Quantum Chemistry PDF

Quantum Chemistry  Evolution of Quantum mechanics 1922: 1925: 1940: Heisenberg Pauli proves...

Quantum Chemistry  Evolution of Quantum mechanics 1922: 1925: 1940: Heisenberg Pauli proves Neils Bohr Uncertainty Spin-Statistic won Nobel principle Theorem Prize in physics Wolfgang Pauli spin number 1932: 1944: 1918: Werner Heisenberg 1905: Max Planck won won Nobel 1936: Pauli won Nobel Prize in Nobel prize in Prize in Schrödinger's Albert proposed the physics physics physics Equation idea photoelectric effect. 1924: 1929: 1933: 1943: 1921: Louis De Broglie 1900: 1913: Won the Nobel electron waves. Louis De Otto Stern Broglie won measures the Otto Stern Nobel Prize in Physics prize in Physics Max Planck the Nobel magnetic Neils Bohr black-body photoelectric Prize. moment of the atomic radiation effect. proton structure  count rate of incident particles Interpretation: Rutherford’s Atomic Model · Since the vast Ernest Rutherford (Cambridge majority of aEngland, University, 1871-1937) particles pass studied through the Au α emission foil undeflected, the from newly- Au atoms are mostly discovered radioactive elements. · "It was A very tiny percentage of a particles almost as massive hit something incredible as if in the youand atom fired a fifteen-inch shell at a piece indicates backscatter (bounce back). This observation of tissuethat paper and most it came of the mass back of and hit you" the atom is concentrated in a very small volume relative to the volume of the entire atom. We now call this the NUCLEUS. · New model for a particles · Rutherford proposed the charge on the nucleus to be positive, since electrons are negatively charged and atom s are neutral. Charge of the electrons in an atom = - Ze Z = atomic number e = absolute value of electron’s charge Charge of the nucleus of an atom = · Rutherford calculated the diameter of the nucleus to be 10 m. His calculation (see below) related the probability of backscattering to the diameter of the nucleus based on the size of the atom and the thickness of the foil.  Rutherford’s Atomic Model Drawback According to classical theory, atoms in the model for Rutherford’s are not stable. The motion of moving electrons should cause them to radiate energy in their orbits and to quickly execute a death spiral and collapse into the nucleus. The time taken for this collapse is estimated around 10−8 s, which is very small. According to the model, every atom will crumple in 10−8 s, but we know this is not reality. If the electrons are not revolving but stationary, still the electrons will fall into the nucleus by the electrostatic force between both. The predicted death spiral of the electron! Hydrogen Atomic Spectrum  Hydrogen Atomic Spectrum When a high-energy discharge is passed through a sample of H2 gas, the H2 molecules absorb energy, causing some of the H-H bonds to break. The resulting hydrogen atoms are excited; they contain excess energy, which they release by emitting light of various wavelengths to produce what is called the emission spectrum of the hydrogen atom., called a line spectrum.  Do the electrons in atoms and molecules obey Newton’s classical laws of motion? It turns out that the laws of classical mechanics no longer work at this size scale. A new kind of mechanics was needed to describe this and other “unsettling” observations.  Classical Mechanics Classically, particles and waves are distinct: – Particles – characterised by position, mass, velocity. – Waves – characterised by wavelength, frequency. 𝜆𝜈 = 𝑐  Classical Mechanics By the 1920s, however, it was becoming apparent that sometimes matter (classically particles) can behave like waves and radiation (classically waves) can behave like particles.  The Photoelectric Effect A beam of light hitting a metal surface can cause electrons to be ejected from the surface. Classical Paradigm: the energy of the ejected electrons should be proportional to the intensity (I) of the light and independent of the frequency (𝝂) of the light. Experiment: the energy of the ejected electrons is independent of the intensity (I) and depends directly on the frequency (𝝂) of the light.  