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Electronic structure and spectroscopy Péter G. Szalay Institute of Chemistry, Eötvös Loránd University, Budapest Course material at: http://www.chem.elte.hu/departments/elmkem/szalay/szalay_files/Pavlodar/ 2013. október 21. 1 1. Today’s concept of the atom Shortly after Dalton’s atomic theory became widely accepted (matter consists of undividable particles called atoms) experiments by physicists indicated that the atoms indeed have some structure. 1.1. Discovery of electron Joseph John Thomson (1856-1940) His famous experiment with the cathode ray tube (1897): In a cathode ray tube, independently of the material of the cathode, the same event can be observed: light spots (flashes) appear on the screen, i.e. a particles leave the cathode and fly against the plate. In an electric field, this particle deviates towards the positive pole, i.e. it must have a negative charge. Its mass (measured by magnitude of the deviation) is much smaller than the mass of the atom (in fact Thomson meassured the ratio of mass and charge). He called it a „Corpuscle”, the name „electron” was later introduced by G. Johnstone Stoney (1853-1928), who worked with electricity and called the unit of electricity as electron. Thomson was also a bit skeptical: "Could anything at first sight seem more impractical than a body which is so small that its mass is an insignificant fraction of the mass of an atom of hydrogen?" (Thomson) He got the Nobel prize in 1906. But how the atom should look like? There is a positive charge with large mass and a tiny electron bearing the negative charge. It can be like a plum pudding; electrons are 2 the plums, and the pudding is the distributed positive charge. 1.2. Discovery of the nucleus Ernest Rutherford (1871-1937) Experiment: alpha particles originating from nuclear decay were used to bomb gold foil (Geiger és Mardsen): 3 The atom must have a nucleus which is much smaller than the atom and it is bearing the positive charge. „It is like to fire with a cannon on a sheet of paper and it would bounce back” Rutherford’s atomic model (1911): an atom consists of a positively charged small nucleus bearing almost all mass of it; the electrons are orbiting around it like the moon around the Earth. There are lots of unanswered question, though. For example, why the electron does not fall into the nucleus? (This system is different from the Earth-Moon system, since it has charges!!!) Today we know that even the nucleus has a structure: it consists of protons and neutrons, and even these can be divided into smaller elementary particles. For chemistry this is, however, not relevant, chemistry in most part can be described by a nucleus „surrounded” by the electrons. This is, on the other hand, already very relevant for chemical analysis! 4 2. Development of quantum mechanical view The atomic theory allowed the development of modern chemistry, but lots of questions remained unanswered, and in particular the WHY is not being explained: • What is the binding force between atoms. It is not the charge since atoms are neutral. Why can even two atoms of the same kind (like H-H) form a bond? • Why atoms can form molecules only with certain rates? • What is the reason of the periodic table of Mendeleev? At the turning of the 19th and 20st century new experiments appeared which could not be explained by the tools of the classical (Newtonian) mechanics. For the new theory new concepts were needed: • quantization: the energy can not have arbitrary value • particle-wave dualism ⇒ development of QUANTUM MECHANICS The new theory was developed along a long root (which in time was not that long at all!). We will follow this route now and stop at the most important steps. 2.1. Introduction: same basic terms related to light In the strict sense, „light” is a narrow range of electromagnetic radiation, what we can sense with our eyes. In physics very often the term „light” is used for the entire spectrum. The electromagnetic radiation consists of oscillating magnetic and electric fileds wich are perpedicular to each other and the direction of its propagation. 5 Basic terms: • ν: frequency of the oscillation [1/s] • ν ∗ : wavenumber [1/m] • λ: wave length [m] • c: speed of light • polarization: there is oscillation only in a plane Important relations: 6 Magspingerjesztés Ranges of the electromagnetic radiation: Ionizáció 1 λ Molekularezgések gerjesztése ν∗ = Molekulákforgásának gerjesztése c ν Elektrongerjesztés Maggerjesztések λ= What is spectroscopy? The matter can absorbe or emit light. The absorbed/emitted light can be divided into its components, and these will be characteristic for the matter. The light, thus, can be divided into its components, for example by a prism. 2.2. 