some Laser Dyes J. Chem. Phys. Edmund N. Harvey 1957 A History of Luminescence American Physical Society Jerzy Karpiuk 1994 The Triplet States in Lactone Form of Rhodamine 101 J. Luminescence T. Karstens, et al. 1980 Rhodamine B and Rhodamine 101 as Reference J. Phys. Chem. 17 Substances for Fluorescence Quantum Yield Measurements L. P. Kazachenko 1965 A Consequence of the Universal Relationship between Absorption and Emission Spectra of Complex Molecules Optics and Spectroscopy Klaus Kemnitz, et al. 1991 Entropy-Driven Dimerization of Xanthene Dyes in Nonpolar Solution and Temperature Dependent Fluorescence Decay of Dimers J. Phys. Chem. B. Klabuhn, et al. 1973 Ein Computer-Verfahren zur Analyse von UV- Spektren Spectra. Acta L. Kozma, et al. 1966 The Yield of Anti-Stokes Fluorescence Optics and Spectroscopy Tomas Kurecsev 1978 Vibronic Analysis of the visible absorption and fluorescence spectra of the fluorescein dianion J. Chem. Educ. Gerald S. Levinson, et al. 1957 Electronic Spectra of Pyridocyanine Dyes with Assignments of Transitions ??? L. V. Levshin, et al. 1961 Dependence of the Association of Rhodamines on the Structure of Their Molecules and the Nature of the Solvent Optics and Spectroscopy F. E. Lytle 1982 Laser Excited Molecular Fluorescence of Solutions J. Chem. Educ. R. Mahiou, et al. 1990 Anti-Stokes Fluorescence of Gd3+ in K2GdF5 Journal of Luminescence Jacob Malkin 1992 Photophysical and Photochemical Properties of Aromatic Compounds CRC Press A. J. G. Mank, et al. 1995 Visible Diode Laser-Induced Fluorescence Detecion in Liquid Chromatography after Precolumn Derivitization of Amines Anal. Chem. Mark Maroncelli 1993 The Dynamics of Solvation in Polar Liquids J. Molecular Liquids A. R. Masri, et al. 1987 "Fluorescence" Interference with Raman Measurements in Nonpremixed Flames of Methane Combustion and Flame R. S. Moog, et al. 1982 Viscosity Dependence of the Rotational Reorientation of Rhodamine B in Mono- and Polyalcohols. Picosecond Transient Grating Experiments J. Phys. Chem. Douglas C. Neckars 1987 The Indian Happiness Wart in the Development of Photodynamic Action J. Chem. Educ. Alexander D. Osborne 1980 Internal Conversion in the Rhodamine Dye, Fast Acid Violet 2R J. C. S. Faraday II