The Photoelectric Effect The following observations characterize the photoelectric effect. 1. No electrons are emitted by a given metal below a specific threshold frequency 𝝂0. 2. For light with frequency lower than the 𝝂0, no electrons are emitted regardless of the intensity of the light. For light with frequency greater than the 𝝂0, the kinetic energy of the emitted electrons increases linearly with the frequency of the light. 3. The number of electrons emitted per second (i.e. the electric current) is independent of light frequency above the 𝝂0 and zero below. 4.For light with frequency greater than the 𝝂0, the number of electrons emitted increases with the intensity of the light. These data were in direct opposition to theQuantization of Light predictions of classical m echanics. In 1905 Einstein analyzed plots of K.E. as a function of frequency for different m etals and found that all of the data fit into a linear form Rb K Na K.E. n0(Rb) n0(K) n0(Na) n y = mx + b -hn0(Rb) The (mof KE vs is h slope slope -hn0(K) 6.626 x 10-34 Js = Planck’s constant = -hn0(Na) y-intercept (b) = - hn0 Einstein could rewrite the equation of the line: Workfunction, ɸ y = mx + b  Quantization of Light These observations can be explained by assuming that electromagnetic radiation is quantized (consists of photons) can describeand that the threshold frequency represents the minimum this mathematically:. energy required or to remove E the electron from the metal’s i surface. te: these are just different form s of the equation K.E = hn - hn ) Minimum energy required to remove an electron, E0 = h0 0 s try some example problems. # of electrons ejected from the surface of a metal is proportional to the hotons absorbed by the metal and not the energy of the photons (assum ing Ei ≥ f). nhc E= (n = 0,1, 2,3,.......)  us, the intensity (I) of the light (energy/ sec) is proportional to the # of photons orbed/ sec and the # of electrons em itted/ sec _____________________________________________________________________________ Topics: I. Light as a particle continued II. Matter as a wave Quantization of Light III. The Schrödinger equation _____________________________________________________________________________ I. LIGHT AS A PARTICLE CONTINUED A) More on the Photoelectric Effect e- ejected e- ejected e- NOT ejected Three photons, each with an energy equal to f/ 2 eject an electron! Waves behaving as Particles Terminology tips to help solve problems involving photons and electrons: · photons: also called light, electromagnetic radiation, etc.  Example What is the energy in joules and electron volts of a photon of 420-nm violet light? What is the maximum kinetic energy of electrons ejected from calcium by 420-nm violet light, given that the workfunction for calcium metal is 2.71 eV?  Example What is the longest-wavelength electromagnetic radiation that can eject a photoelectron from silver? Is this in the visible range? Given that the workfunction of silver is 4.72 eV. Only photons with wavelengths lower than 263 nm will induce photoelectrons. This is ultraviolet and not in the visible range.  Wave Nature of Particles Louis de Broglie postulated that as light has wave-like and particle-like properties, matter must also be both particle-like and wave-like. A particle, of mass m, travelling at velocity v, has linear momentum p = mv. By analogy with photons, the associated wavelength of the particle () is given by: h h de Broglie wavelengthλ = = de Broglie’s equation p mv The fact that particles can behave as waves but also as particles, depending on which experiment you perform on them, is known as the  Bullets are far more massive than the electrons. One can observe them as long as one likes but it would not make any difference to them. The interference wiggles in the case of bullets are so crowded that it is physically impossible to resolve them, one sees an average behaviour The wave properties of matter are only apparent for very small masses of matter.  Wave Nature of Particles: Validation According to classical physics, electrons should behave like particles - they travel in straight lines and do not curve in flight unless acted on by an external agent. In this model, if we fire a beam of electrons through a double slit onto a detector, we should get two bands of "hits”. Can be explained only in terms of waves.  