2.2.1. Observations leading to quantum mechanics Black Body Radiation Planck in 1900 came up with a new, unusual explanation: according to his theory, the energy of the radiation is quantized, it can only be hν, 2hν, 3hν ..., thus it does not change continuously. Here h is the so called Planck constant: h = 6.626 · 10−34 Js (Planck himself did not like his own theory, since it required an assumption (postulate), i.e. the existence of constant h; he wanted to derive this from the existing theory. He was not successful with this; now we know it is not possible to derive since it follows from a new theory. Thus, despite of his genius discovery, he could not participate in further development of quantum mechanics. ) 2.2.2. Heat capacity According to the Dulong-Petit rule, the heat capacity is given by cv,m ≈ 3R, i.e. it is independent of temperature. This is valid at temperatures people could investigate until 7 the 19th century, but then it turned out that at low temperatures it goes to zero: Einstein explained this using Planck’s idea: the matter is also quantized, the oscillators (vibrations) can not have any energy, like the oscillators causing the black body radiation. (The final form of the theory with several oscillators was derived by Debye.) 2.2.3. Photoelectric effect Shining light on the metal plate can result in electric current in the circuit. However, there is a threshold frequency, below this there is no current, irrespective of the intensity of the light, i.e. • below the threshold frequency, no electron leaves the metal plate 8 • increasing the intensity of the light, the energy of the emitted electron does not change, only their number grows. According to the measurements, the following relation exists between the kinetic energy of the electron (Te ) and the frequency of the light (ν): Tel = hν − A where A depends on the quality of the metal plate (called work function). Explanation was given again by Einstein using the quantization introduced by Planck: the light consist of tiny particles which can have energy of hν only. (Note that Planck opposed the use of his „uncompleted” theory!!) 2.2.4. The Compton effect A photon collides with a resting electron, it looses energy. Therefore its frequency also changes. The photon acted as a particle in this experiment!! Note a wave scattering on an object would not change its wave length or frequency!!! 9 2.2.5. Scattering of electron beam Experiment by Davisson and Germer (1927), as well as G.P. Thomson (1928). There are interference circles on the photographic plate, just like in case of X-ray radiation → electron acted as a wave. 2.2.6. The hydrogen atom Hydrogen atom has four lines in his emission spectra in the visible range (experiment by Ångsröm): The position of the lines have been described by Balmer (so called Balmer formula): 1 1 1 = R 2− 2 λ 2 n 10 n = 3, 4, 5, 6 (1) where R is the so called Rydberg constant, λ is the wave length. The energy of the hydrogen atom must be quantized, too!! Explanation by Bohr: in his atomic model, the electron must fullfil some „quantum” relations: • in case of orbits having certain radius, the electron do not dissipate energy; these are the so called stationary states; • if the electron jumps from one orbit to the other, it emits (or absorbes) energy in form of electromagnetic field („light”). • the possible values for the energy are: E = − 1 e2 2n2 a0 n is real number (2) (e is the charge of the electron, a0 unit length (1 bohr)). Gives the Balmer formula back. HOMEWORK: SHOW THAT THIS IS TRUE. However, can not be applied for the He or any other atom!!! 2.2.7. Summary Event New term black body radiation energy quantized (hν) photoelectric effect energy of the light is quantized heat capacity at low temperature matter is quantized goes to zero Compton effect electromagnetic radiation acts like a particle scattering of electron electron acts like a wave 11 Discoverer Planck (1900) Einstein (1905) Einstein (1905), Debye Compton (1923) Davisson (1927), G.P. Thomson (1928) Remember: • ν is the frequency of light • λ is the wave length of light (λ = νc ) • c speed of light • h = 6.626 10−34 Js a Planck constant • h̄ = h 2π Important consequence of all these: particle-wave dualism (dual nature of the matter) The existing theories need to be revised completely! Although Bohr could „fix” this old theory with quantum condition to describe the hydrogen atom, but the theory does not work for other atoms! New theory: • Heisenberg (1925): Matrix mechanics • Schrödinger (1926): Wave mechanics It turned out later that the two theories are equivalent, they use only a slightly different mathematics. Now we call this theory as (non-relativistic) quantum mechanics. 12 2.3. 2.3.1. Basic concepts of quantum mechanics Operators What is an operator? It acts on a function and produces an other function: Âf (x) = g(x) There are special functions called eigenfunctions of an operator: if the operator acts on its eigenfunction, it results in a constant times the same function: Âf (x) = af (x) where a is a constant, the eigenvalue of the operator. Demonstration: Assume that d2 Â = dx2 Then cos(x) is an eigenfunction of this operator since: Â cos(x) = d2 cos(x) = − cos(x) dx2 its eigenvalue being −1. 