Wave Nature of Particles: Validation Davisson and Germer showed that a beam of electrons could be diffracted from the surface of a nickel crystal. Scattered light can interfere constructively (the peaks and troughs of the beams are in phase) to produce a bright area or destructively (the peaks and troughs are out of phase) to produce a dark spot. A diffraction pattern can be explained only in terms of waves. NB: Other “particles” (e.g. neutrons, protons, He atoms) can also be diffracted by crystals.  Wave-Particle Duality Now we have come full circle. Electromagnetic radiation, which at the turn of the twentieth century was thought to be a pure waveform, was found to exhibit particulate properties. Conversely, electrons, which were thought to be particles, were found to have a wavelength associated with them. Energy is really a form of matter, and all matter shows the same types of properties. All matter exhibits both particulate and wave properties.  Putting it All together: The Bohr Atom E4 Bohr proposed a model that included the idea that the E3 E5 E2 electron in a hydrogen atom moves around the nucleus only E1 in certain allowed circular orbits. Bohr assumed that the hydrogen electron could exist only in stationary, non-radiating orbits.  Putting it All together: The Bohr Atom Solving Paradox I: The unstable orbiting electron of Rutherford’s atom. Make the orbiting electron stable by assuming that the electron’s orbit and energy is quantized to certain values and for these values the orbiting electron does not radiate. The electron is stable in these orbits. Thus, the orbits and energies of electrons are quantized. Solving Paradox II: The line spectra of emitting or absorbing atoms. Since only certain energies are allowed for orbiting electrons, only jumps between orbits can be observed. These jumps correspond to discrete frequencies and wavelengths. Thus, line spectra as expected because of the quantized energies of the orbits.  In the Bohr atom, the circular symmetry and the wave property of the electron requires that the electron waves have an integer number of wavelengths. If not, then the waves will overlap imperfectly and cancel out (i.e., the electron will cease to exist). https://www.desmos.com/calculator/gazhf6fafx  The Bohr Atom Light is emitted when an electron jumps from a higher orbit to a lower orbit and absorbed when it jumps from a lower to higher orbit. The energy and frequency of light emitted or absorbed is given by the difference between the two orbit energies, e.g., E(photon) = E2 - E1 (Energy difference) = h𝝂 𝒉𝒄 𝜟𝑬 = 𝒉𝝂 = 𝝀  The Bohr Atom E E2 - E1 = h E 4 3 E5 E2 E1 5 4 3 Photon Absorbed 2 1 410.1 434.0 486.1 656.3 nm nm nm nm Bohr atom: Light absorption occurs when an electron absorbs a photon and makes a transition for a lower energy orbital to a higher energy orbital. Absorption spectra appear as sharp lines.  The Bohr Atom E2 - E1 = h E4 E3 E5 E2 5 E1 4 3 Photon 2 Emitted 1 410.1 434.0 486.1 656.3 nm nm nm nm Bohr atom: Light emission occurs when an electron makes a transition from a higher energy orbital to a lower energy orbital and a photon is emitted. Emission spectra appear as sharp lines.  The Bohr Atom Limitations of Bohr Model At first, Bohr’s model appeared to be very promising. The energy levels calculated by Bohr closely agreed with the values obtained from the hydrogen emission spectrum. However, when Bohr’s model was applied to atoms other than hydrogen, it did not work at all The model only works for hydrogen (and other one electron ions) – ignores e-e repulsion. Does not explain fine structure of spectral lines. Note: The Bohr model (assuming circular electron orbits) is fundamentally incorrect.  Heisenberg’s Uncertainty Principle Bohr orbit: The electron in the orbital is moving around the nucleus in a circular orbit. We can predict the motion of the electron, similar to the motions of particles in the macroscopic world. For example, when two billiard balls with known velocities collide, we can predict their motions after the collision. Werner Heisenberg, discovered a very important principle in 1927—the Heisenberg uncertainty principle.  Heisenberg’s Uncertainty Principle The Heisenberg Uncertainty Principle is a fundamental theory in quantum mechanics that defines why a scientist cannot measure multiple quantum variables simultaneously. Heisenberg made the bold proposition that there is a lower limit to this precision making our knowledge of a particle inherently uncertain. There is a fundamental limitation to just how precisely we can know both the position and the momentum of a particle at a given time. ℏ = ℎ/2𝜋 where Δx is the uncertainty in a particle’s position, Δp is the uncertainty in a particle’s momentum. Note: There is no restriction on the precision in simultaneously knowing/measuring the position along a given direction (x) and the momentum along another, perpendicular direction (z): Δpz. Δx = 0  Making Sense of the Uncertainty Principle To localize a particle in space (i.e. to specify the particle’s position accurately, small Δx) many waves of different wavelengths () must be superimposed  large Δpx (p = ℎ/). The Uncertainty Principle imposes a fundamental (not experimental) limitation on how precisely we can know (or determine) various observables. https://www.youtube.com/watch?v=KT7xJ0tjB4A An electron is confined to the size of a magnesium atom with a 150 pm radius. What is the minimum uncertainty in its velocity? Determine the uncertainty in the position of the ball (163 gm) bowled at with velocity of 111 kph. The hydrogen atom has a radius on the order of 0.05 nm. Assuming that we know the position of an electron to an accuracy of 1% of the hydrogen radius, calculate the uncertainty in the velocity of the electron using the Heisenberg uncertainty principle. Then compare this value with the uncertainty in the velocity of a ball of mass 0.2 kg and radius 0.05 m whose position is known to an accuracy of 1% of its radius. Thus the uncertainty principle is negligible in the world of macroscopic objects but is very important for objects with small masses, such as the electron.  This limitation is so small for large particles such as baseballs or billiard balls that it is unnoticed. However, for a small particle such as the electron, the limitation becomes quite important. Applied to the electron, the uncertainty principle implies that we cannot know the exact path of the electron as it moves around the nucleus. It is therefore not appropriate to assume that the electron is moving around the nucleus in a well-defined orbit as in the Bohr model. Q1: What is the wavelength of electromagnetic radiation that ejects 2.00-eV electrons from calcium metal, given that the work function is 2.71 eV? What type of electromagnetic radiation is this? Q2: An unknown elemental metal has work function of 8.01x10-19 J. Upon illumination with UV light of wavelength 162 nm, electrons are ejected with velocity of at 3.5 x 104 m/s. What is the threshold wavelength? What is the work function in units of eV? Q3: If the position of an electron is within the 10 pm interval, what is the uncertainty of the momentum? Q4: If we know the velocity of an electron to within 3.5 x 10 7 m/s, then what is the uncertainty in its position?  Quantum Mechanics (Wave Mechanics) To Schrödinger and de Broglie, the electron bound to the nucleus seemed similar to a standing wave, and they began research on a wave mechanical description of the atom. The wave model was applied by de Broglie to the Bohr atom by imagining the electron in the hydrogen atom to be a standing wave. Only certain circular orbits have a circumference into which a whole number of wavelengths of the standing electron wave will “fit.” All other orbits would produce destructive interference of the standing electron wave and are not allowed. Quantum Mechanics Schrödinger: If electrons are waves, their position and motion in space must obey a wave equation. Solutions of wave equations yield wavefunctions, , which contain the information required to describe ALL of the properties of the wave.  