2.3.2. Schrödinger-equation The stationary states of a system (e.g. atom, molecule) can be obtained by solving the so called (time independent) Schrödinger-equation: ĤΨ = EΨ (3) with: • Ĥ being the Hamilton operator of the system; • Ψ is the state function of the system; • E is the energy of the system. This is an eigenvalue equation, Ψ being the eigenfunction of Ĥ, E is the eigenvalue. This has to be solved in order to obtain the states of, e.g. molecules. According to Dirac (1929) the whole chemistry is included in this equation: P. A. M. Dirac, "Quantum Mechanics of Many-Electron Systems", Proceedings of the Royal Society of London, Series A, Vol. CXXIII (123), April 1929, pp 714.: „The general theory of quantum mechanics is now almost complete ... The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these equations leads to equations much too complicated to be soluble. It therefore becomes desireable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation." My own interpretation (2013) : 13 • to describe molecules one need quantum mechanics; • we need to develop methods which can give more and more accurate solutions to the Schrödinger equation • we also need approximate methods which support chemical intuition without expensive calculations. In practice: one has to approximate Ψ. Chemists are very good at this! 2.3.3. The Hamilton operator The Hamilton operator of the system (Ĥ) consists of the sum of the kinetic (T̂ ) and the potential energy (V̂ ) operators. Ĥ = T̂ + V̂ (4) the form of T̂ is the same for all systems, while the potential energy represents the molecule by including the interactions between the electrons and nuclei. 2.3.4. State function In quantum mechanics the state of the system is represented by the wave function (or state function) which depends on the coordinates of the particles: Ψ = Ψ(x, y, z) = Ψ(r) (5) Ψ = Ψ(x1 , y1 , z1 , x2 , y2 , z2 , ..., xn , yn , zn ) = Ψ(r1 , r2 , ..., rn ) (6) or in case of n particles: The wave function has no physical meaning, but its square, the so called probability density can be given a probability interpretation: Ψ∗ (x0 , y 0 , z 0 ) · Ψ(x0 , y 0 , z 0 )dx dy dz (7) is the probability of finding a particle at point (x0 , y 0 , z 0 ) (more precisely in the infinitesimal proximity). Shorter notation: Ψ∗ Ψdv or |Ψ|2 dv We have to chose the wave function normalized, otherwise the valószínűség of finding the particle in the whole space would not be one: Z Z Z Ψ∗ · Ψ dx dy dz = 1 14 (8) 2.3.5. Other physical quantities Other physical properties are also described by operators. Most important operators are: • position operator: x̂ (x̂f (x) = xf (x)) ∂ • momentum operator: p̂x = −ih̄ ∂x • kinetic energy operator: T̂ = 1 2 p̂ 2m 2 2 h̄ d 1 = − 2m ≡ − 2m ∆ dx2 • angular momentum operator: ˆl = (ˆlx , ˆly , ˆlz ) • ... Operators of position (x̂) and momentum (p̂x ) do not commute, meaning that [x̂, p̂x ] = x̂p̂x − p̂x x̂ = −ih̄ (9) which means that the two operators can not be interchanged (the results depends on, which operator acts first). Consequence: the two quantities (coordinate and momentum) can be measured in the same time, the product of their uncertainty (∆x and ∆px must be larger the a given value: 1 h̄ 2 This is the famous Heisenberg uncertainty principle. ∆x · ∆px ≥ 2.3.6. (10) „The particle in the box” model This is a very instructive model system which shows nicely the new properties of quantum objects: Hamiltonian: 15 V (x) = 0, 0 < x < L V (x) = ∞, otherwise Within the box of length L the Hamiltonian is equal to the kinetic energy: Ĥ = T̂ +V (x), | {z } 0 The particle can not leave the box, the probability finding it outside the box is zero, therefore the wave function must also vanish there. The keep the wave function continuos, it has to vanish at the walls, as well (boundary condition): Ψ(0) = Ψ(L) = 0 (11) Therefore the Schrödinger equation to solve reads: T̂ Ψ(x) = EΨ(x) (12) After a short (and instructive) calculation one gets the following result: h2 ; n = 1, 2, ... 2 s 8mL 2 π Ψ(x) = sin n x L L E = n2 · The form of the wave function: 16 Notes: • The energy is quantized, it grows quadratically with the quantum number n, it is invers proportional to L2 and m. If L → ∞, E2 − E1 ∼ L = ∞. 22 −12 L2 → 0. This means, the quantization disappears with The same is true for growing mass m → ∞. • There is a zero point energy (ZPE)! The energy is not 0 for the lowest level (ground state). If, however, L → ∞, E0 → 0. Why is ZPE there? This is an unknown term for classical mechanics! It can be explained by the uncertainty principle: ∆x · ∆p ≥ 21 h̄. Since here we have V̂ = 0, E ∼ p2 , i.e. the energy of the particle originates in his momentum only. Assume that E = 0, than p = 0, therefore ∆x = ∞, which is a contradiction since ∆x ≤ L, the particle must be in the box. We conclude that the energy can never get zero, since in this case its uncertainty would also be zero which is possible only for very large box where the uncertainty of the coordinate is large. Or alternatively, one can also say: if L → 0 =⇒ ∆x → 0 =⇒ ∆p → ∞ =⇒ ∆E → ∞. This means the energy of all levels MUST BE larger and larger if the size of the box gets smaller. • Wave function: the larger n is the more nodes are on the wave function: ground state has none, first excited state has one, etc. (Node: where the wave function changes sign). • How does the solution looks like in 3D? π 2 h̄2 E = 2m n2a n2b n2c + 2 + 2 , a2 b c ! where a, b, c are the three measures of the box and na , nb , nc = 1, 2, ... are the the quantum numbers. If a = b = L, then na 1 2 1 nb 1 1 2 E h2 8mL2 2 5 5 We have found degeneracy which is caused by symmetry of the system (two measures are the same). 17