The Schrödinger Equation (1926) The central equation in Quantum Mechanics. Observable = total energy of system. Schrödinger Equation Ĥψ = Eψ Ĥ Hamiltonian Operator E Total Energy where ˆ =T H ˆ +V ˆ and E = T + V. SE can be set up for any physical system. The form of Ĥ depends on the system. Solve SE   and corresponding E.  Time-Dependent Schrodinger Wave Equation Total E term K.E. term P.E. term PHYSICS NOTATION Time-Independent Schrodinger Wave Equation  Wavefunctions In QM, a “particle” is distributed in space like a wave. We cannot define a position for the particle. Instead we define a probability of finding the particle at any point in space. Wavefunction () – a wave representing the spatial distribution of a “particle”. Represents the amplitude of the still vaguely defined matter waves (x) – the magnitude of  at point x. Because  may be imaginary or complex  2 would be negative or complex. For the general case, where  is complex ( = a + ib) then: 2 = * where * is the complex conjugate of . i = −1 probability density (* = a – ib)  Particle Wavefunctions: Examples Examples of (x) y(x) y(x) y(x) x x x The corresponding probability distributions |(x)|2 of these states are: |y|2 |y|2 |y|2 x x x Key point: Particle cannot be associated with a specific location x.  Probability distribution Difference between classical and quantum cases Classical Quantum U(x) U(x) E E x x P(x) P(x) = 2 x x  Wavefunctions The Importance of   completely defines the system (e.g. electron in an atom or molecule). If  is known, we can determine any observable property (e.g. energy, vibrational frequencies, …) of the system. QM provides the tools to determine  computationally, to interpret  and to use  to determine properties of the system.  Wavefunctions The Born Interpretation (1926) “The square of the wavefunction at any point in space is proportional to the probability of finding the particle at that point.” Example: 1-D System If the wavefunction at point x is (x), the probability of finding the particle in the infinitesimally small region (dx) between x and x+dx is: P(x)  (x)2 dx  Normalization of the Wavefunction P(r)  (r)2 d What is the proportionality constant? If  is such that the sum of (r)2 at all points in space = 1, then: P(x) = (x)2 dx 1-D P(r) = (r)2 d 3-D As summing over an infinite number of infinitesimal steps = integration, this means:  2 Ptotal (1D ) =  ψ(x ) dx = 1 −  2    2 Ptotal (3D ) =  ψ(r ) dτ =    ψ(x, y, z ) dxdydz = 1 − − − − i.e. the probability that the particle is somewhere in space = 1. In this case,  is said to be a normalized wavefunction. How to Normalize the Wavefunction 2 If  is not normalized, then:  ψ(r ) dτ = A  1 A corresponding normalized wavefunction (Norm) can be defined: ψ Norm (r ) = ψ(r ) 1 A such that: 2  ψ Norm (r ) dτ = 1 The factor (1/A) is known as the normalization constant (sometimes represented by N).  Hamiltonian Operator Wavefunction () contains all the information we need to know about any particular system. How do we determine  and use it to deduce properties of the system? Operators and Observables If  is the wavefunction representing a system, we can write: Q̂ψ = Qψ where Q – “observable” property of system (e.g. energy, momentum, dipole moment …) Q̂ – operator corresponding to observable Q. d/dx (eax) = a eax d2/dx2 (sin ax) = −a2 sin ax 𝑥ො = 𝑥 Operator for position in the x- Operator for linear momentum in the x-direction: 𝜕 𝑝ෝ𝑥 = −𝑖ℏ direction is just multiplication by x 𝜕𝑥 Constructing Kinetic and Potential Energy QM Operators 1. Write down classical expression in terms of position and momentum. 2. Introduce QM operators for position and momentum. Examples 1. Kinetic Energy Operator in 1-D T̂x px2 ˆ x2 ˆT = p  2  d 2   CM Tx =  QM =− 2m  dx 2  x 2m 2m  2. KE Operator in 3-D T̂ “del-squared” p2 px + py + pz 2 2 2 ˆ2 ˆT = p  2   2 2 2  =−  2 T= = =− + + 2 CM 2m 2m QM 2m 2m  x 2 y 2 z 2   2m partial derivatives operate on (x,y,z) 3. Potential Energy Operator V̂ (a function of position)  PE operator corresponds to multiplication by V(x), V(x,y,z) etc. Examples 1. Particle Moving in 1-D (x) ˆ ψ=T ˆ ψ+V ˆ ψ = Eψ  2   2 ψ  H − + V(x )ψ = Eψ 2m  x 2  The form of V(x) depends on the physical situation: – Free particle V(x) = 0 for all x. – Harmonic oscillator V(x) = ½kx2 2. Particle Moving in 3-D (x,y,z)  2   2 ψ  2 ψ  2 ψ  SE  − + + + V(x, y, z )ψ = Eψ 2m  x 2 y 2 z 2  2 2 or −  ψ + V(x, y, z )ψ = Eψ Note: The SE is a second 2m order differential equation  Particle in a Box model Consider a particle with mass m that is free to move back and forth along one dimension between the values x = 0 and x = L. The potential energy V(x) of the particle is zero at all points along its path, except at the endpoints x = 0 and x = L, where V(x) is infinitely large. Schrödinger equation contains the energy operator Ĥ. The potential energy is zero inside the box, the only energy possible is the kinetic energy of the particle as it moves back and forth along the x axis. →  Particle in a Box model Each solution must be a function whose second derivative has the same form as the original function. One function that behaves this way is the sine function.   2   2 ψ  − = Eψ 2m  x 2  This is a second order differential equation – with general solutions of the form:  = A sin kx + B cos kx   2ψ    = −k 2 (A sin kx + B cos kx ) = −k 2 ψ  x 2      2   2 ψ    2  ( ) SE − = −  − k 2 ψ = Eψ 2m  x 2   2m   k 2 2 (i.e. E depends on k). E= 2m Restrictions on  In principle Schrödinger Eqn. has an infinite number of solutions. So far we have general solutions: – any value of {A, B, k} → any value of {,E}. BUT – due to the Born interpretation of , only certain values of  are physically acceptable: – outside box (xL) V =   impossible for particle to be outside the box   2 = 0   = 0 outside box. –  must be a continuous function  Boundary Conditions  = 0 at x = 0  = 0 at x = L. Effect of Boundary Conditions 1. x = 0  = A sin kx + B cos kx = B 0 1 =0 B=0   = A sin kx for all x 2. x = L  = A sin kL = 0 A=0 ?  sin kL = 0 ?  (or  = 0 for all x) sin kL = 0  kL = n n = 1, 2, 3, … (n  0, or  = 0 for all x) Allowed Wavefunctions and Energies k is restricted to a discrete set of values: k = n/L Allowed wavefunctions: n = A sin(nx/L)  nx  Normalization: A = (2/L)  ψn = 2 L sin    L  Allowed Wavefunctions and Energies  nx  Normalization: A = (2/L)  ψn = 2 L sin    L  k 2 2 n 2 π 2 2 Allowed energies: En = = 2m 2mL2 n 2h2  En = 8mL2 The application of the boundary conditions has led to a series of quantized energy levels. That is, only certain energies are allowed for the particle bound in the box. Quantum Numbers There is a discrete energy state (En), corresponding to a discrete wavefunction (n), for each integer value of n. Quantization – occurs due to boundary conditions and requirement for  to be physically reasonable (Born interpretation). n is a Quantum Number – labels each allowed state (n) of the system and determines its energy (En). Knowing n, we can calculate n and En. Properties of the Wavefunction nx  Wavefunctions are standing waves: ψn = 2 L sin    L  The first 5 normalized wavefunctions for the particle in the 1-D box: Successive functions possess one more half-wave ( they have a shorter wavelength). Nodes in the wavefunction – points at which n = 0 (excluding the ends which are constrained to be zero). Number of nodes = (n-1) 1 → 0; 2 → 1; 3 → 2 …  n 2h2 Energies En = 8mL2 En  n2/L2  En as n (more nodes in n) En as L (shorter box) n (or L)  curvature of n  KE  En E  E 2 node L 1 L1 L2  En  n2  energy levels get further apart as n n E 9h 2 3 E3 = 8mL2 4h 2 2 E2 = 8mL2 h2 1 E1 = h2 8mL 2 ZPE = 0 8mL2 Zero-Point Energy (ZPE) – lowest energy of particle in box: h2 ZPE = E min = E 1 = 8mL2 CM Emin = 0 QM E = 0 corresponds to  = 0 everywhere (forbidden). Density Distribution of the Particle in the 1-D Box The probability of finding the particle between x and x+dx (in the state represented by n) is: Pn(x) = n(x)2 dx = (n(x))2 dx (n is real)  nx   Pn (x ) = 2 L sin 2  dx  L  Note: probability is not uniform – n2 = 0 at walls (x = 0, L) for all n. – n2 = 0 at nodes (where n = 0). Assume that an electron is confined to a one-dimensional box 1.50 nm in length. Calculate the lowest three energy levels for this electron, and calculate the wavelength of light necessary to promote the electron from the ground state to the first excited state. Particle in a 2-D Square or 3-D Cubic Box Similar to 1-D case, but  → (x,y) or (x,y,z). Solutions are now defined by 2 or 3 quantum numbers e.g. [n,m, En,m]; [n,m,l, En,m,l]. Wavefunctions can be represented as contour plots in 2-D  Consider 3-D motion of the electron (-e) relative to the nucleus (+Ze): Hamiltonian Operator ˆ =T H ˆ +V ˆ 2 2 me  m N Kinetic Energy T̂ = −  where =  me 2 me + m N ˆ = V(r ) = − Ze 2 Potential Energy V where 4 0 r r = x2 + y2 + z2 (electrostatic attraction between electron and nucleus). -e    2  2 2  me 2 =  2 + 2 + 2  r  x y z    +Ze mN  Hydrogen Atom The series of atoms/ions H, He+, Li2+, Be3+ … all have: 1 electron (charge –e) nucleus (charge +Ze) To determine the electronic wavefunction ( = e) and allowed electronic energy levels (E), we must set up and solve the Schrödinger Equation for a single electron in an atom. -e me +Ze mN  We must solve the 2nd order differential equation: 2 2 −  ψ + V(r )ψ = Eψ 2 Because of the spherical symmetry of the atom, it is convenient to describe the position of the electron in spherical polar coordinates (r,,), rather than Cartesians (x,y,z). The Schrödinger Equation in Spherical Coordinates The wave function  is a function of r, , . This is a potentially complicated function. Assume instead that  is separable, that is, a product of three functions, each of one variable only:  Due to this spherical symmetry, the wavefunction can be separated into a product of radial and angular components: (r,,) = R(r).Y(,) Imposing boundary conditions  3 quantum numbers (n, , m)  n,,m(r,,) = Rn,(r).Y,m(,) angular radial wavefunction wavefunction (spherical harmonic) Quantum Numbers n,,m(r,,) = Rn,(r).Y,m(,)  depends on 3 quantum numbers (n, , m). 1. Principal Quantum Number (n) Positive integer n = 1, 2, 3 … For H and 1-e ions, the electron energy depends only on n  e 4   Z2  E n = − 2 2    8h    n2   0    2. Orbital Angular Momentum Quantum Number () For a given value of n,  can take the integer values:  = 0, 1, 2 … (n-1) e.g. n = 1 =0 n=2  = 0, 1 n=3  = 0, 1, 2 3. Orbital Magnetic Quantum Number (m) For a given value of , m can take the integer values: m = 0, 1, 2 …   e.g.  = 0 m = 0 =1 m = 0, 1 There are (2+1) allowed values of m. Atomic Orbitals The wavefunctions (n,,m) describing an electron in an atom are known as atomic orbitals. Each atomic orbital (1-e wavefunction) is uniquely defined by 3 quantum numbers (n,,m). The orbital () gives the spatial distribution of the electron (via the Born interpretation of ||2). Orbitals are often drawn as 3-D surfaces which enclose approx. 90% of the probability of finding the electron.  Quantum Numbers and Atomic Orbitals Orbitals differ from each other in size, shape and orientation: – the orbital size is defined by the principle quantum number n – the type of orbital (it’s shape) is defined by the angular momentum quantum number  – the orientation of the orbital is defined by the orbital magnetic quantum number m  Shape of Atomic Orbitals Shape – determined by the angular wavefunction Y,m(,) 1s orbital (n = 1,  = 0) 32 1  1  Normalized wavefunction: (1s ) =     e −r a 0   a0  Has no angular dependence (spherically symmetric, depends only on r). + All s orbitals are spherically symmetrical. 2s orbital ns orbital (2s )  (2 − r ) e − r 2 (ns )  f (r ) e − r n As n, orbitals expand – average radius r 2p orbitals (n = 2,  = 1) There are 3 degenerate orbitals, with 3 different m values (0,1). e.g. m = 0  2pz orbital (pointing along z-axis) 52  1  + (2p z ) = 1      rcos  e − r 2a 0 - + 4 2  a 0  - m = 1  2px, 2py orbitals - + Atomic Orbitals The p orbitals: Three orientations: l = 1 (as required for a p orbital) ml = –1, 0, +1 Atomic Orbitals The d orbitals: Five orientations: l = 2 (as required for a d orbital) ml = –2, –1, 0, +1, +2 Electronic Shells, Sub-shells and Orbitals All orbitals with the same value of n together constitute an electronic shell. n 1 2 3 4 Shell K L M N For each n, orbitals with the same value of  together constitute an electronic sub-shell. Each sub-shell consists of (2+1) orbitals, each with a different m value.  0 1 2 3 Sub-shell s p d f No. orbitals 1 3 5 7  Possible combinations of shells and sub-shells. → 0 1 2 3 … n 1 1s 2 2s 2p 3 3s 3p 3d 4 4s 4p 4d 4f … Total number of orbitals in shell n is n2.  Electronic Shells, Sub-shells and Orbitals 3d orbitals l=2 ml =2 ml =1 ml =0 ml =−1 ml =−2 n=3 3p orbitals l=1 ml =1 ml =0 ml =−1 3s orbital l=0 ml =0 shell sub-shells orbitals  Atomic Orbital Energy Diagram for H Orbital energies depend only on the principal quantum number n. For a given n value (shell), all sub-shells () and orbitals (m) have the same energy – i.e. they are degenerate.  Electron Spin Stern and Gerlach (1921) observed that a beam of Ag atoms is split into 2 beams by an inhomogeneous magnetic field. Dirac (1930) introduced relativistic effects into Quantum Mechanics and showed that to completely describe the state of an electron we must specify: 1. The orbital (n,,m) 2. The spin state of the electron Electron spin is characterised by 2 quantum numbers: – spin angular momentum q. no. s ( = ½ for all electrons) – spin magnetic q. no. ms ( = ½).  Spin angular momentum (S) has magnitude: 3 S = s(s + 1) = 2 The projection of S on the z-axis: Sz = ms = ½  There are two possible electron spin states: ms = +½ “spin up”  ms = -½ “spin down”  Spin is an intrinsic property of the electron and is not connected with orbital motion. Complete specification of an electron in an atom requires 4 quantum numbers (n,,m,ms) – as s is fixed.  The fourth quantum number: Electron Spin ms = +1/2 (spin up) or -1/2 (spin down) Spin is a fundamental property of electrons, like its charge and mass. Quantum Numbers (QN) Principal QN: n = 1, 2, 3, 4…… Angular momentum QN: l = 0, 1, 2, 3…. (n -1) Rule: l = (n - 1) Magnetic QN: ml = …-2, -1, 0, 1, 2,.. Rule: -l….0….+l  Many-Electron Atoms For atoms with 2 or more electrons, the Schrödinger Eqn. must include KE terms for all electrons and PE terms for all e-e and e-n interactions. Example: He (2 electrons) 2 terms in T̂ 3 terms in V̂ -e(1) r1 r12 Z = +2e -e(2) S.E. r2 2 2 2   2 2 2 2e 2e e − 1 − 2 2 + − − +  = E   2m e 2m e  4 0 r1 4 0 r2 4 0 r12  where   2  2  2   i2 = 2 + 2 + 2   x y i z i   i  For N electrons: N = (r1,r2,r3 … rN) For more than 1 e, S.E. cannot be solved analytically – though good numerical solutions may be obtained from computer calculations.  Approximations must be made. The Orbital Approximation The total wavefunction (N) for the N-electron atom is approximated by the product of N 1-e orbitals similar to those of hydrogenic ions: N = (r1,r2,r3 … rN) = (r1)(r2)(r3)…(rN)  Bonding in Molecules Exact solution of the Schrödinger Equation is not possible for any molecule – even the simplest molecule H2+. e rA rB HA RAB HB Full Hamiltonian operator for H2+ : KE PE ( )  1 1  2 2 2 ˆ =−   e e2 H A + B − 2 2 e − 2  +  + 2m p 2m e 4 0  rA rB  4 0 R AB nuclear K.E. electron K.E. e-n attraction n-n repulsion  Application Computational chemistry is a rapidly growing field in chemistry. Computers are getting faster. Algorithms and programs are maturing. Some of the almost limitless properties that can be calculated with computational chemistry are: Equilibrium and transition-state structures dipole and quadrupole moments and polarizabilities Vibrational frequencies, IR and Raman Spectra NMR spectra Electronic excitations and UV spectra Reaction rates and cross sections thermochemical data  Limitations Schrödinger Equation can only be solved exactly for simple systems. Rigid Rotor, Harmonic Oscillator, Particle in a Box, Hydrogen Atom For more complex systems (i.e. many electron atoms/molecules) we need to make some simplifying assumptions/approximations and solve it numerically. However, it is still possible to get very accurate results (and also get very crummy results). In general, the “cost” of the calculation increases with the accuracy of the calculation and the size of the system. Yttrium catalyzed rearrangement of acetylene 40 2A 20 Energy (kcal/mol) 2A’ 0 2B 2 2A -20 1 2B 2A’ 2A 2 2 2 B -40 2 2A 1 -60 Reaction Coordinate Biochemistry applications  Tools There exist a large number of software packages capable of performing electronic structure calculations. MOLPRO, GAMESS, COLUMBUS, NWCHEM, MOLFDIR, ACESII, GAUSSIAN,... The different programs have various advantages and capabilities. We also have available to us Gaussview which is a GUI that interfaces with Gaussian for aiding in building molecules and viewing output.